You’ve done the workshops. You’ve bought the programs. You’ve built the pacing guides, scheduled the coaching cycles, and launched the new initiative with a strong September push.

And yet, somewhere around November, you feel it start to slip.

If you lead mathematics across a school or district, you already know that teaching math is hard. What’s less talked about is why it’s so hard — and why your hardest, most well-intentioned improvement efforts so often fail to stick.

Decades ago, mathematician Hans Freudenthal described the math most students experience in school as the “fossilized remains” of real reasoning — the dried-out skeletons of thinking, stripped of the living process that created them. Years later, Alan Schoenfeld picked up that thread in a paper we’ve referenced often on our podcast: Why Are Learning and Teaching Mathematics So Difficult?

Schoenfeld’s answer is structured as three expanding circles, moving outward from the individual student to the world around them:

1. Thinking Mathematically — what it actually means to do math
2. The Learning Environment — the classroom conditions that let students think
3. The Cultural Surround — the systems, policies, and inequities that shape everything inside

If you read it as a leader, something becomes clear that the paper only hints at: every one of these three difficulties is, at its core, a systems problem. And systems problems can’t be solved with one more initiative, one more PD day, or one more passionate person carrying the work on their back.

They’re solved by building a flywheel.

Let’s unpack Schoenfeld’s three circles, and then look at how each one points toward the same conclusion — that sustainable math improvement is designed, not willed into existence.

What Schoenfeld’s Research Reveals About Why Math Improvement Stalls

Before we get into the three circles, it’s worth naming the trap most leaders fall into.

When math results disappoint, the instinct is to find something new. A new resource. A new framework. A new focus for the year. We pile it onto teachers’ plates because we genuinely can’t tell which of last year’s ingredients made a difference — and we’re terrified that if we remove something, scores will drop.

The result is a system that is perpetually busy but rarely moving forward. As W. Edwards Deming put it, “The system you have is perfectly designed to get the results you’re getting.” If your district keeps restarting every September, that’s not a motivation problem. It’s a design problem.

Schoenfeld’s three circles help explain exactly where those design flaws live.

Circle One: Thinking Mathematically — The Problem Isn’t What Teachers Cover, It’s What They Understand

Schoenfeld’s first and innermost circle is about the nature of mathematical thinking itself. His argument is that most school math teaches students to memorize the products of reasoning — the procedures, definitions, and calculations — while omitting the reasoning that produced them.

He identifies four things that matter when someone actually does mathematics:

• Resources — content knowledge, mathematical processes, and practices
• Problem-solving strategies — the Pólya-style heuristics for tackling unfamiliar problems
• Metacognition — monitoring and regulating your own thinking
• Belief systems — whether a student believes math is sense-making they can participate in

Here’s the uncomfortable question for leaders: how many of those four are actually visible in the classrooms you walk into? For most systems, instruction lives almost entirely in the first bucket — and even then, mostly in content coverage, with mathematical practices treated as an afterthought “if time allows.”

Why This Is a Leadership Problem, Not Just a Teaching Problem

It’s tempting to read Schoenfeld’s first circle as a teacher development issue: teachers need stronger content and pedagogical knowledge. That’s true. But the leadership insight runs deeper.

You cannot ask teachers to teach mathematics as sense-making if they themselves haven’t experienced math as sense-making. Schoenfeld is blunt about this — teachers “may not have had opportunities to develop such understandings themselves.” The supports for building those understandings, he notes, are “few and far between.”

This is where so many improvement plans quietly fracture. A district prioritizes “problem-based learning” or “discourse,” but the professional learning focuses on strategies and routines — the visible moves — without deepening teachers’ own understanding of the mathematics underneath. The result is what we call pedagogy-first fragility: teachers adopt the moves, but because the understanding is shallow, the practice collapses the moment the coach leaves or the year resets.

Sustainable improvement requires building teacher capacity through conceptual understanding, not compliance. That’s not a one-day event. It’s a system — one layer of support connecting to the next, with growth compounding over time rather than fading.

Circle Two: The Learning Environment — Five Dimensions That Don’t Survive Without a System

Schoenfeld’s second circle widens the lens from the individual mind to the classroom environment. As he puts it, mathematics classrooms are “the primary locales in which people’s mathematics identities are shaped.”

Drawing on his Teaching for Robust Understanding (TRU) framework, he identifies five dimensions that characterize classrooms where students emerge as knowledgeable, flexible, resourceful thinkers:

1. The Mathematics — the richness of the content and practices students engage with
2. Cognitive Demand — opportunities for productive struggle in the zone of proximal development
3. Equitable Access to Content — every student meaningfully engaged, not just a confident few
4. Agency, Ownership & Identity — students making the math their own and seeing themselves as mathematicians
5. Formative Assessment — student thinking aired publicly, with the classroom responding adaptively

Schoenfeld is honest about how demanding this is. Orchestrating a classroom that does well on all five dimensions is, in his words, “an extremely demanding task” — and that’s true even under supportive conditions like well-designed curricula, aligned assessment, effective PD, and consistent policy.

The Hidden Word in That Sentence Is “Aligned”

Read that list of supportive conditions again: well-designed curricula, aligned assessment, effective PD, consistent policy messages. Schoenfeld is describing a coherent system — and quietly admitting that most teachers don’t have one.

This is the heart of what we see in district after district. It’s rarely that a system lacks effort or good ideas. It’s that the pieces don’t connect. One school is deep into Building Thinking Classrooms. Another is focused on number talks. A third is mid-way through an assessment redesign. Everyone is working hard. Nobody is working in the same direction.

We call this systemic misalignment — and it’s why the most common barrier we hear from teachers isn’t resistance. It’s “I don’t have time.” But that statement usually means something more precise: “I don’t see how this helps enough to make it worth my time… yet.” That’s a signal that the system around the teacher is asking them to spread thin across competing priorities rather than go deep on a shared one.

Schoenfeld’s five dimensions can’t be sustained by individual teachers heroically holding them up classroom by classroom. They require a system where the curriculum, the PD, the coaching, and the walkthroughs all reinforce the same small set of priorities — so that growth compounds instead of scattering.

Circle Three: The Cultural Surround — Why Improvement Collapses When People Leave

Schoenfeld’s third and outermost circle is the cultural surround: the recognition that classrooms are microcosms of society. Two forces from this outer ring press inward on every classroom.

The first is resource allocation. Schools serving the highest concentrations of historically marginalized students consistently face fewer credentialed teachers, fewer resources, and dramatically higher turnover. Schoenfeld cites a striking figure: teacher turnover rates are 90% higher in the top quartile of schools serving students of color than in the bottom quartile for math and science teachers.

The second is societal bias, which shapes how students get positioned — by peers and teachers alike — and therefore which students get access to rich mathematics and which get sidelined.

For leaders, the third circle surfaces two barriers that quietly undermine improvement plans: a measurement problem and a sustainability problem.

The Measurement Problem: Steering With a Fogged Windshield

Schoenfeld notes that assessment too often “fails to focus on thinking” — and the same is true at the system level. Most districts measure improvement with one instrument: the year-end standardized test.

But those scores arrive late, disconnected, and clouded by variables outside your control — different student cohorts, inconsistent test items, the natural lag between instruction and assessment. Even when scores rise, you can’t pinpoint why. Was it the new approach? The coaching? The curriculum? And when scores drop, it’s just as murky.

This is the trap of measuring only lagging indicators. You’re steering the whole system by looking in the rearview mirror.

The shift is measuring leading indicators — the observable changes in teacher practice and student experience that you can see every week, not every spring. Are problem strings happening weekly? Are PLCs analyzing student thinking? Are coaching cycles being completed? These predict outcomes far more reliably than a year-end score, and they let you adjust during the year instead of after it’s over.

The Sustainability Problem: When the System Depends on a Person

Here’s the barrier Schoenfeld’s research circles but never names directly: most math improvement depends on a person, not a process.

A passionate coordinator. A skilled coach. A principal who gets it. And when that person changes roles — which, given the turnover Schoenfeld documents, happens constantly — the work collapses. The flywheel resets to zero.

Sustainable improvement requires distributed leadership: a structure where math leads, coaches, and administrators each hold defined roles within a documented, repeatable system. New staff can enter mid-year and quickly see where they fit because the supports are embedded, not improvised. The work no longer collapses when someone leaves — it strengthens.

This is the difference between building a flywheel and riding a ferris wheel. A ferris wheel needs constant power and carries the same passengers in circles; the moment the operator steps away, it stops. A flywheel, once turning, carries its own momentum — and anyone can step in to give it the next push.

From Three Circles to One System: Building Your Math Improvement Flywheel

Step back and look at Schoenfeld’s three circles together. Each one names a difficulty:

• Thinking Mathematically reveals that teacher understanding — not just teacher strategies — is what makes instruction durable.
• The Learning Environment reveals that the five dimensions of robust teaching only survive inside a coherent, aligned system.
• The Cultural Surround reveals that without the right measures and without distributed leadership, improvement stays fragile and resets every year.

Notice what they have in common. None of these is solved by working harder. They’re solved by building a system where the right work compounds.

That’s what we mean by a Math Improvement Flywheel. Borrowed from Jim Collins’ Good to Great, a flywheel is a heavy wheel that takes enormous effort to start — but once moving, each consistent push in the same direction builds unstoppable momentum. You don’t push harder in one heroic moment. You push consistently, in the same direction, over and over.

We’ve organized the flywheel into four interconnected components, and you’ll notice each one directly answers a difficulty Schoenfeld names:

Design & Measure — Define what great math teaching looks like and how you’ll measure it with leading indicators, not just test scores. (This answers the measurement problem.)

Align & Sustain — Build the PD, coaching, and PLC structures that keep everyone moving in the same direction so the five dimensions of robust teaching can actually take root. (This answers systemic misalignment.)

Build Capacity — Strengthen teachers’ own mathematical understanding and confidence from within — not just their repertoire of moves. (This answers pedagogy-first fragility.)

Inspire Growth — Celebrate what works, codify it, and distribute leadership so improvement outlasts any single person. (This answers the sustainability problem.)

When these four components operate together, each rotation makes the next easier. Focus drives alignment. Alignment drives meaningful measurement. Measurement drives effective PD. And that learning strengthens leadership capacity, which deepens focus again. The cycle feeds itself.

The result is a district where improvement no longer depends on personality, position, or chance. It depends on a process — a living system that grows stronger with each turn of the wheel.

A Final Reflection for Math Leaders

Schoenfeld ends his paper soberly: “The challenges are clear, and there is much to be done.” He’s right that the challenges are real. The fossilized math, the demanding classroom environment, the inequitable surround — none of it is simple.

But here’s what his research makes clear when you read it as a leader: you don’t have to solve all three circles at once, and you definitely don’t have to solve them through sheer force of effort.

You have roughly 180 instructional days a year. That’s all. Without a system, it’s easy to arrive at next fall in the same place — tired, uncertain, and waiting on test results that won’t arrive until the year is already over.

So before you choose next year’s initiative, sit with one question:

If you stepped away tomorrow, what parts of your math improvement would keep moving — and what would stop?

Whatever would stop is the part of your flywheel that isn’t built yet.

Curious where your system is gaining traction and where it’s still spinning? Take our District Math Improvement Assessment — a short screener that gives you a free, customized plan across the six areas that matter most for sustainable math improvement. 

And if you want the simple decision-making tool we use with districts across North America to keep everyone aligned, grab the free Math Coherence Compass template.