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.
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.
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.
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.
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.
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.
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.
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.)
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.
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.
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.
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!
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.
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.
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.
> 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.'
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.
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)
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.
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?"
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).
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.
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).
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.
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.]
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?
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...).
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"...
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.
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.
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.
[+] [-] forkandwait|15 years ago|reply
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|reply
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.
[+] [-] simpleTruth|15 years ago|reply
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.
[+] [-] timwiseman|15 years ago|reply
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.
[+] [-] boboblong|15 years ago|reply
You mean "narrower" or "skinnier", not "flatter". A flatter bell curve would mean the opposite.
[+] [-] happy4crazy|15 years ago|reply
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.
[+] [-] xal|15 years ago|reply
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...
[+] [-] shasta|15 years ago|reply
[+] [-] raganwald|15 years ago|reply
(My comment sonly apply to standardized education, not to streaming students into different programs such as specialprograms for gifted students.)
[+] [-] yummyfajitas|15 years ago|reply
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.
[+] [-] hackinthebochs|15 years ago|reply
[+] [-] lolcraft|15 years ago|reply
Furthermore, where do you read that the objective is "low variance"? I read "higher average".
[+] [-] dailo10|15 years ago|reply
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.
[+] [-] HockeyBiasDotCo|15 years ago|reply
[+] [-] fname|15 years ago|reply
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.
[+] [-] scott_s|15 years ago|reply
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.
[+] [-] asolove|15 years ago|reply
[+] [-] patrickyeon|15 years ago|reply
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.'
[+] [-] ntoshev|15 years ago|reply
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?
[+] [-] tokenadult|15 years ago|reply
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.
[+] [-] yummyfajitas|15 years ago|reply
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.]
[+] [-] johnzabroski|15 years ago|reply
Wish I could've upvoted this 100,000,000 times.
[+] [-] jobu|15 years ago|reply
[+] [-] pnathan|15 years ago|reply
[+] [-] keeperofdakeys|15 years ago|reply
[+] [-] gersh|15 years ago|reply
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|reply
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...).
[+] [-] forkandwait|15 years ago|reply
[+] [-] RBr|15 years ago|reply
I hope that the Jump system is real and that it solves the problems outlined in the article.
[+] [-] rubergly|15 years ago|reply
[+] [-] crasshopper|15 years ago|reply
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.
[+] [-] ph0rque|15 years ago|reply
[+] [-] scotty79|15 years ago|reply