I am enjoying another Corwin Press book on brain-based teaching: How The Brain Learns Mathematics. In David Sousa’s chapter on identifying math difficulties, he first suggests that teachers analyze the type of math instruction being provided and consider other environmental issues before determining that a child has an actual disability. He describes some of the pitfalls of current math instruction. Sousa’s book provides excellent strategies to support students at different developmental levels. It also describes effective assessment for determining a student’s present level of performance.
Sousa reviews research that describes a continuum of learning preferences for quantitative versus qualitative reasoning. Some mathematical behaviors associated with quantitative reasoning include a proclivity for linear thinking, an emphasis upon the components of problems rather than broader concepts, and difficulty with multi-step problems. Qualitative reasoning is characterized by emphasis upon broader concepts, difficulties with precise calculations, multiple approaches to problem solving, and an enjoyment of geometry.
From Sousa’s description of this continuum, I immediately thought of dyslexic readers, who often have a better grasp of the “gestalt” and experience more difficulty with the “smaller” components of language. However, Sousa cites research indicating that dyscalculia and dyslexia are not genetically linked, although kids may certainly have both impairments.
Research supports the use of a concrete-pictorial-abstract approach to math instruction, which allows students at all levels (including middle and high school) the opportunity to interact with math via manipulatives first. (Check out Nerd in the Brain‘s website for her awesome use of math manipulatives.) During the concrete stage, it is important to link math to real-world problems. Students then transition to pictures which assist students in visualizing the math process. The last step is the use of symbols as a more efficient means of representing mathematical operations. Sousa emphasizes that without a concrete link between symbols and real-world problems, students will simply memorize material and procedures without understanding.
Depending upon the current bandwagon, I’ve seen math instruction stall at the concrete level or actually start at the symbolic level. Over the years, regular classroom math instruction most often whisks students into the symbolic level too quickly. I have also been guilty of rushing students through the concrete level because they may “waste time” or become distracted by playing with materials. It’s a challenge to make math instruction efficient as well as cognitively appropriate. I have found that an effective assessment eliminates some of that pressure because I can better target my instruction. For kids who absolutely cannot touch manipulatives without building towers, there are some good online manipulatives which allow them to experiment without getting mentally lost. Glencoe has a super set of free manipulatives, with creative work mats that kids truly enjoy. The National Library of Virtual Manipulatives is also excellent (and free) but doesn’t have the flexibility of Glencoe’s site for creating real-world problems.