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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

#161

Post by chrisseaton »

SamReidHughes wrote: post #153
chrisseaton wrote: post #138

Earlier quoted context omitted.

This is just re-stating the algorithm for the umpteenth time. I can read the algorithm. I just can't intuit why it works or why it has been designed this way. I mean your very first line has a logarithmic operation that was not mentioned anywhere in your text or your comment! Just pops in there out of nowhere. I guess it's about the number of decimal digits? But why? Why are we doing things in decimal? A dense ten-li…

> But why? Why are we doing things in decimal? We’re subtracting off multiples of 10^N because that’s easy when you write your numbers in base 10. Try subtracting off multiples of 9^N, or multiples of N!, or some other choice, and you’ll see why.

Ok, but do you see what I mean about how that part was just sitting there without any explanation? That's why I find it hard.

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

#162

Post by kasey_junk »

_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…

I was taught math via spatial properties and it wasn’t until I came back to it from a logical perspective that I was able to understand it.

Just another anecdotal data point but my experience was completely opposite yours.

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

#163

Post by matwood »

BrandoElFollito wrote: post #79

I have a hard time explaining to my children how addition of fractions work. Not how to do that, but how to feel the intuitive way. And percentages, to some extend. The sad part is that I have a PhD in physics and an engineering degree in CS. I used to teach physics at the uni and loved it. My children like me to explain them physics because I love it so much and they appreciate the analogies, their limits etc. It is…

> I have a hard time explaining to my children how addition of fractions work.

The best way I found to learn fractions was through carpentry. Measuring and cutting over and over quickly builds the intuitive sense of how fractions work.

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

#164

Post by LifeLiverTransp »

chrisseaton wrote: post #56
_bxg1 wrote: post #54

Earlier quoted context omitted.

Bud, I really hope you can grasp more complicated algorithms than that if you have a PhD in computer science. It's one thing to call it unintuitive and not worth your time, but let's not be hyperbolic.

No I have genuinely tried to learn it and I have no understanding why I am doing any of the steps in the algorithm.

So you know the how, but not the why. The why knowledge is genrally more valuable, because basic principles are reused over and over, while the how changes often. And its hard to recover from a how change, if you do not know the background whys applied. Thats why university focuses so much on the whys, and doesent give a fuck about the hows. Hows can be learned fast. Hows can even be deduced if you know the whys.

If you now a recipe to brew beer - you can create your own brew. If you know about microbiology of fermentation, you can deduce why the recipe sometimes doesent work...

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

#165

Post by rocqua »

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.

Standard deviation is a form of variance that has the same units as the mean. Whereas the variance has square units.

This means the quantity of mean/variance depends on your choice of units, whereas the quantity of mean/standard deviation. Is constant no matter the units.

Why do we care about variance? Because it is a nice and linear property. This means it is easy to calculate and manipulate.

Moreover, the variance / standard deviation very nicely describe a Gaussian distribution (bell curve). This is a very important distribution because of the law of large numbers. Because we see bell curves so often, it is nice to have tools (std.dev) that work well with these curves.

It should be noted that, in optimization problems, there can be reasons to try and minimize something other than variance. We tend to pick the variance / std.dev because it is familiar, easy to work with, and very efficient. Notably, the derivative of variance tends to be linear, which makes it pretty efficient to use in gradient descent.

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

#166

Post by kasey_junk »

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…

Conversely I have been immediately impressed with the way my first grader is being taught math.

It’s completely different than the way I was taught so I have to invest in figuring it out but it’s led them extremely quickly through multiplication & division and into algebra.

I’d characterize it as “multi-algorithmic”. They come at it from a variety of algorithms, assumedly so that each student can find one that works for them.

As someone who had to teach themselves a different algorithm than I was taught many painful years later this resonates.

Perhaps the rift was always there and you ended up in the lucky side of it?

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

#167

Post by garmaine »

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…

It is nothing to do with book publishers, but new federal mandates for "Common Core" curriculum as part of the No Child Left Behind legislation.

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

#168

Post by anewone »

chrisseaton wrote: post #143
marakv2 wrote: post #141

Earlier quoted context omitted.

What? 35/350 (to simplify) Once is 325, twice is 300, X, X, X, we end up at 10 times is 350 divisible by 35. I'm wondering if I'm missing something (and if I am I'll own up to it lol)

Lol I can't understand any of that. 'Once is', 'twice is' what does that mean? Once what is what? Twice what is what? You aren't even writing coherent English phrases how am I supposed to follow that?

They're referring to the number of times to subtract 35 from 350. Do it 10 times and you end up with 0, there is nothing more to divide and so 10 is the answer.

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

#169

Post by rocqua »

chrisseaton wrote: post #19
geomark wrote: post #8

Earlier quoted context omitted.

I don't understand why long division would be a mystery. It's just repeated subtraction - you are finding how many times you can subtract the divisor from the dividend, in a systematic way.

Because the algorithm is so astronomically complicated that people can’t see any pattern or reason in it anymore and it becomes an opaque mystery. I have a PhD in computer science but I cannot understand long division.

It sounds like you've memorized the very specific notational steps. But if you would have to come up with an algorithm for doing division using only integers with multiplication, subtraction and comparisons, you would come up with some equivalent to long division.

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

#170

Post by username90 »

chrisseaton wrote: post #159
username90 wrote: post #154

Earlier quoted context omitted.

Economics of stocks is a good example, when you put a lot of stocks in one portfolio you can calculate its risk (standard deviation) and returns (average value), and see how much each piece of stock contributed to that. This is because the results of many variables looks like a normal distribution with the combined average and standard deviation. Now it gets a bit more complicated in the real world since stocks are n…

> Now it gets a bit more complicated in the real world since stocks are not independent from each other This is also where I get frustrated. I'm taught about the CLT but every place I think I might use it, the samples aren't really IDD when you look hard enough, so it doesn't apply. Computer benchmarking (which is where I'm usually trying to apply statistics) is the classic example. One iteration changes the state of…

Right, that is how statistics usually is done. Note however that you don't need IDD, independence is enough. So you assume independence, from that CLT follows which means that standard deviation is all that matters. Then the real world deviates from this based on how much your distributions fails to be independent.

I'd assume that computer benchmarks have very low correlation between runs, unless you leak ram or your components don't keep a stable temperature. So in that case the CLT seems to apply just fine.

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