By Keith Devlin @profkeithdevlin

“In math you have to remember, in other subjects you can think about it.” That statement by a female high-school student, was quoted by my Stanford colleague Prof Jo Boaler in her 2009 book What's Math Got To Do With It? I took it as the title of my June 2010 Devlin’s Angle post, which was in part a review of Boaler’s book. In a discussion peppered with quotations similar to that one, Boaler describes the conception of mathematics expressed by the students in the schools where she conducted her research. To those students, math was a seemingly endless succession of (mostly meaningless) rules to be learned and practiced. Among the remarks the students made, are (the highlighting is mine):

“We're usually set a task first and we're taught the skills needed to do the task, and then we get on with the task and we ask the teacher for help.”

“You're just set the task and then you go about it ... you explore the different things, and they help you in doing that ... so different skills are sort of tailored to different tasks.”

“In maths, there's a certain formula to get to, say from A to B, and there's no other way to get it. Or maybe there is, but you've got to remember the formula, you've got to remember it.”

Given that is their experience of mathematics, there is no surprise that many students that are taught that way give up and bail out at the first opportunity. In fact, a more natural question is, “Why do a few students enjoy math and do well in it, answering questions at the board with seeming ease.”

The answer is, the students who do well in math and enjoy it, are doing something very different from the activity described in the above quotes. Indeed, one of the things that attracts students to math is that it is the subject where you have to learn the least number of facts or methods, and can spend most of your time in creative thinking. Those “good” students have discovered that, in the math class, there is relatively little you have to remember; most of the time, you can wing it, and you’ll do just fine. It’s not about learning a wide range of formulas and special techniques, the trick is to learn to think a certain way.

For the few who know the “one big trick” professional mathematicians rely on, math class is an engaging and enjoyable creative experience. How those few get to that point seems to be exposure to an inspiring teacher at some point, hopefully before the rot sets in and the student has been completely turned off math, or perhaps some other fortuitous event. Absent such a stimulus, though, it’s no surprise that when fed a steady diet of math classes focused on mastering one concept, formula, or special technique after another, the majority sooner-or-later give up, and simply endure it (in bored frustration) until they are through with it.

Which brings me to the mathematics Common Core State Standards, rolled out in 2009 to guide developments in education required to meet the changing environment and needs of the 21st Century.

If you go onto the CCSS website, you will find a large database of specific standards items, one such (which I picked at random) being

CCSS.MATH.CONTENT.5.MD.C.5.   Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

Parsing out the reference code for this particular standard, it relates to Grade 5, Measurement & Data, Geometric measurement: understand concepts of volume, item 5.

It is important to realize that the Common Core is not a curriculum, nor does it stipulate how any topic should be taught. It is exactly what its name indicates: a set of standards that educators should aim to meet at each stage. But it is tempting to view it as a list of specific topics that a teacher should cover one after another. (Tempting, but not easy if you are working from the website since it isn’t presented as a list. I suspect that is deliberate.) If you do that, then there is an obvious danger that the result will be a continuation of the approach to math education that students experience as a process of learning one little trick after another, and you are back with the situation Boaler catalogued.

But those individual CC items are the terminal-nodes on a branching tree that has a regular structure, and it is in that structure that you see not only order but just a handful of basic principles. It is those basic principles that should guide math instruction. There are just eight of them. They are called the Common Core State Standards for Mathematical Practice. Here they are:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Those eight principles (the website elaborates on each one) constitute the core of the mathematics Common Core. They encapsulate the key features of mathematics learning essential for anyone living or working in today’s world. Notice that there is nothing about having to learn specific facts, formulas, or techniques. The focus is entirely on thinking.

The same is true of the specific items you will find in the rest of the Common Core website. When you drill down, you will find targets to aim for in order to develop thinking, following those eight principles at each grade level.

When mathematics is taught as a way of thinking, along the lines specified in those eight Common Core principles, then along the way, a student will in fact pick up a whole range of facts, and meet and learn to use a variety of formulas and techniques. But the human brain does that naturally, as an automatic consequence of lived experience. We are hardwired that way!

In contrast, learning becomes hard when presented as a sequence of items to be learned and practiced one by one, each in isolation, based on the false premise that you must first learn the “basics” before you can “put them together” to form the whole. The moment you realize that mathematics is about process rather than content, about doing rather than knowing, the absurdity of the “must master the basics first” philosophy becomes apparent.

Notice I am not saying “the basics” are irrelevant. Rather, they are picked up far more easily, and in a robust fashion that will last a lifetime, by using them as part of living experience. For sure, a good teacher can speed the process up by helping a student recognize the used-all-the-time basics, and maybe provide instruction on how they can be used in other contexts. But the focus at all times should be on the thinking process. Because that’s what mathematics is!

If the above paragraph sounds a bit like learning to ride a bike, then all to the good. A child learning to ride a bike will acquire a good understanding of gravity, friction, mechanical advantage, and a host of other physics basics. An understanding that a physics teacher can use to motivate and exemplify lessons in those notions. But no one would say that you cannot learn to ride a bike until you have mastered those basics! Think of doing math as a mental equivalent to riding a bike. (I wrote about this parallel in Devlin’s Angle before, in my March 2014 post. My final point there was somewhat speculative, but as a mathematician who also rides bikes, I claim that the overall parallel between the two activities is very strong and illuminating.)

Now comes the point where I part company with the CCSS, and indeed much of the focus in present-day American mathematics education and standardized math assessment.

Let me ease myself in by way of my cycling. I learned to ride a bike as a child and used a bike to get around throughout my entire childhood up to graduation from high school. I then hardly ever got on a bike again until I was 55 years old and my knees gave out after a quarter century of serious running, and I bought my first (racing-style) road bike. For the first twenty minutes or so, I felt a bit unsteady on my new recreational toy, but I did not need to seek instruction or help in order to get on it that first time and ride. The basic bike-riding skill I had mastered as a small child was still there, available instinctively, albeit a bit rusty and in need of a bit of adjusting for the first few minutes.

Moreover, when I started riding with a local club, my fellow riders gave me lots of tips and advice that made me able to ride more safely and at higher speeds. Some of what I learned was not obvious, and I had to practice. It was not the same as riding a city bike at low speed.

Likewise, when I bought, first, a mountain bike, and then a gravel bike, I had to take my basic bike-riding ability and transfer it to a different device and different kinds of terrain, and, in each case, once again learn from experts how to make good, safe use of my new machine.

The point is, they were all bicycles and it was all cycling. So too with mathematics. Once someone has mastered — truly mastered — one part of mathematics, it is relatively easy to master another. Yes, you will need to learn some new things, including a new vocabulary, some new techniques, and likely a new ontology, and yes you will almost certainly benefit from (and possibly need) help, guidance, and advice from experts in that new area of math. But you already have the one key, crucial ability: you can think like a mathematician.

In terms of learning mathematics, what this means is that it is enough to devote considerable effort to genuinely mastering just one topic — say elementary arithmetic — and then spending some time going through the process of branching out from that one area to a number of others (perhaps algebra, geometry, trigonometry, and probability theory).

In terms of mathematics assessment, it means that it is enough to test students’ mathematical thinking ability focused on just one topic, and then test to see if they have sufficient knowledge of a number of other topics. The advantage of approaching assessment this way is that, at least with current assessment methods, testing thinking is time-consuming and expensive, since it requires a small army of trained human assessors to grade solutions to open-ended questions, often “complex performance tasks,” whereas assessing breadth of knowledge can be done with a variety of machine-grades tests. So there are significant savings in cost and time if assessing thinking ability is done separately on just one topic. Which is absolutely all that is required, since the ability to think mathematically is just like the ability to ride a bike — once someone can ride one kind of bike, they can, with perhaps some adjustment, ride any kind. There is absolutely no need to test for that. As a result of natural selection of many thousands of years, humans can all do it.

The only question that remains is what mathematical topic should we focus on to develop the ability to think mathematically — including, I should add, an understanding of the importance of the precise use of language, the ability to handle abstraction, the need for formal definitions, and the nature and significance of proof.

Well, why not once again take our cue from how most of us learn to ride a bike. What is the equivalent of our first child’s bicycle? Elementary arithmetic.

What’s that you say? “There isn’t enough meat in elementary arithmetic to learn all you need to know about thinking mathematically, with all those bells and whistles I just mentioned.” Think again. Alternatively, check out the article written by mathematics educator Liping Ma in the article she published in the November 2013 issue of the Notices of the American Mathematical Society, titled A Critique of the Structure of U.S. Elementary School Mathematics.

Based on her experience with mathematics education in China, Ma argues forcefully, and effectively, that there is more than enough depth and breadth in “school arithmetic” (as she calls it) to fully develop the ability to think mathematically. True, in the West we don’t teach elementary arithmetic that way; indeed, we present it as a series of basic number facts to be memorized and algorithms to be practiced, as in the Boaler critique. We do so, at some speed I should add, in large part because we are in so much of a hurry to move on to all the other mathematical topics that someone at some time in the past declared were “essential” to learn in school. But as many have pointed out over several decades, the result is that our mathematics curriculum is “a mile wide and an inch deep”, resulting in students leaving school believing that “In math you have to remember, in other subjects you can think about it.”

In the June 2010 Devlin’s Angle post I referred to earlier, where I talked about Boaler’s then-new book, I mentioned Ma, and said I agreed with her argument about using school arithmetic as the topic to develop the ability to think mathematically. I still do.

I also think school arithmetic provides the one topic you need to assess mathematical thinking ability — regardless of whether you are assessing student learning, teacher performance, or district system performance. Given that, assessment of whatever breadth is required can be done relatively easily and cheaply. Because the thinking part is essentially the same, the assessment of the breadth can focus on what is known (rather than what can be done with that knowledge).

And (of course), one really valuable benefit of focusing on school arithmetic is that it provides as level a playing field as you can hope for, with elementary arithmetic the one mathematical topic that everyone is exposed to at an early age.

In a future post, I’ll take this topic further, looking at the implications for teaching, the educational support infrastructure (including textbooks), the effective use of modern technologies, and the educational implications of those technologies.

Also, as the title makes clear, the focus of this article has been systemic mathematics education, the mathematics that states decide is essential for all future citizens to learn in order to survive and prosper and contribute to society. There is a whole other area of mathematics education, where the focus is on the subject as an important part of human culture. That’s actually the area where I have devoted most of my efforts over the years, writing books and articles, giving public talks, and participating in radio and television programs. So I’ll leave that for other times and other places.

Read the Devlin’s Angle archive.