Effective Teaching of Mathematics

The framers of CCSS-M have identified Standards of Mathematics Practice (SMP) based on practices and research on teaching and learning that consistently produce mathematics learners with high achievements. To be an effective teacher, one needs to know the content, pedagogy and models to deliver that content, and understand how students learn.

CCSS-M gives us what is important to teach and learn at what level. It describes the content and levels and nature of content mastery. However, teachers decide how to teach and assess. The goal of the first four to five years of students’ mathematics experience is to become proficient and comfortable in number concept, numbersense, and numeracy. After that, students use their understanding and fluency in numeracy skills to learn mathematics—algebraic and geometric models, and their integration.

The Standards for Mathematical Practice (SMP) describe ways in which students should be engaged in increasingly demanding subject matter as they grow in mathematics expertise and content throughout the elementary, middle and high school years.

To support students’ growth in mathematical maturity, designers of curricula, assessments, and professional development should connect mathematical instructional practices to the mathematics content espoused in the CCSS-M. Without connecting content standards with instruction practice standards, we cannot achieve the goals of CCSS-M. Assessment, on the other hand, is to see whether that connection has been made or not.

The Mathematical objects (e.g., numbers of various kinds—natural to complex; geometric entities—shapes, figures, diagrams, functions; operations and procedures of different types—decomposition/ recomposition, manipulations of numbers of different kinds and forms, e.g., long-division; transformations and functions—static and dynamic, congruence and similarity, matrices and determinants, etc. are examples of components and manifestation of content standards.

Mathematics practice standards, on the other hand, describe what actions teachers are to take so that students make the mathematics content—language, concepts, procedures, and skills their own. Thus while we usually pay attention to nouns in content standards, for practice standards we must pay attention to verbs. The Standards of Mathematics Practice are action steps to make the content of the CCSS-M possible to take hold in the classroom and make students learners of mathematics.

Why SMP?
The National Mathematics Advisory Panel concluded that our students have reasonable, though incomplete, factual and procedural knowledge, but poor conceptual knowledge (e.g., many do not fully understand the base-10 number system, concepts of fractions, how decimals and fractions are related, the differences in arithmetic and algebraic reasoning). This is because of our over emphasis on procedural knowledge throughout children’s mathematics experience.

At the same time, the last 20 years of mathematics education reform indicate that American students, even many college students, neither have automatized fact retrieval nor have achieved fluency with procedures. For example, many students can perform routine procedures (e.g., procedures on fractions) but cannot justify the reasons for the steps involved or provide estimated answers before they execute the procedures.

Our students’ lack of conceptual understanding is a major cause of concern and requires investment after they leave school. For example, many corporations spend large sums of money in training high school graduates in their use of simple arithmetic. The problem has also reached college and university levels. Half of the students at community colleges, and 1 in 5 students at four-year institutions, require remedial courses in writing and mathematics, with community colleges spending more than 2 billion dollars on remediation and four-year colleges $500-million. As one university leader pointed out,

Many of us in higher education have observed an increasing number of students arriving at our doorstep not fully prepared to pursue a college degree. This is our collective problem as a nation. (Chronicle of Higher Education, 2014, June 14)

Framers and supporters of CCSS-M and SMP believe that this latest educational reform, if properly implemented, can alleviate some of these problems and narrow the college and career preparation gap. Their aim is to improve instruction so that students acquire mathematics ideas with conceptual understanding and procedural fluency so they can apply mathematical tools effectively and provide reasons for what they do.

Let us consider an analogy to illustrate this point: You can watch two people swim a length of a pool. They take nearly the same time to swim the same distance, but one of them churns the water more and takes more strokes. When this swimmer gets out s/he is breathing a bit heavier but is in great shape, so it is not too noticeable. The other swimmer took fewer strokes and seemed to glide through the water.

If we assess the performance only by a stopwatch, we will conclude that they are swimmers of the same competence. In actuality, they are not.  As the lengths pile up and the task gets harder, the second swimmer will do much better. No matter what the stopwatch said, s/he is a much better swimmer than the other. And if we ask the first swimmer to swim a long distance over deep water, s/he may very well drown.

Students who only memorize facts, formulas and procedures, without understanding, are like the first swimmer. They churn and work hard, and if they are gifted with an outstanding memory, they can pull it off for a while. These students can – and often do – get by in the early grades when they can rely on their strong counting skills and contextual clues (e.g., concrete materials and pictures) to “find” a fact but they typically hit a wall sometime around 4-5th grade when they have to deal with fractions and decimal numbers with problems involving unfamiliar content and complex mathematics vocabulary words. On the other hand, children who have the rich language, robust conceptual schemas, and clear understanding of and fluency in executing procedures develop into graceful, effortless swimmers in the waters of mathematics.

In cultivating greater conceptual knowledge, effective teachers do not sacrifice procedural or factual knowledge. Procedural or factual knowledge without conceptual knowledge is shallow and unlikely to transfer to new contexts. At the same time, conceptual knowledge without procedural or factual knowledge is ineffectual and inefficient in execution. It needs to be connected to procedures so that students learn that the “how” has a meaningful “why” associated with it. It is more effective to move from conceptual knowledge to procedural knowledge. Increased conceptual knowledge helps students move from competence with facts and procedures to the automaticity needed to be good problem solvers.

When teaching conceptual, procedural and factual knowledge, effective teachers ensure that students gain automaticity. Their students know that automaticity and understanding of procedures and facts is important because it frees their minds to think about concepts and making connections. This requires some memorization and ample practice and the ability to communicate with the support of reason.

What the Research Says
The data from the 13 million students who took the Program for International Student Assessment (PISA) tests shows that the lowest achieving students worldwide are those who use memorization strategies – those who think of mathematics as methods to remember and who approach mathematics by trying to memorize facts and procedural steps. The implication is that we should not encourage rote memorization without understanding (i.e., in dividing fractions, teachers should not continue to use statements such as: “just invert and multiply”) but, instead, we should present appropriate models to create conceptual schemas and arrive at and master the procedure with proper understanding and mastery that is rooted in reasoning.

Because of the emphasis on procedural teaching, the U.S. has more memorizers than most other comparable countries. Perhaps not surprisingly mathematics teachers, driven by narrow and subjective tests (e.g., end of section tests in textbooks that value only the methods considered in the textbook), have valued those students over all others, communicating to other students that they do not belong in the mathematics class. Current research about how we learn and how our brains receive and process information shows that the students who are better memorizers may not have more ability or potential, but, unfortunately, we continue to value the faster memorizers over those who think slowly, deeply and creatively. We need students with mastery or conceptual, procedural, and factual knowledge for our scientific and technological future. Certain things should be memorized to relieve our work memory from mundane fact work to focus on creativity and applications.

Poor conceptual understanding comes at a cost. For example, if a student thinks that an equal sign means “put the answer here in the box,” she will be confused the first time she sees an equation with terms involving variables and multiple operations on both sides of the equal sign. Similarly, when a student first encounters factoring (whether in the case of whole numbers or polynomials), she ought to see its relationship to division and multiplication. But she may not be able to do so unless she has a deep conceptual understanding of multiplicative reasoning—that division and multiplication are inverse operations. She also will be slowed in factoring if she hasn’t memorized the multiplication tables, divisibility rules, prime factorization, and short division.

Factual and procedural knowledge are acquired by practice and stick-to-ness to tasks and practice. But, to be engaged in the task deeply, the task should be meaningful and the student should have understanding of the concepts and skills involved in it.

Of the three varieties of knowledge that students need, conceptual knowledge is difficult to acquire. It is difficult because knowledge is never transferred from one person to another directly nor is it developed without explorations, concrete manipulations, effective reasoning, and questioning. Rather, new concepts must build upon something that students already know when they explore the new concepts and ideas. Examples that are familiar to students and analogous to the current concept are useful to understand the concept.

SMP Principles
The writers of the CCSS-M were careful to balance the development of conceptual understanding, procedural skill and fluency, and application at each grade level. The standards are based on the idea that procedural skill and fluency expectations hinge on conceptual understanding. Fact fluency and procedural fluency help students to develop concepts, make connections, observe patterns, and form relationships between ideas, concepts, skills, and procedures thereby facilitating mathematical thinking. With mathematical thinking, students take interest in mathematics and develop mathematics stamina.

To achieve the different kinds of knowledge, we need to adopt pedagogical principles in every lesson that are informed by the Standards of Mathematics Practice:

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

Each of these standards adds to a teacher’s ability to develop the different components of knowledge, helping children acquire mathematical ways of thinking, creating interest for mathematics, and recognizing the power of mathematics. We need to incorporate these practices in our lessons if we want to have students who enjoy doing mathematics and achieve higher.

Examples of Standards
In future posts, I want to comprehensively develop how to implement each of these standards. At present, I want to consider a few illustrative examples.

Standard number one, for example, deals with developing understanding and engagement with a problem and creating mathematics stamina in solving it.

Solving a problem almost always depends on what tools (linguistic, conceptual, and procedural) one knows and how to connect those tools with the current problem. As students advance and encounter new problems, new concepts will increasingly depend on old conceptual knowledge. For example, understanding and solving algebraic equations depend on the understanding of the concepts of equality, variable, arithmetic operations, operations on fractions and integers, ability to generalize, etc. In solving problems factual, procedural, and conceptual knowledge all go together. And to stay engaged with the problem requires a student to have mastery of these prerequisite tools.

Familiarity is not the only ingredient necessary for successful problem solving. Students are more likely to understand abstract ideas when they see many diverse examples in the classroom that depict the conceptual components, schemas, and constraints of the ideas. In such instances, they can learn the essential properties in the concept of the problem (e.g., in fractions, the division of the object into equal parts and what is equal there) and which properties are incidental (e.g., in fractions, that the resulting parts need to be whole numbers).

Standard number seven: to look for structure in mathematics concepts and procedures is at the heart of understanding mathematics. Students need to realize that mathematics is the study of patterns—the underlying structures. Students frequently fail to understand the concept if they are not helped to discern patterns—to look for the structure—commonalities among examples and what is different in these examples.

Indeed, when the teacher introduces a concept through an abstract definition alone, e.g., the standard deviation is a measure of the dispersion of a distribution, students miss the conceptual understanding of the concept of standard deviation. The standard deviation is dependent on the spread; therefore, our examples should show the impact of that spread on standard deviation: Two groups of people have the same average height, but one group has many tall and many short people, and thus has a larger distribution and standard deviation, whereas the other group mostly has people’s heights right around the average, and thus has a small standard deviation.

If we introduce students to the formal procedure of finding the standard deviation too quickly, they won’t realize the relationship of spread on standard deviation. To realize the importance of such a component in the concept, we should also consider special cases and non-examples: e.g., to find the standard deviation when every piece of the data is the same, the spread has a particular meaning.

The third standard is the key to developing conceptual understanding in students. No conceptual understanding can be developed without emphasizing reasoning. To emphasize the importance of reasoning— concrete and abstract, examples, non examples, and counter examples play a crucial role.

Let us consider the definition of prime number: A number is called prime if it has exactly two distinct factors, namely, 1 and itself. The definition of prime number has two key features: it has factors and there are exactly two of them. 2, 3, 5, etc. are prime numbers as they have two factors only. It is important to give examples of prime numbers and examples of numbers that are not prime and the reasoning behind the choice. Examples emphasize the components of the concept, and “non examples” help students see the subtleties and nuances of the concept. For example, why is 1 not a prime number? (It has only one factor.) Why is 0 not a prime number? (0 is divisible by any non-zero number; therefore, it has more than two factors.)

The discussion to discern similarities and differences—comparing and contrasting examples, helps students to acquire the language and conceptual knowledge with rigor. With the help of appropriate language and transparent models, conceptual knowledge is converted into procedural knowledge. For this to happen, the conceptual (representational models—concrete and pictorial) should be congruent with abstract, procedural models. And with practice, procedural knowledge is then converted into factual knowledge. This practice should take place in a variety of problems and problem situations that are related to the procedure to arrive at the appropriate level of fluency.

If students fail to gain conceptual understanding, it will become harder to catch up, as new conceptual knowledge depends on the old. Students will also become more likely to simply memorize algorithms and apply them without understanding.

Helping Students Learn Concepts and Procedures
In our schools, much is made of the use of manipulatives to help children understand abstract concepts in mathematics, but many manipulatives and models themselves are abstract (students treat them as a symbol for something else), and not all manipulatives help learning—they sometimes impede it. This is most likely when manipulatives are so visually interesting that they distract from their purpose, when their relationship to the concept to be represented is obscure, or when they are used for rote counting. Manipulatives seem helpful because they are concrete; to be helpful, they should satisfy certain properties.

To illustrate the idea of a fraction, one might divide a cookie in two for the purpose of sharing it with a student. The concreteness of this example is likely less important than its familiarity. In contrast, suppose I cut a hexagon into two pieces and said, “See? Now there are two equal pieces. Each one is half a hexagon.” That example is concrete but less effective because it is unfamiliar; the student has no experience with divided hexagons, and the purpose of sharing is also missing.

Concreteness, in itself, is not a magical property that allows teachers to pour content into students’ minds. It is the familiarity that helps because it allows the teacher to prompt students to think in new ways about things they already know. However, familiarity also may create some misconceptions, half a pizza, half a cookie, half a glass are not precise as key characteristics of fractions may be missed. Students know a fraction when they focus on: (a) What is my whole here? (b) How many parts are there in this whole? (c) Are the parts equal? (d) Do all the parts together make the whole? (e) What is the name of each part? (f) How many of these parts will make the whole? And (g) What is the new name of the whole in the light of these parts? The teacher’s language, questions, and sequence of activities with materials transform the concrete models into representations—pictorial and abstract.

A teacher must move from familiar materials and models to the form that shows all the attributes of the concept and then can lead to abstract representations that are congruent to the abstract procedure. As concepts become more complex, it becomes harder to generate familiar examples from students’ lives to generate mathematics conceptual schemas, and teachers may have to use analogies more often. In such cases, a familiar situation is offered as analogous to the concept under discussion, not as an example of the concept.

An Example of Implementing SMP
In order to focus instruction responding to CCSS-M and applying SMP, teachers need to identify the essential language, concepts, procedure and skills in each major standard. SMP does not advocate a “one size fits all” model or a boxed curriculum. Essential Elements of each lesson informed by SMP include: Integration of language, concepts, and procedures, Multisensory (appropriate and efficient models), decomposition/ recomposition of problem components, Synthetic-Analytic (seeing patterns and analyzing the problem), Structure (logical language categories), Sequential (simple to complex), Cumulative (continually making connections), Repetitive, cognitive (meta), Diagnostic and Prescriptive (design lesson to assure progress and plan next lesson around noted errors that need additional reinforcement).

Let us illustrate this process in one specific mathematics standard. Mathematics standard 4.OA.A.3 (CCSS-M, 4th grade) says:

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

This standard includes only a few sentences, but involves several different terms, concepts, procedures, and skills. To make sense of and understand them, students need to know the meaning of these terms and concepts and execute the procedures.

There is a need to delineate these elements, focus on them, teach them, help students master them, connect them with other concepts and procedures, and then assess all of these elements.

There should be a clear understanding of what and how to represent each concept, procedure and the skill involved in this standard. Every concept and procedure involved in this standard should be transformed into a set of concepts and skills to be learned, mastered, and applied by the students. In the context of CCSS-M, teaching should be to acquire understanding; students should arrive at fluency and should be able to apply concepts and skills contextually.

Step # 1
Language and Concepts

  • Know the meaning of each word and term in order to translate from English to mathematical equations
  • Identify the unknowns and understand the role of these unknowns; know the relationship(s) between knowns and unknowns

Step # 2
Language and Concepts

  • Represent terms and words into appropriate mathematics symbols; translate multi-step word problems into/by equation(s)

Skills and Facts

  • Identify the units and the domain and the range of the variable(s) involved in the problem

Step # 3
Procedures

  • Solve multistep word problems by establishing the sequence of arithmetic operations

Skills and Facts

  • Know and apply the properties of equality; mastery of arithmetic facts; execute procedures for whole numbers efficiently; know the order of operations

Step # 4
Concepts and Procedure

  • Assess the reasonableness of the answer

Skills and Facts

  • Numbersense: Use mental computations such as rounding to estimate the outcome of an operation

Step # 5
Language and Concepts

  • Interpret the answer including the remainder if involved; express the division problems in multiple ways

Skills and Facts

  • Add, subtract, multiply and divide whole numbers fluently with understanding; know the role of numbers in each operation, e.g., know the role of remainder in practical situations

Learning with rigor using SMP means that the students not only understand the concept and procedures but also see that a particular method(s) may have limitations and that the context of the problem defines the applicability and efficiency of the method.

 

Effective Teaching of Mathematics

What Does it Mean to Master Arithmetic Facts?

Question: Everyone is focusing on children knowing mathematics ideas. What does it mean to know arithmetic facts? Is mastery different than knowing? Is it important to memorize arithmetic facts?

Answer: In the context of mathematics curriculum, the term mastery is associated with several words: know, understand, master, being fluent, proficient, etc. Although it has been clearly defined in the Common Core State Standard in Mathematics (CCSS-M) documents, in our schools, the definition of mastery of mathematics curricular components still varies with individuals, individual school systems, and textbook series.

Students need to satisfy certain characteristics to have acquired mastery of mathematics curricular elements. Mastery of different curricular elements is complex; it is multi-dimensional.

Components of Mastery
A student has mastered an arithmetic fact when three conditions are satisfied: The child

  • demonstrates the understanding of the concept—can provide an efficient strategy (based on decomposition/recomposition and select the most efficient strategy out of several strategies), without counting, for arriving at the arithmetic fact;
  • demonstrates fluency/automatization (able to produce a fact orally in less than two seconds and in writing in less than three seconds), and
  • can flexibly apply the knowledge of the fact in different situations in solving problems without counting and using concrete or pictorial models.

Learning with rigor (understanding, fluency, and applicability) is at the center of the Common Core State Standards for mathematics (CCSSM). Teaching with rigor produces flexible and long lasting mastery.

Understanding, here, means a student is able to derive a fact or procedure with effective, efficient strategies and able to explain his approach, for example, an addition fact is derived by using decomposition/recomposition of numbers, not just by counting up. This also includes flexible use of these strategies. It is widely acknowledged that practice, drill and memorization are essential if students are to become mathematically fluent. However, practice without understanding is of very little value as the facts and skills are not integrated and applications under such limited level of mastery become difficult.

The concept of “fluency” refers to knowing key mathematical facts and methods and recalling them efficiently. Fluency is automatized mastery of a fact, skill, or procedure.

Understanding and fluency facilitate the application of facts and procedures in problem solving. Applications may be learning other mathematics concepts using those facts. It may be problem solving in other disciplines or extra curricular situations.

Without this level of automatization, children become dependent on concrete or pictorial materials used in instruction such as: counting objects—number line, hash-marks, fingers, TenFrames, discrete objects, etc. Strategies based on counting materials do not help children to achieve either the understanding and fluency or the ability to apply such knowledge.

The elements of rigor (mastery) are inseparable. The absence of any one of them is problematic. For example, when children do not have understanding of the strategy and have not automatized facts, they are not able to apply their knowledge to newer situations effectively. They digress from the main problem to generate the facts needed to be used in solving a problem. Their working memory space is filled mostly in constructing facts. Then it is not available to pay attention to the instruction, to observe patterns, focus on concepts, nuances, relationships, and other subtleties involved in the concepts, procedures, and applications.

When children possess number combination mastery, their achievement increases at a steady rate in arithmetic, whereas children with low mastery in arithmetic facts make little to almost no progress in later grades. When students, without mastery of arithmetic facts, are provided classroom instruction in procedures, although they make good strides in terms of facility with these algorithms and procedures and even solving simple word problems, deficits in the retrieval of basic combinations remain. These deficits inhibit their ability to understand and participate fully in the mathematical discourse and to grasp the more complex multi-step and algebraic concepts later. Failure to instantly retrieve a basic combination, such as 8 + 7, often makes discussions of the mathematical concepts involved in algebraic equations more challenging.

Before we can have effective mathematics teaching and children can achieve higher in mathematics, everyone involved needs to have a well-defined and commonly-agreed upon definition of knowing/mastery for a concept, procedure, or skill—with clear markers for mastery. Then and only then can it be taught well, retained by students, and assessed and monitored effectively.

Essentials for Higher Achievement
To assure the conditions for higher achievement, we first need to
(a) identify non-negotiable concepts, skills, and procedures to be achieved by each child at each grade level,
(b) develop common definitions and criteria for “knowing” concepts, “mastering” skills, and achieving “proficiency” in executing procedures, across the school system, and
(c) identify and discuss the most “effective,” “efficient,” and “elegant” ways of teaching the key developmental mile-stone concepts and procedures at each grade levels. These include: number concept, arithmetic facts, place value, fractions, integers, and algebraic thinking.

When we have identified these, training should be provided for all teachers (all classroom teachers and interventionists of various kinds) and administrators by using content embedded pedagogy. The training should also include how to observe children’s work and learn from the error analysis about their level of mastery.

Observing Children’s Work
In a second grade classroom I observed a teacher assessing children’s mastery of addition and subtraction facts. The purpose of the test was to assess understanding, fluency, and applicability of addition and subtraction. The test had 25 problems on addition and subtraction.

One of the problems on the test was 17 − 9 = ☐. I identified five children from the class to observe work on the problem. Problems were written both in horizontal and vertical forms:

17

 –9

Here are my observations.

Child One
One child read the problem and solved it in about 20 seconds by sequentially counting on his fingers: 10, 11, 12, 13, 14, 15, 16, and 17. Then he recounted the fingers used: 1, 2, 3, 4, 5, 6, 7, 8 to find the answer and wrote the correct answer 8.

Child Two
The second child answered the problem in about 50 seconds. She drew seventeen tally marks in front of 17 and nine tally marks in front of 9. She crossed one tally mark in front of 17 from the top and one from the 9. After all the nine tally marks were exhausted, she counted the remaining tally marks in front of 17 and correctly wrote the answer as 8.

Child Three
The third child answered the problem in about 36 seconds. The child counted: 17, 16, 15, 14, 13, 12, 11, and 10 on his fingers and counted the fingers used. And then he wrote the correct answer in the right place.

Child Four
The fourth child used the number line pasted on his desk. The child located numbers 9 and 17 on the number line and then counted the numbers from 9 to 17 and wrote down the correct answer in about 27 seconds.

Child Five
The fifth child read the problem and thought for a moment and wrote down the answer (8) in 2 seconds in the correct place.

The teacher collected the papers of all the children from the class. During the debriefing, I asked her to check the problem: 17 − 9 = ☐ on these five children’s papers. She did. She put a check mark (✓) in front of the problem on their papers.

I asked her whether these five children “knew” the answer to the problem 17 − 9 = ☐ and did she have enough information to judge the responses to satisfy the criteria for knowing?

She said: “Of course. They have the correct answer on their papers.” I asked other participants to also look at the response of these five children. They agreed with the teacher.

It is true that the children had the correct answers, but they knew the fact 17 − 9 = 8 at different levels of knowing.

The first and the third child knew the fact: 17 − 9 = 8 at the concrete level and used counting as a strategy. The second and the fourth child knew the fact at the pictorial level and also used counting strategy. The second child used a simple one-to-one correspondence counting. Unlike the others, the fifth child answered the problem without any overt strategy and very quickly.

In the case above, only the fifth child, who answered the problem in 2 seconds (within the expected time for response), had automatized this subtraction fact. However, we do not know whether she just memorized the fact in a rote manner or with some strategies. Therefore, her example does not satisfy the definition of knowing.

The four children arrived at the correct answer but used inefficient strategies. They also have not achieved fluency in arriving at the fact. Very few children arrive at fluency using inefficient strategies such as counting.

The best method of automatization of arithmetic facts is to practice the facts first orally. When children have shown the mastery orally using efficient strategies, then the children should practice the facts in writing. A fact is mastered orally when the child can answer the fact in 2 seconds or less. A fact is mastered in written form when the child can answer a fact in 3 seconds or less and able to furnish a strategy when asked.

Assessment of arithmetic facts should be oral first with immediate feedback about the strategy and its efficiency. When children can answer facts orally in the prescribed time, the teacher should ask for the strategy used. Then there should be intentional effort to make children flexible by providing alternate strategies. There should be a class discussion on which strategy is most exact and of all the exact strategies which ones are efficient and then which ones are elegant.

Without focusing on children using efficient strategies and correcting papers only for correctness, we send a message to the children that they can keep using inefficient strategies. When children become fluent in using inefficient strategies, they may become fluent counters.

Several studies found that a significant area of difference between students with number combination mastery and those without was the sophistication of their strategies. The poor combination mastery group in second and third grades continued to use fingers to count when solving problems. In contrast, their peers increasingly used verbal counting or decomposing/recomposing numbers without fingers, which led much more easily to the types of mental manipulations that constitute mathematical proficiency.

In the development of counting knowledge in young children, one can observe that children use an array of strategies when solving simple computational problems. For example, when figuring out the answer to 6 + 8, a child using an unsophisticated, inefficient strategy would depend on concrete objects by picking out first 6 and then 8 objects and then counting how many objects there are all together.

A slightly mature but still inefficient counting strategy is to begin at 6 and “count up” 8. Still more mature would be to begin with the larger addend, 8, and count up 6, an approach that requires less counting. However, all of these are based on counting, and many of these children will not reach fluency in addition and subtraction. Effective teachers promote efficient use of number relationships and ultimately help children transcend counting. They help children acquire efficient number combination strategies based on decomposition/recomposition (e.g., sight facts, making ten, teens numbers, doubles, near doubles, the missing double, etc.).

For example, using decomposition/recomposition, a child might say that 6 + 8 = 6 + 4 + 4 = 10 + 4 = 14 (making ten and teens numbers), or 6 + 8 = 4 + 2 + 8 = 4 + 10 = 14 (making ten and teens numbers). Some children might give this as: 6 + 8 = 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, 6 + 8 = 6 + 6 + 2 = 12 + 2 = 14, or 6 + 8 = 6 + 1 + 7 = 7 + 7 = 14 (decomposition/recomposition and doubles). When children can furnish more than one strategy (other than counting) in arriving at a fact, they show flexibility of thought. Children with flexibility of thought are able to apply their fluency in novel situations and go higher in mathematics.

Some children will simply have this combination stored in memory and remember that 8 + 6 is 14. Ultimately, that is what we want children to reach—a level of automatization. But this should be reached with understanding and with the use of efficient strategies. Memorization of a fact with repeated use remains an isolated fact. Its application also remains isolated. However, it does have some advantages if that fact mastery is used to gain other fact mastery by using decomposition/recomposition. For example, when a child can retrieve some basic combinations (sight facts of a number, say 10 = 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10), then he or she can use this information to quickly solve other problems (e.g., 6 + 5) by using decomposition (e.g., 6 + 4 + 1 = 11).

The ability to derive, store, and easily retrieving information in memory helps students to build both procedural and conceptual knowledge of abstract mathematical principles, such as commutative and the associative laws of addition and mental fluency and numbersense (e.g., 9 + 7 = 9 + 1 & can be extended to 59 + 7 = 59 + 1 + 6; 149 + 7 = 149 + 1 + 6). Immature finger or object counting creates few situations for learning these principles. Research also suggests that maturity and efficiency of strategies are valid predictors of students’ ability to profit from later mathematics instruction.

Fluency
In order to achieve fluency, teachers, parents, and students should understand what fluency looks like. To achieve fluency there is a definite progression. For example, in the case of addition, it involves:

  • mastering number concept—subitizing and visual clusters, decomposition/recomposition, 45 sight facts (what are sight facts, see previous blog on sight facts);
  • mastering addition strategies—commutative property of addition, N + 1, making ten, teens numbers, N + 9, double numbers, near doubles, missing double, N + 2, near tens (9 and 11), and the remaining four facts (8 + 4, 4 + 8, 8 + 5, 5 + 8) using decomposition/recomposition;
  • “working out” using efficient strategies with understanding to efficiently generating an answer (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, 8 + 6 = 4 + 4 + 6 = 4 + 10 = 14, 8 + 6 = 2 + 6 + 6 = 2 + 12 = 14, 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, 8 + 6 = 6 + 8 = 6 + 1 + 7 = 7 + 7 = 14);
  • practicing “rapid recall” and finally,
  • the ultimate goal of “instant recall.”

The time taken and the expectations of efficient recall vary per topic. In mastering addition facts, the role of sight facts, decomposition/ recomposition, making ten and teens numbers is crucial. For example, a child may be able to recall the sight facts of 9 and then practice adding other numbers to it using decomposition/recomposition (9 + 2 = 9 + 1 + 1; 9 + 3 = 9 + 1 + 2, etc., or 9 + 7 = 10 + 7 – 1, etc.). Students need to keep working on decomposition/recomposition strategies till they are able to generate the facts and then keep on practicing automatization.

When a child can read fluently, we are not able to detect the strategies used for acquiring the reading skill. Fluent reading means the child has transcended the strategies and skills used for arriving at that fluency.  Similarly, when we ask a child to find the sum of numbers 8 and 7 and the child counts 9-10-11-12-13-14-15 and says: 8 + 7 = 15, we should not be content with that. Instead, we need to give the child better strategies and work with the child until he or she arrives at fluency. Sequential counting is decoding of numbers; it is not mastery of a fact. By giving the feedback—“good job” to such a strategy, we are sending the wrong message that decoding of numbers is adequate. Knowing addition as “counting up” and subtraction as “counting down” are not strategies.

Accuracy
Mistakes are an integral part of learning, but it is equally important for teachers and children to be aware of the need for accuracy. When a child gives a wrong answer (for example, 8 + 6 = 15), the teacher needs to redirect the child:

  • 8 + 7 = 15, our problem is 8 + 6, can you use that fact to find 8 + 6?
  • what other strategy would you use for finding the answer? How would you make 8 as 10? Etc.
  • can you use another strategy? Etc.

Through these activities—by constantly bringing their attention to the appropriate strategy, the child should derive the fact. Once the child derives the correct answer using an efficient strategy, direct the child to another equally efficient strategy.

After that, observational assessments should be used to ensure that all children are being accurate. Accuracy should be achieved first at oral level with immediate feedback when an incorrect answer or strategy is produced. This involves listening to children’s verbal responses, targeting specific children with differentiated questions and checking responses on whiteboards. Then, children should also be given responsibility for self-assessing their own work for accuracy.

Speed
Automatization and fluency at appropriate level (speed) is achieved by practice. Practice sessions for achieving speed should be brief, paced, and create a buzz of excitement in which children’s recalling and using their knowledge efficiently gives the feeling of achievement (immediate feedback).

It is important, though, to recognize that fluency is not solely about memorizing and recalling facts; it also means being able to work flexibly and choose the most appropriate method for the problem at hand. Children do need drilling in the basics, but this can be delivered in open-ended, rich and engaging ways. The key is to balance the three components outlined above. As is often the case in teaching, getting the balance right is crucial. Fluency of facts is essential, and if we teach in a style and order that suits the development of this fluency, we do not risk sacrificing creativity and contextual richness in mathematics tasks.

A short modeling demonstration by the teacher should be followed by game-like activities lasting between three and four minutes. The fast pace of these activities, combined with the emphasis on aiming to beat the child’s own personal best, makes the session exciting and engaging for the child. Each child should be working on identified facts. For example, The Addition Fact Ladders[1] and Fact War Games[2] are good for achieving accuracy and speed.

Thus, for automatization, first the child must have the understanding, and then accuracy, followed by speed. For that, a great deal of practice is essential.

Difficulty of the tasks (facts to be mastered) should be gradually increased while practicing and testing. The first practice and test should not be difficult, and oral practice with immediate practice is required. Teachers should point out not only the strategy but also the name of the strategy. For example, when a child is practicing a fact, say 8 + 6, the child says: I take 2 from 6 and give it to 8 and then I add 10 and 4. The answer is 14. The teacher should ask: Which strategy did you use? The child should say: “Making ten and then teens’ facts.” The strategies can be shared with parents, and the Addition Ladders can be given to them so they can practice them with their children at home.

In my experience with thousands of children, I have found that children overwhelmingly want more challenge in mathematics; they want mastery, but they also want efficient methods. What is important is giving every child a challenge that is personal to them and is attainable with the right amount of effort and practice. Children like to set their own mathematics targets in consultation with and with scaffolding from their teacher or tutor. These targets should be specific and achievable within a period of a few weeks. This motivates students to work hard and keep pushing themselves towards new goals.

During the “Tool Building Time” of daily lessons, the teacher should identify an ‘arithmetic fact of the day’ or adding a particular number to other 10 numbers (using the Addition Ladder). She should ask that fact orally and in multiple forms from all children. This process should be repeated until the fact(s) has been mastered orally. Then give them 20 facts on a sheet of paper (randomly organized and including the ten facts just practiced interspersed with previously mastered facts). The goal is one minute. At the end of one minute, children put a marker and continue till finished. The teacher reads the answers and children correct their papers. Children, in turn, give a correct strategy for a wrong answer.

One of the important elements in helping a student to memorize arithmetic facts is the questioning process and the type of questions the teacher asks. It works in the following way: Once the teacher has chosen an arithmetic fact, then she asks children the various forms of that fact.

The nature of questions to be asked should be at the level and competence of the individual student. For example, if you ask a student the question: what is 8 + 7? and the next student has difficulty with most addition facts, then the teacher should just ask: “what did the previous student give as the answer for the questions: 8 + 7? “What question did I ask and what was his/her answer?” “You know 7 + 7 is 14, then what is 7 + 8?” “What is 8 + 7?”

If the student is able to repeat it, then ask: if 8 + 7 = 15, then what is 7 + 8 = ? A child who already knows 8 + 7, then the teacher, in the next turn, should ask the child: 18 + 7 = ?, or 48 + 7 = ?, etc. She can then ask that student questions like this, out of turn also. This will help him to participate in the lesson and increase his confidence and deepen his understanding. This is an example of differentiation.

To develop flexibility in arithmetic facts, I suggest the following types of questions to be asked for each arithmetic fact.

7 + 4 = ______; 7 + ____ = 11;        ____ + 4 = 11; ____ + ____ = 11; ____ = 7 + 4; 11 = 7 + ____; 11 = ____ + 4; 11 = ____ + ____

Once a student has mastered the family of facts related to a given arithmetic fact orally, then the teacher could give an Addition Fact Ladder to a pair of students and ask them to practice it orally by quizzing and helping each other.

The speed is also achieved by using the Visual Cluster cards. The teacher displays one card after another and asks questions using different strategies (see the list of Addition Strategies below):

  • What is this card?
  • What is one more than this number?
  • What number will make it 10?
  • What is the double of this number?
  • What is one more than the double of the number? Etc.

This is repeated for the whole deck of Visual Cluster Cards. The same questions are repeated for making ten, doubles, double and one more, two more, making 9, making 11, etc.. After the practice has been done for each strategy individually, the teacher displays two cards and asks:

  • What are these two numbers?
  • What is the sum of these two numbers? If the student gives the correct answer, then ask:
  • What strategy did you use?
  • Can you use another strategy?
  • Any other strategy? Etc. If the child doe not give the correct answer, the teacher helps him to arrive at the appropriate strategy to be used.

When To Automatize Arithmetic Facts?
To achieve mastery of numeracy by the end of fourth grade, mastering of fractions in fifth and sixth grade, and integers by the end of sixth grade, children should have mastered 45 sight facts by the end of Kindergarten; 100 addition facts by the end of first grade, 100 subtraction facts by the end of second grade, 100 multiplication facts by the end of third grade, 100 subtraction facts by the end of fourth grade. This should be with understanding, fluency, and applicability.

[1] Available electronically free from the Center for Teaching/Learning of Mathematics

[2] See Games and Their Uses by Sharma (2008).

What Does it Mean to Master Arithmetic Facts?