Math and science can be hard to learn–and that’s OK. The proper job of a teacher is not to make it easy, but to guide students through the difficulty by getting them to practice and persevere. “Some of the best basketball players on Earth will stand at that foul line and shoot foul shots for hours and be bored out of their minds,” says Williams. Math students, too, need to practice foul shots: adding fractions, factoring polynomials. And whether or not the students are bright, “once they buy into the idea that hard work leads to cool results,” Williams says, you can work with them.
There are plenty of lines to cringe at in Kakaes’ article. PJ Karafiol knocks down most of them in a great post that was eventually syndicated by Slate. (Good for Slate. Good for Karafiol.) Mr. Williams’ metaphor deserves extra scrutiny, though. Here are just two of its most screwy aspects:
- Drills aren’t a basketball player’s first, only, or most prominent experience with basketball. Drills come after a student has been sufficiently enticed by the game of basketball – either by watching it or playing it on the playground – to sign up for a more dedicated commitment. If a player’s first, only, or most prominent experience with basketball is hours of free-throw and perimeter drills, she’ll quit the first day – even if she’s six foot two with a twenty-eight inch vertical and enormous potential to excel at and love the game.
- Basketball players aren’t bored shooting foul shots. Long before “math teacher” was on my resume, I was a lanky high school basketball player trying to get his foul shooting above 50%. I’d shoot for hours but I wouldn’t get bored, as Williams suggests I must have been. That’s because I knew my practice had a purpose. I knew where that practice would eventually be situated. I knew it would pay off in a game where I’d be called to the line for a shot that had consequences.
There is a place for drills and explanation in mathematics, as in basketball. But consider what little good they do in either arena if the student isn’t first made aware of the larger, more enticing purposes they serve.
BTW. The worked examples literature leans heavily on De Groot’s research into chess masters who, it turns out, have memorized an enormous number of board configurations relative to casual players. This is unsurprising in the same way it’s unsurprising that professional basketball players practice their free throws much more often than amateurs. But it doesn’t necessarily follow from either of those facts that the best way to start inducting new members into either of those groups is to force novice chess players to memorize board configurations for hours or new basketball players to shoot hours of free throws from the line.
BTW. Max Ray articulates a strong framework for technology use in the math classroom at the end of his recent post at the Math Forum.
2012 Jul 11. PJ Karafiol follows up.
Featured Comment.
Am I the only one who is reminded of The Karate Kid? Mr. Miagi has Daniel do crazy, de-contextualized drills without knowing their purpose. In the end it works (because it’s a movie), but in the meantime Daniel gets extremely frustrated and wants to give up. Perhaps if Mr. Miagi had made it more clear what the “cool results” would be or how he would be “painting the fence” and “sanding the floor” in a tournament, Daniel-san would have been more than happy to wax all his cars.
34 Comments
Michael P (@mpershan)
July 8, 2012 - 3:22 pm -Reminds me of this:
[Source: Let the Games Begin]
Matt
July 8, 2012 - 3:26 pm -Makes me wonder:
1) What makes someone good at basketball, enjoying/loving/having an emotional attachment to basketball or practicing the mundane aspects of basketball for hours?
2) Can you get good at basketball without loving it?
3) If you were trying to win a basketball game, would you want someone who loved basketball or who was good at basketball?
richkonar
July 8, 2012 - 6:25 pm -It’s difficult to compare sports and academics. Sports are more focused than mathematics. In sports we play basketball, baseball, football, etc. individually, we never play them altogether in a one hour class.
I think the “end result” is a major factor with drilling. In sports, we know if we practice hard enough a certain skill we will get better and we can “see” the result quickly as the previous posters have stated. In math, most people don’t get to calculus to see the end result of algebra & geometry.
For the previous poster,
(1) Its fun for them and they are competitive/don’t like losing.
(2) Yes, many pro and college players state that they don’t love the game and do it because it pays them or got them a scholarship.
(3) you would want a mix of both, just because you love something doesn’t mean you’re good at it; and conversely if you’re good at something doesn’t mean you love it. If the game is on the line, put it in hands of your best players, unless he’s Scotty Pippen.
blink
July 8, 2012 - 6:38 pm -If we only saw the end of the quote — “once they buy into the idea that hard work leads to cool results” — perhaps we would all be on the same page. Yes, the foul shooting analogy is silly and misleading and ought to be called out. Still, if the feuding factions can recognize that they share the same deep goals, this may be a productive step. I hope that these articles point in that direction.
Fritzky
July 8, 2012 - 7:06 pm -This is the kind of post that really hits home. Does anyone else have kids in the halls of their schools practicing multiplication facts on index cards? I’m not sure the kids have really bought into achieve something great. I feel as though most kids are now spectators in both math and sports. Today’s teachers have a difficult responsibility because students do not have experience overcoming obstacles in any walk of life, whether it is sports, relationships or school.
mr bombastic
July 8, 2012 - 8:29 pm -I doubt that telling a lazy high school basketball player that Kobe Bryant shoots hundreds of shots a day is going to provide much motivation. The problem with the sports analogy is that the same motivation problems in the classroom also appear in the locker room. I agree athletes and musicians usually are not “bored out of their mind” when practicing drills, but they often are a little bored and much less focused than they are during a game or concert. Ask a coach at your school if you don’t believe me, or Allen Iverson.
Is Mr. Williams arguing for drills as a math students first, only, or most prominent experience with math?
I think drill has a role, but it is hard to argue that we need more drill in the typical US math curriculum — probably better drills — but not more. Also hard to argue that we shouldn’t at least scrimmage every now and then in math class.
Lance Bledsoe
July 9, 2012 - 4:30 am -Learning basketball is a pretty good analogy for learning math, but it’s certainly not perfect. One big difference between the two is that most basketball players, even young kids, actually want to be on the court practicing and learning the game, at least to some degree; kids who aren’t interested in basketball find something else to do with their time. They don’t show up to practice, and coaches don’t have to spend time and effort trying to entice those kids to love the game. Math is something that kids are forced to learn whether they have any interest in it or not.
I think having a job may be a better analogy for learning math. Some people really love their jobs and they work really hard at them. They’re lucky enough to have found a job in which they get paid to do something they love. Other people have jobs they don’t like very much, or maybe even despise, but they can’t quit because they have to pay the rent and they don’t have any prospects for getting a better job. So they’re just there for the paycheck. They’re not going to start loving their job just because they see why it’s valuable to their company or valuable to society. So they try to figure out how to make their job as bearable as possible, put as little into it as they can, and live for the weekends.
That’s not to say that a good boss won’t try to make the distasteful aspects of a job less distasteful, but some people just don’t like their jobs. There’s only so much the boss, even a really good boss, can do about that.
Lynn
July 9, 2012 - 6:01 am -On the first day of school, I walk into my middle school math class with my golf clubs. We don’t start off talking about rules or going over the syllabus…I just set my bag down and pull out my putter. I explain how it was the first club I learned to use and how it’s a pretty easy club to control. But how good of a golfer would I be if I only knew how to use that one club? Next I take out my 5-iron and explain how it’s my absolute favorite club! I can tee off with it and use it in the fairway, but if I want to get better at my game, I need to expand my use of clubs for different shots. Finally I take out my 1-wood and explain that I have a terrible slice. It’s not my favorite club, but when I hit it just right, it goes twice as far as my 5-iron. As I pull out the clubs and talk about my love for golf, I make some comparisons with math along the way…algebra is like my 5-iron, calculus is my 1-wood, etc. I am a better golfer when I can approach a shot and decide on the best club to use in my bag. I’m better at problem solving when I have a variety of strategies to choose from. One of the students’ first assignments is to do the same presentation with a sport/activity they enjoy. During the presentation, students identify a simple/basic skill in the sport/activity and compare it to something that comes easily to them in math. They also identify more complex skills and identify something they’d like to improve upon in math. I love hearing about their interests and we often refer back to those presentations as we’re learning new things throughout the year.
I don’t know if this is any better than the basketball analogy in this post and I’d be interested in hearing your thoughts on this approach and/or if you have any suggestions for improvement.
Barry Garelick
July 9, 2012 - 7:11 am -I used to enjoy doing factoring problems and the many other types of algebra exercises that people are convinced turn kids off to algebra. I saw the merit of algebra from the get-go, because we used a book that gave us word problems almost from day one, and I could see its usefulness, even if the word problems would be met with disdain by those who align with this blog.
Also, I was intrigued with algebra for years and couldn’t wait until I could take it. The idea of using letters to solve things appealed to me. So, I had no problem doing factoring or any of the other types of drills that many claim turn kids off to math.
I think the baseketball analogy is just fine. I would agree that kids practicing foul shots are not bored because they have an end result in mind.
Also, anyone who knows Vern Williams knows what results he’s been getting for the 30 or so years he’s been teaching math. And I’m not just talking about gifted students.
Kari
July 9, 2012 - 7:12 am -Every one of your posts is thought-provoking, but I must mention that each time you use the pronoun SHE, my heart sings. The absence of this pronoun in much of educational writing is insidious in its invisibility. Thanks!
BalancEdTech
July 9, 2012 - 7:35 am -Nice version of the sports metaphor in David Perkins’ book Making Learning Whole: How Seven Principles of Teaching Can Transform Education.
http://www.uknow.gse.harvard.edu/teaching/TC326.html
Bob Hansen
July 9, 2012 - 9:04 am -My son’s teacher last year was big on times table drill, even to the point of annoying me, but I’d be lying if I said it didn’t pay off. Just the other day I was going over units of measure with him and he was able to skip steps or even skip whole problems and jump right to the end because of his fluency and there is much more to this fluency than the mechanics of calculating. Because of this fluency he is able to recognize and factor arithmetic situations and it shows in our dialogs, not just with units, but with ratios, fractions and decimals as well. I agree that practice alone will not evoke this response but neither will just dialog. I would say that anything near 50/50 is in the zone and can be adjusted as the student desires.
The problem is that the students don’t generally hold up their 50% of the bargain.
Bob Hansen
Lauren Banko
July 9, 2012 - 10:10 am -“But it doesn’t necessarily follow from either of those facts that the best way to start inducting new members into either of those groups is to force novice chess players to memorize board configurations for hours”
This is not the conclusion to be drawn for the literature on chess players. Psychologists here would advocate that such knowledge of chess is acquired by *playing* chess.
Miss_techie
July 9, 2012 - 7:41 pm -I compare math to skateboarding with my students. There will be some cuts, bruises and major falls, but you get right back up and try again. It’s about pushing yourself creatively in math, not in one single remote skill or drill isolated.
P.J. Karafiol
July 9, 2012 - 7:47 pm -Thanks, Dan, for the shout-out. I’ve posted some thoughts about your analysis at http://anglesofreflection.blogspot.com/2012/07/more-kakaes-followup.html
Andy
July 9, 2012 - 8:02 pm -With my students, I compare parts of math to training for football. Most students agree weight lifting is important for football, but it would be dumb to lift weights on the field. Even though these analogies are good for students to hear and understand, I would disagree that teaching math should be done this way. You just miss the beauty of math when you do not connect the topics and look mostly at drills.
Matt
July 10, 2012 - 5:43 am -Additional thought: What if Dan, you have this exactly backwards?
We tend to believe that the love and enjoyment of a pursuit or activity helps us overcome the challenging, mundane or boring practice of skills that almost everyone agrees is necessary to achieve the activity at a high level. But what if the mundane practice is actually what develops our love of the activity? What if the boring repetition allows us to see the beauty in the pursuit? What if it allows us to see the hard work that others put in before us? What if it allows us to come to an understanding of the patterns and deep lying structures that many of us who love math/science see? Shouldn’t the practice come first and the love second?
I think in reality the enjoyment of an activity and the practice of an activity are deeply entwined. The more we practice something, even the boring parts, the better at it we become, and we start to enjoy it more. If we enjoy it more, we practice it more, and so this is a positive feedback cycle.
Jeff de Varona
July 10, 2012 - 7:16 am -Am I the only one who is reminded of “The Karate Kid”? Mr. Miagi has Daniel do crazy, de-contextualized drills without knowing their purpose. In the end it works (because it’s a movie), but in the meantime Daniel gets extremely frustrated and wants to give up. Perhaps if Mr. Miagi had made it more clear what the “cool results” would be or how he would be “painting the fence” and “sanding the floor” in a tournament, Daniel-san would have been more than happy to wax all his cars.
I always tell my classes you can’t factor unless you factor – so we need to practice. But we never practice before we know the purpose. They need to have a reason to factor in the first place. So we start with a complex problem for which factoring will make it possible (or at least easier) to solve.
Just my two cents.
Steve Goldberg
July 10, 2012 - 7:35 am -I am teaching a “math camp” this summer and we are “warming up” before each session with a few rounds of the game 24 — the fractions and decimals edition. I explained to the students that it’s a way to work on basics such as multiplying and dividing fractions, as well as pattern recognition. They seem to like it. And when we solve linear equations later today, it will help to know how to divide fractions. The trick for me is coming up with linear equations that seem worth solving :)
Max
July 10, 2012 - 10:13 am -Thanks for the shout-out. Reading this made me think of the following very cool thing (courtesy of Alexander Bogomolny, @CutTheKnotMath): http://www.quora.com/Mathematics/What-is-it-like-to-have-an-understanding-of-very-advanced-mathematics
It’s a mathematician explaining what his brain feels like doing math, basically. Sort of like hearing Wayne Gretzky describe how he sees the ice, or Yo-Yo Ma explaining how he feels music.
I wonder if one reason it’s hard to scrimmage in math class is because not enough people have felt or even heard articulated the feeling of powerful mathematical playfulness?
Gary Stager
July 17, 2012 - 8:17 am -Dan,
You MUST MUST MUST read this book – http://amzn.to/Mh1SPo
Ben Blum-Smith
July 17, 2012 - 11:21 am -A not-entirely-on-topic riff on the Karate Kid thread:
Actually Mr. Miagi occupies a weird point in the spectrum of mindlessly-drill-vs-see-the-purpose. Daniel does know what the big picture of what he is supposedly learning supposedly looks like. He is supposed to be learning to fight. What he can’t see (because Mr. Miyagi has, seemingly intentionally, hidden it) is the connection between what he’s doing right now (wax on, wax off) and this bigger picture. The frustration that results from this gap is actually (at least according to the film) fertile, because it makes possible what I’d pick as the movie’s most immortal scene: Daniel eventually flips out at Mr. Miyagi for all the menial tasks, and Miyagi shows him (in the most primal way possible) that he has really been learning to fight all along. From this moment on, Daniel belongs to Mr. Miyagi as a learner. Mr. Miyagi can design any learning experience he wants, and Daniel will enter it with trust.
As a grown-up teacher the pedagogy strikes me as suboptimal, mostly because since Daniel doesn’t know what he’s doing, I don’t see any reason to think he’s practicing it right. In fact, Daniel has already been painting the fence all day when Mr. Miyagi comes home and corrects his form. Was all that prior practice wasted?
That said, there’s something about Mr. Miyagi’s manipulation of Daniel’s frustration in order to get a payoff in the form of trust, that seems like an at least interesting pedagogical idea. It’s so not my style that it’s hard for me to imagine how I’d use it; but it does feel to me like there’s something there.
Ms. Warren
July 18, 2012 - 2:45 pm -Great post! I always remind my students that if you persevere through all of you mathematical challenges it will pay off.
Kevin Hall
July 19, 2012 - 8:18 am -I don’t think it’s correct that the worked example literature relies heavily on chess studies. The “experimental support” section and the references list on this page point to peer-reviewed journal articles backing up the worked example effect.
http://learnlab.org/research/wiki/index.php/Worked_example_principle
Kevin Hall
July 19, 2012 - 8:22 am -…and, I should add, none of those studies (that I’m aware of) relate to chess.
Dan Meyer
July 19, 2012 - 9:33 am -Hi Kevin, here is a search in Google scholar for articles that cite De Groot’s Thought and Choice in Chess. You’ll find John Sweller cropping up frequently.
Kevin Hall
July 19, 2012 - 11:39 am -Cool, didn’t know that. I was interpreting your comment as questioning the validity of the worked example effect for novice learners. My point was that enough experimental studies have been done on it in the math domain to address any potential limitations in De Groot’s work (with which I am not familiar). In particular, the worked example effect has so far been shown to be strongest for novice learners and to lose effectiveness as the subject’s skill develops, in what’s known as the expertise reversal effect.
This doesn’t mean it’s okay to start class with a long sequence of boring worked examples. It just means that it’s a good way to speed up and improve student’s performance on practice problems. I’m curious what you think.
Kevin Hall
July 19, 2012 - 1:49 pm -Never mind, I’m new to your blog and just noticed you’ve already had a long conversation on Sweller and motivation, mainly getting at the point the boring nature of studying worked examples. By the way, since I’m new here, I’d like to say NICE BLOG. Oh, and I’m going to try that Simpsons Sunblock activity next year, so thanks to your troll for continually bringing it up so I noticed it :)
I basically agree with the other commenter from your earlier Sweller thread, who said worked examples can be compatible with inquiry and PBL. My reading of the literature is minimally-guided instruction is most effective when it’s followed by drill practice after the inquiry phase.
(For example: Dean, D., & Kuhn, D. (2006). Direct instruction vs. discovery: The long view. Science Education, 91, 384 —397). That’s where worked examples have their place, in my opinion.
In case people think the research by Sweller (1985) is outdated, there are tons of studies since Sweller that also support the worked example effect (e.g., this one from 2010: http://www.springerlink.com/content/r7582501251686l6/ ) .
Anyways, nice blog, and thanks for all the good lesson ideas!