Posted on January 2, 2016

# Why is our Math Instruction Stuck in the 19th Century?

Our math instruction is stuck in obsolescence. There was a time when it was necessary to do mathematics by hand, but that time has passed. The advent of computer technology has opened up new doors to us as teachers of numbers, but we have yet to walk through them.

This is an issue that has been nagging at the back of my consciousness for some time now, but only recently has it been brought to the forefront. Recently I watched as day by day a handful of my students struggled to master the procedures for long division, even when the rest of them had moved on. I began to wonder why; why is this necessary? Why do I bring some of them literally to tears by making them repeat steps for an algorithm I have never used in a practical sense in my entire life? How far do I push them to master this?

The answer, of course, is that I have to. This skill is on the big high stakes standardized test, so I am required to have them master this skill. After all, we are ALL graded by their performance on this test, so they must be competent in all of the skills as laid out by the state and federal government. But what if that test weren’t there? Would I still force them to master this skill?

It wasn’t until watching a Ted Talk by Conrad Wolfam that my thinking began to clear. Wolfram argues that solving a math problem is made up of 4 steps:

1. Pose the right question about an issue

2. Change that real world scenario into a math formulation

3. Compute

4. Take the math formulation and turn it back into a real world scenario to verify it

And while each of these steps are crucial, we as teachers spend an astounding 80% of our instruction time exclusively on step 3: computation. Worse, this step is one that can easily be taken over by computers. In fact, it is often better if computers take over this step since computing numbers is what computers are significantly more efficient and accurate at.

Yet in many cases computation is all we teach. Teachers (myself included) feel that if we can just get our kids to know how to compute well, the other three steps will take care of themselves. In reality, steps 1, 2, & 4 are quite complex and require a different kind of thinking entirely. Knowing how to effectively compute alone will never get students to a place where they can strategically solve complex real world problems. And we wonder why our students are bad at word problems?

Keith Devlin makes a similar argument when he points out that if you “go into most math classrooms and what you see will most likely bring to mind a room full of clerks in the pre-computer age when companies employed large numbers of numerically-able people to crunch their numbers… Which was, of course, what the system was set up to provide.”There was once a time where it was necessary to know how to compute well, just as there was a time where it was important to know how to to hitch a horse to a buggy. And while both of those skills may still be quaint and interesting to know nowadays, neither are essential to functioning in the 2st century. So why do we continue to teach them?

I think the reason is threefold. First we have teachers who feel very strongly that teaching computation gives students a good sense of numbers and how they work. While this is true for many kinds of mental calculations, it does not hold up when it comes to more complex algorithms. Take, for example, long division. The standard algorithm for long division is a complex pattern of steps designed as a shortcut to ease with calculation. These steps walk you through a process which actively circumvent knowledge of place value and division in order to arrive at an answer. Of course, this algorithm is complex, difficult to understand and is easy to make a mistake anywhere along the line that will result in a wrong answer. It is much faster and almost always more accurate to simply use a calculator or spreadsheet when it comes to division. The same can be said for multi-digit multiplication, fractions, linear functions, calculus, etc.

Of course, teachers are not alone in their expectation of calculation as the dominant form of math instruction in the classroom. Parents play an active role as well. Many parents expect daily calculation instruction and homework and can become frustrated when they don’t see that work coming home. Parents expect things to look the same as when they were in school and if it isn’t there something must be wrong. While this is not true of all parents, of course, it is enough to make the job of changing difficult.

The final reason why we are stuck in a computational training system is the big standardized test. If the test requires computation, then it is assured that teachers will teach those skills. Until we recognize that we are testing the wrong thing, we will continue to spend a majority of our time teaching computation.

I think that this issue can best be summed up in an interaction I had with a student a few weeks ago. He was working on a complicated word problem in class. It was the kind of problem that required not only steps 1, 2, and 4, but also a heavy dose of step 3: computation. As he began to delve into the computation I suggested that he use a calculator to both improve his accuracy and to free his brain from the challenge of calculating so that he could focus on the bigger picture. When the steps were finished, the calculations complete, and the problem correctly solved I congratulated him on a very excellent job of thinking through a tough problem. He thanked me, then sighed and said, “yeah, but I cheated.”

“You cheated?” I replied, surprised.

“Yeah, I used a calculator.” He said sadly.

He solved the problem correctly and yet he felt like he cheated because he used a better tool and didn’t solve the problem in the most difficult way possible. The idea of using the best computational tool available didn’t just not occur to him, he actively rejected it because he had been drilled to look at calculators as cheating. How is this helping this student?

This is a massive and difficult shift in thinking for everyone. We have been teaching math the same way for over 100 years. It is difficult to move beyond that. For the sake of our students, however, we must.

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