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The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math

by lmkg 16y ago

The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math is taught as a set of rules that must be followed, rather than as a common-sense way of systematizing simple common concepts. Part of this is that teachers tend to have a poor quantitative background themselves, so they can do math but they don't grok it. I've seen transcripts of grade school teacher math lessons, and it's clear that they perceive math in a legalistic sense, using phrases like "those are the rules, right?" The result is that the students only see a set of rules, and not the reasoning, so the set of rules looks basically arbitrary. If you explain well enough the concepts of place-value and multiplication, most children can independently invent an algorithm for long multiplication. They tend to arrive at similar solutions which aren't as well-organized or space-optimized as the traditional long multiplication algorithm, but they made it themselves so they understand not just how to do it but why it works. The exact algorithms being taught for math education may make a difference in exposing the concepts, but I think the real determiner of math education is not the algorithms but how the material is presented. Addition and multiplication make sense, if they're taught as concepts, but the focus on test scores emphasizes ability to plug-and-chug over ability to understand what's going on under the hood. This is also why everybody hates word problems, and why I think math testing should use more word problems. Math isn't about your ability to perform rote tasks, it's about your ability to apply general concepts to specific situations, and that in itself is as important as the rest of schooling combined (indeed, it makes the rest of schooling useful). Relevant parting anecdote: some school district knew that the Pythagorean Theorem would be on their standardized algebra tests that year, so they drilled all their students on it. The students could use the formula flawlessly. The test rolled around, and surprise! the Pythagorean Theorem question was a word problem, and most of the students bombed the question.

btilly 16y ago
The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math is taught as a set of rules that must be followed, rather than as a common-sense way of systematizing simple common concepts. http://randomgraphs.blogspot.com/2009/06/feynman-on-education.html http://randomgraphs.blogspot.com/2009/06/feynman-on-educatio... is a good cautionary tale on what this tendency leads to. Part of this is that teachers tend to have a poor quantitative background themselves, This is not a new problem. In the 50s a survey of elementary school teachers found that most thought that 3/5 was larger than 2/3. Based on anecdotal experience and the impressions of people I've talked to, I would expect the same result if the same question was ask of today's teachers. Speaking of anecdotes, the first time I won an argument with a teacher was in grade 6. I hadn't done my homework, so I sat reading the textbook while the class went through homework. I understood the textbook's explanation, and realized that the teacher had been teaching us the wrong thing for the last 2 weeks, and that explained why the class had consistently found that the answers in the official answer key were "wrong". It took me 20 minutes of arguing to convince the teacher, and everyone had to relearn that 2 weeks of material. (We had been practicing doing arithmetic in bases other than 10, and he had been teaching us to just calculate the answer in base 10 then convert into the right base. What we needed to do was convert the number into base 10, calculate, then convert back into the right base.)
- JoeAltmaier 16y ago
  What you need to do is, learn to calculate in other bases. It works exactly as well as doing it in base 10, and of course is yards quicker.
  - btilly 16y ago
    If you use another base a lot, then yes. But when you're facing problems in a half-dozen bases at once, then the learning curve of keeping that many times tables separate is not worthwhile.
- eru 16y ago
  You should also read how Feynman helps with choosing school books in California. http://www.textbookleague.org/103feyn.htm http://www.textbookleague.org/103feyn.htm
RiderOfGiraffes 16y ago
> If you explain well enough the concepts of > place-value and multiplication, most children > can independently invent an algorithm for long > multiplication. You're going to stare at your screen in disbelief, my experience suggests otherwise. A small number of gifted children have, in my experience, been able to work out their own methods, and they've become more efficient over time, and eventually converged to something isomorphic to long multiplication. But this is not the norm. Evidence suggests - and my experience agrees - that correctly teaching process creates the ground-work for moving to understanding. Correct methods, practised over time and in different context, result in an internalisation that leads to an ability to generalise and apply in wider contexts. This is my point about research and evidence-based teaching systems. The existing fads are borne of insufficiently researched and poorly developed methods, badly applied. They are often driven by dogma, and then delivered by teachers who don't properly understand what they're doing. There has to be a better way, and unfortunately, nearly everyone has an opinion, and none of them are supported by anything like enough evidence. And so we continue to damage our children. In parting, much of what you say I agree with, but not all. This isn't really the place to have a lengthy debate, but our positions aren't as far apart as my comment might have you think at first sight.
- lmkg 16y ago
  > This isn't really the place to have a lengthy debate To the contrary, this is Hacker News! Unfortunately, the situation that I'm referring to with children inventing their own long-multiplication algorithm was presented at a talk, so I can't point to the specific results, if they're even published yet, but my contention is based on an academic research study. The punchline of the experiment was that a full class of approximately-average grade-schoolers were taught math in a novel method, and developed their own multiplication algorithms, and went on to have above-average improvements in math test scores. The development of those algorithms were probably guided to a certain extent, but they looked to me like someone who was working the problem out from the first principles based on the concept of place value. That said, I'm definitely willing to accept the evidence you claim to have (especially since it leaves open the meaning of "correct"). I definitely agree that education should be based on science and not on dogma. Just because we think something ought to work doesn't mean that it works (it usually doesn't), and education is something you don't want to fuck up because the stakes are high and you only get one chance. But, I'm not sure we're even at the point where we can do research yet. Before you can measure results, you need to decide what results you're looking for and how you're measuring them. I'm a mathematician, and I'll be the first to admit that being able to perform long-multiplication isn't actually that useful of a skill, and yet that's mostly what we measure on standardized tests. Understanding what multiplication means and being generally numerate is far more useful and not measured very often (the best we have is word problems). The more desirable side-effects, such as the ability to abstract concepts, to generalize processes, and to apply knowledge from one domain to another, aren't being measured at all, and those are really the only things from my math education that I use in my day-to-day life. Similar things happen across other subjects--knowing what year the War of 1812 happened in isn't useful, while understanding the social, economic, and political conditions that led to that war is useful for the average citizen in a democracy (but doesn't fit well in a scantron). I think we as a society need to first have a big sit-down and figure out what the point of education is. We know it's important, but we've never coherently stated particularly why, and that lack of explicit direction is what lets politics, dogma, and fads more easily take control of the process. Once we've done that, we can let science guide the structure of education. Or rather, we should. There's a bit of a chicken-and-egg problem that without solid backgrounds in quantitative and logical thinking, most people have difficulty making data-driven decisions, and people in general won't get better at that until the education system improves.
chrischen 16y ago
> The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math is taught as a set of rules that must be followed... Well that's actually how most of the stuff in school is taught. Whenever a kid goes "why am I learning this I'm never going to use this" that's a sign they don't know what they're learning.
blackguardx 16y ago
Actually, word problems were the only thing that I liked about math class, but then again I am an engineer...

by lmkg 16y ago

btilly 16y ago

The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math is taught as a set of rules that must be followed, rather than as a common-sense way of systematizing simple common concepts. http://randomgraphs.blogspot.com/2009/06/feynman-on-education.html http://randomgraphs.blogspot.com/2009/06/feynman-on-educatio... is a good cautionary tale on what this tendency leads to. Part of this is that teachers tend to have a poor quantitative background themselves, This is not a new problem. In the 50s a survey of elementary school teachers found that most thought that 3/5 was larger than 2/3. Based on anecdotal experience and the impressions of people I've talked to, I would expect the same result if the same question was ask of today's teachers. Speaking of anecdotes, the first time I won an argument with a teacher was in grade 6. I hadn't done my homework, so I sat reading the textbook while the class went through homework. I understood the textbook's explanation, and realized that the teacher had been teaching us the wrong thing for the last 2 weeks, and that explained why the class had consistently found that the answers in the official answer key were "wrong". It took me 20 minutes of arguing to convince the teacher, and everyone had to relearn that 2 weeks of material. (We had been practicing doing arithmetic in bases other than 10, and he had been teaching us to just calculate the answer in base 10 then convert into the right base. What we needed to do was convert the number into base 10, calculate, then convert back into the right base.)

JoeAltmaier 16y ago
What you need to do is, learn to calculate in other bases. It works exactly as well as doing it in base 10, and of course is yards quicker.
- btilly 16y ago
  If you use another base a lot, then yes. But when you're facing problems in a half-dozen bases at once, then the learning curve of keeping that many times tables separate is not worthwhile.
eru 16y ago
You should also read how Feynman helps with choosing school books in California. http://www.textbookleague.org/103feyn.htm http://www.textbookleague.org/103feyn.htm

RiderOfGiraffes 16y ago

> If you explain well enough the concepts of > place-value and multiplication, most children > can independently invent an algorithm for long > multiplication. You're going to stare at your screen in disbelief, my experience suggests otherwise. A small number of gifted children have, in my experience, been able to work out their own methods, and they've become more efficient over time, and eventually converged to something isomorphic to long multiplication. But this is not the norm. Evidence suggests - and my experience agrees - that correctly teaching process creates the ground-work for moving to understanding. Correct methods, practised over time and in different context, result in an internalisation that leads to an ability to generalise and apply in wider contexts. This is my point about research and evidence-based teaching systems. The existing fads are borne of insufficiently researched and poorly developed methods, badly applied. They are often driven by dogma, and then delivered by teachers who don't properly understand what they're doing. There has to be a better way, and unfortunately, nearly everyone has an opinion, and none of them are supported by anything like enough evidence. And so we continue to damage our children. In parting, much of what you say I agree with, but not all. This isn't really the place to have a lengthy debate, but our positions aren't as far apart as my comment might have you think at first sight.

lmkg 16y ago
> This isn't really the place to have a lengthy debate To the contrary, this is Hacker News! Unfortunately, the situation that I'm referring to with children inventing their own long-multiplication algorithm was presented at a talk, so I can't point to the specific results, if they're even published yet, but my contention is based on an academic research study. The punchline of the experiment was that a full class of approximately-average grade-schoolers were taught math in a novel method, and developed their own multiplication algorithms, and went on to have above-average improvements in math test scores. The development of those algorithms were probably guided to a certain extent, but they looked to me like someone who was working the problem out from the first principles based on the concept of place value. That said, I'm definitely willing to accept the evidence you claim to have (especially since it leaves open the meaning of "correct"). I definitely agree that education should be based on science and not on dogma. Just because we think something ought to work doesn't mean that it works (it usually doesn't), and education is something you don't want to fuck up because the stakes are high and you only get one chance. But, I'm not sure we're even at the point where we can do research yet. Before you can measure results, you need to decide what results you're looking for and how you're measuring them. I'm a mathematician, and I'll be the first to admit that being able to perform long-multiplication isn't actually that useful of a skill, and yet that's mostly what we measure on standardized tests. Understanding what multiplication means and being generally numerate is far more useful and not measured very often (the best we have is word problems). The more desirable side-effects, such as the ability to abstract concepts, to generalize processes, and to apply knowledge from one domain to another, aren't being measured at all, and those are really the only things from my math education that I use in my day-to-day life. Similar things happen across other subjects--knowing what year the War of 1812 happened in isn't useful, while understanding the social, economic, and political conditions that led to that war is useful for the average citizen in a democracy (but doesn't fit well in a scantron). I think we as a society need to first have a big sit-down and figure out what the point of education is. We know it's important, but we've never coherently stated particularly why, and that lack of explicit direction is what lets politics, dogma, and fads more easily take control of the process. Once we've done that, we can let science guide the structure of education. Or rather, we should. There's a bit of a chicken-and-egg problem that without solid backgrounds in quantitative and logical thinking, most people have difficulty making data-driven decisions, and people in general won't get better at that until the education system improves.

chrischen 16y ago

> The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math is taught as a set of rules that must be followed... Well that's actually how most of the stuff in school is taught. Whenever a kid goes "why am I learning this I'm never going to use this" that's a sign they don't know what they're learning.

blackguardx 16y ago

Actually, word problems were the only thing that I liked about math class, but then again I am an engineer...