Math Education, a Traffic Cop, a Magistrate, and Intuitive Learning Beyond Computation

Last week I decided to watch a sampling of YouTube videos about the calculus of rotations of curves about an axis. The technique is used to find the surface areas and volumes of objects such as round flower vases, rocket cones, or parabolic mirrors. Some videos were made by math tutors. Others were made by college math professors or high school calculus teachers. I found that most videos were intended to help students prepare for traditional exams and were strictly about practicing the steps in a recipe to carry out the computation. The speakers presented a formula, into which they substituted values to find a solution. Only one out of seven videos provided any intuitive information about how we got the formula, and even that video discussed the path to the solution before explaining the logic behind it.  That speaker claimed that he followed that order because that’s what students want: Assistance to get through the exam. Students have evolved to favor this learning pathway, and traditional exams encourage this preference because they favor exercises that assess computation skills. Most students don’t realize how they are being swindled by this approach. They memorize a formula, pass the exam, and then forget about it. Why take the class in the first place? Math courses should express more courtesy in their use of student’s time.

Yesterday, my wife was pulled over by a state policeman, who informed her of a dead headlight. Instead of simply accepting that she would follow his instructions and respond to her own commitment to safety by replacing the bulb, he presented her with a formal ticket. This ticket meant that in addition to replacing the bulb, we would need to show the repaired light to another state police officer, who would provide affidavit that the problem was corrected, to be presented in court within a month. She was required to enter a guilty plea with the magistrate, who would then table the charges after viewing the second officer’s affidavit. Now, don’t get me wrong, if a citizen who is pulled over for a missing headlight is belligerent and insists that he will not comply, the officer has a duty for public safety to follow through with the formal ticketing process. Yet, the vast majority of motorists in this situation are not aware of the problem until the officer tells them. Sometimes lights, even those that were recently replaced, burn out while on the road.  In a misplaced effort to increase public safety, the system has become unnecessarily complicated, wasting the time of the officers, the magistrate, and the citizen. I found that the problem was that the electrical connection to the bulb had jiggled loose. It took me less than a minute to resolve the mechanical problem. I was subjected to more than an hour of run-around Saturday morning, to get the second officer’s affidavit, and the whole process will not even be complete for at least another couple of weeks as the court processes the paperwork. We also were subjected to an unnecessary sense of guilt and risk from having signed a guilty plea, as if we committed a crime because normal vibrations shook the cord loose.

The time we wasted in this quirk of the legal system pales in comparison with time wasted in math education. I think that students, parents, and teachers waste countless hours favoring the processes of computation. Some focus on computation is needed to generate broader understanding and to build full practical ability to apply the techniques, but I think it should occupy less than 20% of learning time. Time should be more heavily weighted toward exercising mathematical intuition: It is more important to understand the concepts behind algorithms than to learn to use the algorithms themselves. When students build intuition around a concept, they are more apt to innovate with the concept, a trait that would better prepare them to solve the presently unknown problems of the future world.

Zack and the Spelling Experiment: An Illustration of How Young Children Learn

Eight-year-old Zack has just woken up on a weekday morning at his leisure. He crawls out of bed, picks up his hand-me-down laptop computer and charger cord, and migrates downstairs to his green-padded cube chair and the makeshift coffee table that he has claimed as his desk. He sets a bowl of cereal next to the computer and opens his first children’s game site for the day. Some playmates are already there, taunting a big ugly boss. He quickly types a message to one of them, abbreviating as needed, leaving off punctuation and capital letters. His typing is fast, but he finds his own non-traditional hand positions. He gobbles some cereal, then stacks his bowl behind the computer, not yet feeling moved to take it to the kitchen, continuing with his game.

Zack is home schooled, and for the most part, he sets his own schedule. He might not have had a formal reading, spelling, or math class from a trained elementary school teacher in his life, and he has little use for curriculum. He knows how to use Google, and he occasionally talks about science, math, and reading with his parents, siblings, and peers in the community. Story time last night was a chapter in the Andrew Clement book Frindle, about a fifth grade student who creates a stir in his community by making up a new word for “pen”. Zack could not help but chuckle at the book’s depiction of elementary school culture, which he has never experienced. He expressed some doubts about whether some of the classroom experiences described in the book could be real. Do other eight-year-old boys really sit at desks that many hours a day, under almost complete control of a teacher? What are vocabulary-spelling lists for, and why does the school system think 8-year old boys need them? Zack has never learned new words or spelling from a list, and he wonders whether his peers in public school really pick up new words that way.  

Now, you might be curious whether Zack himself can spell properly, without ever having the guidance of a trained elementary school teacher?

 Once a year, to fulfill requests of the state to track the progress of home-educated students, Zack takes a standardized test. His parents don’t like the idea of high stakes testing, but their own drive to know of his progress, including curiosity about how he compares academically with his peers, keeps them in favor of some testing. They spread the different short sections of the test out over a few days. Today’s exam is spelling. Zack first takes a practice test. Exercises include a list of words, from which he needs to pick out one that is misspelled. He casually finishes the dozen practice exercises without an error. Then, he takes the formal 25-question exam, correctly responding to 24 of them.

How does he do it? He reads few books on his own, and he certainly does not study vocabulary lists. No one has ever schooled him formally in standard English rules of spelling. The best way his parents can explain his success is that he learns by doing, through his online gaming, and through interaction with his parents and peers. His parents work to put learning experiences in his pathway, providing good environments for learning. He picks some of it up by watching street signs and advertisements while commuting to town. In short, he has become exceptional at spelling in the same way he learned to speak as a baby: by trying it, and by building it actively into his interactions with those around him. What’s most important is that the evidence shows that it works, placing his skills in the upper range of the distribution of his public-schooled peers.

Most systems of primary education around the globe depend on systematic curriculum that exerts controls on young minds through copious repetition, imposed structure, and a teacher who is clearly in charge. Although evidence suggests that some of the public education system’s approaches benefit young children and society as a whole, illustrations of the experiences and successes of countless young children like Zack show that systematic constraints on the learning process, including control by a teacher, are not necessary for their progress. Society expends vast resources in constraining the learning in young children, many of whom evidently could get most of the basics on their own without sacrificing the joys of childhood, including free play, which studies suggest is necessary for them to learn higher social skills and executive functions.

Surely there are better ways we can envision to use the resources of the education system for young children that take advantage of how they actually learn. Why not focus our more systematic curricula on older children who are ready for more advanced academics, and let young children be children? Instead of exerting firm classroom controls on young children, we need to create academically stimulating environments and surround them with conversations, games, and peer group activities that bring joy and a sense of commitment into their lives, letting their curiosity gradually drive them toward increasing appropriate structured curriculum as they mature.