Teaching middle school I often had students who still used their fingers to find the difference between 12 and 7. They would start with 7 and then count 8, 9, 10, 11, 12 while putting up fingers each time they would say a number. When we worked with integers and subtraction the idea that positive 6 minus negative 3 related to how far apart the numbers were was hard for them to understand even though, in their procedure, they were finding out that very thing. Something about the way they thought about subtraction and their procedure did not allow them access to thinking of subtraction in a way that made sense beyond finding the answer. I think because of that the answer didn’t make sense beyond “I did the procedure correctly”.
This way of understanding math could be referred to as the Correct Answer for Some Tasks Expression or CASTE system. Yes, I totally made this up for this blog, but indulge me. The CASTE system of knowing mathematics limits the ability to move into new area of mathematics and expand the contexts. We see this system in math in many forms. It might be the FOIL (first, outside, inside, last) method of multiplying polynomials or Keep, Flip, Change (KFC) for dividing fractions. While these work for the immediate context of math they limit math down the road because the CASTE system focuses on the math of the moment. You are dividing fractions today and you will divide fractions tomorrow. Don’t be concerned if the answer makes sense or the applicability of your method outside of today’s problems.
Fighting the CASTE system in math is a goal of the AIMS Center although it has not been expressed in precisely those terms. The work we have been doing at the center in the units coordinating team in subtraction, composite units and fractions is all aimed at developing ways of attending to student thinking and responding in such a way as to not promote math understanding that will limit students. It is an area of service that I have enjoyed and for which I have a passion. Thanks for indulging me.