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aspiecat
05-23-2014, 01:13 PM
I was marking DS's Geometry today, including, in accordance with how things are done in Math U See, a systematic review of previously-taught material from Algebra I. This is done in every practice lesson to ensure kids don't forget what they have learned previously. Fair enough - I think it's a good idea.

One of the questions asked was the following:

-3^2 = ?

There were three other questions that were similar, but there were parentheses involved in different places, eg, -(12)^2 = ? These questions were obviously checking that the student still knows that parentheses or lack thereof can change how something with an exponent is worked out.

Now, I was taught as a kid that -3 squared is positive 9, but apparently the way kids are taught nowadays is that what they are doing is negating the 3 squared - so the answer is actually -9.

Therefore:

-3^2 = - (3)(3) = -9

I discussed this at length with my DH and FIL, both who agreed with me that -3 ^2 = 9, however, after a while, we decided that what is happening is kids are increasingly being taught and retaught as if they are dumber than hell. Sure, we could see HOW -3^2 is actually interpreted as the negating of -3^2, and the only way to get positive 9 out of this is to have (-3)^2.

(-3)^2 = (-3)(-3) = 9

I clearly remember the concept *I* was taught as -3^2 meant -3 x -3, which equals positive 9. But it was in context, when working something out, definitely NOT a standalone concept that had no relationship to anything else.

After some heated debate, we came to the conclusion that many PS teachers are being taught so simplistically, and they in turn teach kids simplistically. Hence, textbooks are often carrying on this simplistic, "let's assume our kids can't be taught things in context" manner.

While the rules of Order of Operations might well hold firm and true here, we all three of us were shocked that any concept is being taught out of context. It's as if society expects to turn out kids who can't think properly.

Aspie

Solong
05-23-2014, 02:31 PM
Painful. I was taught:

-3^2 = (-3) x (-3) = 9

-(3)^2 = -(3 x 3) = -9

Ran this by my brother and sister-in-law, both high school math teachers... they agreed, but said, "Teaching math isn't as straight-forward as one would assume, and many math teachers aren't actually trained in teaching mathematics." :confused:

MNDad
05-23-2014, 02:38 PM
I'm not sure I agree.

I think the difficulty is that language is an imperfect way of expressing concrete mathematical expressions, even ones as simple as this.

If I asked anyone: "What is -3 squared?" Then answer would be "9". Easy. But the written expression "-3^2", although seemingly the literal translation of that expression, equals -9, because of the PEMDAS order of operations.

You could illustrate the difference in code. Here's a quick Python script:


#!/usr/local/bin/python

y = -3 ** 2
print """The expression '-3 ** 2' equals %d""" % y

y = (-3) ** 2
print """The expression '(-3) ** 2' equals %d""" % y

x = -3;
y = x ** 2;
print """The expression 'x ** 2' equals %d when x = -3""" % y


whose output is:


The expression '-3 ** 2' equals -9
The expression '(-3) ** 2' equals 9
The expression 'x ** 2' equals 9 when x = -3


Here's a reference on order of operations as it relates to exponents of negative numbers: Exponents and order of operations (https://www.cos.edu/Faculty/rossr/Documents/2.6_Exponents-and-Order-of-Operations.pdf).

murphs_mom
05-23-2014, 02:41 PM
...and many math teachers aren't actually trained in teaching mathematics." :confused:

Yep. NCLB made it possible for most anyone deemed 'highly qualified' to teach pretty much any subject where they decide it's difficult to find degreed and licensed educators. (I taught art.) Math and science are two areas where they frequently use untrained educators around here.

For example, if the person was an engineer or computer science major and could provide transcripts to prove that they'd completed the required # of math/science hours, then they could teach that subject for up to 2 years with no one being the wiser. After 2 years, though, that person either has to get licensed or the school district has to notify every parent in writing that they're using a untrained, but highly qualified, person to teach a given subject. The whole system was boogered before the common core insanity came along.

Solong
05-23-2014, 04:33 PM
I'm not sure I agree.

I think the difficulty is that language is an imperfect way of expressing concrete mathematical expressions, even ones as simple as this.

If I asked anyone: "What is -3 squared?" Then answer would be "9". Easy. But the written expression "-3^2", although seemingly the literal translation of that expression, equals -9, because of the PEMDAS order of operations.

You could illustrate the difference in code. Here's a quick Python script:


#!/usr/local/bin/python

y = -3 ** 2
print """The expression '-3 ** 2' equals %d""" % y

y = (-3) ** 2
print """The expression '(-3) ** 2' equals %d""" % y

x = -3;
y = x ** 2;
print """The expression 'x ** 2' equals %d when x = -3""" % y


whose output is:


The expression '-3 ** 2' equals -9
The expression '(-3) ** 2' equals 9
The expression 'x ** 2' equals 9 when x = -3


Here's a reference on order of operations as it relates to exponents of negative numbers: Exponents and order of operations (https://www.cos.edu/Faculty/rossr/Documents/2.6_Exponents-and-Order-of-Operations.pdf).

Be patient with me. I don't do code. So...

-3^2 could be either -9 or 9? Or, it is absolutely -9? I thought the number preceding an exponent was to be used as the number expressed (in this case, -3) when parentheses were not included (imperfectly represented).

(-3)^2 is absolutely 9. So, when in doubt, use the parentheses?

dbmamaz
05-23-2014, 04:45 PM
I just asked my mathematician husband - he agreed. without parenthesis, -3^2 is -9 because of order of operations. I didnt remember it that way, but I trust him. I believe, though, that when we were first learning about negative numbers and exponents, they probably didnt do that? But i dont remember. odd.

aspiecat
05-23-2014, 04:57 PM
What it seems to be is this: -3^2 = -9 because you're "negating 3^2".

It's that bloody simple LOL.

I remember being taught that -3^2 = 9 because it is "negative-three squared, which equals nine", but apparently kids need to be taught that -3 to the power of any number is NOT "negative three to the power of a particular number" - rather, it's negating three to the power of that number. Thereby -3^2 is negating three to the power of 2 which, naturally, turns out an answer of -9.

Of course, it's not so much what -3^2 actually equals - it's about how such a written equation is translated into the working out, and the context of the working out. However, kids are NOT taught this and the mention of there being Math teachers who cannot actually teach only made DH and me shake our heads at the education system. Not that we didn't already know that, heck, we pulled him out of PS to homeschool him!

The whole thing has had me delightfully thinking about Math all day, which I actually enjoy and is only exacerbated by the fact I am married to a man with an IQ that is around 190 (sometimes 170, sometimes over 190...depends on the cognitive test) and who LIVES for Math.

Thanks for the input, everyone!

Solong
05-23-2014, 05:53 PM
Sheesh. Mr. Mellis, wherever you are, you were an imperfect math teacher. At least you graded based on what you taught us! Now absent parentheses are on my shit list, along with absent commas. I'm totally pulling this out at our next dinner party with uni staff.

farrarwilliams
05-23-2014, 06:33 PM
Okay, I just looked it up and it seems that it has maybe changed and that computer algorithms may, in fact, be the reason. (Not undereducated math teachers, though that's a problem for all kinds of reasons.)

Some things in math are absolute. Like, if you have a triangle with two equal length sides, it will also have two equal angles. There's no other way to do that. And no matter how you draw the triangle or what your symbols are, it'll be the same.

But order of operations is not absolute - it's based in our notations system. If mathematicians had decided that the way you do math was to read it right to left every time, then that would be the rule. It wouldn't change the absolutes of math, just the notation of how it works. And this falls in that category.

MNDad
05-23-2014, 07:00 PM
Be patient with me. I don't do code. So...

-3^2 could be either -9 or 9? Or, it is absolutely -9? I thought the number preceding an exponent was to be used as the number expressed (in this case, -3) when parentheses were not included (imperfectly represented).

(-3)^2 is absolutely 9. So, when in doubt, use the parentheses?

Probably not the absolute clearest example; but no the expression -3^2 (or -3 ** 2 in the example) is always -9 because there is an implied 0 in front of the -3, i.e. 0-3^2.

As farrar says, I think the increased emphasis on coding is driving a renewed interest in the order of operations. And agree, as far as I know, it's just a convention. (Also, not a universal one. The BASIC language has messed up order of operations.)

Solong
05-23-2014, 07:10 PM
LOL! What I have learned from this thread... is that I am old. I pre-date code-driven mathematics. (adjusts tin-foil hat with as much dignity as she can muster) Sigh.

dbmamaz
05-23-2014, 07:40 PM
my husband hasnt studied mathematics in almost 30 years, but he knew this right away. I think its more that when we were learning math, they just didnt bother with the parenthesis because its easier to just say -3^2 on the worksheets than to put in the damn parenthesis every time. so the worksheets were wrong out of laziness but the math hasnt actually changed. maybe we're teaching it more accurately now, but i dont think the rules have changed. it wouldnt be the first time schools used inaccurate textbooks . . .

farrarwilliams
05-23-2014, 07:56 PM
That's funny, but I can believe it... That the old books were the wrong ones.

aspiecat
05-23-2014, 08:51 PM
I tend to view newer textbooks as having more errors than older ones. Unless information has changed things, then that's not a matter of error - it's a matter of going with whatever information is available to mankind at the time. I remember being fascinated as a young teen by an old encyclopedia of my father's from when he was in the equivalent of middle school (that would have been late 1930s/early 1940s). It discussed and showed "artists' impressions" of the canals of Mars LOL.

I don't even see the discovery I made today as an error - it does, however, remind me how education is not as it used to be in schools, and too many teachers are far too uneducated themselves nowadays. The number of times my son corrected teachers for teaching the wrong thing...hoo boy...NOT a popular move on his part, but what else could he do? LOL

Aspie

farrarwilliams
05-23-2014, 09:38 PM
I think there have been bad, slapdash textbooks for at least forty or fifty years. I have read things from as far back as the 1970's about poor textbook assembly methods and poor text adoption methods in schools, so it's not a new problem by any means. When you combine that with the fact that things change... not usually math, but common notation can change, as well as expected skills (such as for things like programming)... Well, basically I don't think any textbook is more or less suspect just by its age.

aspiecat
05-24-2014, 09:03 AM
The thing is, mathematical formulae are simply codes for what we need to calculate. So if "the powers that be" decide this:

;) ^2 = 36

...then that is one way to get to 36. If -6^2 = 36, or -6^2 = -36, it's not a matter of which one is correct or incorrect; it's a matter of being taught which one is which, the reason why, and that is that.

A case in point: when I was very young, about seven, I remember learning that 1 billion could be either 1,000,000,000,000 or 1,000,000,000. I recall thinking, "Surely one is right and one is wrong" but of course it was a situation where we had to accept different countries had different interpretations of how to write 1 billion. NZ ended up with it being "1 million x 1 thousand" - in other words, 1,000,000,000. And that is how the majority of the world does now see it.

So if -3^2 does in fact equal -36, I'm fine with that. But given the fact that four adults in one household recall being taught -3^2 = 36 tells us a lot - in fact, it tells us kids are being dumbed down and not trusted to think things through. So what if a kid doesn't show their working out in a way that a teacher understand? So long as the working out is consistent each time and the answers are correct each time, that's what counts. We don't burst into theoretical physicists' offices and inform them the way they work out squaring negative numbers is incorrect, and our kids deserve to be considered the same way. Both my DH and my DS have a tendency to work things out in their heads in ways that are not taught, and get the correct answer every time. Both of them hate to show their working, and both have been given 0 in tests in schools for not showing working, or showing THEIR way of working out rather than how they've been taught.

Aspie

dbmamaz
05-24-2014, 10:18 AM
I really think you have this backwards, though, Aspie. What I think happened is that FOREVER kids have been taught incorrect math because its simpler to teach. If they ever go further, they would have to relearn some of it, because they were taught something which is wrong. And in college level, for example, they would not be wishy-washy about it, there would be a right and a wrong answer, and that would conform to the international mathematical standards.

aspiecat
05-24-2014, 10:46 AM
What I am saying is not there are correct or incorrect ways of having something symbolize something else - it depends entirely on the logic and context. I don't mind that -3^2 = -9 - it was simply a surprise that it is being taught because of the dumbing-down of kids in regular schools. Kids are not being encouraged to work out Math problems in a way that works for them - they are being taught to work out in accordance with what is the easiest path for the education system. The Math U See thing from yesterday was not to say Steve Demme is teaching incorrectly - it simply became apparent that kids are being not trusted to think for themselves and things are taught out of context in many ways.

I could care less what symbolizes what in Math, but if kids are taught "no, your way of working out is WRONG, even though you consistently get the right answer every time" - well, that is a MAJOR failing on the behalf of the education system. And it is, to be honest, one of the reasons why I prefer homeschooling to regular schooling: I get to let my son work things out how he wants, so long as what he does is consistent, efficient enough that it doesn't take all the time in the world, and he gets the answers consistently correct.

If it is easier for DS to write -3^ = 9 for his working out, then that is fine with me. He *does*, I should add, do it the way it seems to be taught now, ie, the answer being -9. So he is in line with what is apparently "correct working out".

My DH and I want DS to know that so long as his working out for anything follows a consistent pattern, and he gets the answers correct each time, then that is how he should do things. At college, my DH was never once told his working out, or his lack of *showing* his working out made him wrong about something - so long as the answer was correct, that was important. In high school, DH was punished for not showing working, or for having his own way of working things out - and that, as it happens, included him writing things like -3^2 = 9, NOT -9.

Insofar as the actual concept of increasing negative numbers by using exponents, the only way that "negative three squared" should equal "negative nine" should be by having parentheses around the 3^2, with the negation on the left of the opening parenthesis. to make it clear from the outset that given ANY context, that number is to be negated AFTER the exponent has affected the number. The rule I am talking about can be found in many places to mean "squaring the number, then negating it" and in as many places to mean "squaring the negative number". Therefore there is clearly an issue with this rule.

Math is not a perfect science. And people are only people (bring on cybernetic enhancements LOL), and people who are involved in education will dictate certain things in Math when they perhaps ought not to.

Solong
05-24-2014, 07:08 PM
It does seem unfair that mathematically-inclined persons are required to communicate each step, where others mightn't be. I never saw a fellow student required to give a detailed explanation on the creative process behind an inspired story... or an amazing drawing... they just DO it. Some people just DO math. Intuitively.

Dd9 hated math in school. Loathed and feared it. It's taken more than two years, but now she has math confidence. She'll ask for mental math problems to MANAGE ANXIETY over more complicated things... like, you know, the rest of life. The teachers at ps obviously hated teaching math, they openly criticized the texts they were forced to use (Jump), and spent the absolute minimum amount of time required on their least favourite subject. I think, in general, kids hate math because teachers hate math.

I can't blame them, though. I spend more time looking for fun/relevant/inspiring math ideas than I spend on all other topics combined. Much more time. Math just isn't that fun or intuitive for most of us. My experience as a hs mum leads me to assume that math is also the hardest subject to teach in ps. I would cling to the teacher's guide like a drowning person to a life preserver too, if I was staring down 30 students each day.

Dd took the provincial G4 exams this year and missed a word problem (number of visitors to the zoo each day, etc), because she drew both a bar graph and a pie chart. They asked for a pictograph, which was covered in G1. She's forgotten most of G1 by now, lol. Big zero - the grader was so, so embarrassed. She wanted to give full credit, since dd communicated the information clearly and correctly. She was required to give a zero, because the test writers feel that following instructions is more important than the math itself. Honestly, I felt worse for the grader than I did for dd.

Anecdotal report - every person I've asked so far has answered "9". Fun :)

aspiecat
05-26-2014, 10:06 AM
The thing about Math is that it is the one and only subject that, without fail, will make you a "top student" if you are good at it. If you excel at Math, you are a brilliant student. If you do well in Math standardised testing, your future looks bright for graduating well, getting a good college degree and landing a fabulous and highly-paid job. You can be mediocre at English, Social Studies and Music, but who care? You're a Math wiz, and that is that.

Conversely, if you suck at Math, but are a natural at English, at Geography, at Art, at Music...you are apparently not academically-inclined and seen as a "meh" student. You can even be a natural as Science, or one field of it, but if your Math is shaky, well, you're not going to get that medical degree you so dearly wish for.

Math is the be-all and end-all of academic competence and success, according to the brain-washing we've undergone for way too many years. Sure, you can't do higher study of science without strong math behind it, because so much of all science, even biology, counts on a certain amount of competence in high school math. Yet we are fooled into thinking that the only way to approach Math as a subject is to do it "this way" or "that way", and to take a different tack, to think outside the box in your calculations, is WRONG. You most certainly cannot NOT show your working and, if you do show it and it's not how you've been taught, well, you are again not doing well in Math and you're labelled an academic failure.

A case in point is the story behind the wonderful film "October Sky", about students in a regional mining town in the 1950s. They decide they want to try rocketry after seeing the televised account of Sputnik being launched, but are met with mocking, derision and a hell of lot of doubt. Part of it is the fact they are nearly young men and they ought to be thinking about their future working in the mines, but another part is that only one of them is seen as having any sort of academic future - because he is very good at - wait for it! - Math. All boys are interested in science, particularly rocket science (at high school, that would be higher physics), but because three of them are not natural mathematicians, they are not seen as having a chance to do anything in the field of science, or anything academic AT ALL.

I won't give too many spoilers here, in case anyone here wants to see this movie, but I will say that with help in Math, the kids begin to excel in their study of rocket science. So clearly, they need the Math to help them in what they wish to achieve. However, this film also gives us an idea that even as far back as the 1950s, before "teaching to the test" became the thing to do in schools, it was Math that denoted whether a student would be a success or not.

Fairielover
05-26-2014, 01:23 PM
I love October Sky. Being a mom of a young Rocketeer we watch it often.

CatInTheSun
05-26-2014, 07:51 PM
Math hasn't changed. The orders of operations hasn't changed. But the assumption about what the question is asking has.

Put another way, if asked:

What is x^2 when x=3? when x=-3? The answer has been the same all along. Same is true for -x^2.

Asking what -3^2 means is a meaningless calculation without reference. Order of operation 30 years ago would have said exponent first, so the answer should have been -9, but (wink, wink) if the question was being asked on a worksheet teaching the concept of squares, it was *implied* that the number being squared was (-3) even without parenthesis. That was good enough in the 60s and 70s, but now, with ubiquitous computers you must follow the rules AT ALL TIMES.

So yes, I would say the books from 40 years ago were wrong if they said -3^2 was 9. I'm a scientist and it is obvious to me that it is -9. But I cannot be caviler about order of operations and I do program. If I wanted to square -3 I would certainly use parentheses! :D

aspiecat
05-26-2014, 08:23 PM
Math hasn't changed. The orders of operations hasn't changed. But the assumption about what the question is asking has.

Put another way, if asked:

What is x^2 when x=3? when x=-3? The answer has been the same all along. Same is true for -x^2.

Asking what -3^2 means is a meaningless calculation without reference. Order of operation 30 years ago would have said exponent first, so the answer should have been -9, but (wink, wink) if the question was being asked on a worksheet teaching the concept of squares, it was *implied* that the number being squared was (-3) even without parenthesis. That was good enough in the 60s and 70s, but now, with ubiquitous computers you must follow the rules AT ALL TIMES.

So yes, I would say the books from 40 years ago were wrong if they said -3^2 was 9. I'm a scientist and it is obvious to me that it is -9. But I cannot be caviler about order of operations and I do program. If I wanted to square -3 I would certainly use parentheses! :D

That's exactly what I have been saying, Cat (another Cat yay!) - one has to look at context when doing any mathematical calculations, not only the equation itself.

I naturally head towards 9 when calculating "negative three squared". But that's the whole thing - I might be MEANING the negation of 3^2, or -3 times -3 when I say that - but which one? You would have to ask for clarification.

At least the issue is not whether 10 is a solitary number, or if there is an infinite number of twin primes! Gosh I love those problems...they cause me fewer sleepless nights.

farrarwilliams
05-26-2014, 10:56 PM
It does seem unfair that mathematically-inclined persons are required to communicate each step, where others mightn't be. I never saw a fellow student required to give a detailed explanation on the creative process behind an inspired story... or an amazing drawing... they just DO it. Some people just DO math. Intuitively.


The thing about Math is that it is the one and only subject that, without fail, will make you a "top student" if you are good at it. If you excel at Math, you are a brilliant student. If you do well in Math standardised testing, your future looks bright for graduating well, getting a good college degree and landing a fabulous and highly-paid job. You can be mediocre at English, Social Studies and Music, but who care? You're a Math wiz, and that is that.

Conversely, if you suck at Math, but are a natural at English, at Geography, at Art, at Music...you are apparently not academically-inclined and seen as a "meh" student. You can even be a natural as Science, or one field of it, but if your Math is shaky, well, you're not going to get that medical degree you so dearly wish for.

I disagree with these assumptions.

I think the idea that being good at math make you a "good" student in the eyes of school and society is not always true. Sometimes it can make you an "egghead" or make you derided. Sometimes people see it as a sort of Rainman thing if you're really into math. And on the flip side, sometimes it's philosophy and Latin and so forth that people find the most intimidating as "academic" subjects and they see math as a simplistic, utilitarian thing - not a true challenge for the mind. I just don't see this as a straight black and white thing.

As for explanations... If you think about the levels of understanding, being able to do the math is a pretty surface level of understanding. Being able to explain it is a much higher standard, and one people should strive to meet. It builds a stronger foundation. I know that a lot of schools have taken this WAY too far - having kids writing these long paragraphs about math problems in elementary school is absurd. However, explanations matter. I'm kinda sick of seeing people say that they have no place in math. The meaning of math and being able to talk about math has no place in math? That defies common sense to me.

As for explaining writing and art... Honestly, if you're good at it, you should be able to talk about it on an academic level. I know someone who has an mfa in poetry. She's a poet. She can pick a poem apart like no one you've ever met. Because that's a deeper level of understanding. When I go to my writing group and we all talk, we talk about writing and process and meaning - because that stuff matters. It's not just a magic thing to write something - it's a process. And the same thing with art. If you want to be an artist, you should be able to talk about color theory and materials and composition and so forth. And if you study to become an artist that's exactly what you study.

The world needs math... and art and writing and so forth. But I guess I think that we all need to have expression skills or the meaning is lost.

Solong
05-27-2014, 12:46 AM
I disagree with these assumptions.

I think the idea that being good at math make you a "good" student in the eyes of school and society is not always true. Sometimes it can make you an "egghead" or make you derided. Sometimes people see it as a sort of Rainman thing if you're really into math. And on the flip side, sometimes it's philosophy and Latin and so forth that people find the most intimidating as "academic" subjects and they see math as a simplistic, utilitarian thing - not a true challenge for the mind. I just don't see this as a straight black and white thing.

As for explanations... If you think about the levels of understanding, being able to do the math is a pretty surface level of understanding. Being able to explain it is a much higher standard, and one people should strive to meet. It builds a stronger foundation. I know that a lot of schools have taken this WAY too far - having kids writing these long paragraphs about math problems in elementary school is absurd. However, explanations matter. I'm kinda sick of seeing people say that they have no place in math. The meaning of math and being able to talk about math has no place in math? That defies common sense to me.

As for explaining writing and art... Honestly, if you're good at it, you should be able to talk about it on an academic level. I know someone who has an mfa in poetry. She's a poet. She can pick a poem apart like no one you've ever met. Because that's a deeper level of understanding. When I go to my writing group and we all talk, we talk about writing and process and meaning - because that stuff matters. It's not just a magic thing to write something - it's a process. And the same thing with art. If you want to be an artist, you should be able to talk about color theory and materials and composition and so forth. And if you study to become an artist that's exactly what you study.

The world needs math... and art and writing and so forth. But I guess I think that we all need to have expression skills or the meaning is lost.

I think it is ok to acknowledge and legitimize a child's innate talent, without requiring them to express their inspiration or process. Totally ok. Most children's communication skills lag behind their natural talents... and most can catch up later. In primary, showing your math work is really about showing that you didn't cheat or just guess right. It would be nice to think it's about developing expression skills. I'm not sure that's true, though.

As adults, I absolutely agree that communication/collaboration skills are far more important than talent. Hands down. All the biologists here answered "9" (ahem, even the young ones), but our "stats guy" at the university answered, "well, it depends...". Yes, he's our guy because he can do what we can't/don't want to do. Mostly, HE's our guy because he's easy to work with and get along with.

As far as math being valued more than other academic areas? I can see that, and I think it's just economics. There are fewer people honing their mathematical skills, so it becomes a demand vs. availability issue. Also, there is very little variability when measuring mathematical skill. If you are good at math, you are good at math. If you are good at poetry, you might just suck at poetry... depending on the audience.

CatInTheSun
05-27-2014, 02:09 AM
As to being expected to explain your thinking and show your work: I think the way the idea of this is written up in the CCSSI as a goal makes sense -- the idea of promoting conceptual over procedural understanding -- but the implementation of the EVALUATION of whether this standard is MET by curriculum publishers and test makers...sucks.

YES, ultimately you should be able to explain and show your work...BUT comparing an 8yo learning basic math to an artist with a masters degree isn't really fair, is it? I would expect an adult with a MFA to be conversant on art, while an 8yo may only be able to tell me they painted the sky green because they thought it was pretty.

I expect my kids to show their work *sometimes*, and I stress the idea that in the workplace (engineering) no one will pay you for solving a problem -- you have to be able to give a solution written out in a way someone else can verify it. Similarly, in the medical field, if it isn't in the chart it didn't happen -- you have to document. That said, IMO this is an area that K-4 at LEAST effective evaluation needs to be done one-on-one to see if a child understands the concepts. Asking a child to your LA skills to explain their math thinking is like asking a child to explain their french vocab in spanish. FUBAR They need to be fluent first.

As to the value of mathematics education. This is always a sore spot, but as far as I know there are a lot fewer unemployed STEM grads than FA majors and they tend to find work in their fields. If we stop equating what someone will pay us with what we are worth as human beings, the point is that math will be more highly valued in a technology driven capitalist society. It's not just rarity (though that helps) but that it is a skill that adds wealth to corporations. Even theoretical math profs have research contracts with tech companies to improve computational algorithms and the like! Those companies use that tech to compete and make money. That's a different world than the poet, whose work is valued in a different way.

Does that mean we should all be mathematicians? Of course not. :) But it does mean life can be harder for those pursuing a FA degree than those pursing an ENGR degree. To have a career with a FA degree takes a lot of commitment, so if your kid has that great. I would not be enthusiastic about a child saying "Hey, maybe I'll go to school and be a poet" nor one saying "Hey, I thought, you know, why not just go and be a doctor" -- both paths require too much passion and hard work and sacrifice to succeed with a cavalier attitude!

It's late. I'm rambling. :p FWIW: don't mean to pick on fine arts. Just using it as the extreme of the liberal arts.

aspiecat
05-27-2014, 09:01 AM
Farrar - I should have pointed out the "if you're good at Math, you're seen as smart" bit is something that has been researched and studied over the years. It's not my personal opinion - far from it! - but I *have* seen it in the schools DS has attended and in the attitudes of people who have one child doing well in Math and another child doing well in one of the Arts. It's amazing to see how, albeit subconsciously, many parents can be disappointed with the academic progress of a child who is getting high grades in English, or the Humanities, or Music, or Art...and how pleased they are of a child doing well in Math and perhaps Math alone. Admittedly, kids who do well in Math *usually* do well in other subjects anyway, statistically-speaking, but I have known many kids who do well in Math and suck at the Arts. The parents of the kids in this situation are oftentimes okay with how their children are doing (I wouldn't be myself), whereas they see their child's report card filled with Cs and Ds in Math and As and Bs in other subjects, and PANIC MODE IS SET.

I love Math. I am not good at the calculations of the higher (ie PG college) stuff, and have to be reminded of them by DH, but my brain cracks onto mathematical concepts very well. People tell me often enough I should be a Math teacher, not an English language one. DH is a brilliant natural mathematician, but he is a writer, and people who graduated with him are so surprised that he isn't involved in Math professionally somehow. He and his family still remember the calls throughout his early college time from the military, trying to get him to become an army or navy surgeon, simply based on his Math scores from high school - not even his Science scores (which were off the scale as well).

I personally believe that focus should be on a child's individual talents, no matter what they are, and the very real view many people - including children - have regarding Math only serves to make education a bit of a joke in the regular institutionalised environment. Thank goodness for homeschooling!

Jen V
05-27-2014, 11:40 AM
omg.. no speaka de maths! What am I doing homeschooling? LOL

farrarwilliams
05-27-2014, 11:47 AM
I'm not saying it's untrue exactly, just that that was never my experience in education. I was always so-so at math, great at writing and I always got a lot of "you're smart" props. And I've seen that with others too.

farrarwilliams
05-27-2014, 11:48 AM
As to being expected to explain your thinking and show your work: I think the way the idea of this is written up in the CCSSI as a goal makes sense -- the idea of promoting conceptual over procedural understanding -- but the implementation of the EVALUATION of whether this standard is MET by curriculum publishers and test makers...sucks.

Exactly. It's fine if the ability to express understanding lags. But it should get there. Our methods of enforcing and evaluating it seem to really suck though.

freerangedad
05-27-2014, 12:56 PM
As to being expected to explain your thinking and show your work: I think the way the idea of this is written up in the CCSSI as a goal makes sense -- the idea of promoting conceptual over procedural understanding -- but the implementation of the EVALUATION of whether this standard is MET by curriculum publishers and test makers...sucks.


I think the fact that it is presented in this way might be why so many parents object to it. Language arts are creeping into our math lessons when we ask our kids to explain their thought precess in english. I think that is OK, but let's call a spade a spade. Let's remember, math IS a language. Some right brain kids must be baffled when asked to explain their work. "Don't you have eyes?" they must think, "My thought process is right there on the paper".

We are fortunate that there are scientists who are able to translate quantum physics into English, but physicists tell us that something is lost in the translation. The only way to truly understand it is to speak the language. When we ask our children explain their thought process into English, we are asking them to translate a mathematical conceptualization into into English. It's a very good skill, but let's remember that is what we are teaching. There are clearly parents who are being frustrated by their children being turned off by math because of the attempt to get them to express themselves in English.

I agree that kids should be able to show their work. That is, they should be able to show their work in mathematical terms while having the freedom to derive the solution in what ever way works for them.

What a great thread. I'm glad it went on long enough for me to understand why the answer is -9. I was clinging to 9 for the longest time. :) I found the explanation that the ubiquity of computers helped change our assumptions when solving the problem to be fascinating.

ikslo
05-27-2014, 12:58 PM
What a great thread. I'm glad it went on long enough for me to understand why the answer is -9. I was clinging to 9 for the longest time. :) I found the explanation that the ubiquity of computers helped change our assumptions when solving the problem to be fascinating.

What he said. :)

hsjar00
05-27-2014, 01:33 PM
I don't have anything new to add to the discussion, but I'm just delighted that there has been a four-page discussion on a math problem! I love these message boards!

farrarwilliams
05-27-2014, 01:49 PM
But I think it's fine for students to be able to explain their process in math as well. I don't think it has to be English. But I don't think at this level that anything is lost if it is. Elementary math and basic algebra/geometry are not higher maths. Really, this is just a new level of "show your work."

freerangedad
05-27-2014, 02:26 PM
But I think it's fine for students to be able to explain their process in math as well. I don't think it has to be English. But I don't think at this level that anything is lost if it is. Elementary math and basic algebra/geometry are not higher maths. Really, this is just a new level of "show your work."

I don't think anything is lost as well. I was just pointing out that the a child's inability to explain their work in English does not mean he/she lacks a conceptual understanding. A conceptual understanding is expressed perfectly well in mathematical terms, so let's let our nonverbal students do so. That doesn't mean stop trying to teach kids to express themselves in English. But let's not frustrate the nonverbal learners; something that is clearly happening more often than it should.

ikslo
05-27-2014, 02:45 PM
My DS(7) had to do one of those "show your work" problems the other day. He looked at me annoyed, and then wrote, "I looked at the picsher." Then he turned to me and said, "Duh. They did it for me." I silently wrote "picture" on the whiteboard, he corrected his spelling, and we moved on. I love my kid. :)

reefgazer1963
05-27-2014, 07:24 PM
This is how I rad it and how I taught my DD.

aspiecat
06-01-2014, 07:06 PM
I am both surprised and delighted that my "conundrum" has brought on such a conversation here in SHS! I usually open my gob and people look at me as if to say "WTF?!?" haha.

I see how -9^2 equates to BOTH 9 and -9. The latter was taught to me at school and even using in context, always ensured I came to a correct answer. However, the explanation behind how the former is "really the answer" is just as viable to me.

You see, it's all about interpretation. More than that, it's all about how we are taught to interpret such a question. If we are taught that 2+2=5, and given the rationale behind why that is so, then it is so. My DH will discuss at length why that equation actually works LOL and I just sit there laughing at him trying to convince me how "two plus two" equals both 4 and 5 at the same time, even when question appears to be the same for each. Of course, they are not, that is easy to fathom, but still...I will continue to sit and laugh and shake my head and say 4 is the easiest answer for me.

Aspie