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It seems likely that no one taught those students how to think about math. I.e. teaching students the steps to solve a math problem is not teaching them how to

It seems likely that no one taught those students how to think about math. I.e. teaching students the steps to solve a math problem is not teaching them how to think about the problem. I instantly knew 12/13 + 7/8 was ~2 because I visualize two pie charts in my head, both of which are mostly full. This is in contrast to the other way to solve the problem, converting the fractions to a common denominator and then dividing by the denominator. It would take me some time to do the latter, whereas I can instantly do the former. I don't think the students who got that wrong (nor some who got it right) do any kind of visualization in their heads. Teachers need to realize that it's the operations in the head that count the most, not rote memorization of steps to solve a problem.

dos1 13y ago
It's always interesting to hear how people go about solving math problems. You mention a pie chart visualization and then the much more labor intensive (but maybe "correct"?) method. I used a third way, which was thinking that 13/13 would be one, so 12/13 is pretty close, so that's ~1. And 8/8 would be 1, so 7/8 is pretty close and also ~1. 1 + 1 = 2 :) I imagine there are myriad other ways people approach estimation problems like this. In response to the rest of your post, I was never taught how to "think" about math. I was educated in a decent school system, but it was all rote memorization of multiplication tables. I think most people who are interested in learning will come up with their own tricks regardless of curriculum. Of course, imagine how much better I'd be at this stuff if I had math teacher's who were competent :)
- SiVal 13y ago
  I use the heuristic that many of us here probably use, consciously or not, after our years of experience with math problems: if it's a math problem, as opposed to problem in some other domain that ends up requiring math (science, accounting, carpentry, etc.), there will be some degree of artifice in the problem. Somehow, the numbers will just happen to end up being integers or perfect squares or exact multiples or whatever, so that there is an easy way to solve this specific problem (not a general problem of this sort but this specific instance). In this case, you examine the numbers and spot that they are both just "one off from one" fractions, so the sum is roughly 1+1. The test givers will then see to it that there is only one answer that matches the result of the "trick" they were testing to see if you could find. Kids who get a lot of math internalize this heuristic, which actually trips them up briefly when they start having real science classes, because they think they've done something wrong if the answer turns out to be 5.6293 or 0.07291 instead of 4 or 9 or 5/8 or sqrt(10). They assume they missed the trick.
  - Periodic 13y ago
    When I did my undergraduate degree in physics I think one of the best things I learned early on was estimation skills. I was used to doing things precisely and finding the tricks. Our professors made jokes about things just needing to be right to "within an order of magnitude", and it wasn't for two years that I internalized that. When you deal with the real world there are always a lot of errors and uncertainty in measurement. Simply being within 10% of the right answer is generally sufficient and quickly getting that answer over getting the 99.99% accurate answer is better if it takes you one-tenth the time.
    - waps 13y ago
      I remember this from my first university physics class. We would derive a movement equation for a cannonball, to find the optimal angle to shoot a cannon for maximum travel. Everybody knew the answer of course, but we'd always just used the formula. This time we'd start with the obvious integration equation, movement + attraction between 2 point masses, integrate over flight time, and find the point where it crosses the ground plane. And then the teacher just took the range from the integration, and the formula, multiplied the two and put a ~= sign between them. I believe I actually stood up and said you can't do that and we had the first of many discussions about exactness. That was scary. That was my first run-in with what I considered the central article of my then faith : that you can derive the structure of the physical world from first principles. Throwing away terms in an equation in order to arrive at correct physics laws, I don't know, I considered it sacrilege or something. Of course I've since learned that deriving all of physics from it's own basic laws doesn't work, and the way we fix that is that we delete "inconvenient" terms in the equations when required. Deriving physics from a few mathematical laws is completely impossible. You can't even correctly derive the (mathematical) fields used in physics, so the very numbers that one uses to do physics aren't actually valid mathematical numbers. So the relation between physics and mathematics is not that one is based on the other, because that was tried and didn't work out, and people have almost completely given up. So it was replaced by a marriage of convenience (this works ! Sure it won't validate mathematically but the numbers look really similar), ignoring at least a dozen elephants that stood in the way, and we just act like they don't exist.
- chacham15 13y ago
  I did it that way too. And while I agree that the best teacher for a person is that person themselves, this is such a basic understanding of math that I think that even the less motivated need to know this.
- pjungwir 13y ago
  As I read the question, I was prepared to work it out the hard way, and when I got to the part about not choosing the exact answer but only the closest, I was really dismayed because I thought I'd also have to work out how close each choice was to the actual value. Then I saw the possible answers and I thought, "Oh, 2."
_ZeD_ 13y ago
this isn't about high level math... this is about penny pinch... doesn't no-one go to open-air market?

dos1 13y ago

It's always interesting to hear how people go about solving math problems. You mention a pie chart visualization and then the much more labor intensive (but maybe "correct"?) method. I used a third way, which was thinking that 13/13 would be one, so 12/13 is pretty close, so that's ~1. And 8/8 would be 1, so 7/8 is pretty close and also ~1. 1 + 1 = 2 :) I imagine there are myriad other ways people approach estimation problems like this. In response to the rest of your post, I was never taught how to "think" about math. I was educated in a decent school system, but it was all rote memorization of multiplication tables. I think most people who are interested in learning will come up with their own tricks regardless of curriculum. Of course, imagine how much better I'd be at this stuff if I had math teacher's who were competent :)

SiVal 13y ago
I use the heuristic that many of us here probably use, consciously or not, after our years of experience with math problems: if it's a math problem, as opposed to problem in some other domain that ends up requiring math (science, accounting, carpentry, etc.), there will be some degree of artifice in the problem. Somehow, the numbers will just happen to end up being integers or perfect squares or exact multiples or whatever, so that there is an easy way to solve this specific problem (not a general problem of this sort but this specific instance). In this case, you examine the numbers and spot that they are both just "one off from one" fractions, so the sum is roughly 1+1. The test givers will then see to it that there is only one answer that matches the result of the "trick" they were testing to see if you could find. Kids who get a lot of math internalize this heuristic, which actually trips them up briefly when they start having real science classes, because they think they've done something wrong if the answer turns out to be 5.6293 or 0.07291 instead of 4 or 9 or 5/8 or sqrt(10). They assume they missed the trick.
- Periodic 13y ago
  When I did my undergraduate degree in physics I think one of the best things I learned early on was estimation skills. I was used to doing things precisely and finding the tricks. Our professors made jokes about things just needing to be right to "within an order of magnitude", and it wasn't for two years that I internalized that. When you deal with the real world there are always a lot of errors and uncertainty in measurement. Simply being within 10% of the right answer is generally sufficient and quickly getting that answer over getting the 99.99% accurate answer is better if it takes you one-tenth the time.
  - waps 13y ago
    I remember this from my first university physics class. We would derive a movement equation for a cannonball, to find the optimal angle to shoot a cannon for maximum travel. Everybody knew the answer of course, but we'd always just used the formula. This time we'd start with the obvious integration equation, movement + attraction between 2 point masses, integrate over flight time, and find the point where it crosses the ground plane. And then the teacher just took the range from the integration, and the formula, multiplied the two and put a ~= sign between them. I believe I actually stood up and said you can't do that and we had the first of many discussions about exactness. That was scary. That was my first run-in with what I considered the central article of my then faith : that you can derive the structure of the physical world from first principles. Throwing away terms in an equation in order to arrive at correct physics laws, I don't know, I considered it sacrilege or something. Of course I've since learned that deriving all of physics from it's own basic laws doesn't work, and the way we fix that is that we delete "inconvenient" terms in the equations when required. Deriving physics from a few mathematical laws is completely impossible. You can't even correctly derive the (mathematical) fields used in physics, so the very numbers that one uses to do physics aren't actually valid mathematical numbers. So the relation between physics and mathematics is not that one is based on the other, because that was tried and didn't work out, and people have almost completely given up. So it was replaced by a marriage of convenience (this works ! Sure it won't validate mathematically but the numbers look really similar), ignoring at least a dozen elephants that stood in the way, and we just act like they don't exist.
chacham15 13y ago
I did it that way too. And while I agree that the best teacher for a person is that person themselves, this is such a basic understanding of math that I think that even the less motivated need to know this.
pjungwir 13y ago
As I read the question, I was prepared to work it out the hard way, and when I got to the part about not choosing the exact answer but only the closest, I was really dismayed because I thought I'd also have to work out how close each choice was to the actual value. Then I saw the possible answers and I thought, "Oh, 2."

_ZeD_ 13y ago

this isn't about high level math... this is about penny pinch... doesn't no-one go to open-air market?