SEDSO Lecture overview

What is School Mathematics?

Hung-Hsi Wu’s answer, and why he thinks the question still needs asking

When
Friday, March 13, 2026
Event
EMS Education Webinar — the third lecture in the European Mathematical Society’s education series
Speaker
Hung-Hsi Wu, professor emeritus of mathematics, University of California, Berkeley
Sources
The recording, and Wu’s own written summary of the talk dated September 20, 2026

A mathematician who spent two decades writing the textbooks he thought were missing asks a question that sounds too simple to need an hour: what, exactly, is the mathematics we teach in school? His answer is that what American schools have taught for seventy years does not deserve the name, that two well-meant reforms failed because nobody had written down the alternative, and that he has now spent 2,402 pages writing it down.

Hung-Hsi Wu Professor emeritus of mathematics at the University of California, Berkeley. He has written on school mathematics and the preparation of its teachers since the 1990s, and from 2000 onwards on the six volumes described below. His page at Berkeley

About this page. This is SEDSO’s overview of the talk, not a transcript of it. The argument, the sequence and the examples are Wu’s; the summarising prose is ours. Passages in quotation marks are his words, taken from the written summary he published alongside the webinar. Where you want his full case, watch the recording or read his summary — both say it better and at greater length.

Words from the SEDSO 300 list are linked, lightly, once each, in the argument below.

Go to the source

The talk itself is on YouTube, hosted by the European Mathematical Society. Wu also wrote the argument out in full, and that paper is the source of every quotation on this page.

Watch the webinar Read Wu’s summary PDF, 16 pages

In brief

Wu names the thing he is arguing against: TSM, Textbook School Mathematics, the mathematics actually found in standard American school textbooks. His charge is that it has little or no mathematical integrity — it defines almost nothing, explains almost nothing, contradicts itself between grades, and never says why any of it is worth learning. He walks through four places this shows: the standard algorithms taught as rote, fractions defined three incompatible ways, the pretence that fractions and whole numbers are unrelated, and a high school geometry course that abandons its own earlier definitions and asks students to prove the obvious. Three reforms since 1950 tried to do better. The New Math overcorrected into abstraction; the NCTM and Common Core reforms asked for coherence and reasoning but left teachers no account of what those words meant in practice. His remedy is mathematical engineering: customise university mathematics, faithfully, until school students and their teachers can use it. What results, if it satisfies five stated principles, he calls PBSM — and his six volumes exist to prove such a thing can be written at all.

The argument

The question, and why Klein did not settle it

The title sounds almost too simple to be worth an hour. Wu begins by granting that: one’s first instinct is to look up Felix Klein, who spent years on what he called the double discontinuity between school mathematics and university mathematics. But Klein, Wu says, was so absorbed in the mathematical preparation of teachers that all he offered on the question itself was this:

If we educate better teachers, then mathematics instruction will improve by itself, as the old consigned form will be filled with a new, revitalized content!

Felix Klein, quoted by Wu

Wu calls this an early version of the American “intellectual trickle-down theory” — the hope that good teachers will produce good school mathematics on their own. He is less optimistic, so he sets out to answer the question directly.

Three reasons to ask it now

  1. The mathematics in American schools has been unlearnable for at least seven decades. Are we willing to call that school mathematics?
  2. The two most recent reforms both assumed school mathematics could be “something better,” but described that better thing only in the barest outline. How do we know it can exist at all?
  3. Education research needs something correct and learnable to study. The unlearnable version has ruled for so long that it is nearly the only mathematics many researchers in the field have ever seen.

TSM: what the textbooks actually teach

Wu gives the thing a name. The mathematics in standard American school textbooks he calls Textbook School Mathematics, or TSM. He then introduces the standard he will measure it against: mathematical integrity, meaning roughly the quality of “being precise, clear, logical, coherent, and always supported by reasoning.”

His summary judgement is one sentence: TSM has little or no mathematical integrity. The reasons he lists are that TSM has few definitions that make sense, so students never learn precisely what they are supposed to learn; that it is largely a collection of unexplained and disconnected procedures, which the human memory is poorly equipped to retain; and that its imprecision confuses students.

Four places TSM breaks down

The standard algorithms. TSM teaches the four algorithms for whole numbers as disconnected skills to be memorised by rote, with no reasoning. What children take away is that mathematics means memorising facts and taking orders. The reasoning that is withheld from them is elegant: the algorithms reduce any whole-number computation, however large, to computations with single-digit numbers. Wu calls this the reduction of the complex to the simple, and notes that it also explains why the multiplication table is worth committing to memory — a question TSM never answers.

Fractions. TSM defines a fraction as “part of a whole,” or “like a piece of pizza,” or a ratio — sometimes all three at once, which he says makes it three times more incomprehensible. None of these tells a child that a fraction is a number, like 8 or 27. And a fraction is the first real abstraction children meet: they can find a whole number on their fingers, but where do they find seven-thirteenths? That is precisely why it needs a definition in terms of what they already know, rather than an analogy.

Coherence. TSM presents fractions as utterly different from whole numbers, so the four operations get taught twice, as unrelated skills. This is false — the operations on fractions are conceptually the same as those on whole numbers — and it raises anxiety for no reason. The same failure recurs with length, area and volume, which are basically the same idea.

High school geometry. Before high school, TSM defines congruence as “same size and same shape.” In the geometry course those definitions are quietly abandoned and replaced, but only for triangles. What congruence means for two ellipses, TSM does not say. Meanwhile students who were never shown a proof of “invert and multiply” are suddenly required to prove everything, including that if AB = CD then AC = BD — a fact a glance confirms, for which the accepted proof runs to seven steps in two columns. What they learn is not reasoning but how to fill in blanks.

TSM has to go.

Hung-Hsi Wu

Two demands, and three reforms that missed

To judge any version of school mathematics, Wu proposes two tests. The substance must be consonant with mathematical integrity. The level of presentation must suit school students. TSM, he argues, ignores the first and pursues the second ruthlessly, stripping mathematics down to procedures and banishing reasoning along the way.

Three reforms have tried to do better since 1950. The New Math of 1957, launched in the panic after Sputnik, was run by mathematicians who did the opposite of TSM: they pressed the first demand at the expense of the second, handing elementary students sets, number bases and symbolic logic. It had all but vanished by the early 1970s. The NCTM reform of 1989 and the Common Core of 2010 both called for conceptual understanding, coherence, and reasoning and proof — but issued standards that said what to teach only in outline. In a world where everyone had been raised on TSM, Wu writes, few teachers or administrators knew what those phrases even meant. They needed mathematical help, and the reforms did not supply it.

Mathematical engineering

His proposed route out borrows from engineering. Electrical engineers customise Maxwell’s equations into cell phones; civil engineers customise Newtonian mechanics into bridges. The same can be done with mathematics: take university mathematics and customise it, faithfully, until it is consumable by school students and their teachers.

His worked example is the rational numbers. University mathematics builds them from the integers as a quotient field, which is far too abstract for a classroom. The engineering decision is to put the integers on a number line as equally spaced points, then add the fractions as the points that subdivide the gaps evenly. Same mathematics, different delivery.

A tenet of engineering is that it must stay faithful to the abstract theory, or bridges collapse. Mathematical engineering is no exception — which is why “mathematical integrity” has to be defined exactly rather than gestured at.

The five principles

This is the heart of the talk, and the most portable part of it. A version of school mathematics has mathematical integrity, Wu says, if it satisfies five fundamental principles:

  1. Every concept is precisely defined, so students know what they are learning.
  2. Every claim is supported by logical reasoning, so students see why it is true.
  3. The language is precise, so the intended meaning cannot be mistaken.
  4. The concepts and skills are shown to be parts of a coherent whole.
  5. The purpose of learning each concept or skill is consistently made clear.

TSM, he argues, violates all five, and he gives an instance of each: “fraction” has no definition; nobody explains why long division works; “same size and same shape” is offered as a definition of congruence; the high school geometry course sticks out from the rest of the curriculum; and no one says why the four standard algorithms are worth learning at all.

PBSM, and six volumes

A version of school mathematics that meets both demands he calls PBSM — Principles-Based School Mathematics. He is careful about what is and is not known here: whether PBSM exists as a coherent body of knowledge across all of K-12, and whether it can survive contact with real classrooms, were open questions. Both the NCTM and Common Core reforms, he argues, quietly assumed it was already out there and merely needed nudging into place.

So he wrote it. Beginning in 2000, and finishing in 2020, Wu published six volumes covering K-12 — 2,402 pages in all, from the American Mathematical Society — each essentially the lecture notes of a professional development course for teachers in one grade band. Their main claim, in his words, is to furnish the first existence proof of PBSM in grades K-12.

He is the first to say this is a beginning rather than an end. Getting PBSM into classrooms is the arduous part, and it will be slow and expensive. He hopes the volumes draw criticism — are they really sound? is the level pitched too high? can they survive the arrival of AI in K-12? — and, more than that, he hopes competing versions of PBSM appear.

His answer

The third reason for asking the question returns at the end, and it is the sharpest. Because TSM has been almost the only school mathematics ever discussed in American schools of education, research on it becomes, in his phrase, an exercise in refuse-reclamation: scholarly effort spent making a bad product slightly more palatable.

Students should be fed healthy food, not reclaimed scraps from garbage piles.

Hung-Hsi Wu

Worse, he argues, studying TSM seriously lends it legitimacy, leaving teachers and students with the impression that it is simply a regrettable fact of life. On that view the NCTM reform of 1989 has an unquestioned merit: it was a revolt begun inside the inner circle of mathematics education. He hopes it finishes the job.

And so his answer to the question in the title, which he concedes may be anticlimactic by this point: “School Mathematics” should be PBSM.