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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

#331

Post by vworldv »

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…

Due to a parity issue (50 here is even) we need to adjust username90's solution. In particular, a sequence of consecutive integers of even length has a half-integer in its middle.

In this case, let's consider the sequence of 50 consecutive integers 16, 17, ..., 40, 41, ..., 65 where 40.5 is in the middle and the sum is 40.5 * 50 = 2025. Subtracting 2009 we get a supposed erased value of 2025 - 2009 = 16 and this indeed correspond to a valid solution:

16, 17, ..., 65 => 17, ..., 65

We also could've considered the sequence 17, ..., 41, 42, ..., 65, 66 centered at 41.5 and this would've led to an erased value of 41.5 * 50 - 2009 = 66 which is another solution:

17, ..., 65, 66 => 17, ..., 65

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

#332

Post by zamfi »

rocqua wrote: post #278
zamfi wrote: post #269

Earlier quoted context omitted.

> The benefits of education go to more than just the (parents of) the recipients of that education. I think this is exactly the intent — get the whole county / district to pay, not just the parents, right? And these payments are usually compulsory regardless of whether you have children or send them to private school...

It should be spread to the state / country level rather than being based on districts is my point.

At least in California, it is indeed apportioned at the state level.

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

#333

Post by chongli »

vworldv wrote: post #331
chongli wrote: post #324

Earlier quoted context omitted.

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…

Due to a parity issue (50 here is even) we need to adjust username90's solution. In particular, a sequence of consecutive integers of even length has a half-integer in its middle. In this case, let's consider the sequence of 50 consecutive integers 16, 17, ..., 40, 41, ..., 65 where 40.5 is in the middle and the sum is 40.5 * 50 = 2025. Subtracting 2009 we get a supposed erased value of 2025 - 2009 = 16 and this inde…

Right, but we are talking about sixth-graders here. If a sixth-grader saw username90's solution and tried to apply it to the adjusted problem, they would not know that the result is incorrect without actually summing up all the numbers to check, an undesirable chore on a math contest with other problems to worry about.

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

#334

Post by d0mine »

chongli wrote: post #300
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…

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…

Given 11 numbers: n, n+1,...,n+10 the sum of 10 of them:

    10*n+sum_conseq=2012
where max(sum_conseq)=0+1+...+10=(0+10)+(1+9)+(2+8)..=55

therefore erased number is n+3 because (2012-55-3) divisible by 10 -> n=196.

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

#335

Post by Falkon1313 »

When I was in school, the problem was the way it was taught. There was no real explanation of anything or questioning that would lead to learning the fundamentals until the higher grades, when kids were already expected to understand things that they'd never been taught.

They could have taught us about arithmetic, but instead it was "Just stare at this table until your attention wanders off, then daydream until the bell rings and hope by chance some bit of it sticks in your memory."

No phonetics or linguistics to explain how things were spelled, instead just "Write this word over and over until your hand cramps, then do the same with the next one." As if by getting enough hand cramps you would magically learn how language works.

Just recite "In 1492 Columbus sailed the ocean blue." Never any question into why the Portuguese, with their lead on exploration and colonization, did not end up with a bigger overseas empire than the British. As if that should be obvious as long as you can recite that rhyme and get the date right.

If the teachers are around my age, most of them were probably taught the same way, which explains why they wouldn't understand math.

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

#336

Post by d0mine »

chongli wrote: post #315
mepiethree wrote: post #314

Earlier quoted context omitted.

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.

Simple examples may be useful. Exploring them, proving stuff about them it is how one may carve a foothold to solve a harder problem.

Example: The Josephus Problem - Numberphile https://youtube.com/watch?v=uCsD3ZGzMgE&t=86

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

#337

Post by scscsc »

adriand wrote: post #184
scscsc wrote: post #181

The author unfortunately has a misunderstanding of equality. Quoting him, "The equal sign is another mathematical concept that’s often misunderstood. It means, of course, that whatever is on either side of the equal sign is equivalent." This is actually wrong. The equals sign (=) is a shorthand for stating not that the two sides are equivalent, but that they are the same (i.e., they are equal). If they were just equi…

(2 + 3) and 5 are not “the same”. If we’re using ordinary English language to discuss this issue I think “equivalent” is a sensible choice.

As expressions, sure, they are not the same. But "2 + 3 = 5" is a statement about numbers, not about expressions. When you talk about numbers (the object of study), you do not (usually) talk about arithmetic expressions. Arithmetic expressions are part of the language used to talk about numbers.

When talking about arithmetic expressions themselves, you would use a meta-language (e.g., English). The arithmetic expressions would then be called the object-language (the language being studied). All of this should be pretty clear after taking a serious course in Logic.

I am not saying the choice of words is not sensible, I'm saying it it factually wrong.

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

#338

Post by anongraddebt »

_bxg1 wrote: post #11
pjscott wrote: post #7

Earlier quoted context omitted.

My parents invented a simplified division algorithm so they could teach me the four basic arithmetic operations by third grade, and it was so much easier than long division that I could never remember how to do things the conventional way. I just pretended to follow the teachers' instructions for the next few years. (All these methods had the same asymptotic complexity. Mine was slower by a small constant factor; on…

I repeatedly got in trouble for not showing my work in the format they wanted, whether because of intuitive leaps or because of tricks I came up with myself.

This worked for me up until Algebra 2. Up through Algebra 1 I'd just do everything in my head, never show my work, and thus solved a lot of problems using non-standard/creative methods on the fly. Ended up hurting myself a lot. I didn't have any muscle memory for solving the trivial parts of problems as I progressed from Algebra 2 through BC Calc. This led to me choosing a non-mathy degree in undergrad.

I could breeze through higher-level logic courses, but was essentially remedial for higher-level math.

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

#339

Post by nwah1 »

abakker wrote: post #90
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

What is short division? I only remember being taught long division and having a calculator. Is there another method?

Probably is that New Math that Tom Lehrer warned us about.

https://www.youtube.com/watch?v=UIKGV2cTgqA

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

#340

Post by notacoward »

_bxg1 wrote: post #54
chrisseaton wrote: post #19

Earlier quoted context omitted.

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.

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.

Some people are just pathological hyperbolists. I think that's really the handicap here, not any facility for understanding algorithms.
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