I am looking at the Kahn discussion from the outside since I have been out of the classroom for several years. I admit my first look was one of dissapointment as it does appear to be procedural. It does a good job of procedural, however. They seem to get the math right. Were it riddled with errors, it would deserve severs criticism.
I am once again impressed by P.J.'s take on this and it got me to thinking about print paterials that are similar.
I am willing to bet, if I beleived in betting, that most of you who are over thirty posses(ed) copies of Schaum's outline for something. Calculus was a big seller. There was not much more than worked out examples, but many students found tham very helpful. I suppose they still exist. They served a purpose. I think there was a similar product for novels, Cliff's Notes, I think. Cliff's Notes presented a summary of books like War and Peace in thirty pages. This was certainly not the same as reading War and Peace, but it did help some students wade through the novel and keep track of who was who and what was going on. Scahum's did the same thing for math and science. No one that I know ever used one as a replacement for a textbook, but thousands of students learned important procedures from them.
Am I too far from being accurate to say that Kahn Academy presentations are a digital version of Cliff's Notes/ Schaum's outline, or am I missing something important?
By the way, I found Scham's usefull as a source of worked out examples. I could give my students one of their problems to work without having to create one and work it out to makes sure it "worked out nice".
Wednesday, March 20, 2013
Tuesday, March 19, 2013
What value Khan?
It's almost become a party game among my math educator friends to talk smack about Khan Academy. "The lessons are just procedural!" (not always true--I've seen some conceptual explanations). "There's no effort to build in the Standards for Mathematical Practice!" (Mostly true.) "Some lessons reinforce common underlying misconceptions." (I haven't seen them, but it's plausible.) And so on.
What follows is an open letter to those friends of mine -- and superstar math educators around the country -- who take these positions. I don't think they're wrong. But they are short-sighted. Read on to find out why.
Friends--
At the risk of stating the obvious, you aren't run-of-the-mill math teachers. Of course you can envision--indeed, give daily--lessons that are more in-depth, challenging, authentic, inquiry-based, etc., than Khan Academy. Indeed, I would be shocked if you couldn't.
But that's not the question. Khan Academy wasn't created for yourstudents. It was created for kids whose teachers, in many cases, don't even know the content, much less how to present it clearly or explain it well. Have you been to the elementary schools in my district? Because (as many of you know) something like half the freshmen who come to my school can't use a protractor to measure an obtuse angle --- they tell me it's 61 degrees or something cockamamie like that --- and they TOOK A TEST to get into my school (indeed, the cutoff score for my school is over 800 out of 900 possible points; we rejected more than 2000 kids out of the 2400 who applied). Those kids will get more effective instruction from Khan Academy than they can get in a regular classroom, because right now they aren't getting effective instruction in their regular classrooms, period. (I'm not blaming anyone in particular here, simply making the tautological claim that instruction that doesn't result in kids being able to do the things they are being instructed in how to do is, by definition, not effective.)
It works for other kids too: my daughter was far ahead of her class last year, and for the first half of the year did worksheets in the back of the room. For the second half of the year, she and two friends got to go on Khan Academy and pick their own lesson every day, and she grew more (as measured by NWEA/MAP scores) in that semester than in the previous 1.5 years combined. And she got about a quarter of the way through a standard Algebra I course.
Finally, I'd say this--about flipped classroom stuff generally and KA in particular. Right now, I'm cooking up a pot of Cincinnati Chili (mmm...can you smell it?). It's delicious, nutritious (yay low-fat turkey!), and my kids love it. But there's a diner down the street from my house, and any day I want, I can go there and get a reasonably tasty, reasonably healthy meal at a reasonably low price. And so every week or two--when I'm too tired, or we have nothing in the refrigerator--we go there for dinner. It's not Tru, or Topolobampo, or any of the other great restaurants Chicago is known for--but it's a reasonable way to get fed once in a while. I think KA and other online videos are like that: not as good as the best (although maybe if you watch the first lecture of the Udacity physics series, on Eratosthenes' measure of the circumference of the earth, you'd be surprised). But KA delivers reasonably clear, correct instruction to people who might not otherwise have access to it. Friends who have expressed skepticism about the "All Khan, All the Time" approach: I agree wholeheartedly. Let's give our kids a balanced diet of different kinds of instruction and different ways of thinking about problems. But I don't think that's a reason to trash on Khan altogether.
At the risk of stating the obvious, you aren't run-of-the-mill math teachers. Of course you can envision--indeed, give daily--lessons that are more in-depth, challenging, authentic, inquiry-based, etc., than Khan Academy. Indeed, I would be shocked if you couldn't.
But that's not the question. Khan Academy wasn't created for yourstudents. It was created for kids whose teachers, in many cases, don't even know the content, much less how to present it clearly or explain it well. Have you been to the elementary schools in my district? Because (as many of you know) something like half the freshmen who come to my school can't use a protractor to measure an obtuse angle --- they tell me it's 61 degrees or something cockamamie like that --- and they TOOK A TEST to get into my school (indeed, the cutoff score for my school is over 800 out of 900 possible points; we rejected more than 2000 kids out of the 2400 who applied). Those kids will get more effective instruction from Khan Academy than they can get in a regular classroom, because right now they aren't getting effective instruction in their regular classrooms, period. (I'm not blaming anyone in particular here, simply making the tautological claim that instruction that doesn't result in kids being able to do the things they are being instructed in how to do is, by definition, not effective.)
It works for other kids too: my daughter was far ahead of her class last year, and for the first half of the year did worksheets in the back of the room. For the second half of the year, she and two friends got to go on Khan Academy and pick their own lesson every day, and she grew more (as measured by NWEA/MAP scores) in that semester than in the previous 1.5 years combined. And she got about a quarter of the way through a standard Algebra I course.
Finally, I'd say this--about flipped classroom stuff generally and KA in particular. Right now, I'm cooking up a pot of Cincinnati Chili (mmm...can you smell it?). It's delicious, nutritious (yay low-fat turkey!), and my kids love it. But there's a diner down the street from my house, and any day I want, I can go there and get a reasonably tasty, reasonably healthy meal at a reasonably low price. And so every week or two--when I'm too tired, or we have nothing in the refrigerator--we go there for dinner. It's not Tru, or Topolobampo, or any of the other great restaurants Chicago is known for--but it's a reasonable way to get fed once in a while. I think KA and other online videos are like that: not as good as the best (although maybe if you watch the first lecture of the Udacity physics series, on Eratosthenes' measure of the circumference of the earth, you'd be surprised). But KA delivers reasonably clear, correct instruction to people who might not otherwise have access to it. Friends who have expressed skepticism about the "All Khan, All the Time" approach: I agree wholeheartedly. Let's give our kids a balanced diet of different kinds of instruction and different ways of thinking about problems. But I don't think that's a reason to trash on Khan altogether.
Sunday, March 3, 2013
What's Wrong With Grade "Inflation"?
At dinner last night, I was talking to a friend involved in education, and she was pressing me hard on what she perceived to be grade inflation at my school. I could have argued the facts more vigorously: the kids she meets are applicants for an ultra-elite college (she's an interviewer), so when they say "I'm getting A's and so are my friends," that's hardly a representative sample of the class. But I admitted that our bell curve is centered on a B--probably a high B--rather than on a C. And then we started talking past each other: her interpretation seemed to be that, because our kids are really smart and do their work, our attitude was something like "They probably deserve A's, so why not just give them A's?" Her response to this hypothetical motive was to ask me "What do you do to differentiate students?"
In fact, it's more complicated than that, and at 24 hours remove, I feel more clearly the need to challenge the entire premise of her argument. (And, to be fair to my friend, this is a common argument. So even if I've misattributed it to her, it's an argument worth discussing.) You see, my goal isn't primarily, or even partially, to differentiate students. I understand that that's something colleges wish I would do, that the entire college admissions system depends on using grades (and test scores) as differentiators. But I'm a teacher, and so my goal is, primarily, to teach. And at some level, that's the opposite goal.
At the beginning of my course, I have to figure out two essential questions. The first--really the a pair--looks at the present and immediate past: what do my students know and what can they do? The second pair looks towards the future: at the end of my course, what do I want my students to know, and what should they be able to do? If I'm waxing philosophical, I add a third essential question: ten years from now, what do I want them to retain of the experience of having been in my class?
Once I've articulated my standards, students who meet those standards get good grades: A's and B's. In fact, when my students get mostly A's, I generally feel like I've done a good job: it means that I've gotten most of my students to master all or almost all of my standards. When my students get C's, D's, or F's, that's supposed to tell them that there's room for improvement. And it tells me that there's room for me to improve, too: because especially when you're working with children, you can't take their attitudes and behaviors as a given. If a kid struggles and doesn't do homework, I wonder what I could do to convince him or her that the homework is worthwhile, that time spent doing these problems will be fruitful in some crucial way.
Now I'm not saying that every time a class gets all A's, everything's hunky-dory. Sometimes it's a sign that the time has come to raise standards, to demand more of students. If you have kids who are currently scoring at the 40th percentile, then when they get all A's, you have external evidence that they have room to grow (that 40th percentile score), and internal evidence that they have the capacity to do it (they're doing everything you ask and are being successful). So then you raise standards. The same might be true if your kids are scoring at the 80th percentile, or even at the 90th--it depends on what your overall goals are. You might change your standards: spend less class time on the stuff they're clearly mastering, and add in projects or more exploratory work. (Some of these changes might decrease test scores but reflect important long-term goals that standardized tests rarely assess.) Partly it depends, too, on what it takes for your students to meet those standards: are they getting A's easily, or are they doing multiple retakes, asking lots of good questions, etc.?
Where do classroom standards come from? In many cases, the common core, or state directives. In the case of the class I'm teaching, they come from my reading of comparable classes taught at the University of Chicago and the University of Illinois. I have external validation--from those schools--that the things I want my kids to be able to do are reasonable goals for honors first-year math majors. I have internal validation that few, if any, of my students find the coursework easy; they all make some mistakes, and many come back for multiple retakes of my quizzes and tests. So yeah, if at the end of the semester almost all of my kids can do those things, then almost all of my kids will get A's. I'm not going to run around trying to ratchet up standards and lower the number of A's. I'm going to be glad that they, and I, have done our jobs. And if colleges can't tell the difference between them without actually reading the two-page single-spaced recommendations I write for the majority of my students--well, that's their problem.
In fact, it's more complicated than that, and at 24 hours remove, I feel more clearly the need to challenge the entire premise of her argument. (And, to be fair to my friend, this is a common argument. So even if I've misattributed it to her, it's an argument worth discussing.) You see, my goal isn't primarily, or even partially, to differentiate students. I understand that that's something colleges wish I would do, that the entire college admissions system depends on using grades (and test scores) as differentiators. But I'm a teacher, and so my goal is, primarily, to teach. And at some level, that's the opposite goal.
At the beginning of my course, I have to figure out two essential questions. The first--really the a pair--looks at the present and immediate past: what do my students know and what can they do? The second pair looks towards the future: at the end of my course, what do I want my students to know, and what should they be able to do? If I'm waxing philosophical, I add a third essential question: ten years from now, what do I want them to retain of the experience of having been in my class?
Once I've articulated my standards, students who meet those standards get good grades: A's and B's. In fact, when my students get mostly A's, I generally feel like I've done a good job: it means that I've gotten most of my students to master all or almost all of my standards. When my students get C's, D's, or F's, that's supposed to tell them that there's room for improvement. And it tells me that there's room for me to improve, too: because especially when you're working with children, you can't take their attitudes and behaviors as a given. If a kid struggles and doesn't do homework, I wonder what I could do to convince him or her that the homework is worthwhile, that time spent doing these problems will be fruitful in some crucial way.
Now I'm not saying that every time a class gets all A's, everything's hunky-dory. Sometimes it's a sign that the time has come to raise standards, to demand more of students. If you have kids who are currently scoring at the 40th percentile, then when they get all A's, you have external evidence that they have room to grow (that 40th percentile score), and internal evidence that they have the capacity to do it (they're doing everything you ask and are being successful). So then you raise standards. The same might be true if your kids are scoring at the 80th percentile, or even at the 90th--it depends on what your overall goals are. You might change your standards: spend less class time on the stuff they're clearly mastering, and add in projects or more exploratory work. (Some of these changes might decrease test scores but reflect important long-term goals that standardized tests rarely assess.) Partly it depends, too, on what it takes for your students to meet those standards: are they getting A's easily, or are they doing multiple retakes, asking lots of good questions, etc.?
Where do classroom standards come from? In many cases, the common core, or state directives. In the case of the class I'm teaching, they come from my reading of comparable classes taught at the University of Chicago and the University of Illinois. I have external validation--from those schools--that the things I want my kids to be able to do are reasonable goals for honors first-year math majors. I have internal validation that few, if any, of my students find the coursework easy; they all make some mistakes, and many come back for multiple retakes of my quizzes and tests. So yeah, if at the end of the semester almost all of my kids can do those things, then almost all of my kids will get A's. I'm not going to run around trying to ratchet up standards and lower the number of A's. I'm going to be glad that they, and I, have done our jobs. And if colleges can't tell the difference between them without actually reading the two-page single-spaced recommendations I write for the majority of my students--well, that's their problem.
Wednesday, February 27, 2013
Math contests - continued
I feel a need to further P.J.'s comments on math contests, because I think he indadvertedly overlooked the most important part of math competition, and that is practice. Math contests are not sports. It matters not if you win or lose, what matters is that you try your best to do as well as you can. for most students, and teams hopefully, this does NOT involve endless drill and practice, nor does it have anything to do with memorizing endless formulas. Granted some teams do this, and they are often "winners" but they have missed the entire point of having a math team.
Math team is a place where students who like math and are interested in math, and may be talented in math, have a chance to gather together and work on interesting problems together. It is a chance to share creative insights and to learn about things that are not usually taught in the classroom. Math team practice is where I can ask the problems that are too hard to ask in class , and, yes, if no one can solve the particular problem, then take it home and work on it. The message I hope most coaches send is that we are here to learn math, to work on interesting problems and to have fun doing it. The contest itself is the carrot, but by no means is it why we spend so much time doing this. Out hope is that some of our mathletes will find that math is not about grades, nor is it about winning. It is about working on interesting problems with others who also think it is fun to work on interesting problems.
One of my favorite stories about math team, one I have told countless times, and one that convinced me that math contests are wonderful things, involves a young student named Eric Winfree. It was clear that Eric was brilliant but definitely not fast. He could solve the problems but often not within the artificial constraints set up by contests. One week he failed to make the Chicago area ARML team and did not qualify for theAIME. I talked to him on Friday of that week, and expected to find him dejected. Instead, he was cheerful, even robust. I conversed with him and he said I got all of the problems this week. I asked him what he meant, and he said that he had correctly solved all of the ARML tryout problems and all of the AMC problems by Thursday night, and three of them were really cool. Let me show you what I learned.
Eric is now a working scientist, mathematician and CalTech. . He attributes math contests as one of the many important parts of his education. I always tried to get our math team students to understand Eric's attitude about math contests. I hope most math team coaches see it this way, as that is why I spent considerable time and effort during my career promoting math contests, and coaching teams that did well but didn't win very often. I count my time spent with this as some of the most important successful time I spent doing anything, because many students were drawn into the math fold in part because of math contests.
Math team is a place where students who like math and are interested in math, and may be talented in math, have a chance to gather together and work on interesting problems together. It is a chance to share creative insights and to learn about things that are not usually taught in the classroom. Math team practice is where I can ask the problems that are too hard to ask in class , and, yes, if no one can solve the particular problem, then take it home and work on it. The message I hope most coaches send is that we are here to learn math, to work on interesting problems and to have fun doing it. The contest itself is the carrot, but by no means is it why we spend so much time doing this. Out hope is that some of our mathletes will find that math is not about grades, nor is it about winning. It is about working on interesting problems with others who also think it is fun to work on interesting problems.
One of my favorite stories about math team, one I have told countless times, and one that convinced me that math contests are wonderful things, involves a young student named Eric Winfree. It was clear that Eric was brilliant but definitely not fast. He could solve the problems but often not within the artificial constraints set up by contests. One week he failed to make the Chicago area ARML team and did not qualify for theAIME. I talked to him on Friday of that week, and expected to find him dejected. Instead, he was cheerful, even robust. I conversed with him and he said I got all of the problems this week. I asked him what he meant, and he said that he had correctly solved all of the ARML tryout problems and all of the AMC problems by Thursday night, and three of them were really cool. Let me show you what I learned.
Eric is now a working scientist, mathematician and CalTech. . He attributes math contests as one of the many important parts of his education. I always tried to get our math team students to understand Eric's attitude about math contests. I hope most math team coaches see it this way, as that is why I spent considerable time and effort during my career promoting math contests, and coaching teams that did well but didn't win very often. I count my time spent with this as some of the most important successful time I spent doing anything, because many students were drawn into the math fold in part because of math contests.
Sunday, February 24, 2013
Why compete?
My friend Cathy has written extensively about why math contests suck, and there's a lot to what she says. Many--I'd even concede "most"--contests encourage high-speed, single-step problem-solving rather than thoughtful analysis, or the kind of synthetic work that leads to new ideas and big theorems. They also discourage kids who don't have the particular skill- and mind- sets that lead to success on math contests:
- "Pyrotechnic" problem-solving ability.
- Near-perfect recall for theorems and situations that come up often.
- Ability to calculate by hand, flawlessly (sadly, because of the huge inequities and arms races that can result, many math contests--including my own ARML--don't let participants use calculators).
- Willingness to push forward with a single solution rather than considering all options carefully.
The real misfortune is when kids lacking in these mind/skill-sets "learn", not just that they're not good at math contests, but that they're "not good at math," period.
So is there a benefit to competition?
I would suggest three:
- The opportunity for talented math students to learn that they have a lot to learn. Even our strongest students wind up doing not-so-well on contests once in a while, and it's a reminder that math isn't always easy--and won't be.
- The opportunity for talented math students to learn that hard work can pay off. For many strong math students, the experience of math class is that what they need to learn to do well comes naturally--so they don't learn to connect effort with success in the context of mathematics. Our arch-rival Whitney Young is currently the second-ranked high school math team in the state, not because their kids are smarter than everyone else's, but because they work extremely hard--an hour every day, several hours virtually every Saturday. (Our team mostly feels like we've reached an optimal point on the effort-reward curve: we work pretty hard, practicing 3-4 hours per week, and make it in the top ten of the state; our students have more time and freedom to do other activities. But I digress...) They know it; their students know it; and what they've learned is more useful than any theorem or formula.
- The opportunity for students to learn to lick their wounds. My son plays competitive chess, and one of the hardest parts of chess tournaments is their duration: you play four or five games in a day, and if you lose one, you still have two or three ahead of you. I've seen Jonah learn to dust himself off after a loss and go back in swinging (okay, that's a metaphor). And I've seen the same thing in math team: students screw up a contest, and instead of saying "We're dumb" or "the questions were dumb," students can be taught to go back, study the contest, and practice hard for the next one. That's another powerful lesson.
I wish there were more other ways for kids to learn these lessons--by doing authentic mathematics research at an age-appropriate level, for example (New York and Chicago now have math fairs that do for mathematics what science fair do for science), or just by having that scrape-your-knuckles-and-try-again experience in math classes. But the lessons I cite are important ones, and important ones to learn about math.
Wednesday, February 20, 2013
Preparing Students for College
Again this week, I heard the justification for an absurd policy (in this case: if you make up a standards-based quiz, you can't get as high a score as you would have if you had passed it the first time) being that it "prepares students for college, where you don't get second chances." (I've also heard the same thing said about the real world.) This may be the stupidest justification for educational malpractice I've heard, for two reasons.
- In college, and even more, in the real world, people are mostly reasonable. How many times have you handed in some paperwork late for your job, or forgotten some important thing, and had people essentially say "It's okay, don't do it again" -- or even (gasp) not mention it at all? In college, I memorably missed a make-up test for my German class because (soooo embarrassing) I mistranslated my teacher's instructions about time (given in German) and showed up an hour late. I didn't fail, or get a zero, or even points off. He laughed, said he'd wondered where I was, and then gave me the test anyway.
The world itself is not reasonable; it obeys the laws of physics, which are notoriously amoral. So the water in the pipe to my outside spigot really will freeze if I forget to drain it before a cold snap. But if I forget to renew my license plate sticker, I pay a fine and get on with my life. Sure, there are exceptions, and we love to spread those stories around--they're like fishing stories, only in reverse. In real life, people get second chances: they're accepted back into colleges, or even elected vice-president (for two terms) after being caught red-handed plagiarizing.
In fact, I've noticed that (at least for me) the circles I inhabit have gotten more reasonable: while I have many notable frailties, I've learned workarounds over time (email myself any important information, have a phone that gets email, etc.), and I'm good at enough things that the people around me are willing to put up with the things I'm not good at. I think that's true of most professional people: we work our way into niches where we get to spend most of our time doing things that we're either pretty good or trying to get better at, and only a small fraction on things that we truly dislike and are abysmal. Real life is not high school. To paraphrase Dan Savage, "it gets better." - EVEN IF college were the one-strike-you're-out system these teachers say it is, it seems obvious to me that the number one way to prepare kids for college is to actually teach them the academic skills and habits they will need to be successful there. Sure, it's important to get things in on time, and to do well on quizzes the first time you take them. But "on time" won't save a literary analysis paper that's an incoherent mash of plot summary and personal reflection: you have to be able to read and write, too. And it's hard to learn calculus without a solid grounding in functions, graphs, and algebra. So high schools should adopt policies that encourage students to go on developing those core skills, and recovering from their mistakes, rather than telling them that mistakes are insurmountable. And isn't that exactly what "no late work" or "no retake" policies say?
The point of draconian policies in high school is to discourage kids from making mistakes. But everyone makes mistakes; the point is to learn from them. So I'm not saying there should be no penalties, ever, for late work, or for screwing up an assignment the first time (I rather like "you have to show me you can do it right."). But those penalties should give kids incentives to learn, not teach them the mostly-false lesson that you can't recover from your mistakes. Because in the real world, people make mistakes all the time, and learn from them. Wouldn't teaching kids how to do that be the best preparation of all?
Monday, February 11, 2013
At last, an iPad game that engages mathematics in an interesting way!
OK, so that title is a little overstated. After all, I (and my kids) love WolframAlpha, and especially the WolframAlpha fractals app--basically an interactive dictionary of fractals, but oh-so-cool!
No, what I'm talking about is about the world of "math games" that really just consist of drills of basic facts, sussed up so that adding numbers blasts an alien or something. Math games, in other words, that practice very low-level skills and that don't work to develop problem-solving, thoughtful application of skills, multiple pathways, etc.... in other words, math games that turn math into a kind of spelling practice.
And then there's Dragon Box.
In Dragon Box, you drag cards around a two-sided playing field (looks like a tennis court, behaves like an _ _ _ _ _ _ _ ) to isolate the "dragon box", which is represented by a picture of a sparkling crate [screenshot from the PC version, because it's easier for me to use with Blogger]:
In the picture above, I've just added a purple fly to the left side to cancel out the black fly -- totally allowed, although unorthodox -- and the computer is prompting me to deal out another purple fly on the right side. By merging the flies together, I can make a "galaxy" (zero) and isolate the box!
As you go through the levels, two things happen. First, the tasks get more complicated: you start having to flip the cards to their opposites before adding them (in this example, I was given the opposite card); the cards at the bottom include ones you don't need to use; the game adds representations of multiplication and division. (Yes, as with addition, the program forces you to distribute correctly, at least in the first 60 or so levels that I've worked.) Second, the program spirals in and out of representations that look much more like traditional algebra:
No, what I'm talking about is about the world of "math games" that really just consist of drills of basic facts, sussed up so that adding numbers blasts an alien or something. Math games, in other words, that practice very low-level skills and that don't work to develop problem-solving, thoughtful application of skills, multiple pathways, etc.... in other words, math games that turn math into a kind of spelling practice.
And then there's Dragon Box.
In Dragon Box, you drag cards around a two-sided playing field (looks like a tennis court, behaves like an _ _ _ _ _ _ _ ) to isolate the "dragon box", which is represented by a picture of a sparkling crate [screenshot from the PC version, because it's easier for me to use with Blogger]:
In the picture above, I've just added a purple fly to the left side to cancel out the black fly -- totally allowed, although unorthodox -- and the computer is prompting me to deal out another purple fly on the right side. By merging the flies together, I can make a "galaxy" (zero) and isolate the box!
As you go through the levels, two things happen. First, the tasks get more complicated: you start having to flip the cards to their opposites before adding them (in this example, I was given the opposite card); the cards at the bottom include ones you don't need to use; the game adds representations of multiplication and division. (Yes, as with addition, the program forces you to distribute correctly, at least in the first 60 or so levels that I've worked.) Second, the program spirals in and out of representations that look much more like traditional algebra:
Notice the c and -c cards?
And this brings me to a big-picture musing. My son, in third grade (and not yet doing official Algebra) loves this game. He's learning a set of rules, without worrying too much about the conceptual underpinnings. In recent decades, progressive teachers have moved away from this model: everything should be explained when it's introduced, so that it feels natural. And yet, I'm not so sure. If we wait until after Jonah has played 200+ levels of this game and does the right moves every time to explain the underlying algebra--what have we lost?
My experience in learning mathematics was that often self-contained systems only made sense later. And I don't think that's a terrible thing. The sum and difference identities for trig functions? The first important thing for 99% of students is just to know and be able to apply those identities--and whom do we serve if we spend the first twenty minutes of the period in a lengthy derivation that leaves students angry and confused? And anyway, some of the proofs I've seen -- using matrix multiplication and rotation formulas, for example, as my own UCSMP book does -- aren't necessarily that convincing.
Of course, I know that it's possible to motivate these formulas in lots of ways. My own class on sin(A + B), for example, started with the example of sin(A + π/2) ≠ sin
A + sin π/2 because of all the cool ways we could see that the "identity" isn't true; then, graphing on the calculator, we came up with a better answer that we could justify using the unit circle. But then I went ahead and taught the "right" rules, and we spent the rest of the period learning how to apply them. The following day was when we went back and came up with some proofs (using geometry, actually).
There are lots of questions, besides "why is it true?", that engage a student in understanding a formula:
- What happens when we swap the variables around or change them in some other way?
- What versions of this formula correspond to cases we already know? Does this formula yield the results we've come to expect?
- What about extreme cases?
For example, when I teach Hero's formula in Geometry--which, despite the fact that the proof is beautiful, I do without proof, we discuss the following:
- Does changing the order of the sides change how the formula works? The area of the triangle?
- What if a = b = c, so that the triangle is equilateral?
- What happens if a + b = c, so that the triangle is degenerate?
- Why are there four terms under the square root sign? What are the units of the variables and of the answer?
For the vast majority of my honors geometry students, these questions provide a lot of thinking--a lot more food for thought than the kneejerk "Why is it true?"
Obviously, we do a terrible disservice to kids when we say "Math is a collection of rules; your job is just to learn them." The thing most of my strongest students like about math is that there's so little to memorize. But I think it's okay to say "Here, we're going to learn to play with this set of rules for now, and then we'll figure out why they work or what they apply to." At least once in a while.
Agree? Disagree? Let me know. But if you have kids -- and even if you don't -- get DragonBox. What a blast!
Subscribe to:
Posts (Atom)