Working Memory: Role in Mathematics Learning (Part Two)

Markers of Working Memory Problems in Learning Mathematics
Because working memory is involved in so many activities, there are many different indications of working memory problems. Teachers, parents, and students should be aware of situations where working memory lapses may be evident. Here are some possible signs:

  • Trouble remembering the components of a multi-step task, like standard procedures (loss or delay in recall, failure to follow instructions, place-keeping errors, and frequent task abandonment)
  • Trouble remembering the sequence of the task elements
  • Trouble concentrating on a task or instruction
  • Easily distracted from a task
  • Forgets what s/he is doing in the middle of a task
  • Trouble staying on the same topic when talking
  • Forgets instructions easily
  • Trouble doing more than one thing at a time
  • Skipping or repeating steps in a task
  • Trouble with reading comprehension or understanding instructions
  • Difficulty prioritizing multiple demands posed by the task(s)

All or many of these signs may be implicated in a child having trouble with mental arithmetic and in mathematics problem solving.  It is important, however, to recognize that everyone experiences some of these difficulties some of the time; that is normal. But if these kinds of problems or incidents are frequent and/or severe, they may be an indicator of a working memory problem. Because it is difficult to determine a working memory problem informally, formal testing is required.

There are several reasons for a person’s working memory to be taxed: task complexity, lack of understanding of the task (due to poor reading skills, poor language of mathematics, lack of concepts and skills, etc.), poor teaching, and personal reasons (emotions, stress, and math anxiety). For example, when a person is stressed, the pressure of anxiety blocks the working memory; in such situations even the facts with which people are familiar cannot be recalled because the “mind has gone blank.” This is the impact of stress blocking the working memory. Even more importantly, math anxiety influences those with high rather than low amounts of working memory—precisely those students who have the greatest potential to take mathematics to high levels. When students who experience stress in timed conditions cannot access their working memory, they underachieve, which causes them to question their math ability and, in many cases, develop further stress and anxiety.  Several studies have demonstrated that the link between the working memory and mathematics difficulties is stronger in children than adults and therefore places greater limits on the computational performance of children than young adults.

Working Memory’s Influence on Arithmetic Problem Solving
Problem solving in any setting is a complex cognitive activity. However, it is more demanding and involved in mathematics settings as most of the activity is being accomplished at abstract and symbolic levels. A multitude of lower and higher order cognitive thinking skills (e.g., pattern analysis), language processing (decoding, comprehending, meaning making, etc.), visuo-spatial (e, g., space organization and orientation, visualization, etc.), and reasoning (deductive and inductive) skills are called upon. To be proficient in solving an arithmetic or mathematics problem, an individual would have to able to

(a) focus attention on each component of the information about and in the problem as it is presented (vocabulary, syntax, format, etc.) to understand it (as against reading in areas other than the sciences, where one can still have some understanding of the problem without close attention to the specific language, word for word, used in the problem);

(b) hold the relevant information in the working memory (to receive, comprehend the meaning and concept involved, and to translate from native language to math language—quantitative and spatial, etc.);

(c) scan the long-term memory to find related language, concepts, or procedures, formulas, definitions, and skills;

(d) mix, relate, and reformulate the new information with the relevant information from the long-term memory—reshaping existing conceptual schema by consolidating partial schemas, extending previous schemas, and even abandoning previous schemas in the light of new information and understanding;

(e) manipulate the problem information by mentally performing the required operations (attach the appropriate conceptual schema to the information);

(f) selectively maintain some of the information (most recent outcome of the operation) in a temporary mental storage (buffer between working and long-term memories); and

(g) complete all of these tasks within the span of a few seconds to minutes (each component of the operation is identified and acted upon).

All of this is performed multiple times during the problem solving process. Moreover, this process is iterative in every multi-step procedure/algorithm/ problem-solving situation in mathematics. For example, in executing the standard long division algorithm, the student needs to have a mastery (conceptual understanding, fluency, and applicability) of the related concepts and procedures (multiplication tables and procedure, place value, subtraction, and estimation), and the prerequisite skills (following directions, spatial orientation/space organization, visualization, following the pattern—estimate, multiply, subtract, bring-down, etc.).

In a world problem, understanding the language (vocabulary and syntax) for identifying the variables and the problem, seeing their relationships, forming expressions and equations (translating from native language to mathematics), then solving them (mastery of concepts and procedures) and finding the relation of the solution to the problem (translating from mathematics symbols to native language: does the answer make sense?) also involve an iterative process. The same process happens when we are finding the factors of a whole number or an algebraic expression (e.g., trinomial).

The Demands of Procedures on Working Memory
Iterative processes heavily tax the working memory. However, concrete and pictorial models used in arriving at these iterative processes help minimize the impact on the working memory. For example, the use of area model in arriving at multiplication and division procedures whether involving whole numbers, fractions, decimals, or binomial expressions helps see the sequence of steps and connections between concepts. In the case of finding the factors of a number the proper use of empty number line helps.

Similarly, to find the factors of 72, if we represent the factors on the number line by their location, it is easier to understand the interrelationship between the different factors and can extend to finding the greatest common factor (GCF) or least common multiple (LCM). For example, the factors 1 and 72 are located on the number line. Then we ask what the next number that divides 72 is. Using the divisibility rules, we find 2 divides 72 as 72 is even. And factors 2 and 36 are identified (by performing the short division). Now, 2 is placed next to 1, and 36 is placed in the middle, in its right location, on the number line. The empty space (between 36 and 72) indicates that there are no factors between 36 and 72. The divisibility rule is used again and we find 3 and 24 are the new factors.  They are located in their appropriate places.  By the same argument we arrive at the conclusion that there are no factors between 24 and 36, between 18 and 24, and 9 and 18.

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To find the greatest common factor (GCF) of two numbers (say, 48 and 72), it becomes visually clear to a student as the common factors will occupy the common locations, thus, the greatest of the common factors of 48 and 72 can be found easily, as:

capture2

On the other hand, to find the least common multiple (LCM) of the two numbers we write the multiples of the two numbers, identify the common multiples, and then identify the least of them.  This means, we need to employ two processes (one for GCF and the other for LCM with finding the multiples of the numbers may involve several calculations); these procedures are cumbersome and not generalizable to algebraic expressions. This puts more demands on the working memory.

However, the successive prime division (SPD) method connects GCF and LCM better and makes fewer demands on the working memory. For example, the first prime factor is found by asking the question: “What is the first prime number that divides 72 and 48?” By the help of the divisibility test one knows the first prime number is 2 and the corresponding factors (36 and 24) by the help of short-division are found. The same process of successive prime division is repeated to find all the common factors.

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The first column gives us the common factors of 72 and 48, therefore, the greatest common factor (GCF) of 72 and 48 = 2×2×2×3 = 24.Similarly, applying the definition of the least common multiple of 72 and 48 (LCM), we find that it should have factors 2, 2, 2, 3, 3, 2 and removing any one of them will not satisfy the property of multiples of 72 and 48. Thus, the LCM of 72 and 48 = 2×2×2×3×2×3 = 144.

The visual representation of the successive prime division method visually reduces the demands on the working memory and GCF and LCM are found together. This process reduces the work involved in adding and subtracting fractions more effectively using the successive prime division to find the least common denominator (LCD).

Similarly, in the case of finding the factors of a trinomial (e.g., x2 +6x +5), the use of algebra tiles or Base Ten blocks helps students to see (x + 3) and (x +2) as the factors, represented by the sides of the rectangle and its area as their product—the trinomial, x2 +6x +5.

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The model facilitates the visualization of the multiplication of binomials, and even the division of the trinomial by a binomial becomes clear—the vertical side as the divisor, the area as the dividend and the horizontal side as the quotient. This visual modeling helps students to hold the visual representation in the working memory’s sketchpad. For this reason, the procedures we use and how we derive them play an important role in the task-load on the working memory.

Engagement and Its Role on Working Memory
Engaging the learner, particularly those who experience difficulty in learning mathematics because of ADD or ADHD and working memory issues, in a mathematics learning or problem-solving task is the most important goal of a teacher, tutor, or interventionist. To effectively and meaningfully engage students in the learning tasks requires proficiency in:

(a) knowing the trajectory of the development of the mathematics concept or procedure at hand, that means knowing: (i) what secondary and primary concepts and prerequisite skills are involved, (ii) how much and the level of language required, (iii) how to establish the sequence of the tasks, and, (iv) to determine the cognitive complexity of each task involved, etc.,

(b) understanding the learner characteristics—cognitive preparation, mastery of language, reading and comprehension levels, basic skills level, presence or absence of conceptual schemas, nature of mathematics learning problem/disability, limitations imposed by the nature of the learning disability (e.g., the level and condition of the learning disability places limitations on the prerequisite skills—following directions, pattern recognition, spatial orientation/space organization, visualization, estimation, deductive, or inductive reasoning); and

(c) acquiring competence in effective, efficient, and elegant (generalizable to abstract concepts and procedures) pedagogical approaches—models, questioning, instructional materials, sequencing of tasks, tool building, appropriate and timely reinforcement.

Working memory functions require the learner, in mathematics problem solving and conceptualizing mathematics ideas, to simultaneously attend, store, and mentally process a rather large amount of information within a relatively short period of time. These functions demand a higher level of active involvement on the part of the child. This is particularly difficult if a child has organic/neurological reasons for his/her ADD or ADHD. It requires the teacher or interventionist to be extra vigilant, creative and aware of the child’s strengths and weaknesses.

Information Load on Working Memory
Information load on working memory in the problem solving process is a major factor in determining task complexity and difficulty and completion of the task by the child. The task difficulty arises, for example:

(a) in elementary school, in the use of effective strategies for learning facts (additive and multiplicative—sight facts, decomposition/ recomposition, development and execution of sequence of steps in procedures and mastering them)[1],

(b) in upper elementary and middle school, in discerning and using patterns in arriving at conceptual schemas and standard procedures (multi-digit multiplication and division) and in pre-algebra (fractions, integers, concept of equations, etc.)

(c) during high school, in deductive and inductive reasoning in understanding concepts and procedures in algebra, and in deriving definitions, relationships between different geometrical entities, development and execution of proofs—use of spatial, deductive, and syllogistic reasoning, in geometry.

In all of these complex concepts and procedures, the working memory is highly involved, and this involvement is essential for understanding and applying concepts and ideas.

Whenever a child faces a new concept, particularly a secondary concept (a concept involving several primary concepts, multi-directions, etc.), s/he faces an overload on the working memory. The more a child is free from constructing basic arithmetic facts when needed, the more the child is able to devote the limited working memory resources on learning the new language, concept, or procedure and their relationships. Automatizing basic arithmetic facts, therefore, is important for two reasons: (i) the student is able to discern patterns in number relationships and therefore make more connections and (ii) the working memory is freed from constructing these facts every time there is need for them.

On the other hand, using ineffective strategies such as “counting up or down” for arithmetical operations (counting up addition, counting down for subtraction, skip counting forward for multiplication, skip counting backward for division) takes up all the available space in the working memory.

Similarly, giving children the multiplication tables or addition charts does not reduce the load on the working memory. The students will not comprehend the problem without understanding the language and conceptual schemas related to the problem and the isolated facts retrieved from these sources or generated on calculators do not provide the opportunity of making connections. These approaches, therefore, are not “real” answers to the problem. However, when the students have acquired the language, concepts, and strong numbersense, then giving them calculators or computer for executing procedures or problem solving is good use of assistive technology.

How to manage the workload on working memory is the joint work of the teacher and the learner. Better understanding of task analysis (by the teacher and then by students) and the development of metacognition in students are essential. Effective task analysis is dependent on understanding the concept. The teacher should share how to do the task analysis (e.g. “notice to multiply 124×8,one should know place value—expanded form, multiplication table of 8, distributive property of multiplication over addition, and then adding) and then during the “tool building time” of the lesson, she should be developing these tools and their integration. Each lesson should have three components: tool building (25% of the lesson time), main concept/ procedure (50% of the lesson time), and reinforcement—guided, differentiated and supervised practice to master the concept/procedure in order to convert it into skill-set.

The teacher should anticipate the level of information load on children’s working memory before assigning a task. When a teacher has a better understanding of the task analysis of concepts, procedures, and problem solving process and the student has a higher level of metacognition, the student is better able to manage information. Therefore, helping students to become aware of their own understanding of their learning, strategies, and awareness of the task analysis prepares them for better engagement in the tasks and enables them to manage the task-load on the learning system, including the short-term, working and long-term memories.

As children develop, they use their working memory in different ways and for different purposes. It is critical for learning the alphabet and number concept in the first few years (Pre-K, K, and 1), for reading comprehension and mental arithmetic in elementary school, and for completing homework independently, solving multi-step problems, and completing projects in upper elementary and middle school. In high school, working memory is essential for writing essays and reports, proving theorems, solving problems. Working memory performance is crucial on tests such as the SAT and ACT: one has to keep all four multiple-choice options in mind and decide which is best! In college, working memory helps students maintain their focus during long lectures, complete papers, lab reports, and study for exams.

As students reach higher grades, the complexity of mathematics content and the related higher order reasoning skills needed place extra demands on the learning systems. Mathematics arguments in proving theorems in geometry, multi-step problem solving, and multi-step equations invite reasoning about relationships that involve two or more concepts and ideas, for example, “all rectangles are closed figures” and “all squares are rectangles.” The student needs to hold this information to derive an inference from these two statements.  If this information is provided only orally, it is difficult to hold that much information in the working memory and also to focus on each statement simultaneously. It taxes the working memory and overwhelms the thinking process for those who have limited working memory space. In such a situation, however, if the same information is visually presented (e.g., Venn diagrams, graphic organizers, flow-charts, etc.), then the two premises are constantly available for examination and therefore the load is reduced. Overloading this fragile mental workspace can lead to significant loss of information from the working memory and a feeling of being lost and bewildered.

Teachers and textbooks sometimes inadvertently present information in a manner that may unduly strain the processing capacity of a student’s working memory. To help children learn arithmetic and mathematics, teachers and interventionists need to understand:

  • the role of working memory in learning and teaching of arithmetic and mathematics,
  • how the working memory components can be formally and informally assessed,
  • how limitations in the working memory contribute to the development of mathematics difficulties and disabilities,
  • what kinds of instructional interventions or remedial approaches are available for mitigating the detrimental effects of the working memory limitations in mathematics achievement, and,
  • what kinds of interventions can improve the working memory space and its components.

Memory Functioning and Learning
Research points towards the feasibility of working memory strategy training in enhancing numbersense. In fact, all three types of memories can be improved with the use of efficient models, effective strategies, proper practice, and an enabling sequence of questions that help a child to see patterns and relationship. A strong positive relationship between the working memory and mental arithmetic competence is evident.

Of course, memory (short-, working, and long-term, explicit and implicit, episodic and procedural, etc.) plays important roles in any learning. Poor long-term memory has impact on learning. If a child is a fluent reader and has good comprehension, s/he is capable of mastering the basic arithmetic facts (addition and subtraction—sums up to 20 and decomposition/recomposition, multiplication and division—tables up to 10 and distributive property of multiplication over addition). However, the child’s ability is dependent on whether effective strategies have been used in teaching them. Poor strategies (addition is counting up and subtraction is counting down) do not leave residue from a learning experience. Most of the time, therefore, it is not poor long- term memory, but poor strategies of learning a concept or skill and lack of proper and efficient practice that are responsible for children not remembering their facts.

There are ways that long-term memory can be improved. The two most effective learning techniques are distributed practice and practice testing. Distributed practice is spreading out practice sessions over time. So instead of spending a long time on the same subject/topic/same procedure, one studies for small segments of time each day. For example, rather than reviewing the previous grade’s material for the first two months of the new grade, the teacher devotes a few minutes each day as a tool building session in each lesson. Or, rather than devoting a great deal of time on mastering facts, the student masters a strategy and applies it right away. For example, learn the table of 8 and then simplify fractions that involve only the multiples of 8, divide a multi-digit number by 8, or multiply a six-digit number by 8.

Practice testing is trying to answer a question without looking at the solution. That means doing a new math problem to learn a math concept or answering a fact related question to remember a fact rather than just repeatedly reading that material or repeating that fact. Arriving at facts by counting again and again does not help children to automatize them. Using properties of operations and definitions to deriving the facts: For example, using distributive property and decomposition/recomposition to derive facts.

Distributed practice is effective because of the spacing effect. Spreading exposure to multiple sessions, separated in time, will have a better long-term impact on memory.  In order to turn an experience into long-term memories, the short-term exposures to information need to be consolidated. Spending more time with the material, but not allowing space between for consolidating the information may mean some of the extra exposure time is wasted. Each time we instigate the old information as we start using the information, our brain needs to activate the context of memories it is a part of. Each time we activate this context, we strengthen our ability to do it in the future. Studying in one batch only needs to load the context once, so it doesn’t strengthen as much as having to recall from scratch multiple times.

Distributed practice does not mean cutting practice sessions into tiny slivers to maximize the spacing effect, which has the unrelated downside of making focus very difficult to accomplish.

A better way to implement distributed practice is to review older units, chapters, tests, facts, or definitions on a regular basis. The teacher should not make learning the current lesson as the only goal for the children. By devoting a little extra time to review, the accumulated reviews will make far more impact than a cram session. For example, each homework assignment should include problems from previous topics on a regular basis.

Many teachers and students do not see the difference between recall and recognition. For example, when we see some material again and want to answer questions related to it, we are testing only recognition.  But, recognition is usually easier than recall. The ability to recognize information can give a false sense of confidence about the subject. One may feel that one “knows” but cannot recall it when one needs it on a test or in a real situation, leading to frustration and failure. Recall is an indication of mastery, whereas recognition shows only superficial mastery. If one knows something, one should be able to recognize that it is the right answer and should be able to recall the answer if someone asks the question. Properly designed multiple-choice questions can test both recognition and recall.

Memory research gives evidence that recognition and recall may involve different cognitive and psychological processes. And even if the two processes do share a common mechanism, it is not the case that being able to recognize a piece of knowledge is equivalent to or results in being able to recall it. Being able to remember something has two parts: first you need to have the knowledge represented in your brain. This representation, initially calls for visualizing that information and then constructing a conceptual schema for that information. Understanding the language associated with the concept and then constructing the schema (a working memory activity) helps to send it to the long-term memory. But then, crucially, you also need to be able to find it at the right moment and in acceptable time.

Practice testing gets around this since it forces the student not just to store information but also to develop strategies to search for it at the correct time. Practice and reinforcement using effective, efficient strategies help place the information in multiple places. Recall is facilitated when there are multiple contexts to search in the long-term memory.

The best way to apply practice testing is to avoid its opposite: passive learning. Students should not re-read again and again the notes unless they are searching for the answer to a particular question. Instead, they should cover up the notes and see if they can describe the concept or repeat the definition or reproduce the procedure without looking at the solution. This also means that after the students have understood the concept or procedure, they should have access to large sets of practice questions on that topic, and they should be encouraged and helped to practice them. Nothing will serve the students better than doing large numbers of practice questions as the foundation to learning a subject.

Let us consider an example: learning the long-division algorithm. At first, the student might not get the procedure correct all the time. There are several concepts, skills, and steps and a lot to think about, so she may forget to apply a step as the memory for it as a whole may not be that strong yet. To master this procedure (estimation, multiplication, and subtraction), the teacher should write 10 to 20 problems and make the first step to estimate the answer in all of these problems and check it by actually multiplying the estimated quotient with the divisor, and then the next task for all of these problems, and so on. If we sampled the student performance on a test with different component tasks/questions, we could measure how much better she gets at doing the test over time. The score will steadily get better as she gets more practice.

Eventually, however, the student will reach a plateau. After this point, she mostly stops getting better at the test. This could be because she is scoring 100% every time as the familiarity of the tests sets in. Now, we could make the test difficult enough so that it is impossible not to make a few silly mistakes, leaving her with 95% or something similar. The question is: Should she keep on practicing beyond this point? Is there any benefit in continuing to practice? The answer, surprisingly, is yes.

Continuing to practice after this point has a different effect. This effect, called overlearning, doesn’t affect her test scores (that’s already at a maximum). Instead, it helps with the longevity of the memory.

In the beginning, practice improves performance. Later, when performance is maximized, it continues to improve longevity of the memory, and in the process fluency is also achieved. Overlearning something, therefore, is the strategy for remembering it permanently.

Is Overlearning Worth It?
The answer is, no doubt, yes. It is essential for automatization, fluency, and freeing the working memory to acquire higher order thinking. In every facet of our lives we do it, so how do we achieve it?

Strategy #1: Learning +1
The first approach comes from studies where the effects of learning a skill are assessed at different intervals. For example, in a study in which people were given an algebra test after a class, and then tested again decades later, to see how much they had remembered.

Interestingly, the people who did best on the first test didn’t have more durable memories than those who did poorly. Of course, if you remembered more for the first test, you’d remember more for the second. But the *rate* of forgetting in both cases was the same. But one group of students did not see their memories decay: students who went on to study calculus. In the process of learning calculus they used algebraic skills routinely and therefore reinforced the algebraic skills. Learning a subject above one’s current level forces one to overlearn the basics of the previous subject. If one wants to make memories last longer, one should apply the skills to problem situations and to more advanced topics, and in the process one learns the previous skills and retains them longer and stronger.

Strategy #2: Immersive Overlearning
Immersion method of learning languages makes a great use of overlearning. It can only partly be used in learning mathematics. Complete immersion is possible if the students’ learning is project-based learning where they see mathematics and all other skills for extended periods, integrate mathematics ideas with language, representation, discerning and forming relationships as mathematics expressions and equations, and then solve them. True and complete immersion in mathematics is practiced only by professional students of mathematics (mathematicians), physicists, engineers, technologists and mathematics teachers. However, the new crop of STEM courses is a good attempt to have the kind of immersion that helps students to engage in learning.

Strategy #3: Practice Makes Perfect
Practicing a test repeatedly is not a good idea, but practicing a different test on the topic is good even after scoring 100%, particularly those concepts and skills that you want to make permanent. For example, mastery of arithmetic facts should be made permanent as they are used in all aspects of life. The key is that one needs to be selective—overlearning every possible fact will limit time to learning new things. However, if a certain set of knowledge is essential for students to have at their fingertips, we should make sure that they overlearn it. We should practice it until they get it perfect and then practice it some more.

Working Memory Can be Strengthened
Though working memory has been studied for decades, it has only recently been proven to be a plastic function of the brain, able to be strengthened through rigorous training and effective and efficient teaching strategies. The brain is capable of enormous change through experience. This capacity of the brain is called neuroplasticity. With appropriate activities and exercises, we can improve our working memory to be better equipped to meet challenges. Like a muscle, it can be improved through certain types of exercises.

Researchers have used neuroimaging techniques to explore the neural basis of working memory plasticity. Their results show that, through working memory training, the activity of the brain areas related with working memory can be enhanced. For example, although the working memory capacity of seven-year-olds is smaller than that of older children and adults, their attentional processes are just as efficient—so long as their smaller working memory capacity is not exceeded by overloading it with extraneous and irrelevant information. When their working memory is overloaded, attentional efficiency declines, suggesting that intervention aimed at enhancing working memory will in turn improve attentional efficiency. In general, children’s attention to relevant information can be improved by (a) minimizing irrelevant objects or information cluttering working memory, (b) training them in efficient learning strategies, and (c) using effective and efficient learning models and materials.

Studies show a positive relationship between brain plasticity and learning. For example, just like the reading brain is different than the non-reading brain, the experience of learning math facts actually changes the memory patterns and neural connections and in turn aids in learning more facts. These connections become more stable with skill development. So, learning addition and multiplication tables and having them in rote memory (of course, after understanding them) helps develop the capacity to learn not just the content but the ability to learn more.

Nevertheless, the memorization of facts should be practiced first with the simple facts children already know (e.g., table of 1, 10, 5, 2, 4, 9 must be mastered before others and the 45 sight facts of addition, commutative property, making ten, and teens numbers should be learned before any other addition or subtraction facts)[2], then one should show them how to construct new ones using decomposition/recomposition for addition and multiplication facts. Similarly, in preparation for mastering tables, we should quiz children their multiplication tables in different order, for example, ask: 8×1, 8×10, 8×5, 8×2, 8×9, and then 8×6, etc. If the child does not respond within a few seconds, help her to break it into two multiplication facts: 8×6 = 8×5 + 8×1, then help the child to combine them. Then practice them so they really remember and do not have to think it through. Similarly, to find 8 + 7, we should ask what number will make 8 as 10. The answer: 2 (application of the strategy of making 10 or sight facts of 10). Then, what is left in 7 after 2 is used up to make 8 as 10. Answer: 5 (Sight facts of 7). Then what is 10 + 5. The answer: 15 (making teen’s numbers).  So what is 8 + 7? The answer: 15. So, the arithmetic facts should be automatized using efficient strategies. For some children, initially, the construction of facts in their minds may still be slow, but it will be faster than if it is not automatized or they derive them by counting.

With effective strategies (e.g., for addition—decomposition/recomposition, mastering making ten, teens’ numbers), rehearsals, and usage, facts become automatic.

The feeling of automaticity is a result of brain circuitry that’s been strengthened through repetition.  When we have automatized basic facts, the brain doesn’t have to work as hard on simple math. It has more working memory free to process the teacher’s new lesson on more complex math, and more patterns can be seen and more connections are made. The novice and the person who has not mastered facts use, for example, the memory system differently. For instance, most adults don’t use their memory-crunching hippocampus in the same way as novices. Retrieving six plus four equals 10 from long-term storage for them has become almost automatic, instead of a great effort.

Learning, for example the 45 sight facts of addition, calls for the integration of information and the function of the four components of the working memory—executive function (allocator of resources like attention and focus), phonological memory (holding linguistic information and its rehearsal), the visual/spatial sketchpad and memory (holding visual spatial information), and the episodic buffer. It is important, therefore, to use materials and strategies that help the child to enhance the working memory and learn, in this case, the sight facts.

Visual Cluster Cards (VCC) and Cuisenaire Rods (CR), because of their color, shape, size and patterns, force the executive function to attend to and direct it to the task. Further, they engage the visual-spatial sketchpad. The language used by the teacher is supported by the ability of the VCC and CR to attract the attention of the phonological loop. Thus, the child can hold that information in the working memory, manipulate it, and send it to the long-term memory.

The teacher’s questions, commentary and associated language with VCC and CR activities, student decision making, the patterns of the VCC, lengths, and the colors of the CRs enhance the phonological loop, visual/spatial sketchpad, and executive functions. As a result, these materials not only help children to learn the sight facts but also integrate them with existing concepts and schemas. Thus, these materials strengthen the different components of the working memory. The auditory information gains obligatory access to the phonological store: we do not have to do anything to create a phonological record. However, nameable visual inputs such as pictures, written letters or written words, must first be ‘recoded’ into a phonological form in order to gain access. Concepts presented through these materials provide access (as the color, length, and pattern invoke words) to the phonological store and, therefore, the learning is optimized.

Since VCC and CR have quantitative information, it is easy to create a phonological and visuo-spatial record as efficient and elegant concrete models invoke language naturally. The mix of visual-spatial information from VCC and CR and vocabulary forms number relationships and creates a higher possibility of a stay in the episodic buffer before searching for the related facts and concepts, making the number relationships and connections in the long-term memory.

Improving Working Memory
It is possible to significantly improve working memory through training, practice, and the effective use of efficient models and materials.

Using brain research, gaming experts have created computerized programs to improve working memory through exercises and training. Some examples are computerized working memory-training programs such as: Cogmed Working Memory Training and Lumosity. These programs include a series of engaging working memory games and challenges. Like most popular videogames, these games get progressively harder as players’ skills and capacity improve. The programs claim that working memory improves and that the improvements last long after the training ends. Their research seems to show that the programs are effective for working memory deficits that accompany ADHD, stroke, and aging. They advise that an adult oversee the training process when children are doing the training – sitting with the child, encouraging him or her when the tasks get difficult, witnessing and appreciating his or her efforts, hard work, and successes.

In general, certain principles about this research on learning are useful in day-to-day interactions in the classroom and at home:

Learning means new connections in the brain
The brain works like an electrical circuit. Just as an electrical current travels through a circuit, signals are transmitted from one group of neurons to another. Every time one learns something new connections form between neurons in the brain. The stronger the electrical signal, the stronger the connection between the neurons. The stronger the connections, the greater an individual’s ability to form and retrieve facts from memory easily and fluently.

More connections mean more effective learning
Just as the strength of neural connections plays a role in learning, so does the quantity and quality of these connections. Learning is not just acquiring knowledge or facts; it is linking them and freely connecting old and new knowledge. An isolated fact can be tough to remember or recollect, unless it is overlearned, connected, or accompanied with a strategy.

One can make learning a fact easier by relating it to other networks of information in one’s brain. This is the idea of constructivism – that we are able to place Lego bricks of knowledge into our long-term memory, use them, and build on them when and where we need to. The more ways we construct (this happens in the working memory), the more places we place (this happens in the long-term memory), the more connections we make (more residue in the long-term memory), the more is the flexibility of thought, and the easier it is to retrieve the information.

For example, when we derive a fact (e.g., 8 + 6) in multiple ways (e.g., (a) 8 + 6 = 8 + 2 + 4; (b) 4 + 4 + 6; (c) 2 + 6 + 6; (d) 8 + 8 – 2; (e) 7 + 1 + 6; and (f) 3 + 5 + 5 + 1) and with multiple strategies such as: making ten [(a), (b), and (f)]; doubles [(c), (d), (e), and (f)] not only do we make many more neural connections and reinforce previous facts, but we also place the facts in different files, by making more connections, in the long-term memory. The quantity, quality, and strength of these connections are the memory traces. Memories can be procedural, emotional, semantic, and some other kinds. Thus, to summarize:

 capture4

What Can Teachers Do to Enhance the Working Memory?
The teacher must monitor the child’s performance and be on a lookout to recognize working memory related failures and lapses. She should ask parents, the child’s other teachers, and the child for examples of such working memory lapses. For example: “she lost her place in a task with multiple steps,” “raised his hand to answer a question, but when called upon he had forgotten his response,” “raises his hand, but then responds by saying never mind,” etc.

Developing the ability to observe, discern, expand, create, and apply patterns is an anti-dote to working memory failures and the means to enhance the working memory. For example, asking children to solve addition and subtraction problems (any level from whole numbers to algebraic expressions) using visual-spatial (continuous) materials that emphasize patterns, color, size, shape, etc. (Visual Cluster Cards, Cuisenaire rods, Base-Ten blocks, TenFrames, Fraction strips, Algebra Tiles, Invicta tiles, etc.) rather than just sequential discrete (discontinuous) materials (e.g., fingers, counting blocks, number line, hash marks, etc.). Compared to discrete discontinuous material the visual-spatial (continuous) materials because of their pattern forming ability linger in the working memory longer and their representations easy to recall from the long-term memory.  Discrete discontinuous models should be used only initially to introduce an idea, but should move to more efficient continuous models.

In planning their lessons, teachers should aim at reducing unnecessary working memory loads and include activities that enhance the working memory. Working memory load should be varied according to the task either verbally or visually (e.g., long definitions; premises in syllogistic reasoning tasks are presented visually so that the information is continuously available for inspection). Arithmetic procedures should be arrived at by using efficient models and materials (e.g., multiplication and division of numbers, from whole numbers to algebraic expressions should be arrived by using Cuisenaire rods, fraction strips and BaseTen blocks and employing the area model) and appropriate language and questioning. Heavy loads are caused by lengthy sequences of information, unfamiliar and meaningless content, and demanding mental processing activities. The teacher should evaluate working memory loads in tasks before assigning them to children.

The teacher should repeat important information to help children process and should ask diagnostic questions to ascertain whether the information has been received correctly. The teacher or students nominated as memory guides can supply the repetition. Load on the working memory can also be minimized if important information can be identified with the student who makes/arrives at the conjectures. Conjectures named after the students are remembered better as they serve as “pegs” for memory. Similarly, the teacher should name each strategy derived and used (e.g., making ten, divisibility rules, successive prime division method, vertex form of the equation, etc.) and should remind them when they forget the name (e.g., missing double strategy, etc.).  When they use a strategy, students should use the name of the strategy, why they chose to use it, and describe it (e.g., in combining integers, if the signs are same, then keep the sign and add the numerals and if the signs are different, then keep the sign of the larger numeral and subtract the smaller numeral from the larger numeral).

Teachers should encourage use of memory aids only after understanding of the concept or procedure so that memory aids do not tax the memory. These include wall charts and posters, word wall, display of important definitions and formulas, use of graphic organizers, frequent use of visualization exercises, etc. Complex concepts place heavier loads, but use of efficient materials mitigates that load. For example, use of Visual Cluster cards and Cuisenaire rods is more efficient in developing decomposition/recomposition, properties of numbers, and properties of operations—commutative, associative, distributive properties; properties of equality and inequalities. Efficient use of multi-sensory materials helps minimize the working memory loads.

The load on working memory and retrieval from long-term memory are reduced when children develop their own effective and efficient strategies. These include asking for help, frequent rehearsals, productive self-testing, and effective note-taking—designated math notebook (graph papers—where left pages is for note-taking and the right side page for student work, the right page is divided vertically into two parts—2/3 vs. 1/3, left part for problems and right part for calculations and subsidiary work), efficient organizational strategies, graphic organizers, mnemonic devices, etc. Visual and auditory information is presented for a subject-determined time and material.

Emotion and Working Memory
Emotional control, or the ability to regulate one’s emotional responses and staying focused on a goal, helps determine success. Research shows that emotional control and working memory rely upon some of the same brain areas, including the fronto-parietal area and the amygdala. Since these two functions share brain pathways, strengthening one could strengthen the other. At the same time, certain parental and teacher behaviors—meaningful praise, affection, sensitivity to the child’s needs, and encouragement, along with intellectual stimulation, support for autonomy, help in organization, task analysis, and well-structured and consistent rules—can help children develop well functioning working memory and robust executive function skills.

[1] See an earlier blog on Sight Facts and Sight words.

[2] Sequence of strategies for teaching addition and multiplication facts in How to Teach Arithmetic facts Effectively and Easily (Sharma, 2008). List of strategies is available free from the Center (www.mathematicsforall.org).

Working Memory: Role in Mathematics Learning (Part Two)

Working Memory: Role in Mathematics Learning (Part One)

During my yearly clinical course, where many of the participating children have learning problems in mathematics (for example, specific learning disabilities, dyscalculia, dyslexia, etc.), the question of the role of working memory and executive function in mathematics learning kept coming up from teachers and parents.

The terms working memory and executive function are seen as important components in human development and learning and are implicated in many learning problems. Working memory is at work not just in formal learning. Working memory is one of our most crucial cognitive capabilities, essential for countless daily tasks like following directions, remembering information momentarily, complex reasoning, or staying focused on a project.

Understanding the importance of working memory can provide great hope to people who suffer from working memory deficits, including those with attention problems (ADD or ADHD), learning disabilities, or injury to the brain. Children with attention problems often have working memory deficits; however, in some cases, poor working memory may be the cause of certain second order attention problems. Deficits in working memory can affect an individual’s ability to focus attention, control impulses and solve problems. Someone with a working memory deficit or limitation can have difficulty attaining proficiency in mathematics, particularly problem solving as working memory load is a major factor in determining task difficulty. They may lose focus frequently when reading and solving a mathematics problem. While the connections between working memory deficits and mathematics performance seem clear, it is not certain whether these deficits cause attention deficits behaviors.

Executive functions and working memory differ between low achieving and typically achieving children not only in acquiring reading skills but also in mathematics achievement. The working memory and its functioning are heavily taxed in academic subjects such as reading and computational mathematics. Researchers have found strong involvement of working memory and executive function in mathematics learning and difficulties. For example, an increasing number of studies show executive functions as predictor of individual differences in mathematical abilities. Deficits in different components of executive function can be seen as precursors to math learning disabilities in children. For example, number concept, numbersense, and numeracy, implicated in dyscalculia and learning these concepts, are highly dependent on working memory and executive functioning.

Numbersense is an intuitive understanding of numbers, their magnitude and inter-relationships. The cognitive mechanism that helps strengthen numbersense is working memory. Furthermore, when the predictive value of working memory ability is compared to preparatory mathematical abilities, there is a definite relationship between them. Performance on working memory tasks predicts math learning abilities and disabilities, even over and above the predictive value of preparatory mathematical abilities. Strong and efficient use of working memory has been linked to higher academic success, including mathematics.

There are many reasons children may fail to learn or experience difficulty in learning arithmetic—number concept, numbersense, and numeracy. And these arithmetic difficulties, in turn, contribute to difficulties in learning other mathematics concepts and procedures. Apart from environmental factors such as poor instruction (teaching ineffective and inefficient strategies), lack of skill experience and reinforcement/practice, poor expectations, other examples for difficulty in learning include anxiety about mathematics, lack of experience and poor motivation, reading difficulties, neuropsychological deficits and damage, and cognitive delay and deficits. Arithmetical learning difficulties can be associated with cognitive deficits.

The cognitive deficits have a long list; however, they are exemplified by poor memory—short-term, long-term and/or working memory, lack of flexibility of thought (centeredness), lower levels of abstract thinking, visual perceptual deficits, and inefficient language development.

One component of cognitive ability is the size and working of working memory. More working memory space and flexible and effective usage by individuals mean greater potential for academic success, including mathematics.

Working memory functioning improves throughout childhood, adolescence, and adulthood and can be strengthened through intensive practice and training. George Miller was the first psychologist to attempt to quantitatively measure the working memory’s capacity. Miller coined the term ‘magical seven’– the idea that working memory could hold seven plus or minus two items. These items can be digits, letters, words, phonemes, bits, or small groups.

We can fool this limitation by chunking the information. Chunking is grouping and organizing discrete pieces of information into smaller groups/clusters. Through chunking, we organize and collect information and relate ideas. By chunking related pieces of information we can fit more into our working memory. Our ability to chunk together different kinds of information allows us to carry out a complex practical task without being overwhelmed.

We can easily overfill our working memory. When we do, we induce cognitive overload. Students who struggle with chunking new information become overloaded and cannot fit more information in their working memory, not without discarding something else. Cognitive overload can create misconceptions and muddy previously clear concepts.

Definition of Working Memory
Working memory (WM) refers to the capacity to store information for short periods of time when engaging in cognitively demanding activities. Whereas the short-term memory is like a relay station—the information is constantly coming and going, WM plays a more influential role in learning and academic performance, including mathematics. This is because mathematics tasks involve multiple steps with intermediate solutions, and children need to remember those intermediate solutions as they proceed through the tasks.

One of the areas affected by poor working memory is attention. But what we rate as inattention has nothing to do with actual attention.  Actual attention is hard to assess (you have to control for motivation, competence, reward, relevance, etc.). We generally settle for sustained engagement as an alternative and call it attention. The prevalent thinking is that if you didn’t keep going, you probably lost attention, which is usually untrue.  Most kids stop trying because the task is difficult (they do not understand it and do not have the skills for it), it has no intrinsic or extrinsic value (it is boring or unrewarding for them), or they do not see the purpose or connection of it with anything they know (the teaching was not engaging). It usually has nothing to do with attention even though many parents and teachers consider boredom and switching from one task to another as a sign of inattention.  The question is: Did they just drift away or were they looking for something more fun and interesting? In other words, is their attention poor or is their tolerance for boredom and frustration poor?

Most current research explores the dependence of mental calculation on working memory and how the limited-capacity system of working memory affects keeping track of temporary information during ongoing processing of mental calculations. Empirical studies tend to support the view that it is the limited capacity of the working memory that is responsible for inattention.

Not all models and pedagogy in use when planning activities in schools pay enough attention to the role of working memory in learning arithmetic. In this post, we explore answers to questions such as:

  • How is working memory related to learning arithmetic, therefore, mathematics?
  • How does working memory support numeracy, particularly, calculations?
  • What can we do to help children with poor working memory?
  • What teaching/intervention strategies and models can enhance and support working memory for all children?

Components of Working Memory
In the late 19th century, the American psychologist William James first proposed the distinction between a “primary” memory with a limited capacity and a long-term memory. British psychologists Baddeley and Hitch added a third element—working memory, to the learning cycle consisting of short-term and long-term memories. They postulated working memory as a temporary storage of information between the short- and long-term memories.

This model has three components: a central executive component, a phonological loop, and a visuo-spatial sketchpad. The phonological and visual components are referred to as ‘slave’ systems given that they hold specialized information for short periods of time. Working memory, thus, is a multifaceted function that captures visual, spatial, kinesthetic, and auditory information, directs attention to it, and coordinates processes to deal with its components, nature, and functioning. Much of the research in the cognitive psychology of working memory has been influenced by this multi-component model of working memory. It is time to bring it to the classroom and tutorials.

The central executive component has five capacities: to (1) coordinate and monitor input from the two slave systems, (2) shift attention, (3) focus on one stimulus, inhibit and/or enhance it, (4) hold and manipulate information from short- and long-term memories, and (5) update information.

The phonological loop is further divided into two sub-processes: a phonological input store and an articulatory rehearsal process. The articulatory rehearsal process refreshes verbal input. It focuses on the auditory and linguistic input.

The visuo-spatial sketchpad is devoted to visuo-spatial input. It pays attention to color, shape, texture, size, patterns, etc. The phonological loop is devoted to processing verbal speech input and is part of the rehearsal process for visual input as well.

Working memory is the ability to maintain and manipulate information temporarily. Despite its limited capacity, with effective materials and efficient strategies, an individual is able to perform complex cognitive tasks. It is the core of high-level cognitive activities and an essential component in the processes of learning, comprehending, reasoning, problem solving and intelligent functioning.

Whereas the short-term memory is a unitary storage and a passive place, the working memory is a multi-modal, multi-component, and multi-function place where temporary storage takes place before the information is intentionally transferred to long-term memory – if it is not transferred, it escapes. It is an active system that provides the basis for complex cognitive abilities. In working memory, we consciously process selective information; therefore, working memory is linked to attention control.

In the working memory we store small amounts of information in order to use that information to complete a task at hand—e.g., create a new conceptual schema and learn or form a new idea. In the working memory we bring information from the long-term memory and mix it with the incoming information from the short-term memory to comprehend, to learn, to solve problems, complete tasks, manipulate and see relationships, and make connections.

Short-term Memory
Short-term memory is a relay station—information enters involuntarily and leaves. Either involuntarily or voluntarily information is transferred to either long-term memory or working memory. It goes to long-term memory when (a) it connects to some related information that overlaps with it, (b) we connect it with what we already have in store, (c) it has novelty, or, (d) it is emotionally charged.

The information from the short-term memory goes to the working memory when we consciously begin to work on it—(a) mix it with information from long-tem memory, (b) reorganize or represent it to construct new information, and (c) rehearse it.

Construction of Concepts
The comprehension of incoming information takes place in the working memory. We classify it, represent it, organize/reorganize it; we transform it into a word(s), a graphic, a conceptual schema, a strategy, or a procedure and then by understanding it and rehearsing it send it to the long-term memory. The process of construction, in the working memory, can be self-initiated by reflection either on a recent event or in the past. It may also be instigated by concrete or visual models, words, diagrams, metaphors, similes, analogies, or some information that can be accessed from the long-term memory. For example, a child is asked:

What is 6 + 8?

Situation 1: Answer: 14. The problem 8 + 6 (presented orally or symbolically) invokes the response instantly, if the child has mastered it before (it already resides in the long-term memory). If the answer is affirmed, the memory traces: 8 + 6 = 14 is strengthened.

Situation 2: 6 + 8? Child manipulates the information: I take 4 from 8 and give it to 6 and then I have 10 and 4.  That is equal to 14. This is happening in the working memory. He has made connections and the working memory and long-term memory both are strengthened.

Situation 3: Oh 6 + 8 = 8 + 6. I take 2 from 6 and give it to 8 and then I have 10 and 4.  That is equal to 14. This is again happening in the working memory. He has made connections and the working memory and long-term memory both are strengthened.

Situation 4: 6 + 8. I think of 6 as 8 and I know 8 and 8 is 16 then I take 2 away from 16 and I have 16 – 2 = 14. This is happening in the working memory. He has made connections and the working memory and long-term memory both are strengthened.

Situation 5: 6 + 8. I take 1 from 8 and I have 7 and 7.  That is 14. So 6 + 8 = 14. This is happening in the working memory. He has made connections and the working memory and long-term memory both are strengthened.

Situation 6: 6 + 8. I can find the answer in several ways. I have found it before and now I know it right away.  This is where we want to ultimately arrive in learning: understanding, fluency, and applicability.

Teaching students effective and efficient strategies using effective instructional models makes lower demands on their working memory; it also facilitates recalling information from the long-term memory, holding the information, and manipulating it in the working memory. In the process they improve their working memory.

Notice all the components of the working memory system are being called upon and strengthened.

Situation 7: When the child does not know the fact, the responses are: (a) repeats the question, once or several times. Let me see: 6 + 8?  A child with a poor strategy and/or poor teaching (adding is counting up) counts 7, 8, 9, 10, 11, 12, 13, and 14 either in his head or on his fingers.  Let me see I have counted 8 or not? And he verifies by counting.  Notice, he needs to maintain 16 numbers (1-7, 2-8, 3-9, 4-10, 5-11, 6-12, 7-13, 8-14) simultaneously in his head (working memory) and that is difficult. In this process none of the components of the working memory system are used and strengthened. The activity will not leave the trace of the final result in the memory, and the same process will be repeated next time the same problem is presented.

Ineffective strategies place a higher demand on the working-memory and create frustration that further diminishes the functioning of the working memory. When facts are not mastered using effective strategies, situations 1 to 6 do not happen.

When effective strategies are learned, one can extend them to develop and strengthen mental math—an activity that is dependent on a strong working memory. For example, when one tries to find the sum: 58 + 17, the child may think of the following operations in his mind. 58 + 2 is 60 by taking 2 from 17 so 2 less than 17 is 15, so I add 15 to 60 and I get 75, so 58 + 17 is 75. All of this is taking place in the working memory. However, to come to this level of mental calculation, efficient concrete materials and pictorial representations help. Here the child had experience in using Cuisenaire rods (58 = 5 orange rods and the brown rod—the 8-rod and 17 = 1 orange rod and the black rod—7-rod.). Similarly, using the Empty Number Line (ENL), one can create images in the working memory and, therefore, develop mental math—holding numerical information in the working memory and manipulate it.

image-1

The color and size of the Cuisenaire rods engage the slave systems—articulatory loop (as we read the rods as numbers), articulatory rehearsal (the presence of the rods keeps the information alive in the memory), and visuo-spatial memory (the size and color of the rods). Even children with poor working or short-memories are able to achieve more. In the process they improve their working memory and create and leave a residue of the experience in the long-term memory. When concrete models are supported by efficient and elegant representations, the images are further strengthened.  For example, one can go easily from Cuisenaire rods to Empty Number line.

Cuisenaire rods, Visual Cluster cards, and Empty Number Line help children to acquire mental math competence.

Recognition and Comprehension
Many studies have examined the relationships between working memory and word recognition. The same system is involved in the recognition of visual clusters and large numbers (place value)—essential elements for the development of numeracy. However, a problem like the one mentioned above also involves comprehending and understanding that system and manipulating the numbers using strategies. Good readers allocate more working memory resources to text comprehension than to word recognition when compared to poor readers. Good readers produce more integrative inferences than poor readers, who are constrained by their working memory processing capacities when building mental models of texts. Similarly, low achieving students, because of poor strategies (addition is counting up and subtraction is counting down) and ineffective instructional models (counting objects, number line, and lack of patterns and color) face heavier loads on their working memories. They allocate more working memory space for deriving arithmetic facts by counting, and little space in working memory is left for seeing patterns, relations, and making connections. As a result, they miss developing understanding, fluency, and mental math and have difficulty in applying their knowledge and skills to problem solving.

Working memory, effective strategies, efficient models, and residues in the long-term memory interact with and influence each other throughout the learning process.

We use the working memory as the sketchpad and working place for thinking—as the brain’s conductor. It allows us to hold onto information, for a short time, and then to work with and manipulate that information. So for example, when we speak or do mental calculations, the working memory brings the words, arithmetic facts, concepts, and relevant procedure that we know together and connects them into a coherent sentence or outcome—a calculated answer.

The phonological loop is specialized for the storage and rehearsal of speech-based verbal information (notice the language used in the six situations of 6 + 8 above) whereas the sketch-pad is specialized for holding visual and spatial material (the equation formed by the Cuisenaire rods, graphic organizers, Empty Number lines, Invicta Balance, Algebra Tiles, etc.). They constantly interact with each other.

Most of the time, the working memory is the connecting link between the short- and long-term memories and plays a crucial role in sending information to the long-term memory. What goes in the long-term memory is dependent on what is being worked on in the working memory. One can greatly enhance the capacity, the nature, and functioning of the long-term memory by improving the working memory system. For example, when we want to calculate the product 222×3, we bring the relevant information from the long-term memory, such as 222 = 200 + 20 + 2 (understanding of place value), 2×3 = 6, 20×3=60, 200×3=600 (the facts and the distributive property of multiplication over addition) to the working memory and then mixing this information, we get 232×3 = (200 + 30 + 2)×3 = 200×3 + 20×3 + 2×3 = 600 + 60 + 6 = 666.

Different types of information are brought from the long-term memory to the working memory—from specific to general (recognizing 5 as a prime number) and from general to specific (which rule of exponents to apply in evaluating a3×a4). Children compute with mathematics facts—such as those required in timed tests—by recalling them from the long-term to working memory and using them in computations/procedures in paper-pencil situations or mental math.

Formation of Memories
In living and learning, we rely on two types of long-term memories: explicit and implicit memories. Explicit memory is the “conscious” memory for specific facts and events, as opposed to “subconscious” implicit memory.  Remembering a fact (8 + 6 = 14) or concept one relies on explicit memory (multiplication means repeated addition, groups of, an array, or the area of a specific rectangle). Remembering how to add (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14), in general, relies on implicit memory. Similarly, remembering the concept of multiplication as repeated addition relies on implicit memory whereas the ability to remember that 7 × 8 is 56 or am × an = am+n relies on explicit memory.

The concrete models and particular learning activities are the means for students to create conceptual schemas, associated visual representations, verbal discussions (development of language containers, and words and instructions for language rehearsals), and cues (mnemonic devices). These processes help us remember the information later. They are responsible for the formation of explicit memories. However, activities such as projects, explorations, experimentation, and problem solving situations are helpful in forming implicit memories. Proficiency in mathematics calls for both. Implicit memories build understanding and comprehension and context for a particular concept and explicit memories help in developing fluency, skills, and procedural competence. The integration of the two types of memories helps us in the applications and problem solving processes.

Components and Related Tasks
Different components of the working memory have specialized roles in learning arithmetic. For example, the phonological loop appears to be involved in arriving at facts by using a variety of strategies: (a) counting: when a child tries to find 8 + 6 by counting up and says: 9, 10, 11, 12, 13, and 14 and at the same time keeps track of the addend 6 as 1, 2, 3, 4, 5, and 6; (b) decomposition/recomposition: (i) when the child says: 8 + 2 is 10 and then 4 more is 14; (ii) when he says: 6 + 4 is 10 and 4 more is 14; (iii) using properties of mathematical entities in holding information involved in complex calculations: to find the answer for 23 × 7 mentally, the child first thinks of 23 as 20 + 3, calculates 20 × 7 and thinks 2 × 7 = 14 so 20 × 7 = 140 and 3 ×7 = 21, and 140 + 21 = 161, so 23 × 7 = 161 (also known as distributive property of multiplication over addition); and verbal rehearsal of the problem to keep it current: to find 23 × 7, the child keeps repeating the problem or the components of the problem. The actions in (b) rely heavily on visual/spatial working memory.

Children with poor arithmetic have normal phonological working memory but have impaired spatial working memory and some aspects of executive processing. Compared to ability-matched controls, they are impaired only on one task designed to assess executive processes for holding and manipulating information in the long-term memory. These deficits in executive and spatial aspects of working memory seem to be important factors in poor arithmetical attainment.

The visuo-spatial sketch pad appears to be involved in operations involving multi-digit problems where visual and spatial knowledge of column positioning is required, relationships between positions of digits in multi-digit numbers, and location and position of objects in visual clusters. For example, mentally locating numbers on the Empty Number Line as in when we find the difference 91 – 59, one thinks of 59 on the number line and then takes a jump of 1 to reach 60 and then a jump of 31 to reach 91, arriving at the answer of 91 – 59 = 32, and spatial representations of individual numbers. All of these actions take place in the working memory’s visual-spatial sketch-pad.

The role of the central executive is noted in many situations in learning and mastering arithmetic language, concepts, operations, and procedures. The central executive processor is responsible for identifying, initiating and directing processing, symbol and word recognition, comprehension and understanding, and retrieval of relevant information from the long-term memory. For example, all the decisions in estimating and computing the answer for 23 ´ 7 mentally are executive functions of the central processor and are being processed in the working memory. To estimate, the student first thinks of 23 as about 20 and 7 as either 5 or 10. And converts the problem mentally as 20 ×10. Then he thinks 20 × 1 = 20, so 20 × 10 = 200, or he thinks of 20 × 7. This is possible, of course, if he knows the concept: What happens when you multiply a number by 10, place value, and the table of 2. Of course, there are other routes. However, a child who thinks of 23 is made up of 2 and 3 will never be able to estimate the answer. This is an interplay between executive function in the working memory and the information being brought from the long-term memory. To compute 23 × 7, the child first thinks 23 as 20 + 3, therefore, 23 ×7 is thought as (20 + 3) × 7, and then to calculate 20 ×7 he thinks 2 ×7 = 14 so 20 × 7 = 140 and 3 × 7 = 21, and 140 + 21 = 161, so 23 × 7 = 161. All of these decisions involve the central executive.  When a student has mastered the operation of multi-digit multiplication with understanding, he can apply the procedure mentally.  That will again take place in working memory with help from long-term memory.

Appropriate and precise language, effective concrete and pictorial models, and efficient strategies are important not only for learning quality content but also for improving student learning capacity, including the working memory.

(Part Two: How to Enhance Working Memory for Mathematics Learning)

 

 

 

 

Working Memory: Role in Mathematics Learning (Part One)

Look for and Express Regularity in Repeated Reasoning: Concepts to Procedures

It has been well said that the highest aim in education is analogous to the highest aim in mathematics, namely, not to obtain results but powers, not particular solutions, but the means by which endless solutions may be wrought. George Eliot, 1885

In the previous standard (Look for and use the structure of mathematics), the framers of CCSS-M and SMP placed an emphasis on students learning to observe and appreciate the structure of mathematics—paying attention to its unique properties and systems, big ideas, and organizing principles, appropriate to their age and grade level. The goal of this standard is to help students to see the nature and appreciate the power of mathematics.

The eighth standard: Look for and express regularity in repeated reasoning promotes the ability to recognize quantitative (numeric) or spatial (geometric) patterns, make connections, and use those patterns to arrive at an understanding of mathematics structures—the general methods, the reasons behind the efficiency and elegance of algorithms/procedures, work out effective strategies, properties of numbers and operations at their grade levels.

In the previous standard, students looked for and used the structures in mathematics to apply them to learn more mathematics and solve problems. In this standard, the aim is to develop those structures from concepts and even construct new ones—new to students or even new to mathematics at their grade level. This process provides skills for students to own, retain and communicate, efficiently, accurately, and with rigor, the understanding and mastery of mathematics content. It also means developing the awareness of their own learning and the nature of mathematics conceptualization—the patterns. But most importantly, the objective is to help students to create interest in mathematics and to continue it with accuracy, confidence, and enthusiasm beyond school.

Mastery of this standard allows students to complete the cycle of learning mathematics: try problems, recognize a pattern in the solution process, discern a pattern(s), see the mathematics structure in the pattern, generalize the terms of that structure, generate a conjecture, test the conjecture, and then arrive at a key concept, procedure, strategy, or relationship between different elements of mathematics.

What is a pattern?
A pattern is a relationship between many of pieces of data, that is on going, something that is repeating—a data or a relationship between data, and can be predicted either explicitly or implicitly. Can one have a pattern when there are one, two, or three pieces of data? One piece of data is an incident, two are a coincident, with three there is a possible beginning of a pattern. If there is definite relationship between these pieces of data, then one needs a fourth one to verify that pattern.

Observing patterns and regularity in processes is the key to learning and using the prerequisite skills for mathematics learning. Students experience and develop affinity for mathematics when they also develop the prerequisite skills for mathematics learning. These prerequisite skills include: ability to follow sequential directions, classification, organization, quantitative and spatial reasoning, visualization, deductive and inductive reasoning. In designing lessons, therefore, teachers must also include the development of these prerequisite skills in addition to introducing interesting and relevant content. However, they should be integrated rather than isolated exercises.

A central idea here is that mathematics is drawing general results (or at least good conjectures for the moment) by trying examples and looking for regularity in the results generated by these patterns. In this standard, we want students to experiment with examples and arrive at patterns and then generalize those patterns to arrive at workable rules and procedures.

When students have explored several problems related to a concept, a few inevitably say, “I see a pattern here, so do I have to do all this work?” This is where the teacher moves the learning into generalizing mode—using the pattern to arrive at a conjecture, testing it and then arrive at the procedure or rule. She can’t stop at the pattern level. Students need to test their pattern as well as to recognize it in diverse situations and construct counter examples or non-examples. For example, instead of teaching just the rules for combining integers, students should look at patterns of combining numbers to generalize and come up with the rule. In the number -3, the digit 3 is the numeral and -3 is the number. Thus, a number is a directed numeral.

Mathematically proficient students develop two opposing but intertwined abilities: persevering in solving problems and looking for generalizable, efficient methods (i.e., short-cuts). These students notice patterns in calculations, use those patterns to create shortcuts, and learn multiple algorithms for the same idea and choose the appropriate one for the context. This helps them to do more interesting mathematics with less effort, thus generating an inclination for mathematics.

On the other hand, low achieving students lack both these qualities. As a matter of fact, low achieving students work harder in mathematics classes as they persevere using inefficient and long-drawn out methods, for example, they will use counting for learning addition and subtraction facts rather than efficient strategies based on decomposition/recomposition. Partly, it is not their fault; they have been instructed to do so. Because of these laborious methods, they give up easily. For them mathematics becomes hard, uninteresting, and a collection of unrelated procedures because they are not given help to develop efficient strategies and practice in seeing patterns and mastering strategies.

Solving many problems develops students’ inductive reasoning, and then justifying and communicating their thinking develops deductive reasoning. Inductive and deductive thinking are essential components of the mathematical way of thinking. The SMP classrooms are expected, through these practices, to develop habits of higher order thinking skills. Synthesizing concrete models and examples into symbolic and abstract ideas, at the elementary school, are good examples of applications of inductive reasoning. In contrast, most middle school and high school concepts and procedures are based on deductive reasoning or analysis.

Habits of mathematical thinking (observing patterns, looking for structure and regularity, repeated reasoning, making conjectures, providing examples, and counter examples, etc.) enrich learners’ problem solving experiences. By looking for patterns, developing conjectures, making discoveries and inferences, generalizing results, monitoring the solving process, checking the reasonableness of answers, and devising new avenues to explore to make learning mathematics engaging, they develop the habit of persevering (increasing stamina) for problem solving and communicating what they know.

Mathematical practices such as constructing reasoning and critiquing the reasoning of others are examples of deductive reasoning; however, this particular standard is best developed when students have extensive experiences in inductive reasoning. Observation of regularity, patterns, structure, and symmetry develops inductive reasoning. Further, practice of formal mathematics logic and reasoning such as supporting arguments and formal proofs develops rigor and a deeper understanding of mathematics. Deeper understanding in mathematics is an integration of deductive and inductive reasoning: analysis and synthesis in the process of solving problems.

Being a student of mathematics means balancing both qualities. Sometimes one needs to persevere and to stick with a set of strategies that will definitely work, given time. Putting in the effort, even though it is hard, is necessary. But, the effort must be put in developing and using efficient methods and strategies. Getting stuck doing the same inefficient methods is not what being a mathematician is about. Mathematicians explore new approaches and look for efficient and elegant solutions. They want methods to be powerful and generalizable. They effectively use what they know and have done to become efficient, more powerful problem solvers. They look for and represent regularity in repeated reasoning in order to develop efficient shortcuts or notations. This standard urges us, as teachers, to prepare and remind students that mathematicians move from painstaking but reliable problem-solving methods to efficient procedures via looking for regularity in their repeated reasoning.

Look for and express regularity in repeated reasoning.
Regularity and repeated reasoning abound in mathematics. Many facts, diverse terms, concepts and processes reoccur in the development of mathematical ideas. In this standard, the key words are: “look for” and “express” regularity, which means the realization on the part of students that learning mathematics is looking for and seeking regularity, recurrence, and patterns, the role of regularities and patterns, and describing these with mathematical symbols and expressions. Experiencing regularity and repeated reasoning—the processes and procedures, that sometimes may be generalized and even created into shortcuts, may lead to developing a rule, creating a formula, or arriving at an efficient and elegant procedure.

For example, the idea of decomposition and recomposition of numbers recurs whenever we add natural numbers (58 + 64 = 50 + 8 + 60 + 4 = 50 + 60 + 8 + 6 = 50 + 60 + 8 + 2 + 4 = 50 + 60 + 10 + 2 = 50 + 50 + 10 + 10 + 2 = 100 + 20 + 2 = 122, 58 + 2 + 62 = 60 + 62 = 60 + 60 + 2 = 122 or 58 + 64 = 50 + 60 + 8 + 4 = 110 + 12 = 122) to rational numbers. Consider the example,
A recipe calls for 1cups of milk. Larissa has 3cups of milk. How much milk will be left for the cake recipe, she plans to make?

Using the decomposition/recomposition, one observes:
1⅞ +⅛ =2 and then 2 + 1⅛ = 3⅛.
Therefore, to go from 1to reach 3⅛, we took jumps of cap1and or cap2        

Both of these examples can be translated into a powerful method called Empty Number Line (ENL). Several other methods have emerged in the same way (e.g., the Bar Model, used in Singapore and other Asian countries).

The area of a rectangle is used in arriving multiplication procedures from two-digit by two-digit numbers to multiplication of complex numbers. This recurrence is the basis of making connections between concepts and procedures.

For example, consider the instructional sequence, in a problem in which students study the concept of rate and ratio in a mixture of juice (NCTM Practice Illustration, 2013).

For every 5 cups of grape juice, mix 2 cups peach juice. Students are asked to develop a general mathematical relationship between the amounts of juices (say x and y, representing the two juices) needed to create a particular amount of mixture with the same taste.
cap3

To graph this data, we begin with (0,0) and one finds that for each 1 unit one moves to the right, one moves up of a unit. Thus, when one goes 2 units to the right, one goes up 2· units. Similarly, when one goes 3 units to the right, one goes up 3· units and when one goes 4 units to the right, one goes up 4· units. Observing this regularity, one can generalize this to:
                        When one goes x units to the right, one goes up x· or (x) units.
Therefore, starting from (0, 0), to get to a point (x, y) on the graph, one goes x units to the right, then one goes up units. Therefore, one can relate the two variables, x and y by the expression: y = = (x)

While students may immediately notice some regularity, it is the process of expressing—from noticing a pattern in the observations about the data in the table—the pattern into statements like “if we increase the grape juice by 1 cup, we must increase the peach juice by of a cup to have the same” in language and ultimately to translating this statement into writing the equation y = (x) which constitutes the bulk of the mathematical work of the task. The formulation of CCSSM, at appropriate places, provides language to discuss this kind of expressive mathematical work—a result from the regularity of the data.

Mathematically proficient students notice if calculations are repeated. They look for general methods (by observing specific examples, they extend to generalized situations) and for shortcuts (from general cases, they arrive at efficient applications to specific cases). The observation of regularity and repeated reasoning is an important aspect of both deductive and inductive reasoning.

In the early grades, students notice repetitive actions in counting and computation, etc. For example, all addition and subtraction facts can be derived by using the strategies that employ the property of making ten and decomposition and recomposition of numbers (9 + 7 = 9 + 1 + 6 = 10 + 6 = 17, 9 + 7 = 6 + 3 + 7 = 6 + 10 = 16). Similarly, when children have multiple opportunities to add and subtract―ten and multiples of ten, they notice the pattern and gain a better understanding of place value. And then decomposition/recomposition is extended to multiplication in the form of distributive property of multiplication over addition or subtraction.

For example, an addition fact-practice exercise might result in seeing patterns and generalization: ask children to choose three consecutive whole numbers (e.g., 5, 6, and 7) and compare the double of the middle number to the sum of the two outer numbers. In this example, the two sums are 12 and 12. After they do this for several such triplets of numbers, they are likely to conjecture a pattern that allows them to state that when the two numbers are two apart, their sum is double of the middle number, 6 + 8 = 7 + 7, 19 + 21 = 20 + 20, etc. In later grades, they will realize that the average (mean) of three consecutive numbers is the middle number and later they will generalize the result to the triplet: (N -1), N, and (N+1) with the equation: (N-1) + (N+1) = N – 1 + 1 + N = N + N = 2N.

After several examples with addition, some students may ask what will happen when we ask the same question about multiplication about this triplet. Teacher asks children to compare the middle number times itself to the product of the two outer numbers in the triplet of the consecutive whole numbers. The two products are 36 and 35. After they do this for several triplets of whole numbers, they are likely to conjecture a pattern that allows them to multiply 9 × 11, 29 × 31, 99 × 101, etc. mentally because they expect it to be one less than 10 × 10, 30 × 30, 100 × 100 which they can do in their heads. Seeing this regularity is typically easy for fourth and fifth graders, but expressing it clearly and formally is harder. They do not have the language for communicating.

In later grades, students are able to understand the idea of naming the numbers. A simple non-algebraic “naming” scheme can be used to describe the pattern: the numbers are named “middle/inner” and “outer.” In this form, the triplet can be seen as “middle minus one”, middle, “middle + 1” or (middle – 1), middle, (middle + 1). Then they can attempt to state the result as: (middle – 1) × (middle + 1) = (middle)2 – 1. The step from this statement to standard algebra is just a matter of adopting algebraic conventions: naming numbers with a single letter m instead of a whole word like “middle,” and omitting the × sign: m2 – 1 = (m + 1)(m – 1). It is revisited in algebra as we teach this pattern in factoring. It is generalized into: x2 – 1 = (x + 1)(x – 1). Let’s take the first problem: if x = 10, then 10 × 10 = 100 and 9 × 11 = 99. If x is 10, then x2 – 1 = 99 and (x + 1)(x – 1) = 100 – 1= 99. It helps students to begin with number patterns before moving into using the concept of variables to describe the pattern.

Upper elementary students, for example, might notice that when dividing 25 by 3, after the initial step, they are repeating the same calculations and conclude they have a repeating decimal. In order to arrive at these general observations, the teacher needs to give examples of this kind of division to show that the repetition of digits is dependent on the number used for the divisor. For example, when the divisor’s prime factorization involves only the prime numbers 2 and 5, the resulting decimal will be terminating as 10 and 10n, for every positive integer n have a prime factorization as 2m5p, for some non-negative integers m and p. For example:
cap4

On the other hand, when prime factorization of the divisor involves some other prime numbers or does not have 2 and/or 5 as prime factors only, the division results in a non-terminating, repeating decimal. The repetition will occur before the number of digits is less than the divisor. For example, in the case of ⅔, the repetition will take place with less than three digits as the divisor is 3. In the case of these fractions cap5 the repetition will take place before in number of digits is 7 or less than 7 and 11 or less than 11 digits, respectively.

Middle school students might abstract the equation (y – 2)/(x – 1) = 3 by paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3. Let (x, y) be any point on the line passing through (1, 2) with slope 3, then applying the definition of slope formula using two points (x, y) and (1, 2), one has the equation:
cap6
This converts into the form: y 2 = 3(x − 1), or by simplifying the equation, we get the equation in the y = 3x −1 in the slope-intercept form. Therefore, in general, the equation of the line passing through the point (x1, y1) and slope m is: (y – y1) = m (x –x1).

Earlier we observed that (x – 1) (x + 1) = x2 – 1.   Similarly, noticing the regularity in the way terms cancel when expanding (x1)(x + 1) = x2 + x – x – 1 = x2 − 1, (x – 1)(x2 + x + 1) = x3 − 1, and (x – 1)(x3 + x2 + x + 1) = x4− 1, etc., we can generalize it to:
                                       (a – b) (a2 + ab + b2) = a3 – b3
and by symmetry, we have:
                                       (a + b) (a2 – ab + b2) = a3 + b3.

However, if we continued the repetition and the pattern leads to the general formula for the sum of a geometric series:
Let S = 1 + x + x2 + x3 +… + xn-1 + xn.
Then, multiplying both sides by (x − 1)(assuming, of course, if (x − 1) ≠ 0, or x ≠1, as we can only divide or multiply by a non-zero number), we have
(x − 1)S = (x − 1)(1 + x + x2 + x3 + … + xn)
(x − 1)S = xn− 1
S = (xn− 1)/(x− 1).
Therefore, we have:
(x − 1)(1 + x + x2 + x3 + … + xn) = xn− 1
And, when |x|<1, xn approaches 0 for n is very large, then the above relation gives us the sum for a geometric series:
1 + x + x2 + x3 +… + xn + …
S = 1/(x−1).

Regularity also means looking for similarity and symmetry. The development of divisibility rules is a good example of regularity, symmetry, and repeated reasoning. For example, when we consider the divisibility test for 2, we look for the one’s place being even as the value of digits in other places (10’s, 100’s, etc.) are already divisible of 2 and will not leave any remainder when divided by 2 as its value is a multiple of 10n for every natural number n which in turn is a multiple of 2. Thus, a six digit number: 135,798 is divisible by 2 as 135,798 = 100,000 + 30,000 + 5,000 + 700 + 90 + 8, each one of the numbers is divisible by 2, therefore, their sum is also divisible by 2. Therefore, a number is divisible by 2 if the digit in the one’s place is an even number.

Similarly, when we consider the divisibility test for 4, we look for the number formed by the ten’s and one’s place digits being divisible by 4 as all the other digits in the number will not yield any remainder as their value is a multiple of 10n, for every natural n greater than 1 is a multiple of 4 (i.e., 102 =100, 103 = 1,000, etc.). Therefore, a number is divisible by 4 if the number formed by the ten’s and one’s digits is divisible by 4. Example: In the number 333,324, the ten’s and one’s digit form the number 24, that is divisible by 4, so 333,324 is divisible.

Now, we can develop a test for divisibility by 8 by following this pattern, e.g., 111112 is divisible by 2 (look at one’s place); 111132 is divisible by 4 (look at 32—the number formed by the ten’s and one’s place); and 111128 is divisible by 8 (look at 128—the number formed by hundred’s, ten’s, and one’s place). Thus, for the divisibility by 8, we look at the number formed by the 100’s, 10’s, and 1’s digits as all other digits in the number will have values as multiples of 10n for every natural number n > 2, and when divided by 8 will not leave any remainder. Thus, a number is divisible by 8 if the number formed by the hundred’s, ten’s and one digit is divisible by 8. Or simply, if a number is divisible by 2 and then the new quotient is again divisible by 2, then the given number is also divisible by 4 and if the number is divisible by 2 and then by 2 and then again by 2, then the number is divisible by 8. For example, 1448 is divisible by 2, 4, and 8, so is 111,448.

For elementary and middle school children above results are achieved by observing the nature of many numbers and actually performing the division or the teacher playing a game of seeing the pattern in the divisibility and then arriving at the tests. 

Many middle school and almost all high school students should be given experiences in deriving these results. In general, the number N = an an-1 an-2 ……. a0 (In the expanded form it is N =10n an +10n-1 an-1 + ……+ 102 a2 +101 a1 + 100 a0 ) is divisible by 2, if a0 is an even digit and an, an-1, an-2, ….., a1 can be any digit. N is divisible by 4 if the number the number a1a0 (which in the expanded form is 101 a1 + 100 a0) is divisible by 4. Similarly, N is divisible by 8 if the number a2 a1 a0 (which in the expanded form is 102 a2 +101 a1 + 100 a0) is divisible by 8 as 10n an, 10n-1 an-1, 10n-2 an-2, …103 a3 are all divisible by 2, 4, 8 for any n ⫺ 3.

The divisibility by 5 and 10 is quite clear: the one’s place should have 0 or 5 in the case of divisibility by 5 and 0 in the case of divisibility by 10 as all other places are multiples of 10n, where n ⫺ 1.

Let us consider the divisibility by 3. We can see a pattern in the numbers that are divisible by 3. For example, in the numbers 33; 24; 234; 111; 232323 the sum of the digits in these numbers is a multiple of 3 and we observe that the numbers are divisible by 3. Once again, the teacher gives the examples of numbers that are divisible and not divisible by 3. And create two columns. One for the numbers divisible by 3 and the other with those not divisible by 3. By observing these numbers and share their observations. The teacher provides examples, counter examples, and non-examples to test their conjectures. Students arrive at a pattern. Thus, by observation they arrive at a conjecture (a conjecture is an observed pattern) for the divisibility of 3: A number is divisible by 3 when the sum of its digits is a multiple of 3.

In a general case, we can use mathematical reasoning to prove this statement. The proof for high school students is a very good example of understanding place value, laws of exponents, and the concept of division. Let us consider a six digit number: N = abcdef where a, b, c, d, e, f are digits in the number and can assume values between 0 and 9, except a ≠ 0, otherwise it will not be a six digit number (we could easily extend it to a more general number and any number of digits, N = an an-1 an-2 ……. a0 =10n an +10n-1 an-1 + ……+ 102 a2 +101 a1 + 100 a0, where the coefficients are digits ).
N = abcdef = a×105 + b×104 + c×103 + d×102 + e×101 + f×1
= a×100000 + b×10000 + c×1000 + d×100 + e×10 + f×1
= a×(99999+1) + b×(9999+1) + c×(999+1) + d×(99+1) + e×10 + f×1
Now, if we divide the number by 3, the numbers 99,999; 9,999; 999; 99; and 9 will not have any remainder. When divided by 3 each one of the expressions (99999+1), (9999+1), (999+1), (99+1), and (9+ 1), and 1 will have a remainder of 1. Thus, the expressions a×(99999+1), b×(9999+1), c×10(999+1), d×(99+1), e×10, f×1when divided by 3 will have remainders a, b, c, d, e, and f respectively. Therefore, the right hand side of the expansion of abcdef will have only a + b + c + d + e + f as the remainder. Thus, when the number N= abcdef is divided by 3, then the remainder is a + b + c + d + e + f. Therefore, the number N = abcdef is divisible by 3 if the sum of the digits a + b + c + d + e + f is divisible by 3. Thus, the divisibility test for 3 is proven. Similarly, the test for divisibility by 9 can be developed using the same reasoning: A number is divisible by 9 if the sum of its digits is a multiple of 9.

The number 6 is a product of 2 and 3. The test for 2 and 3 combined works for 6. Thus, a number is divisible by 6 if it is divisible by 2 and 3. This test works because the numbers 2 and 3 have an interesting relationship. 6 = 2×3 and 2 and 3 are relatively prime (two numbers are called relatively prime when their greatest common factor is 1).

This suggests that this divisibility test can be generalized to any whole number. For example, a number is divisible by 24, if it is divisible by 8 and 3; since, 24 = 8×3 and 8 and 3 are relatively prime number. For example, 144 is divisible by 24 because 144 is also divisible by 8 and 3. However, a number may not be divisible by 24 if the number is divisible by 12 and 2 as 12 and 2 are not relatively prime numbers (2 is their greatest common factor). For example, 36 is not divisible by 24 although 36 is divisible by12 and 2 and 24 = 12×2. The condition of two factors being relatively prime is important. Thus, in general, a number n is divisible by another number c, if n is divisible by a and b, where c = a×b, and a and b are relatively prime. Thus, for most purposes—learning operations on fractions, simplifying radical expressions, etc., the divisibility test for up to ten are adequate. However, there is not a “good” test for the divisibility by 7. There is a procedural test for the divisibility by 7 developed in the 7th century.

A great deal of research shows that one can predict which students will do well in higher mathematics are those who are well-versed and fluent in long division and operations on fractions. This is particularly so in a course on algebra. And the four major prerequisite skills for understanding and mastering the concept of fractions and operations on them are:

  • Multiplication tables
  • Divisibility rules
  • Short-division, and
  • Prime factorization.

These are the tools of success in mastering fractions and then algebraic operations. These skills are best derived by patterns and then practiced using these patterns.

To make the observation of regularity, pattern, repeated reasoning, and structure possible, both teachers and students need to pose a series of questions such as:

  • Is there a process—operation, property, condition that I am repeating?
  • Can I use this repeated idea to create a pattern and then to develop a conjecture?
  • Can I make a generalization of the conjecture?
  • Can I express this abstractly based on this repeated reasoning?
  • Does this abstraction result into a procedure, method, or strategy?
  • Are there exceptions to this generalization/abstraction?
  • Can I make this method, procedure more efficient and elegant (this is like editing a piece of writing)

Teachers can ask questions such as:

  • You claim that you observed a pattern, how do you know it is a pattern?
  • Can you describe your pattern, conjecture, or method?
  • Explain why your method makes sense?
  • Can you describe your method in formal mathematics terms and reasoning?
  • How would you explain why it works?
  • Have you checked whether the answer seem reasonable?
  • What have you learned about ….?
  • How would this work with other numbers or situations? Does it work all the time? How do you know?
  • What do you notice when …?

Mathematics is about constructing and solving problems, and interesting problems must be the focus of students’ mathematical experiences; they should be the content of daily mathematics lessons. Problems should be accessible to children, yet they should be modestly challenging and at times creatively frustrating. These kinds of problems should be the focus when students and their teachers are engaged in sharing their thinking process. These thinking processes— having ideas, not having ideas, seeing relationships, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work are the back-bone of the standards of mathematics instruction.

Standards of mathematics practice—techniques and methods, when routinely practiced, will implement the content advocated in the CCSSM. Arising naturally out of practice, content will not be in isolated from; it will be an organic outgrowth of the problem background. When a concept and procedure are learned, they should be practiced till they convert into skills and tools for doing meaningful mathematics. Doing mathematics is different than knowing mathematics. However, practice is a good reinforcement tool if students have learned the content first.

The Standards of Mathematics Practices (SMP) aim to develop mathematics content, the mathematical way of thinking, and mastery of the content with deeper understanding. As students work on solving problems, they are guided to observe regularity and symmetry, discover relationships, and make conjectures. They are asked to evaluate the reasonableness of their intermediate results and arrive at mathematical concepts. They articulate these results and then prove or disprove their results using formal reasoning and their prior knowledge of related concepts. The success of this evaluation process is dependent on several skills that can be developed with the following activities:

  • Encourage students to share multiple methods for solving the same problem and then request that students use someone else’s method to solve a similar problem.
  • After sharing and naming multiple methods for a problem, tweak the parameters and conditions of the problem and ask students “Which method would you use to solve the problem if it were like this?”
  • Playfully and explicitly add an element of time to students’ problem solving, once they have had experience developing methods for a particularly problem structure. For example, if students have painstakingly drawn pictures and calculated the sum of the interior angles of rectangles, rhombuses, pentagons, and hexagons, and shared their different methods for doing so, ask them to think about ways they could get more efficient. Then ask them to make a bid for how many polygons they could find interior angle sums for in 5 minutes. Pass out the requested number of puzzles to each team, and start the clock. While you’ll have to pay attention to group dynamics and support students to be good team members even under pressure, you’ll also hear good ideas about mathematical shortcuts that didn’t come up when students didn’t have any reason to need more efficient methods.

Another purpose of this standard is to make sense of the formulas we use by finding a pattern or relationship in numbers generated during an exploration of the topic. It is to understand that there are purposeful connections between procedures and concepts or between the manipulation of tools and a procedure. For example, students can explore the area of a triangle by building rectangles, drawing the diagonal, and realizing that the area of a triangle is half of the area of a rectangle formed with the same base and same height—thus the formula is developed and then relate to the parallelograms of the same base and height.

It is important to design lessons where patterns are revealed and lessons that encourage students to make generalizations. Some of these tasks involve students working with prior knowledge in a non-routine way.

The learning that takes place through these lessons is incredible. Using the best-designed lessons is not the only necessary component in developing this practice in students, so be sure to facilitate conversations that include comments and questions such as:

  • Notice repeated calculations and reasoning
  • What generalizations can you make?
  • Look for general methods and shortcuts
  • What mathematical consistencies do you notice?
  • How would you prove that …?
  • What would happen if …?
  • What predictions or generalizations can this pattern support?
  • How would this strategy work with another number? Does it work all the time? How do you know?
  • Maintaining over sight of process while looking at details evaluate reasonableness of results
  • Finding new structures/methods, generalizing
  • Can you find a shortcut to solve the problem? How would your shortcut make the problem easier?

 

 

 

 

 

 

 

Look for and Express Regularity in Repeated Reasoning: Concepts to Procedures

Look For and Make Use of Structure: Understand the Nature of Mathematics

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. G.H. Hardy

The Common Core State Standards for Mathematics (CCSS-M) include both content standards and standards for mathematical practice (SMP). The content standards define “what students should understand and be able to do.” The standards for mathematical practice describe “varieties of expertise that mathematics educators…should seek to develop in their students.” The “what” part encourages students to amass a body of content whereas the “why” part develops students’ mathematical way of thinking. These practices help them become better learners of mathematics and problem solvers. The why part in teaching adds value to both student learning and formative assessment by the teacher. It informs the teacher and the students. Unless we give students opportunities to work on tasks that target the standards for mathematical content and require students to explain their reasoning with models, diagrams, equations, or oral and written explanations of the structure of the mathematics of the task, we might find ourselves with a limited or false sense of student understanding.

Looking for patterns in information and making use of the structure in mathematics ideas is a fundamental process of mathematics learning and an important component of the mathematical way of thinking. We must present students with tasks that address the content standards with rigor, using as many of the standards of mathematics practices as possible. It is crucial for students to look for and make use of a structure because this practice requires them to reason about the underlying mathematical structure and unity of mathematics ideas.

Children naturally seek and use structure. They learn their native language by observing patterns and then extending them. If the extension works, they become bold and create language expressions. Mathematics has far more consistent structure than our language, but too often it is taught in ways that don’t make that structure easily apparent. If, for example, students’ first encounter with the addition of same-denominator fractions drew on their well-established spoken structure for adding the counts of things—three books plus four books make seven books, three hundred plus four hundred make five hundred, and three globs plus four globs make seven globs, no matter what a glob might be—then they would already be sure that three ninths plus four ninths makes seven ninths. Developing the linguistic structure first is important so that we add or subtract only if the two “things” we are adding are of the same type or have some common property or common characteristic. Instead, children often first encounter the addition of fractions in writing, as 3/9 + 4/9, and they therefore invoke a different pattern they’ve learned—add everything in sight—resulting in the incorrect and nonsensical 7/18.

Structure defines a language and form defines ideas. Mathematics is a language and collection of wonderful ideas. First we need to acquire the structure and form of the language and then we can play with it. Creativity is part of all learning and it takes place in organized chaos. In the chaos of problems, in the midst of new information, students need to see the pattern—organization and structure of the problems in order to understand them and enter into the solution process.

“Look for” “patterns,” and “structure” are key phrases in the seventh standard of math practice. It calls for going beyond the given information in the problem and shifting the perspective to discern relationships between pieces of information either explicitly or implicitly or predict the structure in the information. The skills involved in the process are:

  • Look for relationships (explicit and implicit) between the information given in the problem and also what is hidden
  • Look for pattern/structure in the problem or concept under discussion
  • Step back for an overview/shift perspective
  • See something as a whole or as combination of parts
  • Using familiar/known structures to see something in a different way

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, using Cuisenaire rods might notice that the 3-rod and 7-rod joined together are the same length as the 7-rod and 3-rod joined, therefore, infer that three and seven more is the same amount as seven and three more and that, with a few more example, results in the generalization: commutative property of addition (3 + 7 = 7 + 3; light green + black = black + light green; lg + bk = bk + lg).

Children may sort a collection of shapes according to their different attributes, for example, of how many sides the shapes have and arrive at a definition of that category of objects: A triangle is a three straight-sided closed figure or three sided polygon; a quadrilateral is a four straight-sided closed figure or four-sided polygon; a decagon is a ten straight-sided closed figure or ten-sided polygon; and a n-gon is a n straight-sided closed figure or n-sided polygon. The concepts are clearer and learned better when the structure is emphasized. The examples below, one from geometry and one from arithmetic make the structure of mathematics evident.

Example 1: Structure of Classification
Capture1

Example 2: Decomposition/Recomposition and Distributive Property
Students see that a rectangle made using 7 brown Cuisenaire rods (8-rods) defines the multiplication fact 7 × 8 (area model) and then they create two rectangles from it: 7 by 5 and 7 by 3 of known facts 7 × 5 = 35 and 7 × 3 = 21, therefore they know 7 × 8 equals the well remembered 7 × 5 + 7 × 3, which when several such examples are used is generalized into the distributive property of multiplication over addition. In fact, the standard procedure/algorithm of multiplying two multi-digit numbers is derived from the distributive property of multiplication over addition.
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The properties of operations (addition and multiplication) such as associative, commutative, and distributive properties and properties of equality (=), such as reflexive, symmetric, transitive, and equivalence are indispensable in achieving fluency with computations and algebraic manipulation. Without them many students waste time in simple computations such as:
Compare the two expressions
2×3×2×5×2 and (b) 3×2×5×3×2×2
What is the relationship between the two expressions?

Students invariably compute both of them rather than just noticing that both of them have a common factors: 2, 2, 2, 3, and 5 and the second expression has an extra factor of 3; therefore, the second expression is 3 times the first product. And during the computation they will compute from left to right rather than recognizing that two of the factors are 2 and 5; therefore, it is easier to multiply them first and then the others. This shows they are not looking for the inherent structure of mathematics (commutative and associative properties and the pattern of multiplying by 5 and then multiplying by 10).

Let us consider a third grade problem to see the importance of this structure.
Robert has 48 candies with 6 pieces of candy in each bag. How many pieces of candy does Robert have? Jessica has 16 bags of candy with 3 pieces of candy in each bag. How many pieces of candy does Jessica have?

Compare Robert and Jessica’s bags of candy. Who has more and how do you know? Make a diagram or write an equation that explains how you know who has the most and why?

Some students claim Robert and Jessica have the same amount of candy even though one of them has 8 bags and the other has 16 bags. Explain with a diagram or an equation why you agree or disagree.

This simple task asks students to work with several content standards:

  • Interpret products of whole numbers. Students wrestle with the meaning of the factors 8 × 6 and 16 × 3 in a multiplication problem.
  • Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities. Students compare how much candy Robert and Jessica have.

This simple task also asks students to use a variety of mathematical practices:

  • Students must make sense of the problem and persevere as they attempt to determine the way to represent both students’ amounts of candy with a diagram and equations.
  • Students must reason abstractly and quantitatively because the task is a contextual situation and students are required to write an equation to represent each student’s candies. When students re-contextualize the algorithms in the context of the situation and explain the meaning of the expressions, they will demonstrate if they can work quantitatively.
  • Construct viable arguments and critique the reasoning of others. Students are likely to reason that Jessica and Robert have the same amount of candy because both have 12 pieces of candy, thus constructing a viable argument.
  • Model with Mathematics. Students’ equations, diagrams of the bags of candies and their written explanation will let us know if they have a means of modeling with mathematics.

However, by asking the last part of the question, they are engaged in examining the structure of multiplication. Without the final prompt, we may fail to find out if students really understand the reason why both expressions equal 12 candies. The focus on 2 bags versus 4 bags draws students’ attention to the number of equal groups in relationship to the number of items in each of the groups. Ideally, students will explain that the two students’ amounts are equivalent because although Jessica has more bags, he has fewer candies in each whereas Robert has fewer bags but each bag has more candies so, in the end, the total number of candies are the same.

The distributive property derived using the area models of multiplication is then applied to multiplication of binomials and relates to factoring of trinomials. Let us consider the following example:
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With a few more examples, students see the pattern and (x + 2)(x + 3) = x2+ x(2 + 3) +2Ÿ3 = x2 + 5x + 6 can be generalized into (x + a) (x + b) = x2+ x(a + b) + ab.

Every problem should be examined, even after it has been solved, from different perspectives to get depth, to make connections, and to gain efficiency in the solution process. The reexamination is to step back for an overview and shift perspective.

Mathematically proficient students look for the overall and inherent structures and patterns in mathematics. For example, they can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects (or a compound figure as one geometric entity and at the same time composed of several other figures with specific properties). They also observe the nature of numbers in complex expressions. For example, they can see 5 – 3(xy)2 as 5 minus a positive number times a perfect square (which in the case of a real number is also positive) and use that to realize the value of the expression cannot be more than 5 for any real numbers x and y, and later they realize that in the expression f(z) = 5 – 3(xy)2, f(z) has a maximum value of 5 for all real values of x and y.

Mathematically efficient students observe that patterns and repeated reasoning can help them solve more complex problems. For young students this might be recognizing fact families, that arithmetic operations have their inverse operations, properties of operations (commutative, associative, and distributive properties), of numbers (prime, even, divisibility, etc.), and as students get older, they can break apart problems and numbers into familiar relationships. For example, in a Kindergarten class I asked them to calculate the expression: 5 + 1 + 7 + 5 + 8 + 2 + 3 + 9. The students who knew how to make 10 and the commutative property right away were able to give the answer: 40.

There are specific ways mathematics ideas are learned by students. They learn them through

  1. Using concrete models,
  2. Applying analogies,
  3. Seeking and extending patterns, and
  4. Applying formal logic and reasoning.

In the other standards, there was an emphasis on using concrete models, analogies in solving problems, and logic and reasoning. However, students learn quickly that finding a pattern simplifies the logic and makes access to the solution approach and solving the problem easier. Recognition of patterns helps students realize effective conceptualization. Larger/complex concepts can be seen as the generalization and extensions of simpler and primary concepts. As a result, the bigger task becomes simpler, and the custom command can be used to solve the “small problem.”

Knowledge is cumulative. What a student is capable of learning depends upon what she already knows. We can see more patterns when we have more learning in store.

To observe and seek structure, we may assume that many or most students are properly equipped to make sense of new information and are able to observe patterns. We need to constantly help students to learn and amass more skills and more knowledge. Students need to be guided down the path of their learning—to seeing patterns and structures. Teachers should remain central to the activity of imparting knowledge to students.  Learning is most effective and enjoyable when it is carefully sequenced and scaffolded by the teacher and when students see structure and patterns in the information, they make connections.

Structure is everywhere in mathematics, and when we understand the structure embedded in a concept like addition/subtraction or multiplication/ division, we have a deeper understanding of the concept.

There are deep ideas that nourish the different branches of mathematics. One can think of specific mathematical structures of and patterns in (a) Number systems, (b) Algorithms, (c) Shapes and relationships, (d) Data, and (e) Relationships, such as rate of change or ratio. These structures describe and give rise to attributes such as: (a) Linear or Non-linear, (b) Periodic or Random, (c) Symmetric or Chaotic, (d) Continuous or Discontinuous/ discrete, (e) Exact or Approximate, and (f) Maximum vs Minimum.

The skills of using models, applying analogous thinking and conditions, and taking advantage of pattern analysis are as important in mathematics learning as the use of logic and reasoning. The most important of these is pattern analysis.  Therefore, a central idea of mathematics is the study of patterns—to discern, extend, create, apply in order to make conjectures based on these patterns and then to arrive at general results, principles, and procedures. By using formal logic one converts the conjectures arrived by discerning patterns, if possible, into well-founded results—concepts, definitions, and procedures/algorithms. Seeing and revealing hidden patterns are what mathematicians do best. That applies to both developing and accomplished mathematicians.

We need to help our students see and reveal patterns in the given information. The guide to this growth is not just calculations and formulas but an open ended search for pattern. Pattern analysis involves four sub-skills: (i) discerning and recognizing patterns to understand the problem, (ii) extending patterns to enter into the solution process, (iii) creating new patterns to verify the validity of the solution process and generalize the approach, and (iv) applying patterns to discover and develop procedures.

Mathematically proficient student look for patterns and structure in mathematical situations. Students who are proficient at making use of structure are able to identify and create viable and efficient ways of looking at and using patterns. Students may then extend the pattern and derive the mathematical expression or equation related to that pattern. Once a pattern is understood, students can explain the process to their peers.

 As students look for and make use of structure, some of the questions they may ask include

  • Is there a pattern in the problem situation?
  • Can I state the pattern or structure as a rule?
  • Is this a rule that holds true every time?
  • If there the rule does not work every time, can I adapt it to work every time?
  • When does it not work?

Mathematically proficient students use structure to make connections and deepen conceptual understanding. The understanding of connections also enables students to shift perspectives and see the pattern from a different view.

To increase the likelihood of student pattern recognition, teachers can incorporate work on pattern analysis throughout the year. For example, a morning mathematics meeting framework, such as “calendar time” in K-2 grade classes is an excellent time to incorporate extra math pattern/structure exposure.

In the early grades, students can recognize, identify and apply pattern knowledge through calendar work, weather graphing and exploring daily numbers (e.g., date, number of days in school, and the number of the day) to incorporate extra math/structure exposure. For example, one of the activities that I have found to be useful in developing patterns and increasing children’s numbersense is “skip counting.” Skip counting should be done in each grade, even if it is just for a few minutes. For example, in Kindergarten: counting backward and forward by 1, 2, and 10 starting from any number; First grade: by 1, 2, 5, and 10; Second grade: 1, 2, 5, 10, and 100; Third grade: by 1, 2, 5, 10, 100, 1000, and unit fraction (e.g., 1/3, 1/5, etc.); Fourth grade: by proper fraction (e.g., 2/7, 3/10, etc.); Fifth grade through eighth grade: by mixed fraction (e.g., 1⅜,Capture4 , etc.).  Here is an example of counting activity in a second grade classroom.

Procedures

  1. Oral only
  • Announce the counting number and beginning number
  • Ask who has the next one, several times till all children had a chance
  • Once the pattern is established, continue around the room asking each student
  1. Oral and written
  • Announce counting number and beginning number
  • Record on the board to show the pattern, both vertically and horizontally
  • The number of entries in the columns changes everyday. For example, in the above example, the number of entries in each column is 4. Next day, there may be 8 entries in each column.
  • Ask students to write the next 5 numbers on their papers

Example:
149     169  189
154     174  194  ___  ___          ___   ___   ___
159     179  199
164     184  ___                                 ___

As children are writing their numbers, work with the student(s) who are having difficulty in counting (e.g., 199 + 5 = ?).

Teacher: What should you add to 199 to make 200?
Student: 200.
Teacher: Great! What more is left from 5 to be added?
Student: 4.
Teacher: What is 4 more than 200?
Student: 204.
Teacher: Now use the same strategy for the next number,

  1. Discussion

Once children have generated the five numbers, ask them to check each other’s work and then the teacher records them in the columns. After the entries have been recorded, identify an empty place in the column 4 row 2 (see the empty place 1), discuss with your partner to find what number will fit in this place. When you have the number raise your hand, both persons from the team hands must be raise their hands for me to recognize the team. One hand will not be recognized. Write all the responses on the side of the board and ask them to justify their answers. If the number is wrong, locate it in the right place with a comment: it belongs in this place. Do not disclose the pattern. If a team finds the correct answer and in their explanation discloses the pattern that is fine. Now identify another place (see column 5 row 2). Continue for several examples. A variety of questions can be formed, for example,

  • Ask students to fill in the places you have identified on the board.
  • Ask students to calculate the difference between 2 of the identified numbers. For example, what is the difference between 184 and 169? Etc.

The same process can be repeated in different grades with different numbers. For example, the counting will be by mixed fraction (say 1 starting from (say 7). The same kinds of questions are asked.

Mathematics is learned by doing and it should always include discovering patterns, developing and crafting conjectures, and providing correct and incorrect, rough and elegant, emerging, elegant, and meaningful explanations to understand the inherent structure of mathematics. Patterns abound in information generated in any field. Mathematics is no longer just the language of physics and engineering but now an essential tool for banking, manufacturing, social science, and medicine. Active mathematicians seek pattern wherever they arrive. For example, when mathematicians look at the numbers on any random page of a newspaper they discovered a pattern, the first digit of numbers, 1 occurs more often (about 30% more), 2 comes next (about 18% more) and so on. This phenomenon (Banford’s Rule) occurs often wherever random data is involved.

Mathematicians train their eyes to see patterns. “I see” has always had two distinct meanings for them: to perceive with the eye and to understand with the mind. Learning to see and recognize patterns and symmetry trains the mathematical eye. This “seeing” is the outcome of actions, such as:

        (a) Experimenting,

        (b) Discovering quantitative and spatial data,

        (c) Representing in language, pictures, diagrams, words, symbols,

        (d) Visualizing,

        (e) Classifying and organizing,

        (f) Identifying and defining,

        (f) Modeling,

        (g) Computing/approximating/estimating,

        (i) Generating conjectures,

        (i) Verifying, and

        (j) Proving.

These actions and processes result in: (a) generations of and representations in symbols, such as number symbols, variables, infinity, etc., (b) development of relationships—terms and expressions, (c) arriving at formulas, (c) representations through forms, models, figures, diagrams, and (e) structures, processes and algorithms.

The processes create structures in mathematics; however, some of these may have dichotomies. For example, numbers can be discrete or continuous; figures can be regular and irregular; relationships can be equations or inequalities; forms can be linear or non-linear; formulas can be explicit (closed) or recursive; etc. Students should be able to understand and apply dichotomies such as, (a) Discrete vs. continuous vs. chaos, (b) Finite vs. infinite, (c) recursive vs. explicit, (d) Algorithm vs. existential, (d) Stochastic vs. deterministic, (e) Exact vs. approximate. A sound education in mathematics requires student encounters with all of these.

At the elementary level, these relationships and structures are evident in numbers and their operations, in geometry, and in daily activities. The structure of mathematics gives the power to the learner. A procedure arrived at in some specific situations is applicable to infinitely many situations. Consider the normal distribution curve (Gauss Curve, Bell Curve), which was arrived at in a few specific situations, but it is now applicable to a myriad of situations—where there is randomness, there is the Bell curve and the results of normal distribution are applicable from IQ scores to population distribution, agriculture production to manufacturing widgets, and dispensing of coffee from the coffee machine to the distribution of the cars by age on the highway. Such structures abound in mathematics.

As an example of structure, consider the concept of place value: when children are asked to make a number, say 127, they are being asked to decompose 127 into hundreds, tens and ones in all the possible ways they can find. There are two kinds of structures: canonical decomposition: 127 = 100 + 20 + 7; or non-canonical decompositions: 6 tens and 67 ones; 12 tens and 7 ones; 1 hundred, 2 tens, and 7 ones; 1 hundred and 27 ones; etc.  In order to be efficient or complete in arithmetic operations, students need to use exchanges—of a ten for a group of ten ones or a hundred for ten tens—systematically, etc. The canonical decomposition and recomposition is at the heart of our base-ten number system. That is, there is a structure of systematic exchanges, which students must look for and make use of whether it is addition or subtraction procedure.  We can say that this task invites students to engage through both its “look for” and “make use of” structure. Children should explore the activity before systematic exchanges are suggested by the task or by the teacher. Unless they experiment first, the practice won’t be fully engaged in. Once a child can read a 3-digit number, she can read any number. Similarly, to read a multi-digit number (say, 9876543210), we explore the patterns to understand the structure of place value. First, we group them in 3 digits as 3-digit numbers, starting from right to left:

109,876,543,210.

Then, we name these groups; the second group (543) is called the thousand’s group. The “commas” in the number represent the name of the group and their names are called when you reach that comma. Since 543 is a three digit number it will be read as five-hundred-fort-three and it is the thousand’s group, it will be read as: five-hundred-fort-three thousand. Here, since the digit 5 in 543 is in the hundreds place, the value of digit 5 in 543 is 500 and value of digit 5 in 543,210 will be read as: five-hundred thousand. The next group is one (mil) group (llion) away from thousand so will be called group of millions. 876 being a 3-dgit number will be read as eight-hundred-seventy six. Therefore, the 876,543,210 will be read as: eight-hundred-seventy-six million… The next group is two (bi) groups (llion) away from thousand so will be called group of billions. 109 is a 3-didit number so will be read as one-hundred nine. Therefore, the 109, 876,543,210 will be read as: one-hundred billion, eight-hundred-seventy-six million… The place value of the digit 1 in 109 is one-hundred and the place value of 1 in 109,876,543,210 will be 100 billion. The next group will be trillion as it is 3 (tri) groups (llion) away from thousand, etc. The next group is quadrillion (quadri-four, llion-group), pentillion, sextillion, septillion, octillion, nonilllion, decillion, and so on. There is a pattern that defines the structure. This helps children to remember the place value.

To be useful, mathematical concepts, like power,  must be understood in their components, including their strength, reach and limits. It is when we understand the primary components of a concepts and their integrated relationships—the bonds of structure between them, that we understand the power and limitations of the methods and procedures derived from the concepts. For example, the power of the ‘long division procedure’ is really understood when we understand the structure of place value, estimation, and the four arithmetic operations that build the structure of the procedure. The long division procedure is such a powerful arithmetic achievement that depriving children of the economy, the fluidity, and the power of this procedure is to deprive them of many other related ideas, for example, the division of polynomials by binomials or the structure and behavior of the curves described by these polynomials.

Mathematically proficient students discern patterns, generalize ideas from specific situations, and then apply general mathematical rules to specific situations. Young students, for example, might notice and understand that our number system is based on the decimal system and certain numbers play an important role: 1 (iterative nature of natural numbers—1 more than a number is the next number), 10 (after a group of ten numbers we move to the next decade), and the teens numbers (10 + 1 = 11, 10 + 2 = 12, 10 + 3 = 13, etc. Therefore, it is important that children understand and master the structure of the number 10 (9 + 1, 8 + 2, 7 + 3, 6 + 4, 5 + 5), 1 more than a number, and the teens numbers. If the students know: (a) 1 more or one less than a number, (b) Making 10, (c) Teens numbers, and (d) commutative property, they can easily learn all the other addition and subtraction facts. All of the addition and subtraction strategies (number + 9; doubles, near doubles, missing double, near ten, etc.)[1] are extensions of understanding the making ten (structure) and decomposing/recomposing number patterns. For example, 8 + 7 = 8 + (2 + 5) = 10 + 5 = 15, or 8 + 7 = (5 + 3) + 7 = 15, or 8 + 7 = (3 + 5) + (5 + 2) = 3 + 10 + 2 = 10 + 3 + 2 = 10 + (3 + 2) = 10 + 5. And, then they can extend it to the procedure of addition: 28 + 37 = 20 + 8 + 30 + 7 = 20 + 30 + 8 + 7 = 50 + 15 = 65. This regrouping is a part of the fundamental structure of arithmetic. Almost all of the arithmetic procedures are derived from this structure of number 10.

Similarly, in the process of developing a proof for a theorem in geometry, they recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.

How to Help Students Look For, Use, and Reinforce Structures in Mathematics
This is mostly a learned behavior. In order that students look for and use structures of mathematics in learning new concepts and problem solving, teachers should point out the structures and patterns by asking enabling questions. Teachers can instill and then observe the same in students’ work.

  • What observations do you make about this data, this figure, these numbers, etc.?
  • Can you transform this expression into another form?
  • What do you notice when you simplify, transform…the …?
  • What different components, ideas, and concepts make this expression, equation, figure, problem, etc?
  • What parts of the problem might you estimate or simplify first?
  • What property of numbers, quadrilaterals, graph, … can you apply?
  • How do you know if something is a pattern?
  • What patterns do you find in…?
  • Are the conditions of a pattern satisfied here?
  • What are the characteristics of the pattern in this problem?
  • What ideas that we have learned before were useful in solving this problem?
  • What are some other problems that are similar to this one?
  • How does this relate to …?
  • In what ways does this problem connect to other mathematical concepts?
  • Can we generalize the results from this problem/situation?
  • Can you make a conjecture from this situation?
  • Does everyone agree with this conjecture?
  • If you do not agree with the conjecture, can you give a counter example?

The most important mathematics activity that this standard asks students to demonstrate is to see relationships between mathematical objects and the relations they can form between different elements. They should know what a conjecture is (e.g., an observed pattern). They should learn to express conjectures linguistically and symbolically: How to make them. How to justify them. How to disprove by providing a counter example. What a counterexample is.

  • Can you describe the pattern using your own words?
  • Can you generalize a pattern?
  • Can you describe this pattern using mathematical symbols as an expression, inequality, or a formula?
  • Can you give an example for which this conjecture does not hold good?

Making conjectures, giving examples and counter examples demonstrate the mathematical way of thinking—the understanding and knowledge of structure. For example, ask students to observe the patterns in the following numbers and articulate their conjectures:

 Capture5

In the first group of numbers, the “first difference” becomes constant. It means that the next number is 3 more than the previous one.  In other words, the numbers can be described as:

1st     2nd                                                   100th term               nth term

1, 1 + 3(1), 1 + 3(2),  1 + 3(3), …, …, 1 + 3(100 – 1), …,        1 + 3(n-1) + …

If we call the value of the term as y and the term as x, then the relationship between y (nth term) and x (1 + 3(n -1) is expressed as: y = 1 + 3(x – 1) or y = 3x – 2. In other words, one level of difference gives rise to a linear relationship, where the rate of growth is 3.

In the second group of numbers, the second difference becomes constant. The intuition based on the first set of numbers seems to indicate the relationship may be a second degree (quadratic) expression. It means if call the value of term is denoted by y ad the terms by x, then it suggests:

y = ax2 + bx + c

Assuming that this is the case, when we substitute x = 1, x = 2, x = 3, e get

     9 = a×12 + b×1 + c or 1 = a + b + c

          12 = a×22+ b×2 + c or 4 = 4a + 2b + c

          17 = a×32+ b×3+ c or   9 = 9a + 3b +c

When we solve these 3 simultaneous linear equations in a, b, and c, we get a = 1, b = 0, and c = 8. The sequence defines a relationship; the sth-term y can be defined in terms of the position of the term x as y = x2+8. Thus, a second level difference defines a quadratic relationship. 

A pattern begins to develop since in the next group of numbers, we find that the third level difference is constant. We can, therefore, conjecture that this defines a cubic relationship, and when we actually calculate, it is a cubic relationship. The fourth group, therefore, is a quartic relationship.

Procedural mathematics, particularly arithmetic, can be taught with or without attention to pattern. The CCSS-M acknowledges that students do need to know arithmetic facts at an automatized level, but random-order fact drills rely on memory alone, whereas patterned practice can develop a sense for structure as well. Learning to add 8 to number—not just to single digit numbers—by thinking of it as adding 10 and subtracting 2 can develop just as fast recall of the facts as random-order practice but it also allows students to generalize it and therefore be able to add 18 or 28 to any number mentally. The structure is a general one, not just a set of isolated memorized facts, so students can use it to add 19 or 39, or 21 or 41, to any number, too. With a bit of adjustment, they can use the same thinking to subtract mentally. This is, of course, exactly the way we hope students will mentally perform 350 – 99 (1 more than 99 is 100, so 350 – 100 = 250, therefore, 350 – 99= 251).

In elementary school, attention to structure also includes the ability to defer evaluation for certain kinds of tasks. For example, when presented with 29 + 15 ☐ 29 + 14 and asked to fill in <, =, or > to compare the two expressions, first and second graders are often drawn—and may even be explicitly told—to perform the calculations first. But this is a situation in which we want the students’ attention on the structure, ✪ + 15 ☐ ✪ + 14 or even u + 15 ☐ u + 14, rather than on the arithmetic. This can then be generalized later in middle and high school to:

For what value of x, the x2 + 2x + 9 > x2 + 2x + 7 is true?

This same skill of deferring evaluation—putting off calculation until one sees the overall structure—helps students notice that they don’t have to find the common denominator for 1¾ – ⅓ + 3 + ¼ – ⅔ but can simply rearrange the terms to make such a trivial computation that they can do it in their heads. Similarly, to find the value of the expression
Capture6
they do not have to make any calculations, if they know that Capture blog 6.02is equal to 1, if a = b and b ≠0. Those who do not see the structure, will go through the whole calculation as a large number of high school students actually did on a state test. They found the sum of the numerator and denominator by finding the common denominator and then performed division and multiplication.

When students begin to solve algebraic equations, the same idea will help them notice that 3(5x – 4) + 2 = 20 can be treated as “something plus 2 equals 20” and conclude, using common sense and not just “rules,” that 3(5x – 4) = 18. And 3 times something is 18 means that 5x – 4 = 6. From such reasoning, they can learn to derive rules that make sense.

Similar thinking is present in higher mathematics. For example, use a graphing tool to graph each of the following equations. Describe what shape you see in your graph and what is changing on the graph each time.

a.  x2 + y2 = 1,    b.  x2 + y2 = 4,    c.  x2 + y2 = 9
d. 
(x + 2)2 + y2 = 4,    e. x2 + (y + 3)2 = 4,    f. x2 + (y – 3)2 = 4

How might this help you to describe the graph of x2 + y2 = 16 and x2 + (y – 6)2 = 4 ? And what is the relationship between the two curves? In what ways are they similar and different?

The structure means some properties, some relationships, some characteristics are present in a particular problem, in a particular concept, and then they are present in a global situation—they are present in a class of problems. In pattern analysis, we are dealing with inductive reasoning. In pattern analysis, we move from particular situations to a general situation whereas in looking for structure we move from general to specific. Looking for and making sense of structure means we are using deductive reasoning. In most of elementary mathematics, we use less of deductive reasoning, but more of inductive reasoning. Mathematics is all about patterns and structure—inductive and deductive reasoning, so it isn’t something that should be taught as a single standard but rather as a practice that we use when thinking mathematically.

These types of conversations need careful facilitation. We shape students’ thinking by the way in which questions are posed during instruction as well as by how questions are written on assessments. I end with a list I like to incorporate as I support students in becoming more proficient with this mathematical practice:

  • Why does this strategy work, and can a solution be found using this strategy?
  • What pattern do you find in ___?
  • What are other problems that are similar to this one?
  • How is ____ related to ____?
  • Why is this important to the problem?
  • What do you know about ____ that you can apply to this situation?
  • In what ways does this problem connect to other mathematical concepts?
  • How can you use what you know to explain why this works?
  • What patterns do you see?
  • Is there a structure? How can you describe the structure?

Some educators argue that we should give students as many tasks as possible that draw on or require using many mathematical practices when determining the solution path for a task that targets a mathematical concept. Some argue that by continually giving students high-level tasks, they will eventually engage with all the math practices. I argue that we must design strategically to engage students in all the mathematics practices regularly.

[1] See the list of Strategies for Mastering Addition Facts and how to teach them in How to Master Arithmetic Facts Easily and Effectively (Sharma, 2008). List is available free from the Center.

Look For and Make Use of Structure: Understand the Nature of Mathematics

Attend to Precision: The Foundation of Mathematical Thinking

The sixth of the Standard of Mathematics Practice (SMP) in Common Core State Standards (CCSS-M) is: Attend to Precision. The key word in this standard is the verb “attend.” The primary focus is attention to precision of communication of mathematics—in thinking, in speech, in written symbols, in usage of reasoning, in applying it in problem solving, and in specifying the nature and units of quantities in numerical answers and in graphs and diagrams. With experience, the concepts should become more precise, and the vocabulary with which students name the concepts, accordingly, should carry more precise meanings.

The word “precision” calls to mind accuracy and correctness—accuracy of thought, speech and action. While accuracy in calculation is a part, clarity in communication is the main intent of this standard. The habit of striving for clarity, simplicity, and precision in both speech and writing is of great value in any discipline and field of study. In casual communication, we use context and people’s reasonable expectations to derive and clarify meanings so that we don’t burden our communication with too many details that the reader/listener can surmise anyway. But in mathematics (thinking, communicating, and writing), we base each new idea/concept logically on earlier ones; to do so “safely,” we must not leave room for ambiguity and misconceptions.

Students can start work with mathematics ideas without a precise definition. With experience, the concepts should become more precise, and the vocabulary with which we name the concepts can, accordingly, should carry more precise meanings. But we should strive for clarity and precision constantly. Striving for precision is also a way to refine understanding. By forcing an insight into precise language (natural language or mathematical symbols), we come to understand it better and then communicate it effectively. For example, new learners often trip over the order relationships of negative numbers until they find a way to reconcile their new learning (–12 is less than –6) with prior knowledge: 12 is bigger than 6, and –12 is twice –6, both of which pull for a intuitive feeling that –12 is the “bigger” number. Having ways to express the two kinds of “bigness” and the sign defining the direction helps distinguish them. Learners could acquire technical vocabulary, like magnitude or absolute value, or could just refer to the greater distance from 0, but being precise about what is “bigger” about –12 helps clarify thinking about what is not bigger. With such a vocabulary, one can express the relationship between the two numbers more precisely.

The standard applies equally to teachers and students and by extension to textbooks, modes and purpose of assessments, and expectations of performance. To achieve this, teachers need to be attentive to precision in their teaching and insist on its presence in students’ work. They should demonstrate, demand and expect precision in all aspects of students’ interactions relating to mathematics with them and with other students. Teachers must attend to what students pay attention to and demonstrate precision in their work, during the learning process and problem solving. This is not possible unless teachers also attend to the same standards of precision in their teaching.

Teachers, while developing students’ capacity to “attend to precision,” should focus on clarity and accuracy of process and outcomes of mathematics learning and in problem solving from the beginning of schooling and each academic year. For example, teachers can engage their students in a “mathematics language talk” to describe their mathematics activity. The emphasis on precision can begin in Kindergarten where they talk about number and number relationships and continues all the way to high school where they furnish mathematics reasoning for their selection and use of formulas and results.

Attention to precision is an overarching way of thinking mathematically and is essential to teaching, learning, and communicating in all areas of mathematical content across the grades.

For the development of precision, teachers should probe students to defend whether their requirements for a definition are adequate as an application to the problem in question, or whether there are some flaws in their group’s thinking that they need to modify, refine and correct. Just like in the writing process, one goes through the editing process, students should come to realize that in mathematics also one requires editing of expressions to make them appealing, understandable and precise.

However, communication is hard; precise and clear communication takes years to develop and often eludes even highly educated adults. With elementary school children, it is generally less reasonable to expect them to “state the meaning of the symbols they choose” in any formal way than to expect them to demonstrate their understanding of appropriate terms through unambiguous and correct use.

The expectations according to the standard are that mathematically proficient students

  • communicate their understanding precisely to others using proper mathematical terms and language: “A whole number is called prime when it has exactly two factors, namely 1 and itself” rather than “A number is called prime if it can be divided by 1 and itself.
  • use clear and precise definitions in discussion with others and in their own reasoning: e.g. “A rectangle is a four straight-sided closed figure with right angles only” rather than “A four-sided figure with two long sides and two short sides.”
  • state the meaning of the symbols they choose, use the comparison signs ( =, >, etc.) consistently and appropriately, for example, the names of > and < are not greater and smaller than respectively, but depend on how we read them: x > 7 is read as: x is greater than 7 or 7 is less than x; 2x + 7 = -5 + 3x is bidirectional (2x + 7 => -5 + 3x and 2x + 7 <= -5 + 3x).
  • are careful about the meaning of the units (e.g., “measure of an angle is the amount of rotation from the initial side to the terminal side” rather than “measure of an angle is the area inside the angle or the distance from one side to the other”), identifying and specifying the appropriate units of measure in computations, and clearly labeling diagrams (e.g., identify axes to clarify the correspondence with quantities and variables in the problem, vertices in a geometrical figure are upper case letters and lengths are lower case letters, and the side opposite to the <A in ΔABC is denoted by “a”, etc.).
  • calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context (e.g., the answer for the problem: “Calculate the area A of a circle with radius 2 cm” is A = 4π sq cm not A = 12.56 sq cm; if x2 = 16, then x = ± 4, not x = 4, whereas √16 = 4, etc.).
  • know and state the conditions under which a particular expression, formula, or procedure works or does not work.

Beginning with the elementary grades, this means that students learn and give carefully formulated explanations to each other and to the teacher (at Kindergarten level it may mean that the child explains her answer for 8 + 1 = 9 as “I know adding by 1 means it is the next number. I know 9 is next number after 8” or can show it concretely as “Look here is the 8-rod add the 1-rod and I get the 9-rod.” By the time they reach high school, they have learned to examine claims—their own and others’ in mathematical conversations, make explicit use of definitions, formulas, and results, and proper and adequate reasoning. At the high school level the explanations are rooted in any or more of these:

  • demonstrating it concretely,
  • showing by creating and extending a pattern,
  • application of analogous situation, or
  • logical reasoning—proving it using either deductive or inductive reasoning or using an already proved result.

What Does the “Attention to Precision” Look Like?
Effective mathematics teachers who use precision and efficiency in their teaching and encourage precision in their classrooms produce mathematically proficient students. Mathematically proficient students understand the role of precision in mathematics discourse and learning. They understand that mathematics is a precise, efficient, and universal language and activity. Precision in mathematics refers to:

Language

  • Appropriate vocabulary (proper terms, expressions, definitions), syntax (proper use of order of words), and accurate translation from words to mathematical symbols and from mathematical symbols to words.
  • Knowledge of the difference between a pattern, definition, proof, example, counter example, non-example, lemma, analogy, etc. at the appropriate grade level.
  • Reading and knowing the meaning of instructions: compute or calculate (4 × 5, √16, etc., not solve), simplify (an expression, not solve), evaluate (find the value, not solve), prove (logically, not an example), solve (an equation, problem, etc.),
  • Know the difference between actions such as: sketch, draw, construct, display, etc.
  • Precise language (clear definitions, appropriate mathematical vocabulary, specified units of measure, etc.).

Teacher instruction about vocabulary must be clear and correct and must help children to understand the role of vocabulary in clear communication: sometimes formal terms and words distinguish meanings that common vocabulary does not, and in those cases, they aid precision; but there are also times when formal terms/words camouflage the meaning. Therefore, while teachers and curriculum should never be sloppy in communication, we should choose our level of precision appropriately. The goal of precision in communication is clarity of communication and achieving understanding.

A teacher can use familiar vocabulary to help specify which object(s) are being discussed—which number or symbol, which feature of a geometric object—using specific attributes, if necessary, to clarify meaning. Actions such as teaching writing numerals to Kindergarten by “song and dance” is a good starting point, but ultimately the teacher should use the proper directional symbols, e.g.,

  • To write number “4” the teacher first should point out the difference between the written four (4) and printed four (4). Then she needs to show the direction of writing (start from the top come down and then go to the right and then pick up the pencil and start at the same level to the right of the first starting point and come down crossing the line).
  • When discussing a diagram, pointing at a rectangle from far away and saying, “No, no, that line, the long one, there,” is less clear than saying “The vertical line on the right side of the rectangle.”
  • Compare “If you add three numbers and you get even, then all the numbers are even or one of them is even” with “If you add exactly three whole numbers and the sum is even, then either all three of the numbers must be even or exactly one of them must be even.”
  • Compare giving an instruction or reading a problem as “when multiply 3 over 4 by 2 over 3, we multiply the two top numbers over multiply two bottom numbers” to “find the product of or multiply three-fourth by two-third, the product of numerators is divided by the product of denominators.”

Elementary school children (and, to a lesser extent, even adults) almost never learn new words effectively from definitions. Virtually all of their vocabulary is acquired from use in context. Children build their own “working definitions” based on their initial experiences. With experience and guidance, the concepts should become more precise, and the vocabulary with which children name the concepts will carry more precise meanings. Formal definitions generally come last. Children’s use of language varies with development but typically does not adhere to “clear definition” as much as to holistic images. If the teacher and curriculum serve as the “native speakers” of clear Mathematics, young students, who are the best language learners around, can learn the language from them.

Quantities
Accuracy (know the difference between exact, estimate, approximation and their appropriateness in context) and appropriate level of precision in use of numbers (level and degree of estimation, significant digits, significant powers, units of measurement), correct classification and location of number on the number line (e.g., to locate ⅞, one divides the unit segment into halves and then each half into fourths, and then each fourth into eighths and then locates ⅞ rather than arbitrarily divide the unit segment into eight parts), correct relationships between numbers (e.g., √(140) is between 11 = √(121) and 12 = √(144), because, we have 121 < 140 < 144, therefore, √(121) < √(140) < √(144), but √(140) much closer to 12 as 140 is much closer to 144 than 121), selection of appropriate range and window on graphing calculator, tool selection (when to use what tools–paper-pencil, concrete models, diagrams, abstract, or calculator), and appropriate meaning of numbers in the outcome of operations (what role do the quotient and remainder play in the outcome from the long division algorithm, etc.). Precise numbers (calculate accurately and efficiently; given a context, round to an appropriate degree of precision)

Teachers should use written symbols correctly. In particular, the equal sign (=) is used only between complete expressions and signals the equality of those two expressions. To explain one way to add 42 + 36, we sometimes see it written (incorrectly) this way: 40 + 30 = 70 + 2 = 72 + 6 = 78. This is a correct sequence of calculator buttons for this process but not a correct written mathematics expression: 40 + 30 is not equal to 70 + 4; only the last equals sign is correctly used. We need the = sign to have a single, specific meaning. Also, the equal sign should not be misused to mean “corresponds to”: writing “4 boys = 8 legs” is incorrect.

Models
Appropriate choice of concepts and models in the problem solving approach: choice of strategy in addition/subtraction (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, 8 + 6 = 4 + 4 + 6 = 10 + 4 = 14, 8 + 6 = 2 + 6 + 6 = 2 + 12, 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, or 8 + 6 = 7 + 1 + 6 = 7 + 7 = 14 rather than “counting up” 6 from 8 or 8 from 6), appropriate multiplication/division model (the only models of multiplication work for fraction multiplication are “groups of” or “area of a rectangle” not “repeated addition” and the “array” models), which exponential rule, which rule of factoring, which rule for differentiation, what parent function to relate to, what formula to use, etc.

Reasoning, Symbols, and Writing Mathematics
Appropriate and efficient use of definitions, reasons, methods of proof, and order of reasoning in solving problems and explanations. For example, children should know the reasons for using the “order of operations” or that the solutions of equations have domains and range. Precise usage of symbols and writing:

  • Choose correct symbols and operators to represent a problem (knowns and unknowns; constants and variables),
  • State the meaning of the symbols and operations chosen appropriate to the grade level (multiplication: 4×5, 45, 4(5), (4)5,(4)(5), a(b), (a)b, (a)(b), ab),
  • Label axes, shapes, figures, diagrams, to clarify the correspondence with quantities in a problem, location of numbers,
  • Show enough appropriate steps to communicate how the answer was derived,
  • Organize the work so that a reader can follow the steps (know how to use paper in an organized and systematic form—left to right, top to bottom),
  • Clearly explain, in writing, how to solve a specific problem,
  • Use clear definitions in discussion with others and in reasoning
  • Specify units of measure and dimensions,
  • Calculate accurately and efficiently.

At the elementary level, even the simplest of things such as: the proper way of forming numbers and mathematical symbols, writing the problems solving steps in a sequence: ([3(4 + 8) – (4 ÷ 2)] = [3(12) – (2)] =[36 − 2]= 34 rather than 4 + 8 = 12 × 3 = 36 −2 = 34). Similarly, clarity in reading numbers and mathematical symbols needs to be  emphasized from the beginning (e.g., ¾ is read as “3 parts out of 4 equal parts” rather than “3 out of 4,” “3 divided by 4” rather than “3 over 4.”

It is difficult to change inappropriate and incorrect habits later on. For example, when elementary grade teachers do not emphasize the importance of aligning multi-digit numbers in their appropriate place values, this creates problems for children later. New symbols and operations are introduced at each grade level, so it is important for the teacher to introduce them correctly and then expect precision in their execution.

Similarly, when middle and high school students are not instructed to write fractions properly, it creates problems. The following high school lesson illustrates the point. The problem on the board was:

27.1

To solve the equation, in order to eliminate fractions in the equation, the student suggested we multiply the whole equation by the common denominator of all the fractions in the equation (a correct and efficient method). When I asked for the common denominator, the student said: 9x because the denominators are 3x, 9 and 3. The error is purely because of lack of precision in writing fractions in the equations.

Precision often means including units when specifying numerical quantities. But not always. The purpose of precision is never to create work, only to create clarity. Sometimes a number is clear by itself, other times a unit is needed, sometimes a whole sentence is required: the situation determines the need. For the same reason, label graphs and diagrams sufficiently to make their meaning and the meanings of their parts clear.

Exposure and consistent questions from the teacher such as the following help students to be accurate, precise and efficient:

  • Is this the right way of writing the expression (number, symbol, etc.)?
  • Does the diagram you have drawn show the elements asked for or given in the problem?
  • Is this the right unit for the quantities/numbers given in the problem?
  • What mathematical terms apply in this situation?
  • Is the term you used the right one in this situation?
  • How do you know your solution is reasonable and accurate?
  • Explain how you might show that your solution answers the problem?
  • How are you showing the meaning of the quantities given in the problem (e.g., problem says: “the length of the rectangle is 3 more than twice the width)? Does your rectangle demonstrate the right dimensions? Your rectangle looks like a square.
  • What symbols or mathematical notations are important in this problem?
  • What mathematical language, definitions, known results, properties, can you use to explain ….?
  • Can you read this number (symbol, expression, formula, etc.) more efficiently?
  • Is ___ reading (saying, writing, drawing, etc.) correctly? If not, can you state it correctly and more efficiently?
  • How could you test your solution to see if it answers the problem?
  • Of all the solutions and strategies presented in the classroom, which ones are exact/correct?
  • Which one of the strategies is efficient (can achieve the goal more effectively)?
  • What would be a more efficient strategy?
  • Which one is the most elegant (can be generalized and applied to more complex problems) strategy? Etc.

The number and quality of questions in a classroom bring the attention of students to appropriate and precise conversation. In a fourth grade geometry lesson, I had the following exchange with the students: 

Sharma: Look at this rectangle (I was holding one of the 10 by 10 by 1 rectangular solids in my hand) while touching the 10 by 10 face, I asked: What are the dimensions of this rectangle?

A student raised his hand and said: “That is not a rectangle. It is a square.”

I said: “yes, it is a square. Can you also call it a rectangle? Is it also rectangle?”

“No!” He declared emphatically.

I asked the class: “How many of you believe that it is not a rectangle?” Almost every hand went up.

When I asked them what the definition of a rectangle was, almost all of them said: “A rectangle has two long sides and two shorter sides.” I drew a quadrilateral with 2 long sides and 2 short sides that did not like a rectangle.

Another student said: “The sides are parallel.” I drew a parallelogram.

The student said: “No! That is not what I mean. Let me show you what I mean.” He drew a correct rectangle.

One student said: “A rectangle has four right angles and 2 longer sides and 2 shorter sides. Like this.” He drew a correct rectangle.

We had a nice discussion and came to the conclusion that a rectangle is: A straight-sided closed figure with four right angles. I also emphasized the meaning of the word “rectangle.”  It is made up of two words “recta” and “angle.” The word “recta” means right.  Therefore, a rectangle has only right angles. With this discussion and the precise definition, they were able to accept and see the face of the object I was showing as a rectangle.

This episode, in one form or the other, is repeated in many classes, from urban to rural classrooms, in many elementary schools. The same misconception is present even in many classrooms in many middle and high schools students. This is an example of lack of precision in teaching and, therefore, lack of precision in student understanding and expression.

There are many examples of such misconceptions. For instance, children often misunderstand the meaning of the equal sign. The equal sign means is “the same as,” “equal in value” “equal in some specified characteristic—length, area, quantity, volume, or weight,” but most primary students believe the equal sign tells you that the answer is coming up to the right of the equal sign. When children only see examples of number sentences with an operation to the left side of the equal sign and the answer on the right, this misconception is formed and generalized. Teachers should, therefore, emphasize the true meaning of the equal sign. From the very beginning—Kindergarten children should be shown that the equal sign “=” is a two-way implication. For example, Kindergarteners should be shown and know the simple facts as: 2 + 8 = 10, 8 + 2 = 10 & 10 = 2 + 8, 8 + 2 = 10 and first graders need to see equations written in multiple ways, for example 5 + 7 = 12, 7 + 5 = 12, 12 = 5 + 7, 12 = 7 + 5, and 5 + 7 = 2 + 10, 5 + 7 = ☐+10, ☐ + 2 = 9 + ☐. Although most above average and many average children are able to realize this level of understanding of the concept of equal or equal sign, there are many average and children with learning disabilities who have difficulty in reaching that level of understanding. This level of precision in understanding can be achieved by using Cuisenaire rods, the Invicta math balance for teaching arithmetic facts, and proper and appropriate language usage and questioning by teachers.

If students are taught using imprecise language, they will necessarily learn imprecise language and concepts, because language is the basis of mathematics learning. Later, they will not only resist when asked to use precise language in mathematics, but they will also have difficulty applying the concepts. A sequence of ideas begins to take place in students’ mind when we ask questions and emphasize language.
Questions instigate language.
Language instigates models.
Models instigate thinking.
Thinking instigates understanding.
Understanding produces conceptual schemas.
Conceptual schemas produce competent performance.
Competent performance produces long lasting self-esteem.
Self-esteem produces willingness to inquire and learn.

With proper language and conceptual models a great deal can be achieved. It is not too late to instill precision even at the high school level; however, if it is not emphasized at the elementary and middle school levels, it is much more difficult to do so. This does not mean we give up; it only means we redouble our effort and find better ways of doing it, such as using concrete models, patterns, and analogies when we are introducing new mathematics concepts and procedures.

As students progress into the higher grades, their ability to attend to precision will expand to be more explicit and complex if we constantly use proper language and symbols.

As students develop mathematical language, they learn to use algebraic notation to express what they already know and to translate among words, symbols, and diagrams. Possibly the most profound idea is giving names to objects. When we give numbers names, not just values, then we can talk about general cases and not just specific ones.

Correct use of mathematical terms, symbols, and conventions can always achieve mathematical precision but can also produce speech and writing that is opaque, especially to learners, often to teachers, and sometimes even to mathematicians. Good mathematical thinking, therefore, requires being correct, but with the right simplicity of language and lack of ambiguity to maintain both correctness and clarity for the intended audience. If we are particular about this in the first few grades, it becomes much easier to attend to precision in later grades.

Attend to Precision: The Foundation of Mathematical Thinking

Use Appropriate Tools Strategically – Part II

Teachers’ Role in Tool Building and Using
Teachers play a critical role in the development of the strategic use of tools. First, they make a diversity of tools available to students. From the beginning of the year, students should know where the math tools are in the room and how they will be used throughout the class. From the beginning, a teacher should declare: “Just like many, you play a sport of your choice and get better at it with practice. Similarly, each one of you should become an expert in a mathematics tool or strategy and its usage in mathematics concepts and procedures.” Then the teacher should present varieties of situations or problems where that tool is applicable and effective (e.g., different types of rules or protractors). Then she works on the strategic use of the tools. The first step is modeling their appropriate use.  One can begin by using phrases like “I bet a ruler would help me divide this into even pieces” or “I wonder if using a graph paper would help me organize my work” or “I think I could use a calculator to double check my accuracy on this one.” The key factor in getting students to use mathematical tools efficiently is exposure—multiple and varied exposures.

Students need to see how teachers make decisions about using tools so they know what appropriate use is. Using a calculator to solve 50 + 50 is not appropriate—but it is appropriate to check a complicated computation. A teacher should not admonish students who choose tools inappropriately. Instead, she should ask them to share their reasoning for using a particular tool in a specific way. They should also make explicit how a particular tool or approach will connect to an abstract mathematics idea.

As the year continues and new tools are introduced, students will be able to apply their current knowledge of mathematical tools to the new ones. It is important to plan tasks that will require multiple learning tools.

In order for students to be proficient they need to start using the tools independent of the teacher. They will then pick tools based on the needs of the problem and plan. They will also be able to visualize the results after using the tool.

Introductory Part of the Lesson (Teacher Directed: Didactic and Socratic Roles)
For Teachers: Decisions

  • What are the goals of this lesson?
  • What language, concept, procedures, and skills do I want students to develop?
  • What activities and tools are best suited for this purpose?
  • What tools and methods my students are already familiar with?

Middle Part of the Lesson (Socratic and Coaching Roles)

  • Have the students acquired the concept/procedure using this tool, what tools can further expand, enhance, or deepen these ideas?
  • Can they apply these tools in problem solving with my support and in collaboration with their colleagues?

Last Part of the Lesson (Coaching and Supporting Roles)

  • What problems can I assign students that will provide opportunities of applying these and previous tools in solving them?
  • Can they create/construct problems that can be solved by these tools?
  • Discussion to establish the efficiency of tools and develop proficiency and competence in the use of tools and integrate these tools with earlier tools.

Teachers must recognize that tools do not produce understanding, problem solving, and solutions. These come when teachers ask questions and make connections between the tool and the concept and when students do the same. Providing students with protractors does not ensure that they will measure the angles and find angle sums of triangles with accuracy. Similarly, a graphing calculator doesn’t consider user error or misconception when graphing a linear equation. The teacher should therefore bring to students’ attention the strength and limitations of the tool and its usage.

User error (including a broken ruler) can occur with a basic ruler or a calculator. If we have the concept and understanding, we adjust. We try the tool again, maybe a little differently.

Initially it is the teacher’s questions that help students in the tools’ usage, but then the teacher needs to transfer their usage to students and the questioning process to facilitate this. The questions we want students to ask when selecting and using tools include:

  • Do I need a tool in this situation?
  • Which tool will work for this situation?
  • Is this the right tool?
  • How does it work?
  • What tool is the best to use in this situation?
  • Do the results align with what I was expecting?
  • Do the results make sense?

Most importantly, when children select, use, make decisions, compare the usage of tools, they think and develop metacognitive processes. As a result, their cognitive ability and potential increase. They become better learners. This development is centered on teachers facilitating the process by asking questions and creating cognitive dissonance in students’ minds. Thus, we encourage the thinking behind the tool as well as the procedure for using the tool. We should require our students to predict what their findings might be prior to using the tool and then require them to reflect on the results and if the answers make sense.

For Students: Problem Solving
The standard says:
[S]tudents consider the available tools…. Proficient students…make sound decisions about when each of these tools might be helpful.
[They]… use technological tools to explore and deepen their understanding.

These phrases focus on the student. The goals for students are: Use available tools recognizing the strengths and limitations of each. Use estimation and other mathematical knowledge to detect possible errors. Identify relevant external mathematical resources to pose and solve problems. Use technological tools to deepen understanding of mathematics.

Students learn through the questions teachers pose to help them think and sort through the ideas that are forming. Helping students to generate questions results in their seeking and using tools.

  • What information do I have?
  • What is stated in the problem?
  • How will I represent the information in the problem?
  • What tool(s) can help me visualize and represent this information to understand its nature?
  • What do I know that is not stated in the problem?
  • What approach should I consider trying first?
  • What other mathematical tools could I use to visualize and represent situations and conditions in the problem?
  • What is the expected range of the answer for this problem?
  • Should I make an estimate for the answer?
  • What estimate can I make for the solution? Should I change the numbers for that purpose?
  • What will be the unit of my estimate?
  • In this situation, would it be helpful to use…a graph, number line, ruler, diagram, calculator, or a manipulative?
  • Why was it helpful to use…?
  • What can using a ____ show us that ____ may not?
  • In what situations might more information be helpful …?

The following problem illustrates students’ reasoning:
Three-fourth of the yard was converted into a vegetable garden. Two-third of the garden is used for gardening herbs. What fraction of the garden is herbs?

Student One: I am going to assume that the garden is a rectangular shape. Let me represent the rectangle as my whole. The herb garden is going to be ⅔ of. Approach One: I can see my problem as coloring first ¾ of the garden representing the garden and then I color ⅔ of the ¾ in another color. That will represent the herb garden as: ⅔ of ¾ = ¼+¼=½.
blog26.1

That is a correct approach to get the answer; however, it does not help us to arrive at the procedure.

Student Two: I am going to assume that the garden is a rectangular shape. Let me represent it as a 1 by 1 rectangle. The herb garden is going to be a ¾ × ⅔ or ⅔ × ¾ rectangle. By definition ¾ × ⅔ is the area of the rectangle with the dimensions of ¾ and ⅔. Therefore, the vertical side of the rectangle is divided into four equal parts and the horizontal side into three equal parts and the ¾ × ⅔ rectangle is formed.
blog26.2

As the diagram suggests, this area is 2 by 3 parts out of 3 by 4 parts and that is represented as (2×3) out of (3×4) or blog26.3. In other words, we have
blog26.4

The representational tool “area as multiplication” is more efficient and strategic compared to the “groups of” tool for multiplication. Although at the whole number level, the four models of multiplication: repeated addition, groups of, an array, and the area of rectangle are equally good as an introduction, the most efficient is the area model.

When students are sufficiently familiar with the tools, the teacher should pose problems that are tool specific and help students to sharpen their tools by practicing them daily till they are proficient, a better tool is available, or a better way of using that tool is possible.

I suggest teachers devote about 20% to 25% of time in every lesson on tool building (achieving proficiency in the use of physical tools and fluency in the use of thinking tools). For example, arriving at a particular definition relating to a mathematical idea is part of conceptual development. But, mastery in using definitions, concepts, and procedures is tool building. As students get older, they need to add more versatile tools or create new tools by combining tools to their mathematical toolbox.

The teacher starts with definitions, develops conceptual understanding and arrives at the procedure, but the children should not remain at the definition and procedural level; they should solve problems—solving problems builds tools and their efficient use builds mathematics stamina. Proficiency in the use of tools and concepts are built when we use them regularly and apply them in different settings. Some concepts become tools and then these tools are used to learn new concepts and new tools. This iterative process continues in learning mathematics. When we use tools, we learn new mathematics. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a whole number and concluded that it must be a different kind of number.

 

 

Versatile tools (exact, efficient, elegant) build mental schemas/models that last. What makes a tool like the Cuisenaire rods, Base Ten blocks, Invicta balance, Empty Number line, Bar Model or the area model truly powerful is that it is not just a special-purpose trick or temporary crutch for a particular type of problem but is faithful to the mathematics and is applicable to many domains and concepts. Because these tools help students make sense of mathematics, they last. And that is also why the CCSS and SMP mandates them.

Mathematically proficient students are able to identify relevant external mathematical resources such as people, books, digital content on a website and use them to pose or solve problems. They use technological tools to explore and deepen their understanding of concepts and make connections between concepts and procedures.

Technology Tools
Hands-on tools are useful; however, as we prepare students for the world of work and the power of technology, students need to understand the range of use, strengths, and limitations of these tools. Today, technological computations tools (e.g., calculator) are common outside the classroom, so the classroom needs to reflect this reality. With the different technologies—calculators, smart phones, tablets, and laptops, the question of when and how to use technology becomes even more important.

Technology can allow greater opportunities to visualize, explore, predict, and compare mathematical ideas. A parallel practice for teachers, therefore, is to augment the use of appropriate technological tools for mathematics instruction at the appropriate time. The use depends on teachers deciding first the mathematical goals of instruction and then which tools may be most effective in accomplishing them.

The question of when to use technology tools is a question about the most productive use of valuable classroom time. The answer to that question may lie in the reasons we teach mathematics today:

  • to understand numbers and patterns found in nature (number concept, numbersense, numeracy)
  • to acquire math tools, know when and how to use those tools
  • to make fast and accurate predictions and check the reasonableness of answers
  • to grow and maintain mental power
  • to identify unknowns in a situation, represent and deal with them
  • to think logically and clearly when solving problems
  • to feel comfortable with quantitative and spatial thinking demands in a technological world

Using a calculator as a tool should be a strategic decision. Calculators should be used with caution in elementary school (that means very carefully). Calculators should be available as computational tools, particularly when many or cumbersome computations are needed to solve problems. However, when the focus of the lesson is on developing computational skills or algorithms, the calculator should not be used. It should be a tool that provides access, simplifies the task, or confirms accuracy. It doesn’t make sense for a fifth grader to use a calculator for 8 + 13. However, it may make sense for a first grader to confirm the sum that way. But, for 18.1759 + 27.19427, use the calculator.

In Elementary School, the following conditions are satisfied then and only then I will allow the use of calculators: (a) student knows the arithmetic facts, (b) good in estimation, (c) understands the concept/procedure for which the calculator is being used (e.g., in how many 4 are there in .04, the student knows which one is the divisor and the dividend). Teachers should encourage students in all contexts to estimate first and then if necessary to use calculators. Students who know “about” how much an inch is can tell when they use a ruler if their answer is reasonable. Students who understand how to round and estimate when multiplying money know if what they plug into a calculator makes sense.

Students should have the opportunities to discuss when they might use tools (e.g., concrete, technology, paper/pencil, mental math), and they should know when and where tool use is appropriate. Tool use is not an ideological decision: it is neither dedication to “times tables,” “long division,” and “repetition and memorization” nor is it allegiance to “fuzzy math” reform—preaching concept over content, exposure over mastery, insight over “right.” Tool use is the judicious decision of both and is the integration of the linguistic, conceptual, and procedural components. This can happen if we:

  • allow students to spend less time on tedious calculations and more time first on language development, conceptual understanding and solving problems
  • help students develop better number sense (number concept, arithmetic facts, and place value)
  • allow students to study mathematical concepts they could not attempt if they had to perform the related calculations themselves and use the tool to develop the concept by seeing the patterns
  • allow students who would normally be turned off by math because of frustration or boredom to increase their mathematical understanding and help them to acquire fluency of arithmetic facts by seeing number relationships
  • simplify tasks while helping students determine the best methods for solving problems
  • make students more confident about their math abilities, once a problem is solved using a calculator, make the number manageable and help them solve without the calculator.

While few educators deny the usefulness of calculators at the high school level, we need to rethink their use in the lower grades. Inappropriate use of calculators prevents students from seeing the underlying structure and beauty in mathematics, inhibits them from seeing mathematical relationships, and gives them a false sense of confidence about their math ability. Students who do not do long division, who quickly pull out their calculator to find the answer, do not understand the underlying principle of division. For example, when I asked eleventh grade students: What is the largest remainder you could expect when you divide a six-digit number by 11? most took out their calculators and started calculating by choosing random numbers. Many gave unreasonable answers.

If properly used, technology is an important means to achieve the goals of instruction. However, specific examples of technology use do not start appearing in the content standards (CCSS-M) until the middle school grades, and most appear at the high school level.

There are elements of mathematics at all levels (not just elementary) where some basic facts simply must be memorized (of course, after understanding and using them in problems). For example, to succeed in algebra, one should have mastered operations on fractions and integers. And success in mastering fractions is dependent on the mastery of:

  • understanding, fluency, and applicability of multiplication tables,
  • divisibility tests,
  • prime factorization, and
  • short division.

Similarly, at each level new skills and relationships emerge (e.g., laws of exponents, etc.), which need to be automatized (once again first understanding and then using them in problem solving situations) and without rote.

Many teachers, not just at third, fourth, and fifth grade levels, but at the high school and college levels, report that students do not know their basic facts or the concept of fractions. This is due to the ineffective use of tools. That is, students are overly dependent on concrete models, counting, number line, calculators, graphic organizers, multiplication charts, etc.

Others disagree, however, claiming that calculators help younger students grasp the underlying principles of mathematics by allowing them to spend time on those principles rather than on the rote computations necessary to solve them.

To build a beautiful piece of furniture, a carpenter does not use only one tool but recognizes the relationship between the object to be made and the tools needed. It is up to individual teachers, of course, to find the right way to achieve the many and complex goals of mathematics learning. And most teachers strive to do just that.

Examples of Tool Usage

Common Core Curriculum Standards (CCSS-M, 2010) call for specific situations and context to use specific tools. For example, the following geometric standard calls for the use of classic geometric tools:
Make formal geometric constructions with a variety of tools and methods (compass     and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle, bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

This high school standard on “Interpreting Categorical and Quantitative Data” suggests the use of several appropriate tools. Here’s an example:
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

The high school standards for number and quantity include this paragraph: Calculators, spreadsheets, and computer algebra systems can provide ways for students to become better acquainted with these new number systems and their notation. They can be used to generate data for numerical experiments, to help understand the workings of matrix, vector, and complex number algebra, and to experiment with non-integer components.

In high school algebra, the standards suggest uses for spreadsheets of computer algebra systems: A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave. Here’s an example:
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Kindergarten through Second Grade The focus of these three years is to master Additive Reasoning and its applications and identifying, recognizing, drawing, and using 2 and 3-dimensional shapes and figures. To achieve these goals, the tools may include counters, Cuisenaire rods, Invicta Balance, place value (base ten) blocks, hundreds number boards, number lines, and concrete geometric shapes (e.g., pattern blocks, 3-d solids). Students should also have experiences with educational technologies such as virtual manipulatives and mathematical games and toys that support conceptual understanding, but calculator is not advisable. During classroom instruction, students should have access to various mathematical tools as well as paper (for applying alternative methods of adding and subtracting—concrete models, Empty Number Line, Decomposition/recomposition, transforming a problem by translating, by place value methods, and standard procedure), and determine which tools are the most appropriate to use. For example, find the difference: 93 – 46.

Here are the methods in order of efficiency.
Concrete Tools
Cuisenaire rods, BaseTen blocks

Pictorial/Representational Tools
Hundreds’ Chart, Empty Number Line (small-big, big-small, small-big-small jumps)

Abstract/Symbolic/Procedural Tools
Decomposition/ recomposition methods:
26.12

The standard procedure:
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The fundamental concept in these grades is decomposition/recomposition. Decomposing and recomposing numbers should be done with manipulatives and models until it becomes something students can do mentally. Then we should go to the standard algorithms and mnemonic devices.

Third and Fourth Grade The focus of these two years is to master Multiplicative Reasoning and its applications; identifying, recognizing, drawing, and using 2 and 3-dimensional shapes and figures; and introducing the concept of fractions. To achieve these goals, the tools may include Cuisenaire rods, Invicta Balance, place value (base ten) blocks, hundreds number boards, number lines, concrete geometric shapes (e.g., pattern blocks, 3-d solids), and factions strips. By the end of fourth grade, mathematically proficient students have mastered numeracy skills (ability to execute the four whole number operations correctly, consistently, fluently, with understanding in the standard form), use available tools (including estimation) when solving problems and decide when certain tools might be helpful. For instance, they may use graph paper to find all the possible rectangles that have a given perimeter and area. Or they may use graph paper or a number line to represent and compare decimals and protractors to measure angles. They use other measurement tools to understand the relative size of units within a system and express measurements given in larger units in terms of smaller units.

Fifth and Sixth Grades The focus of these two years is on Proportional Reasoning, introduction to integers and equations, and their applications to quantitative and geometrical situations. Mastery of the concept of and operations on fractions in different forms—parts to whole, comparison of quantities (ratio, rate, etc.), comparison of a quantity with a standard (decimals and percents), comparison of comparisons (proportions), concept of integers and its applications; identifying, recognizing, drawing, and using 2 and 3-dimensional shapes and figures and their relationships. To achieve these goals, the tools may include Cuisenaire rods, Invicta Balance, place value (base ten) blocks, hundreds number boards, number lines, fraction and decimal strips, geoboard, concrete geometric shapes (e.g., pattern blocks, 3-d solids), and factions strips. Technological tools—calculators, Apps, Geometric Sketch pad, etc. Mathematically proficient students consider the available tools (including estimation) when solving problems and decide when certain tools might be helpful. For instance, they may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions. They use graph paper to accurately create graphs and solve problems or make predictions from real world data.

At any level, what is critical is that students are given opportunities to use each tool and to learn when its use is appropriate. For example, is it better to use a tape measure or a ruler to measure the length of a room? Why? In what situation would you use a protractor? Why would pattern blocks be a tool for helping students understand the need for a common denominator when adding or subtracting fractions? Is this the only tool students should experience with this specific content? Questions such as these will help teachers determine how tools foster mathematics learning most effectively.

Use Appropriate Tools Strategically – Part II

Use Appropriate Tools Strategically: Right Tools for the Right Job – Part I

One of the Standard of Mathematics Practice (SMP 5, CCSSI 2010, p. 7) calls for selecting appropriate tools and using them strategically. The two words “appropriate” and “strategically” apply to students as well as teachers. What does appropriate and strategic mean in the use of a tool? The answer depends on our interpretation of tools, our expectations for using them, and their role in gaining mathematical maturity for our students.

Simply, a tool is anything that aids in accomplishing a task—learning a concept/procedure. It is appropriate if it makes the concept transparent and provides the learner access to the concept. A tool is an appropriate tool in the context of what it is for and who is using it and for what purpose. Appropriateness of a tool, thus, is a function of the concept, the user, and the standard of mastery expected. A tool is appropriate if it helps the student learn the concept at the expected level.

Without strategic use, any tool, including an appropriate one may be ineffective and may not produce optimal results. However, we need to have a common definition of “using a tool strategically.” If the tool produces optimal results—develops language, concepts, and procedures with rigor and efficiently, the tool is being used strategically.

The number line is sometimes regarded just as a visual aid for children—as a physical tool. It is, in fact, a sophisticated image used even by mathematicians; it is a thinking tool. For young children, it helps develop early mental images of addition and subtraction that connect arithmetic with measurement, mental arithmetic, and standard algorithms. Rulers are just number lines built to specifications. In Kindergarten and first grade, it is the starting of solving a problem like 9 – 5 = ?

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This number line image shows “the distance from 5 to 9.” It gives visual and conceptual richness to the problem and in extension the flexibility of thought. For example, if children are given the problem:

My team scored 9 points on Monday and 5 points on Tuesday. Then,

  • How many more points did they score on Monday than Tuesday?
  • How many fewer points did they score on Tuesday than Monday?
  • What was the difference between the scores on Monday and Tuesday?
  • How many more points should they have scored on Tuesday so that their score was the same as on Monday?
  • How many less points should they have scored so that their score would have been as on Tuesday?
  • How many extra points did they score on Monday if the goal of the game was to score only five points?

The number line can answer all of the questions raised in diverse contexts. Children who see subtraction that way can use this model to see the problems with larger quantities and different numbers. For example, let us consider the problem: 63 –27 as “the distance between 28 and 63.” To do so without crossing out digits and borrowing and following a rule, they may only barely understand.

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But this number line easily explains the procedure and extends to mental calculations and applications in real life situations. In fact, it leads them to forming mental models of subtraction and helps achieve fluency in problem solving—both addition and subtraction. The number line model also extends naturally to decimals, fractions, integers, and elapsed time. For example, when students are asked to solve the problem:
If the temperature in the morning was -20 and reached 50 at noon time, what was the change in the temperature?

Many students answer it as 30, showing that they do not have the conceptual understanding and visual image in their mind for the problem. However, with the use of the number line, they can see that the distance from -2 to 5 is the number we must add to -2 to get 5: From -2 to 0 is 2 units and from 0 to 5 as 5 units, therefore, the total distance from -2 to 5 as 7, and they can generalize to solve a problem like: 42 – (-36), which can also be seen as distance from -36 to 42, using the Empty Number Line as the sum of distances from -36 to 0 (=36) and then from 0 to 42 as (=42) or 36 + 42.

Number line, on one hand, unifies arithmetic, making sense of what is otherwise often seen as a collection of independent and hard-to-remember rules and, on the other hand, it is generalizable and one can leap into algebra. The number line remains useful as students study data, graphing, and algebra: two number lines, at right angles to each other, label the addresses of points on the coordinate plane.

To find the difference 231 – 197 by counting on the number line by tens or ones is an inefficient use of the number line. But treating the problems as an addition problem and using the number line as Empty Number Line is effective and efficient as it improves numbersense and mental math.

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A tool by itself is neither appropriate nor strategic. It is its use that determines whether it is appropriate and strategic.

Tools are meant to help teachers and students make sense of mathematics and its role in the world around us. They are to make teaching efficient and to support accurate, rigorous, and proficient learning. It is, therefore, our responsibility to know how to select, understand, and use the tools strategically to develop our students’ proficiencies in learning and their competence in mathematics. Ultimately, the strategic use of tools is when teachers are able to transfer the control of the use of tools to students and they use them strategically.

For students it means that they acquire the facility to use appropriate tools strategically in learning and solving problems in mathematics. It is one of the important skills of mathematically proficient students.

Teachers’ Role in Using Tools Strategically
An important element of the strategic use of tools depends on the goals of instruction. A teacher first considers mathematical goals of her instruction and then decides which tools may be most effective in accomplishing them. It means to select tools to get the concept across the students and then to use them with optimal results in learning and achievement.

Appropriateness of a tool means that the concept becomes transparent to the students and they can see the congruence of representations through the tools and the abstract/symbolic form. To achieve this, the teacher asks:

  • Does it show the concept exactly—is the representation transparent?
  • Is it efficient to demonstrate the concept or procedure?
  • Is it easy to work with, to manipulate?
  • Does the student see it efficiently and clearly?
  • Can this tool be used to extrapolate, generalize, and abstract the concept from the current manifestation?
  • Is it available to them?
  • Will the student be able to use this tool easily and effectively?

Effective, appropriate, and strategic use of tools is important at all grade levels, but the types of tools and how they are used can differ. Golf players know when to use which “iron.” They constantly practice their usage of the tools. The same is true for a “budding” mathematician and a real mathematician alike. In developing students’ capacity to “use appropriate tools strategically,” teachers make clear to students why the use of tools will aid their problem solving processes.

Proficient students are sufficiently familiar with tools appropriate for their grade to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.

Using Appropriate Tools Strategically
Mathematically proficient students consider the available tools when solving a problem and they use them strategically. The framers of CCSS-M seem to refer to two kinds of tools: physical and thinking tools. In the case of physical tools, one is looking for proficiency, and in the case of thinking tools, one wants fluency.

The physical tools (commercially prepared or constructed by teachers and students) might include pencil and paper, concrete manipulative models (sundry counting objects, fingers, TenFrames, Cuisenaire rods, Algebra tiles, Base-Ten blocks, Invicta Balance, fraction strips, games and toys, straight edge, rulers, diagrams, two-way tables, graphs, graphic organizers, protractor, compass, calculator, spreadsheet, computer algebra systems, statistical package, or dynamic geometry software, geometry sketch pad, geogebra, iPad apps, Smart Phone Apps, Graphing Calculator, Algebra Computer System, Statistics Package, Spread Sheet, etc.).

Manipulatives are objects that appeal to several senses and that can be touched, moved about, rearranged, and otherwise handled by children. Using manipulatives in the early grades is one way of making mathematics learning more meaningful to students as they are used to make abstract ideas more concrete and transparent. Modeling with manipulatives is the first step in creating an environment where students can begin to understand abstract mathematical concepts in a variety of contexts and ways. For example, an elementary teacher might have students select different color tiles to show repetition in a patterning task. A middle or high school teacher might have established norms for accessing tools during the students’ group learning and problem solving processes to make things and see geometrical relationships from them.

However, a manipulative does not by itself carry the intended meaning and does not guarantee that mathematical understanding will result from use. It is the expertise of the teacher in the use of manipulatives and the amount of time and experiences students are given to interact with the manipulatives that lead to increased achievement.

Counters of many kinds, Base-10 blocks, Cuisenaire rods, Pattern Blocks, measuring tapes, spoons or cups, and other physical devices are all, if used strategically, of great potential value in the elementary school classroom. They are the “obvious” tools. But, physical tools should satisfy the following properties: they should (a) be exact and transparent, (b) be efficient, and (c) be elegant. The physical tools serve three purposes:

(a) generate the language of that mathematical idea,
(b) help develop the conceptual schema of the idea, and
(c) derive the procedure related to the idea.

Concepts must be developed and reinforced by the tool. The use of the tool itself should support reasoning rather than mere procedure. Reasoning develops understanding. And understanding develops mental math and strategies. The idea of understanding holds true for other tools and transfers to paper/pencil as well. With understanding, physical tools develop into thinking tools. For example, the practice of making ten by the help of Cuisenaire rods develops the mental math strategies suing making ten, for example, 8 + 6 = 8 + 2 + 6 or 4 + 4 + 6; 17 – 9 = 7 + 10 – 9 = 7 + 1, etc.

Understanding helps students realize accuracy and proficiency. Consider 9.1888 + 11.1020. If I use a calculator, I should know that my sum will be in the neighborhood of 20. I need to reconsider if my calculator result is dramatically different. This transcends grade level. For example, if I determine that my slope is negative and my line rises from left to right, then something is not right.

In other words, students must derive and understand outcomes of operations with and without a calculator but also reinforce this understanding while using the calculator. If the Sin of an angle comes out to be more than 1, then, there is something wrong. Similarly, using a protractor in measuring an angle, it is more than just “lining it up the right way.” Understanding enables them to be proficient in diverse situations and even with diverse protractors.

When we have developed the language, concept, and procedure, using physical tools, students should convert them into thinking tools and then practice the procedure and the skills related to that idea. The physical tool should always be converted into thinking tools.

The thinking tools refer to vocabulary, written or mental strategies (decomposition/recomposition, properties of operations, etc.), conceptual schemas (e.g., area model of multiplication), approaches (e.g., prime factorization for LCM, etc.), skills (e.g., facts, translation from native language to math symbols, etc.), and procedures (standard or alternative). The mathematical thinking tools deal with intellectual and cognitive skills.

Cognitive/Learning Skills
A major outcome of using concrete materials as tools for mathematics is the development of prerequisite skills to anchor mathematics ideas.

  • Following sequential directions: every procedure and task analysis is dependent on this skill,
  • Pattern analysis: mathematics is the study of patterns in quantity and space; recognizing, identifying, extending, creating, and applying are integral part of tool usage,
  • Spatial orientation/space organization: observing and identifying spatial orientation, organization, and relationships is essential in tool usage,
  • Visualization: holding and manipulating information are essential for mathematics, particularly for mental math and planning problem solving and selecting tools. Tools that have patterns, color, shape, and size (e.g., visual cluster cards, Cuisenaire rods, etc.) develop visualization and therefore enhance working memory.
  • Estimating: along with number concept, numbersense, the key skill implicated in dyscalculia is estimation; using appropriate concrete tools (non-counting materials) help develop estimation,
  • Deductive and inductive reasoning: The development of formal/ abstract/logical reasoning begins when children use concrete tools effectively,
  • Collecting/classifying/organizing: These are developmental concepts; children begin at concrete level and then are transitioned to abstract/formal levels (e.g., collecting data—look up information on Internet, in a book, in one’s notes, and read teacher comments on home work and tests, etc.),
  • Metacognition: Learning about one’s learning—what works and does not work.

Mathematical Skills
The purpose of many physical tools is to acquire abstract/formal tools to prepare students for college and careers. This is achieved when they have these tools:

  • linguistic: read the problem (e.g. focus on instructions), know the vocabulary, rewrite the problem in one’s own words, underline and understand the key words, recall and define the key terms, translate terms from English language to mathematical language and symbols, ask questions, etc.;
  • conceptual: describe what the problem means, identify what mathematical concept is involved, what the unknowns are, what the knowns are, draw diagrams/figure/curve, make tables, create relationships between knowns and unknowns, write mathematical expressions, equations/inequalities, see patterns, solve a special case, recall an analogous situation or problem, consult a related solved problem, generalize, etc.;
  • arithmetic: know decomposition/recomposition of numbers, master arithmetic facts, understand place value, describe the relationship between the quantities, estimate the outcome, create an empty number line, make a bar model, make a concrete model, draw a picture, create or use a graphic organizer, etc.;
  • algebraic (identify the variables, write a formula, equation, or inequality, construct a table, chart, graph, or diagram, sketch the function, identify the parent function, create a prime factor tree or successive prime division chart, use a graphic organizer, etc.),
  • geometric tools (draw a figure or diagram, classify data or information, look for spatial relationships, etc.);
  • probabilistic and statistical (draw a Venn diagram, make a graph, create a tree-diagram, make lists, make a model, consult result tables, guess and check, etc.)

Mathematically proficient students gain entry to the problem situation and the solution process by using appropriate physical tools, manipulative materials—such as Cuisenaire and BaseTen blocks, for example, at the elementary school level, fraction strips, fraction bars, algebra tiles at higher grades or thinking tools (writing relationships between knowns and unknowns) to model a problem. For example, mathematically proficient high school students analyze graphs of functions and solutions and their behaviors with a graphing calculator and realize that technology can enable them to visualize the results of different assumptions on the conditions of the problem, explore their consequences, compare predictions with data, and the role of assumptions and constraints on the solution process.

Thinking tools also develop the ability to make sound decisions about when each of these tools might be helpful and gain the insight from their optimal use and also their limitations. This certainly requires that students gain sufficient competence with the tools to recognize the differential power and efficiency they offer.

It also requires that their learning include opportunities to decide for themselves which tool serves them best and why. In order for students not to become dependent on a particular tool and strategy and to develop flexibility of thought, it is important that the curriculum and teaching include the kinds of problems that involve the use of different tools. Students use tools efficiently and deepen their understanding by using different tools to solve the same problem. For example, from time to time, a particular tool is used until students develop a competency that would allow them to make sound decisions about which tool to use. The proficiency in the use of a tool is developed when we use it frequently, discuss its use from different perspectives, and apply it in several problems. In many situations, paper and pencil are inefficient and using them is not strategic. We must therefore develop the notion that mental computations are possible, reliable, and often more efficient. However, students should have skills to detect possible errors by strategically using estimation and other mathematical knowledge.

Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful (e.g., the flat piece in the BaseTen blocks kit represents 10×10 =102 at the third grade level, 1×1 = 12 at fifth grade level, and a×a = a2 at seventh grade level).

As we explore the connections between different types of concepts (e.g. numbers relationships) to use them flexibly, we need to explore the similar interactions between different types of tools to be able to use them flexibly and strategically.  For example, using BaseTen blocks for place value or for addition and subtraction operations encourages children to count, but combining BaseTen blocks and Cuisenaire rods precludes that possibility. Similarly, learning how to solve linear equations can follow the sequence for the strategic use of several tools:

  • Invicta balance to derive and learn the properties of equality,
  • Cuisenaire rods, BaseTen blocks, and Algebra tiles to learn arithmetic and algebraic manipulations and then to arrive at the procedures, and properties of operations,
  • Paper and pencil to record these activities and procedures, then practice these operations formally,
  • Graphing tool to see the behavior of the equations, functions, and solutions,
  • Using computer algebra system (CAS) to take more complex equations and see their relationships and behaviors. To have proficiency in the strategic use of tools, the role of questions and classroom discussions is critical. The teacher can ask questions to help students to identify, select and use tools effectively.

To have proficiency in the strategic use of tools, the role of questions and classroom discussions is critical. The teacher can ask questions to help students to identify, select and use tools effectively.

Use Appropriate Tools Strategically: Right Tools for the Right Job – Part I