Bravo Math News

(5) Three Revolutionary Hypotheses

A startup is largely a collection of hypotheses in a founder’s head, a bunch of ideas that, if true, can revolutionize an industry.

Here are mine. They are the path to building the best damn math course ever.

Hypothesis 1: Bottleneck Content

Students, teachers, and parents typically insist I help them with schoolwork. This is quite difficult when they resist learning 6 – 8 = ___ because their schoolwork is 6f(x) – 8f(x) = ___ . In fact, it’s a vicious cycle that ensnares millions and releases virtually none.

To hell with such schoolwork, even if it means refusing the students and parents’ revenue. Forget the student’s age. All students should learn the same bottleneck content first.

  1. Whole number sense: +, −, ×, and ÷, what they mean, when to use them, and place value.
  2. Rational number sense. This is mainly extending whole number sense to fractions, decimals, and percents.
  3. Introductory algebra. This includes algebra’s generality, representing relations, and some solving.

With these bottlenecks relieved, there’s little to stop a grade 5 student from mastering grade 12 math. This would transform almost any kid’s life.

And that’s Hypothesis 1.

If repeated at scale, it would be an earthquake in education.

Disproof of the hypothesis would be students who have mastered rational number sense and introductory algebra, but are still struggling badly in math otherwise. In 10 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!

Hypothesis 2: How to Engineer Learning 

The next question becomes how to relieve those bottlenecks. My answer:

Rapid Prototyping
User-Experience Testing
Cognitive Science
+Diverse Team
Pedagogical Breakthroughs

Or in more detail:

VariableStatus QuoBravo Math
ContentTeach the student the government’s prescribed learning outcomes, especially what’s being done in school now.Focus on the bottlenecks of K-12 math regardless of student age or current school work. Work on limiting factors only.
WhoTeacher works largely alone, inputting craft knowledge and hard work. Best practices rarely spread. Diverse team of teachers, tutors, students, researchers, parents, etc. design learning activities inputting hard work, craft knowledge, cognitive science, UX, etc.
Data SourcesTeacher’s memory and relatively static student output, such as tests, quizzes, projects, presentations, etc.Use all of that (left) plus videos of the student learning. Use talk-aloud protocols and visible assessment to help track student cognition.Long-term follow up on the lesson.
Radically superior data.
Feedback LoopA teacher administers 1 or 2 versions of a lesson per calendar year.Test and iterate 4 versions of a lesson per day. Accelerate feedback loop at least 500-fold.
AssessmentRigor means seeking evidence of learning.
Confirmation (bias).
Rigor means seeking evidence the student has not learned.Seek early indicators of trouble.
Output, AssetsA teacher produces lessons, materials, craft knowledge, skill sets, etc. for herself.The team produces “Open-and-Go” lesson plans and resources that are so good that anyone who went through the lesson can administer it.

The company is nowhere near ready for this now, but within a few years I hope this is what a typical workday looks like.

9amOne employee administers Version 1 of a lesson.
Rest of the team observes and records on video.
10amEveryone watches the video of Version 1 together.
Everyone collaborates to develop Version 2 of the lesson.
11:30amOne employee administers Version 2 of the lesson.
Rest of the team observes and records on video.
1:00pmEveryone watches the video of Version 2 together.
Everyone collaborates to develop Version 3 of the lesson.
2:30pmOne employee administers Version 3 of the lesson.
Rest of the team observes and records on video.
3:30pmEveryone watches the video of Version 3 together.
Everyone collaborates to develop Version 4 of the lesson.
etc.Continue the above cycle until the lesson and resources are blazingly awesome.

Even a 5% improvement to each version would be a revolutionary advancement in education. In just a few dozen versions, lgo from grade 6 math to calculus is absolutely possible.

This is how we relieve the bottleneck.

Hypothesis 3: Bravo Math Graduates Will Be Amazing Tutors

To grow exponentially, requires a different format of education: peer-to-peer (P2P) instruction.

Most people teach the way they were taught. If Bravo Math teaches our students well, then when they graduate from the program, they will have excellent teaching instincts, and some will be worthy of hiring. I’ve already begun this by hiring a former student and it’s going great! [That’s why I might hire you.]

And it’s going well without the kind of world-leading math materials I want to build.

If the average Bravo Math Graduate passes on what they’ve learned to two students each year, then 5000 kids doing calculus will be just the beginning.


In summary:

Hypothesis 1: Students who learn basic number sense and algebra are ready for calculus regardless of their age or what’s happening in school.

Hypothesis 2: A team that rapidly tests and iterating learning activities will learn to teach students that number sense and algebra in under a year.

Hypothesis 3: Students who experience that will become excellent peer tutors, growing our impact exponentially.

Those are the hypotheses Bravo Math needs to test.

Bravo Math News

(4) The Great Stagnation, Part 3: Sabotage and Gridlock

Imagine an education system that squeezes out a 1% improvement each year and causes, say, today’s high school graduates to learn 50% more than graduates from 50 years ago.

Imagine a university that squeeze out a 1% improvement in their teaching each year so that, over the course of, say, 100 years, they could teach 100% of high school graduates.

Have you ever heard of such systemic improvements?

Of course not. The field of education is so stagnant that even a tiny stream of 1%/year improvements are unheard of at the system level.

Blame the status quo. It is all but rigged to slow the rate at which teachers improve their craft.

  • Teachers-in-training do not learn the most proven techniques for learning. We assume we should trust our intuition when cognitive scientists adamantly say not to.
  • Teachers work alone even though innovation tends to come from diverse teams. Teachers learning by observing and collaborating with other teachers is rare, unpaid, logistically difficult, and a violation of norms.
  • Teaching can be tougher than rocket science, more demanding than being a first-year lawyer, and so relentless that teachers can’t find time to go to the bathroom.
  • Teachers in much of the world are paid so little that they can’t work on their craft because they work a second or third job and are too busy selling their blood plasma.
  • Teachers work without useful feedback and without useful research. Are you in touch with your kindergarten teacher? Do you know exactly what long-term effect their pedagogy had on you? How much do you know about the long-term effects your teaching has had? Can you point to evidence to support your guesses on the matter? Are such guesses based on a large collection of double-blind, randomized controlled field trials with big samples that measured how each lesson changed students’ opportunities, work habits, and attitudes over a 30-year time frame? [The answer is ‘no’ because such a collection does not exist!] The evidence for many educational practices is scant and rarely causal because there is very little money spent on educational research. The pharmaceutical industry invests about 17% of revenues in R&D. The education sector has comparable revenues, yet invests about 0.1% of them in R&D. Guess which sector is known for innovation?

So, next time you want to criticize a school teacher, a corporate trainer, or an educator of any type, please consider how rigged the system is against their improvement.

At an institutional level, the forces of stagnation are just as powerful, as documented by Larry Cuban.

For example, let’s say you are a brilliant educator and administrator ready create a self-paced high school program with an interdisciplinary approach that kids love.

You replace English 12 and a bunch of other courses with a multi-disciplinary course of history/literature/civics/drama. Will your students meet the government’s graduation standards? (Probably not.) Will students or parents be OK with that? (No.) Will school administrators and boards be OK with that? (No.) Will typical post-secondary institutions recognize your course? (No.) Will employers recognize that course? (No.) Major reforms to high schools are virtually impossible because the governmental graduation standards, post-secondary admission requirements, tradition, and employer interpretations would all have to shift simultaneously. You’ll also have to survive angry parents, skeptical school board members, media griping, and indignant politicians lecturing you on what a “real school” is.

You’re in a for a multi-year battle on many fronts and your “side” is nearly certain to crumble because staff turnover in education is very high.

Good luck with that.

Even the intuitive appeal of self-pacing – every child learning at their own pace, mastering content before moving on – fades very quickly in practice. A “self-paced” course means the student is working alone, often in a solitary manner. The student is not working with other learners, not sharing or learning from positive social norms of classmates, not learning to communicate or collaborate. The weaker and less motivated students fall just as far behind as they would in a collectively-paced class. Self-paced programs also suffer from rampant cheating because having 200 students all writing an algebra test at 200 different times guarantees some will tell others too much about the test in advance. Students say they miss interacting with the passionate expert.

Self-pacing lacks an impressive track record.

The difficulties of improving pedagogy and the gridlocked nature of educational reform may be daunting obstacles. But they are not immovable. They are not invincible.

Coming soon: How Bravo Math will bust through the status quo.

Another barrier is socioeconomic status, a daunting obstacle in the way of improving educational outcomes. But we know that it is not the only barrier to educational revolutions for two reasons.

  1. Organizations that basically exclude the underprivileged, such as many employers and universities, do not claim big advances in teaching either. Not even educational publishing companies claim these kinds of improvements.
  2. Schools vary dramatically in how they help (or harm) low-income students and this has been measured by random assignment data.

I will write about how Bravo Math will tackle this problem in a later post.

Bravo Math News

(3) The Great Stagnation, Part 2: 8 Beliefs & Barriers

Part 1 established education’s as civilization’s most stagnant sector.

Here in Part 2, I’ll enumerate some myths that prevent major improvements in education.

Let’s start with “Two Truths and a Lie” but with 8 statements.

  1. Teachers should organize their lessons around students’ learning styles.
  2. Continuous feedback supercharges learning.
  3. Learning is all about recognizing innate abilities and talents.
  4. We don’t need to remember facts any more since we can look them up so easily.
  5. “This technology will revolutionize education.”
  6. Learners should master one skill or one concept at a time.
  7. Busting teacher unions will unleash educational progress.
  8. The key to learning is doing. If my students do X easily, it proves they’ve mastered X.

Do you consider each of those to be true or false? Take a minute to write down your answers and a brief explanation.

Ready for the answers?

All of those statements vary from false to terribly misleading.

Let’s take another look, but this time with a tiny sample of citations coming from ~unanimous conclusions of researchers.

  1. “Teachers should organize their lessons around students’ learning styles.” There is literally no evidence for the existence of learning styles. We misattribute to learning styles what is better explained by prior knowledge. Yet overwhelming majorities of people believe in learning styles. This isn’t even debated among psychologists. Whoops!
  2. “Continuous feedback supercharges learning.” Wrong. If you provide feedback continuously or even at a very high frequency, you do not know what the learner is doing or thinking independently and what the learner is doing only because of the feedback. The best outcome is that nobody knows what the learner’s best effort looks like, but the more common outcome is perceiving mastery instead of reliance on external feedback. Continuous feedback is a path to delusion, not a supercharger of learning.
  3. “Learning is all about recognizing innate abilities and talents.” Actually, most evidence shows that most people can reach high levels in most disciplines with the right training. This is yet another damning indictment of our educational system.
  4. “We don’t need to remember facts any more since we can look them up so easily.” Actually, what we can learn next depends almost entirely on our prior knowledge – that is, facts, procedures, and concepts that we remember. Facts like phone number do not need to be memorized because further learning does not depend their memorization. But if an engineer is looking up grade 5 math right before designing a bridge, would you drive over it? Could that engineer just look up “Advanced Dynamics” this afternoon and design a suspension bridge this evening? Of course not. There is no way around it – expertise requires vast amounts of facts, algorithms, and concepts in long-term memory.
  5. “This technology will revolutionize education.” From radios, to Thomas Edison’s motion picture, to computers, to the internet and MOOCs, we’ve seen every generation of information technology promise and fail to transform education. Virtually none make a compelling case for why they’re better than a textbook in terms of the mental effort elicited from students.
  6. “Learners should master one skill or one concept at a time.” Another pillar of educational delusions. A math worksheet entitled “Division Word Problems” cannot tell you if the student is doing division because they recognize the concept of division in the word problem or if they’re just responding to the title of the worksheet. It certainly does NOT give them practice in making such distinctions. The worksheet should have been called “Word Problems – But Only Solve the Problems that Require Division”. Learners need to mix it up! This interleaving might be the most powerful pedagogical technique discovered in decades and it damns the one-concept-at-a-time organization of virtually every textbook you’ve ever used and every course you’ve ever taken.
  7. “Busting teacher unions will unleash educational progress.” LOLz
  8. “The key to learning is doing. If my students do this task easily, it proves they’ve mastered the task.” This is doubly wrong and a gigantic mental trap. How might this thinking go wrong in a physics laboratory? A physics teacher designs an experiment with tether ball so that students will engage with the laws of physics. During the experiment, students are supposed to think about circular motion, the definition of velocity and acceleration, the interaction with gravity, and how you can use those to predict where the ball will go. But they they often actually think “Step 1 says to put the pole here and step 2 says to press that button on the camera and step 3 says to lift the ball this high and step 4 says to record a measurement in cell D13 of the spreadsheet and step 5 says to multiply that number by 9.8…” This latter mental sequence causes no learning because it is devoid of physics yet it is the normative response to many supposedly active learning methods. Second, even if the students were trying to think in terms of scientific principles, were they doing so productively? Or were they entrenching misconceptions? How much will they retain? Will they be able to transfer their learning? None of this is knowable from what students do physically because the proximate cause of learning is what students do mentally.

These beliefs are mainstream and false and they underlie many of the failed attempts to improve education.

But as harmful as they are, they are not the only culprit in the stagnant world of education.

There’s more.