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Many elementary teachers don’t understand math, and it makes them anxious

latimes.com

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Re: Many elementary teachers don’t understand math, and it makes them anxious

#321

Post by username90 »

chongli wrote: post #319
username90 wrote: post #317

Earlier quoted context omitted.

You wondered how a sixth grader could solve it, I showed you how. Being able to do it using formal methods is good, but my point is that sixth graders are still pretty smart and can use more creative methods which still works very well as I showed. Also ny solution wasn't brute force or trial and error, there are very few possible sequences of 11 numbers around 201, so you are guaranteed to find the solution quickly.

I'm a pure math student and when I first found the problem I sent it to my friend who is a stats student. She solved it very quickly, but admitted to using a calculator. None of my exams permit calculators, so I've gotten used to never using them. Perhaps the elementary school math contest permits the use of a calculator? That would make it very quick for sixth graders to solve it using your method.

You could do it without a calculator like this:

Assume the middle of the sequence is 201 since 2012 / 10 ~= 201. Then the sum of the 11 numbers between 195 and 206 would be 11 * 201 = 2211. Subtract the sum of the values we didn't remove to get 2211 - 2012 = 199. 199 is between 195 and 206 proving that it is a valid answer.

Edit: Also I did my masters in pure math, doesn't mean that I cannot use common logic without the formalism.

Re: Many elementary teachers don’t understand math, and it makes them anxious

#322

Post by lotu »

keenmaster wrote: post #15
_bxg1 wrote: post #2

I still don't know why long-division works. I don't think I even remember how to do it as an adult. I've always had a very intuitive grasp of math, which meant I excelled at solving things in my head but struggled with (and resented) having to show my work in the arbitrary algorithms we were taught. I understand division despite my education. They really should try to come up with algorithms that are more intuitively…

That’s a good point. You know what, it’s way easier to teach how to solve for x when y + x = 3x + 3 and y = 4 than it is to teach long division. As far as I could tell, that basic algebra is also much more important than long division, so why not teach it at an earlier age? The symbology of algebra is probably great for developing minds. Just show long division for “proofs” while introducing the concept of division,…

Generally a lot of the formalism of Algebra requires specific brain development milestones dealing with abstraction to be reached. That’s why it is taught at the end of middle school. Now could some concepts be introduced earlier? Yes a common exercise in elementary school is to solve questions in the form of 7 + _ = 15. In this case x is replaced with a _ because variables are hard to grasp.

Re: Many elementary teachers don’t understand math, and it makes them anxious

#323

Post by leetcrew »

fractaled wrote: post #256
leetcrew wrote: post #228

Earlier quoted context omitted.

the way I learned it, it's just a notational difference. either way, you are dividing the divisor into each place of the dividend (starting with the most significant digit) and carrying over the divisor to the next place. short division is just a more implicit way of doing this that saves space. the shorthand version doesn't really work for divisors with more than one digit though.

What doesn't work with short division and multi-digit divisors?

it's not that the algorithm "breaks"; it's just that it stops being a convenient notation for humans.

the "short" notation relies on the ability to squeeze the remainder in between the digits of the dividend. if you have a divisor greater than ten, it's possible to have a remainder with more than one digit. two or more remainder digits is a lot harder to write legibly between the digits of the dividend. you also end up subtracting larger numbers in your head, which is error prone. finally, most people only memorize the multiplication table up to 12x12 or so. once they can't simply do a lookup into their memorized table, they switch to more of a "guess and check" approach, where they will inevitably have to cross out or erase their previous work.

in the end, it introduces a lot of unnecessary opportunities to make mistakes and hides stuff that people can't do reliably in their heads.

Re: Many elementary teachers don’t understand math, and it makes them anxious

#324

Post by chongli »

username90 wrote: post #321
chongli wrote: post #319

Earlier quoted context omitted.

I'm a pure math student and when I first found the problem I sent it to my friend who is a stats student. She solved it very quickly, but admitted to using a calculator. None of my exams permit calculators, so I've gotten used to never using them. Perhaps the elementary school math contest permits the use of a calculator? That would make it very quick for sixth graders to solve it using your method.

You could do it without a calculator like this: Assume the middle of the sequence is 201 since 2012 / 10 ~= 201. Then the sum of the 11 numbers between 195 and 206 would be 11 * 201 = 2211. Subtract the sum of the values we didn't remove to get 2211 - 2012 = 199. 199 is between 195 and 206 proving that it is a valid answer. Edit: Also I did my masters in pure math, doesn't mean that I cannot use common logic without…

199 is between 195 and 206 proving that it is a valid answer.

Try the same trick with this modified problem:

Fifty consecutive positive integers are written on a board. Maria erases one of the numbers. If the sum of the remaining numbers is 2009, what number did Maria erase?

You'll find that it does not work. The trick you did relies on the specific problem as stated. It is not true in general.

2009 / 49 = 41. 41 * 50 = 2050.

2050 is not the sum of the numbers 17..66, 2075 is, and subtracting 2009 from 2050 just gets you 41 again, which is not the correct answer.

Re: Many elementary teachers don’t understand math, and it makes them anxious

#325

Post by sicariusnoctis »

chrisseaton wrote: post #33
dboreham wrote: post #32

Earlier quoted context omitted.

>I have a PhD in computer science but I cannot understand long division I think you're trolling us a little: you could sit down with pencil and paper and think about how it must work and 20 minutes later you'd understand it, surely.

I have no intuitive understanding or mental model of the algorithm. I could follow the instructions if they’re written in front of me but I have never managed to understand why they work or to memorise them. I have a similar mental block for standard deviation - I always seem to have another ‘but why?’ question that eventually people answer with ‘because!’ and then we’ve both given up.

1. Long division uses a similar idea to Euclid's algorithm for GCD. Are you familiar with that? Regardless, both algorithms are divide and conquer algorithms.

6240 / 5

= divide(6240, 5)

= divide(1240, 5) + 1000

= divide(240, 5) + 1000 + 200

= divide(40, 5) + 1000 + 200 + 40

= 1000 + 200 + 40 + 8

= 1248

----------

2. Regarding standard deviation, one of these bullets might help:

- The normal distribution has exactly one shape, centered at x=0. But it's useful to apply two transformations to it: translation and horizontal stretch. To translate a distribution left/right, change its mean. To horizontally stretch a normal distribution, change its standard deviation.

- Mean has units of length. Standard deviation has units of length. They tell you where the normal distribution is offset, and how wide it is. Mean and standard deviation are just measuring sticks/rulers for normal distributions.

- When people talk about the standard deviation with any arbitrary data, they're usually assuming the data is normally distributed. If the data is not normally distributed, standard deviation no longer refers to the width of the normal distribution, so we lose that visualization.

- With non-normally distributed data, the standard deviation is still useful as an analytical tool because taking the (sqrt of the) summed squares still gives us a number that grows as the data spreads further apart or if the distribution grows wider. There's a center point for the data (the mean), so to ensure you're measuring the overall spread of the data, you subtract the center point off each data point before squaring them. In other words, you're squaring deviations from the mean. And unlike the sum of absolute differences, the sum of squared differences (variance) is differentiable. Differentiability is a great property, so this is the standard way to compute a sum of deviations from the center.

Re: Many elementary teachers don’t understand math, and it makes them anxious

#326

Post by tacomonstrous »

nopinsight wrote: post #227
nopinsight wrote: post #115

For over a decade, I have run a successful tutoring center focused on math and science for elementary school students in a developing country. Our students won numerous awards both at the national and international levels. We usually employ teachers with degrees in math or science, or sometimes engineering. Occasionally for early elementary levels (grades 1-3), some teachers have a degree in another discipline. Regar…

For those curious, below are some kinds of primary math questions we model after to test teacher applicants. In my experience, many kids are much more enthusiastic about this kind of "challenging" problems than the drills in many standard textbooks, as long as the problems are chosen to match their level. They definitely learn a lot more as well. Note that although they do require a little arithmetic to solve, the ch…

I'm a working mathematician and would need a non-trivial amount of thought to get any of these problems.

I was also pretty poor at this kind of thing in high school and college, but fortunately the skills involved are almost but not quite disjoint from what is needed to do research on math.

Re: Many elementary teachers don’t understand math, and it makes them anxious

#327

Post by taesko »

chongli wrote: post #324
username90 wrote: post #321

Earlier quoted context omitted.

You could do it without a calculator like this: Assume the middle of the sequence is 201 since 2012 / 10 ~= 201. Then the sum of the 11 numbers between 195 and 206 would be 11 * 201 = 2211. Subtract the sum of the values we didn't remove to get 2211 - 2012 = 199. 199 is between 195 and 206 proving that it is a valid answer. Edit: Also I did my masters in pure math, doesn't mean that I cannot use common logic without…

199 is between 195 and 206 proving that it is a valid answer. Try the same trick with this modified problem: Fifty consecutive positive integers are written on a board. Maria erases one of the numbers. If the sum of the remaining numbers is 2009, what number did Maria erase? You'll find that it does not work. The trick you did relies on the specific problem as stated. It is not true in general. 2009 / 49 = 41. 41 * 5…

I solved it like this: The numbers are x, x+1..., x+10 = 11x + 55. One is erased therefore 10x + 55 - y = 2012. 10x - y = 1957. We know X is whole so we need a number between 1957 and 1967 that divides 10 and that is X. 1960 so X = 196 and therefore y = 3. Answer is X+Y = 199.

I think a school student can do this math manually. The hardest part is the sum of the first N(in this case 10) numbers and then finding the divisor. Yes there might be a better solution.

But I think a kid would think up a brute force algorithm like this and it is easier for small N. The computation is only in O(N) complexity haha. And if an answer exists it will always be found - so it is a real solution. Might be the first algorithm you write if you are tasked to do this with software.

Re: Many elementary teachers don’t understand math, and it makes them anxious

#328

Post by optimiz3 »

xiphias2 wrote: post #320
optimiz3 wrote: post #18

Routinely we have to explain to users in a crypto related business that 0.00835 Satoshis is the same as 0.008350 Satoshis. Adult humans don't handle decimals.

Why does it even matter? Sub-Satoshis are used only as transaction fees in lightning network, and even there 1 milli Satoshi is the smallest amount that's allowed for now AFAIK.

It's a hypothetical example based on actual customer cases. Change the units to BTC/LTC/whatever if you'd like to make it more precisely simulate the user interaction.

Re: Many elementary teachers don’t understand math, and it makes them anxious

#329

Post by novok »

tacomonstrous wrote: post #326
nopinsight wrote: post #227

Earlier quoted context omitted.

For those curious, below are some kinds of primary math questions we model after to test teacher applicants. In my experience, many kids are much more enthusiastic about this kind of "challenging" problems than the drills in many standard textbooks, as long as the problems are chosen to match their level. They definitely learn a lot more as well. Note that although they do require a little arithmetic to solve, the ch…

I'm a working mathematician and would need a non-trivial amount of thought to get any of these problems. I was also pretty poor at this kind of thing in high school and college, but fortunately the skills involved are almost but not quite disjoint from what is needed to do research on math.

What are the working skills needed for research in math?

Re: Many elementary teachers don’t understand math, and it makes them anxious

#330

Post by nopinsight »

tacomonstrous wrote: post #326
nopinsight wrote: post #227

Earlier quoted context omitted.

For those curious, below are some kinds of primary math questions we model after to test teacher applicants. In my experience, many kids are much more enthusiastic about this kind of "challenging" problems than the drills in many standard textbooks, as long as the problems are chosen to match their level. They definitely learn a lot more as well. Note that although they do require a little arithmetic to solve, the ch…

I'm a working mathematician and would need a non-trivial amount of thought to get any of these problems. I was also pretty poor at this kind of thing in high school and college, but fortunately the skills involved are almost but not quite disjoint from what is needed to do research on math.

I know some students who excelled at this kind of problems when they were in elementary school and now possibly on path to be good researchers in math and theoretical CS.

One is doing math PhD at Harvard; another received a PhD in theoretical CS from Berkeley with stellar track record (over 20 papers when he graduated after just 4 years in PhD--many/most in top venues, as well as perhaps the most prestigious best paper award for CS Theory PhD student when he was in 2nd year). Not all went on the research path though; some are working in applied software engineering. A few who did math undergrad degrees at places like MIT but I'm not sure where they are going next.

Another conspicuous example: Terrence Tao was also the youngest ever International Math Olympiad gold medalist (he won it when he was 12-13!).

There appears to be a significant correlation between subsets of people who are good at certain types of math and theoretical CS research and at solving this kind of problems when younger.

I read that there are mathematicians who focus on solving problems and those who build frameworks. Perhaps the skills for solving these problems are closer to the former?

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