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1. My personal experience as a teacher. 2. The average state of math knowledge of contemporary american high school graduates, coupled with statistics on neede

1. My personal experience as a teacher. 2. The average state of math knowledge of contemporary american high school graduates, coupled with statistics on needed remedial education for college entrants with high school diplomas. Teaching math is easy. I can teach any kid math. It's not a problem at all because we know how to do it. Many countries do it successfully. Observing what schools are doing, the textbooks, the approaches, and the level of math competency of elementary school teachers completely explains what is going wrong. It's not a mystery. Also explains why kids homeschooled by parents, even parents who were high school drop outs, end up with considerably better math skills than most public school graduates, and even many private school graduates. Typical high school graduates don't understand fractions, percentages or decimals. They can't do division. They don't understand algebra at all. We know this is true because of college placement tests. This is absurd. 100% of high school graduates should have passed differential and integral calculus. That should be the minimal acceptable standard. That it is not, and that graduates fail at even moderate arithmetic skills, shows the complete failure of the system. Downvote all you want while shouting "the dedicated teachers are not the problem, the students and their parents are!" That won't change a thing. Meanwhile, those who utilize proven approaches succeed and those who make excuses and follow fads fail.

graeme 15y ago
What effect do you think calculators have? I do some SAT tutoring, and I find that students are helpless at basic arithmetic if they can't use a calculator. My suspicion is that calculators prevent students from developing mental math skills, and that this hurts conceptual understanding across the board. But I can only see the results of the system, so I don't know what actually produced the types of students I'm seeing. I may be wrong about calculators - from what you say, it could simply be that the system in general is bad.
- tel 15y ago
  I grew up learning math from calculators. I still today am terrible at mental math, but having immediate, accurate graphing capabilities from precalc onward was an incredible boon. Even before that, mechanical skills such as polynomial factorization or algebraic reorganization ("fiddly symbol manipulation") were actually, in my opinion, aided by having a calculator which could perform them for me. By releasing the burden of actually performing the computations, I was focused on larger patterns which made the computation meaningful and possible. There are two camps here, of course. As an engineer or mathematician, doing mental arithmetic is a dead skill---all the effort comes from modeling and understanding the structure of math. However, "practical" math such as computing tips or adding up grocery bills---and this was the kind of math application stressed in my public school education---probably benefits much more from rapid mental sums. So, ironically, I'd support the heavy involvement of calculators (even ones as powerful as Mathematica itself) in education if it's to be aimed to produce engineers, scientists, or mathematicians. Otherwise, they're probably damaging.
  - moped 15y ago
    WolframAlpha is a good visual tool also, and I often use it when I tutor math.
  - lnguyen 15y ago
    The issue is whether you learned the actual skill (graphing, factorization, etc.) or just learned how to get the calculator to do it for you. In many cases, the end results appear identical. However, I'd rather the upcoming engineers, scientists and mathematicians be banned from using calculators for anything more than basic arithmetic until they prove they do understand the concepts.
    - gergles 15y ago
      Factorization and graphing aren't actual skills, though. They are mechanical computations that basically scream "please use a tool to do this". We don't pine for the days when people had to hand-calculate logarithms, because we've come up with better ways of using that tool. CASes and graphing calculators are the same thing.
      - gmichnikov 15y ago
        This is not true. Calculating a log by hand would be the same process every time. Factoring quadratics is mostly the same every time, while factoring in general is not, and graphing definitely is not. Graphing, in particular, is a crucial math skill, because it forces you to understand what is actually going on with each function. As a rule, students who have no idea what a graph is going to look like before they press "graph" also don't understand many important underlying fundamentals. There is no reason middle/high school students should be using graphing calculators on a regular basis. Scientific calculators for things like logs are fine, but beyond that the calculator replaces actual learning.
      - lnguyen 15y ago
        You do realize that the tools have to be created based on some underlying process. If you have an understanding of that process you'll have a better understanding (and appreciation) of that tool. And maybe be able to come up with a better one.
- droithomme 15y ago
  I agree with you about calculators. People need an intuition about numbers that comes from actually thinking about them rather than pushing buttons in order to be able to notice when they make serious errors. Calculator trained cashiers take in a $10 bill on a $5 purchase and then return $95 in change because the computer told them to.
xiaoma 15y ago
Out of curiosity, why is it that you would prioritize differential and integral calculus? I studied calculus very early myself, but outside of later classwork, I've only found it useful on rare occasions. I surely would have used it more if I had gone into a research job or into material sciences, but I estimate that very, very few careers are like that. What would have been much more useful for me would have been a stronger grounding in statistics.
- droithomme 15y ago
  I didn't mean to prioritize it, I'm saying that level of understanding should the goal of average achievement for high school graduates. It is also really really far from what is being accomplished on average. High schools have seniors who can't do division and don't understand operations on fractions. Calculus is something anyone can understand, but it is seen by teachers as some super advanced mysterious stuff for rocket scientists while they struggle along to teach subtraction to 18 year olds. Imagine a physical education fitness curriculum whose goal after 13 years of training was for 18 year olds to be able to turn over by themselves and begin to crawl. That's what we are doing, being satisfied with goals for graduates that shouldn't be challenging even for 6 year olds. It shouldn't be tolerated at all. Schools that are graduating students with no math skills have had 13 years of failed instruction with these students. Not just one teacher along the way. All of them. The system should be burned to the ground and started over. That will never happen though, it's too corrupt and incompetent. Myself I've given up hope reform of the system is possible.
  - saulrh 15y ago
    Speaking as a recent high school graduate, please, please don't give up. I made it out of school knowing calculus and more because I had one physics teacher and one math teacher that hadn't given up. They wrote their homework themselves, used college textbooks and twenty-year-old texbooks they'd collected from eBay, and railed constantly at the district's purchasing people. They weren't able to change the system all by themselves, but they still made - and continue to make - all the difference for dozens or hundreds of students. You can do the same. Please.
  - 15y ago
aik 15y ago
Do you think the Singapore method at one point was a fad (or new math teaching methodology) and if not, why or how not? Secondly -- You say "Teaching math is easy." What does that mean? Just from personal experience, 10 years or so ago I took Calc 1 and 2 in high school, and calc 2 again in college, and did well in all, and know how to integrate and differentiate and use various formulas due to a recognition of certain inculcated patterns, however would I say I am skilled at actually using calculus as any sort of real life thinking/problem solving tool? Absolutely not and not even close. Is this a success? Absolutely not. One of the most important things was completely overlooked by the "traditional" method I was taught in.
derleth 15y ago
> 100% of high school graduates should have passed differential and integral calculus. I'd argue that a similar pass rate for introductory statistics is vitally important for our continued viability as a society. Calculus is, frankly, less important than knowing when a politician is lying to you, or whether that test result means you need to draw up a living will now.

graeme 15y ago

What effect do you think calculators have? I do some SAT tutoring, and I find that students are helpless at basic arithmetic if they can't use a calculator. My suspicion is that calculators prevent students from developing mental math skills, and that this hurts conceptual understanding across the board. But I can only see the results of the system, so I don't know what actually produced the types of students I'm seeing. I may be wrong about calculators - from what you say, it could simply be that the system in general is bad.

tel 15y ago
I grew up learning math from calculators. I still today am terrible at mental math, but having immediate, accurate graphing capabilities from precalc onward was an incredible boon. Even before that, mechanical skills such as polynomial factorization or algebraic reorganization ("fiddly symbol manipulation") were actually, in my opinion, aided by having a calculator which could perform them for me. By releasing the burden of actually performing the computations, I was focused on larger patterns which made the computation meaningful and possible. There are two camps here, of course. As an engineer or mathematician, doing mental arithmetic is a dead skill---all the effort comes from modeling and understanding the structure of math. However, "practical" math such as computing tips or adding up grocery bills---and this was the kind of math application stressed in my public school education---probably benefits much more from rapid mental sums. So, ironically, I'd support the heavy involvement of calculators (even ones as powerful as Mathematica itself) in education if it's to be aimed to produce engineers, scientists, or mathematicians. Otherwise, they're probably damaging.
- moped 15y ago
  WolframAlpha is a good visual tool also, and I often use it when I tutor math.
- lnguyen 15y ago
  The issue is whether you learned the actual skill (graphing, factorization, etc.) or just learned how to get the calculator to do it for you. In many cases, the end results appear identical. However, I'd rather the upcoming engineers, scientists and mathematicians be banned from using calculators for anything more than basic arithmetic until they prove they do understand the concepts.
  - gergles 15y ago
    Factorization and graphing aren't actual skills, though. They are mechanical computations that basically scream "please use a tool to do this". We don't pine for the days when people had to hand-calculate logarithms, because we've come up with better ways of using that tool. CASes and graphing calculators are the same thing.
    - gmichnikov 15y ago
      This is not true. Calculating a log by hand would be the same process every time. Factoring quadratics is mostly the same every time, while factoring in general is not, and graphing definitely is not. Graphing, in particular, is a crucial math skill, because it forces you to understand what is actually going on with each function. As a rule, students who have no idea what a graph is going to look like before they press "graph" also don't understand many important underlying fundamentals. There is no reason middle/high school students should be using graphing calculators on a regular basis. Scientific calculators for things like logs are fine, but beyond that the calculator replaces actual learning.
    - lnguyen 15y ago
      You do realize that the tools have to be created based on some underlying process. If you have an understanding of that process you'll have a better understanding (and appreciation) of that tool. And maybe be able to come up with a better one.
droithomme 15y ago
I agree with you about calculators. People need an intuition about numbers that comes from actually thinking about them rather than pushing buttons in order to be able to notice when they make serious errors. Calculator trained cashiers take in a $10 bill on a $5 purchase and then return $95 in change because the computer told them to.

xiaoma 15y ago

Out of curiosity, why is it that you would prioritize differential and integral calculus? I studied calculus very early myself, but outside of later classwork, I've only found it useful on rare occasions. I surely would have used it more if I had gone into a research job or into material sciences, but I estimate that very, very few careers are like that. What would have been much more useful for me would have been a stronger grounding in statistics.

droithomme 15y ago
I didn't mean to prioritize it, I'm saying that level of understanding should the goal of average achievement for high school graduates. It is also really really far from what is being accomplished on average. High schools have seniors who can't do division and don't understand operations on fractions. Calculus is something anyone can understand, but it is seen by teachers as some super advanced mysterious stuff for rocket scientists while they struggle along to teach subtraction to 18 year olds. Imagine a physical education fitness curriculum whose goal after 13 years of training was for 18 year olds to be able to turn over by themselves and begin to crawl. That's what we are doing, being satisfied with goals for graduates that shouldn't be challenging even for 6 year olds. It shouldn't be tolerated at all. Schools that are graduating students with no math skills have had 13 years of failed instruction with these students. Not just one teacher along the way. All of them. The system should be burned to the ground and started over. That will never happen though, it's too corrupt and incompetent. Myself I've given up hope reform of the system is possible.
- saulrh 15y ago
  Speaking as a recent high school graduate, please, please don't give up. I made it out of school knowing calculus and more because I had one physics teacher and one math teacher that hadn't given up. They wrote their homework themselves, used college textbooks and twenty-year-old texbooks they'd collected from eBay, and railed constantly at the district's purchasing people. They weren't able to change the system all by themselves, but they still made - and continue to make - all the difference for dozens or hundreds of students. You can do the same. Please.
- 15y ago

aik 15y ago

Do you think the Singapore method at one point was a fad (or new math teaching methodology) and if not, why or how not? Secondly -- You say "Teaching math is easy." What does that mean? Just from personal experience, 10 years or so ago I took Calc 1 and 2 in high school, and calc 2 again in college, and did well in all, and know how to integrate and differentiate and use various formulas due to a recognition of certain inculcated patterns, however would I say I am skilled at actually using calculus as any sort of real life thinking/problem solving tool? Absolutely not and not even close. Is this a success? Absolutely not. One of the most important things was completely overlooked by the "traditional" method I was taught in.

derleth 15y ago

> 100% of high school graduates should have passed differential and integral calculus. I'd argue that a similar pass rate for introductory statistics is vitally important for our continued viability as a society. Calculus is, frankly, less important than knowing when a politician is lying to you, or whether that test result means you need to draw up a living will now.