**Mahesh C. Sharma **

Children’s lower achievement—lack of arithmetic competence and difficulties in learning mathematics concepts, procedures and skills—and learning problems are often explained in terms of memory deficits (short-term, working, and long-term), problems with pre-requisite skills in learning mathematics (e.g., ability to follow directions), and information processing issues. In most studies, these factors are implicated alone as well as in combinations.

Behavioral and neurological evidence from the last 30 years of research has demonstrated the complex network of skills involved in reading, writing and math achievement. Researchers have illustrated the innate processing differences among students with and without disabilities in different aspects of skill learning (e.g., reading, language, etc.). However, their role is less explored and understood in the area of mathematics learning difficulties.

When a mathematics problem is posed, before linguistic, quantitative and spatial processes are applied in solving the problem, children and adults alike go through a combination of conscious and unconscious processing and decision-making protocols. A set of skills: processing skills, executive functions (working memory and flexibility of thought) are called upon to make sense of the information in the problem and then solving it.

**A. Working Memory**

Mathematics learning is a complex phenomenon, both in content and involvement of learning processes. One of the important aspects of this learning is holding and manipulating information during reading a definition, solving a problem, or seeing connection between different pieces of information. This takes place in the memory complex, particularly, in the working memory. How do we keep everything in mind when solving a simple or complex mathematics problems? When you read a word problem, listen to the teacher or have a conversation about a concept or problem–how does our brain hold onto all that information? It takes place in the **working memory**.

Unlike long-term memory (that is where what you have mastered resides—arithmetic facts, procedures, etc.), working memory is not about remembering the facts, formulas, and procedures already learned. Instead, it’s about holding together the current information (received from short-term memory, brought from long-term memory, generated by reflecting on these, and by visualizing) in our mind so we can learn, make decisions and solve problems.

Working memory (WM) iscalled the workbench/sketch-pad, working band-width of the mind. Working memory allows us to store useful bits of information for a few seconds and use that information across different brain areas to help solve problems, plan or make decisions.^{[1]}Working memory has a very high correlation with learning, in general, and mathematics, in particular. Working memory skills are frequently utilized in almost every area of mathematics for holding information in the mind temporarily while simultaneously performing specific operations, in order to comprehend the problem, manipulate the information, and possibly produce a correct answer.

Working memory is a key aspect of ** mathematical way of thinking **(creating ideas, discerning patterns, seeing relationships, making connections, modeling ideas, etc.)

**Much of our mathematics learning depends on working memory. Think of the last time you followed a higher mathematics class. In the beginning, you might have kept up fine. But eventually it became harder and harder to understand what the teacher was saying. Even though you tried your best to pay attention, you left feeling confused and frustrated. The concepts and procedures being discussed required your working memory to process too much new and old information at the same time. As a result, the system became overwhelmed and broke down. This happens to our students, with and without poor working memory, in most of our mathematics classes.**

*.*One leading hypothesis contends that working memory works by far-flung brain areas firing synchronously. When two areas are on the same brain wavelength, communication is tight, and working memory functions seamlessly. This is particularly important in mathematics concept/ procedures/problems. Most concept involves various parts of the brain concurrently: *quantitative (*counting, sequential procedural steps in left hemisphere*) *and *spatial information*(e.g., integration of algebraic and geometrical concepts in right hemisphere)),*discrete*and *continuous *information (e.g., visual cluster and number line in learning number concept), *linguistic*and *conceptual*part of a mathematics idea (e.g., solving a word problem). In such problems, there are higher demands made on the working memory to relate and integrate different components of the information.

Working memoryinvolves not only in receiving, retaining and manipulating mathematics information in auditory and visual processing, but also in *monitoring attention*, *mental concentration*(inhibition control), *organization*, and*reasoning*—all skills closely related to learning and achievement in mathematics.

Researchers are trying to understand why this ability is poor in some children and fades as we age and whether we can improve it and slow, or reverse, that decline. Studies have examined the degree of overlap between executive functions and processing speed at different preschool age points; and (2) determine whether executive functions uniquely predicts children’s mathematics achievement after accounting for individual differences in processing speed.

For example, when a child with slow processing speed sees the letters that make up the word *house*, she may not immediately know what they say. She has to figure out what strategy to use to understand the meaning of the group of letters in front of her. It is not that she cannot read. It is just that a process that is quick and automatic for other kids her age takes longer and requires more effort for her.

Similarly, when a student, with or without processing deficit) sees 8×7, she may not know what it is. She has to figure it out. Without the appropriate language and efficient strategies, she may take longer and can easily forget. Let us assume, she even had the language associated with it (8 groups of 7 or 8 sevens)—counting 8 groups of 7 takes a long time and possibility of error. In both cases, she may appear to have poor processing speed.

When a student has poor processing speed, it is important to help student acquire easily accessible, and efficient strategies so that she can take action right away. For example, when a student cannot recall a multiplication fact, she can decompose one of the factors (8×7), 7 in this case into 5 and 2 and then apply decomposition/recomposition (8×7 = 8×5 + 8×2 = 40 +16 = 56.). However, these strategies must be based on strong conceptual base and easy to use.

Research demonstrates that age-related increases in processing speed result in increases in working memory as faster processing may facilitate the formation of connections between the current incoming information from short-term memory and resident information in the working and long-term memory spaces. The formation of more connections between different elements of the mathematics curricula at any grade level, in turn, results in age-related increases in working memory, processing, fluid reasoning and crystalized learning.

**B. Processing Speed**

It is a cognitive ability that could be defined as the time it takes a person to do a mental task. It is related to the **speed** in which a person can understand, react, and respond to the information they receive, whether it is visual (words, letters, numbers, symbols, representation), *auditory*(language—words, instructions, questions, expressions), or*kinesthetic*(psycho-motoric—touch, movement, concrete manipulation). Only through the response—the time it takes, the nature of the response one determines whether the slow response is due to organic reasons or it is lack of knowledge.

Many times, deficits in processing speed and learning and attention issues coexist. Slow processing speed is not a learning or attention issue on its own. But it can contribute to learning and attention issues like ADHD, dyslexia, dyscalculia and auditory processing disorder. It can also impact executive functioning skills—*working memory, inhibition control, organization, and flexible thinking.*

However, having slow processing speed has nothing to do with how smart a student is—just how fast can he/she take in and use the information.

For example, giving too many instructions with advanced and highly complex content with multiple directions may produce slower processing speed. For example, the problem

“*In a collection of 4 consecutive even integers*, *twice the sum of first two consecutive even integers is 3 more than the sum of the last two consecutive integers.*”

contains several concepts, specialized language, and multiple steps. It is difficult to hold this information in the working memory to process the relationship between numbers and then relationship between the phrases. It is not uncommon such situations that a student in spite of having no processing speed deficit problems may have difficulty processing and expressing it.

Whereas, when a student is given the problem:

“*Find the least common multiple of 12 and 20.*”

First, the student will not respond for some time, as the answer is not forth coming, and then will respond: “*I do not know*.” When you insist that they try, most students, in this case, focus on “*least*” first and, then “*common*” and then “*multiple*,” therefore, they give a wrong answer.

The wrong answer is neither because of poor processing, poor working memory, or any other cognitive deficits. It is purely because of the language of mathematics. The order of words (*syntax*) of the phrase and their relationship are quite demanding. Before introducing this concept and the procedure for finding it, I always give a parallel statement:

*John is an intelligent, handsome, tall boy*.

What is the relationship of these words with each other? Where do you start? What do you understand? Students invariably say:

*“John is boy, then John is a tall boy, then John is a handsome tall boy, and then last: John is an intelligent, handsome, tall boy.”*

Here, they can concretize it and can visualize the problem; therefore, they can process it.

Processing Speedis the ability to quickly and correctly scan the information, discriminate visual and auditory components, receive it, and sequence tasks to be performed, perform the immediate relevant mental tasks, and communicate the received information in the appropriate form, contextually. At the lowest level, it involves short-term visual and auditory memory, attention, and visual motor coordination. It requires the student to plan and carry out some instructions given by the problem quickly and efficiently. Research shows that when reading ability is controlled for, arithmetic ability is best predicted by processing speed, with short-term memory accounting for no further unique variance.

Most mathematics tasks rely heavily on visual processing and in the initial stages of learning mathematics concepts (number concepts and number relationships) are dependent on fine-motor skills when children interact and use concrete materials and in the process develop language and conceptual schemas. Later on, processing tasks involve receiving and representing the information visually—either as a diagram, writing equation(s), language and mathematics expression(s), and executing procedures.

Students with poor fine motor skills processing problems do slightly better when there is less of a motor component required in learning and problem solving, whereas, the student with strength in processing in tactile/ kinesthetic modality can compensate for other processing deficits by relying on and using fine motor skills.

Research suggests that we integrate both visual (non-verbal activity—watching lips and facial expressions) and auditory cues when processing speech. Similarly, in learning mathematics, multiple cues (language, visual representations, tactile, and auditory) are used in learning. Therefore, multisensory approaches to instruction have proven beneficial because such instruction provides more cues upon which to build a representation.

The following examples demonstrate the importance of the integration of these phenomena in mathematics learning. Let us consider few examples of such mathematics tasks, at different levels of the mathematics curriculum, to demonstrate the involvement of these processes.

*Find the sum: 8 + 6*.

To find this sum, the student goes through a script (consciously and subconsciously) and asks himself:

- Do I know the answer? Searches the
memory store for the answer. “No!”*long-term*

The child with math anxiety or previous unsuccessful experiences gives up. Others try.

- Do I know how to find the answer? Searches
memory for a strategy. “Yes!”*long-term*

“I can use *Cuisenaire *rods. Places brown (= 8) and dark green rods (= 6) end-to-end. It is more than ten. I place 10-rod parallel to the two rods. I can place the 4-rod in the empty space and it becomes 10 +4. So, 8 + 6 = 10.”

- Do I know any other way of finding the answer? “Yes!”

“I can use *Empty Number Line*(*ENL*). I begin with 8 on the number line and 2 to make it 10. The 2 came from 6 so I add the remaining 4 to 10 to get 10. So, 8 + 6 = 10.”

- Can I do it without any materials? “Yes!”

“The bigger number is 8. I will make it 10. I know how to make ten. I need 2 more to make 8 as 10. I will break 6 into 2 and 4. Then, I have 8 + 2 = 10, and then I add 10 + 4 = 14. So, *8 + *6 =14.”

It should be noted that the strategies at all levels (Concrete, pictorial, visualization, and abstract) use a script and the script provides cues to hold and manipulate the information in the mind’s eye. They provide the student actions to take rather than feel over-whelmed and feel defeated. The Cuisenaire rods, ENL, and decomposition/recomposition help in all aspects of learning.

Most of this information and activity is being processed in the ** working memory**. The child produces the result at each stage and communicates it to him/herself or to others. When this process is repeated, each time it leaves a trace of this action—as a residue in the long-term memory, and ultimately the fluency in constructing it is achieved. Then, the fact and the related strategy go to the long-term memory and get connected with other facts in the old “fact-file” and the “strategies file.” At the same time, a new “file” is also opened. This new fact is at the “constructed stage.” It needs reinforcement—reconstruction and practice. The presence of information in multiple files results in flexibility of thought. Then, the information is processed much faster. Even the processing of new related facts gets easier.

This process reinforces the previous knowledge (making ten, sight facts, teens’ facts, decomposition/ recomposition of number, following a sequence of steps, etc.) and also develops the stamina for constructing facts. This also gives the student a possibility of trying other strategies.

On the other hand, if the student derives the answer by counting: *“I start after 8 and count 6 numbers. 9-10-11-12-13-14. 8 + 6 = 14.”* The student arrived at the answer. But, after finding the answer, the answer is forgotten. No relationships are built; no patterns are discerned. The task is repeated next time when the sum of 8 + 6 is asked for. No reinforcement of working memory and long-term memory takes place. There is very mental processing or flexibility of thought is developed. No mathematical way of thinking is developed or reinforced. This kind of process does not help the student to become a better learner. The deficit remains.

In the absence of an efficient strategy, the student, because of poor strategy—such as counting, takes much longer and to the observer this delay looks like a case of slow processing. And the counting process does not leave any residue in the working memory, so no connections are made.

Most times, the slow processing is misdiagnosed as a result of lack of knowledge both of the content and the absence of efficient strategies. It does not mean there are no cases of slow processing, but these are far fewer than reported.

Efficient strategies help develop and strengthen their ability in learning processes—memory system, processing, organization, and flexibility of thought.

To find this quotient, one decides to apply the process of executing standard long-division procedure. Writes the problem in procedural form:

Learning long division procedure is important as once, learned properly, later in algebra, a similar problem—division of a polynomial by a binomial can be understood and executed. The division of a polynomial by a binomial

requires similar cognitive skills although with a different content. Processing and executing the long-division procedure involves:

(a)** Content retrieval**:

- The steps of the procedure to be executed—
*divide, multiply, subtract, bring-down*, repeatedly applied; - Concepts —
*place value, multiplication, subtraction, spatial orientation/space organization*; - Language—how many groups, is it about right, less than, more than, what is left; and arithmetic facts from long-term memory;

(b) ** Actions**:

- Holding and manipulating new and retrieved information in the working memory space;
- Integrating retrieved information with new information being generated in the process in the working memory;
- Expressing it orally (asking questions to execute the procedure—how many time does 21 divide into 45, etc.);
- Recording (in writing) – the outcome of these steps on paper;
- Making decisions, and,
- Evaluating the impact of these decisions on each step and on the outcome—is the product about right, does the answer make sense.

Long division procedure/algorithm is a complex process and most students have difficulty learning, executing, and applying it, particularly, children with poor fact mastery. And, indirectly, those children with poor working memory and processing issues. The possibility of things going wrong is very high because of any of the memory, thinking, and processing deficits.

All multi-step procedures, thus, place heavy demands on the *working memory*(to hold partial solutions, intermediate steps, and facts in the working memory), *processing speed*(several complex tasks to be executed fairly quickly and sequentially in order not to lose the information), *flexibility of thought*(multiple tasks to be performed presented in multiple forms, decisions to be made and evaluated for usage and efficiency). This creates problems for students with *limited working memory capacity*, *slow processing speed *and*rigidity in thought and action*.

Except the simplest of problems—involving one-step action or primary concepts, every other concept and similar issues affect procedure in arithmetic, geometry, and algebra. For example, *solving multistep equations, writing proofs in geometry, solving a word problem*where language processing and several concepts and operations involve the interaction of memory systems, processing speed, and executive functions.

The underlying causes of the breakdown of memory systems and processing deficits are unclear. However, some researchers suggest a slow articulation rate as a cause.

*Explanation*: Slow articulation makes increased decay of information during recall and interaction.

Others suggest limited space in the working memory.

*Explanation:*This prohibits a learner to hold the information in the working memory, thereby inability in manipulating the information. This, in turn, may prohibit formation of connections between the incoming information from the short-term memory and long-term memory. These preclude seeking, observing, and inability in extending patterns between quantities and shapes. That limits the ability for learning mathematics concepts. This is particularly so as mathematics is the study of patterns in quantity and shapes and their relationships.

Others offer an explanation in terms of slow speed of item identification.

*Explanation:*This creates difficulty in retrieving relevant information stored in long-term memory to the working memory.

General processing speed is also related to measures of short-term memory.

Another factor that is responsible in heightening the effects of deficits in working memory, slow processing speed is not new to learning, but has acquired higher significance. It is the prevalence of distractions and shorter attention spans. They continue to be a growing challenge for students.

*Explanation*: Children, today, are exposed to a lot of technology — iPhones, iPads, and, therefore, higher degree of vicarious socialization. It is changing the architecture and the behavioral manifestation of the brain, so much so that applying themselves to a task that requires concentration is becoming very difficult. It is part of the reason, that math-tutoring programs have become increasingly popular in the last decade.

The key to math success includes teaching a critical combination of both understanding concepts along with improving working memory, processing speed, and flexibility of thought, on one hand, and drills and repetition exercises, on the other, to increase confidence and aptitude.

Children with arithmetic difficulties have problems specifically in automating basic arithmetic facts that may stem from a general speed-of-processing deficit. Strategies such as counting and lack of decomposition/ recomposition may further slow down the processing speed. Moreover, the demands of understanding and meanings to be derived from mathematics terms, symbols, equations, and inequalities place heavier demands on the working memory, processing speed, and flexibility of thought than reading process and comprehension. That is one of the reasons, processing speed plays a very big role in mathematics achievement.

Achieving mathematics understanding of concepts and procedures is more demanding than reading comprehension. In the case of reading, some meaning and a level of comprehension can be derived just from context, but understanding of a mathematical idea cannot be derived just from context.

Children’s information processing, particularly its speed is a driving mechanism in cognitive development that supports gains in executive processes—working memory, inhibitory control, organization, flexible thinking, and associated cognitive abilities. Accordingly, individual differences in early executive task performance and their relation to mathematics may reflect, at least in part, underlying variation in children’s processing speed and therefore achievement. Processing Speed, working Memory, and fluid Intelligence (Fry and Hale 1996) are highly correlated with each other and are essential elements for learning. Deficiency in anyone of these skills may have impact on learning. Slow processing speed may have impact on many of the mathematics processes. Slow processing speed is not a learning or attention issue on its own.

Slow processing speed impacts learning mathematics at all stages. It can make it harder for young children to master the basics of mathematics language, writing numbers and symbols, understanding number and related numbersense. For example, let us consider, finding and learning the multiplication fact: 8×7*= ?*

In the absence of automatic recall, a student plans to construct it using a strategy based on decomposition/recomposition, The student decides to break the problem as: 8 ×7 = 8 ×5 + 8 ×2 (because he knows the sight facts of number 7 and the distributive property of multiplication over addition) = 40 + 16 (he knows the tables of 5 and 2) = 40 + 10 + 6 (he knows his teen’s numbers) = 50 + 6 = 56 (he knows his place value of two-digit numbers).

As one can see, by task analysis, one can determine the component primary concepts: recognition of related sight facts (including the sight facts of 10), decomposition/recomposition of numbers, decomposition/ recomposition of teens’ numbers, place value of numbers, and tables of 5 and 2. Mastery of these component skills makes finding and mastering multiplication facts/tables is easier. Slow processing, in such a situation does not debilitate student progress in learning arithmetic facts. The same process works in intervention/remediation situations with older students. Using the same efficient strategies, they can acquire the ability to master arithmetic facts quickly and accurately.

There is research and training efforts to improve auditory and visual processing to mitigate the deficits as related to reading and language development. However, none exist to minimize the impact of processing deficits for improving mathematics learning. We believe that first we need to differentiate the poor performance in mathematics due to poor processing or poor strategies. We should differentiate between appearance of poor processing and true deficit in processing.

We have been working with children with and without learning disabilities and some with processing deficits and the choice of instructional materials, questioning processes, and training in visualization. We have observed that improvement in strategies helps children improve their processing speed, working memory, and flexibility of thinking. These, in turn, also have impact on executive function that is essential for learning mathematics.

Training in executive function and processing does have impact in all areas of learning. For example, studies show that infant attention skills are significantly related to preschool executive function at age three and even later. Higher attention span in infancy may serve as an early marker of later executive function, processing speed, and attention to learning.

Two very powerful tools, we have used to improve working memory and processing speed are: Efficient and effective ** Concrete/pictorial**models, and

**for streamlining tasks. Concrete models should be selected with following characteristics in mind: color, shape, size, pattern, and generalizability. Similarly, understanding the trajectory of the development of a concept and the related task analysis must develop scripts for task implementation.**

*developing scripts***Auditory Processing Speed**

Auditory process is making sense of information received from the auditory channels. When there is delay or deficit in processing this information it is called Auditory Processing Disorder (APD). It is also called Central Auditory Processing Disorder (CAPD), and Specific Learning Disability/ Disorder with impairment in listening. It is quite common in **Dyslexia and dyscalculia**.

Increasing evidence suggests that some children with developmental dyslexia and/or dyscalculia exhibit a deficit not only at the segmental level of phonological processing but also, by extension, at the supra-segmental level. Thus these children when confronted with mathematics language in the context of conceptual development and word problems exhibit some of the difficulties related to processing speed.

The APD brain requires 2 to 5 times longer registering a speech sound. This is particularly so when the context and the complex and specialized vocabulary creates even more problems. The normal rate of speech is too fast for the APD brain to perceive and process all of the information heard. The result is a person who appears to have a long delay between what they hear and their response to it. When they do respond, the response may be inappropriate or may clearly indicate that they did not comprehend the information heard. They cannot accurately repeat auditory information. Parts are missing. It is not effective to give them spoken instructions because they require lots of repetition and redirection. Nor, it is not enough to by-pass the auditory channel altogether. It should be multi-sensory. For example, in a word-problem situation, ask the students to read the problem out loud and supplement it by asking a great deal of scaffolded probing questions. Create a script and ask them to repeat the script as apply the instructions from the script.

In reading fluently, processing speed plays a keyrole. The goal of improving reading rate and fluency is to positively impact reading comprehension; however, it is unclear how fast students with learning disabilities (LD) need to read to reap this benefit. There is a point of diminishing return for students who are dysfluent readers. In other words, it is important to determine where the linear relation between reading rate and comprehension breaks down for LD students: the rate at which getting faster no longer contributes clearly to reading comprehension improvement. For dysfluent readers, improving reading rate improves comprehension only in the bands between 35 and 75 words correct per minute in second grade and between 40 and 90 words correct in fourth grade. Reading at faster rates reveals no clear advantage for reading comprehension, beyond that point.

In mathematics the role of processing speed is more complex; more demanding than in acquiring language and reading skills. For example, demands made on fluency while reading a word problem is sufficient about what one reads in the band between 50 and 90 words. However, because of specialized vocabulary and syntax, skills needed to understand the content of word problems, the level of reading comprehension, is much higher. Similarly, when it comes to constructing or producing a fact or a formula, and executing a standard procedure, we need speed, therefore, a higher level of processing speed is needed.

On the other hand, when we are solving a problem we are not necessarily looking for speed, there we need to discern and discover patterns and relationships. For that, we need strong working memory and fluidity of thought. Mathematics learning (integration of language, concepts, and procedures), therefore, is a partnership between processing speed and working memory. These two together give us fluid intelligence or flexibility of thought.

Mastering arithmetic facts, formulas, and sequential steps involved in executing mathematics procedures calls for high level of processing speed, e.g., What are the factors of 48? It calls for knowing the multiplication facts that result in 48. It calls for automatization of multiplication facts. In the absence of this facility, one should know certain pre-skills: (a) divisibility tests, (b) short division, and (c) place value.

The divisibility tests of 2 and 3 (the first prime number is 2 and it divides 48 as it is even number; 3 divides 48 as the sum of its digits is 12; 4 divides 48 as 40 is made up of 40 and 8, and both are divisible by 4; since 2 and 3 divide 48, therefore, 6 divides 48. And, since, the only number between 6 and 8 is 7, and that does not divide 48. We have found all the factors of 48.

Knowing the short division process facilitates the process of dividing by 2, 3, 4, and 6. In addition to this, one should know the organization of these numbers on the number line as follows.

**48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.**

Writing the factors in their appropriate place on the number line shows that there are no factors of 48 between 24 and 48; between 16 and 24, between 8 and 12, and the only number between 6 and 8 is 7 and that does not divide 48. It also is very convenient when we are looking for the greatest common factors (GCF) of two numbers. For example, to find the greatest common factor of 48 and 40, we have:

Once, can easily notice the common factors {1, 2, 4,and 8} and the greatest common factor (8).

Another effective strategy for finding GCF and least common multiple (LCM) of two numbers is the use of prime factorization.

**GCF (48 and 40)** = 2×2×2 = 8 (the product of the prime factors in the first column)

**LCM** (**48 and 40**)= 2×2×2×6×5 = 240. (the product of factors in the left-most column and the last row. As one can see, LCM has both 48 and 40 as its factors, it is a multiple of both, and it is the least such multiple.

These efficient strategies and visual representation aid in *discerning patterns,** making connections, improving processing speed *and

*. Therefore, development of*

**the working memory****. With enough practice of efficient strategies and developed mathematical way of thinking, one can make up for slow processing speed and working memory deficits and in the process improve both of them.**

*mathematical way of thinking*In the absence of appropriate processing speed one may not arrive at a fact easily and efficiently. A student may take two hours to do math homework that takes others only 20 minutes. This means that the studentoften does poorly on tests even though she may know the material. The student knows the procedures, but is not able to follow the multi-step directions, suchas executing the long-division algorithm, solving a system of simultaneous linear equations, adding two fractions with different denominators, and applying the order of operations in a complex numerical or algebraic expression. This is especially so when there is not much time to get the task done?

While there are many possible reasons for these struggles, slow processing speed may be a factor. Having slow processing speed has nothing to do with how smart kids are—just how fast they can take in and use information. It may take kids who struggle with processing speed a lot longer than other kids to perform tasks, both school-related and in daily life.

Lack of ability, fluency in decomposing and recomposing numbers, and poor processing speed interfere with learning number relationships.

Comprehending mathematics concept involves integrating three actions—recognizing grapheme (the orthographic mathematics symbols), the idea it represents (concept), and the meaning and action associated with it (knowing the associated skill or procedure). For example, recognizing a cluster of objects (*quantity*—5 objects in a cluster), giving it a number name (*numberness*—five), and writing its orthographic image (*5*) results in number concept. Again, practice with instructional materials such as: Visual Cluster cards, Dominos, dice, Cuisenaire rods, TenFrames, etc. facilitate this integrative process and increase processing speed.

Processing, integrating, and communicating these three items (whether in written, oral, or action forms) takes time, thus affecting learning. Many teachers, particularly interventionists/special educators, with good intensions of helping students and aware of their deficits, try to short change it by giving “cooked” procedures with short cuts. Many interventionists seek simplistic instructional/interventional programs that lack this integrative process. In the math education and special education circles this has created the “Math wars”—whether it is important to give students just simplistic mathematics procedures or they should have the conceptual understanding and leave the procedures to technology-assisted methods.

This clash in opinion about mathematics instruction is most often expressed as a divide between “*back-to-basics*” and “*reform *or*discovery math*.” Advocates of back-to-basics tend to think of math as a subject that is fact-focused, with emphasis on memorization of these facts, fluency and speed, teacher-centered and test-heavy. These are precisely opposite descriptions to learning concepts and procedures with understanding and applications. That approach is concept-driven, learner-centered, problem-rich, involves explorations and experimentation, and emphasizes learning generalizable strategies. Just like we settled a similar debate in reading education by recognizing that decoding/encoding (fluency) and comprehension are both necessary elements in a “*balanced*” approach to teaching reading.

Fortunately, the demands of a technological knowledge-based society call for the two sides to be closer to each other. There are signs of that. On one side, few educators would now say that learning math is just a matter of memorizing facts and rules, similarly, most educators now recognize that mastery of basic facts and rules facilitates higher-order thinking and seeing patterns and connections. On this count, for the most part, both sides have compatible intentions.

However, challenges arise when discussions shift from what to teach to how to teach children, with or without learning disabilities. This is where the debate between “direct instruction” and “inquiry learning” continues. Educators are often asked to choose between teacher-driven explanations of isolated topics and learner-driven explorations of whole concepts in rich settings.

Each side seems to have a compelling argument for its view. Proponents of direct instruction assert that mathematics is well defined and unambiguous, and so it should be delivered efficiently and with fidelity. They argue that it is ridiculous to expect high school students to “discover” concepts that eluded all but the best-prepared minds until quite recently. Reform advocates counter by noting that learning is not about “*acquiring*” objects of knowledge. Understanding cannot pass from teachers’ instructions to learners’ minds. Rather, learning is about building understandings from personal experiences. For instance, when asked for a quick definition of “number,” most people respond in terms of counting. That interpretation is limiting, and learners who haven’t incorporated additional meanings (for example, size, distance, location) are disadvantaged as early as upper elementary. Or, when they see multiplication only as “groups of,” “repeated subtraction,” or “an array.” Each of one of these linguistic and conceptual models is limited in scope. The “repeated subtraction” model does not work if both factors are fractions. The “groups of” model does not work when dealing with multiplication of algebraic numbers and expressions. The array model does not work when the factors are not discrete quantities. Whereas, when students by the end of third grade are not shown that all of these models can be generalized into “the area of a rectangle” model, they have difficulty conceptualizing multiplication of fractions, decimals, and algebraic expressions, etc.

The raw materials for enriched understandings are found in personal action and interpretation of experiences and when the experiences are diverse, multi-faceted, and effective. It is the role of the teacher to provide opportunities for these rich interactions with language, concepts, materials, and human resources. That is why reform advocates contend that presenting mathematics as a purified or standardized form of knowledge risks making it meaningless while suppressing curiosity and motivation.

Mathematical knowledge is not an “*object*” or collection of facts that a teacher can simply hand off to students, but neither is it a web of concepts that are self-evident or inherently embedded in experience; mathematics is a complex, evolving combination of what we might call “principles” and “logics.” Mathematics knowledge, mathematics understnading, and mathematical way of thinking has to be constructed and practiced. Teachers, effective teaching, and efficient teaching resources are the bridging processes between experiences and the outcomes—mathematics way of thinking, mathematics content, and process outcomes in attitudes and interest. There are four prspectives and approaches to learning mathematics, where we combine basic principles into more abstract and more powerful principles using combinations of these perspectives:

- Through concrete experimentation, working on projects, and problem solving, we observe, develop, and relationships between ideas. Concrete experimentation helps build visaulization–a key ingredient in strengthening working memory and processing.
- By discerning and extending patterns and regularities, we make conjectures. Conjectures invite us to establish them in to elements of mathematical way of thinking. They build intuition and store of mathematics content knowledge.
- By reasoning analogically (such as noticing that different physical processes, such as movement in a straight line, elapsed time and growth can all be interpreted in terms of addition), we connect ideas and understanding. Using analogous thinking from multiple settings, we see realtionships and willingness to explore. Processes, such as analogies, metaphores, and similies build understanding and comprehension.
- Using various logic—deductive and inductive reasoning, through sequenced chains of argument (such as, if
*a < b*, and*b< c*, then*a< c*; one is less than two and two is less than three; so one must be less than three).

An effective teacher will offer exercises and activities that channel learners’ attentions to relevant principles and encourage appropriate use of different logics by systematically juxtaposing clusters of such experiences.

We need to use most effective and appropriate approaches to teaching mathematics to all children, with and without learning disabilities. An analogy might be drawn from research into how people move from novice to mastery stage; what actions and processes are involved. For example, how people learn to play chess at competitive levels. Intuition might suggest that playing game after game is key, but cognitive scientists have found players advance more quickly by doing exercises that offer incremental challenges but that do not overwhelm working memory. The tasks that are within their capacity and yet moderately challenging. Tasks that build new insights, skills, and stamina. Both automaticity and strategy can be effectively developed through, for instance, playing a mini-game with just a few pieces, analyzing a single position in-depth and studying sequences of moves by master players. Such conclusions aren’t specific to chess. Research on expert performance across domains consistently reveals that people learn best when they are engaged in a way that doesn’t exceed the limits of their working memories and are exposed to moderately challenging experiences. The task must fall in the Zone of Proximal Development (ZPD). This represents a very specific zone of difficulty, which looks different for every student.

Specifically, the ZPD is the area between comfort and frustration — a student has not completely mastered the material yet, nor are they frustrated with its difficulty. The teacher with her own skills, insights, and adaptive technology helps each student accesses this key area, maximizing learning efficiency and continuing that process. Of course, choice of instructional materials is the key here. Even during the time, the studnet is using the adaptive technology, the teacher must ask a great deal of questions to make sure the student is understanding the language and concepts behind the tasks they are involved with; pushing buttons, midlessly is not the role of adaptive technology.

Such research highlights that each side of the math wars is correct about something. You cannot have one without the other. Math is not just a collection of facts just to be memorized; some of it has to be memorized by doing it, using it. It is a complex system. Each part of it is related to other parts of the system. Skills and concepts in algebra and geometry give rise to coordinate geometry, trigonometry, calculus, and probability and statistics. Mathematics understanding is constructed. Mathematics is the study of patterns in quantity and spatial relationships and their integrations. Mathematics needs to be presented in a curriculum that is less cluttered, yet connected and more clear, but meaningful. We need to emphasize “non-negotiable concepts and skills” at each grade and age level for all children.^{[2]}

Traditionalists are right to argue for carefully structured learning situations and ample supervised and independent practice. And reformists are right to insist that powerful learning is more likely when practice is embedded in rich and personally meaningful situations and efficient strategies. Together, these insights point to the need for fine-grained analyses of mathematical concepts, skilled design of learning situations to help learners notice and make connections. It calls for constant awareness of students’ evolving understanding, skills, attitudes, and flexibility to adapt teaching approaches depending on the content and how students are progressing. At the same time, we should be cognizant of the learning differences and their nature and how that impact learning.

Improved model of teaching for students having problems in learning mathematics not just with dyscalculia and/or dyslexia, but also children with processing speed can be built on math learning that focuses on engagement, when teachers check in with students regularly to ensure they understand and provide appropriate support and practice. In this method the teacher presents only one concept at a time, only enough that a student’s working memory can handle and ensure every student understands before they move on. And, then make sure appropriate connections with other concepts, procedures, and processes are made.

This involves: (a) selecting a key concept to learn; (b) task analysis for unwinding the fine details of the concept (taking it apart with students and then putting it back together again); (c) engaging students in effective concrete models and strategies for learning the concept; (d) helping students observe patterns and make conjectures; (e) plan and execute supervised and supported practice; (f) provide success at each junction; (f) make conncetions with other cocnepts and procedures–both vertically and horizontally, for example, how multiplication manifests itself at different grade levels and how each model of multiplicaiton is related with different modesl of division; and, (g) check in with students or “notice” whether they understand the relationship between the strategies and their success. This process helps students become better learners and acquire meaningful and appropriate mathematics content.

^{[1]} See several posts on this blog about working memory and mathematics learning.

^{[2]} See posts on *Non-negotiable Mathematics Skills*at Different Grade levels in this Blog.