From manipulatives to abstraction: learning math at Chugach

I sat down with Linda Biddle, my son Joey’s teacher, to ask how she teaches math in a classroom of students in charge of their own learning. Here is what she told me.
–Charles Wohlforth

Most Chugach Optional students learn math differently than their parents did. A Chugach parent may wonder when his or her child will adopt certain familiar algorithms, or procedures, for doing arithmetic, or even become nervous if it doesn’t happen ‘soon enough.’

At Chugach, ‘soon enough’ is different for each child. Our students are introduced to math through visual representations and manipulatives. Each child transitions to abstract computation methods when ready as an individual. As long as he or she understands and can accurately complete mathematical problems, that’s success.

Math concepts begin with manipulatives, such as interlocking blocks, strips, or grids written on paper that represent units and groups of units. Being able to see the units helps children understand the concepts behind operations and develop number sense and mathematical intuition.

Most adults initially learned pencil-and-paper processes and memorization. These techniques are abstract representations of underlying mathematical concepts. It’s possible to be a whiz at computation without really understanding what the numbers mean. That’s what we don¹t want.

Students with a deeper understanding of mathematics often grow more easily into the advanced math concepts they will encounter in algebra, geometry and calculus. They also are more likely to recognize calculation errors or improbable results, because they have a feel for numbers.

The switch from concrete manipulation to the use of abstract computation could come at any age. We introduce a variety of abstract methods and let the children decide which strategies to use and when. Sometimes they make the connection and adopt the shortcuts right away, sometimes they wait, and sometimes they invent their own algorithms that work better in their own
minds.

Linda’s son made it in high school mathematics while still using visual methods of strips and mats, arranging them in his mind’s eye rather than with physical materials. A high school teacher thought he was cheating because she couldn’t see how he got his results. He proved in a parent-teacher conference that he was solving the problems, but doing it in his own way.

My older son and daughter also have excelled at math in high school and college. Their computation seemed slow and painful at times in elementary school, but they emerged with a strong understanding of math concepts that have made higher level work a breeze.

Parent should have confidence even if the switch to traditional computation methods takes time. Chugach works!