8 ms·

A better way to teach math

167 points by dctoedt 16y ago

asolove 16y ago
Marvelous. I am glad to see someone emphasizing structure and practice in early math curricula. While it is possible to overemphasize memorization and rote practice, the pendulum has certainly swung too far the other direction as a reaction against the Victorian knuckle-rapping methods.
- patrickyeon 16y ago
  > While it is possible to overemphasize memorization and rote practice, the pendulum has certainly swung too far the other direction Throughout my grade school and high school years (born mid-80s, Canadian), near everything was memorization. Memorize multiplication tables instead of learn to do arbitrary multiplications quickly. Memorize your table of elements instead of emphasis on what it means. Memorize how to find the roots of a quadratic function, the rules for arithmetic with fractions, physics equations... (I'm pretty bad at memorizing, so this doubly infuriated me because I had to look for the underlying relationships to do well). Even after my province introduced a new curriculum which was supposed to stop all that and focus away from memorization. Anecdotally, this is only slowly changing if at all, as new teachers come in and old ones who won't give up on rote memorization retire. I know this is all just anecdotal, but in my (ongoing) university education, I still see a lot of students put their emphasis on memorization, which suggests to me that they were taught that is what it means to 'learn.'
  - hugh3 16y ago
    Memorize multiplication tables instead of learn to do arbitrary multiplications quickly. How can you do arbitrary multiplications quickly without memorizing your times table? You can't quickly compute 47*36 without being able to reel off "six times seven is forty-two, seven times three is twenty-one, four times six is twenty-four, four times three is twelve", can you? I hated learning my times table (in fact I hated mathematics when mathematics was just arithmetic) but a certain amount of memorization is necessary and unavoidable if you're going to be able to do basic mental arithmetic, which remains an important life skill even if you do have a calculator.
    - marklubi 16y ago
      > You can't quickly compute 47*36 without being able to reel off "six times seven is forty-two, seven times three is twenty-one, four times six is twenty-four, four times three is twelve", can you? My mental process for solving your example equation was to reduce it to 50x36-3x36 which is IMHO a much simpler process. I don't know where I learned that process (probably from my dad), but simplifying the values to easily calculated approximate values and then adjusting that value for more precision seems to make much more sense than what was taught to me in school. It also caused a lot of frustration between myself and my teachers who demanded that all of the micro steps be written out instead of making what seemed to me as logical jumps. Edit: formatting
    - nantes 16y ago
      Completely off-topic, but that sounds very similar to approach taken by RISC CPU's. Be very good at doing smalls things very, very fast and then compose those operations in to higher order operations. Just saying.
fname 16y ago
The money quote: Teachers tell me that when they begin using Jump they are surprised to discover that what they were teaching as one step may contain as many as seven micro steps It's about finding out how to simplify what some may see as the easiest step. Think about the lowest common denominator and build from there.
happy4crazy 16y ago
It sounds trivial, but the idea of breaking things down into micro-steps is incredibly useful. I think it's fundamental to effective abstract thinking, and yet most people (including me!) don't consciously think this way. As an example, I was a physics major in college, so I'm used to thinking of myself as being pretty numerate. And yet, I've noticed that I'm not a very efficient learner of higher math; I enjoy it, which keeps me chugging along, but I often find myself getting discouraged when my brain doesn't automagically internalize new abstractions. Instead of approaching a new abstraction as a bundled collection of less-abstract micro-steps, I think "hmm, if I were really smart this would just sink in." Sometimes the new abstraction does just sink in, but it often doesn't, and then I feel briefly bummed about not being the radiant genius I thought I was. This is a dumb attitude! An abstraction is like a steak; you can eat it, just maybe not in one bite. And the incredible, amazing thing about abstraction is that as you get better at it, you get to take bigger bites! This all reminds me of something Kent Beck says in his TDD book: It is not necessary to work in such tiny steps as these. Once you've mastered TDD, you will be able to work in much bigger leaps of functionality between test cases. However, to master TDD you need to be able to work in such tiny steps when they are called for. Being able to do things in tiny steps is a skill, and being willing to do things in small steps when you choke on a big step, rather than feeling dumb and giving up, is the key to learning just about anything.
gersh 16y ago
This goes against math tradition where classical math texts say things like "this is obvious and left as an exercise to the reader". Implicitly, I think many mathematicians believe students should be made to struggle for their own good. The problem is that this approach requires patience, persistence and hard work. Do this help students in the long-term? If you learn math without struggling, will you learn to think in the same way? Is this teaching to the test over teaching you how to think?
- icegreentea 16y ago
  You really don't see that type of text appear until university these days. Most grade school, and highschool math texts that I have seen attempt to show you where most things come from, and attempt to show the steps. Now, this doesn't mean that they do so in an effective way (there's often that random ass step which doesn't make any sense at all), but they do try. The struggle mostly comes from situations where a) people just can't follow the explanations at all, since the reasoning is not explained well enough, or there are jumps too large, or b) people who have trouble adapting past strategies and concepts to more novel situations. For example, they might know how to solve some class A of word problems, but once you switch the word problems to solving the equation in the other direction, they just get lost. I think most of the 'left as an exercise for the reader' stuff appears once you get to a high enough level that you a) assume that the reader is proficient enough and b) that the reader actually cares enough about the subject to be able to do so. For example, my friends taking a bunch of pure math subjects get that crap all the time. "So we've proved X theorem for Y case, Z will be left as an exercise", and they eat it up (partly because it really is an exercise). Now, I still think a lot of the time, it's inappropriate, and just used cause the writer is lazy, or has used up too much space on diagrams (especially in physics textbooks...).
  - gersh 16y ago
    In high level texts, the "obvious" problems often aren't easy. I think there is a strong element of mathematician arrogance. If you can't figure it out, you don't deserve to know. This may go all the way back to Pythagorus, who ran a secretive cult. There are actual exercises, where solving it is just a matter of applying some concept. However, a lot of math books have things where you have beat your head against the wall to figure them out. Although, the internet has made it easier to lookup, nowadays. School textbooks often try to make things simpler. However, when these texts are written based off of older texts, which skip steps, they may inherit the style of the older works.
keeperofdakeys 16y ago
Relevant, http://www.maa.org/devlin/LockhartsLament.pdf http://www.maa.org/devlin/LockhartsLament.pdf
shasta 16y ago
Getting the students at the low end of the spectrum to "get math" is certainly a noble goal, but in order to have such a low variance in math ability at the end of the year, there's another required component - you have to keep the high end students below their potential. As far as I'm concerned, anyone who claims otherwise has a high burden of proof. Go take a remedial math class and turn it into a winning USAMO team and then get back to me. High end math education (at least in the US) is just as poor as the low end, but because the scores are acceptable not as many people care. Talk of "evening things out" is misguided.
- HockeyBiasDotCo 16y ago
  Your assumption does not hold water where my kids go. ...Fortunately They are in small classes and the school has 3 math levels per grade; one for the advanced students, one for the struggling students, and one for kids that fall between these 2 camps. It is great!
  - shasta 16y ago
    So your kid's school has the kids segregated by ability... and that contradicts some assumption I've made?
- raganwald 16y ago
  What if we discover some magic new program that improves every student's scores, but it improves the poorest students the most and the gifted students the least relative to our existing programs? Such a program wouldn't be holding the gifted students back relative to existing education, it would improve them, and it would still reduce the variance. (My comment sonly apply to standardized education, not to streaming students into different programs such as specialprograms for gifted students.)
  - shasta 16y ago
    I don't know why you have to postulate "magic". I didn't say that a program that helps each kid progress as quickly as possible wouldn't reduce the variance. It probably would, as the kids on the bottom have more room for improvement. But I would still expect a large amount of variance. Actually, I just noticed that the "after" graph is clustered near 100%, so it's possible that there still is a large variance, but it's just not measured by the tests that were administered. These are probably standardized test scores and don't really address higher end performance. So I guess my points are: 1. I'm doubtful that the "paint by numbers" approach is what the young Picasso needs. 2. Maybe I'm wrong about (1), but we certainly shouldn't subject the advanced students to this method until higher end performance has been measured. 3. I find the attitude of the author, who used the phrase "even things out" as if it was a good thing, concerning. Evening things out should be an explicit non-goal of education.
RBr 16y ago
This article is very interesting. However, while reading, it sounded like a late night infomercial. There may be real value in this style of teaching Math, but as my first introduction, this Times article feels like the sponsored hooks used for products such as those baby reading flash cards and acne medication. I hope that the Jump system is real and that it solves the problems outlined in the article.
forkandwait 16y ago
It might be paranoid, but I think the old style of teaching math -- a few big examples, let the student find their way on their own -- is actually designed to create an artificial hierarchy and ranking. Rich kids and lucky kids figure out they have to break things down into micro steps, either do that on their or hire a tutor (have a parent) to do it for them, but in class they just seem to be "bright". (I have been that tutor, and totally changed several kids from "I'm/ he is just not good at math" to "Oh -- math is straightforward if you break it down and work your ass off, like almost everything else" This is basically how Law School Confidential thinks law school manages to keep the supply of good lawyers low: fool the students into thinking that talent and intuition teach you law, and let a few students who outline and memorize like crazy "fall up" through the cracks. I say "designed", but in that mostly subconscious way that we "choose" to do most things -- it feels right, it is "obvious" because it is how we learned originally, and it doesnt lead to weird results (a world full of talented math users which might cheapen the supposed "talent" of the lucky ones). In my experience, public school teachers absolutely love to categorize kids into stupid and smart, with disastrous results unless you are one of the lucky ones (which probably has as much to do with good looks, high-prestige parents, and social skills). I wouldn't say there is NO bell curve, but it could be much, much flatter if it we wanted it to be -- but it's no fun to be an officer without a bunch of stupid enlisted men to boss around, and public school wastes a lot of taxpayers money to create those stupid people.
- halostatue 16y ago
  I'm not sure what your experience is with public school teachers, but I suspect that very few of them do that. None of the teachers that I know, including the one I'm married to, do what you're suggesting. Inasmuch as any teacher categorizes the students, it's to provide skills-appropriate challenges and assistance. In my experience, most peoples' experience with public school teachers is formed around their own educational experiences, which is essentially no experience at all. Being the parent of a student in public school isn't necessarily a useful experience, either, as the teacher may be trying to tell you that little Johnny isn't as bright as you think he is, but you're still insisting that he's the smartest kid in the class. My wife has had students that simply don't want to learn. They're smart kids, but they spend all their energy coming up with excuses for why they're not doing their schoolwork. My wife has had students that want to learn but are so angry at the world because of what's going on in their home life (abusive parents or siblings or just living in the projects). My wife has had a fourth grader who understood 'quarant' (French for 40) in terms of 'quarantine' and made connections that way. Public school teachers, by and large, love to teach and love to see kids learn. Yes, there's some really bad teachers out there, but there are also really bad designers and really bad software developers. For the most part, though, we (as designers and software developers) don't have to work in thoroughly dysfunctional workplaces.
ph0rque 16y ago
Breaking down the problem into steps so small that each is trivial to do: is there anything this technique can't solve?
- scotty79 16y ago
  Can you break into trivial steps the problem of how to break problems into trivial steps?
  - ph0rque 16y ago
    it's easy if you understand recursion...
    - hugh3 16y ago
      What's the first step to understanding recursion?
      - discreteevent 16y ago
        The Little Schemer. Plus it must be one of the canonical examples of how to teach by breaking things into the smallest steps you can imagine.
tokenadult 16y ago
International comparisons such as the TIMSS and the PISA studies have already been showing for more than a decade that school systems in North America (the United States and Canada) have been underperforming and failing to serve most students well. Chapter 1: "International Student Achievement in Mathematics" from the TIMSS 2007 study of mathematics achievement in many different countries includes, in Exhibit 1.1 (pages 34 and 35) http://timss.bc.edu/PDF/t03_download/T03_M_Chap1.pdf http://timss.bc.edu/PDF/t03_download/T03_M_Chap1.pdf a chart of mathematics achievement levels in various countries. Although the United States is above the international average score among the countries surveyed, as we would expect from the level of economic development in the United States, the United States is well below the top country listed, which is Singapore. An average United States student is at the bottom quartile level for Singapore, or from another point of view, a top quartile student in the United States is only at the level of an average student in Singapore. That the UPPER range of students in the United States is poorly served by current school mathematics instruction in the United States is shown by a careful analysis of the PISA studies of developed countries around the world. PISA's own analysis refers to specific instructional practices in different countries and other differences in country conditions that make a difference in educational outcomes. http://www.oecd.org/document/2/0,3343,en_32252351_32236191_39718850_1_1_1_1,00.html http://www.oecd.org/document/2/0,3343,en_32252351_32236191_3... Some bloggers in the United States persist in blaming these outcomes on the ethnic diversity of the United States (ignoring the ethnic diversity of Singapore and other countries that outperform the United States). Eric A. Hanushek, Paul E. Peterson, and Ludger Woessmann point out in their analysis of the PISA data, "U. S. Math Performance in Global Perspective: How well does each state do at producing high-achieving students?" http://www.oecd.org/document/2/0,3343,en_32252351_32236191_39718850_1_1_1_1,00.html http://www.oecd.org/document/2/0,3343,en_32252351_32236191_3... that the real problem in United States mathematics education is leaving behind too many of the high-ability students, of whatever ethnicity, compared to many other countries. A specific response about what is wrong with mathematics teaching in United States classrooms comes from Patricia Clark Kenschaft in the Notices of the American Mathematical Society volume 52, number 2 (February 2005). http://www.ams.org/notices/200502/fea-kenschaft.pdf http://www.ams.org/notices/200502/fea-kenschaft.pdf Most elementary school teachers in the United States, repeated studies of the issue have shown, http://www.nctq.org/resources/math/ http://www.nctq.org/resources/math/ have poor mathematics preparation in their own higher education and little mathematics knowledge when they enter the classroom. They then are directed by their school districts to use textbooks that are ineffective for primary mathematics instruction, so it is no wonder that most pupils in the United States (and the same applies to Canada) finish primary schooling with poor preparation for higher mathematics study. I speak and read Chinese and have lived in various parts of the Chinese-speaking world. I have Chinese-language textbooks of mathematics at home from more than one country. I am confident that young people of any ethnicity in North America can learn math well if they are taught with materials like those, because I am a math teacher by occupation and my classes include a very ethnically diverse group of students, who thrive in the classes and far exceed the meager expectations of United States classrooms.
ntoshev 16y ago
I was also wondering recently why illiteracy carries social stigma, while innumeracy does not. The article asserts everyone just considers math ability to be innate, but I'm not sure this is a good explanation. Our daily life seems to rely on literacy much more than it relies on numeracy. You can only know how to add and subtract (mostly money) and you will still function all right in your daily life and a great variety of jobs. Not so much if you have difficulty reading. This seems to be slowly changing in the future, but if you consider the learning curve, mathematics has a much steeper one compared to reading and writing well. Does anyone have a better explanation?
rubergly 16y ago
The philosophy of Jump reminded me a lot of the philosophy behind Khan Academy, but with that philosophy used to adapt the curriculum instead of to create a new way of digesting knowledge and exercising it. While they're similar, the biggest advantage I see for Jump is that a school adapting their curriculum to Jump would be much easier than adapting their curriculum to Khan Academy.
xal 16y ago
The problem with math education is a lot more basic then everyone pretends: there are two variables, mastery and time and we made the wrong one static. This was a global decision born out of necessity. Because there aren't as many teachers as students we had to make time static and mastery variable. You take a class for a set period of time and then you get a grade based on how much of it you understood. A B means you understand the topic about 80%. In an ideal world every student would always get 100% but simply move at a different pace with the best students simply consuming more material throughout their school career (calculus, linear algebra, etc). The tremendous news is that technology can and will turn this on it's head. The Kahn Academy does this already with tremendous success and it's the single most important thing that has happend to education in a long time. For getting a full idea of the scope and vision, whats Sal's ted talk: http://www.ted.com/talks/salman_khan_let_s_use_video_to_reinvent_education.html http://www.ted.com/talks/salman_khan_let_s_use_video_to_rein...
scott_s 16y ago
I took a similar approach when I taught programming to non-CS majors last summer. The difficulty is that it required significant one-on-one time with each student that needed help, and it required that they came to me. Luckily, my students weren't shy and I made myself readably available to them almost every day. But college students are generally better motivated to seek out help than high school, middle school and elementary students. What took so long with each student was, based on their original question, systematically figuring out what their real misunderstanding was. This could be very time consuming, and took enormous patience on my part. It required much back and forth with the student so I could build a mental model of their mental model. Then I had to figure out how to build a bridge from their mental model to the correct one. That also was time consuming, because I sometimes had to build several bridges before finding one that clicked with the student. Usually helping the Nth student on a project was quicker than helping the first because by that time I recognized what were common problems and built a bag of tricks to explain them. For example, it took me a surprisingly long time to realize the students had no practice running through an algorithm on paper, then translating that to code. (This was partially because the algorithm was so simple I hadn't even realized it was an algorithm.) So one technique I used was to, on paper, set up what was needed for the algorithm to work (list of numbers, table of results, etc.), then make them tell me what to do to get the correct result. I was acting as, basically, an intelligent computer that could be programmed in English. After doing this a few times, the students could finally "see" the algorithm, but it took a lot of time and effort with each student.
crasshopper 16y ago
Here's a peep at the JUMP curriculum: http://jumpmath.org/TM%20for%20Introductory%20Unit%20Using%20Fractions.pdf http://jumpmath.org/TM%20for%20Introductory%20Unit%20Using%2... Emphasize the positive. If a child gets three out of four questions wrong, I will mark the question that is correct first and praise them for getting the correct answer. Then I will say, “I think you didn’t understand something with these other questions” or “You may have been going too fast,” and then I will point out their mistake – or ask them to find it themselves! I’ve found that if I start by mentioning the mistakes, a weaker student will sometimes simply give up or stop listening.

167 points by dctoedt 16y ago

Marvelous. I am glad to see someone emphasizing structure and practice in early math curricula. While it is possible to overemphasize memorization and rote practice, the pendulum has certainly swung too far the other direction as a reaction against the Victorian knuckle-rapping methods.

patrickyeon 16y ago
> While it is possible to overemphasize memorization and rote practice, the pendulum has certainly swung too far the other direction Throughout my grade school and high school years (born mid-80s, Canadian), near everything was memorization. Memorize multiplication tables instead of learn to do arbitrary multiplications quickly. Memorize your table of elements instead of emphasis on what it means. Memorize how to find the roots of a quadratic function, the rules for arithmetic with fractions, physics equations... (I'm pretty bad at memorizing, so this doubly infuriated me because I had to look for the underlying relationships to do well). Even after my province introduced a new curriculum which was supposed to stop all that and focus away from memorization. Anecdotally, this is only slowly changing if at all, as new teachers come in and old ones who won't give up on rote memorization retire. I know this is all just anecdotal, but in my (ongoing) university education, I still see a lot of students put their emphasis on memorization, which suggests to me that they were taught that is what it means to 'learn.'
- hugh3 16y ago
  Memorize multiplication tables instead of learn to do arbitrary multiplications quickly. How can you do arbitrary multiplications quickly without memorizing your times table? You can't quickly compute 47*36 without being able to reel off "six times seven is forty-two, seven times three is twenty-one, four times six is twenty-four, four times three is twelve", can you? I hated learning my times table (in fact I hated mathematics when mathematics was just arithmetic) but a certain amount of memorization is necessary and unavoidable if you're going to be able to do basic mental arithmetic, which remains an important life skill even if you do have a calculator.
  - marklubi 16y ago
    > You can't quickly compute 47*36 without being able to reel off "six times seven is forty-two, seven times three is twenty-one, four times six is twenty-four, four times three is twelve", can you? My mental process for solving your example equation was to reduce it to 50x36-3x36 which is IMHO a much simpler process. I don't know where I learned that process (probably from my dad), but simplifying the values to easily calculated approximate values and then adjusting that value for more precision seems to make much more sense than what was taught to me in school. It also caused a lot of frustration between myself and my teachers who demanded that all of the micro steps be written out instead of making what seemed to me as logical jumps. Edit: formatting
  - nantes 16y ago
    Completely off-topic, but that sounds very similar to approach taken by RISC CPU's. Be very good at doing smalls things very, very fast and then compose those operations in to higher order operations. Just saying.

fname 16y ago

The money quote: Teachers tell me that when they begin using Jump they are surprised to discover that what they were teaching as one step may contain as many as seven micro steps It's about finding out how to simplify what some may see as the easiest step. Think about the lowest common denominator and build from there.

happy4crazy 16y ago

It sounds trivial, but the idea of breaking things down into micro-steps is incredibly useful. I think it's fundamental to effective abstract thinking, and yet most people (including me!) don't consciously think this way. As an example, I was a physics major in college, so I'm used to thinking of myself as being pretty numerate. And yet, I've noticed that I'm not a very efficient learner of higher math; I enjoy it, which keeps me chugging along, but I often find myself getting discouraged when my brain doesn't automagically internalize new abstractions. Instead of approaching a new abstraction as a bundled collection of less-abstract micro-steps, I think "hmm, if I were really smart this would just sink in." Sometimes the new abstraction does just sink in, but it often doesn't, and then I feel briefly bummed about not being the radiant genius I thought I was. This is a dumb attitude! An abstraction is like a steak; you can eat it, just maybe not in one bite. And the incredible, amazing thing about abstraction is that as you get better at it, you get to take bigger bites! This all reminds me of something Kent Beck says in his TDD book: It is not necessary to work in such tiny steps as these. Once you've mastered TDD, you will be able to work in much bigger leaps of functionality between test cases. However, to master TDD you need to be able to work in such tiny steps when they are called for. Being able to do things in tiny steps is a skill, and being willing to do things in small steps when you choke on a big step, rather than feeling dumb and giving up, is the key to learning just about anything.

gersh 16y ago

This goes against math tradition where classical math texts say things like "this is obvious and left as an exercise to the reader". Implicitly, I think many mathematicians believe students should be made to struggle for their own good. The problem is that this approach requires patience, persistence and hard work. Do this help students in the long-term? If you learn math without struggling, will you learn to think in the same way? Is this teaching to the test over teaching you how to think?

icegreentea 16y ago
You really don't see that type of text appear until university these days. Most grade school, and highschool math texts that I have seen attempt to show you where most things come from, and attempt to show the steps. Now, this doesn't mean that they do so in an effective way (there's often that random ass step which doesn't make any sense at all), but they do try. The struggle mostly comes from situations where a) people just can't follow the explanations at all, since the reasoning is not explained well enough, or there are jumps too large, or b) people who have trouble adapting past strategies and concepts to more novel situations. For example, they might know how to solve some class A of word problems, but once you switch the word problems to solving the equation in the other direction, they just get lost. I think most of the 'left as an exercise for the reader' stuff appears once you get to a high enough level that you a) assume that the reader is proficient enough and b) that the reader actually cares enough about the subject to be able to do so. For example, my friends taking a bunch of pure math subjects get that crap all the time. "So we've proved X theorem for Y case, Z will be left as an exercise", and they eat it up (partly because it really is an exercise). Now, I still think a lot of the time, it's inappropriate, and just used cause the writer is lazy, or has used up too much space on diagrams (especially in physics textbooks...).
- gersh 16y ago
  In high level texts, the "obvious" problems often aren't easy. I think there is a strong element of mathematician arrogance. If you can't figure it out, you don't deserve to know. This may go all the way back to Pythagorus, who ran a secretive cult. There are actual exercises, where solving it is just a matter of applying some concept. However, a lot of math books have things where you have beat your head against the wall to figure them out. Although, the internet has made it easier to lookup, nowadays. School textbooks often try to make things simpler. However, when these texts are written based off of older texts, which skip steps, they may inherit the style of the older works.

keeperofdakeys 16y ago

Relevant, http://www.maa.org/devlin/LockhartsLament.pdf http://www.maa.org/devlin/LockhartsLament.pdf

shasta 16y ago

Getting the students at the low end of the spectrum to "get math" is certainly a noble goal, but in order to have such a low variance in math ability at the end of the year, there's another required component - you have to keep the high end students below their potential. As far as I'm concerned, anyone who claims otherwise has a high burden of proof. Go take a remedial math class and turn it into a winning USAMO team and then get back to me. High end math education (at least in the US) is just as poor as the low end, but because the scores are acceptable not as many people care. Talk of "evening things out" is misguided.

HockeyBiasDotCo 16y ago
Your assumption does not hold water where my kids go. ...Fortunately They are in small classes and the school has 3 math levels per grade; one for the advanced students, one for the struggling students, and one for kids that fall between these 2 camps. It is great!
- shasta 16y ago
  So your kid's school has the kids segregated by ability... and that contradicts some assumption I've made?
raganwald 16y ago
What if we discover some magic new program that improves every student's scores, but it improves the poorest students the most and the gifted students the least relative to our existing programs? Such a program wouldn't be holding the gifted students back relative to existing education, it would improve them, and it would still reduce the variance. (My comment sonly apply to standardized education, not to streaming students into different programs such as specialprograms for gifted students.)
- shasta 16y ago
  I don't know why you have to postulate "magic". I didn't say that a program that helps each kid progress as quickly as possible wouldn't reduce the variance. It probably would, as the kids on the bottom have more room for improvement. But I would still expect a large amount of variance. Actually, I just noticed that the "after" graph is clustered near 100%, so it's possible that there still is a large variance, but it's just not measured by the tests that were administered. These are probably standardized test scores and don't really address higher end performance. So I guess my points are: 1. I'm doubtful that the "paint by numbers" approach is what the young Picasso needs. 2. Maybe I'm wrong about (1), but we certainly shouldn't subject the advanced students to this method until higher end performance has been measured. 3. I find the attitude of the author, who used the phrase "even things out" as if it was a good thing, concerning. Evening things out should be an explicit non-goal of education.

RBr 16y ago

This article is very interesting. However, while reading, it sounded like a late night infomercial. There may be real value in this style of teaching Math, but as my first introduction, this Times article feels like the sponsored hooks used for products such as those baby reading flash cards and acne medication. I hope that the Jump system is real and that it solves the problems outlined in the article.

forkandwait 16y ago

It might be paranoid, but I think the old style of teaching math -- a few big examples, let the student find their way on their own -- is actually designed to create an artificial hierarchy and ranking. Rich kids and lucky kids figure out they have to break things down into micro steps, either do that on their or hire a tutor (have a parent) to do it for them, but in class they just seem to be "bright". (I have been that tutor, and totally changed several kids from "I'm/ he is just not good at math" to "Oh -- math is straightforward if you break it down and work your ass off, like almost everything else" This is basically how Law School Confidential thinks law school manages to keep the supply of good lawyers low: fool the students into thinking that talent and intuition teach you law, and let a few students who outline and memorize like crazy "fall up" through the cracks. I say "designed", but in that mostly subconscious way that we "choose" to do most things -- it feels right, it is "obvious" because it is how we learned originally, and it doesnt lead to weird results (a world full of talented math users which might cheapen the supposed "talent" of the lucky ones). In my experience, public school teachers absolutely love to categorize kids into stupid and smart, with disastrous results unless you are one of the lucky ones (which probably has as much to do with good looks, high-prestige parents, and social skills). I wouldn't say there is NO bell curve, but it could be much, much flatter if it we wanted it to be -- but it's no fun to be an officer without a bunch of stupid enlisted men to boss around, and public school wastes a lot of taxpayers money to create those stupid people.

halostatue 16y ago
I'm not sure what your experience is with public school teachers, but I suspect that very few of them do that. None of the teachers that I know, including the one I'm married to, do what you're suggesting. Inasmuch as any teacher categorizes the students, it's to provide skills-appropriate challenges and assistance. In my experience, most peoples' experience with public school teachers is formed around their own educational experiences, which is essentially no experience at all. Being the parent of a student in public school isn't necessarily a useful experience, either, as the teacher may be trying to tell you that little Johnny isn't as bright as you think he is, but you're still insisting that he's the smartest kid in the class. My wife has had students that simply don't want to learn. They're smart kids, but they spend all their energy coming up with excuses for why they're not doing their schoolwork. My wife has had students that want to learn but are so angry at the world because of what's going on in their home life (abusive parents or siblings or just living in the projects). My wife has had a fourth grader who understood 'quarant' (French for 40) in terms of 'quarantine' and made connections that way. Public school teachers, by and large, love to teach and love to see kids learn. Yes, there's some really bad teachers out there, but there are also really bad designers and really bad software developers. For the most part, though, we (as designers and software developers) don't have to work in thoroughly dysfunctional workplaces.

ph0rque 16y ago

Breaking down the problem into steps so small that each is trivial to do: is there anything this technique can't solve?

scotty79 16y ago
Can you break into trivial steps the problem of how to break problems into trivial steps?
- ph0rque 16y ago
  it's easy if you understand recursion...
  - hugh3 16y ago
    What's the first step to understanding recursion?
    - discreteevent 16y ago
      The Little Schemer. Plus it must be one of the canonical examples of how to teach by breaking things into the smallest steps you can imagine.

tokenadult 16y ago

International comparisons such as the TIMSS and the PISA studies have already been showing for more than a decade that school systems in North America (the United States and Canada) have been underperforming and failing to serve most students well. Chapter 1: "International Student Achievement in Mathematics" from the TIMSS 2007 study of mathematics achievement in many different countries includes, in Exhibit 1.1 (pages 34 and 35) http://timss.bc.edu/PDF/t03_download/T03_M_Chap1.pdf http://timss.bc.edu/PDF/t03_download/T03_M_Chap1.pdf a chart of mathematics achievement levels in various countries. Although the United States is above the international average score among the countries surveyed, as we would expect from the level of economic development in the United States, the United States is well below the top country listed, which is Singapore. An average United States student is at the bottom quartile level for Singapore, or from another point of view, a top quartile student in the United States is only at the level of an average student in Singapore. That the UPPER range of students in the United States is poorly served by current school mathematics instruction in the United States is shown by a careful analysis of the PISA studies of developed countries around the world. PISA's own analysis refers to specific instructional practices in different countries and other differences in country conditions that make a difference in educational outcomes. http://www.oecd.org/document/2/0,3343,en_32252351_32236191_39718850_1_1_1_1,00.html http://www.oecd.org/document/2/0,3343,en_32252351_32236191_3... Some bloggers in the United States persist in blaming these outcomes on the ethnic diversity of the United States (ignoring the ethnic diversity of Singapore and other countries that outperform the United States). Eric A. Hanushek, Paul E. Peterson, and Ludger Woessmann point out in their analysis of the PISA data, "U. S. Math Performance in Global Perspective: How well does each state do at producing high-achieving students?" http://www.oecd.org/document/2/0,3343,en_32252351_32236191_39718850_1_1_1_1,00.html http://www.oecd.org/document/2/0,3343,en_32252351_32236191_3... that the real problem in United States mathematics education is leaving behind too many of the high-ability students, of whatever ethnicity, compared to many other countries. A specific response about what is wrong with mathematics teaching in United States classrooms comes from Patricia Clark Kenschaft in the Notices of the American Mathematical Society volume 52, number 2 (February 2005). http://www.ams.org/notices/200502/fea-kenschaft.pdf http://www.ams.org/notices/200502/fea-kenschaft.pdf Most elementary school teachers in the United States, repeated studies of the issue have shown, http://www.nctq.org/resources/math/ http://www.nctq.org/resources/math/ have poor mathematics preparation in their own higher education and little mathematics knowledge when they enter the classroom. They then are directed by their school districts to use textbooks that are ineffective for primary mathematics instruction, so it is no wonder that most pupils in the United States (and the same applies to Canada) finish primary schooling with poor preparation for higher mathematics study. I speak and read Chinese and have lived in various parts of the Chinese-speaking world. I have Chinese-language textbooks of mathematics at home from more than one country. I am confident that young people of any ethnicity in North America can learn math well if they are taught with materials like those, because I am a math teacher by occupation and my classes include a very ethnically diverse group of students, who thrive in the classes and far exceed the meager expectations of United States classrooms.

ntoshev 16y ago

I was also wondering recently why illiteracy carries social stigma, while innumeracy does not. The article asserts everyone just considers math ability to be innate, but I'm not sure this is a good explanation. Our daily life seems to rely on literacy much more than it relies on numeracy. You can only know how to add and subtract (mostly money) and you will still function all right in your daily life and a great variety of jobs. Not so much if you have difficulty reading. This seems to be slowly changing in the future, but if you consider the learning curve, mathematics has a much steeper one compared to reading and writing well. Does anyone have a better explanation?

rubergly 16y ago

The philosophy of Jump reminded me a lot of the philosophy behind Khan Academy, but with that philosophy used to adapt the curriculum instead of to create a new way of digesting knowledge and exercising it. While they're similar, the biggest advantage I see for Jump is that a school adapting their curriculum to Jump would be much easier than adapting their curriculum to Khan Academy.

xal 16y ago

The problem with math education is a lot more basic then everyone pretends: there are two variables, mastery and time and we made the wrong one static. This was a global decision born out of necessity. Because there aren't as many teachers as students we had to make time static and mastery variable. You take a class for a set period of time and then you get a grade based on how much of it you understood. A B means you understand the topic about 80%. In an ideal world every student would always get 100% but simply move at a different pace with the best students simply consuming more material throughout their school career (calculus, linear algebra, etc). The tremendous news is that technology can and will turn this on it's head. The Kahn Academy does this already with tremendous success and it's the single most important thing that has happend to education in a long time. For getting a full idea of the scope and vision, whats Sal's ted talk: http://www.ted.com/talks/salman_khan_let_s_use_video_to_reinvent_education.html http://www.ted.com/talks/salman_khan_let_s_use_video_to_rein...

scott_s 16y ago

I took a similar approach when I taught programming to non-CS majors last summer. The difficulty is that it required significant one-on-one time with each student that needed help, and it required that they came to me. Luckily, my students weren't shy and I made myself readably available to them almost every day. But college students are generally better motivated to seek out help than high school, middle school and elementary students. What took so long with each student was, based on their original question, systematically figuring out what their real misunderstanding was. This could be very time consuming, and took enormous patience on my part. It required much back and forth with the student so I could build a mental model of their mental model. Then I had to figure out how to build a bridge from their mental model to the correct one. That also was time consuming, because I sometimes had to build several bridges before finding one that clicked with the student. Usually helping the Nth student on a project was quicker than helping the first because by that time I recognized what were common problems and built a bag of tricks to explain them. For example, it took me a surprisingly long time to realize the students had no practice running through an algorithm on paper, then translating that to code. (This was partially because the algorithm was so simple I hadn't even realized it was an algorithm.) So one technique I used was to, on paper, set up what was needed for the algorithm to work (list of numbers, table of results, etc.), then make them tell me what to do to get the correct result. I was acting as, basically, an intelligent computer that could be programmed in English. After doing this a few times, the students could finally "see" the algorithm, but it took a lot of time and effort with each student.

crasshopper 16y ago

Here's a peep at the JUMP curriculum: http://jumpmath.org/TM%20for%20Introductory%20Unit%20Using%20Fractions.pdf http://jumpmath.org/TM%20for%20Introductory%20Unit%20Using%2... Emphasize the positive. If a child gets three out of four questions wrong, I will mark the question that is correct first and praise them for getting the correct answer. Then I will say, “I think you didn’t understand something with these other questions” or “You may have been going too fast,” and then I will point out their mistake – or ask them to find it themselves! I’ve found that if I start by mentioning the mistakes, a weaker student will sometimes simply give up or stop listening.