A better way to teach math

http://opinionator.blogs.nytimes.com/2011/04/18/a-better-way-to-teach-math/

by dctoedt • 15 years ago

167 71 15 years ago

opinionator.blogs.nytimes.com

asolove• 15 years 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• 15 years 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.'

hugh32• 15 years 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.

fname• 15 years 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.

happy4crazy2• 15 years 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.

gersh2• 15 years 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• 15 years 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...).

gersh2• 15 years 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.

forkandwait4• 15 years ago

When I teach math, I make sure the students understand smaller concepts, then I give them problems that integrate the concepts so they can practice that. The old style just gave the integrative problems without the more basic problems, and only worked for students who were highly motivated and/ or had outside help (cliff notes have always been popular, even though brown-nosing students never admit to using them). The old style was good at "weeding out"...

keeperofdakeys• 15 years ago

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

shasta3• 15 years 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• 15 years 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!

shasta3• 15 years ago

So your kid's school has the kids segregated by ability... and that contradicts some assumption I've made?

raganwald2• 15 years 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.)

shasta3• 15 years 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.

yummyfajitas6• 15 years ago

Based on the graphs given, it looks like the low variance may simply be a result of having the x-axis be percentile.

Consider Round 2, where sigma=1.2%, mu=98%. Suppose hypothetically that 90% of students are clustered below 100 (an absolute measure of performance), 96.8% of students are below 150 pts, and 99.2% of students are below 300 pts.

In this case, the absolute variation is huge (300 vs 150). But because only 2.4% of students score between 150 and 300 pts, on a percentile graph, it looks like sigma has been reduced.

I think it's unlikely that this has occurred, but the graphs given don't preclude it.

hackinthebochs4• 15 years ago

You're missing an important point: there is a built-in ceiling to how good a child can be in a particular math class. There is only a finite amount of material taught and a finite amount of time. So any program like this is inevitably going to greatly reduce the variance as everyone gets pushed towards this ceiling. This isn't keeping gifted students below their potential. It's just not giving them special treatment by accelerating the class material for their sake. Which is the same as before.

dailo10• 15 years ago

This approach seems to benefit everyone, including the top students. If you look at the graphs, in Round 1 the Max went from 80% to 99% and in Round 2 the Max went from 75% to 99%. This occurred not in a remedial class, but in an independent, unscreened school.

Perhaps one side-effect of the overall improvement is that the teacher doesn't have to spend as much time with struggling students and can devote more time helping high end students unlock their potential.

lolcraft• 15 years ago

Are gifted children that much helped by the current system? For all I know, they are just ignored.

Furthermore, where do you read that the objective is "low variance"? I read "higher average".

RBr• 15 years 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.

forkandwait4• 15 years 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• 15 years 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.

forkandwait4• 15 years ago

I think most public school teachers are soldiers in a system they don't even know is there, which includes "kids who don't want to learn", parents who don't give a shit, administrators who piss off parents, etc, etc. While I took a dig at them in my post, I dont actually think it is their fault, they just go with the flow, and the flow is toward barely conscious re-creation of social hierarchy.

My experience of public school teachers has been from volunteering regularly as a tutor, being in the periphery of the educational system as an Upward Bound teacher and tutor. The public school teachers I have met generally taught to the "talented and driven" in their class, did a decent job, but were incredibly naive about the social system in which they did their supposedly idealistic work. And to a one, they all believed in the myth of talent and a wide bell curve (and had never heard of the pygmalion effect...)

simpleTruth4• 15 years ago

To play the devils advocate: IMO, it's the focus on the micro steps that's the problem.

HS level math is vary simple. You can write a single textbook that covered the full range of math from preschool to calculus, but the focus on 3 days of instruction, a day of review, and then a quiz or test slows things down. It can be easier to skip ahead and then go back and review than try and approach math in tinny nibbles. It's like spending a full year going over cement foundations before you mention that the goal is to put a house on top of that flat slab.

Personally, I used to do other classes homework assignments in my math classes. I can recall getting in about 4 seconds a new topic that the class spend a full week going though in minute detail. It was so bad I once accidentally did the next chapters review vs the assigned homework and did not even notice at the time.

PS: I am all for better instruction, but perhaps we could consider going a little further. We could probably get the average 10th grader to really understand Calculus, but I think the goal should be to dive into DifEq and number theory etc.

kenjackson4• 15 years ago

* To play the devils advocate: IMO, it's the focus on the micro steps that's the problem.*

I don't think so. Everyone builds on a solid foundation and microsteps. I think what tends to happen though, and is discussed in the article, is that by the time kids reach HS they either have a very solid foundation, or an extremely poor one. The kids with the poor foundation will have trouble understanding things regardless of how long you spend on it. They've developed gaps over the past nine years of math education.

I think if we address the gaps upfront in elementary school that in HS math will move much quicker. Not because the kids are smarter, but simply because you aren't compensating for 30 kids in a class who all struggle with a variety of concepts they should have learned in elementary school.

tokenadult2• 15 years ago

IMO, it's the focus on the micro steps that's the problem.

It would be very problematic, and this happens, if teaching consisting of showing micro steps is followed up by homework and assessments that also focus narrowly on micro steps. That serves up exercises for students, but it doesn't allow students the learning opportunities developed by working on actual problems. The rest of this comment is a FAQ file I send to families of new students in my math classes about the distinction between "exercises" and "problems."

PROBLEMS VERSUS EXERCISES

I frequently encounter discussions among parents about repetitive school math lessons, so a few years ago I prepared this Frequently Asked Question (FAQ) document about the distinction between math exercises (good in sufficient but not excessive amount) and math problems (always good in any amount).

Most books about mathematics have what are called "exercises" in them, questions that prompt a learner to practice the concepts discussed in the mathematics book. By reading one mathematics book, and then several more, I learned that some mathematicians draw a distinction between "exercises" and "problems" (which is the terminology generally used by the mathematicians who draw this distinction). I think this distinction is useful for teachers and learners to consider while selecting materials for studying mathematics, so I'll share the quotations from which I learned this distinction here. I first read about the distinction between exercises and problems in a Taiwan reprint of a book by Howard Eves.

"It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy.

"It is impossible to overstate the importance of problems in mathematics. It is by means of of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician."

Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix.

I have since read about this distinction in several other books.

"Before going any further, let's digress a minute to discuss different levels of problems that might appear in a book about mathematics:

Level 1. Given an explicit object x and an explicit property P(x), prove that P(x) is true. . . .

Level 2. Given an explicit set X and an explicit property P(x), prove that P(x) is true for FOR ALL x [existing in] X. . . .

Level 3. Given an explicit set X and an explicit property P(x), prove OR DISPROVE that P(x) is true for for all x [existing in] X. . . .

Level 4. Given an explicit set X and an explicit property P(x), find a NECESSARY AND SUFFICIENT CONDITION Q(x) that P(x) is true. . . .

Level 5. Given an explicit set X, find an INTERESTING PROPERTY P(x) of its elements. Now we're in the scary domain of pure research, where students might think that total chaos reigns. This is real mathematics. Authors of textbooks rarely dare to pose level 5 problems."

Graham, Ronald, Knuth, Donald, and Patashnik, Oren (1994). Concrete Mathematics Second Edition. Boston: Addison-Wesley, pages 72-73.

This digression becomes the subject of a, um, problem in Exercise 4 of Chapter 3: "The text describes problems at levels 1 through 5. What is a level 0 problem? (This, by the way, is NOT a level 0 problem.)"

"First, what is a PROBLEM? We distinguish between PROBLEMS and EXERCISES. An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. . . .

"A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it."

Zeitz, Paul (1999). The Art and Craft of Problem Solving. New York: Wiley, pages 3 and 4.

". . . . As Paul Halmos said, 'Problems are the heart of mathematics,' so we should 'emphasize them more and more in the classroom, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are.'

"The problems we have selected are definitely not exercises. Our definition of an exercise is that you look at it and know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises."

Andreescu, Titu & Gelca, Razvan (2000), Mathematical Olympiad Challenges. Boston: Birkhäuser, page xiii.

"It is easier to advance in one topic by going ahead with the more elementary parts of another topic, where the first one is applied. The brain much prefers to work that way, rather than to concentrate on ugly technical formulas which are obviously unrelated to anything except artificial drilling. Of course, some rote drilling is necessary. The problem is how to strike a balance."

Lang, Serge (1988), Basic Mathematics. New York: Springer-Verlag, p. xi.

hackinthebochs4• 15 years ago

>IMO, it's the focus on the micro steps that's the problem.

No, the micro-steps are the solution, not the problem. Those who "get" math intuitively break problems down into smaller steps, often unconsciously. For those that can't intuitively do this (most students), they need to be taught explicitly the micro-steps. The micro-steps build the foundation for the higher level steps and the problem solving.

Learning the micro-steps well is like learning an abstraction in a program. Once you learn the abstraction to the point that your understanding of it is unconscious, then your thought processes are lifted to a higher level. This is true understanding of math.

mattdeboard• 15 years ago

Congratulations, you internalized the micro steps by repetition and critical analysis. Not everyone gets there on their own for a million different reasons. I do not think there should be a 'focus' per se on 'micro steps' but there does need to be encouragement to kids to think about this and find solutions on their own.

Case in point, I was made a very proud dad today. My first-grade daughter and I were going over her math homework, which was flash cards with addition equations on them. This week the focus in her class is the numbers 4 and 5, so on each flash card, at least one of the terms was 4 or 5.

I was drilling her on them, and every time there was an equation the sum of which was >10, she'd mutter under her breath something about subtraction that I couldn't quite discern. I finally stopped at the "5+7" flash card, and asked her to walk me through her process for solving that particular problem. Her response:

"Well, I know 7 is minus three [from ten], so I do minus 3 from 5 which is 2, then I put a 1 in front of that, and that's 12."

In other words, she's doing | (10 - 7) - 5 | + 10. Or, basically, 7+3 = 10, 5-3 = 2, 10+2 = 12. I asked her, "Emma, is that how they teach you to add at school?"

"No," she said, "I just think about it like that."

Is that the absolute easiest way to solve that problem? Well, maybe; I don't know. I'm like 99% sure, however, I do that exact same process when I do mental calculations. That doesn't matter though. What matters is that she's analyzing problems, identifying and generalizing patterns, then applying those patterns to NEW problems. She is numerically literate! (http://en.wikipedia.org/wiki/Numeracy) This is especially gratifying for me, since we spend a few hours at home every week going over more advanced math topics like fractions and multiplication. She doesn't quite grok it yet, but at least when she sees it in school it won't be a total surprise.

In the end, she figured out a system to solve problems on her own that works 100% of the time. Critical thinking skills are paramount. Learn those and everything else comes eventually.

dataduck• 15 years ago

I've been tutoring maths at primary to A-level in the UK for more than ten years now. In that time, the most critical problem I've seen is when children (usually in the 12-16 range) see mathematics as a plethora of small, arbitrary techniques to solve specific problems, and can't see that they all are related together under a few key concepts. This means they hit a brick wall when they are assumed to have mastered and internalised the previous techniques. For instance, many fail to connect multiplication, division, addition, subtraction, integers and scale in order to construct an intuitive sense of rational number. This makes further work with rationals an uphill struggle. I've found going back to the beginning and putting together whatever concepts are lacking using proof and demonstration creates an almost digital change in confidence and proficiency: before it's a mystery, after it's simple. I'm pretty sure that the way schools break down mathematics into tiny portions and call each one a separate topic is to blame here.

While breaking things up into micro steps is the easiest way to solve a particular mathematical exercise (and important in solving problems), joining things up into a single overarching entity is how you actually understand the subject. So breaking stuff up is really useful within a problem, but on its own doesn't give you the long-term preparation you need, and is inappropriate over an entire curriculum. I think any mathematics curriculum needs to have both, but it's the joined-up approach that's more often lacking in schools.

I haven't been able to get the JUMP curriculum (site down?), but I'd really like to see how it deals with these issues.

boboblong• 15 years ago

"I wouldn't say there is NO bell curve, but it could be much, much flatter if it we wanted it to be"

You mean "narrower" or "skinnier", not "flatter". A flatter bell curve would mean the opposite.

timwiseman• 15 years ago

but it's no fun to be an officer without a bunch of stupid enlisted men to boss around

I do not know how you intended this to sound, but it comes off in a way that is horribly inaccurate of both officers and enlisted men.

The majority of officers I know have gone out of their way to help the enlisted learn and did what they could to ensure they had access to colleges. Many officers were ready to help any enlisted person with a real desire to do so to get their education finished and get a commission themselves.

On the other side of it, most of the enlisted men I knew were at or above average intelligence. Especially in certain fields such a communications and intelligence it was very common to find people who already had degrees when they enlisted or finished them quickly after enlisting.

forkandwait4• 15 years ago

I don't know if you will read this, but I was just being flip, and meant nothing specifically about officers or enlisted men beyond the fact of a hierarchy that might be more based on opportunity and family than anything else. I have immense respect for all the military professionals I have knows (not that many, but still).

ph0rque2• 15 years ago

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

scotty79• 15 years ago

Can you break into trivial steps the problem of how to break problems into trivial steps?

ph0rque2• 15 years ago

it's easy if you understand recursion...

tokenadult2• 15 years 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

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_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_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

Most elementary school teachers in the United States, repeated studies of the issue have shown,

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.

jobu• 15 years ago

It looks like the Jump Math site is down(jumpmath.org). Does anyone have links with more info about the program or a cached copy of the site?

johnzabroski• 15 years ago

This was the single most amazing reply to a HN story I've ever seen. Then I read your bio. Go figure :)

Wish I could've upvoted this 100,000,000 times.

yummyfajitas6• 15 years ago

Which bloggers?

Incidentally, I think "ethnic diversity" is a red herring. Any bloggers who talk about "ethnic diversity" are merely trying to couch their conversation in PC language to avoid ad-hominem accusations of racism.

The issue is not "ethnic diversity". The issue is that certain ethnicities underperform. Specifically, African Americans and Hispanics (30% of the US, 0% of Singapore) underperform. Nonhispanic whites (about 65% of the US and close to 0% of Singapore) achieve mid-level performance. Asians (about 4% of the US and close to 100% of Singapore) overperform.

These effects are HUGE in comparison to inter-country effects. Asian Americans score about 10 pts below Singapore. The EU15 scores about 50 pts below Asian Americans, and a couple of points above All Americans.

http://super-economy.blogspot.com/2011/01/how-well-do-above-...

http://super-economy.blogspot.com/2010/12/amazing-truth-abou...

It is simply incorrect to pretend that ethnic gaps in education do not exist, or to pretend that they do not explain a large portion of the gap between the US and the rest of the [edit: first] world.

Also, it would be helpful if you were a little more specific in your citations. Citing a gigantic PISA reports is much less useful than citing a specific table or figure. I'd love to learn more, but I don't have time to read the whole thing, and I have no idea which parts you are referring to.

[edit: wanted to clarify that I don't think the gaps between the US and poor locations, e.g., rural inland China or India, are primarily due to ethnicity. But gaps between wealthy first world countries do seem well explained by such factors.]

kenjackson4• 15 years ago

Unfortunately, you can't disaggregate in the way suggested above. Blacks and Hispanics are also far less wealthy than whites in this country (which has a larger impact than even income). Once you control for wealth a lot of the gap disappears.

Ethnic gaps may exist, but they probably are symptomatic of gaps in wealth more than anything else. But who konws. There are so many other factors at play for people of darker skin in general in society that I'd hesitate to subscribe to one theory.

With that said, it does point to the fact that US public schools are probably better than most believe them to be.

pnathan• 15 years ago

Anecdote: most pre-teachers I met in college were pretty academic crappy. Many were in it for the MRS degree, or so it felt.

ntoshev• 15 years 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• 15 years 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• 15 years 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_rein...

scott_s• 15 years 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• 15 years ago

Here's a peep at the JUMP curriculum: 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.

A better way to teach math

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