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9-12 math graphingOver the past few weeks, I have had the privilege of working with some fourth- and fifth-grade students to learn more about their mathematical thinking. It has been fun and informative. I am an “old dog” and resort to my “old tricks,” so hearing some of their approaches to solving a math problem was nothing short of enlightening.

As we looked at various problems, I learned so many ways to solve them. For example, when given the problem 1000-998, one student did the following: 1000-900=100; 100-90=10; 10-8=2. Another approached it in a similar fashion, saying 900+100=1000; 100-98 =2. And yet another student counted up from 998 to 1000, realizing that they only had to count up 2 numbers. Each student arrived at the correct answer; each answer was the result of a different approach. They explained their thinking, and it was as varied as they are.

The thing that fascinates me about this is that when I was a student in upper elementary school, there was only one way to approach that problem: you set it up vertically, canceled the zeroes, borrowing from each previous digit and arrived at the solution of 2. There you have it! There was one way to do it and one correct answer. For everyone. No leeway allowed. And, boy, could those zeroes be tricky.

One of the greatest strides made in education is teaching concept development. The recognition that skills can build while working to build a given concept has opened many doors to learning. Yes, children still learn their ABCs and 123s. They also learn the sounds the letters make and the ways in which those sounds go together. They learn about place value and what it means to cancel those zeroes that I “got rid of.” What was I borrowing from? How does our number system work and why? At that age, I hadn’t a clue, but I could usually arrive at the correct solution.

I am delighted when I get a firsthand look at the processes students employ to solve problems, any kind of problems, math or otherwise. Thinking is the substance of learning. Learning one way to solve a problem doesn’t involve a lot of thinking. Learning from these students is eye opening and reassuring for our future and theirs. When was the last time you approached a problem in a new or varied way?