I’ve just finished reading this paper by Richard Skemp (nicely republished here on the Republic of Mathematics site) about the difference between instrumental and relational understanding when it comes to mathematics education. My first thought was, “this is what sets Montessori maths education apart from traditional mainstream.” The Montessori mathematics materials lay a groundwork for relational understanding. Eric, who’s 7, not only knows how to calculate the area of a rectangle (width x height) but why that calculation works, because he spent lots of time building rectangles with wooden cubes, and learning through building that the “area” of shape is derived from the number of cubes inside.
A few years ago, when Rachel was still in preschool, Gary & I attended a parent education evening where her teacher showed us the “binomial cube” material that she was learning. Of course, at age 4, she did not learn the binomial equation, but by manipulating the pieces of wood to fit them properly into the cube, she was gaining understanding that, years later, she could use to learn exactly how to solve the algebraic formula (a+b)³. The cubes and prisms in the binomial cubing material represent the factors of the equation.
I sat there and thought, “I remember learning the binomial cube expression!” and if I thought hard enough, I might have actually come up with the solution. Gary, on the other hand, picked up the material and immediately sorted out all the pieces and proceeded to show me how the equation was constructed based on those pieces. In case you’ve forgotten: (a + b)³ = a³ + 3a²b + 3ab² + b³
I understood it instrumentally, but he understood it relationally.
Dr Skemp, in his paper, argues (in a nutshell) that relational understanding of mathematics is better. I would have to agree.
