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By Kristina Scott, Ed. D.

We probably know someone who has uttered the phrase, “I hate math.” Math phobia affects many individuals, and we know from the data that there are significant gaps in math performance when we compare students with a disability to their peer group without a disability (NAEP, 2017). Struggles in math can occur when trying to apply procedural knowledge (the formulas and math rules that students learn) into real-world application, or can occur at the executive functioning level where a student needs to use working memory, organizational skills, connect conceptual knowledge to real-world practice, and take risks. When you add in a reading component to this math language (word/story problems), learning disabilities and struggles become even more prominent.

One way to approach teaching students that have math phobia or significant difficulties with math is to have them engage with math in a structured format moving from the concrete to a representation level and then progressing towards the abstract. This means that all new math begins with physical manipulatives. Then it moves from these manipulatives to visual representation (or pictures) of the concept. Students need to meet mastery at each of these stages before seeing the concept or formula in just abstract terms. A simple illustration of this approach, known as graduated instructional sequence, would be as follows:

In teaching subtraction, students would first work with physical manipulatives for the subtraction problem 7-3. They would be using physical objects to display and solve the problem through kinesthetically moving three objects away from the seven. Once students master being able to execute this concept, we then move to picture drawings (visual representation). Students in this stage would draw seven objects (cubes for example), and then illustrate subtracting three from this seven by crossing out cubes in their drawing. Students again would have to show mastery at this level before moving to just seeing the abstract problem (7-3) with no cueing or visual supports.

This same approach, although often associated with early elementary grades, should be used as students advance in grades and are exposed to new concepts. Physical manipulatives should be used in all grades when new math concepts are being taught!  Let’s look at a more complex algebra example (algebraic expressions) to show how this same concrete-representational-abstract (CRA) instructional sequence should be used:

For this example we will use the problem 6x-3=, when x=2. Students are first going to use concrete algebra tiles, by using 6 green algebra tiles to represent 6X in this equation, and 3 red tiles to represent -3 in this equation. We then need to replace the green algebra tiles with our x-value (2 blue squares for each tile). Essentially there will be 12 blue squares and 3 red squares in this equation. We would then match each red square with a blue square until we are left with 9 blue squares, which is the answer to this problem. Here is a video that demonstrates the use of algebra tiles to solve algebraic expressions: This physical conceptual understanding would be practiced until mastery—moving next to picture representation, and finally the abstract problem (6x-3) without visual supports.

The CRA instructional sequence helps link abstract expressions to concrete concepts, supporting the application of procedures and procedural knowledge to real-world use—an area identified as sticking point for those with math phobia. This approach through the use of manipulatives and gradual scaffolds calls for explicit instruction, modeling and guidance, and supported independent practice which are all research-based approaches in teaching. Using this approaches and moving in a gradual release model ensures mastery of conceptual knowledge to support real-world application.

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