Revolutionary Hypotheses

August 2026

I dream of thousands of formerly normal ten year olds begging for more calculus.

In Education’s Great Stagnation, I discussed how little education has improved over the decades and why. Here I discuss the core hypotheses that must be true if I’m going to build the best damn math course ever and make that dream come true.

Hypothesis 1: A Narrow Content Bottleneck Controls the Path to Calculus

As a math tutor, students, teachers, and parents typically insist that I help them with schoolwork. Students resist learning

6−8=‾6-8=\underline{\hspace{2em}}

because their schoolwork is

6f(x)−8f(x)=‾6f(x)-8f(x)=\underline{\hspace{2em}}.

Of course, the student might cheat or rote memorize their way to a correct answer, but they learn nothing but worse habits while falling farther behind, creating a vicious cycle that ensnares millions and releases virtually none.

Many other students bring me trivial work, such as creating bar graphs for data, that they could get 92% with 65 minutes of work on their own, but they want 100% in 55 minutes with me.

What can be done?

My answer: To hell with schoolwork! Forget the student’s age and grade and their teacher’s classwork. Refuse the business of students and parents who wrongly insist the above is the way to succeed in math.

Instead, focus on the critical path.

From kindergarten to calculus, there is a sequence of bottleneck content, which, if mastered, will allow anyone to learn introductory calculus, including plenty of 10 year olds. This would transform nearly every kid’s life.

That bottleneck content is:

  1. Whole number sense: \+, −, ×, and ÷, what they mean, when to use them, how they relate, base 10, and negative integers.
  2. Rational number sense, mostly fractions, decimals, and percents.
  3. Introductory algebra. This includes algebra’s generality, basic solving, and a few types of functions.

That’s it.

Notice that it excludes probability, geometry, time, shape, space, measurement, money, statistics, and all other aspects of math. It excludes trendy generic skills such as collaboration, communication, critical thinking, and creativity.

Why?

Because those other aspects do not constrain grade 5 students from doing calculus. Only the bottleneck content does.

Anyone that masters the bottleneck content can learn introductory calculus regardless of age, maturity, and other branches of school mathematics.

That’s Hypothesis 1.

Disproof of the hypothesis would be students who have mastered rational number sense and introductory algebra, but are still struggling badly in calculus. In 17 years in the math education field, I have never met or even heard of such a student, but if you have, please let me know!

The next question is how to relieve those bottlenecks.

Hypothesis 2: Better Learning Can Be Engineered

Many inputs impact the design and effectiveness of learning activities, so it’s ridiculously hard to get teenagers excited about logarithms. But I think it can be done. Here’s how.

Status Quo Bravo Math
WHO: Teachers plan and prepare alone. Good practices spread slowly if at all. WHO: Diverse team of educators, current and former students, researchers, psychologists, parents, etc. design learning activities together.
CONTENT: Schoolwork. Government curriculum. CONTENT: Bottleneck content only.
DATA: Teacher memory and student work. DATA: Student work plus videos of the students learning with talk-aloud protocols.
FEEDBACK LOOPS: Teachers try 1 or 2 versions of an activity per year. FEEDBACK LOOPS: Test 4 versions per day.
ASSESSMENT: Seek evidence of learning, risk confirmation bias. No shared definition of learning. ASSESSMENT: Seek evidence of non-learning. Falsification over confirmation.
Define learning as sustained abstraction and automaticity.
OUTPUT, ASSETS: A teacher produces materials and craft knowledge for themselves. OUTPUT, ASSETS: The team produces such great resources that graduates use them to help teach the next cohort.
BEHAVIOUR MANAGEMENT: Brief discussion of rules at the beginning. Focus on intrinsic motivation almost exclusively. BEHAVIOUR MANAGEMENT: Teach behaviour with explanations, examples, experiences, feedback, practice, marks, etc. Aim for intrinsic motivation but use external motivators of any magnitude when necessary.

Even a tiny improvement to each version would be a revolutionary advancement in education. In just a few dozen versions, every lesson could leave the students thrilled and begging for more as they learn and love more math than they ever thought possible.

Specifically, I’m aiming to shorten the path from addition facts to calculus to under 100 lessons, including summative assessments.

This is how I think we can relieve the content bottleneck.

I think this would thrill and delight many students and parents.

But only impact at scale would ignite a real revolution in education.

Hypothesis 3: Exceptional Learning Will Spark a Feedback Loop

Many people parent as they were parented. Many teachers teach as they were taught.

Bravo Math can take advantage of this with the following loop.

  1. Bravo Math teaches our students superbly. Think of the 10-year-old doing calculus.
  2. Two consequences of such teaching:
    1. Word of mouth will create a huge demand.
    2. Many course graduates will develop excellent teaching instincts. This is loosely analogous to a small share of 12-year-old campers becoming camp counselors a few years later.
  3. A subset of those graduates can be rapidly trained to become superb teacher assistants then superb teachers.

This loop is the path to exponential impact.

Hypothesis 3 is falsified if any of the following occur:

  1. The outlier performance of our students does not create enormous demand.
  2. Bravo Math is unable to recruit enough graduates to become teachers
  3. Too few of the recruits become effective teachers.

To recap:

As far as I know, there is little evidence as to these hypotheses, but if you know of any, please share. If false, Bravo Math will have to make some big changes. If true, Bravo Math will spark a revolution.

Contact me to learn more.