Dyscalculia refers to one's inability to effectively learn mathematical skills.
Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and may have problems learning number facts and procedures.
Even if they produce a correct answer or use a correct method, they may do so hesitantly and without confidence" (DfES, 2001). There is a high co-morbidity rate for children with developmental dyscalculia and dyslexia. Between 60% and 100% of dyslexics have difficulty with certain aspects of mathematics (Miles, 1993 & Joffe, 1990).
While there is not a single indicator of a disability in math, poorly-developed number concepts in kindergarten are strongly correlated with poor math performance in later grades (Jordan, Kaplan, Locuniak, & Ramineni, 2007; Mazzocco & Thompson, 2005). All children benefit from number-rich learning environments, and students with poor number concept can benefit greatly from early and intensive intervention. Instruction for number sense should concentrate on number skills, symbolic skills, and the ability to compare and estimate quantity (Mazzocco, 2011).
Just as fluency is an essential skill for proficiency in literacy, fluency is also foundational for proficiency in math (Powell et al., 2011). Number combinations can refer to basic addition, subtraction, multiplication, and division operations that undergird algebra, geometry, trigonometry, etc. Students who struggle to quickly and accurately do number combinations are severely handicapped when it comes to grasping later mathematical concepts. Four recent studies conducted with third graders with math difficulties yielded positive results with drills for math facts (through a variety of formats), conceptual instruction, and counting strategies, (Fuchs et al., 2010). It was important to not only increase the fluency for math facts, but to also provide students with strategies for solving number combinations when they could not retrieve the math fact (e.g., if a student incorrectly answered a question, he solved the problem with a strategy). The correct answer was always achieved to solidify the math fact in the student’s memory. The frequency of practice was viewed as an important component of mastery.
While there is no "cookbook" approach for teaching math, some basic principles have been suggested by the contributors of "Dyslexia and Mathematics-Second Edition" edited by Miles and Miles (2004).
- Request a comprehensive math assessment to determine what a student already knows and what he or she needs to learn. Several questions that should be considered when selecting a test include:
- How does the test look? For example, how closely spaced are the items?
- How complex is the language? Are you allowed to read the paper to the pupil?
- Are there diagrams to illustrate the problems?
- How many items to increase the grade-equivalence by one year (Most tests have about 4-5 items)?
- What are the details of the sample used for the standardization of the test? Did it include any dyslexic pupils?
- Does the test match your teaching program? Note: you may also request a curriculum-based math assessment.
- What diagnostic information can you extract from the test?
- For pupils who have attention-span problems, how long does the test take? Is it timed?
- Is there a parallel form for re-testing?
- What is the age range of the test (for monitoring a student longitudinally)?
- Obtain standardized information as well as a classroom assessment of math performance (comparing the student to his own peers). An error analysis should be conducted by asking the student, "How did you do that problem?" This can provide valuable information about where the breakdown is occurring.
- Establish a positive rapport between the teacher and student with dyscalculia and dyslexia. Since anxiety and fear are often present, the teacher must help to identify the specific breakdowns, learning styles, and strategies to provide successful experiences with math. A teacher must have confidence in her ability to help the student, which will yield confidence in the student.
- Point out the importance of the position of numbers in powers, fractions, equations, etc.
- Teach students about flexibility with regard to directionality. For example, in some operations, such as addition, subtraction, and multiplication, the student must begin with the right-hand column, whereas the student moves from left-to-right in long division. In an equation, on the other hand, the student needs to solve it in either direction, depending on what he needs to do.
- Introduce small, incremental steps for each new type of problem.
- Determine the student’s preferred thinking style to assess how he or she approaches a problem, solves a problem, and evaluates the outcome. The Test of Thinking Style in Mathematics (Chinn, 2003) can assess whether a student uses a more inchworm (bottom-up) approach or a grasshopper (top-down) approach to math. Good math students are able to move flexibly between these two approaches according to Kruteetskii (1976). Students with dyslexia may "get stuck" using one cognitive approach, which does not serve them well in learning the curriculum. Note: many curriculums teach more to one style than the other. The Inchworm/Grasshopper Thinking Styles can make learning more explicit for all students in the classroom.
- Model linking language to the actions (using the objects) to facilitate language growth, problem solving, memory, and self-monitoring. For example, use Cuisenaire rods to do a subtraction task: "I take one ten from the tens-position and exchange it for ten units. I put the ten units in the units-position and exchange it for ten units. I put the ten units in the units-position, and then I take away the six."
- Explicitly teach math terms. Associations and mnemonic strategies will foster greater memory.
- Instruct students in the meanings of symbols such as: / < > = + % ( ) and *. Make a reference chart with the definitions of each symbol.
- Alert the student to the myriad mathematical synonyms, such as the following terms for addition: "altogether," "sum," "total," "plus," "add," and "and."
Teaching to your student's preferred learning style
- Use multi-sensory methods of instruction such as: Cuisenaire rods, Dienes blocks, and multimedia calculators.
- Repeat verbal-kinesthetic procedures until the manipulatives can be faded out entirely.
- Make a script of the procedure and kit of materials that travels home with the student to increase consistency.
- Teach for mastery before moving on to new information. Students with dyscalculia need to over-learn each skill and this necessitates working at their own pace.
- Improve memorization of times tables using "Gypsy Math" where the student can use his or her fingers to quickly remember how to multiply by 6, 7, 8, 9, and 10 (this requires no immediate knowledge of a times table higher than four). A complete description and mathematical proof of this strategy can be found in Dyslexia and Mathematics 2nd Edition edited by Miles and Miles.
- Besides difficulty with awareness of numbers, many students with dyslexia have difficulty with working memory. Working memory is the ability to attend and hold multiple things in the forefront of your mind while analyzing or manipulating them during the course of a few seconds. Since math is replete with problem solving and analysis, this is a major stumbling block for students. While some students may develop their own strategies for compensating for weak working memory, other students may be confounded and unable to move past these difficulties. You have the potential to lighten their cognitive load by using the following principles in your instruction.
- Identify a student struggling with working memory. Your student may seem to "stall" in the middle of completing their work. They will struggle with multi-step directions and operations. They may give up, demonstrate place-keeping errors, and have trouble with recall.
- Analyze the task for working memory and cognitive demands. Consider the number of steps in a calculation and directions. Also, factor in how well the student knows the concepts.
- Scaffold the task so that the student is doing it along with you (e.g., in a hands-on apprenticeship).
- Talk to your student and assess his meta-cognition and ability to self-monitor. Improve these executive skills and focus on getting your student to ask for help when he gets "stuck."
- Provide a sequence or a pattern for problem solving. If a task appears "familiar," your student will more likely "stick with it" and will tend to be more accurate. Your student may need an approach to problem solving.
- Give your student examples to look at in order to decrease the cognitive load.
- Teach for mastery. Don’t move on until your student has achieved competency or you will be setting him up for failure.
- Model an approach for breaking apart multi-step directions and teach your student a strategy that works for him. He may benefit from drawing hash lines between each step, or highlighting, crossing out, or numbering each step.
- Make visual aids for number facts available, including multiplication grids, number lines, and manipulatives.
- Allow your student to use a calculator. (Make sure to include this accommodation in the 504 or IEP so that it can be used on standardized achievement tests.)
- Consider the role of anxiety, since it can hinder memory, focus, and attention to a task. See the section on motivation below.
- Read more on the role of working memory in math in "Working Memory Limitations in Mathematics Learning: Their Development, Assessment, and Remediation" (Berch, 2011).
A classroom study by Chinn (1995) suggests that the errors made by dyslexics on an untimed test were not significantly different to those made by non-dyslexics, with one notable exception, the error of no-attempt. The hypothesis is that the dyslexic pupil looks at the item and, if he feels he may not get the right answer, he simply does not begin. Avoidance is one symptom of lack of motivation. Becoming the class clown or making comments like, "I hate math," or, "My teacher doesn’t like me," are indicative of motivational difficulties for math.
Numerous factors may contribute to a lack of motivation for math, such as repeated failure; a mismatch in the student’s learning style and manner of instruction; teachers’ and parents’ comments; feeling overwhelmed or confused; or fear of embarrassment/failure in front of peers, to name but a few. It may be helpful to interview your student to get a sense of what the underlying cause is. A helpful assessment that yields qualitative, quantitative, and standard scores is The Survey of Teenage Readiness and Neurodevelopment Status (STRANDS). This test may be administered by any professional that works with students ages 13-19. This test directly assesses meta-cognitive skills by asking the student for his perception of his learning and learning environment. Once you have unearthed the root causes, you may elect to do work with the student, modify his environment, or do a combination of both.
Strategies for Working Directly with the Student
- Explore the value of math with your student. Find ways to make it relevant for his daily life (e.g., calculating a sale price, following a recipe, or planning a party). Point out the social consequences and costs of not being proficient in math.
- Relate your student’s goals for the future to being "mathematically literate." For example, if your student wants to pursue a career that requires a college degree, point out that he will be required to take college level math courses.
- Develop your student’s internal locus of control. Illustrate the importance of locus of control in a game requiring some level of strategy. Connect that with your student’s beliefs about math. Take an experimental approach to illustrate the direct relationship between hard work, strategy development, and improved outcomes (as opposed to blaming the teacher or stating that the material is too hard).
- Sample a variety of strategies for your student’s learning style and develop a "toolbox of strategies" that your student can utilize. For more on teaching strategies for your student’s learning style, see (link to article on meta-cognition).
- Teach your student self-advocacy skills such as seeking help and accommodations from teachers, seeking alternative ways to learn math (e.g., computer based models) ,or tutoring. For more on teaching self-advocacy for your student’s learning style, see Teaching Self-Advocacy.
Modifying the Environment
- Structure goals for mastery. This allows your student to see the result of his practice and effort and promotes a feeling of success. It will require your student to learn at his own pace.
- Minimize social comparison for grades and reward effort and attitude as well.
- Find a match between your student’s learning style and the teaching style.
- Allow for more individualized math instruction so that your student can learn at his own pace and develop the needed strategies, working memory skills, and motivation.
- Select a teacher that is effusive in his or her enthusiasm for math. These attitudes are contagious.
- Use reinforcement for both effort and mastery. Allow the student to select his reinforcement.
- Seat your student where he can see the chalkboard/screen and teacher. Ensure that there are not many other distractions (social or otherwise) nearby.
- Read more on the role of motivation in math in "Motivating Students Who Struggle with Mathematics: An Application of Psychological Principles" (Hanich, 2011).
If you are interested in more examples and evidence-based suggestions for improving math in your students, see Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. Another excellent resource is the book Dyslexia and Mathematics edited by T.R. Miles and Elaine Miles (2004). You may also be interested in the spring 2011 edition of IDA’s Perspectives on Language and Literacy: Mathematical Difficulties in School Age Children.
Even if you do not hold an advanced degree in mathematics, you have much to offer your student who struggles with math. You can elucidate the underlying causes of the struggle, recommend a thorough assessment, and highlight the type of learning environment your student needs. You can help your student develop the requisite language, motivational, and meta-cognitive skills to achieve fluency in math. Success starts here!
Berch, D. (2011) "Working Memory Limitations in Mathematics Learning: Their Development, Assessment, and Remediation," Perspectives on Language and Literacy, vol. 37 no. 2, 21-25.
Chinn, S.J. (1995) "A pilot study to compare aspects of arithmetic skills," Dyslexia Review, 4, 4-7.
Chinn, S.J. (2003). Test of Thinking Style in Mathematics, Belford, Ann Arbor.
The National Numeracy Strategy: Guidance to Support Pupils With Dyslexia and Dyscalculia, DfES 1051 212001
Fuchs, L.S., Powell, S. R., Seethaler, P.M., Cirino, PT., Fletcher, J.M., Fuchs, D. & Hamlettt, C.L. (2010). The effects of strategic counting instruction, with and without deliberate practice, on number combination skill among students with mathematical difficulties. Learning and Individual Differences, 20, 89-100.
Hanich, Laurie, B. (2011). "Motivating Students Who Struggle with Mathematics: An Application of Psychological Principles" Perspectives on Language and Literacy, vol. 37 no. 2, 41-45.
Joffe, L.S. (1990) The Mathematical Aspects of Dyslexia: A Recap of General Issues and Some Implications For Teaching, Links, 15 (2), 7-10
Jordan, N.C., Kaplan, D., Locuniak, M.N., &Ramineni, C. (2007). Predicting first-grade math achievement from developmental number sense trajectories. Learning Disabilities Research and Practice, 22(1), 36-46.
Krutetskii, (1976) in Kilpatrick and Wirszup (eds) The Psychology of Mathematical Abilities In School Children, Chicago, University of Chicago Press.
Mazzocco, M.M. M., & Thompson, R. E. (2005). Kindergaten Predictors of Math Learning Disability. Learning Disabilities Research & Practice, 20, 142-155.
Mazzocco, M.M. M. (2011). Number Matters. Perspectives on Language and Literacy, vol. 37 no. 2, 47-49.
Miles, T.R. (1993) Dyslexia: The Pattern of Difficulties, London, Whurr.
Powell, S.R., Fuchs, L.S. & Fuchs, D., Number Combinations Remediation for Students with Mathematics Difficulty. Perspectives on Language and Literacy, vol. 37 no. 2, 11-15.