How We Stopped Learning Math

A slow-motion train wreck people have been warning about for centuries.

1. The Concise Book of al-Jabr and Muqābalah

One of the most successful math textbooks of all time (maybe second-most, after Euclid’s Elements) was written by al-Khwārizmī circa 820 CE. The language we use to talk about math pays homage to its impact. “Algebra” comes from the title. “Algorithm” comes from the author’s name. It starts by explaining digits, or what we now call “Arabic numerals,” not because they’re actually Arabic but because this book was. In Romance languages, the cognate to “algorithm” means digit, because al-Khwārizmī is who taught people what a digit was.

It starts with a concise author’s preface, which could be even more concisely summarized¹ like this:

I’ve tried to include only the easiest and most widely useful mathematical topics.

Science is always evolving. We can feel gratitude towards the science writers of the past while still being open to correcting their mistakes or improving their clarity. I hope future readers will have that attitude towards me.

To compare and contrast, here’s an excerpt from the preface to a modern al-Jabr textbook:

Progress in Mathematics, now in its sixth decade of user-proven success, is a complete basal mathematics program. Written by experienced teacher-authors, it integrates a traditional course of study and today’s academic Standards [sic] with the most up-to-date methods of teaching.

Progress in Mathematics is designed to meet the individual needs of all learners. Teachers who use Progress come to understand that students may progress as quickly as they can or as slowly as they must.

The chief similarity for me as an English speaker is that I’m not sure how to pronounce either title. Is Progress meant to be read as a verb or a noun? Either way, it’s clear in context that we’re talking about the progression of the individual student in mastering the art, not the progression of the art itself. Progress in Mathematics confidently announces itself to be “user-proven” and “complete.” This is everything you need to learn about math, presented in the best way possible, and you need to learn it all, no matter how long it takes.

The other difference that jumps out is that al-Khwarizmi’s preface is addressed to the student, while the modern preface is addressed to the teacher. In the modern textbook, the closest we get to a preface for the student is this:

Progress in Mathematics includes a “handbook” of essential skills, Skills Update, at the beginning of the text. These one-page lessons review skills you learned in previous years. It is important for you to know this content so that you can succeed in math this year.

If you need to review a concept in Skills Update, your teacher can work with you, using manipulatives, which will help you understand the concept better.

The use of the jargon word manipulatives, which in English means “things,” makes me doubt that even this is actually aimed at non-educators. But taken at face value, it’s notable that it doesn’t try to sell you on what you’re studying. “It is important for you to [learn math] so that you can [learn math].”

Al-Khwarizmi’s pitch is that he’s researched what math people actually use most often, and included only that. He doesn’t exhort you to make sure to learn everything in the book. Everything in it is useful for something, but some of those things are learning more advanced math, or digging canals, or calculating a worker’s salary. If you’re not expecting to need a section, obviously you just skip or skim it. His guide to determining when Jewish holidays fall this year was published separately.

The pitch in Progress is that they’ve accumulated sixty years of “user-proven success.” The users succeeding here are not canal-builders, proven to be better at designing canals after taking this course. Their definition of success is that children who study the curriculum can pass a test on the content of the curriculum.

Part of the difference here is cultural—the tendency of writers in the Abrahamic faiths, Islam in particular, towards humility, versus corporate-speak’s requirement to show total confidence at all times. But it’s mainly practical. There’s no longer a need to pitch algebra to the students—they’re being required to learn it. There’s not even a need to pitch it to the teachers—they’re being required to teach it. So why waste space?

All we need to do is remember to check in every so often to make sure what we’re teaching actually stays with the students. And, sure enough, we can see that…Wait. Uh-oh.

These are college students. They have spent, over their lives, about 3000 hours studying math. They’ve “succeeded,” even, at studying it. More than a quarter of students in that remedial class, reports Kelsey Piper, had gotten a perfect 4.0 GPA in high school math. All of them got into University of California-San Diego, which rejects 75% of its applicants. Our user-proven, state-of-the-art math education system worked perfectly for them, except for the part where you come out of it knowing any math.

2. The First And Ruling Defect

I keep seeing people respond to the UC story, and stories like it, by calling for stricter math education standards, in one way or another. I agree that the story is horrifying, but I’m going to advocate basically the opposite response.

This debate has a long history, but given that modern math classes, even in Baghdad, are typically taught in English, maybe I can get away with going back a mere three centuries to when people started really arguing about it in English.

On the pro-standardized-education side, here’s clergyman and social commentator John Brown in the mid-1700s, listing the reasons the Roman Republic was doomed from the start:

The first of these was the Neglect of instituting public laws, by which the education of their children might have been ascertained.

England, he wrote, was experiencing the same decline and fall into decadence and effeminacy, for many of the same reasons. Instead of modeling ourselves after Rome, he argued, we should follow the example of ancient Sparta, which was manly and strong and uniform and manly. English education was more similar to that of the weak and dissolute Athenians.

The first and ruling Defect in the Institution of this Republic seems to have been “the total Want of an established Education, suited to the Genius of the State.” There appears not to have been any public regular or prescribed appointment of this Kind beyond what Custom had accidentally introduced.
…
Parents were much at Liberty to do as seemed good to them. Hence a dissimilar and discordant System of Manners and Principles took Place; while some youthful Minds were imbued with proper and virtuous Principles, some with no Principles, and some with vicious Principles; with such as must therefore on the Whole tend to shake the Foundation of true Freedom.

(His second “ruling defect,” incidentally, was that too many people had the right to vote.)

The scientist Joseph Priestley rebutted clergyman Brown at length.

In conducting my examination of these sentiments, I shall make no remarks upon any particular passages in the book, but consider only the author’s general scheme, and the proper and professed object of it. And as the doctor has proposed no particular plan of public education, I shall be as general as he has been, and only shew the inconvenience of establishing, by law, any plan of education whatever.

Priestley’s core argument² was that establishing legal standards of education would make it harder for education to evolve over time. Education, like any science, benefits from having different people try different things so you can find out what works best. Education is hardly a mature art, he says, and it won’t get better without more data and intellectual diversity. (Which, of course, is why Athenian ideas have been so much more influential than Spartan ones—you tend to get more ideas when it’s legal to have ideas. Euclid was not from Sparta.)

Imagine, he writes, unintentionally breaking my heart, what would have happened if we’d had standardized education a few centuries ago, when everybody was wrong about everything. We’d be so much more ignorant today.

And for the same reason, were such an establishment to take place in the present age, it would prevent all great improvements in futurity.

3. The Past Warns; The Present Mourns

Priestley, of course, lost the argument. In some ways, the consequences were less dire than he predicted—educational philosophy has evolved. A little. But not always for the better. Brown worried that a bad education would harm a child’s morality, and Priestley worried that it would harm a child’s intellect. But, today, we worry that a bad education will harm a child’s education. If you don’t learn algebra, you can’t learn calculus. If you can’t learn calculus, you won’t be prepared for college. If you’re not successful in college, you’ll have trouble getting a good job.

Those are the sort of worries people express when they see a tweet about students not knowing how to solve 7+2 = □ + 6. Nobody’s saying “Oh no! Now when they encounter a white square in real life, they won’t know what number to draw on it!”

That’s one of the subtler dangers of uniformity—it fosters these closed, self-reinforcing loops. Employers could just stop paying attention to your college background, but it wouldn’t be wise, because “smarter people get into better colleges” is still a nigh-universal rule. Colleges could stop hyper-focusing on calculus, but when everybody entering college has been told that they already know calculus, how can you not build on that? Public school curricula could change, but then their kids wouldn’t get into good colleges.

The more blatant problem is that tomorrow’s elementary school math teachers come from today’s elementary school students. So if we accidentally stopped teaching actual math, how would we know?

Say we accidentally stopped teaching what a quadratic equation is and what it means to “solve” it and why you might want to. Somebody’s cat deleted a huge swath of text from the Quadratic Equations chapter. The chapter’s still there, but all that’s left is the sentence “x equals minus b plus or minus the square root of b squared minus four a c, all over 2 a,” and then a note about how that sentence tells you what arithmetic problems you need to solve when you see math questions formatted a certain way. Well, that’s a hard and tedious thing to memorize, but some kids would succeed at it, and those would be the ones teaching the next generation, not knowing anything was missing.

According to a widely-circulated 2002 essay by research mathematician and math teacher Paul Lockhart, popularly known as “Lockhart’s Lament,” that is very much not a hypothetical.

It’s a great read³, and it doesn’t pull any punches. In part, it’s Priestley’s argument again with the tenses changed.

The truly painful thing about the way mathematics is taught in school is not what is missing— the fact that there is no actual mathematics being done in our mathematics classes— but what is there in its place: the confused heap of destructive disinformation known as “the mathematics curriculum.” It is time now to take a closer look at exactly what our students are up against— what they are being exposed to in the name of mathematics, and how they are being harmed in the process.

The most striking thing about this so-called mathematics curriculum is its rigidity. This is especially true in the later grades. From school to school, city to city, and state to state, the same exact things are being said and done in the same exact way and in the same exact order. Far from being disturbed and upset by this Orwellian state of affairs, most people have simply accepted this “standard model” math curriculum as being synonymous with math itself.

This is intimately connected to what I call the “ladder myth”— the idea that mathematics can be arranged as a sequence of “subjects” each being in some way more advanced, or “higher” than the previous. The effect is to make school mathematics into a race— some students are “ahead” of others, and parents worry that their child is “falling behind.” And where exactly does this race lead? What is waiting at the finish line? It’s a sad race to nowhere. In the end you’ve been cheated out of a mathematical education, and you don’t even know it.

Lockhart argues that we should have more amateur mathematicians, in the way we have amateur artists and amateur musicians. Amateurs aren’t just a feeder system for the pros. Just as amateur artists make society more artistic, amateur mathematicians would make society more mathematical. Also, they’d be having fun, and fun is good.

4. Negative Predictive Weight

One might reasonably object that clearly some people still know what a quadratic equation is (me, maybe, or Lockhart). Those people can point out the problem and get the missing bits restored to the curriculum. So we’re hardly trapped, right? I must be overstating the problem.

Nah, I’ve been understating it. See, actual math is in fact useful for tasks other than passing tests and teaching math. One of those tasks is setting education policy. So once one generation grows up not knowing math (but thinking they do), we lose much of our ability to reform the system.

For example, let’s pick on UCSD some more. The whole UC system recently finished phasing out the use of SAT scores in their application process—admissions are now “test-blind.” The key decision was made in 2020, in a 23-0 vote by the California Board of Regents headed by Janet Napolitano. It followed a similar decision by California Governor Gavin Newsom to veto allowing the SAT to be used in public high schools. Napolitano and Newsom, who each have a Bachelor of Science in Political Science from Santa Clara University, gave similar explanations: the SAT simply wasn’t a good predictor of college success.

This surprising claim was backed by research, namely Geiser and Studley (2007), who collected data on UC students and ran a regression analysis.

They explained their results as follows.

As Table 5 demonstrates, the data provide no support for the hypothesis that the SAT I is a better predictor of freshman grades than the SAT II in certain academic disciplines than others [sic]. In fact, in the physical sciences and engineering, which are among the most competitive academic disciplines at UC, SAT I scores have negative predictive weight within a regression equation that simultaneously takes into account HSGPA and SAT II scores.

Amazing, right? Higher scores on the SAT I, half of which is a math test, actually mean you’re worse at math. It’s right there in the data. Or it looked to lead researcher Saul Geiser (PhD in Sociology, UC Berkeley) like it was.

No. This is so elementary a statistical error that it is literally a textbook example. When teaching Berkson’s Paradox (aka range restriction bias), people often use analyses of college success predictors just like this one. (Kind of telling that to give a “real world example” that students will relate to, our go-to example is from education.) It was one of my 26 examples when I explained it, incorrectly thinking it was a hypothetical straw man.

So yes, some people still know math. Some of them immediately spotted the problem and wrote about it. But then they died of old age. The past 20 years have consisted of Geiser’s claim being repeated as fact in mainstream media while people like Freddie DeBoer scream ineffectually at them. See, you also need to know math in order to know which math is the good math. Even if some expert told you that Geiser’s paper has a range restriction error, if you then went to the paper and searched for the words “range restriction,” you’d find a footnote explaining how it’s totally fine in this case.⁴ The explanation is nonsense, verging on gibberish, but it sure looks mathy.

The point here isn’t really about the SAT. The point is that the process by which California educators decide how to structure college admissions is broken, because every major figure involved is a product of California’s standardized math programs.

5. Why We Should Start Teaching Math Again

Fun fact! There are areas of policy other than educational policy! They also require math in order to be done well! They are not currently being done well!

Say you’re picking the next director of the Center for Biologics Evaluation and Research, the branch of the FDA in charge of approving new gene therapies, among other things. Who looks better to you?

1. Vinay Prasad, oncologist and professor of epidemiology and biostatistics at UCSF, who has published many papers in major journals that look like this:

   A total of 187 trials leading to 176 approvals for 75 distinct novel anticancer drugs by the FDA were evaluated. Sixty-four (34%) were single-arm clinical trials, and 123 (63%) were randomized clinical trials. A total of 125 (67%) had at least 1 limitation in the domains of interest; 60 of the 125 trials (48%) were randomized clinical trials. Of all 123 randomized clinical trials, 37 (30%) lacked overall survival benefit, 31 (25%) had a suboptimal control, and 17 (14%) used crossover inappropriately.

2. Peter Marks, oncologist and private clinical researcher, who publishes papers that look like this:

   During the next pandemic, there will be an impetus to test CP in RCTs, but such trials should not commence until there is information on dosage and optimal timing of antibody therapy since it is impossible to design good studies without that knowledge (25). Information on optimal dose and timing can come from registries. Furthermore, given a safety record extending over more than one century, the deployment of CP should not be delayed since it is likely to be the only therapy available in the early days of a future emergency. The argument that deployment of CP without RCT data will preclude the completion of such trials was refuted by the experience in the United States where the CONTAIN trial was completed even when CP was available under Emergency Use Authorization (26).

You don’t need to be a full-on biostatistician to know which paper is making an actual argument, and which paper is just assuming the conclusion in a really complicated way. You just need to not have your eyes glaze over when you try to skim through them, a test that almost everybody fails. If you read Prasad’s work critically, you will notice that at no point does he compare two comparable numbers. You are not actually doing statistics if your conclusion doesn’t rely, ultimately, on observing that X > Y⁵. Marks isn’t doing statistics here either, but he’s also not pretending to. He doesn’t think he is. He’s making a qualitative argument.

The topic of both of those linked papers is whether we should authorize experimental treatments before conducting or completing randomized controlled trials under certain conditions. Marks, who held the position for a long time, is arguing the pro side. Prasad, who took over a few months ago after Marks was forced out, is…well…he’s killing people for incoherent reasons and sounding very mathy while he does it.

You’ll probably have to take my word for that. And if I’m wrong, I’ll probably never know.

This example shows that our problem isn’t really that our math education standards aren’t high enough. In order to avert a Prasad using standardized education, we would need to build a world where most people can read biostatistics papers with a critical eye. This is not a realistic goal. It’s not even really a desirable goal! Most people have better things to do.

The actual fix would involve having more people learning math, and fewer people learning “math.” If our future statisticians were getting a better math education, there would be fewer bad ones. In an ecosystem with more amateur mathematicians, we’d have an easier time spotting the bad ones.

6. Can we get there from here?

The present day has lost the meaning of algebra. Literally. We’re not sure what al-Jabr originally translated to. But our best-guess translation of al-Jabr and Muqābalah is something like “restoring the broken and finding the balance.”

We can’t get better at teaching kids math right now. There are not enough adults who know math, and fewer still who have both that knowledge and all of the other stuff you need to be a good teacher.

We can’t even get systematically better at teaching math teachers math. In a study published in 2016, researchers looked at what happens to students when their K-8 math teacher takes 93 hours of math class. According to their metrics, anyway, the teachers ended up learning more math and being better teachers. The richness of their math content improved dramatically. What didn’t improve, at all, were the students’ test scores. Good luck persuading decision-makers that the problem is with the tests, not your expensive personal development program.

Here’s what we can do. Though we can’t start teaching math yet, we can stop teaching not-math. That’s not quite the same thing, but maybe it’ll help. It’s often useful, as an intermediate step, to set one term of an equation to zero. There is no net benefit to standardized math classes, at least ones that go beyond the first page of al-Khwārizmī’s book. They are resulting in a general population that literally hates math. That is much, much worse than nothing.

The only educational standard we should have is that every student has the opportunity to fall in love with various kinds of mathematics, as a future career or as a hobby. There shouldn’t be any demographic, any school district, where nobody learns algebra. That’s an achievable and measurable goal.

It’s when we tell school districts that all of their students should learn algebra that we’re making a colossal mistake. They’re not teaching algebra. They’re teaching kids to hate math, or how to fake it.

1. From the preface as translated by Friedrich Rosen in 1831:

The learned in times which have passed away, and among nations which have ceased to exist, were constantly employed in writing books on the several departments of science and on the various branches of knowledge, bearing in mind those that were to come after them…

Some applied themselves to obtain information which was not known before them, and left it to posterity; others commented upon the difficulties in the works left by their predecessors, and defined the best method (of study), or rendered the access (to science) easier or placed it more within reach; others again discovered mistakes in preceding works, and arranged that which was confused, or adjusted what was irregular, and corrected the faults of their fellow-labourers, without arrogance towards them, or taking pride in what they did themselves.

That fondness for science, by which God has distinguished the Imam al Mamun…has encouraged me to compose a short work…confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law-suits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned—relying on the goodness of my intention therein, and hoping that the learned will reward it, by obtaining (for me) through their prayers the excellence of the Divine mercy…

2

Education is as much an art (founded, as all arts are, upon science) as husbandry, as architecture, or as ship-building. In all these cases we have a practical problem proposed to us, which must be performed by the help of data with which experience and observation furnish us. The end of ship-building is to make the best ships, of architecture the best houses, and of education, the best men. Now, of all arts, those stand the fairest chance of being brought to perfection, in which there is opportunity of making the most experiments and trials, and in which there are the greatest number and variety of persons employed in making them. History and experience show, that, cæteris paribus, those arts have always, in fact, been brought the soonest, or the nearest to perfection, which have been placed in those favourable circumstances. The reason is, that the operations of the human mind are slow; a number of false hypotheses and conclusions always precede the right one; and in every art, manual or liberal, a number of awkward attempts are made, before we are able to execute any thing which will bear to be shown as a master-piece in its kind; so that to establish the methods and processes of any art, before it have arrived to a state of perfection (of which no man can be a judge) is to fix it in its infancy, to perpetuate every thing that is inconvenient and awkward in it, and to cut off its future growth and improvement. And to establish the methods and processes of any art when it has arrived to perfection is superfluous. It will then recommend and establish itself.

…

To shew this scheme of an established method of education in a clearer point of light, let us imagine that what is now proposed had been carried into execution some centuries before this time. For no reason can be assigned for fixing any mode of education at present, which might not have been made use of, with the same appearance of reason, for fixing another approved method a thousand years ago. Suppose Alfred, when he founded the university of Oxford, had made it impossible, that the method of instruction used in his time should ever have been altered. Excellent as that method might have been, for the time in which it was instituted, it would now have been the worst method that is practised in the world. Suppose the number of the arts and sciences, with the manner of teaching them, had been fixed in this kingdom, before the revival of letters and of the arts, it is plain they could never have arrived at their present advanced state among us.

SALVIATI: We are losing so many potentially gifted mathematicians— creative, intelligent people who rightly reject what appears to be a meaningless and sterile subject. They are simply too smart to waste their time on such piffle.

SIMPLICIO: But don’t you think that if math class were made more like art class that a lot of kids just wouldn’t learn anything?

SALVIATI: They’re not learning anything now! Better to not have math classes at all than to do what is currently being done. At least some people might have a chance to discover something beautiful on their own.

SIMPLICIO: So you would remove mathematics from the school curriculum?

SALVIATI: The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas.

SIMPLICIO: But how many math teachers know enough about their subject to teach it that way?

SALVIATI: Very few. And that’s just the tip of the iceberg…

3. Here’s his summary of the current state of standardized algebra education.

ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation. The smooth narrative thread that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no characters, plot, or theme. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.

Statistical techniques can be used to “correct” observed correlations for range restriction using national data on SAT takers, but those techniques depend on assumptions that cannot be directly verified, such as the assumption that the relationship between test scores and college grades is linear and identical across the observed and unobserved ranges of the data, or that the conditional variance is homogeneous across the observed and unobserved ranges. Moreover, for purposes of comparing the predictive validity of the SAT I and SAT II, there is no straightforward method to “correct” the observed relationships, since the national populations of SAT I and SAT II takers are so different (the population of SAT II takers is much smaller and includes a much larger proportion of high-achieving students than the SAT I population). Among students who enroll at UC, in contrast, the variances of SAT I and SAT II scores are very similar (see www.ucop.edu/sas/researchandplanning), and range-restriction effects therefore do not account for the predictive superiority of the SAT II shown in the UC data. For all of these reasons, it is appropriate to use “uncorrected” regression data when comparing the relative predictive power of the SAT I and SAT II at UC, and only observed statistical relationships are presented in this paper.

4. No matter your level of math knowledge, the only correct response to a paragraph like this is one that would get you a bad test score: “Huh? I don’t understand.” Maybe you could get extra credit by trying on your own to figure out whether it was true that similar variances among two variables meant you could safely ignore range restriction bias. You could create a toy example: “Suppose I had a dataset that consisted only of people whose height in inches, added to their age in years, was between 80 and 100. That constraint could cause there to be similar variances in each variable within the test data, even though age is higher-variance in general, because it would reduce the age variance by filtering out young people and old people. And that dataset would probably show a strong negative correlation between age and height—the average 20-year-old would be 70 inches tall, while the average 30-year-old would be only 60 inches tall. Ergo, the average person shrinks ten inches between the ages of thirty and forty.”

5. A hypothetical valid version of Prasad’s argument could include stats showing that approvals based on one study design are more likely to be reversed later than approvals based on another. Or, even better, present a model wherein the expected deaths from approving too many experimental treatments are higher than the expected deaths from delaying approval of experimental treatments for years until RCTs can confirm the observational data. But I dream.