A Mathematical Jeremiad

I hated math in elementary school. At least, I thought I hated math. It took years before I discovered that what I really hated was arithmetic, not math. Arithmetic is only barely math at all, yet it’s the entry point to mathematics for all elementary school students.

Elementary school arithmetic is a pernicious barrier that prevents students from entering STEM fields later in life. Arithmetic isn’t outright biased against students because of their gender, race, ethnicity, or religion. Instead, arithmetic instruction discriminates against different types of student, types of students that are found in every gender, race, ethnicity, and religion.

Arithmetic instruction discriminates against the smart kids. Arithmetic instruction convinces the creative kids to do something other than math. Arithmetic instruction shunts the sort of young people we need to be dreaming of ways to cure various types of cancer into other pursuits where math is not a requirement.

Even at a young age, a budding scientist is going to be wondering “why?” and a natural-born engineer is going to be asking “how?” Arithmetic answers neither of those questions. Arithmetic just declares the answer to be what it says it is and then demands that the answer be recalled. Children inclined to ask “why?” or “how?” will naturally tune arithmetic out. Getting those students to tune into math again once they (maybe) progress to the kind of math they will enjoy (and that they will need if they are to be scientists or engineers) is somewhere between difficult and impossible. If those students don’t tune back into math, they are lost to STEM fields forever. Fixing elementary school math and de-emphasizing arithmetic will bring more of everyone into math, boys and girls alike.

I discovered arithmetic to be a never-ending series of pointless worksheets and timed tests back when I was a kid, and it hadn’t improved by the time my daughters were in elementary school. Everything about arithmetic is focused on mentally calculating answers as quickly as possible. Of course, the fastest way to calculate is to memorize a lot of math facts, so the entire enterprise quickly becomes a challenge to commit math facts to memory. Rapid mental calculation has almost no application for anyone anymore. Yet, somehow most people seem to think that the ability to calculate super fast in your head is the mark of a talented mathematician and a prerequisite for studying any STEM field in the future.

The depth of this misconception was made clear to me at an elementary school parent-teacher conference over a decade ago. There was a new math curriculum that year, and my wife and I found it to be much better than we had expected. We thumbed through the math books as soon as they were distributed, and we were thrilled to see the workbook gently introducing the concept of Riemann sums by asking students to draw rectangles in “ponds” to estimate the area of a curved shape. To our dismay, as the first quarter wore on those pages of the workbook were omitted in favor of extra multiplication worksheets from other sources. In the parent-teacher conferences at the end of the first quarter we asked, as politely as we could, why the math program was replacing such an important bit of conceptual learning with yet more memorization work.

Our daughter’s teacher looked at us with an air of authority and explained that “some of these students are going to go on to be scientists and engineers.” Her tone indicated that it was quite clear to her that our daughter was not in the future scientist or engineer category. To be fair, it was certainly true that our daughter did very poorly on all of those damn worksheets. The teacher went on to explain that those future scientists and engineers “will need to know their multiplication tables. They’re never going to need to estimate the area of some weird shape by drawing rectangles in it.”

Of course, estimating the area of weird shapes by drawing rectangles in them is a pretty good layperson’s definition of integral calculus. Good luck building a bridge, or studying epidemiology, or learning any advanced physics if you don’t understand how to “estimate the area of some weird shape by drawing rectangles in it.”

Calculus is one of the most exciting and important innovations humanity has made in all of the time that humanity has existed. Calculus is both elegant and useful. It stirs our souls and unlocks the mysteries of the stars. It is the very foundation of most branches of science and engineering. No one ought to expect an elementary student to be ready to grasp calculus in a deep sense, but why drop math lessons that build calculus intuition in favor of memorizing math facts?

The answer to that question, of course, is that I’m sure my daughter’s teacher really thought that memorizing math facts was the most important thing for future scientists and engineers to do in elementary school. She certainly reacted with surprise tinged with anger when we pointed out how important estimating areas of curved shapes using rectangles would be for her future scientists and engineers later in their education and work. I don’t think she believed us at first, so my wife and I did something we really didn’t want to do. We pulled rank on her in the form of our pair of MIT degrees. Faced with two MIT graduates insisting that memorization was a waste of time and that estimating the areas of ponds was important, she at least stopped arguing with us.

I don’t think the teacher ever believed us, though, because the stupid math worksheets kept being assigned. The only change was that they now came into our home with the special proviso that our daughter was excused from working any math assignment that we deemed pointless. At the time, I chafed with the realization that while we’d helped our little girl, we hadn’t helped any other students in the class. In retrospect, though, I am pretty sure that other parents — parents without a clue about math — would have been in that classroom complaining if worksheets focusing on memorization weren’t being sent home. No doubt parents would have complained that they wanted their son to be an engineer, so he needed to be forced to memorize multiplication tables against his will. The teacher wasn’t the only one who didn’t understand math.

I feel more than a little bad to be writing this mathematical Jeremiad at all. It’s just another complaint on a towering stack of laments about the state of math education. If you spend 15 seconds with an internet search engine you can find countless similar complaints. While there are some wonderful nonprofits helping as many kids as possible learn math, this problem requires a change in culture even more than an army of tireless workers. I know that this post isn’t going to be the one that tips the scales.

On the other hand, this topic is too important to not write about, too vital to not talk about. I’m sure that I’ll write about it again. If you understand that turning math into a memorization exercise destroys mathematical learning, I hope that you will take that message to your community as well. This problem is bigger than one teacher, or one textbook, or one set of educational standards. The plague of elementary school arithmetic is a cultural phenomenon, and it takes a lot of people to change the culture.

Oh, one last thing. If you’re worried that the little girl who wasn’t included in the “future scientists and engineers” was discouraged by all of this, you wouldn’t be wrong. Of course the never-ending worksheets and lack of confidence from her teacher was discouraging. The discouragement didn’t stop her, though. She’s now a professional physicist. She sometimes sends me interesting papers (some written by her) with lots of integral calculus in them. I guess that estimating the area of weird shapes by drawing rectangles in them was useful after all.

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