On making omelets and learning math

As the old saying goes, “You can’t make an omelet without breaking eggs.” Similarly, you can’t learn math without bruising your ego. Learning math is inescapably difficult, frustrating, and painful, requiring high tolerance of failure. Good teachers have long known this, but the message has never managed to get through to students and parents (and it appears, many system administrators who evaluate students, teachers, and schools).

The parallel (between making omelets and learning math) plays out in the classroom in a manner that many students and parents would find shocking, were they aware of it. It’s this.

All other factors being equal, when you test how well students have mastered course material some months or even years after the course has ended, students who do well in courses, getting mostly A’s on assignments and exams, tend to perform worse than students who struggled and got more mediocre grades at the time.

Yes, you read that correctly, the struggling students tend to do better than the seemingly “good” students, when you see how much they remember, and how well they can perform, months or even years later.

There is a caveat. But only one. This is an “all other things being equal” result, and assumes in particular that both groups of students want to succeed and make an effort to do so. I’ll give you the lowdown on this finding in just a moment. (And I will describe one particular, highly convincing, empirical demonstration in a follow-up post.) For now, let’s take a look at the consequences.

Since the purpose of education is to prepare students for the rest of their lives, those long term effects are far more important educationally than how well the student does in the course. I stressed that word “educationally” to emphasize that I am focusing on what a student learns. The grade a student gets from a course simply measures performance during the course itself. 

If the course grade correlated positively with (long-term) learning, it would be a valuable measure. But as I just noted, although there is a correlation, it is negative.  This means that educators and parents should embrace and celebrate struggle and mediocre results, and avoid the false reassurance of progress that is so often the consequence of a stellar classroom performance. 

Again, let me stress that the underlying science is an “all other things being equal” result. Assuming that requirement is met, a good instructor should pace the course so that each student is struggling throughout, constantly having to spend time correcting mistakes.

The simple explanation for this (perhaps) counter-intuitive state of affairs is that our brains learn as a result of trying to make sense of something we find puzzling, or struggling to correct an error we have made. 

Getting straight A’s in a course may make us feel good, but we are actually not learning something by so doing; we are performing. 

Since many of us discover that, given sufficient repetitive practice, we can do well on course assignments and ace the final exam regardless of how well we really understand what we are doing, a far more meaningful measure of how well we have learned something is to test us on it some time later. Moreover, that later test should not just be a variant of the course final exam; rather we should be tested on how able we are in making use of what we had studied, either in a subsequent course or in applying that knowledge or skills in some other domain.

It is when subjected to that kind of down-the-line assessment that the student who struggled tends to do better than the one who performed well during the course.

This is not just some theoretical idea, removed from reality. In particular, it has been demonstrated in a large, random control study conducted on over 12,000 students over a nine-year period.

The students were of traditional college age, at a four-year institution, and considerable effort was put in to ensuring that all important “all other things being equal” condition was met. I’ll tell you about the study and the institution where it was carried out in a follow-on post to this one. For now, let’s look at its implications for math teaching (for students of all ages).

To understand what is going on, we must look to other research on how people learn. This is a huge topic in its own right, with research contributions from several disciplines, including neurophysiology.

Incidentally, neurophysiologists do not find the negative-correlation result counter-intuitive. It’s what they would expect, based on what they have learned about how the brain works. 

To avoid this essay getting too long, I’ll provide an extremely brief summary of that research, oriented toward teaching. (I’ll come back to all these general learning issue in future posts. It’s not an area I have worked in, but I am familiar with the work of others who do.) 

Learning occurs when we get something wrong and have to correct it. This is analogous to the much better known fact that when we subject our bodies to physical strain, say by walking, jogging, or lifting weights, the muscles we strain become stronger—we gain greater fitness.

The neurophysiologists explain this by saying that understanding something or solving a problem we have been puzzling over, is a consequence of the brain forming new connections (synapses) between neurons. (Actually, it would be more accurate to say that understanding or solving actually is the creation of those new connections.) So we can think of learning as a process to stimulate the formation of new connections in our brain. (More accurately, we should think of learning as being the formation of those new connections.)

Exactly what leads to those new connections is not really known—indeed, some of us regard this entire neurons and synapses model of brain activity as, to some extent, a scientific metaphor. What is known is that it is far more likely to occur after a period in which the brain repeatedly tries to understand something or to solve the problem, and keeps failing. (This is analogous to the way muscles get stronger when we repeatedly subject them to strain, but in the case of muscles the mechanism is much better understood.) In other words, repeatedly trying and failing is an essential part of learning.

In contrast, repeatedly and consistently performing well strengthens existing neuronal connections, which means we get better at whatever it is we are doing, but that’s not learning. (It can, however, prepare the brain for further learning.) 

Based on these considerations, the most effective way to teach something in a way that will stick is to put students in a position of having to arrive at the best answer they can, without hints, even if it’s wrong. Then, after they have committed, you can correct, preferably with a hint (just one) to prompt them to rectify the error. Psychologists who have studied this refer to the approach as introducing “desirable difficulties.” Google it if you have not come across it before. The term itself is due to the Stanford psychologist Robert Bjork. 

For sure, the result of this approach makes students (and likely their parents and their instructor) feel uncomfortable, since the student does not appear to be making progress. In particular, if the instructor gauges it well, their assignment work and end-of-term test will be littered with errors. (Instructors should grade on the curve. I frequently set the pass mark around 30%, with a score of 60% or more correct getting an A, though in an ideal world I would have preferred to not be obliged to assign a letter grade, at least based purely on contemporaneous testing.)

Of course, the students are not going to be happy about this, and their frustration with themselves is likely to be offloaded onto the instructor. But, for all that it may seem counterintuitive, they will walk away from that course with far better, more lasting, and more usable learning than if they had spent the time in a feelgood semester of shallow reinforcement that they were getting it all right. 

To sum up: Getting things right, with the well-deserved feeling of accomplishment it brings, is a wonderful thing to experience, and should be acknowledged and rewarded—when you are out in the world applying your learning to do things.  But getting everything right is counterproductive if the goal is meaningful, lasting learning. 

Learning is what happens by correcting what you got wrong. Indeed, the learning is better if the correction occurs some time after the error is made. Stewing for a while in frustration at being wrong, and not seeing how to fix it, turns out to be a good thing. 

So, if you are a student, and your instructor refuses to put you out of your misery, at least be aware that the instructor most likely is doing so because they want you to learn. Remember, you can’t learn to ride a bike or skateboard without bruising your knees and your elbows. And you can’t learn math (and various other skills) without bruising your ego. 

Cracking your ego is an unavoidable part of learning.

What topics should be covered in school mathematics?

On May 25, Jo Boaler and I had a public conversation at Stanford about K-12 mathematics. (An edited video recording will be made available on the youcubed and SUMOP websites as soon as it is ready.) Our conversation was live-tweeted by @AliceKeeler, and that led to a lively twitter debate that lasted several days, with much of the focus on what topics should be taught.

Having been a professional mathematician for close on fifty years (first in pure mathematics, then in the world of business and government service), my take, which I articulated in my conversation with Jo and you can find in the Blue Notepad videos on this site, is somewhat unusual. I actually believe it is (in many ways, but not all) not important what is taught, but rather how it is taught.

My recent post in my monthly Devlin’s Angle for the Mathematical Association of America explains why I have that view. In a follow-up post next month, I will connect the argument I present this month with the discussion Jo and I had and the ensuing twitter debate.

New video posted in the VIDEOS section

The interview was recorded in February last year, in Belgrade, Serbia, but was only recently published on Youtube. You can find it on the site VIDEOS page. The accompanying text translates more or less as follows:

“An exclusive interview given in February last year to the 12th edition of the Elements magazine, published by the Center for the Promotion of Science.
In a comprehensive interview with Aleksandar Ravas and Tijana Markovic, published in the 12th Element, Devlin spoke about mathematical thinking, mathematics education, predicting the future, the difference between false stories in mathematics and the reality, and many other interesting and socially engaged topics.”

New podcast on mathematical outreach

BBVA tweet, May 9, 2019

The latest podcast in the “We learn together” (Aprendemos juntos) series sponsored by the BBVA (Banco Bilbao Vizcaya Argentaria, a multinational bank) features an interview they recorded with me a few months ago. This is actually the audio track from a video that will be published shortly.

Much of the interview focuses on my work on mathematical outreach, including what led me to devote so much effort to that enterprise.

— KD

The Misconceptions About Math That Are Keeping Students From Succeeding

That’s not my title above. It’s the headline to an article published by Forbes two days ago. It’s a must read for all students, teachers and parents.

SUMOP addresses the biggest of the myths. (Actually, all of them, but the biggest one jumps right out.) Not by saying it is myth. Not by explaining why it’s a myth. Rather, by showing it’s a myth. With relevant, contemporary, real-world examples.

Pacioli’s “Summa” in San Francisco

Setting up to give the talk under the watchful eye of Picasso

On April 24, SUMOP visited San Francisco’s new arts complex in the old dockland area to give a short talk about the background and importance of Luca Pacioli’s 1494 book Summa de arithmetica, geometria, proportioni et proportionalita. The occasion was a public showing of one of the few remaining first edition copies of the book by Christie’s, who are auctioning the book in New York in June. Giving a talk with a life-sized painting of Picasso behind me and original paintings by Renoir and Monet to my left (the total estimated value of all three at auction is $27M) was a new experience for me.

But I got an even greater thrill from spending a few moments before the event taking a close look at the book that initiated the modern commercial world by providing business leaders and financiers with the modern accounting methods necessary to run large organizations. Without the methods described in Pacioli’s book (which Christie’s expects to fetch between $1M and $1.5M), there would not be a financial system to give those paintings their much greater financial value. (Clearly, while I share with many the appreciation of fine art, the value I put on brilliant mathematical innovations is not mainstream.)

For a greatly expanded, written version of my talk, see the forthcoming May 1 issue of my regular Devlin’s Angle commentary for the Mathematical Association of America. (The April 1 post had a preview of the Christie’s event.)

Here are a few photos from the event. Two of the pictures show me examining one of the examples of double-entry bookkeeping in the textbook that first taught that method to the world.