"one key difference between kids who excel at math and those who don't"
November 12, 2013 7:52 AM Subscribe
Ever since my school days, I've known plenty of people who just shrugged and said, "Oh, I just can't read/spell/write/sing/etc." But because society doesn't value those things as much as math, the shrug is accepted and nobody writes articles about it.
I think the difference is less about what society values and more about math being much easier to measure. If you say "I can't do algebra," it's pretty simple for someone to show you, "Here's what algebra is, now let's do some, look! You can do algebra." But if you say "I can't write," it's more difficult for a teacher to say, "Okay, let's compose a short story that grips the reader and maintains a coherent voice."
posted by Etrigan at 8:12 AM on November 12, 2013 [2 favorites]
I think the difference is less about what society values and more about math being much easier to measure. If you say "I can't do algebra," it's pretty simple for someone to show you, "Here's what algebra is, now let's do some, look! You can do algebra." But if you say "I can't write," it's more difficult for a teacher to say, "Okay, let's compose a short story that grips the reader and maintains a coherent voice."
posted by Etrigan at 8:12 AM on November 12, 2013 [2 favorites]
I've never seen Dweck's worth referred to as specific to math. In the 2007 Scientific American article about her work, "Raising Smart Kids", they referred to IQ tests:
posted by daveliepmann at 8:18 AM on November 12, 2013 [3 favorites]
In studies involving several hundred fifth graders published in 1998, for example, Columbia psychologist Claudia M. Mueller and I gave children questions from a nonverbal IQ test. After the first 10 problems, on which most children did fairly well, we praised them. We praised some of them for their intelligence: “Wow … that’s a really good score. You must be smart at this.” We commended others for their effort: “Wow … that’s a really good score. You must have worked really hard.”(Also, there's a typo in the article: it's Josh Waitzkin, not Weitzkin. Sorry, it jumped out at me.)
We found that intelligence praise encouraged a fixed mind-set more often than did pats on the back for effort. Those congratulated for their intelligence, for example, shied away from a challenging assignment—they wanted an easy one instead—far more often than the kids applauded for their effort. (Most of those lauded for their hard work wanted the difficult problem set from which they would learn.) When we gave everyone hard problems anyway, those praised for being smart became discouraged, doubting their ability. And their scores, even on an easier problem set we gave them afterward, declined as compared with their previous results on equivalent problems. In contrast, students praised for their effort did not lose confidence when faced with the harder questions, and their performance improved markedly on the easier problems that followed.
posted by daveliepmann at 8:18 AM on November 12, 2013 [3 favorites]
But if you say "I can't write," it's more difficult for a teacher to say, "Okay, let's compose a short story that grips the reader and maintains a coherent voice."
Maybe, but the teacher could start with "Let's compose a short story that follows a narrative structure, made up of sentences where the tenses and pronouns all agree."
posted by The Underpants Monster at 8:19 AM on November 12, 2013 [8 favorites]
Maybe, but the teacher could start with "Let's compose a short story that follows a narrative structure, made up of sentences where the tenses and pronouns all agree."
posted by The Underpants Monster at 8:19 AM on November 12, 2013 [8 favorites]
I lived in Japan, taught for many years in Japan, ran, together with my wife and some talented people, a "cram school" that focused on math skills, in Japan.
A few years ago, when our son was in the fourth grade here in Canada, he came home with a "performing under expectations" for math on his report card.
When we asked the teacher about it, she said "Oh, well, kids develop at different speeds. Ideally he will do better in a few months."
This attitude infuriated my wife, who sat down and proceeded to teach our son math. By the next report card he was "exceeding expectations" and we never settle for anything less than 100% on a math test. Why? Because one mistake means you don't completely understand the concept.
I think the biggest challenge with learning math in North America is that elementary school teachers may not always be good at teaching it (it is very difficult to teach), and low expectations of students.
posted by KokuRyu at 8:22 AM on November 12, 2013 [11 favorites]
A few years ago, when our son was in the fourth grade here in Canada, he came home with a "performing under expectations" for math on his report card.
When we asked the teacher about it, she said "Oh, well, kids develop at different speeds. Ideally he will do better in a few months."
This attitude infuriated my wife, who sat down and proceeded to teach our son math. By the next report card he was "exceeding expectations" and we never settle for anything less than 100% on a math test. Why? Because one mistake means you don't completely understand the concept.
I think the biggest challenge with learning math in North America is that elementary school teachers may not always be good at teaching it (it is very difficult to teach), and low expectations of students.
posted by KokuRyu at 8:22 AM on November 12, 2013 [11 favorites]
Ever since my school days, I've known plenty of people who just shrugged and said, "Oh, I just can't read/spell/write/sing/etc." But because society doesn't value those things as much as math, the shrug is accepted and nobody writes articles about it.
Really? My experience has been the opposite: that various really smart, well-educated people have no problems announcing "Oh, I'm just hopeless at math" when they would be (justly) ashamed of saying "Oh, I just can't do books, ugh." I've met "non-math" people who even seemed a little proud of it-- as though ineptitude with numbers signals that your intelligence is just too creative or intuitive or nuanced for all that jazz.
posted by Bardolph at 8:26 AM on November 12, 2013 [20 favorites]
Really? My experience has been the opposite: that various really smart, well-educated people have no problems announcing "Oh, I'm just hopeless at math" when they would be (justly) ashamed of saying "Oh, I just can't do books, ugh." I've met "non-math" people who even seemed a little proud of it-- as though ineptitude with numbers signals that your intelligence is just too creative or intuitive or nuanced for all that jazz.
posted by Bardolph at 8:26 AM on November 12, 2013 [20 favorites]
we never settle for anything less than 100% on a math test. Why? Because one mistake means you don't completely understand the concept.
Or, you know, you transposed a sign and didn't notice before the clock ran out.
posted by vogon_poet at 8:33 AM on November 12, 2013 [34 favorites]
Or, you know, you transposed a sign and didn't notice before the clock ran out.
posted by vogon_poet at 8:33 AM on November 12, 2013 [34 favorites]
Because one mistake means you don't completely understand the concept.
I think I understand what you're getting at, but in my case, my problems in math boil down to 100% simple arithmetic errors. When I was studying for the GRE, I took inventory of every math question I got wrong, categorizing my errors as conceptual or mechanical. It was all dumb, dumb, dumb arithmetic.
At a certain point, my arithmetic errors were so absurd that I chose to do every single calculation using a calculator -- even if it was 7+1 or 3x2. My practice test scores improved dramatically, because my conceptual grasp was fine.
posted by overeducated_alligator at 8:38 AM on November 12, 2013 [8 favorites]
I think I understand what you're getting at, but in my case, my problems in math boil down to 100% simple arithmetic errors. When I was studying for the GRE, I took inventory of every math question I got wrong, categorizing my errors as conceptual or mechanical. It was all dumb, dumb, dumb arithmetic.
At a certain point, my arithmetic errors were so absurd that I chose to do every single calculation using a calculator -- even if it was 7+1 or 3x2. My practice test scores improved dramatically, because my conceptual grasp was fine.
posted by overeducated_alligator at 8:38 AM on November 12, 2013 [8 favorites]
If elementary school teachers were good at math to begin with, their textbooks wouldn't need answer keys.
posted by Renoroc at 8:47 AM on November 12, 2013 [2 favorites]
posted by Renoroc at 8:47 AM on November 12, 2013 [2 favorites]
I have a lot of friends who're firmly in the I Can't Do Math category and I think it's tragic. Like, I get that lots of people will not be able to wrap their heads around multivariate calculus without more effort than they're ever going to put into it, but small children can do basic algebra before they can multiply. It's very doable, but not if you've already told yourself you can't.
Singing is similar, although there's a lot of other stuff built into that. My grandmother can't sing, but she's also nearly deaf even with a hearing aid, and it's very difficult to replicate a pitch you can't hear; some people might not be able to make the notes because of vocal chord issues. But otherwise, practice is most of it. Drawing, very much based on practice. Writing, to the point where I wish we could make all high school kids write fanfic or something because just writing occasional essays is not sufficient practice to develop competency. But oh lord, for the three of those, I am so in the midst of that gap that Ira Glass talked about where my taste sorely outstrips my skill and it's disheartening. At least in math, I don't have an aesthetic sensibility to offend while I'm learning.
posted by Sequence at 8:51 AM on November 12, 2013 [7 favorites]
Singing is similar, although there's a lot of other stuff built into that. My grandmother can't sing, but she's also nearly deaf even with a hearing aid, and it's very difficult to replicate a pitch you can't hear; some people might not be able to make the notes because of vocal chord issues. But otherwise, practice is most of it. Drawing, very much based on practice. Writing, to the point where I wish we could make all high school kids write fanfic or something because just writing occasional essays is not sufficient practice to develop competency. But oh lord, for the three of those, I am so in the midst of that gap that Ira Glass talked about where my taste sorely outstrips my skill and it's disheartening. At least in math, I don't have an aesthetic sensibility to offend while I'm learning.
posted by Sequence at 8:51 AM on November 12, 2013 [7 favorites]
A lot of the "hard work" is actually just busy work, test prep and gaming the system though, rather than developing real skills and knowledge. If an English student knows the grading rubric for an essay assignment inside and out, they can make sure that they hit all of the requirements and squeeze every last point out of it. Whereas another student who is actually a better writer or understood the material better might get a lower grade because they didn't game the scoring system as hard. And if a math student drills on every single math concept over and over, they'll do better on a test than a student who uses their natural problem solving skills to work out solutions to questions they aren't as familiar with.
In most cases, students are rewarded for overfitting their performance for the very specific material they are being tested on, rather than trying to become more well-rounded and ready to wing it when they have to. There is some hard work involved with actually learning the material, but most of the extra effort involved with getting good grades is more about making sure that your actual deficiencies on the material don't have any impact on how you're scored.
The bottom line is that unless your grading system is designed to have everyone on the high side of the grading curve, there's always going to be a gaming the system arms race, and the higher that escalates the more grading hinges on preparation rather than ability. If you look at a lot of the Asian countries mentioned in the article as models for what the US system should do, the prepping arms race has gotten so out of hand that it's nearly impossible to receive the highest grades without spending a massive amount of time and resources on extensive test prep and drilling.
posted by burnmp3s at 8:52 AM on November 12, 2013 [5 favorites]
In most cases, students are rewarded for overfitting their performance for the very specific material they are being tested on, rather than trying to become more well-rounded and ready to wing it when they have to. There is some hard work involved with actually learning the material, but most of the extra effort involved with getting good grades is more about making sure that your actual deficiencies on the material don't have any impact on how you're scored.
The bottom line is that unless your grading system is designed to have everyone on the high side of the grading curve, there's always going to be a gaming the system arms race, and the higher that escalates the more grading hinges on preparation rather than ability. If you look at a lot of the Asian countries mentioned in the article as models for what the US system should do, the prepping arms race has gotten so out of hand that it's nearly impossible to receive the highest grades without spending a massive amount of time and resources on extensive test prep and drilling.
posted by burnmp3s at 8:52 AM on November 12, 2013 [5 favorites]
I think the biggest challenge with learning math in North America is that elementary school teachers may not always be good at teaching it (it is very difficult to teach), and low expectations of students.
Or perhaps it reflects a class size larger than one and includes students without math teacher parents?
posted by srboisvert at 8:53 AM on November 12, 2013 [3 favorites]
Or perhaps it reflects a class size larger than one and includes students without math teacher parents?
posted by srboisvert at 8:53 AM on November 12, 2013 [3 favorites]
If elementary school teachers were good at math to begin with, their textbooks wouldn't need answer keys.
Please please please let's not do this. I used to be an elementary school teacher and I'm actually really, really good at math but when you have to correct a million tests or whatever and you're as short on time as teachers are (there are many, many, MANY things to do) it's really nice to have some assistance, and I say this as someone who had to give my kids (second grade) standardized tests for which I did not receive an answer key. I also made all my own assignments (classwork, homework, tests, and fast finisher work) and it took a really, REALLY long time and ideally I wouldn't have had to do that.
Making the teacher go through and do every problem on every test/homework assignment/classwork assignment/whatever when s/he could be grading or planning or talking to families or organizing field trips or attending professional developments or improving his or her practice in some other way is a really poor use of time.
posted by Mrs. Pterodactyl at 8:55 AM on November 12, 2013 [27 favorites]
Please please please let's not do this. I used to be an elementary school teacher and I'm actually really, really good at math but when you have to correct a million tests or whatever and you're as short on time as teachers are (there are many, many, MANY things to do) it's really nice to have some assistance, and I say this as someone who had to give my kids (second grade) standardized tests for which I did not receive an answer key. I also made all my own assignments (classwork, homework, tests, and fast finisher work) and it took a really, REALLY long time and ideally I wouldn't have had to do that.
Making the teacher go through and do every problem on every test/homework assignment/classwork assignment/whatever when s/he could be grading or planning or talking to families or organizing field trips or attending professional developments or improving his or her practice in some other way is a really poor use of time.
posted by Mrs. Pterodactyl at 8:55 AM on November 12, 2013 [27 favorites]
Re-reading the thread, I'm finding some of the comments really frustrating and representative of a trend throughout life/the internet (not just on MetaFilter) of being down on teachers and claiming they are ineffective or incompetent. This kind of thinking leads to fewer smart people becoming teachers because who wants to be stereotyped as bad at math?
Respectfully, some of the comments in here exemplify some of the reasons that I often skip education-related threads on MetaFilter entirely because there is often a lot of being down on teachers and people suggesting "obvious" solutions that may not be practical in a classroom. People have strong opinions on education, and that's great! Education is really important! But at the same time, because people are invested and have experience with the field either as a student or a parent it can often lead to a very narrow focus or a misunderstanding of the challenges and plenty of suggestions that don't really work and can even be kind of insulting.
I get that people care, and it's really awesome, but if we could maybe sometimes step back and think about all the different factors going on in any one classroom and then about the vast array of challenges and differences faced across the US (which is just one country! There are many others!) and recognize that maybe it's not easy and if there were a simple answer some of the very, very smart people who work on education would have thought of it that would be great.
I like MetaFilter and I like reading what people here have to say on many topics and I am interested in reading about people's experiences as parents/students/stakeholders in schooling but if we could maybe please be respectful of how very hard this is that would be awesome.
posted by Mrs. Pterodactyl at 9:04 AM on November 12, 2013 [22 favorites]
Respectfully, some of the comments in here exemplify some of the reasons that I often skip education-related threads on MetaFilter entirely because there is often a lot of being down on teachers and people suggesting "obvious" solutions that may not be practical in a classroom. People have strong opinions on education, and that's great! Education is really important! But at the same time, because people are invested and have experience with the field either as a student or a parent it can often lead to a very narrow focus or a misunderstanding of the challenges and plenty of suggestions that don't really work and can even be kind of insulting.
I get that people care, and it's really awesome, but if we could maybe sometimes step back and think about all the different factors going on in any one classroom and then about the vast array of challenges and differences faced across the US (which is just one country! There are many others!) and recognize that maybe it's not easy and if there were a simple answer some of the very, very smart people who work on education would have thought of it that would be great.
I like MetaFilter and I like reading what people here have to say on many topics and I am interested in reading about people's experiences as parents/students/stakeholders in schooling but if we could maybe please be respectful of how very hard this is that would be awesome.
posted by Mrs. Pterodactyl at 9:04 AM on November 12, 2013 [22 favorites]
And the engineer-y types generally scoff at all that foofy Arts nonsense, if they are scoff inclined people.
Yeah, but I think the point is that innumeracy is embraced at a far more basic and worrisome level than people embrace not-liking-foofy-Arts. Mathematical thinking isn't a specialized skill, it's a basic human way of interacting with and processing the world, just like reading. An engineer might say, "Yeah, I don't really like a lot of poetry," but they wouldn't say, "I've never been able to read." You might well admit that you don't grok impressionist painting or that you can't carry a tune in a bucket, but to say, "I don't like looking at things" or "I hate music"... isn't that a little weird?
Human beings are programmed to think numerically and quantitatively almost from the womb forwards. It's a shame to have cultural structures in place that validate people's voluntarily closing off that part of their cognition, and I support the dismantling of those structures.
posted by Bardolph at 9:05 AM on November 12, 2013 [7 favorites]
I was half-wrong in my earlier comment. Topynate's kind link to the study shows that this particular study is indeed specific to mathematics. It seems like she doesn't restrict herself to math in her other studies on the subject of innate-versus-learned-intelligence, however.
posted by daveliepmann at 9:11 AM on November 12, 2013
posted by daveliepmann at 9:11 AM on November 12, 2013
Thank you so much for posting this article. 3 of my children were tested and staffed into gifted programs in either kindergarten or 1st grade and 2 of them went completely off the rails as soon as they moved from elementary school to middle school. My youngest (also in the gifted program) son is in 5th grade and we're working super hard to try to make sure he doesn't do the same thing. One thing my husband and I have dealt with is teachers telling the kids that traits are "gifted kid things" such as ADHD and terrible handwriting. Well, maybe so, but it gives my son an excuse for inattentive behavior and rushing through assignments and not even trying to neaten his writing, and believe me he uses that.
Also, I have made the same comments to all my kids about "you're so smart!" when they do well on assignments-but I think I'll go with the "you really worked hard/studied effectively/went above the minimum" when he does well on things. I don't think teachers are lazily saying the wrong things to my son, but saying the natural thing to a frustrated student that you know has the ability to do better.
posted by hollygoheavy at 9:15 AM on November 12, 2013 [2 favorites]
Also, I have made the same comments to all my kids about "you're so smart!" when they do well on assignments-but I think I'll go with the "you really worked hard/studied effectively/went above the minimum" when he does well on things. I don't think teachers are lazily saying the wrong things to my son, but saying the natural thing to a frustrated student that you know has the ability to do better.
posted by hollygoheavy at 9:15 AM on November 12, 2013 [2 favorites]
Yeah, as someone who reads a thesaurus for fun but cannot perform even the simplest arithmatic without a calculator due to dyscalculia, it's been really exasperating to be repeatedly informed that my lifelong inability to do any sort of math means I'm either being willfully ignorant or too precious. But fifteen years out of high school, I took some college entrance exams on a whim -- I tested completely out of English... and into the absolute lowest level math class available: middle school algebra. At age 31. I skipped a grade in elementary school and tanked at math for the rest of my scholastic career.
When I try to describe the degree to which I cannot even really comprehend numbers, let alone the fact that my brain transposes basically all multi-digit numbers as soon as they hit my eyes (i.e. I see 32 and immediately say/think twenty-three), I am told that I'm not trying hard enough. But I still have to count on my fingers! I still have to get other people to confirm that I've managed to leave a 20% tip when all you're supposed to have to do is move the decimal point and then double it! If I'm asked to calculate something on the fly, my brain freezes up so completely that I can't even speak. Estimating quantity is impossible; I cannot meaningfully perceive a difference between seeing 50 and 500 of something. And geometric proofs actually made me question my sanity. I couldn't even do my assignments using the answer key. Man, I've tried.
I realize that the very sentiment irks people who were born with inherent number sense, but I have been trying and failing to grasp numbers since I was a kid and I appear to altogether lack the ability to move from rote memorization into real understanding. I used to think it was because my mind was so language-focused that it just ignored numbers out of convenience, but it turns out that some people literally can't successfully complete tasks that require basic numeracy. It doesn't make us stupid or unwilling to engage; it just makes us bad at math!
posted by divined by radio at 9:16 AM on November 12, 2013 [10 favorites]
When I try to describe the degree to which I cannot even really comprehend numbers, let alone the fact that my brain transposes basically all multi-digit numbers as soon as they hit my eyes (i.e. I see 32 and immediately say/think twenty-three), I am told that I'm not trying hard enough. But I still have to count on my fingers! I still have to get other people to confirm that I've managed to leave a 20% tip when all you're supposed to have to do is move the decimal point and then double it! If I'm asked to calculate something on the fly, my brain freezes up so completely that I can't even speak. Estimating quantity is impossible; I cannot meaningfully perceive a difference between seeing 50 and 500 of something. And geometric proofs actually made me question my sanity. I couldn't even do my assignments using the answer key. Man, I've tried.
I realize that the very sentiment irks people who were born with inherent number sense, but I have been trying and failing to grasp numbers since I was a kid and I appear to altogether lack the ability to move from rote memorization into real understanding. I used to think it was because my mind was so language-focused that it just ignored numbers out of convenience, but it turns out that some people literally can't successfully complete tasks that require basic numeracy. It doesn't make us stupid or unwilling to engage; it just makes us bad at math!
posted by divined by radio at 9:16 AM on November 12, 2013 [10 favorites]
Worse, you may be helping to perpetuate a pernicious myth that is harming underprivileged children—the myth of inborn genetic math ability.
As someone who was never good at math, I took my Dad's advice to heart that I should stop trying to understand it and just learn to do it, and that got me through engineering school so this article is preaching to the converted. OTOH this flies in the face of "Lazy and Shiftless" being a invalid excuse for failure. Whatever your innate ability, hard work can make you better at anything, and good enough for most things.
posted by three blind mice at 9:17 AM on November 12, 2013 [1 favorite]
As someone who was never good at math, I took my Dad's advice to heart that I should stop trying to understand it and just learn to do it, and that got me through engineering school so this article is preaching to the converted. OTOH this flies in the face of "Lazy and Shiftless" being a invalid excuse for failure. Whatever your innate ability, hard work can make you better at anything, and good enough for most things.
posted by three blind mice at 9:17 AM on November 12, 2013 [1 favorite]
An engineer might say, "Yeah, I don't really like a lot of poetry," but they wouldn't say, "I've never been able to read."
Maybe not an engineer. But lots of working- and lower-middle class-folks.
posted by The Underpants Monster at 9:56 AM on November 12, 2013
Maybe not an engineer. But lots of working- and lower-middle class-folks.
posted by The Underpants Monster at 9:56 AM on November 12, 2013
Whereas another student who is actually a better writer or understood the material better might get a lower grade because they didn't game the scoring system as hard.
Agreeing with this. Any test for material that is not direct real-world performance (which for obvious reasons you're not going to be able to use in schools) is going to get gamed in one way or another. Mandatory testing will ultimately mean teaching to the letter of the test to the exclusion of anything else, because that's the results that will be the most visible to a cursory glance. Hence, an industry built around doing well at the SATs and GREs, as opposed to doing well in the classes these tests are supposed to see if you're prepared for.
Why? Because one mistake means you don't completely understand the concept.
Others have responded to this adequately I think and I'm not trying to pile on. Suffice to say, this isn't true.
posted by JHarris at 10:07 AM on November 12, 2013 [1 favorite]
Agreeing with this. Any test for material that is not direct real-world performance (which for obvious reasons you're not going to be able to use in schools) is going to get gamed in one way or another. Mandatory testing will ultimately mean teaching to the letter of the test to the exclusion of anything else, because that's the results that will be the most visible to a cursory glance. Hence, an industry built around doing well at the SATs and GREs, as opposed to doing well in the classes these tests are supposed to see if you're prepared for.
Why? Because one mistake means you don't completely understand the concept.
Others have responded to this adequately I think and I'm not trying to pile on. Suffice to say, this isn't true.
posted by JHarris at 10:07 AM on November 12, 2013 [1 favorite]
Having taught both math and computer programming for years, I believe that there are people for whom mathematics does not come at all easily - whereas so far I haven't really found anyone who was at all interested in computer programming who couldn't "get it".
That said, as people above have written, I still think you can force yourself to "get through it" without loving it.
divined by radio:
> When I try to describe the degree to which I cannot even really comprehend numbers, let alone the fact that my brain transposes basically all multi-digit numbers as soon as they hit my eyes (i.e. I see 32 and immediately say/think twenty-three), I am told that I'm not trying hard enough.
Not at all - you have a failure of mechanism, not of industry or intelligence. I do honestly believe that with considerable effort but more important, with a new vision of how it works, in a few years you could get over this.
Here's an analogy from music. When I have trouble with a musical passage, I break it down into the smallest parts that I have trouble with. If I have to practice TWO notes over and over again, I'll do it - then move to three, four, etc. I'm so used to this that EVERY time I make a mistake, I stop and slowly do two, three...
Take the transposition issue. You've identified a clear problem - and a small, self-contained one. So start with that! Do exercises where you, for example, go to a web page and look at the numbers, then look away and try to reproduce them - first one, and then two...
Take your time!
Another way to get better is to learn to love the numbers for themselves. Both 32 and 23 have their own intimate characteristics - to me, mistaking one for the other is like mistaking chocolate and cheese.
32 ends with a 2 - it's an even number - in fact, it's just 2*2*2*2*2 - it's "2 to the 5th power". And note that 3 + 2 is five... a coincidence but one I know (by now) without even thinking about it.
On the other hand, 23 is an odd number. Indeed, you can't divide it by anything except itself and 1 - it's a prime number. It's also the number in the center of the Illuminatus! Trilogy, and also the number of chromosomes in (the vast majority of) humans - thus the name of the company 23AndMe.
Take your time. Learn to love each number.
Learning magnitudes is the same. Don't understand the difference between 50 and 500? Then get 500 of (some small item). Work out a shopping list - how much music can you buy for $50, or for $500? How much food?
I'd estimate this as follows - I can get an mp3 track for roughly $1, and it might last 4 minutes. An hour is 60 minutes. 60 minutes / 4 minutes is 15, so to purchase an hour of music is (very very roughly) $15. So $50 gets me about 3 and a bit hours of music (50/15) but $500 gets me about 33 hours of music - over a day!
I make these steps without thinking about it. But there was a point where this took me a long time. Take your time! Play around, and get your work checked!
You won't ever be a great mathematician - but if you started now, and spent three hours a week on this, on playing with numbers until you loved them, you would certainly be able to do things like "20% of the tip" without stressing.
posted by lupus_yonderboy at 10:11 AM on November 12, 2013
That said, as people above have written, I still think you can force yourself to "get through it" without loving it.
divined by radio:
> When I try to describe the degree to which I cannot even really comprehend numbers, let alone the fact that my brain transposes basically all multi-digit numbers as soon as they hit my eyes (i.e. I see 32 and immediately say/think twenty-three), I am told that I'm not trying hard enough.
Not at all - you have a failure of mechanism, not of industry or intelligence. I do honestly believe that with considerable effort but more important, with a new vision of how it works, in a few years you could get over this.
Here's an analogy from music. When I have trouble with a musical passage, I break it down into the smallest parts that I have trouble with. If I have to practice TWO notes over and over again, I'll do it - then move to three, four, etc. I'm so used to this that EVERY time I make a mistake, I stop and slowly do two, three...
Take the transposition issue. You've identified a clear problem - and a small, self-contained one. So start with that! Do exercises where you, for example, go to a web page and look at the numbers, then look away and try to reproduce them - first one, and then two...
Take your time!
Another way to get better is to learn to love the numbers for themselves. Both 32 and 23 have their own intimate characteristics - to me, mistaking one for the other is like mistaking chocolate and cheese.
32 ends with a 2 - it's an even number - in fact, it's just 2*2*2*2*2 - it's "2 to the 5th power". And note that 3 + 2 is five... a coincidence but one I know (by now) without even thinking about it.
On the other hand, 23 is an odd number. Indeed, you can't divide it by anything except itself and 1 - it's a prime number. It's also the number in the center of the Illuminatus! Trilogy, and also the number of chromosomes in (the vast majority of) humans - thus the name of the company 23AndMe.
Take your time. Learn to love each number.
Learning magnitudes is the same. Don't understand the difference between 50 and 500? Then get 500 of (some small item). Work out a shopping list - how much music can you buy for $50, or for $500? How much food?
I'd estimate this as follows - I can get an mp3 track for roughly $1, and it might last 4 minutes. An hour is 60 minutes. 60 minutes / 4 minutes is 15, so to purchase an hour of music is (very very roughly) $15. So $50 gets me about 3 and a bit hours of music (50/15) but $500 gets me about 33 hours of music - over a day!
I make these steps without thinking about it. But there was a point where this took me a long time. Take your time! Play around, and get your work checked!
You won't ever be a great mathematician - but if you started now, and spent three hours a week on this, on playing with numbers until you loved them, you would certainly be able to do things like "20% of the tip" without stressing.
posted by lupus_yonderboy at 10:11 AM on November 12, 2013
> Because one mistake means you don't completely understand the concept.
Numerous famous mathematicians notoriously can't add reliably. In an exam, where you have limited time to check and recheck, mistakes are inevitable, and some people will make more mistakes than others without having less understanding of the concepts involved.
posted by lupus_yonderboy at 10:13 AM on November 12, 2013 [2 favorites]
Numerous famous mathematicians notoriously can't add reliably. In an exam, where you have limited time to check and recheck, mistakes are inevitable, and some people will make more mistakes than others without having less understanding of the concepts involved.
posted by lupus_yonderboy at 10:13 AM on November 12, 2013 [2 favorites]
Worse, you may be helping to perpetuate a pernicious myth that is harming underprivileged children—the myth of inborn genetic math ability.
I wish people would stop framing these things so simply. In the article itself, it admits that not everyone is born with the same genetic capacity for math ability - it just states that those inborn abilities should not affect whether or not you can pick up high school math.
It seems that when these things come out, there's a certain amount of "Ha-ha, we're all equal, it's all about hard work/preparation" which is not actually true. It seems much more accurate to say, "No, everyone doesn't start out at the same level, but everyone can improve themselves no matter where they start out; and just having a propensity for a gift in a subject does not mean you will be a golden god in it without work, while a lack does not mean you're forever damned."
posted by corb at 10:20 AM on November 12, 2013 [5 favorites]
I wish people would stop framing these things so simply. In the article itself, it admits that not everyone is born with the same genetic capacity for math ability - it just states that those inborn abilities should not affect whether or not you can pick up high school math.
It seems that when these things come out, there's a certain amount of "Ha-ha, we're all equal, it's all about hard work/preparation" which is not actually true. It seems much more accurate to say, "No, everyone doesn't start out at the same level, but everyone can improve themselves no matter where they start out; and just having a propensity for a gift in a subject does not mean you will be a golden god in it without work, while a lack does not mean you're forever damned."
posted by corb at 10:20 AM on November 12, 2013 [5 favorites]
we never settle for anything less than 100% on a math test. Why? Because one mistake means you don't completely understand the concept.
This is absolutely wrong, and a horrible recipe for eventual burn-out once you encounter sufficiently advanced enough mathematics that you can't completely understand the concept, at least not within the limited confines of a class.
I think the biggest challenge with learning math in North America is that elementary school teachers may not always be good at teaching it (it is very difficult to teach), and low expectations of students.
I think it's just the opposite. Our expectations are too high and our demand for perfection pushes kids into 'cramming' mathematics and learning it as a collection of rote procedures and instructions instead of a system of interrelated concepts from which new truths and applications can be derived.
posted by RonButNotStupid at 10:22 AM on November 12, 2013 [8 favorites]
This is absolutely wrong, and a horrible recipe for eventual burn-out once you encounter sufficiently advanced enough mathematics that you can't completely understand the concept, at least not within the limited confines of a class.
I think the biggest challenge with learning math in North America is that elementary school teachers may not always be good at teaching it (it is very difficult to teach), and low expectations of students.
I think it's just the opposite. Our expectations are too high and our demand for perfection pushes kids into 'cramming' mathematics and learning it as a collection of rote procedures and instructions instead of a system of interrelated concepts from which new truths and applications can be derived.
posted by RonButNotStupid at 10:22 AM on November 12, 2013 [8 favorites]
Renoroc: "If elementary school teachers were good at math to begin with, their textbooks wouldn't need answer keys."
You've never had to grade multiple stacks of papers from multiple classes, have you?
(And, having taken graduate-level physics courses from fairly proficient mathematicians... vetted keys are far more reliable than any person's on-the-spot work.)
posted by IAmBroom at 10:36 AM on November 12, 2013 [2 favorites]
You've never had to grade multiple stacks of papers from multiple classes, have you?
(And, having taken graduate-level physics courses from fairly proficient mathematicians... vetted keys are far more reliable than any person's on-the-spot work.)
posted by IAmBroom at 10:36 AM on November 12, 2013 [2 favorites]
An engineer might say, "Yeah, I don't really like a lot of poetry," but they wouldn't say, "I've never been able to read." You might well admit that you don't grok impressionist painting or that you can't carry a tune in a bucket, but to say, "I don't like looking at things" or "I hate music"... isn't that a little weird?
I've met engineers who happily announce that they "don't read fiction."
posted by naoko at 10:49 AM on November 12, 2013 [2 favorites]
I've met engineers who happily announce that they "don't read fiction."
posted by naoko at 10:49 AM on November 12, 2013 [2 favorites]
Different kids with different levels of preparation come into a math class. ...The unprepared kids, not realizing that the top scorers were well-prepared, assume that genetic ability was what determined the performance differences.
This actually happened to me, but not until college. I did fine-but-not-outstanding in high school math and then got to college, and immediately concluded that I was too dumb to really "get" math because I didn't perform very well right out of the gate. After all, nobody else seemed to even be working very hard!
Later I came to realize that this was at least partly because I was surrounded by people who were, e.g., sandbagging multivariable (which they'd taken at fucking Stuyvesant or TJ or whatever other magnet school) for an easy A. The other part was that there was a lot of social pressure to look like you weren't trying too hard - in fact, effort was often taken as evidence of instrinsic unsuitability for a given field. Totally backwards. It was only after college that I really came to understand that people who seemed to be intrinsically good at something were often just much better prepared, and that no amount of "intrinsic" smarts would help you finish a god damn project anyway.
posted by en forme de poire at 10:59 AM on November 12, 2013 [3 favorites]
This actually happened to me, but not until college. I did fine-but-not-outstanding in high school math and then got to college, and immediately concluded that I was too dumb to really "get" math because I didn't perform very well right out of the gate. After all, nobody else seemed to even be working very hard!
Later I came to realize that this was at least partly because I was surrounded by people who were, e.g., sandbagging multivariable (which they'd taken at fucking Stuyvesant or TJ or whatever other magnet school) for an easy A. The other part was that there was a lot of social pressure to look like you weren't trying too hard - in fact, effort was often taken as evidence of instrinsic unsuitability for a given field. Totally backwards. It was only after college that I really came to understand that people who seemed to be intrinsically good at something were often just much better prepared, and that no amount of "intrinsic" smarts would help you finish a god damn project anyway.
posted by en forme de poire at 10:59 AM on November 12, 2013 [3 favorites]
I chose to do every single calculation using a calculator -- even if it was 7+1 or 3x2. My practice test scores improved dramatically, because my conceptual grasp was fine.
This plus a million. I'm a highly intelligent, extremely logical thinker. I think in sets and steps and blocks, where both order and precision are necessary parts. My career as an application analyst is directly a result of this. I'm currently overseeing a 250 million dollar software implementation that manages approximately $1B a day in transactions.
I can't balance my checkbook for shit.
Not on that little ledger, not with a calculator, not on my banking site, not with an app specifically for that sort of thing. I absolutely have to put them into a plain jane excel spreadsheet because that's what my brain needs to understand everything. Once I do that everything "clicks" and I every piece is a part of the whole and it all makes sense.
As such, I was shit at math in school.
Not because they didn't have excel spreadsheets, not because of genetics, and more than likely not because of the teachers I had or the methods they did/didn't use. But because I never got to see the "whole picture" as both a before and after of the parts and pieces. Figuring out tan and sin or even r-squared made no sense, because I never saw what it would be used for. But knowing how something interacts with the things around it, and being able to interact with the parts and pieces to see how they affected the whole made sense to me.
tl;dr: being "smart" and understanding something are two completely different things. Even of the thing that is/is not being understood is what they're teaching in school. Which, I think, is the antithesis of what this article proposes.
posted by Blue_Villain at 11:10 AM on November 12, 2013 [3 favorites]
This plus a million. I'm a highly intelligent, extremely logical thinker. I think in sets and steps and blocks, where both order and precision are necessary parts. My career as an application analyst is directly a result of this. I'm currently overseeing a 250 million dollar software implementation that manages approximately $1B a day in transactions.
I can't balance my checkbook for shit.
Not on that little ledger, not with a calculator, not on my banking site, not with an app specifically for that sort of thing. I absolutely have to put them into a plain jane excel spreadsheet because that's what my brain needs to understand everything. Once I do that everything "clicks" and I every piece is a part of the whole and it all makes sense.
As such, I was shit at math in school.
Not because they didn't have excel spreadsheets, not because of genetics, and more than likely not because of the teachers I had or the methods they did/didn't use. But because I never got to see the "whole picture" as both a before and after of the parts and pieces. Figuring out tan and sin or even r-squared made no sense, because I never saw what it would be used for. But knowing how something interacts with the things around it, and being able to interact with the parts and pieces to see how they affected the whole made sense to me.
tl;dr: being "smart" and understanding something are two completely different things. Even of the thing that is/is not being understood is what they're teaching in school. Which, I think, is the antithesis of what this article proposes.
posted by Blue_Villain at 11:10 AM on November 12, 2013 [3 favorites]
This is absolutely wrong
To play devil's advocate, this is basically how programs like Kumon work, right? You don't advance to the next section until you get a perfect score.
I mean, one problem with math education is that if you get, say, a B in math in junior high, it seems totally fine - a B is a totally okay grade, right? Except that it really means you didn't grasp around 10-15% of the material. Which in turn means you are guaranteed to be screwed down the line when that material becomes the foundation for something else.
I do think needing a perfect 100% every time is putting a lot of unnecessary stress on a student because the other part of programs like Kumon is that they are self-paced, so if you make a silly error you can just try the test again until you get everything perfect; that opportunity obviously isn't given in the normal math curriculum. And of course, this stops applying once you get into college and you get exams where it's not actually possible to get a perfect score (e.g., an A is ~60-70% and the mean grade is more like a 40).
posted by en forme de poire at 11:11 AM on November 12, 2013
To play devil's advocate, this is basically how programs like Kumon work, right? You don't advance to the next section until you get a perfect score.
I mean, one problem with math education is that if you get, say, a B in math in junior high, it seems totally fine - a B is a totally okay grade, right? Except that it really means you didn't grasp around 10-15% of the material. Which in turn means you are guaranteed to be screwed down the line when that material becomes the foundation for something else.
I do think needing a perfect 100% every time is putting a lot of unnecessary stress on a student because the other part of programs like Kumon is that they are self-paced, so if you make a silly error you can just try the test again until you get everything perfect; that opportunity obviously isn't given in the normal math curriculum. And of course, this stops applying once you get into college and you get exams where it's not actually possible to get a perfect score (e.g., an A is ~60-70% and the mean grade is more like a 40).
posted by en forme de poire at 11:11 AM on November 12, 2013
Or, you know, you transposed a sign and didn't notice before the clock ran out.
The students here in Canada are being assessed on how well they understand the concept, not how fast they finish a test. It's holistic, and the assessment is not based on one test.
posted by KokuRyu at 11:11 AM on November 12, 2013
The students here in Canada are being assessed on how well they understand the concept, not how fast they finish a test. It's holistic, and the assessment is not based on one test.
posted by KokuRyu at 11:11 AM on November 12, 2013
Numerous famous mathematicians notoriously can't add reliably.
As I tell my kids, I don't care about anyone else. I just care about how they are doing. We're talking about the foundational stuff here. It is not hard.
I don't know why we have to make excuses for why kids cannot learn math. Why is it politically correct to say "maybe they couldn't finish the test" or "don't worry, that hypothetical imaginary person also didn't do well at math"?
Then again, we're exceedingly proud that our son went from underperforming (and being allowed to underperform) to exceeding expectations. We don't really worry about what other people expect of their kids.
posted by KokuRyu at 11:14 AM on November 12, 2013 [1 favorite]
As I tell my kids, I don't care about anyone else. I just care about how they are doing. We're talking about the foundational stuff here. It is not hard.
I don't know why we have to make excuses for why kids cannot learn math. Why is it politically correct to say "maybe they couldn't finish the test" or "don't worry, that hypothetical imaginary person also didn't do well at math"?
Then again, we're exceedingly proud that our son went from underperforming (and being allowed to underperform) to exceeding expectations. We don't really worry about what other people expect of their kids.
posted by KokuRyu at 11:14 AM on November 12, 2013 [1 favorite]
We're talking about the foundational stuff here. It is not hard.
I can't add reliably. At all. Suck at it.
I've also passed a couple of Actuary exams, and have very little problem doing calculus.
Hard is relative.
posted by MisantropicPainforest at 11:18 AM on November 12, 2013 [3 favorites]
I can't add reliably. At all. Suck at it.
I've also passed a couple of Actuary exams, and have very little problem doing calculus.
Hard is relative.
posted by MisantropicPainforest at 11:18 AM on November 12, 2013 [3 favorites]
Decimals and integers and basic numeracy are pretty useful for basic day-to-day tasks, like going to the grocery store.
posted by KokuRyu at 11:21 AM on November 12, 2013 [1 favorite]
posted by KokuRyu at 11:21 AM on November 12, 2013 [1 favorite]
Is this where I use the adage that Einstein couldn't tie his own shoes?
I mean, it's false... but it illustrates the point that not everything regarding knowledge is linear.
posted by Blue_Villain at 11:26 AM on November 12, 2013
I mean, it's false... but it illustrates the point that not everything regarding knowledge is linear.
posted by Blue_Villain at 11:26 AM on November 12, 2013
Decimals and integers and basic numeracy are pretty useful for basic day-to-day tasks, like going to the grocery store.
Thanks. I know. I'm sure everyone here knows that too. No one is talking about the utility of any one type of knowledge.
posted by MisantropicPainforest at 11:28 AM on November 12, 2013 [1 favorite]
Thanks. I know. I'm sure everyone here knows that too. No one is talking about the utility of any one type of knowledge.
posted by MisantropicPainforest at 11:28 AM on November 12, 2013 [1 favorite]
Well then, what's your point? You agree that foundational math skills are important. We are talking about math in this thread.
I cannot believe that an actuary cannot balance his/her own chequebook, by the way.
posted by KokuRyu at 11:47 AM on November 12, 2013 [1 favorite]
I cannot believe that an actuary cannot balance his/her own chequebook, by the way.
posted by KokuRyu at 11:47 AM on November 12, 2013 [1 favorite]
I don't know why we have to make excuses for why kids cannot learn math. Why is it politically correct to say "maybe they couldn't finish the test" or "don't worry, that hypothetical imaginary person also didn't do well at math"?
It's not about some kids not learning math at all, it's about some kids not learning math at the same pace as others. It's not realistic to expect every student to score 100% on every assignment, because even if that was the case the assignments would no longer be measuring the relative performance of the top performing students anymore.
Then again, we're exceedingly proud that our son went from underperforming (and being allowed to underperform) to exceeding expectations. We don't really worry about what other people expect of their kids.
Except that the whole concept of underperforming is relative to the overall expectations of the peer group. If all of the other parents spent even more time tutoring their kids to perform better than your kid, then your kid would be back to underperforming. Not everyone can be above average.
posted by burnmp3s at 11:56 AM on November 12, 2013 [3 favorites]
It's not about some kids not learning math at all, it's about some kids not learning math at the same pace as others. It's not realistic to expect every student to score 100% on every assignment, because even if that was the case the assignments would no longer be measuring the relative performance of the top performing students anymore.
Then again, we're exceedingly proud that our son went from underperforming (and being allowed to underperform) to exceeding expectations. We don't really worry about what other people expect of their kids.
Except that the whole concept of underperforming is relative to the overall expectations of the peer group. If all of the other parents spent even more time tutoring their kids to perform better than your kid, then your kid would be back to underperforming. Not everyone can be above average.
posted by burnmp3s at 11:56 AM on November 12, 2013 [3 favorites]
My point is that difficulty is not linear and some things are hard for some people and some things are not hard for some people.
I cannot believe that an actuary cannot balance his/her own chequebook, by the way.
You're extrapolating my comment about how I find simple math difficult to the conclusion that I cannot balance my checkbook?
If you're not going to believe people's self-reporting of their own lived experience then I don't know why you are trying to converse with others.
posted by MisantropicPainforest at 12:05 PM on November 12, 2013 [1 favorite]
I cannot believe that an actuary cannot balance his/her own chequebook, by the way.
You're extrapolating my comment about how I find simple math difficult to the conclusion that I cannot balance my checkbook?
If you're not going to believe people's self-reporting of their own lived experience then I don't know why you are trying to converse with others.
posted by MisantropicPainforest at 12:05 PM on November 12, 2013 [1 favorite]
>If all of the other parents spent even more time tutoring their kids to perform better than your kid,
Or all kids would master the concepts and be at the same level. Master is absolute at this age level, and can be achieved.
I tend to agree that different kids learn at different paces. The youngest children in the cohort typically lag behind children born earlier in the year.
But that's also a symptom of the fucked-up way we approach school (class starts with a bell at 8:42, kids sit in rows, a bell rings after 40 minutes and we then learn something else; no eating or drinking during class; no moving around, no talking, the school day ends at 2:57).
posted by KokuRyu at 12:05 PM on November 12, 2013
Or all kids would master the concepts and be at the same level. Master is absolute at this age level, and can be achieved.
I tend to agree that different kids learn at different paces. The youngest children in the cohort typically lag behind children born earlier in the year.
But that's also a symptom of the fucked-up way we approach school (class starts with a bell at 8:42, kids sit in rows, a bell rings after 40 minutes and we then learn something else; no eating or drinking during class; no moving around, no talking, the school day ends at 2:57).
posted by KokuRyu at 12:05 PM on November 12, 2013
You're extrapolating my comment about how I find simple math difficult to the conclusion that I cannot balance my checkbook?
If you're not going to believe people's self-reporting of their own lived experience then I don't know why you are trying to converse with others.
We're talking about the basics. You told me that, as a professional, the basics aren't even important for you (you're an actuary who can't add).
The stuff kids are learning in elementary school and to some extent middle school, are foundational skills - numeracy - needed for everyday life.
That's what our son's teacher had given up on.
posted by KokuRyu at 12:07 PM on November 12, 2013
If you're not going to believe people's self-reporting of their own lived experience then I don't know why you are trying to converse with others.
We're talking about the basics. You told me that, as a professional, the basics aren't even important for you (you're an actuary who can't add).
The stuff kids are learning in elementary school and to some extent middle school, are foundational skills - numeracy - needed for everyday life.
That's what our son's teacher had given up on.
posted by KokuRyu at 12:07 PM on November 12, 2013
You told me that, as a professional, the basics aren't even important for you
I believe you are confusing me with someone else here.
posted by MisantropicPainforest at 12:10 PM on November 12, 2013
I believe you are confusing me with someone else here.
posted by MisantropicPainforest at 12:10 PM on November 12, 2013
But that's also a symptom of the fucked-up way we approach school (class starts with a bell at 8:42, kids sit in rows, a bell rings after 40 minutes and we then learn something else; no eating or drinking during class; no moving around, no talking, the school day ends at 2:57).
I don't know about Canada but (as someone with a Master's Degree in Elementary Education) I can tell you that this is explicitly NOT how you are supposed to teach and in many districts you would get fired if an evaluator came in and saw that your students were not allowed to move or talk and just had to sit in rows all day.
Also, in terms of the math thing, I think part of the problem is that a lot of people have decided that As are the only acceptable grade, which means that there's not really a great way to stretch people. In theory, "C" is average, so if you're getting 75% on the test you're getting what you need. If we expect all students to get 100% on everything there's no growth possible; can't it be set up so that everyone needs to get 75% but there's some advanced stuff on there that is maybe slightly beyond what is considered "age appropriate", like "stretch work"? If every student is getting 100% on every test then it might mean that the material is too easy for them.
posted by Mrs. Pterodactyl at 12:19 PM on November 12, 2013 [1 favorite]
I don't know about Canada but (as someone with a Master's Degree in Elementary Education) I can tell you that this is explicitly NOT how you are supposed to teach and in many districts you would get fired if an evaluator came in and saw that your students were not allowed to move or talk and just had to sit in rows all day.
Also, in terms of the math thing, I think part of the problem is that a lot of people have decided that As are the only acceptable grade, which means that there's not really a great way to stretch people. In theory, "C" is average, so if you're getting 75% on the test you're getting what you need. If we expect all students to get 100% on everything there's no growth possible; can't it be set up so that everyone needs to get 75% but there's some advanced stuff on there that is maybe slightly beyond what is considered "age appropriate", like "stretch work"? If every student is getting 100% on every test then it might mean that the material is too easy for them.
posted by Mrs. Pterodactyl at 12:19 PM on November 12, 2013 [1 favorite]
we never settle for anything less than 100% on a math test. Why? Because one mistake means you don't completely understand the concept.
And
We're talking about the foundational stuff here. It is not hard.
With all due respect...as someone who has extreme difficulty with even the most basic mathematic concepts (not for lack of trying) I find this attitude both familiar and dismaying. The extreme smugness of people who understand math is a large part of what makes me constantly feel like a fool for not understanding what seems self-evident to others.
Feeling like a fool despite trying to understand basic concepts that everyone else understands is not a nice feeling. To have someone (in the past it was a teacher, these days it is other adults) lord over me telling me that "the foundational stuff" is "not hard" does not help me to understand the concepts.
Then I usually get "I'm breaking it down for you as far as I can go - how do you not understand this."
Guess what. Sometimes I still don't understand it. Despite trying as hard as I can. Maybe it is the way math was taught to me but it always seemed like the "steps" were not sufficiently broken down for me to grasp. And always, always, always I had to deal with, and still deal with, the sort of angry smugness of "why aren't you getting this?"
It's disgusting.
I work with computers and have taught myself CS and some programming to the extent that I need too as a VFX artist - but math is still somewhat of a black box to me. You know what I can do though? I can draw. I can draw well. The analogy I think of is that telling me math is "not hard" is like telling someone with no artistic ability that drawing is "not hard" - why don't you just draw well? Its easy. Just look at what you see and draw it. Why aren't you getting this? Your lines are wrong. You don't understand the concept.
And you know what? Doing it over and over won't help if the person doesn't understand how to see. How to see and how to turn that into 2d on paper.
Some minds require a unique key to unlock in order for them to be receptive to certain concepts, certain fields. The onus in that respect is on the instructor - so long as the student genuinely is trying.
Math is still taught by rote. By "fear tactics." If we really are mathematically inclined as humans I'd like to see a more organic approach. A more human approach.
For me even the most basic explanation of "what are numbers" makes no f*cking sense. I can't wrap my brain around it. I'm glad your child improved by drilling concepts into his head. But I was in their shoes as a child (had 3 math tutors and had to do my homework with the math teacher) and it never worked for me.
posted by jnnla at 12:30 PM on November 12, 2013 [12 favorites]
And
We're talking about the foundational stuff here. It is not hard.
With all due respect...as someone who has extreme difficulty with even the most basic mathematic concepts (not for lack of trying) I find this attitude both familiar and dismaying. The extreme smugness of people who understand math is a large part of what makes me constantly feel like a fool for not understanding what seems self-evident to others.
Feeling like a fool despite trying to understand basic concepts that everyone else understands is not a nice feeling. To have someone (in the past it was a teacher, these days it is other adults) lord over me telling me that "the foundational stuff" is "not hard" does not help me to understand the concepts.
Then I usually get "I'm breaking it down for you as far as I can go - how do you not understand this."
Guess what. Sometimes I still don't understand it. Despite trying as hard as I can. Maybe it is the way math was taught to me but it always seemed like the "steps" were not sufficiently broken down for me to grasp. And always, always, always I had to deal with, and still deal with, the sort of angry smugness of "why aren't you getting this?"
It's disgusting.
I work with computers and have taught myself CS and some programming to the extent that I need too as a VFX artist - but math is still somewhat of a black box to me. You know what I can do though? I can draw. I can draw well. The analogy I think of is that telling me math is "not hard" is like telling someone with no artistic ability that drawing is "not hard" - why don't you just draw well? Its easy. Just look at what you see and draw it. Why aren't you getting this? Your lines are wrong. You don't understand the concept.
And you know what? Doing it over and over won't help if the person doesn't understand how to see. How to see and how to turn that into 2d on paper.
Some minds require a unique key to unlock in order for them to be receptive to certain concepts, certain fields. The onus in that respect is on the instructor - so long as the student genuinely is trying.
Math is still taught by rote. By "fear tactics." If we really are mathematically inclined as humans I'd like to see a more organic approach. A more human approach.
For me even the most basic explanation of "what are numbers" makes no f*cking sense. I can't wrap my brain around it. I'm glad your child improved by drilling concepts into his head. But I was in their shoes as a child (had 3 math tutors and had to do my homework with the math teacher) and it never worked for me.
posted by jnnla at 12:30 PM on November 12, 2013 [12 favorites]
can't it be set up so that everyone needs to get 75% but there's some advanced stuff on there that is maybe slightly beyond what is considered "age appropriate", like "stretch work"?
I like this in theory but in practice I think this just ends up benefiting the kids whose parents can spend the extra time and/or money to tutor them. It's already an open secret that parents of A students already spend a huge amount of extra time coaching their kids (and, for example, doing stencil lettering and drawing horses on their fifth-grade book report posters, which mostly get graded on how precise and beautiful they look). Young students shouldn't be penalized for not having those resources available to them.
posted by en forme de poire at 12:42 PM on November 12, 2013
I like this in theory but in practice I think this just ends up benefiting the kids whose parents can spend the extra time and/or money to tutor them. It's already an open secret that parents of A students already spend a huge amount of extra time coaching their kids (and, for example, doing stencil lettering and drawing horses on their fifth-grade book report posters, which mostly get graded on how precise and beautiful they look). Young students shouldn't be penalized for not having those resources available to them.
posted by en forme de poire at 12:42 PM on November 12, 2013
Young students shouldn't be penalized for not having those resources available to them.
Yeah, I do agree with this and I will say that that's not how I actually graded my students, but I think it's something to consider. I also think part of the problem, of course, is that young students are ALREADY penalized by not having those resources available to them in ways that are more serious than their letter grades.
posted by Mrs. Pterodactyl at 12:47 PM on November 12, 2013 [2 favorites]
Yeah, I do agree with this and I will say that that's not how I actually graded my students, but I think it's something to consider. I also think part of the problem, of course, is that young students are ALREADY penalized by not having those resources available to them in ways that are more serious than their letter grades.
posted by Mrs. Pterodactyl at 12:47 PM on November 12, 2013 [2 favorites]
various really smart, well-educated people have no problems announcing "Oh, I'm just hopeless at math" when they would be (justly) ashamed of saying "Oh, I just can't do books, ugh."
A lot of people have an incredibly bad understanding of the process of learning. Schools (for a variety of reasons) often don't teach you HOW to learn; they just do their best to teach you other stuff, and for some people they aren't even able to do that terribly well.
Teaching skiing, I found an alarming number of adults who came along thinking that there is some innate talent at skiing that they might have; and that they can come to the ski slope and have a quick go in order to find out. When they very first slide down a hill and discover they can't really ski yet, they see this as evidence that the innate ability is missing and they may as well give up. "Oh I knew I would be no good at this!"
Naturally, people have varying levels of co-ordination and balance and fitness and so forth, but 99.9% of people have no fundamental physical problem in the way of me teaching them to ski (some quicker than others!).
I have lost count of the number of people who, after a lesson, have expressed utter surprise that I was able to teach them (whatever we did) and that they were able to learn it.
I often wonder whether this new discovery ever crosses into other aspects of people's lives and leads them to try and learn other things they think they can't do.
posted by emilyw at 12:51 PM on November 12, 2013
A lot of people have an incredibly bad understanding of the process of learning. Schools (for a variety of reasons) often don't teach you HOW to learn; they just do their best to teach you other stuff, and for some people they aren't even able to do that terribly well.
Teaching skiing, I found an alarming number of adults who came along thinking that there is some innate talent at skiing that they might have; and that they can come to the ski slope and have a quick go in order to find out. When they very first slide down a hill and discover they can't really ski yet, they see this as evidence that the innate ability is missing and they may as well give up. "Oh I knew I would be no good at this!"
Naturally, people have varying levels of co-ordination and balance and fitness and so forth, but 99.9% of people have no fundamental physical problem in the way of me teaching them to ski (some quicker than others!).
I have lost count of the number of people who, after a lesson, have expressed utter surprise that I was able to teach them (whatever we did) and that they were able to learn it.
I often wonder whether this new discovery ever crosses into other aspects of people's lives and leads them to try and learn other things they think they can't do.
posted by emilyw at 12:51 PM on November 12, 2013
I also think part of the problem, of course, is that young students are ALREADY penalized by not having those resources available to them in ways that are more serious than their letter grades.
Totally fair. Depressing how many things basically come back around to e.g. systemic income inequality in the end.
posted by en forme de poire at 1:03 PM on November 12, 2013
Totally fair. Depressing how many things basically come back around to e.g. systemic income inequality in the end.
posted by en forme de poire at 1:03 PM on November 12, 2013
We're talking about the basics. You told me that, as a professional, the basics aren't even important for you (you're an actuary who can't add).
Crawling on the floor is a "basic" skill that's completely unnecessary to drive a car. Arithmetic makes for a convenient bootstrap to introduce higher mathematical concepts, but being able to add arbitrary pairs of large numbers in your head isn't all that important in the grand scheme of things.
posted by RonButNotStupid at 1:43 PM on November 12, 2013 [1 favorite]
Crawling on the floor is a "basic" skill that's completely unnecessary to drive a car. Arithmetic makes for a convenient bootstrap to introduce higher mathematical concepts, but being able to add arbitrary pairs of large numbers in your head isn't all that important in the grand scheme of things.
posted by RonButNotStupid at 1:43 PM on November 12, 2013 [1 favorite]
The analogy I think of is that telling me math is "not hard" is like telling someone with no artistic ability that drawing is "not hard" - why don't you just draw well? Its easy. Just look at what you see and draw it. Why aren't you getting this? Your lines are wrong. You don't understand the concept.
And you know what? Doing it over and over won't help if the person doesn't understand how to see. How to see and how to turn that into 2d on paper.
I think about this from time to time. I was in kindergarten, and one day the teacher had taped a bunch of pictures onto the blackboard, and we each had to draw one. And there was this really cool bald eagle partial profile with a tree in the background or something, and I really wanted to draw that. So I tried outlining the eagle but I just couldn't make my hand do what I wanted it to do. I mean I could see in my eye what my crayon drawing would look like, and I knew it wouldn't look exactly like the picture, but I could kind of see how it would differ. So after a few minutes, I raised my hand, and when the teacher came around, explained my situation. She said, "Oh, birds are easy!" and took my crayon and drew the standard upsilon-looking bird in flight. And I was stupefied. No, that wasn't what I was asking at all, but who was I to correct a teacher?
Every time I ever tried to learn to draw, I was met with this sort of disbelief and non-answer ("you just draw, what's so hard about it?"). Yeah, I grokked math at an early age, and I doubt I'd be a great artist. But maybe I'd be a decent artist instead of my sketches looking like xkcd drawn during an earthquake. And there's no reason I couldn't have done that AND still be an engineer.
posted by disconnect at 1:48 PM on November 12, 2013 [1 favorite]
And you know what? Doing it over and over won't help if the person doesn't understand how to see. How to see and how to turn that into 2d on paper.
I think about this from time to time. I was in kindergarten, and one day the teacher had taped a bunch of pictures onto the blackboard, and we each had to draw one. And there was this really cool bald eagle partial profile with a tree in the background or something, and I really wanted to draw that. So I tried outlining the eagle but I just couldn't make my hand do what I wanted it to do. I mean I could see in my eye what my crayon drawing would look like, and I knew it wouldn't look exactly like the picture, but I could kind of see how it would differ. So after a few minutes, I raised my hand, and when the teacher came around, explained my situation. She said, "Oh, birds are easy!" and took my crayon and drew the standard upsilon-looking bird in flight. And I was stupefied. No, that wasn't what I was asking at all, but who was I to correct a teacher?
Every time I ever tried to learn to draw, I was met with this sort of disbelief and non-answer ("you just draw, what's so hard about it?"). Yeah, I grokked math at an early age, and I doubt I'd be a great artist. But maybe I'd be a decent artist instead of my sketches looking like xkcd drawn during an earthquake. And there's no reason I couldn't have done that AND still be an engineer.
posted by disconnect at 1:48 PM on November 12, 2013 [1 favorite]
A lot of the "hard work" is actually just busy work, test prep and gaming the system though, rather than developing real skills and knowledge. If an English student knows the grading rubric for an essay assignment inside and out, they can make sure that they hit all of the requirements and squeeze every last point out of it. Whereas another student who is actually a better writer or understood the material better might get a lower grade because they didn't game the scoring system as hard.
In other words, it's a lot like real life?
posted by ZenMasterThis at 6:07 PM on November 12, 2013 [1 favorite]
In other words, it's a lot like real life?
posted by ZenMasterThis at 6:07 PM on November 12, 2013 [1 favorite]
> > I cannot believe that an actuary cannot balance his/her own chequebook, by the way.
> You're extrapolating my comment about how I find simple math difficult to the conclusion that I cannot balance my checkbook?
Before this gets more heated, I'm absolutely fascinated with this part. It seems impossible to me! (And I mean that in a good way, like, "Cool! A new specimen!")
As someone who tries to teach this stuff, can you tell me more about how you see numbers?
Do you have a feel for magnitudes? If someone says "You get 7.5% of $320", does that ring up for you at all?
I assume you use a calculator for work (as everyone should). What goes wrong when you try to add? What do you remember about how you learned arithmetic the first time?"
posted by lupus_yonderboy at 6:43 PM on November 12, 2013
> You're extrapolating my comment about how I find simple math difficult to the conclusion that I cannot balance my checkbook?
Before this gets more heated, I'm absolutely fascinated with this part. It seems impossible to me! (And I mean that in a good way, like, "Cool! A new specimen!")
As someone who tries to teach this stuff, can you tell me more about how you see numbers?
Do you have a feel for magnitudes? If someone says "You get 7.5% of $320", does that ring up for you at all?
I assume you use a calculator for work (as everyone should). What goes wrong when you try to add? What do you remember about how you learned arithmetic the first time?"
posted by lupus_yonderboy at 6:43 PM on November 12, 2013
If you think this is interesting you should check out Seymour Papert's Mindstorms, which Bret Victor recommends frequently. It's about teaching with computers, but he spends much of the book (pretty much all that I've read so far) arguing how teaching that some people are "good at math/writing" whatever is a fundamentally wrong way of thinking and causes a lot of harm to individuals and society. A lot of his ideas on education without reference to computers come from Piaget, who's also worth looking into.
Mindstorms also reminds me of Prometheus Rising surprisingly often, which was a link I didn't expect.
posted by 23 at 7:05 PM on November 12, 2013
Mindstorms also reminds me of Prometheus Rising surprisingly often, which was a link I didn't expect.
posted by 23 at 7:05 PM on November 12, 2013
I teach remedial math to kids whose performance on state tests is in doubt, and I have undergraduate majors in math and computer science that I didn't undertake thinking I'd go into teaching public school. A few responses to what's been said in this thread:
if a math student drills on every single math concept over and over, they'll do better on a test than a student who uses their natural problem solving skills to work out solutions to questions they aren't as familiar with.
Not so much any more, or at least in the not-too-distant future in the US. The Common Core Standards emphasize deeper, more conceptual thinking, and the PARCC assessments that are going to be implemented in the coming years are going to be more geared toward critical thinking.
For awhile, some teachers and principals have known that "item teaching*" was basically educational malpractice but the high stakes tests that came out to fulfill No Child Left Behind could be gamed in this fashion and a lot of school districts, at least in my state of NJ, tolerated or encouraged item teaching because it got results.
That approach isn't going to cut it when it comes to PARCC assessments. And some of the teachers I know are shitting bricks about this.
I am fucking psyched for it, though.
I can't add reliably. At all. Suck at it.
I've also passed a couple of Actuary exams, and have very little problem doing calculus.
This I totally get. When I think about math, I think about relationships between numbers, relationships between those relationships, and proving that those relationships are true. Numbers, to me, and I think to people who have studied a lot of math, are the particulars. But they're not as important because the relationships we study don't depend on the particulars, and because a machine can do those calculations.
A machine can't really do the thinking that shows that a particular calculation is an important one to carry out, though. And it can't do the theorem proving and problem finding that we can. That's the real math. Kids are going to get a better look at the deeper aspects of math in the coming years in the US.
*item teaching basically means that you're preparing students for a test by giving them a practice test where the only difference with the actual test is that the numbers are different. It's a phrase used in a widely-cited paper on ethical instruction and is worth Googling.
Watch out if your kids have a teacher who does this. It's a sign that things aren't going so well for that teacher, though that teacher may not realize that they're doing something wrong and is probably frustrated by the low grades that keep coming in despite the fact that they're practically giving the test away. But memorization is a shitty substitute for understanding.
posted by alphanerd at 7:45 PM on November 12, 2013 [1 favorite]
if a math student drills on every single math concept over and over, they'll do better on a test than a student who uses their natural problem solving skills to work out solutions to questions they aren't as familiar with.
Not so much any more, or at least in the not-too-distant future in the US. The Common Core Standards emphasize deeper, more conceptual thinking, and the PARCC assessments that are going to be implemented in the coming years are going to be more geared toward critical thinking.
For awhile, some teachers and principals have known that "item teaching*" was basically educational malpractice but the high stakes tests that came out to fulfill No Child Left Behind could be gamed in this fashion and a lot of school districts, at least in my state of NJ, tolerated or encouraged item teaching because it got results.
That approach isn't going to cut it when it comes to PARCC assessments. And some of the teachers I know are shitting bricks about this.
I am fucking psyched for it, though.
I can't add reliably. At all. Suck at it.
I've also passed a couple of Actuary exams, and have very little problem doing calculus.
This I totally get. When I think about math, I think about relationships between numbers, relationships between those relationships, and proving that those relationships are true. Numbers, to me, and I think to people who have studied a lot of math, are the particulars. But they're not as important because the relationships we study don't depend on the particulars, and because a machine can do those calculations.
A machine can't really do the thinking that shows that a particular calculation is an important one to carry out, though. And it can't do the theorem proving and problem finding that we can. That's the real math. Kids are going to get a better look at the deeper aspects of math in the coming years in the US.
*item teaching basically means that you're preparing students for a test by giving them a practice test where the only difference with the actual test is that the numbers are different. It's a phrase used in a widely-cited paper on ethical instruction and is worth Googling.
Watch out if your kids have a teacher who does this. It's a sign that things aren't going so well for that teacher, though that teacher may not realize that they're doing something wrong and is probably frustrated by the low grades that keep coming in despite the fact that they're practically giving the test away. But memorization is a shitty substitute for understanding.
posted by alphanerd at 7:45 PM on November 12, 2013 [1 favorite]
> The analogy I think of is that telling me math is "not hard" is like telling someone with no artistic ability that drawing is "not hard" - why don't you just draw well?
Almost anyone can learn how to draw. If I could eventually learn to do it (I still can't keep it up at all without huge effort) pretty well anyone can.
But math is even harder in many ways than drawing. It's purely abstract, just manipulation of symbols.
For some people it's just plain hard. Still, I'd claim that it's perfectly well within the reach of anyone able to post coherently on Metafilter - if you had a good and patient teacher who was willing to go through it all step at a time for you (which might even be "you") and were willing to devote the time to do it - which might be considerable.
To prove my theory, I offer to personally teach one Metafilter person who cannot do arithmetic at all but is willing to put enough time into it, and have them report back to the group on success or failure.
posted by lupus_yonderboy at 7:47 PM on November 12, 2013
Almost anyone can learn how to draw. If I could eventually learn to do it (I still can't keep it up at all without huge effort) pretty well anyone can.
But math is even harder in many ways than drawing. It's purely abstract, just manipulation of symbols.
For some people it's just plain hard. Still, I'd claim that it's perfectly well within the reach of anyone able to post coherently on Metafilter - if you had a good and patient teacher who was willing to go through it all step at a time for you (which might even be "you") and were willing to devote the time to do it - which might be considerable.
To prove my theory, I offer to personally teach one Metafilter person who cannot do arithmetic at all but is willing to put enough time into it, and have them report back to the group on success or failure.
posted by lupus_yonderboy at 7:47 PM on November 12, 2013
By the next report card he was "exceeding expectations" and we never settle for anything less than 100% on a math test. Why? Because one mistake means you don't completely understand the concept.
I'm quite good at math (I was taking differential equations in high school) but the "no mistakes" attitude my mom had wasn't responsible for that. The "no mistakes" attitude more contributed to deep depression and resulted in psychological and academic breakdown that I'm only now recovering from in my late 20s. So much for that PhD by age 25, Mom!
Sometimes a mistake is a mistake, not an indication of personal failure or lack of understanding.
posted by Anonymous at 8:59 PM on November 12, 2013
I'm quite good at math (I was taking differential equations in high school) but the "no mistakes" attitude my mom had wasn't responsible for that. The "no mistakes" attitude more contributed to deep depression and resulted in psychological and academic breakdown that I'm only now recovering from in my late 20s. So much for that PhD by age 25, Mom!
Sometimes a mistake is a mistake, not an indication of personal failure or lack of understanding.
posted by Anonymous at 8:59 PM on November 12, 2013
Hear, hear, divined by radio. I suspect I have dyscalculia and that it runs in my gene pool (my mom and aunt are also terrible at math), though not super bad by comparison, I suppose. But I got tutored all the fucking time and would forget everything they told me by the next day. I can't picture numbers in my head or do math in my head, even if it's easy math. I could do geometry and quadratic equations, oddly enough, but everything else I had to do involving algebra, I was terrible at. Too bad almost every year of school was algebra, algebra, algebra. Could I do any now? HAHAHAHAHAHAHAHAH NOOOOOOOOOO. I am grateful that my college didn't have rigid math requirements for my majors at the time because fuck if I know how I would have gotten out of college when after years and years of algebra, I still didn't get it. It was so nice to finally not have to try to be "well rounded." I don't round, okay? I'm a deflated basketball, a "smart kid" who was so clearly DUMB in this subject and nobody could understand why I was so dumb at it.
Thing is, you learn after years and years of literal bean counting in first grade, after crying in front of everyone in fourth grade because you don't understand what "two fifths of the whole" is in fractions, after being moved into dumber and dumber math classes with the school delinquents, that you're DUMB and it's genetic and you can't help it, because you have so many years of failure informing you that you're not good at this. Telling me how fucking smart I was didn't help this one bit.
posted by jenfullmoon at 10:49 PM on November 12, 2013 [2 favorites]
Thing is, you learn after years and years of literal bean counting in first grade, after crying in front of everyone in fourth grade because you don't understand what "two fifths of the whole" is in fractions, after being moved into dumber and dumber math classes with the school delinquents, that you're DUMB and it's genetic and you can't help it, because you have so many years of failure informing you that you're not good at this. Telling me how fucking smart I was didn't help this one bit.
posted by jenfullmoon at 10:49 PM on November 12, 2013 [2 favorites]
A machine can't really do the thinking that shows that a particular calculation is an important one to carry out, though. And it can't do the theorem proving and problem finding that we can. That's the real math. Kids are going to get a better look at the deeper aspects of math in the coming years in the US.
That would be excellent. My fingers are crossed that we can finally bury the phrase 'word problem' and stop acting like they're some separate branch of mathematics.
posted by RonButNotStupid at 4:55 AM on November 13, 2013 [1 favorite]
That would be excellent. My fingers are crossed that we can finally bury the phrase 'word problem' and stop acting like they're some separate branch of mathematics.
posted by RonButNotStupid at 4:55 AM on November 13, 2013 [1 favorite]
"Temporarily embarrassed mathematicians"
posted by This, of course, alludes to you at 6:39 AM on November 13, 2013
posted by This, of course, alludes to you at 6:39 AM on November 13, 2013
lupus_yonderboy,
Thanks for your inquiry. I think it has a lot to do with memorization. I can remember things that I memorize, but I suck at specifics. I cannot, for the life of me, remember a line from a movie or book or song exactly. I think memorizing multiplication tables kinda screwed me up because I draw on that memorization, which I suck at, rather than reason or see numbers out. But when I do double intergrals, there isn't a lot of memorization going on. Unless of course I'm using a formula to integrate, which I then have to write out before plugging it in.
I allways screw up signs too, when I integrate exponential functions. That's why tabular integration is a lot easier for me.
posted by MisantropicPainforest at 6:52 AM on November 13, 2013
Thanks for your inquiry. I think it has a lot to do with memorization. I can remember things that I memorize, but I suck at specifics. I cannot, for the life of me, remember a line from a movie or book or song exactly. I think memorizing multiplication tables kinda screwed me up because I draw on that memorization, which I suck at, rather than reason or see numbers out. But when I do double intergrals, there isn't a lot of memorization going on. Unless of course I'm using a formula to integrate, which I then have to write out before plugging it in.
I allways screw up signs too, when I integrate exponential functions. That's why tabular integration is a lot easier for me.
posted by MisantropicPainforest at 6:52 AM on November 13, 2013
Slightly tangentially, I'd like to see a study for what happens if you try to convince people that they're in the talented group. "B+? That's one of the highest results! You were unlucky on Q7, though". I predict they'll work harder if they don't think of it as swimming against the tide.
Also, I'm someone who can do all of the bill splitting, chequebook balancing, vector calculus and sundry calculations no problem, but I can't do timezones to save myself. It's not a terrible admission of defeat to just not be good at something! My intuition, based on having met hundreds of mathmos, physicists, and fellow travellers, is that the qualities that get you through a highly technical degree are not the qualities which help you to teach someone who's having trouble with the a's and b's and x's at school. It's obvious to you, and it may not be obvious to you why it's not obvious to someone else.
If that's not too obvious.
posted by Wrinkled Stumpskin at 12:29 PM on November 14, 2013
Also, I'm someone who can do all of the bill splitting, chequebook balancing, vector calculus and sundry calculations no problem, but I can't do timezones to save myself. It's not a terrible admission of defeat to just not be good at something! My intuition, based on having met hundreds of mathmos, physicists, and fellow travellers, is that the qualities that get you through a highly technical degree are not the qualities which help you to teach someone who's having trouble with the a's and b's and x's at school. It's obvious to you, and it may not be obvious to you why it's not obvious to someone else.
If that's not too obvious.
posted by Wrinkled Stumpskin at 12:29 PM on November 14, 2013
An engineer might say, "Yeah, I don't really like a lot of poetry," but they wouldn't say, "I've never been able to read."
Maybe not an engineer. But lots of working- and lower-middle class-folks.
If someone cannot read they are not any kind of middle class, "lower" or not.
posted by atrazine at 3:58 AM on November 18, 2013
Maybe not an engineer. But lots of working- and lower-middle class-folks.
If someone cannot read they are not any kind of middle class, "lower" or not.
posted by atrazine at 3:58 AM on November 18, 2013
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posted by The Underpants Monster at 8:08 AM on November 12, 2013