I wonder if teaching math should be made to look more like recess. Now that spring has sprung and the grass has dried out, my kids are doing lots of football and baseball after lunch. I've always been fascinated at the range of behaviors you can observe when a ball comes out .
Some boys never engage in these things. Others will only get involved if the ball hits them. Still others only engage when a group of similarly skilled boys coalesce. These kids aren't great at it but they have a good time with each other and almost never engage with the experts. The experts are at the top of this food chain. They exhibit skills that are simply amazing, given their young age (12-14)
The experts are usually (>80%) engaged and the others of course trickle down to no participation at all. There's a natural filter at work. The best are amazing, exhibiting a fluidity and naturalness that you can just tell is extracting the absolute best out of them. The worst (sorry) throw like girls. Actually I have a few girls who throw really well but they've never developed the requisite full motion skills to catch well. Might have to do with breaking a nail, sigh.
What's key is that everybody can throw and catch but there is an elite that clearly will go on to become accomplished athletes in football or baseball (or both). These kids will eventually get formal training in the underlying theory of what is making them excel naturally. For now they just excel and presumably they got that way from huge amounts of play (practice) that they chose (value proposition) because they were good (wired for it) at it.
You could easily conjure up a scenario where if bureaucrats were to see this (fortunately they mostly stay inside) they would be appalled at the disparities on display. They would retire to their offices and write a standard to ensure that no child is left behind in playing catch. Then they would set up classroom training to guarantee that every student has access to the new standardized curriculum. They would introduce nerf balls so that nobody gets hurt or breaks a nail. Then they could train us teachers to introduce mixed ability groups so that the experts are forced to play with the geeks. They would want to ensure that all sports were covered so they would schedule a new game every week in an elaborate multiyear spiral.
Before long every kid in school would hate recess as much as math. The playing field would be leveled and we would get the miserable equal outcomes that we are getting in the academic disciplines. Pretty far fetched, eh?
Well what if you thought of academics more like sports. In math maybe you work with calculators from day one and you make sure that every kid can do the kinds of things we all do with calculators, mastering every day mathematical application. As this process evolves you would see who has the passion and talent to go deeper. Along the way, kids who can't cut it get to drop out of the pursuit of theory and depth but they do so with a repertoire that ensures they can at least use the gadgets in their lives effectively. You could continue this natural filter throughout the K-12 years. Wouldn't every kid get what they need?
Right now we have a tendency to turn off kids at every step in the process by delivering theory first. Eventually the experts get what they need but only after suffering through 12 years of torture playing catch with people who can't catch. Many of the non-experts get turned off before they get the basics of application and leave school unable even to use a calculator properly. Wouldn't it be exciting to demonstrate some neat result and have kids jaws drop and quiz you on why that works? Wouldn't that lead to more interest than theory first?
Isn't it true that history and science are taught this way now? Two subjects that my middle school students love! Hmmmmmm.
There are two key differences between sports and academics. First, only a very small fraction of people will become professional athletes, and many of the remaining 99.9% go on to successful lives and careers in other domains. Therefore, you can say "most people don't have the talent to be pro athletes" without creating a firestorm -- you are not seen as condemning kids to failure. Regarding academics, however, there aren't many high-paying jobs that don't require a college degree (some that do, such as working on an oil rig, are arduous), so you WILL provoke a firestorm if you say "most people aren't smart enough to go to college", and an even a bigger one if you say that many people are not high school material (as I would).
ReplyDeleteThere's also a racial aspect. No one complains that professional athletes are disproportionately black, but a racial make-up in colleges or in honors classes in high school that differs from the general population is considered a social "problem" to be solved. It is assumed that if blacks and Hispanics are under-represented, a predominantly white society has failed them. The success of Asians is usually ignored. If you do in academics what coaches do in sports -- weed out the poor performers -- you will have a "disparate impact". I say so be it -- it's the result of the Bell Curve -- but that is a minority position, at least in public discussion.
Gosh, I have to delete my whole post because Bostonian beat me to it.
ReplyDeleteI'll add one more bit:
In addition to weeding out the poor performers, no one questions the value of coaches forcing the strong players to practice basic skills every day. Drill and kill is seen as a good thing, because winning matters. Constructivist guided discovery does not exist in sports coaching, because efficiency matters, because winning matters. And in the cases where some kids don't care about winning, then they have something called PLAY. They can just PLAY.
Honestly Paul, you've probably just voiced 50% of the arguments in Real Education now.
As I said in the EP&P post, I haven't read it. Guess I'd better go buy it to see if I've been prescient or robbed :>{
ReplyDeleteAnyway, what do you think about teaching math backwards? Applications first, to tweak interest, then theory second to explain why. Can you see it as an effective method?
The chips will fall where the chips will fall won't they? The thing is, we're failing ALL the kids now by making them feel useless early and often. In a way, catering to the politically correct approach has ensured failing the most children possible, most of the time, in most places.
As for academics being remarkably different than athletics, I would disagree. You could say in both cases that only a small percentage will achieve hyper success. Only 1% of athletes become superstars, true. But how many people with science degrees become nationally renowned scientists? How many people continue in athletics as just very good players for a lifetime? How many science degreed professionals become very good scientists?
In the case of athletes, we're conditioned to think of the superstars because they're so much in our face, but drive around any small town in America and count the number of pickup games going on in every sport imaginable. It's not just an either or proposition is it?
"Anyway, what do you think about teaching math backwards? Applications first, to tweak interest, then theory second to explain why. Can you see it as an effective method?
ReplyDeleteI'm not so sure about this. The issue I've been having with my DD in our homeschool is applying what she's learned. She's very good at solving straight forward equations and explaining the underlying concepts but give her a word problem and all of a sudden it's like she's brain dead. So I don't think that starting with the applied math would work for her.
Oh yeah! Word problems are a tough nut to crack but not because of the math. It's an interdisciplinary challenge isn't it?
ReplyDeletePutting that aside here's an example. I can teach kids to solve any problem that can be tackled with first year algebra by using Singapore Math bar models and never going near an equation.
In fact this is not a bad transitional strategy for word problems. The first step always is to convert the words to the bar model in order to visualize the problem.
For many, many kids this will be enough for a life time of problem solving. Others might want to take a next step and see the model turned into an algebraic solution. This is backwards from what is traditionally done. Actually most programs never get to a model they start with expressions/equations. Theory first! Then they just jump to solutions using algebraic methods.
One of the great strengths of Singapore (IMO) is the integration of bar models in even the earliest levels. We were visual animals long before we were brainy.
Allison is absolutely right on the skills and drills emphasis in sports, especially at the most competitive levels, which also include tactical drills as the kids mature. Not only are kids expected to work hard at 3-5 practices per week, they are expected to work on both skills and conditioning on their own, every day. Elite players, even those as young as 9, have to try out for their position on the team every year. Failure to work on their own means their skills, tactics and conditioning drop in relation to others on the team and those coming for tryouts. If you're not in the top 11-14 (team limit is 18 for soccer) you'll be cut, and will have to try out for a lower-level team. There's no secret about it and there's no apology made. The area where my kids played had 5 divisions of 10 teams each at the travel level (promotion/relegation every season for the top/bottom 2 teams), dozens of somewhat competitive teams and leagues and literally hundreds of rec-level teams. Everyone ended up on a team that met their skill and interest/time commitment level.
ReplyDeleteMy understanding is that similar expectations/levels exist in the performing arts. Only in academics is knowledge and skill disdained.
You know - there actually is a whole theory of coaching sports that I **think** is constructivist or close-to.
ReplyDeleteIt's called "Hard First."
I love that name because that's exactly what constructivist schools are like: hard first.
Research in Kindergarten, writing papers in 1st grade, etc.
The woman who may have invented it has a Ph.D. in educational psychology.
"Decision Training (DT) is a new, research-based approach to coaching that increases the opportunities your athletes have to make decisions in training similar to those encountered in competition."
ReplyDeleteIt may not be constructivist - but it certainly has common themes.
I can imagine that you might talk about an application to get them interested, and go back and work the theory BUT this will only work if you really do cover the theory.
ReplyDeleteYou mention science as a case where they do this. One problem I have with K-12 science education is that they only do the fun stuff and never get to the hard, abstract stuff. I get college students who "love science", but when you ask them what they want to do, they describe being a docent (which is fine, but not a way to make a living and not what scientists do). They want to share the love of science, but don't actually know what science is. They often aren't interested in the abstract, or in the hard work necessary to really be a scientist, and they get frustrated and drift out of STEM fields.
A while back, I read an article about high schools in disadvantaged areas called "Education for Unemployment." The over-emphasis on fun stuff in science education can have a similar impact. Students like what they've done, but don't realize it isn't real science.
I'm not saying that applications first (as a carrot) can't work, but be aware it isn't the solution some educators think it is.
The argument that most people should not be taught "higher math" such as algebra is an old one, according to A Brief History of American K-12 Mathematics Education in the 20th Century , which I quote below. I think that everyone who is capable of learning calculus and wants to should have the opportunity to do so, but the question is what fraction have the capability and the desire (and what methods of teaching can increase that fraction).
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Reflecting mainstream views of progressive education, Kilpatrick rejected the notion that the study of mathematics contributed to mental discipline. His view was that subjects should be taught to students based on their direct practical value, or if students independently wanted to learn those subjects. This point of view toward education comported well with the pedagogical methods endorsed by progressive education. Limiting education primarily to utilitarian skills sharply limited academic content, and this helped to justify the slow pace of student centered, discovery learning, the centerpiece of progressivism. Kilpatrick proposed that the study of algebra and geometry in high school be discontinued "except as an intellectual luxury." According to Kilpatrick, mathematics is "harmful rather than helpful to the kind of thinking necessary for ordinary living." In an address before the student body at the University of Florida, Kilpatrick lectured, "We have in the past taught algebra and geometry to too many, not too few."9
Progressivists drew support from the findings of psychologist Edward L. Thorndike. Thorndike conducted a series of experiments beginning in 1901 that cast doubt on the value of mental discipline and the possibility of transfer of training from one activity to another. These findings were used to challenge the justification for teaching mathematics as a form of mental discipline and contributed to the view that any mathematics education should be for purely utilitarian purposes.10 Thorndike stressed the importance of creating many "bonds" through repeated practice and championed a stimulus-response method of learning. This led to the fragmentation of arithmetic and the avoidance of teaching closely related ideas too close in time, for fear of establishing incorrect bonds. According to one writer, "For good or for ill, it was Thorndike who dealt the final blow to the 'science of arithmetic.'"11
Kilpatrick's opinion that the teaching of algebra should be highly restricted was supported by other experts. According to David Snedden, the founder of educational sociology, and a prominent professor at Teachers College at the time, "Algebra...is a nonfunctional and nearly valueless subject for 90 percent of all boys and 99 percent of all girls--and no changes in method or content will change that."12 During part of his career, Snedden was Commissioner of Education for the state of Massachusetts.13
In 1915 Kilpatrick was asked by the National Education Association's Commission on the Reorganization of Secondary Education to chair a committee to study the problem of teaching mathematics in the high schools. The committee included no mathematicians and was composed entirely of educators.14 Kilpatrick directly challenged the use of mathematics to promote mental discipline. He wrote, "No longer should the force of tradition shield any subject from scrutiny...In probably no study did this older doctrine of mental discipline find larger scope than in mathematics, in arithmetic to an appreciable extent, more in algebra, and most of all in geometry."15 Kilpatrick maintained in his report, The Problem of Mathematics in Secondary Education, that nothing in mathematics should be taught unless its probable value could be shown, and recommended the traditional high school mathematics curriculum for only a select few.
What puzzles me about this story is what sort of examples this perosn is thinking about. How can you demonstrate a neat result in maths to kids who don't know any mathematical theory?
ReplyDeleteI think the proof that there is no largest prime is pretty neat, but that requires knowing what a largest prime is. Pythagoras's theorem requires knowing what a square is, and a triangle. The proof that the square root of 2 is an irrational number is gorgeous, but requires knowing what a square root and an irrational number are.
Is there something I'm missing?
i don't think he was talking about theories about primes or the Pythagorean theorem. I think he meant things like double digit addition: teach the kids to add one column at a time, and then after they have some practice doing this, go back and show them how it works.
ReplyDeleteHere's an example which I've used in an afternoon math club...
ReplyDeleteThere's a game (I don't know its name) based on the Fibonacci series. If you know the basis for the game you can play an opponent and beat them every time. I do this with kids until they start to show some sign of frustration with me. Then I expose the secret.
I usually dispatch this player (who knows the secret now) to go find a sucker in the room to repeat the process. At the end of an hour everybody knows the secret but a few are really sucked in to the series, fascinated with it.
Weeks later I can ask these kids to write out a Fibonacci and they do it with ease. The others, not so much.
With Pythagoros you could have kids feed you right triangle leg dimensions while you mysteriously spit out correct hypotenuse dimensions. Make it a game. Get them sucked in, then the reveal. Some will want to know how it works. Some will want to know why it works. Some will want to know how to derive it. Those are the layers you see at recess. The best are the ones that want to derive it. As a teacher you want to find some value for each layer that kids can take away.
I realize that David Snedden was a product of his times, but that quote makes me want to smack him. Boys may somewhat outnumber girls at the high end of the math ability spectrum, but nowhere near 10:1. I think the recent data coming out of Johns Hopkins' CTY is something like 2:1.
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