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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

#311

Post by chongli »

malaya_zemlya wrote: post #305
chongli wrote: post #300

Earlier quoted context omitted.

Eleven consecutive positive integers are written on a board. Maria erases one of the numbers. If the sum of the remaining numbers is 2012, what number did Maria erase This is a very challenging problem. Without a calculator (or computer), it took me a system of an equation with an inequality in 2 variables, using the fact that the solutions are integers, to solve it. I would be very surprised (and excited) to meet an…

try performing all calculations mod 10.

I’m not sure how to make use of your hint.

I solved the problem for n, the largest number in the sequence of eleven, and a, the number Maria erases, by the following:

n - 11 11(n + n - 10)/2 = 2012 + a

I’m interested to know what other methods may be used to solve it (apart from brute force).

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

#312

Post by chongli »

username90 wrote: post #308
chongli wrote: post #300

Earlier quoted context omitted.

Eleven consecutive positive integers are written on a board. Maria erases one of the numbers. If the sum of the remaining numbers is 2012, what number did Maria erase This is a very challenging problem. Without a calculator (or computer), it took me a system of an equation with an inequality in 2 variables, using the fact that the solutions are integers, to solve it. I would be very surprised (and excited) to meet an…

You don't need algebra. You know that 10 numbers are 2012, so on average 201, so the middle of the numbers needs to be around there. After that you just try a few sequences around 201 and you'll find it pretty quickly, most can add 11 numbers pretty fast in 6'th grade. Forcing yourself to only rely on the solution strategies taught by school makes you very bad at problem solving.

I specifically set out to find a solution that doesn’t involve any brute force (trial and error). Trial and error can waste all of your time and get you nowhere on nontrivial problems.

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

#313

Post by chongli »

abdullahkhalids wrote: post #310
chongli wrote: post #300

Earlier quoted context omitted.

Eleven consecutive positive integers are written on a board. Maria erases one of the numbers. If the sum of the remaining numbers is 2012, what number did Maria erase This is a very challenging problem. Without a calculator (or computer), it took me a system of an equation with an inequality in 2 variables, using the fact that the solutions are integers, to solve it. I would be very surprised (and excited) to meet an…

The eleven numbers are n,n+1,..., n+10. Their sum is 11n + 55. Let the number removed be n + k, then the sum of the remaining is just 10n + 55 -k = 2012. Or 10n-k = 1957. Then it is obvious that k=3 so that n is an integer. Hence, n=196 The eleven numbers are 196,197,..., 206, out of which 199 was removed.

That’s a very good solution. Thanks.

I began with n being the largest of the sum so that 11n - 55 = 2012 + k. I solved for k in terms of n and then used an inequality to find n which gave me k.

Now I see that the method I used relies on math beyond elementary school, which makes the problem a bit more complicated than it ought to be. On the other hand, the elementary school only method relies on the fact that you have 10 numbers remaining and so you can easily see the remainder. It wouldn’t generalize as well, especially in cases where there may be multiple solutions and you’re asked to find all of them.

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

#314

Post by mepiethree »

chongli wrote: post #312
username90 wrote: post #308

Earlier quoted context omitted.

You don't need algebra. You know that 10 numbers are 2012, so on average 201, so the middle of the numbers needs to be around there. After that you just try a few sequences around 201 and you'll find it pretty quickly, most can add 11 numbers pretty fast in 6'th grade. Forcing yourself to only rely on the solution strategies taught by school makes you very bad at problem solving.

I specifically set out to find a solution that doesn’t involve any brute force (trial and error). Trial and error can waste all of your time and get you nowhere on nontrivial problems.

OTOH, puzzling something out through trial and error fosters curiosity and teaches a "fail fast" mindset. For a sixth grader, this "play" may be a more beneficial lesson than only being able to tackle problems for which you already know the optimal strategy (in this case, algebra)

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

#315

Post by chongli »

mepiethree wrote: post #314
chongli wrote: post #312

Earlier quoted context omitted.

I specifically set out to find a solution that doesn’t involve any brute force (trial and error). Trial and error can waste all of your time and get you nowhere on nontrivial problems.

OTOH, puzzling something out through trial and error fosters curiosity and teaches a "fail fast" mindset. For a sixth grader, this "play" may be a more beneficial lesson than only being able to tackle problems for which you already know the optimal strategy (in this case, algebra)

I disagree. The hard problems in math, science, and engineering are not amenable to a "fail fast" mindset. They take years, decades, or sometimes centuries of study to solve.

This sort of puzzling through play is good for solving math contest problems that have been specifically designed for the purpose, and little else.

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

#316

Post by voxl »

cptskippy wrote: post #155
_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…

Then you never knew it? Perhaps what you were taught wasn't long division. My kids were not taught the same math that I was taught in the 80s. My 17 year old was struggling with the division algorithm they were teaching in Elementary school at the time (late 2000s). I can't remember the details but it was something I had never seen before that was a step up from throwing poop at the wall. I showed her how to do long…

Anyone who legitimately thinks common core is worse than how math was previously taught does not understand math at all. It is at worst just different with the dream of being better. There is no conspiracy theory about wanting to sell more books or to "separate the masses".

If you actually understand the math picking up the common core methods is minutes of work at worst.

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

#317

Post by username90 »

chongli wrote: post #312
username90 wrote: post #308

Earlier quoted context omitted.

You don't need algebra. You know that 10 numbers are 2012, so on average 201, so the middle of the numbers needs to be around there. After that you just try a few sequences around 201 and you'll find it pretty quickly, most can add 11 numbers pretty fast in 6'th grade. Forcing yourself to only rely on the solution strategies taught by school makes you very bad at problem solving.

I specifically set out to find a solution that doesn’t involve any brute force (trial and error). Trial and error can waste all of your time and get you nowhere on nontrivial problems.

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.

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

#318

Post by renaudg »

Research also shows that, compared with other college students, future elementary teachers are especially prone to math anxiety —apprehension about doing math that’s so severe it interferes with actually doing it. That anxiety remains once they are in classrooms, and studies show that students learn less math from a math-anxious teacher.

Reading this paragraph gives me a strong hunch that it could be gender related. I'm wondering what the gender skew is like in elementary school teaching, similar to nursing ?

While research hasn't shown any difference in STEM ability between genders, it most definitely has when it comes to intrinsic interest (and I know this is controversial, but contrary to what is commonly said it's not because of some evil patriarchal plan to keep girls out : gender-related "things vs people" affinity has been found in baby monkeys too, and seems correlated to pre-natal testosterone exposure !)

One can easily imagine a bunch of people-orientated young female students going into this field for the kids and people interaction not being too thrilled with the topic.

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

#319

Post by chongli »

username90 wrote: post #317
chongli wrote: post #312

Earlier quoted context omitted.

I specifically set out to find a solution that doesn’t involve any brute force (trial and error). Trial and error can waste all of your time and get you nowhere on nontrivial problems.

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.

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

#320

Post by xiphias2 »

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.

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