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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

#171

Post by olooney »

Perhaps base 10 is the problem. In base 2, you can learn the "multiplication table" in 10 seconds (0 X n=0, 1 X n=n, we're done.) The long multiplication algorithm for n and m is just multiply by n by the least significant bit of m, add to the accumulator, left shift, repeat. You can do subtraction by adding the two's complement (just flip every 1 to a zero and every zero to a one - trivial for a child to learn) instead of using the "borrowing" algorithm. You never have to "guess" when doing long division in base 2. All of these operations can be justified in terms of the Peano axioms with very short arguments: prove the associative and distributive properties, and the multiplication algorithm is obvious, etc.

Arithmetic in base 2 is so easy, you can literally teach your pet rock to do it (assuming your pet rock is a reasonably pure silicon wafer with scattered boron and phosphorous impurities shot through with thin veins of copper.)

And no, I'm not advocating teaching kids to convert between different bases, which they almost universally find confusing - I'm saying we should teach kids arithmetic purely in base 2, and only expose them to base 10 once they understand the fundamental concepts.

Am I being serious? Well, it's certainly a lot easier to dismiss this approach out of hand then to pinpoint what - if anything - is actually wrong with it.

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

#172

Post by chrisseaton »

username90 wrote: post #170
chrisseaton wrote: post #159

Earlier quoted context omitted.

> 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…

I can't understand this. Surely either samples are independent, or they are not?

And they are not in a computer program. The state of, to just give one example, the processor cache depends on what happened in the previous run.

We have all this rigour in how statistical techniques are applied, and we make black or white decisions based on the numbers that come out of them, but at their very foundation we're saying 'whatever I'll just pretend this rule applies it's probably fine'. How can we build on that flawed foundation? I don't get it!

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

#173

Post by hashmush »

jeltz wrote: post #136
mavhc wrote: post #81

Earlier quoted context omitted.

What none of them can tell you is why they've been taught long division, when short division is faster. And it's because you need long division to do division in algebra

Yeah, when we were taught short and long division I had the same question and it was not until quite late in high school we got the explanation. I went to school in Sweden where we do teach all this things, but in a strange order.

Went to school in Sweden too in the 90-00s. I wasn't taught long division until I had to divide polynomials in my last year of high school, and that course was an elective.

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

#174

Post by spodek »

Or: why do so few people who understand math teach elementary school?

Once someone learns math, what incentives do they have to choose teaching elementary school over alternatives?

I could imagine that the skills to learn math don't overlap with the skills to teach elementary school and that time spent one detracts from the other. If mastery in each takes a decade or so and few people dedicate themselves to double time in school for a lifetime in communities that misunderstand them, I could see teaching k-12 math as a career and life disaster.

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

#175

Post by barry-cotter »

_7fvc wrote: post #21

The author of this article does not understand math :). Learning math is difficult. There're two things you need to learn: fundamental concepts and skills. These two things are related and commingled. Mastering both requires extraordinary effort. I come from a skill-focused math education. I was pretty good with skills. But I didn't understand fundamental concepts. I still remember my struggle with prime numbers. Tho…

Being able to mechanically perform an algorithm is the first step on the road to in some sense actually understanding it. If you think you understand the fundamental concept but you can’t solve a problem using the algorithm you probably don’t.

> Young man, in mathematics you don't understand things. You just get used to them. - John von Neumann

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

#176

Post by koonsolo »

olooney wrote: post #171

Perhaps base 10 is the problem. In base 2, you can learn the "multiplication table" in 10 seconds (0 X n=0, 1 X n=n, we're done.) The long multiplication algorithm for n and m is just multiply by n by the least significant bit of m, add to the accumulator, left shift, repeat. You can do subtraction by adding the two's complement (just flip every 1 to a zero and every zero to a one - trivial for a child to learn) inst…

The real world depends on base 10, and so it's way more practical to learn calculations in base 10. I said calculations, not math.

I think there is a big confusion between math and calculations. Kids learn calculations, not math. Learning the multiplication table is just a way of how to do quick calculations inside your head.

I need to pay 35.15 euros, so how much do I get back? That is calculation not math. In my opinion the most practical thing that kids need to learn first. It's more about remembering than logic.

When the get older, math comes into the picture. They can leave calculations up to calculators or computers, and learn the language of math.

So in my opinion, there needs to be a more clear distinction between calculations and math, because they are definitely not the same.

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

#177

Post by dannypgh »

_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 still don't know why long-division works

I was taught long division in the 4th grade (age 9) but wasn't taught a proof of why it works until a college algebra course (age 19).

For whatever it's worth a different type of long division is taught now under common core. The common core math stuff that I've seen is impressive - there is more focus on teaching things in a way where comprehension of deeper concepts should be possible. It's a shame there's so much fear of it from parents who are upset kids are learning differently than they did.

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

#178

Post by billfruit »

geomark wrote: post #8
_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 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.

Well if long division is unintuitive, the procedure to compute square-roots by hand is even more mysterious, as if it has transcended dealing with numbers and instead become a symbolic manipulation. Thankfully it is something I never had to apply, even once in adult life till now.

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

#179

Post by barry-cotter »

matwood wrote: post #160
Gunax wrote: post #67

Earlier quoted context omitted.

There seems to be a culture of proud ignorance around math more-so than other subjects. While I am sure there is some engineering student out there griping about having to take a psychology course, they don't seem to complain as much as when non-mathy people have to study math. I think it is a mistake for our education system to treat math as something superfluous and abstract from daily life. While neither Laplace t…

> While I am sure there is some engineering student out there griping about having to take a psychology course, they don't seem to complain as much as when non-mathy people have to study math. This is the underpinning of every conversation where engineers say college is a waste of money. If anything, I see engineers complaining more loudly about taking any course that isn't directly related to engineering. This not o…

> Learning Shakespeare is typically tied to learning how to communicate through reading, writing, and comprehension. IMO, that is much more generally useful than learning anything about a Laplace transform.

Learning to do a poor quality imitation of early twentieth century literary criticism is at best a really inefficient way to teach communication through reading, writing and comprehension. Vocational pseudo reasons to teach the humanities are and always have been horse shit, detracting from the pleasures of those who are actually interested, whether as teachers or students.

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

#180

Post by dannypgh »

spodek wrote: post #174

Or: why do so few people who understand math teach elementary school? Once someone learns math, what incentives do they have to choose teaching elementary school over alternatives? I could imagine that the skills to learn math don't overlap with the skills to teach elementary school and that time spent one detracts from the other. If mastery in each takes a decade or so and few people dedicate themselves to double ti…

I would love to teach math to elementary students but I make a base-10 order of magnitude more as a software engineer for a FAANG. I'd take a pay cut of a factor of 4 or 5 to do it in a heartbeat... but having to go back to school to get a teacher's certificate (and likely a master's degree) to get paid 1/10th as much is challenging. I may end up doing it in a couple of years when I'm even more solidly financially independent, but we as a society should probably just be paying teachers a lot more across the board.

I guess there's more money in advertising than educating the next generation.

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