Math, She Rote

My friends often have different educational backgrounds than mine. Some of them are younger, but even if they aren’t, they’re often from urban areas that had moved to more modern educational curricula before my school system had. The way I learned basic arithmetic remained unchanged from how it was taught from the early 1980s by the time I learned it in the late 1980s and early 1990s because that’s when our books dated from.

I learned during an interesting period in mathematical education history. It represented a kind of educational interbellum—a bit after the “New Math” of the 1960s and 1970s1 but before the “math wars,” instigated by the 1989 Curriculum and Evaluation Standards for School Mathematics. The latter 1989 publication has been called “reform mathematics,” which emphasizes processes and concepts over correctness and manual thinking. In other words, the educators promoting reform mathematics began to believe that the path students took toward the answer mattered more than whether they got the answer right. Many states’ standards and federally funded textbooks followed reform mathematics in the 1990s and beyond.

Reform mathematics emphasized constructivist teaching methods. Under this approach, instead of prescribing to students the best way how to solve a problem, teachers pose a problem and allow the student to surmount it by building on their own knowledge, experiences, perspective, and agency. The teacher provides tools and guidance to help the student along the way. Constructivist approaches involve experiments, discussions, trips, films, and hands-on experiences.

One example of a constructivist-influenced math curriculum, used in elementary school to teach basic arithmetic, was known as Investigations in Numbers, Data, and Space. It came with a heavy emphasis on learning devices called manipulatives, which are tactile objects which the student can physically see, touch, and move, to solve problems. These are items like cubes, spinners, tapes, rulers, weights, and so on.

As another example, someone I know recently described a system they learned in elementary school called TouchMath for adding one-digit numbers, which makes the experience more visual or tactile (analogous to manipulatives). They explained that for each computation, they counted the “TouchPoints” in the operands to arrive at the result.

I had never heard of TouchMath. In fact, I never solved problems using manipulatives, nor any analogue of them. I had little experience with this form of math education. We were given explicit instructions on traditional ways to solve problems (carrying, long division, and so on). Accompanying drawings or diagrams rarely became more elaborate than number lines, grids, or arrangements of abstract groupings of shapes which could be counted. They served only as tools to allow students to internalize the lesson, not to draw their own independent methods or conclusions.

I contrasted my friend’s experience with TouchMath to my experience. To add or subtract one-digit numbers, we merely counted. We were given worksheets full of these to do, and since counting for each problem would have been tedious and impractical, memorization for each combination of numbers would become inevitable. Given the expectations and time constraints, I’m certain rote memorization was the goal.

In a couple of years, we were multiplying and dividing, and we were adding and subtracting two- or three-digit numbers using carrying—processing the numbers digit-wise. At the same time, we were asked to commit the multiplication tables to memory. These expectations came in third grade, and it would be nearly impossible to make it out of fourth grade without committing the multiplication table and all single-digit addition and subtraction to memory (the age of ten for me).

Our teachers did not bother to force us to memorize any two-digit arithmetic operations. But I have some recollection a lot of years ago of my grandma telling me she had most two-digit additions and subtractions still memorized. It was just an offhand remark—maybe something she said as I was reaching for a calculator for something she had already figured out. Maybe we were playing Scrabble.

For context, she would have gone to school in rural Georgia in the 1940s and 1950s, and she graduated high school. (In that time and place, it was commonplace for many who intended to do manual, trade, or agricultural work not to continue through secondary school.)

I remember feeling incredulous at the time about the number of possible two-digit arithmetic operations that would imply memorizing. Of course, many would be trivial (anything plus or minus ten or one, or anything minus itself); others would be commonplace enough to easily memorize, while still others would be rare enough to ignore. But that still leaves several thousand figures to remember.

The more I thought about it, the more I saw that, in her world, it would make better sense to memorize literally thousands of things rather than work them out over and over. She had no way of knowing that affordable, handheld calculators would exist in a few decades after she graduated from school, after all. Each time she memorized a two-digit addition or subtraction, she saved herself from working out the problem from scratch over and over again for the rest of her life. This saved her effort and time every time she

  • balanced her checkbook,
  • filled out a deposit slip at the bank,
  • calculated the tax or tip on something,
  • tallied up the score for her card game,
  • totaled up a balance sheet for her business,
  • made change for a customer, or
  • checked that the change she was being given was correct,

to say nothing of all the hundred little things I can’t think of. She married young and has run small businesses for supplemental income all her life, so managing the purse strings fell squarely into her traditional gender role. Numbers were part of her daily life.

So for the first half of her life, none of this could be automated. There were no portable machines to do the job, and even the non-portable ones were expensive, loud, slow, and needed to be double-checked by hand.2

I don’t believe she remembered these all at once for a test, the way I learned the multiplication tables in third grade. It seems likely she memorized them over time. It’s possible that expectations in her school forced a lot of memorization that I didn’t experience when I went many decades later, but maybe she was just extra studious.

I recall, as I went through school, having to rely more on a calculator as I approached advanced subjects. Before calculators became available to students, appendices of lookup tables contained pre-calculated values for many logarithms, trigonometric functions, radicals, and so on. Students relied on these to solve many problems. Anything else—even if it were just the square root of a number—came from a pen-and-paper calculation. (Many of my early math books did not acknowledge calculators yet, but this changed by the 1990s.)

Charles Babbage reported that he was inspired to attempt to mechanize computation when he observed the fallibility of making tables of logarithms by hand. He began in the 1820s. After a hundred and fifty years, arithmetic computation would become handheld and affordable, fomenting new tension around what rote memorization plays in both learning and in daily life.

Today, we’re still trying to resolve that tension. Memorization may feel like it has a diminished role in a post-reform education environment, but it’s by no means dead. Current U.S. Common Core State Standards include expectations that students “[b]y end of Grade 2, know from memory all sums of two one-digit numbers,” and, “[b]y the end of Grade 3, know from memory all products of two one-digit numbers.” That sounds exactly like the pre-reform expectations I had to meet.

All this means is that there has been neither a steady march away from rote memorization nor a retreat back to it. Research is still unclear about what facts are best memorized, when, or how, and so there’s no obvious curriculum that fits all students at all ages. For example, the Common Core Standards cite contributing research from a paper3 which reports on findings from California, concluding that students are counting more than memorizing when pushed to memorize arithmetic facts earlier. The paper reasons this is probably due to deficiencies in the particulars of the curriculum at the time of the research (2008).

I’m not an expert, and I don’t have easy answers, but my instinct is that rote memorization will always play an inextricable role in math education.

Having learned about the different directions in which the traditional and reform movements of math education have tugged the standards over the years, I tend to lean more traditional, but I attribute this to two things. One is that I was educated with what I remember to be a more traditional-math background, and though I didn’t like it, it seems serviceable to me in retrospect.

The other reason is that, for me, memorization has always come easily. I don’t really know why this is. It’s just some automatic way I experience the world. Having this point of view, though, I can easily see how beneficial it is to have answers to a set of frequent problems ready at hand. It’s efficient, and its benefits never cease giving over time. The earlier you remember something, the more it helps you, and the better you internalize it. Even for those who can’t remember things as easily, the returns on doing so are just as useful.

I do completely agree with the underlying rationale of the constructivist approach. Its underpinnings are based on Piaget’s model of cognitive development, which is incredibly insightful. It seems useful to learn early to accommodate the discomfort of adapting your internal mental model to new information by taking an active role in learning new ideas in order to surmount new problems.

I don’t necessarily believe that a constructivist learning approach is intrinsically at odds with rote memorization—that is to say, that memorization necessarily requires passive acquisition. In fact, the experience of active experimentation and active role may help form stronger memories. It’s more likely they compete in curricula for time. It takes longer to mathematically derive a formula for area or volume by independent invention, for example, than to have it given to you.

In fact, constructivist learning works better when the student has a broader reservoir of knowledge in the first place from which to draw to begin with when trying to find novel solutions to problems. In other words, rote memorization aids constructivist learning, which then in turn aids remembering new information.

My feeling is that math will always require a traditional approach at its very heart to set in place a broad foundation of facts, at least at first, before other learning approaches can have success. Though the idea of critical periods in language acquisition has detractors and heavy criticism, there is a kernel of truth to the idea that younger minds undergo a period of innate and intense linguistic fecundity. Maybe as time goes by, we can learn more about math acquisition and find out which kinds of math learning children are more receptive to at which ages. Until then, I feel like we’re figuring out the best way to teach ourselves a second language.

I am grateful to Rachel Kelly for her feedback on a draft of this post.