Abandon early math instruction!

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This text is part of: "I would never send my kids to school" by Piotr Wozniak (2017)

The clash of two genius minds

My life has been changed permanently by Larry Sanger, the inventor of Wikipedia. We have a 17 year old history of disagreements with Larry, who I love dearly for his intellect and a fiery defense of his positions. When Peter Gray published an article about Benezet experiment (1929-1935), I considered it a rare specimen find with powerful inspirational value. Gray concluded that early years of math at school may do more harm than good. Larry, in contrast, accused Peter of anti-intellectualism. In Larry's mind, Gray jumped to conclusions too fast on too little evidence.

Crowd wisdom at Wikipedia

Back in 2001, when writing about sleep at Wikipedia, Larry Sanger asked me to stick with scientific consensus at Wikipedia. I found it intolerable because his version of consensus aligned with the level of Encarta that carried an article by Dr Jerome Siegel, a famous "memory processing in sleep" denier. Today, having seen lovely Wikipedia articles about sleep, I am proud to have been vindicated. I still prefer to stick with my own wiki and write about neural optimization in sleep using my own understanding of the process. That's liberating. I believe a similar vindication might take place in 10-20 years in reference to schooling. Larry is one of the most famous infovores on this planet. This is admirable. At the same time, he presents adult-centric point of view on education. He is a fantastic homeschooling dad, who still believes that schools can be reformed. I here claim without hesitation: very few functions of today's school will survive in the future. Warehousing function might be hardest to replace. Otherwise, all educational aspects of schooling will drift in the direction of the employment of the learn drive in respect for the fundamental law of learning. All news from neuroscience support Peter Gray. Wikipedia chapter on unschooling will evolve from a radical idea today, to an obvious remedy to all ailments of education in a decade or two.

Early math instruction is bad

Peter Gray is right to point out the evils of schooling. Schooling leads to coercion which leads to toxic memories. Forgetting curves for little children illustrate the absence of learning under instruction (see: SuperMemo does not work for kids).

Gray is unquestionably right that all coercion should be eradicated. Giving up on teaching math in early grades would be a simple step in that direction. Formal math could easily be replaced with playing with math concepts. I believe that Larry's point of view on education is distorted by his own homeschooling success. He believes in the magic of instruction. He believes in the power of young brains in tackling adult problems. I would urge him to introspect his own intellectual success, and how much of that success stems from school, and how much comes from his own passions that have no end. This type of introspection was very helpful in my own case. I spent 22 years in the system and it took me nearly 3 more decades to realize that school did more damage than good. In other words, the cultural imprint of the value of schooling kept me blindfolded for most of my adult life despite my deep involvement in the problem of efficient learning!

Academic instruction is a self-perpetuating ailment

Larry is right that the literal interpretation of Gray, i.e. removing math from school, might indeed backfire. The problem with schooling is that it conditions kids to follow instructions and contribute little of their own. This is why kids locked in the prison of school without math instruction might indeed make little progress in math. How could they possibly progress if all their best brainwork window is covered by "boring" school. Their mental energy for the rest of the day would be drained. I have never seen kids do intellectual pursuits with passion after school. Videogames and YouTube dominate their evening interests at the primary school level. I recall that I had some scraps of passion for writing SuperMemo in 1987, but I was of college age, and I did not have to attend all classes or get up early.

Gray says that we do more damage than good. This is certainly true with poor quality of teaching. However, it might also be true if we employed good teaching (as opposed to just providing good inspiration). The ultimate outcome may depend on what happens after the first 3-5 years of schooling. Those kids who continue on a schooled track, may need those first years of introduction to continue rolling on the predetermined math path. On the other hand, those who are unschooled would be more likely to produce future Einsteins and Turings, who incidentally both hated the mind-shrinking rigidity of schooling.

In the end, both Gray and Sanger might be right. Gray is right that free kids master math basics on their own in a vast majority of cases. Those kids can easily leave behind those who had been subjected to early academic instruction. On the other hand, removing math from school could be a bad idea. We would rather need to remove school altogether from the equation. The experience of democratic schools says that kids need long stretches of detox to regain their own initiative in self-learning after many years of coercive schooling.

The sad truth is that once committed to schooling, kids need to be lead on a leash. That is the price we pay for the invention of the Prussian education system

For an illustration of a sad outcome of early math teaching see: Videogames are better than teachers.

Peter Gray's Opus Vitae

The whole misunderstanding might have been born in Larry's reading Gray without the full context. It is a bit as if an observer complained of the enthusiastic Egyptologist for seeing a world of implications in a 3 cm clay shell uncovered in the desert. To an observer, the shell has little meaning. To an Egyptologist, it may change his take on the course of human history.

Where Larry complains of Gray's assumptions, Gray states facts of well-proven science. As befits a blog entry, Gray just did not bother to mention all references. To anyone familiar with Gray's ginormous opus vitae, it is not necessary to bring up the topic of unschooling and all its ramifications in each piece of his writing. Larry might have missed that context. As Larry is a great enthusiast of early reading, he must have already discovered long ago his common denominator with Gray:

All forms of coercion in education should be eliminated

With that basic agreement established, we should free kids to do their own learning. We should replace teaching with assistance. Flip the classroom. Use play and creativity. Current system damages and robotizes math reasoning in kids. The evidence is overwhelming and Benezet example is just a spicy illustration of how radical ideas in education can surprise an unprepared mind.

Who is truly radical?

Let's now turn tables on Larry's tough accusation:

Radical proposals demand rock-solid proof. [...] It is surprising that a distinguished scholar would make such a dramatic proposal on such a slender basis

I would say that if we have a small sample to prove that unschooling is superior to schooling, Larry's sample is equally small in reverse. What is worse, the small-sample points to superiority of unschooling. To me, schooling is more radical than unschooling. Teaching math before the age of 12 would be radical before the times of Pestalozzi. All best math in this world came from the minds of free thinkers who spent years pondering questions key to the queen of sciences. There is not a single case of modern Newton coming from a cram school. Curricula and testing are far more radical than free thought. They are modern radical inventions. In that sense, Larry's basis for his even more radical proposal is even more slender. The only evidence in support stems from a cultural myth: school is good.

For thousands of years, humans learned math by counting sheep. Today, we send rockets to the moon by perfecting the art of self-learning in the area of mathematical explorations. Calculus at school is just a whip for weaker students. Future rocket engineers will do their own passionate calculations of their own accord with little attention paid to algorithmic prescriptions of the curriculum. The radical experiment of hammering math into every child's head has failed on a massive scale. The evidence of failure is overwhelming and omnipresent. The evidence that things can improve after two centuries of trying is slender.

Schools kill the love of math

Sanger adds:

Gray seems to assume that the math-untrained sixth graders were enthusiastic about math at the end of their year of math

Gray does not assume that untrained kids would be enthusiastic. He only states a well-proven fact that schools suppress the learn drive. Untrained kids would simply be more likely to engage in math problems of their own volition. They will employ their reasoning skills rather than a well-drilled algorithm.

My guess--and the author provides no reason to think this is not the case--is that in the long run, the students who started earlier, and especially the ones like the Doman kids who started thinking about and memorizing facts about numbers as very young children, will be quicker with math

Larry's guess is wrong. Students who start early will on average be much weaker at math. They will simply get love of math hammered out of their heads. The more rigorous the instruction, the more hate it generates. There will always be one nimble exception to perpetuate the illusion that academic instruction works. The rest will fall far behind those natural passionate mathematical tinkerers.

See also: Early Academic Training Retards Intellectual Development

Power of unschooling

It’s easy to agree that kids will not receive a high-quality math education in most classrooms in most public schools in the U.S.–no thanks to all the half-baked failures of the education professors and educational psychologists, who are so often not interested in what has actually been proven to maximize student knowledge

After those words, I know what's coming next: schools are bad, but we can design them to work better. This comes from the most damaging illusion about education. We need to keep re-iterating the fact that schools are inherently unreformable because they violate the fundamental law of learning. If they attempt to stay compliant with the law, they will drift in the direction of unschooling! QED. Gray has actually spent all his later years investigating the superiority of unschooling. He truly knows what maximizes quality knowledge.

Larry took his kids away from school, and for this his boys should remain grateful for life. Still, Larry would want to look for the most effective curriculum verified by test scores. In that he is as wrong as Bill Gates. This is exactly what I would call an assumption based on brainwashing we get from school. Curriculum is bad because it inevitably leads to coercion. Testing is bad because it inevitably leads to cramming. Education should be all about creativity and self-development. Otherwise, we will turn into a society of obedient robots.

The drama of a math joke

One event in my investigations settled my mind about teaching math. All the evidence in the world is not as convincing as a personal visceral experience. Little drama in the playground:

The old math joke goes like this: The ship carries 20 sheep and 16 goats. How old is the captain? When I found out that a large proportion of kids answer this question with 36, I was in disbelief. I struggled with empathy for a brain that would be so badly derailed. When I jokingly tested this question on a first grader, the conversation was not too surprising. The answer was 50. The justification was: "My mom is 25. I think that captain should be a bit smarter. I think 50 would be just fine". I was amused with kid's self confidence, but noticed that his mom started being uncomfortable. As if she feared she would be next. This required further testing. I ruthlessly asked the math question. I was in shock when mom delivered that infamous verdict: 36. I instantly knew this could only be explained by toxic memories. The lady had been paralyzed by the fear of math and, with a knee jerk reaction, assumed the answer must be in numbers in the question. Like a well-schooled robot, she provided the answer. This is what poorly delivered math training does to young minds: useless fear! Fear with no purpose at the cost freedom and long hours of drilling in a school bench. Interestingly, the lady is a shopping clerk and she seems to be pretty fluent with numbers. It is that toxic memory of a typical question delivered by a math teacher that jumbled her mind and destroyed the pleasure of the day. Contrast that with an undamaged kid who hypothesized fearlessly. I can only hope his free thinking does not get damaged at school

This is how I see the events occurring in a panicked math brain. It all begins with a recognition of a typical math task. This instantly triggers a toxic memory that associates math with the state of anxiety. The fear of math paralyzes all intellectual capacities that might lead to a rational solution. In neural terms, well-polished networks trigger fast, high performance, high stability circuits that instantly take over the job of finding the solution. Those circuits are characterized by low coherence and do not integrate well with the world knowledge. They are targeted at providing a robotic solution by employing fast thinking. The algorithm for finding the solution might have, over years, through interference, lost all its vestiges of responding to the actual logical input involved in the task. Instead, the brain strips the problem to bare bones and follows the algorithm:

  • if two numbers appear in a math task, employ 4 basic operators: +, -, *, :, and choose the result that is most plausible
  • if there is little time left (for solving the test), pick addition, which is easiest. In a multiple-choice test, it still provides 25% chance of success
  • if numbers are too big to employ the operator, give up and jump to the next test to solve. In real life, play a diversion game, and cover up your tracks. Be sure you do not get caught with hot ignorance in your hands. Don't let anyone see you with your math pants down

That last part of "covering up" may compound toxic memories that lead to math anxiety.

Contrast this with a toddler who plays Lego bricks, and will keep re-shuffling that wall of bricks until it gets even and provides room for moving to the next level. The little snot will internalize number sense and coherently and consistently integrate it with his world knowledge using his learn drive and his knowledge valuation network.

Outwardly, our abstract adult brain can see the same math problem to solve. Our empathy is too weak to escape the abstraction. On the other hand, for an immature brain, the same powerful learning mechanisms can be employed for two entirely different jobs: (1) solving a real life math problem, or (2) surviving in a classroom.

Misemployment of the learn drive, the genius neural mechanisms developed in the course of evolution, is the key tragedy of modern mass education

I learned no math at school

Here comes my own personal bias:

Personal anecdote. Why use anecdotes?
I have spent many years in a classroom with a math teacher and boring equations on a blackboard that I could not even see due to my shortsightedness. All math that I learned in my life, I learned on my own. All the best tools I mastered required long hours of tinkering on paper or in the computer. I wish all kids had more freedom, time, and mental energy for such explorations. For that, we cannot lock them in the prison of school. Time permitting, I plan to write a longer text about learning math, where I show the difference between academic instruction and real learning

Fallacy of the genius mind

This is no coincidence that the smartest guys on the planet seem to be most wrong about education. One of the biggest obstacles on the way to the Grand Education Reform is the illusion harbored by the adult brain that the child brain is similar and works along similar principles. A genius finds it hard to empathize with ordinary folks. It is even harder to emphasize with a growing brain that may be at the proliferation stage, or at the pruning stage. This growth processes are a serious handicap and neglecting the fundamental law of learning is particularly dangerous. It takes 30 years to grow a strong abstract brain, and another 20-30 to bring it to maximum heights of human wisdom levels. This strongly abstract adult brain may set abstract goals and devour abstract knowledge at will. This is not possible in younger brains that cannot be motivated extrinsically and struggle with abstract motivation. Knowledge must be its own reward at first. Then the brain can slowly drift to rewards based on goals, before, decades later, the rational brain can pick goals on a purely rational basis. Even then, a child lurks inside an old man's brain, and we pick and choose on the basis of fads, emotions, minor rewards, habits, weaknesses, skewed preferences, etc. Bill Gates wonders why a lovely crystallography lecture has a high drop out rate. Larry Sanger insists that early academic instruction in math might provide an advantage. Even Peter Gray is guilty of ignorance. Before his son sent him to hell in his school refusal tantrum, Gray took on a standard culturally imprinted assumption: "school is good". My own transformation started from long conversations with kids who hate school. For years, my default thinking was: "it is just a bad teacher. it is just a bad school", etc. I never stopped encouraging kids. I always asked them to think about college. It took me years of detailed analysis of the knowledge acquisition process before I could see the light. We need to realize that genius mind grows organically. It cannot be imposed from above. It cannot be trained. Freedom to learn and think is the first key initial step. Adult genius is no excuse. It also needs to back off and let the kids grow on their own terms.

Schooling warps reality

To explain the problem of schooling, I try to depict human knowledge in touch with reality as a surface. That surface touches the real world which forms its own surface of exposure. When the two surfaces get in touch, learning occurs. The shape of the surface of contact is determined by the brain's goals. As all knowledge valuations are filtered through the knowledge valuation network, goals determine the surface of human knowledge in its connection with the real world. A change in goals is like an invisible painter that re-paints and re-colors the reality. When goals change from just learning to tests, grades, and scores, schooling warps the reality and dramatically changes the brain development trajectory.

The brain that goes through schooling is equipped with a fraction of knowledge that the learn drive in rich environments could provide. Moreover, that school-delivered knowledge is characterized by low coherence that undermines its stability, increases forgetting, and increases interference with other areas of knowledge. Most importantly, the associative power of knowledge that underlies creativity is diminished. This has a dramatic impact on the capacity for problem solving.

The brain that goes through years of schooling is poorly prepared for problem solving

Early academic instruction in math is just a beginning of a major disruption in the natural development process. However that early intervention might rank among those that incur most damage.

Monuments for Gray and Sanger

There will be monuments for both Larry Sanger and Peter Gray in the future. Sanger would love to see a new intellectual planet. His approach is a bit too conventional though to succeed big time. On the other hand, Gray is not anti-intellectual. Schools are! Gray is one of very few people on this planet who truly understand the mechanics of the learn drive and its value for the future of mankind.

The path to better education will be long and meandering. Our biggest enemy is the current system of education that imprints rich mythology in our minds. Without good support from neuroscience, even the brightest mind can stand in opposition to the truth. For details see: Problem of schooling