For many years, researchers have been exploring the mystery why certain people struggle greatly with mathematics even though they are motivated to learn it and have access to learning resources. Many people, because of these struggles, have given up on learning mathematics—even the simplest of concepts, procedures and skills. The research by cognitive scientists has focused on whether mathematics learning disability (MLD) could be due to a brain region or function that has developed differently in people who struggle with mathematics.
On the other hand, mathematics educators have focused on exploring different ways of teaching mathematics to students with and without mathematics learning disabilities. They have wondered whether the learning resources and experiences provided to them, in the classroom or in intervention settings, are appropriate to their needs.
As a result, if we want to reach these students, we need to examine the interaction of cognitive skills, instructional strategies, and instructional materials in designing diagnosis, instruction (initial and remedial), and assessments. In other words, we need to examine what instruction suites the needs of a student with particular deficits and strengths. To answer some of the questions related to this, research in the field of dyslexia might be instructive.
A. Processing and Mathematics Learning
Research in the field of dyslexia has shown that typically developing children and children with dyslexia differ in all temporal processing (TP) skills. It is also further shown that cross-modal TP also contributes independently to character recognition (i.e., mathematics symbols and characters in ideographic languages such as Chinese, Hindi, Japanese, etc.) in children if the significant effects of phonological awareness, orthographic knowledge, and rapid automatized naming (RAN) are also considered. In multiple situations, it is shown that visual and cross-modal TP skills contribute to mathematics learning in addition to content related skills such as number concept and numbersense. In other words, the poor TP skills have higher impact when numerical skills are poor to start with. That means poor TP skills exacerbate the poor number skills and the difficulty in learning them. For example, visual and cross-modal TP skills have direct effects on character and symbol recognition and reading in the group with deficits such as dyslexia and/or dyscalculia, but, somehow they do not have impact on those who have better content skills in these areas.
Research findings also suggest that TP is more important for reading in children with dyslexia or dyscalculia than in typically developing children, and that TP plays an important role in dyslexia and dyscalculia. One of the ways TP plays role in mathematics learning is through its affect on working memory. In several postings, on working memory, on this blog, I have discussed the various roles working memory plays in learning mathematics and its role when there is deficit in it.
B. Types of Working memory and their role in mathematics disabilities
Developmental learning disabilities such as dyslexia and dyscalculia have a high rate of co-occurrence in early elementary populations, suggesting that they share underlying cognitive and neurophysiological mechanisms and processes. Multiple cognitive and neuropsychological skills, such as executive function and processing skills have been implicated in the incidence of dyslexia and dyscalculia. For example, dyslexia and other developmental disorders with a strong heritable component have been associated with reduced sensitivity to coherent motion stimuli—an important component of visuo-spatial processing and an index of visual temporal processing. It affects focus, inhibition control, visual processing skills, and spatial orientation/space organization—critical components for mathematics learning. Children with mathematics skills in the lowest 10% in their cohort are less sensitive than age-matched controls to coherent motion, but research shows that they have statistically equivalent thresholds to controls on a coherent form control measure. Research has also shown that deficits in sensitivity to visual motion are evident in children who have poor mathematics skills compared to other children of the same age.
Children with mathematics difficulties tend to present similar patterns of visual processing deficit as other developmental disorders suggesting that reduced sensitivity to temporally defined stimuli such as coherent motion represents a common processing deficit apparent across a range of commonly co-occurring developmental disorders, including mathematics disabibities.
However, the term mathematics difficulties and their behavioral skills markers across studies, across curricula, and across interventions are quite diverse. This is so because mathematics is so diverse in nature, content, concepts, and thinking that comparing mathematics difficulties with other learning disabilities is artificial. Mathematics learning ranges from mastery of numerical reasoning—number concept, numbersense, and numeracy (study of patterns in number—arithmetic), spatial reasoning—spatial sense and various geometrical objects and thinking (study of patterns in shapes and their relationships in diverse situations), generalization of arithmetic and sense of variability(study of patterns in variability—algebra thinking), integration of algebraic and geometrical thinking and modeling(coordinate geometry, trigonometry—for every geometrical figure there is an equation or system of equations and for every equation there is a geometrical representation), proportional reasoning(study of patterns in the rate of change—calculus), etc. All of these call for a complex of cognitive, neuropsychological, spatial, logical and linguistic skills at variety of levels. Any deficit in one or more of these skills could be the cause of mathematics difficulties in a various disciplines/fields of mathematics.
Although developmental and classification models in different fields of mathematics have been developed (for example, Geary and Hoard, 2005; Desoete, 2007; von Aster and Shalev, 2007), to our knowledge, no single framework or model can be used for a comprehensive and fine interpretation of students’ mathematical difficulties across different disciplines of mathematics. This is true not only for scientific purposes—diagnosis, behavioral markers, research, prevention, but also for informing mathematics and special educators, and for designing appropriate instruction—appropriate to the content, appropriate to the learner, and appropriate to the condition of his/her mathematics disability.
As an educator1, I believe that reaching a model that focuses on a definite and important aspect of mathematics that combines, elaborates and enhances on existing hypotheses on MLD, based on known cognitive processes and mechanisms, could be used to provide a mathematical profile of a student. Since, not all students reach calculus, just like not all students become professional writers, in such a situation, it is important to focus on the fundamental components of mathematics, such as: numeracy as a focus of study to define the parameters of MLD. Therefore, in this discussion, we will limit MLD to disabilities in learning numeracy: number concept, numbersense, place value, numerical procedures.
Most commonly, MLD has been linked to problems with working memory, i.e. the brain’s ability to hold and manipulate information over a short period of time. Students with poor working memory cannot hold and manipulate information (oral, visual, tactile, or combination of them) in the mind’s eye (working memory space).
Working memory was initially thought to be domain-general, meaning that its importance is the same regardless of whether the content is related to mathematics, reading, or some other subject. However, some studies have shown that working memory may be domain specific—different subtypes of working memory operate for different types of tasks. Research has also shown that there are more than just one cause for MLD by showing that the MLD participants could be divided into at least two groups in which each had a different type of working memory deficit.
One group with MLD shows poor reading skills and scores poorly on verbal working memory tasks (tasks involving holding in memory and manipulating verbal information). The other group has purer deficits in mathematics and scores poorly on visuo-spatial working memory tasks (tasks involving holding and manipulating spatial and visual information). This seems to indicate that there could be at least two different causes of MLD, both of which are subtypes of working memory.
The opposite of this phenomenon is also true: There is a continuum of mathematics learning personalities ranging from quantitative mathematics learning personalities(with strength in sequential processing, procedures, etc.) toqualitative mathematics learning personality(with strength in visuo-spatial processing and pattern recognition, etc.). As a result, quantitative mathematics learning personality students perform better in arithmetic procedures and algebra and have difficulty in conceptual aspects of mathematics, geometry and problem solving (e.g., properties of numbers, word problems, proofs in geometry and algebra, etc.). Qualitative mathematics learning personality students, on the other hand, do better on conceptual aspects of mathematics, geometry, and problem solving and find it difficult to execute multi-step procedures in arithmetic (e.g., long-division, solving systems of equations, etc.). (Sharma, 2010)
Mathematics requires an extensive network of brain activities and that a problem with any one of the two types of memories could lead to MLD. Since verbal processing seems to be required for the brain to conduct mathematics, underdeveloped verbal processing could lead to MLD. In the same way, the brain also to use visuo-spatial processing for mathematics, so a deficit in this area could also lead to MLD. Let us look at the group that performs poorly on both verbal and visuo-spatial working memory. In such a situation, we can see problems in earlier mathematics (numeracy) as well as early reading in lower grades.
Whilst the symptoms of MLD can look similar, the problem may arise from different sources. Since mathematics is such a rigorous (requiring a lot of mental resources) and exacting (requiring one perfect answer in many cases) discipline, it could reveal brain deficits that would not show up with other disciplines. This could be why participants who score poorly on visuo-spatial working memory tasks can do well in other subjects besides mathematics. The fact that MLD can arise because of multiple distinct deficits may be of great importance both to mathematics teachers and researchers investigating causes of MLD in children.
It is, therefore, natural to raise the question: Is it possible to use instructional materials and mathematics specific pedagogy that can enhance not only the mathematics content, but also improve the working memory and processing functions, thereby improving students’ learnability? To find the answer, let us look at a very specific concept and various ways children arrive at the answer as an outcome of instruction and their understanding of the nature of the concept, use of instructional material, language, and expectations from them. Children’s responses to this problem show, in many cases, the weaknesses not because of their own working memory, but imposed by and outcome of inefficient instruction, poor instructional materials, and lower expectations. Such difficulties can be removed with proper instruction, effective, efficient, and elegant instructional materials, rich language, and higher expectations.
For example, let us examine the approaches children are exposed to a range of instructional materials that are used in classroom and the strategies that are developed as a result of this instruction, from pre-Kindergarten to third grade. We want to examine, whether some of them have the potential to improve students’ learning not only the mathematics content and processes, but also the learnability (ability to learn—cognitive skills, for example improving working memory, visual processing, generalizing, etc.). To find the answer to this question, let us consider a simple addition problem and the related strategies.
C. Problem:Find the sum 9 + 7.
To find this sum, there are many options, approaches and strategies, a child may use to arrive at the answer. However, an average or below average child has only those choices that are possible by the instructional material used, strategies possible using those materials, the approach the teacher introduced, or helped children develop in the classroom. The following descriptions are of approaches, I have seen children use in different schools, different settings, and different countries. We want to examine them from the perspective of effectiveness, efficiency, and elegance.
For example: A child:
(a)Counts 9 items and, then 7 items from a collection of random objects and gets the answer. Our early childhood classrooms abound with these kinds of materials—beans, shells, pennies, gummy bears, etc.
(b) Counts 9 and 7 cubes (Unifix cubes, Centi-cubes, inter-locking cubes, wooden cubes, BaseTen cubes, etc.) from the cubes collection, in the classroom.
(c) Counts 7 fingers and then 9 fingers to find the answer.
(d) Decides that 9 is the bigger number, he counts 7 numbers after 9 and finds the answer.
(e) Counts 9 beads on one TenFrame and 7 on another TenFrame and then counts them all, sequentially from the first TenFrame and then the second TenFrame.
(f) Recognizes that 9 is the bigger number, he makes the number 7 configuration by touch points on the paper or on his body part and counts them after 9 to find the answer.
(g) Counts 7 objects (cubes, fingers, marks, etc.) after the number 9 and reaches 16, and declares the answer.
(h) Places 9 unit-rods and 7 unit-rods end-to-end, from the BaseTen set to make a train and then replace the 10 unit-rods by the 10-rod and then place the 10-rod and 6-unit rods end-to-end to make a train and the child counts 10-11-12-13-14-15-16 as he touches each unit-cube, and arrives at the answer.
(i) Locates 9 and 7 on two Ten frames (5 and 4 on one) and (5 and 2 on the second one) and then adds 5 and 5 and 4 and 2, and declares the answer as 16;
(j) Places a 9-rod and 7-rod, from the Cuisenaire rods collection, end-to-end to make a train and then experiments which rod along with the 10-rod is equal to the 9-rod and 7-rod train. The child finds that the train made with the10-rod and 6-rod placed end-to-end is of the same length as the 9-rod plus 7-rod train. The child finds the answer.
(k) Places a 9-rod and 7-rod, from Cuisenaire collection, end-to-end to make a train and realizes that the train is longer than the 10-rod. He places the 10-rod parallel to the 9-rod + 7-rod train. He observes that the 10-rod is 9 + 1 and realizes that there is 6 left from the 7 rod. Now he adds 10 and 6 to get 16.
(l) Places a weight of 1-unit at the peg at 9 and another 1-unit weight at the peg at 7 on the left arm of the Invicta Balance and, then a weight of 1 unit at the peg at 10 and experiments with placing weights on other pegs in order to balance the arm. In experimenting in placing unit weights at the peg at 6, the arm balances. Since, the arm is balanced (is horizontal), the child realizes that 9 + 7 = 10 + 6 = 16.
(m) Makes 9 and 7 marks (⁄, ×, ✓, , etc.) with pencil-on-paper and counts them to find the answer.
(n) Locates a number line (on the wall, on the desk, or in the book) and counts numbers, first the 9 and then 7 more on the number line and finds the answer.
(o) Locates a number line (on the wall, on the desk, or in the book) and first locates number 7 on the number-line, and then counts 9 more numbers after 7 and ends on 16 and declares the answer as 16;
(p) Locates a number line (on the wall, on the desk, or in the book), then locates the number 9 and then counts 7 numbers after 9, and ends at 16 and declares the answer;
(q) Locates 9 beads on a TenFrame, and then realizes he can count 1 more on the first TenFrame and counts 6 on the second Ten frame. He says: 9 + 1 is 10 and 6 more is 16.
(r) Selects a 9-card and 7-card from the Visual Cluster cards collection and visually move the one pip from the 7-card to 9-card to make it a 10-card and then visually realizes that the 7-card has become a 6-card, so the sum 9 + 7 becomes10 + 6. Declares the answer as 16.
(s) Locates 9 and 7 on two Paper form of the Ten frames (5 and 4 on one and 5 and 2 on the second one) and then adds 5 and 5 and 4 and 2, and declares the answer as 16;
(t) Draws an Empty Number Line (ENL) and locates 9 on the left of side of the line and then takes a jump of 1 from 9 and lands at 10 and then takes a jump of 6 from 10 and then lands at 16. Declares the answer as 16.
(u) Places a 9-rod and 7-rod, from Cuisenaire collection, end-to-end to make a train and realizes that the train is longer than the 10-rod. Places the 10-rod parallel to the 9-rod + 7-rod. He observes that the 10-rod is 9 + 1 and realizes that there is 6 left from the 7 rod. Then, combines 10 and 6 to get the answer. At this point, he transfers the learning from this model to visualization and then abstract form: 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16.
(v) Pictures, in his mind, a 9-card and 7-card from the Visual Cluster cards collection and visually moves the one pip from the 7-card to 9-card to make it a 10-card and then visually realizes that the 7-card has become a 6-card, so the sum 9 + 7 becomes10 + 6. Declares the answer as 16.
(w) Pictures, in his mind, a 9-rod and 7-rod, from Cuisenaire collection, thinks of them together, visually takes 1 from the 7-rod and gives it to 9 and then realizes he has 10 + 6. Then, combines 10 and 6 to get the answer. At this point, he has transferred the learning from the concrete model to visualization of 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16. Teacher’s questions and scaffolding makes it possible.
(x) Draws and Empty Number Line in his mind and locates 9 on the line and then starts with 9 and takes an imaginary jump of 1 and lands at 10. He knows that the 1 came from 7 so he takes another imaginary jump of 6 from 10 and lands at 16. Declares the answer is 16. With few practices, he generalizes the strategy into: 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16.
(y) Begins with the problem 9 + 7 = ?, Using the decomposition/ recomposition and commutative property of addition, and decomposition/ recomposition strategies converts the problem into any of the following convenient forms and finds the sum:
(i) 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16;
(ii) 9 + 7 = 6 + 3 + 7 = 6 + 10 = 16;
(iii) 9 + 7 = 2 + 7 + 7 = 2 + 14 = 16;
(iv) 9 + 7 = 9 + 9 – 2 = 18 – 2 = 16;
(v) 9 + 7 = 7 + 9 = 7 + 10 – 1 = 17 – 1 = 16; or,
(vi) 9 + 7 = 5 + 4 + 2 + 5 = 5 + 6 + 5 = 5 + 5 + 6 = 16.
D. Analysis of Strategies Used by Children:
1. Counting based approaches
The approaches from (a)through (i) involve the instructional materials (concrete in nature) that promote the strategy of counting objects. This results in a child conceptualizing addition as the result of counting. The only strategy these approaches develop is counting. The counting process does not develop relationships and patterns between numbers. It does not leave any residue of number relationships on the memory. The activity, thus, does not build up any working memory. Moreover, consistent use of these materials makes the child dependent on concrete materials and, then remains functioning at the concrete level of knowing.
These children, generally, have great deal of difficulty in automatizing addition facts, if at all they reach that level. Many of them remain at the concrete level, even at the high school level. And, when these children are referred to intervention, special education classes, or one-to-one support, they may get the same approach (i.e., addition is counting up). They do not make much progress. Their achievement in mathematics continues to be deficient.
The counting strategy helps them derive the sum, but the mathematics content such derived is very limited, mathematics strategies are inefficient, they cannot be generalized, and the resultant cognitive learning is very limited. These materials and the counting strategies derived from this process are ineffective and inefficient and the resultant arithmetic methods of teaching and learning are limited. Since, there is no improvement in working memory space, the inadequacy in learning continues.
2. Length-based approaches
The approaches from (j)and (k) are also concrete, but the strategy derived is not counting based. Here the number magnitude is being associated with length of the rods.Children begin to see numbers as groups rather than one-to-one. Just like a word is a group of letter and has its own entity, similarly, when we consider number as the representation of a particular length and combining two or more numbers into a new numbers, they get the same feel as making a word from different letters.
This transition from one-to-one counting to forming a group is a ‘cognitive jump’for a child. As we will see, the rods model has the potential to be converted into effective and efficient strategies, not only for addition, but also for other arithmetic facts. Because of the color and size (lengths) of the Cuisenaire rods, one can visualize the numbers (by remembering the rods) and can see the equation in the mind’s eye and, therefore, remember the equation easily. This enhances working memory, visual memory, visual-perception, and pattern recognition. They can transit from addition to subtraction easily.
3. Weight-based approach
The approach in (l)is based on the weight aspect of number. It is not a very transparent strategy. Answer is found easily, but it does not result in a strategy. However, it is the best approach to develop the concept of equality. Later, we can use this instructional material to derive the properties of equality, and deriving the procedure for solving equations. The instructional materials that are very efficient in developing the concept of equality are: Visual cluster cards, Cuisenaire rods, and the Invicta balance.
4. Approaches based on pictorial representation
The approaches from (m) through (q) are also based on counting of objects. The strategy used is still seen as conceptualizing addition as “counting up.”Children are using pictorial representation of objects (pictorial/ representational level of knowing), thereby, are functioning at a higher level than the concrete level. However, conceptually, it is still counting. Once again, generally, they do not automatize addition facts and their achievement in mathematics continues to be deficient. This strategy helps them derive the sum, but neither leaves any residue in the memory nor it develops and increases the working memory space. These materials and strategy derived from this process are still inefficient. This does not help in improving a child’s cognition.
5. Approaches based on decomposition/recomposition
The approaches from (r) through (w) whether derived using Ten Frames, Visual Cluster cards, Cuisenaire rods, or Empty Number Line are helping children to reach abstract level knowingeasier and quicker. These instructional materials are helpful in developing both kinds of working memory skills: verbal working memory and visuo-spatial working memory. The strategies are effective, efficient, and elegant. These children will be able to master arithmetic facts easily. These strategies help children improve their working memory, cognition and ability to generalize their learning. These strategies become more effective when they can see the relationship across instructional materials, as they help integrating the two kinds of working memories.
Concrete models, such as: TenFrames, Visual Cluster cards, Cuisenaire rods, with the help of (a) the teacher’s scaffolded questions; (b) sight facts, (c) decomposition/recompostion; (d) making ten; and (e) knowing teens’ numbers develop strategy and the script for finding the addition fact. For example, the following is the script and the strategy for (i) in approach (y). “I want to find: 9 + 7 = ?. In 9 + 7, 9 is the bigger number. So, I want to make it 10 by adding 1 to it ( 9 + 1 = 10). Number 1 comes from 7, (7 = 1 + 6) . Now I have 10 and 6. I know 10 + 6 is 16. So, 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16. Therefore, 9 + 7 = 16. And, by commutative property of addition, 9 + 7 = 7 + 9, so 7 + 9 = 16.” Similar scripts can be developed and used for other strategies from (ii) to (vi) in approaches in (y).
Teacher’s questions, consistent practice of visualization of these strategies, with constructive feedback to performance, by reminding children of the use of efficient strategies, and use of scripts that empahsize these strategies help children in masteringof addition facts. Mastery here refers to: accuracy, understanding, fluency, and applicability. When addition facts are mastered all other arithmetic facts are eaasy to master.
6. Behavioral markers of mastery (CPVA)
The approaches in section (x) transcend counting of individual numbers and demonstrate the integration of the key prerequisite skills necessary for mastery of addition facts: (i) understanding and using decomposition/ recomposition of number,(ii) mastery of 45 sight facts, (iii) making ten, (iv) making teens’ numbers, (v) visualization (working memory space), and (vi) the concept of equality. In the hands of an effective teachers, the instructional materials: Ten Frames, Visual Cluster cards, Cuisenaire rods, and Empty Number Line (ENL), in this order, when assisted by her appropriate questions and proper language become ideal materials for helping children to transcend from concrete (C) to pictorial (P) to visualization (V) and then abstract (A).
The formation and use of mathematics scripts for arriving at addition facts provides a student practice in oral temporal processing, therefore, their oral working memory is improved. On the other hand, the use of concrete materials such as: TenFrames, Visual Cluster cards, and Cuisenaire rods because of their patterns, color, shape, size, and organization help students to visualize the mathematical actions. This, in turn, improves visuo-spatial processing and working memory. These materials, therefore, not only improve students’ mathematics content mastery, but also working memory and other executive function components, and learning potential. When they feel successful and realize that they are able to find the answers without counting, they develop positive feelings toward mathematics and increased engagment.
With consistent practice, children master these facts (e.g., they have understanding of the concept/strategy, can derive fact accurately, can visualize the relationship/script, automatize the fact, and apply to another fact or problem solving).
All of the major strategies of deriving addition facts (finding the sums up to 20): are based on the above prerequisite skills:
(i) M + N = N + M (commutative property of addition operation), (ii) 9 + N (number), (ii) N + N (doubles), (iii) N + N + 1 (doubles plus 1), (iv) N + N – 1 (doubles minus1), (v) (N – 1) + (N + 1) (numbers that are 2 apart), (vi) Remaining facts can be derived using these strategies. At the same time, these strategies develop and use the major elements of executive function: (i) working memory, (ii) inhibition control, (iii) organization, and most importantly, (iv) flexibility of thought.
Above strategies are aimed at developing the concept of addition, mastering addition facts, and at the same time, developing children’s learning ability to learn future concepts. In other words, the learning disability of a child is not a limiting condition forever. With proper methods, one can improve learning skills and manage the limitations, if they are not improved.