October 18, 2012 9:32 PM Subscribe

"Milgram and Bishop are opposed to reforms of mathematics teaching and support the continuation of a model in which students learn mathematics without engaging in realistic problems or discussing mathematical methods. They are, of course, entitled to this opinion, and there has been an ongoing, spirited academic debate about mathematics learning for a number of years. But Milgram and Bishop have gone beyond the bounds of reasoned discourse in a campaign to systematically suppress empirical evidence that contradicts their stance. Academic disagreement is an inevitable consequence of academic freedom, and I welcome it. However, responsible disagreement and academic bullying are not the same thing. Milgram and Bishop have engaged in a range of tactics to discredit me and damage my work which I have now decided to make public." Jo Boaler, professor of mathematics education at Stanford, accuses two mathematicians, one her colleague of Stanford, of unethical attempts to discredit her research, which supports "active engagement" with mathematics (aka "reform math") over the more traditional "practicing procedures" approach.
posted by escabeche (119 comments total)
22 users marked this as a favorite

This is interesting, thanks for posting.

posted by Blazecock Pileon at 9:49 PM on October 18, 2012

posted by Blazecock Pileon at 9:49 PM on October 18, 2012

What a frustrating situation this must be. The evidence looks pretty damning for Milgram and Bishop, and are pretty in-line with what my female colleagues in academia have told me. Sadly, this seems to be the norm.

posted by spiderskull at 9:55 PM on October 18, 2012

posted by spiderskull at 9:55 PM on October 18, 2012

It is fucking unbelievable how loathsome and hatefully offensive some men find women.

posted by Pope Guilty at 9:59 PM on October 18, 2012 [5 favorites]

I'm not completely well informed on the entirety of the situation, but I will tell you that the Everyday Math curriculum is bullshit and nonsense. I have tutored plenty of kids who were straight up failing under Everyday Math who succeeded enormously as soon as they were given more traditional materials. I have a friend right now who is working hard to get her school district to change curricula after her daughter had the same experience. I mean, the method they teach for long division doesn't even get the right answer half the time.

posted by KathrynT at 10:01 PM on October 18, 2012 [6 favorites]

posted by KathrynT at 10:01 PM on October 18, 2012 [6 favorites]

Okay I looked at one of Milgram's papers where he critiques one of Jo Boaler's exam problems. It is full of nerd rage. This sucks.

posted by polymodus at 10:03 PM on October 18, 2012

posted by polymodus at 10:03 PM on October 18, 2012

Milgram says "that since it was only those issues (it was too easy to do a Google search on some of the quotes in the paper and thereby identify the schools involved) that prevented publication, his critique was in fact peer-reviewed, just not published."

No, that's not how it works. If your paper is rejected for publication, you can't say it was 'peer-reviewed'. I can't even imagine how many science papers are rejected for publication for supposedly 'just this or that one issue'. It doesn't give you the right to treat it like a peer respected paper, and certainly not the right to use the rejected paper to tear down a colleague.

KathrynT, if your experience is common, rather than an exception, then it should be fairly simple for a researcher to do a study showing this. And if the study is done well, it would actually get published.

I'm not saying that parents, teachers, or tutors can't make their own evaluations based on their experience, just that researchers have to jump higher hurdles, and Milgram clearly hasn't.

posted by eye of newt at 10:08 PM on October 18, 2012 [3 favorites]

No, that's not how it works. If your paper is rejected for publication, you can't say it was 'peer-reviewed'. I can't even imagine how many science papers are rejected for publication for supposedly 'just this or that one issue'. It doesn't give you the right to treat it like a peer respected paper, and certainly not the right to use the rejected paper to tear down a colleague.

KathrynT, if your experience is common, rather than an exception, then it should be fairly simple for a researcher to do a study showing this. And if the study is done well, it would actually get published.

I'm not saying that parents, teachers, or tutors can't make their own evaluations based on their experience, just that researchers have to jump higher hurdles, and Milgram clearly hasn't.

posted by eye of newt at 10:08 PM on October 18, 2012 [3 favorites]

This is pretty damning, if true. And the allegations are very specific and seem like the sort of things that could in principle be conclusively investigated (and some of them are matters of public record) so Dr. Boaler would be crazy to be just making this stuff up. It's a one-sided account, but I have a hard time seeing how the kinds of behaviors committed by Milgram and Bishop could be OK under any context.

If even half of this stuff is true then these guys need to be thrown out on their ear from wherever it is they are working. Nasty stuff.

Escabeche, or anyone else here who is a professional mathematician: would you care to venture an opinion as to the likelihood that these allegations will stand up, and the consequences for Milgram and Bishop if so?

posted by Scientist at 10:11 PM on October 18, 2012

If even half of this stuff is true then these guys need to be thrown out on their ear from wherever it is they are working. Nasty stuff.

Escabeche, or anyone else here who is a professional mathematician: would you care to venture an opinion as to the likelihood that these allegations will stand up, and the consequences for Milgram and Bishop if so?

posted by Scientist at 10:11 PM on October 18, 2012

Wait, they teach a method which,

posted by Justinian at 10:12 PM on October 18, 2012 [4 favorites]

I found this article about "Everyday Math" (it seems to be a kind of reform maths). I think the key sentence is "it is not an easy curriculum to teach and teachers must undergo extensive training to teach it successfully" - it sounds like it could be really effective when taught well, but much worse than a more traditional approach when taught badly.

posted by A Thousand Baited Hooks at 10:14 PM on October 18, 2012 [2 favorites]

I am not a math teacher, but I am an educator and former teacher and have done instructional design work. Don't damn an entire method based on a handful of outcomes: getting fidelity to the model is HUGELY challenging due to all kinds of variables: scheduling, resources, support, training, etc. Districts have been known to pay millions for an intervention with solid empirical evidence supporting its efficacy only to implement it incorrectly.

posted by smirkette at 10:26 PM on October 18, 2012 [6 favorites]

posted by smirkette at 10:26 PM on October 18, 2012 [6 favorites]

I know that in Seattle, when the Everyday Math curriculum was adopted, math fluency in schoolchildren PLUMMETED. It got to the point where UW professors were trying to figure out why they suddenly had freshmen in their classes who couldn't do basic math.

I'm not saying Milgram and Bishop are good people, and they certainly do not appear to be motivated by high-minded scholarship. Before this post, I had never heard of either them or Dr. Boaler. I am a big believer in educational reform, and if something truly works better for children, I'm all for it. But Everyday Math is a disaster.

On preview:

*Wait, they teach a method which, when executed properly gives the wrong answer?*

The method of division in ED is called "partial quotients," it's based on rounding and getting an answer that is close enough. So, yeah.

posted by KathrynT at 10:27 PM on October 18, 2012 [9 favorites]

I'm not saying Milgram and Bishop are good people, and they certainly do not appear to be motivated by high-minded scholarship. Before this post, I had never heard of either them or Dr. Boaler. I am a big believer in educational reform, and if something truly works better for children, I'm all for it. But Everyday Math is a disaster.

On preview:

The method of division in ED is called "partial quotients," it's based on rounding and getting an answer that is close enough. So, yeah.

posted by KathrynT at 10:27 PM on October 18, 2012 [9 favorites]

Is the unreliable long division method the partial quotients method? (an explanation on google docs.) That actually seems like a really straightforward and intuitive way to do long division, and I don't see how it could be inaccurate if you get the steps right. I never learned how to do long division at school, because I could never work out how the steps in the procedure related to the end result, but if they'd taught the partial quotients method I bet I would have been fine.

posted by A Thousand Baited Hooks at 10:27 PM on October 18, 2012 [2 favorites]

posted by A Thousand Baited Hooks at 10:27 PM on October 18, 2012 [2 favorites]

Well, because it gives you a whole number answer with a remainder, not an exact decimal answer. That's what I mean by "close enough." But in actual math applications, a whole number + a remainder is often useless.

posted by KathrynT at 10:30 PM on October 18, 2012 [1 favorite]

posted by KathrynT at 10:30 PM on October 18, 2012 [1 favorite]

I thought this was awesome, from the linked PDF:

And despite getting only one side of the story here, it all dovetails so neatly with what I've seen and heard (from friends and family working in academia) about old entrenched male professors at universities that it would not surprised me at all if Boaler were found to be 100% true.

posted by barnacles at 10:32 PM on October 18, 2012 [1 favorite]

I don't know anything about "Everyday Math", and I've been out of a school math classroom for a long time now, but if this sort of pedagogy can get kids interested and open to math and to the fact that math can be learned and shouldn't be shied away from, well, that's just fantastic. More power to them.In the year 3 questionnaires, we offered the statement “Anyone can be really good at math if they try” 84% of Railside students agreed with this, compared with 52% of students in the traditional classes.

And despite getting only one side of the story here, it all dovetails so neatly with what I've seen and heard (from friends and family working in academia) about old entrenched male professors at universities that it would not surprised me at all if Boaler were found to be 100% true.

posted by barnacles at 10:32 PM on October 18, 2012 [1 favorite]

My email to her-

As an educator I cannot thank you enough for your research. You have given me another powerful piece of ammunition in my growing arsenal w/r/t the outdated and ineffective "Prussian" school system. Keeping children of differing skills (but of the same age) locked in concrete cages for 8 hours a day, only to send them away with homework that only the most intelligent and dedicated can possibly hope to competently complete (let alone excel) is a recipe for utter failure.

If I could choose one government program to abolish, it would be my own job- public schools have become nothing more than holding pens for the independent, and day care centers for all but the most obedient.

Thank you from the bottom of my (broken) heart.

posted by dickfitz2 at 10:32 PM on October 18, 2012 [3 favorites]

As an educator I cannot thank you enough for your research. You have given me another powerful piece of ammunition in my growing arsenal w/r/t the outdated and ineffective "Prussian" school system. Keeping children of differing skills (but of the same age) locked in concrete cages for 8 hours a day, only to send them away with homework that only the most intelligent and dedicated can possibly hope to competently complete (let alone excel) is a recipe for utter failure.

If I could choose one government program to abolish, it would be my own job- public schools have become nothing more than holding pens for the independent, and day care centers for all but the most obedient.

Thank you from the bottom of my (broken) heart.

posted by dickfitz2 at 10:32 PM on October 18, 2012 [3 favorites]

KathrynT: "*Well, because it gives you a whole number answer with a remainder, not an exact decimal answer. That's what I mean by "close enough." But in actual math applications, a whole number + a remainder is often useless.*"

From some quick googles, it looks like this method is for showing kids (this curriculum is apparently aimed at K-6!) that long division isn't arcane magick, that there are some simple steps involved in the process, and that it's not rocket science and can be learned. I mean, "actual math applications"? We're talking about elementary school children, here! If they get comfortable with long division before going into middle school, I think that leaves plenty of time for mastering exact decimal answer division and math down the road.

posted by barnacles at 10:36 PM on October 18, 2012 [1 favorite]

From some quick googles, it looks like this method is for showing kids (this curriculum is apparently aimed at K-6!) that long division isn't arcane magick, that there are some simple steps involved in the process, and that it's not rocket science and can be learned. I mean, "actual math applications"? We're talking about elementary school children, here! If they get comfortable with long division before going into middle school, I think that leaves plenty of time for mastering exact decimal answer division and math down the road.

posted by barnacles at 10:36 PM on October 18, 2012 [1 favorite]

Yes the UW's Cliff Mass is a very vocal critic of the Discovery method here in Seattle: Here is an example post from his blog. I have kids that have gone through the curriculum and their experience tends to validate his views. This won't help matters.

posted by Mei's lost sandal at 10:38 PM on October 18, 2012

posted by Mei's lost sandal at 10:38 PM on October 18, 2012

KathrynT: "*Well, because it gives you a whole number answer with a remainder, not an exact decimal answer. That's what I mean by "close enough." But in actual math applications, a whole number + a remainder is often useless.*"

A whole number answer with a remainder is an exact answer, it's not always a useful format (although it can be trivially converted to a whole number and a fraction), but it's not an approximation. It's even fairly straightforward (if you understand place value) to calculate a few decimal places with the same method if you need them (just multiply by a power of ten before you start, then divide by the same power at the end).

posted by Proofs and Refutations at 10:41 PM on October 18, 2012 [10 favorites]

A whole number answer with a remainder is an exact answer, it's not always a useful format (although it can be trivially converted to a whole number and a fraction), but it's not an approximation. It's even fairly straightforward (if you understand place value) to calculate a few decimal places with the same method if you need them (just multiply by a power of ten before you start, then divide by the same power at the end).

posted by Proofs and Refutations at 10:41 PM on October 18, 2012 [10 favorites]

Our son is in Grade 5, and over the years some of the math homework he has brought home has been incomprehensible - it's been really difficult to figure out what the point is of some of the exercises.

Anyway, I think someone said it upthread already, but not all elementary school teachers can teach math, and it's a real problem. And there is also a laisse-faire attitude toward math - it's hard! Not everyone can do it! We've run out of time today!

But what math needs is a mastery approach to teaching and learning, because really, anything less than 100% means there is a broken cog someplace, or faulty logic that will affect the student's numeracy for life.

posted by KokuRyu at 10:42 PM on October 18, 2012 [9 favorites]

Another explanation of the partial quotient method. I was predisposed to dislike it, because I learned math by traditional methods, got it, and liked it. But this partial quotient method makes sense to me—and it's basically what I do if I have to do a long division problem in my head: test a series of approximations until I can zero in on the answer. (Give me paper and pencil, and I'll do long division the way I was taught in school.) Also, I don't see that remainders are a problem with the partial quotient method; I remember that we were taught to write down the remainders from traditional long division when we first learned it, too. Later on we got to fractions and decimals.

This is not to endorse the whole Everyday Math curriculum, which I'm not familiar with. Just saying that the partial quotient method of division doesn't seem unreasonable. And it apparently assumes that students have learned to do quick, accurate multiplication in their heads, so I take it they're not throwing out the multiplication tables with the bathwater. (The article linked by A Thousand Baited Hooks confirms that Everyday Math students learn multiplication tables, although they do so later than in the traditional curriculum.)

posted by Orinda at 10:50 PM on October 18, 2012 [6 favorites]

This is not to endorse the whole Everyday Math curriculum, which I'm not familiar with. Just saying that the partial quotient method of division doesn't seem unreasonable. And it apparently assumes that students have learned to do quick, accurate multiplication in their heads, so I take it they're not throwing out the multiplication tables with the bathwater. (The article linked by A Thousand Baited Hooks confirms that Everyday Math students learn multiplication tables, although they do so later than in the traditional curriculum.)

posted by Orinda at 10:50 PM on October 18, 2012 [6 favorites]

The conflict is pretty transparent, really. On the one hand you have the traditional teaching methods. They work well because mainstream society privileges that system of learning. You weed out students and the ones that survive, let alone thrive, become our future STEM labor force. This is the kind of math that parents want, because it appeals to things like job security and monetary wealth—and yet most of these parents don't actually have half a clue about what it takes to do mathematics. So there is an implicit ideology behind criticism of New Math, and the viciousness of people like Milgram and Bishop stems from being threatened by the potential revolution not just in math but in the underpinning values of society.

I hate to sound like a crank, but the "times table", both the object-in-practice and the very phrase, are actually representative of why our society is doing it wrong. Symbolically it is completely against the ideals of critical thinking and inquiry. The best learning happens through curiosity and making mistakes, and authoritarianism in schools—standardized testing as a simple example—whittle away at that. Yes, the Math Ed community has their heads in the clouds, and the packages they've developed and disseminated thus far like "Discovery" or "Everyday Math" are clearly premature and disruptive in a bad way, screwing over the competence of a whole sequence of cohorts. But philosophically, they're in the right place. I don't work in this area, but after googling the news articles, I find myself rooting for them, not the traditionalists.

posted by polymodus at 11:01 PM on October 18, 2012 [3 favorites]

I hate to sound like a crank, but the "times table", both the object-in-practice and the very phrase, are actually representative of why our society is doing it wrong. Symbolically it is completely against the ideals of critical thinking and inquiry. The best learning happens through curiosity and making mistakes, and authoritarianism in schools—standardized testing as a simple example—whittle away at that. Yes, the Math Ed community has their heads in the clouds, and the packages they've developed and disseminated thus far like "Discovery" or "Everyday Math" are clearly premature and disruptive in a bad way, screwing over the competence of a whole sequence of cohorts. But philosophically, they're in the right place. I don't work in this area, but after googling the news articles, I find myself rooting for them, not the traditionalists.

posted by polymodus at 11:01 PM on October 18, 2012 [3 favorites]

> Well, because it gives you a whole number answer with a remainder, not an exact decimal answer.

I have a degree in mathematics - and you have it exactly backward. The whole number with the remainder is the exact, correct answer to the division of whole numbers - the (finite) decimal answer is an approximation, unless it happens that the divisor is only divisible by powers of two and five (a rare case), in which case there is a finite decimal that exactly represents the answer.

Reading about the "partial quotient" method is interesting. Given the universal ubiquity of calculators, I think it's a better way to go, because it gives you an understanding of how the process works, but better, gives you good fast estimates.

I mean, I learned how to find square roots using the algorithm here and I thought it was the coolest thing. As an adult who actually uses square roots in his day-to-day life, I would always use "successive approximations" from that same page (though I might called it "Newton's method") - at least partly because I never deeply understood that algorithm (I worked out once why it works but it's fiddly and it never sank in), but mainly because successive approximations will give me a good answer almost instantly so I can blurt it out and impress my friends at parties.

posted by lupus_yonderboy at 11:05 PM on October 18, 2012 [12 favorites]

I have a degree in mathematics - and you have it exactly backward. The whole number with the remainder is the exact, correct answer to the division of whole numbers - the (finite) decimal answer is an approximation, unless it happens that the divisor is only divisible by powers of two and five (a rare case), in which case there is a finite decimal that exactly represents the answer.

Reading about the "partial quotient" method is interesting. Given the universal ubiquity of calculators, I think it's a better way to go, because it gives you an understanding of how the process works, but better, gives you good fast estimates.

I mean, I learned how to find square roots using the algorithm here and I thought it was the coolest thing. As an adult who actually uses square roots in his day-to-day life, I would always use "successive approximations" from that same page (though I might called it "Newton's method") - at least partly because I never deeply understood that algorithm (I worked out once why it works but it's fiddly and it never sank in), but mainly because successive approximations will give me a good answer almost instantly so I can blurt it out and impress my friends at parties.

posted by lupus_yonderboy at 11:05 PM on October 18, 2012 [12 favorites]

This is not correct. Mathematicians tend to despise decimal answers precisely because they're almost never exact. A whole number answer with a remainder is always exact (when dividing rational numbers), and -- with conversion to a mixed or improper fraction -- far more suitable for advanced mathematics than a decimal.

I was not familiar with the partial quotient method until now, but a quick glance at it makes it very clear to me that it is extremely simple and will always give a correct answer when executed correctly. Whether it is pedagogically better that the traditional table long division method, the more traditional galley division method, or the far more traditional abacus/counting board method is something I don't have data to evaluate and not something that I particularly care about, but partial quotients is mathematically correct.

I think the more reasonable explanation for "crappy method" reports is a phenomenon that I see all the time: parents with poor understanding of mathematics are uncomfortable with methods that they did not learn in school themselves, regardless of their quality, because they lack the mathematical sophistication necessary to evaluate the correctness of their children's work in an unfamiliar method when asked for help.

Less common among parents, but there is another phenomenon that I see that (mostly among mathematicians) that explains another "crappy method" position: that people who have attained high mathematical achievement find it difficult to imagine how students could have difficulty with "simple" problems that are easy for the mathematician, and blame a change in method taught as the most visible factor rather than taking into account that different people are different -- in other words, mathematicians tend to forget that they are the 1% (of top standardized test takers) and that the other 99% of the population have different experiences, interests, and struggles.

posted by yeolcoatl at 11:07 PM on October 18, 2012 [11 favorites]

I really need more information to evaluate this.

I can't figure out the two sides and I have opinions.

As a child I was enrolled in a number of radically progressive private schools. They tried all sorts of weird methods. One of which I distinctly remember hating, and remember the administration distinctly loving. It was a sort of applied concrete mathematics. I forget the name of the series. But they always involved estimating the depth of staircases, the path of an ant, obscure trigonometry exercises involving theoretical labyrinths, and the precise height necessary to build a fence to stop horses from jumping. So, facing such irrelevant nonsense I decided I had not the mind for mathematics and embraced liberal arts. And so I graduated college without taking a single mathematics course. Hooray, The Evergreen State College.

But then, unemployed and with a few months left on Amazon Prime, I learned how to study proofs thanks to a book by Velleman, and then I learned the common useful formulas thanks to the two Engineering Mathematics books. Apparently, what I had hated all this time was arithmetic, and those super-particular applied math problems.

This leaves me confused. I don't know which side I am on. And who is winning? Is the new math abstract or is the old math? There is a change of guard but I can't tell from their big tall fuzzy hats.

My personal opinion is that the harder abstract mathematics is easier to understand than any dishonest attempt at making mathematics appear immediately useful. You might be able to trick interest out of the student, but in the end you are still having them solve King's Quest puzzles where they don't necessarily know what will work because they don't understand how anything works. And as we know, at least us Adventure Game Nerds, they'll eventually stop buying it.

This comment might label me a conservative or a radical. I'll leave it up to the thread to decide. All I know is that I had GPA sundering trouble with the "real world" problems. But when I approached math as a silly nerdy game and assume it had zero application to the real world, ala Chess or Magic: The Gathering, then I could very readily swallow the nonsense and try my hand at it. And now I am studying tessellation in my free time and boring my friends about Wang Tiles. So, wait, maybe this isn't a good story. Maybe we shouldn't teach mathematics at all, after all it is ruinous to one's social life. Americans have less friends now than ever; an engaging mathematics course might be the straw that breaks our social backs.

posted by TwelveTwo at 11:10 PM on October 18, 2012 [7 favorites]

I can't figure out the two sides and I have opinions.

As a child I was enrolled in a number of radically progressive private schools. They tried all sorts of weird methods. One of which I distinctly remember hating, and remember the administration distinctly loving. It was a sort of applied concrete mathematics. I forget the name of the series. But they always involved estimating the depth of staircases, the path of an ant, obscure trigonometry exercises involving theoretical labyrinths, and the precise height necessary to build a fence to stop horses from jumping. So, facing such irrelevant nonsense I decided I had not the mind for mathematics and embraced liberal arts. And so I graduated college without taking a single mathematics course. Hooray, The Evergreen State College.

But then, unemployed and with a few months left on Amazon Prime, I learned how to study proofs thanks to a book by Velleman, and then I learned the common useful formulas thanks to the two Engineering Mathematics books. Apparently, what I had hated all this time was arithmetic, and those super-particular applied math problems.

This leaves me confused. I don't know which side I am on. And who is winning? Is the new math abstract or is the old math? There is a change of guard but I can't tell from their big tall fuzzy hats.

My personal opinion is that the harder abstract mathematics is easier to understand than any dishonest attempt at making mathematics appear immediately useful. You might be able to trick interest out of the student, but in the end you are still having them solve King's Quest puzzles where they don't necessarily know what will work because they don't understand how anything works. And as we know, at least us Adventure Game Nerds, they'll eventually stop buying it.

This comment might label me a conservative or a radical. I'll leave it up to the thread to decide. All I know is that I had GPA sundering trouble with the "real world" problems. But when I approached math as a silly nerdy game and assume it had zero application to the real world, ala Chess or Magic: The Gathering, then I could very readily swallow the nonsense and try my hand at it. And now I am studying tessellation in my free time and boring my friends about Wang Tiles. So, wait, maybe this isn't a good story. Maybe we shouldn't teach mathematics at all, after all it is ruinous to one's social life. Americans have less friends now than ever; an engaging mathematics course might be the straw that breaks our social backs.

posted by TwelveTwo at 11:10 PM on October 18, 2012 [7 favorites]

Is mysogyny really a motivating factor? I didn't see it mentioned in her report, unless I missed it.

posted by Brocktoon at 11:14 PM on October 18, 2012 [1 favorite]

posted by Brocktoon at 11:14 PM on October 18, 2012 [1 favorite]

The main problem we had as parents trying to help with kid's Discovery homework was that there was rarely an example problem or method spelled out for a particular task that you could look to for guidance. It was pretty frustrating and I am not really a slouch at math.

posted by Mei's lost sandal at 11:16 PM on October 18, 2012

posted by Mei's lost sandal at 11:16 PM on October 18, 2012

I tend to agree with you, although I see nothing wrong with STEM or wanting job security for one's kids. But yeah, this is a societal problem. For some reason we want to warehouse our kids in big boxes during the day, under the supervision of "experts" who teach a curriculum determined by the state. Some students thrive, some merely survive, and others do not.

If you really wanted to improve education outcomes, family would be more involved in education. As it is, children who come from homes that have books (hard to believe some do not) tend to do perform better at school. Students whose parents are actively engaged in learning at home (our son went from 60% to 100% in math) do better.

Critical thinking and inquiry have to be cultural norms - you can't just leave it up to the state or the education system to do it for parents.

posted by KokuRyu at 11:18 PM on October 18, 2012 [4 favorites]

I hate to break it to you, but all math curricula start with arithmetic. If you like proofs, you're out of luck. Students very rarely get into proofs before the upper undergraduate level. You would probably enjoy reading Lockhart's Lament.

Neither. "New Math," the movement from the 60s, was very abstract, but it is no longer taught because it doesn't really work. The failure of "New Math" is pretty much the only thing that both sides of the math wars agree on. It turns out that 4 year olds are equipped to count to 10, but not really well equipped for the set theoretic foundations of arithmetic until they get quite a bit older. Modern traditional and modern reform movements all start with arithmetic and applied problems, although reform movements tend to lean more applied, more theoretical, and less arithmetical than traditional.

So which side are you on? I don't think you have a side.

posted by yeolcoatl at 11:22 PM on October 18, 2012 [1 favorite]

Well, if I don't have a side, at least I have a point?

posted by TwelveTwo at 11:24 PM on October 18, 2012 [8 favorites]

posted by TwelveTwo at 11:24 PM on October 18, 2012 [8 favorites]

Until you learn that the EDM kids are taught to disregard (or in many cases

Certainly, none of the kids I tutored could multiply 3x4 or 8x7 in their heads until I gave them flash cards, and none of them had ever seen multiplication tables in their memory.

I suppose EDM can be a great curriculum if the teacher really takes the time to make sure that everyone understands what's going on. But my experience is that because of the pressure of collaborative group learning, my students didn't want to hold the other kids back or look stupid in front of their peers, so they nodded their heads and pretended to understand stuff that they really didn't. Because there aren't a lot of practice problems, there was no easily available check to show that they didn't have mastery of the concepts being discussed, and so they just got farther and farther behind and scared.

Let me say again -- I'm not saying that all reform math is awful. I'm just speaking to my experience with this specific curriculum, which I find to be horribly deficient.

posted by KathrynT at 11:41 PM on October 18, 2012 [5 favorites]

I read all the links, both papers, the commentary. Here's what I think is really going on. The female "protagonist" has adopted the notion of peer and group teaching methods, to tackle math, a subject which students learn to hate, because math is used as a talent filter without telling them, or their parents.

The approach Boaler advocates, which is not new, proposes to socialize math, and essentially force the math-inclined students to convey their teenage insight to their peers who probably wouldn't talk to them elsewhere. Peer teaching is actually a very good method to solve any mental problems with, and it helps in troubled schools and crowded classrooms.

The problem with Boaler's public relations is that she is claiming two successes, one claming better test scores, essentially beating the old school at its own game. Anyone should be wary here, because it smells like magic to people who know they get math and that everyone else doesn't. Moreover, these tests are also a traditional method, so Boaler is having her cake and eating it too, while touting the "equity" of the results no less. This is all done by her studies based on anonymous schools, requiring a trust that wasn't earned.

Conclusion: Boaler is sitting people down in bean bag chairs and having them take turns explaining the same problem and never feeling the stress that math requires to become an engineer. On the other hand, the "traditionalists" will always be succeeding at their straightforward approach, because math never was for everyone, and it exceeds most people's ability to major in it, and thus the "traditionalists" are holding the trump card of knowing that math ability to succeed is mostly genetic within a moving target zone. It will always be a talent. Even if you make it easier to learn the jobs that require any of it will demand more of it to get the best candidate.

The win-win here is to acknowledge that dental hygienists don't need pre-calculus unless we're looking for a way to restrict program sizes, and that most students probably shouldn't take courses that teach them to hate the subject matter when they are very young, because they may hate any job that requires it.

posted by Brian B. at 11:43 PM on October 18, 2012 [4 favorites]

The approach Boaler advocates, which is not new, proposes to socialize math, and essentially force the math-inclined students to convey their teenage insight to their peers who probably wouldn't talk to them elsewhere. Peer teaching is actually a very good method to solve any mental problems with, and it helps in troubled schools and crowded classrooms.

The problem with Boaler's public relations is that she is claiming two successes, one claming better test scores, essentially beating the old school at its own game. Anyone should be wary here, because it smells like magic to people who know they get math and that everyone else doesn't. Moreover, these tests are also a traditional method, so Boaler is having her cake and eating it too, while touting the "equity" of the results no less. This is all done by her studies based on anonymous schools, requiring a trust that wasn't earned.

Conclusion: Boaler is sitting people down in bean bag chairs and having them take turns explaining the same problem and never feeling the stress that math requires to become an engineer. On the other hand, the "traditionalists" will always be succeeding at their straightforward approach, because math never was for everyone, and it exceeds most people's ability to major in it, and thus the "traditionalists" are holding the trump card of knowing that math ability to succeed is mostly genetic within a moving target zone. It will always be a talent. Even if you make it easier to learn the jobs that require any of it will demand more of it to get the best candidate.

The win-win here is to acknowledge that dental hygienists don't need pre-calculus unless we're looking for a way to restrict program sizes, and that most students probably shouldn't take courses that teach them to hate the subject matter when they are very young, because they may hate any job that requires it.

posted by Brian B. at 11:43 PM on October 18, 2012 [4 favorites]

I do not doubt that this is true, but I would suggest that this is a failing of the teacher or teachers implementing the curriculum, not necessarily a failing inherent to the curriculum. As mentioned up thread, elementary teachers usually don't like mathematics very much and don't know it very well. The problem that you're describing -- teaching students that the remainder is unimportant -- could happen in any curriculum if the teacher doesn't understand remainder very well. Now it's certainly possible that the book explicitly says that remainders are unimportant but I don't know the level of the student's you're talking about or their place relative to the study of fractions, so I can't really comment on that. It may be that there are situations very early in learning division where that's the right choice. Some traditional curricula teach remainders before fractions, but a remainder can't mean anything until after fractions are learned, so remainder does get neglected for a while even in traditional curricula.

As a counter example, one of my (very traditional style) elementary teachers tried to teach me that parentheses were just decorative and had no effect on order of operations until my father put a stop to that, but nobody would say that this was the fault of the curriculum. Rather it was a case of a teacher who did not understand parentheses mis-teaching the curriculum.

posted by yeolcoatl at 12:05 AM on October 19, 2012 [1 favorite]

We thought about enrolling our son in the New Tech High School which opened here, but researched the math aspects, including Project Based Learning, which Boaler advocates. Decided not to. Last year, New Tech received:

F (academic probation) — New Tech High School.

which was based on the math scores.

One factor to note was that many teachers in the school district were RIF'd, including some trained in the New Tech technique. The current math teacher is not the one originally slotted for the spot. So was it a technique issue or a resource issue? My opinion is that it doesn't matter, when the push is on to make students commodities, it's the students who get remaindered.

posted by dragonsi55 at 12:08 AM on October 19, 2012 [1 favorite]

F (academic probation) — New Tech High School.

which was based on the math scores.

One factor to note was that many teachers in the school district were RIF'd, including some trained in the New Tech technique. The current math teacher is not the one originally slotted for the spot. So was it a technique issue or a resource issue? My opinion is that it doesn't matter, when the push is on to make students commodities, it's the students who get remaindered.

posted by dragonsi55 at 12:08 AM on October 19, 2012 [1 favorite]

There is something repulsive about supporting someone for being in the right place philosophically even if they screw over the competence of entire cohorts, as if the latter is somehow less important than the former.

posted by atrazine at 1:15 AM on October 19, 2012 [5 favorites]

It doesn't matter how good a curriculum is when taught correctly, if the majority of teachers can't use it that way. (I can't say whether or not this is the case with "Everyday Math".)

I have a STEM education and work in a technology field and there have been almost no times in my career when I've needed precalculus, let alone more exotic things like Green's functions. I can't imagine not having learned these things (and for that matter, I enjoyed learning them). But, yeah.

As someone who was successful with the traditional techniques, I don't understand this at all. With what would you replace the "times table"? Or are you asserting that with modern tools there's no need to be able to perform the mechanics of arithmetic provided one understands the concepts?

posted by Slothrup at 3:33 AM on October 19, 2012 [1 favorite]

This reminds me of the snark videos and other bashing of Khan Academy. I really wish I had a better understanding of the different positions in play here.

posted by humanfont at 3:43 AM on October 19, 2012

posted by humanfont at 3:43 AM on October 19, 2012

Dental hygienists wouldn't have a job without calculus.

posted by atrazine at 3:46 AM on October 19, 2012 [14 favorites]

Well, technically it would still be 'peer-reviewed'. Your peers reviewed it, and declared that it sucked.No, that's not how it works. If your paper is rejected for publication, you can't say it was 'peer-reviewed'.

It's not "long" division, it's a way to estimate the result of a division operation. It doesn't matter if you always get the right answer, because if you need the right answer, you can use a calculator.Wait, they teach a method which, when executed properly gives the wrong answer? Or it's just a crappy method which people use incorrectly half the time? Because I find it hard to believe there can be actual methods of teaching mathematics in which finding the correct answer isn't the only judge of success?

The problem is a lot of people are attracted to the idea of teaching math today the same way it was taught before calculators and computers existed. In the 1960s lots of people need to do math with a pencil and paper for their jobs. And that math needed to be exactly correct.

Today, that's just not true. Nowadays, for the most part math on the job isn't even done with calculators; it's done with spread sheets and computer programs. You just enter the data. For some jobs, you need to know how to make a new spreadsheet, or a new program, and for other jobs you just enter the stuff.

The reality is, pencil and paper arithmetic is not a skill anyone needs for any job, or any other aspect of life, as far as I know.

What's important is understanding mathematical concepts, having a number sense, being able to do math in your head is useful, but it doesn't need to be 100% perfect the way people needed pencil/paper math to be perfect in the past.

What is an "actual math application" that you think someone might have to do without access to a calculator or computer?Well, because it gives you a whole number answer with a remainder, not an exact decimal answer. That's what I mean by "close enough." But in actual math applications, a whole number + a remainder is often useless.

Also (as others have said) not is integer + remainder an exact answer, in many cases a decimal result is

Also, if you want you can just leave a number un-divided and you still have an exact result "271/9643" is a perfectly valid "number"

---

Anyway, that said when I was in school I always found the whole "real world" math examples really boring. I definitely preferred leaning about math in its "pure" form, totally abstract and unrelated to anything in the real world.

Obviously learning to map real world things to mathematical concepts is an important

What? I'm pretty sure they had people who's job was to clean people's teeth before Newton.Dental hygienists wouldn't have a job without calculus.

posted by delmoi at 4:01 AM on October 19, 2012 [1 favorite]

Well, you can break up numbers into their prime factors. So 7*8 can be broken down into 7*2*2*2, and then you just double it three times 14, 28, 56. Or in some case there are simple formulas for some things - like 9*n or 5*n, so you don't need to 'memorize' those because you can answer those quickly.As someone who was successful with the traditional techniques, I don't understand this at all. With what would you replace the "times table"? Or are you asserting that with modern tools there's no need to be able to perform the mechanics of arithmetic provided one understands the concepts?

Anyway, these things are pretty easy to figure out even if you don't have the exact product memorized. However, memorizing the times table is helpful for doing math quickly in your head with more digits. If you're trying to figure out, say 273*444 in your head, it's better if you know 7*4 off the top of head rather then doubling 7 twice.

posted by delmoi at 4:13 AM on October 19, 2012

Joke explained

posted by edd at 4:15 AM on October 19, 2012 [7 favorites]

A problem I find with those efforts to map maths to "real world things", is that the examples they come up with often seem so trivial and ridiculous. They were when I went to school, and it looks like they still are now.

The example given in one of those links - something like "9 students need to share 407 minutes of computer time. How much time does each student get to use the internet?". My first thought was, why 407 minutes? Who came up with that ridiculous number? I mean, I understand the purpose is to teach what division is used for, but after a couple of examples like that surely we can cut the bullshit and just start asking 407/9=?

I don't know what the answer is. I don't know enough about the technical aspects of the alternative techniques. Is the main complaint about traditional math teaching that it's boring, and makes students scared of math? I learnt long division and it didn't hurt me. On the other hand, I wouldn't have a clue how to do it any more, I haven't used it once in real life since school, and I can't recall if learning the technique actually gave me a deeper understanding of numbers and how they work.

As for kids turning up to university, unable to do basic sums?*Really*? I assumed all this new math teaching was aimed at primary school level, and once you got into the later years of highschool, if you were looking at going on to higher education, you'd be doing specialized classes on calculus etc. Is this no-longer the case? Even advanced students who *want* to learn math, who are aiming at studying science or engineering, are missing out on this stuff?

posted by Jimbob at 4:16 AM on October 19, 2012

The example given in one of those links - something like "9 students need to share 407 minutes of computer time. How much time does each student get to use the internet?". My first thought was, why 407 minutes? Who came up with that ridiculous number? I mean, I understand the purpose is to teach what division is used for, but after a couple of examples like that surely we can cut the bullshit and just start asking 407/9=?

I don't know what the answer is. I don't know enough about the technical aspects of the alternative techniques. Is the main complaint about traditional math teaching that it's boring, and makes students scared of math? I learnt long division and it didn't hurt me. On the other hand, I wouldn't have a clue how to do it any more, I haven't used it once in real life since school, and I can't recall if learning the technique actually gave me a deeper understanding of numbers and how they work.

As for kids turning up to university, unable to do basic sums?

posted by Jimbob at 4:16 AM on October 19, 2012

tl;dr New method is better IF teachers are well trained. If your teachers are badly trained (see horror stories about bad schools), the old method is better.

posted by EnterTheStory at 4:32 AM on October 19, 2012 [1 favorite]

posted by EnterTheStory at 4:32 AM on October 19, 2012 [1 favorite]

hey, I have some anecdata.

I have a phd in mathematics and young homeschooled children. my 8 year old is working through "Saxon" math, which is a neo-traditionalist sequence of elementary math textbooks. He's doing fair number of long division problems. If he asks me for help, I lead him through what is essentially the "method of partial quotients" because I'll be damned if I have to actually go through and do long division, it's more fun to work it out in my head. The problem with partial quotients is that estimating the multiple is intellectual challenging for you kids i.e. for 11/128, which multiples of 11 are less than 128. my son can actually totally do it, but in the end he's more comfortable with the long division algorithm, not because it is conceptually clearer, it's mostly opaque, but because it's an actual algorithm, rather than a sort of higher level framework for implementing lower level algorithm. the point being that adults are much more comfortable starting which an intellectual framework and working down to the little pieces than kids...

there is a general problem with this sort of math ed, which is that conceptually understanding mathematics, even arithmetic, is actually really hard, much harder than implementing an algorithm. and, it tends to rely on obtaining a kind of "total" view of the subject which allows you to deal with specific problems. people actual understanding of the world (IMHO) is made up of lots of different fragments that we can't necessarily piece together to make a total picture of our understanding.

there is another general problem with math ed which is that it asks teachers not to teach what they know, but "implement" some arbitrary curriculum for which they have to be "re-educated." the idea that a teacher is just someone who can be "trained" to implement arbitrary teaching methods is evil.

posted by ennui.bz at 4:43 AM on October 19, 2012 [6 favorites]

I have a phd in mathematics and young homeschooled children. my 8 year old is working through "Saxon" math, which is a neo-traditionalist sequence of elementary math textbooks. He's doing fair number of long division problems. If he asks me for help, I lead him through what is essentially the "method of partial quotients" because I'll be damned if I have to actually go through and do long division, it's more fun to work it out in my head. The problem with partial quotients is that estimating the multiple is intellectual challenging for you kids i.e. for 11/128, which multiples of 11 are less than 128. my son can actually totally do it, but in the end he's more comfortable with the long division algorithm, not because it is conceptually clearer, it's mostly opaque, but because it's an actual algorithm, rather than a sort of higher level framework for implementing lower level algorithm. the point being that adults are much more comfortable starting which an intellectual framework and working down to the little pieces than kids...

there is a general problem with this sort of math ed, which is that conceptually understanding mathematics, even arithmetic, is actually really hard, much harder than implementing an algorithm. and, it tends to rely on obtaining a kind of "total" view of the subject which allows you to deal with specific problems. people actual understanding of the world (IMHO) is made up of lots of different fragments that we can't necessarily piece together to make a total picture of our understanding.

there is another general problem with math ed which is that it asks teachers not to teach what they know, but "implement" some arbitrary curriculum for which they have to be "re-educated." the idea that a teacher is just someone who can be "trained" to implement arbitrary teaching methods is evil.

posted by ennui.bz at 4:43 AM on October 19, 2012 [6 favorites]

I see what you did there.

posted by CheeseDigestsAll at 5:06 AM on October 19, 2012 [2 favorites]

I hated memorizing the times table myself, but it has to done to get to the next level. I can't see how you can work around it.

The reality is, pencil and paper arithmetic is not a skill anyone needs for any job, or any other aspect of life, as far as I know

Scratch arithmetic is useful in carpentry and other building trades. Ironically, as an assistant trade worker I could also have done my job fine if I was functionally illiterate.

posted by ovvl at 5:10 AM on October 19, 2012 [1 favorite]

I went and read about some Everyday Mathematics methods and I sure can attest to feeling the "wait, that's screwed up" reaction that many others in this thread have had—I can wrap my head around why all the addition algorithms are the same, but I really prefer the way I learned ("A fast method (traditional)").

On the other hand, when I got down to the Partial Quotients method of division, I went: wow, that's so much better than traditional long division (as at each step you only need to compare against the easy-to-compute N, 2N and 5N instead of any of N through 9N). So I'm obviously at the opposite end of some scale from KathrnyN and child.

One last thought: when children are working these problems in after-school programs or at-home tutoring by parents, it's not only the teachers' preparedness to teach these methods, but also the tutors and parents. OK, so your new method tests better when everyone instructing the student understood and was using the same method. But how does it fare when the kid goes home to dad who doesn't care to try to understand the method being taught, and simply teaches a traditional method instead? (on rereading, I see that this point was already made by yeolcoatl)

posted by jepler at 5:12 AM on October 19, 2012

On the other hand, when I got down to the Partial Quotients method of division, I went: wow, that's so much better than traditional long division (as at each step you only need to compare against the easy-to-compute N, 2N and 5N instead of any of N through 9N). So I'm obviously at the opposite end of some scale from KathrnyN and child.

One last thought: when children are working these problems in after-school programs or at-home tutoring by parents, it's not only the teachers' preparedness to teach these methods, but also the tutors and parents. OK, so your new method tests better when everyone instructing the student understood and was using the same method. But how does it fare when the kid goes home to dad who doesn't care to try to understand the method being taught, and simply teaches a traditional method instead? (on rereading, I see that this point was already made by yeolcoatl)

posted by jepler at 5:12 AM on October 19, 2012

My takeaway from this is that the reason I was such a constant dismal failure in school is that other kids got help from their parents with homework.

posted by idiopath at 5:41 AM on October 19, 2012 [3 favorites]

posted by idiopath at 5:41 AM on October 19, 2012 [3 favorites]

Back in the 1980s I was enrolled in an elementary teacher education program at a state college in upstate NY. I was part of a cohort of around a hundred students. After I'd heard several of my classmates say they wanted to be teachers because they loved kids and they hated math I did an informal survey of members of the cohort. I think about 75% of the students said they hated math, or sucked at math, and many mentioned that they could not get into other majors because their math skills were so poor.

These are the same people who are supposed to be turning young children on to math. If they hate math, regardless of how well trained they are in methods of teaching math they're going to pass some of that onto their young students.

The solution: if we respected elementary and early childhood education teachers more, if we really valued the critical work they do in turning young children on to math, science, reading, etc., we would pay them more! This way we could attract brighter students to teaching. Indeed, until the last few decades it was often the best and the brightest who entered teaching- the women and the members of minority groups who were excluded from other professions either de jure or de facto.

posted by mareli at 5:47 AM on October 19, 2012 [6 favorites]

These are the same people who are supposed to be turning young children on to math. If they hate math, regardless of how well trained they are in methods of teaching math they're going to pass some of that onto their young students.

The solution: if we respected elementary and early childhood education teachers more, if we really valued the critical work they do in turning young children on to math, science, reading, etc., we would pay them more! This way we could attract brighter students to teaching. Indeed, until the last few decades it was often the best and the brightest who entered teaching- the women and the members of minority groups who were excluded from other professions either de jure or de facto.

posted by mareli at 5:47 AM on October 19, 2012 [6 favorites]

I don't know if they are using the same curriculum, but on Radio 1 in the UK they invited some adults who had struggled in math to try the new methods being used in schools there (and which are also controversial).

They all found the new method to be more understandable.

As for problems: math should be taught as it's used. Some of the best math classes I ever had were learning to figure out mortgages and fill out tax forms - in grade six. Later, I was all excited about learning calculus so I could figure out how far a cannon ball would fly, but they had dropped the real world examples by then.

posted by jb at 5:58 AM on October 19, 2012 [2 favorites]

They all found the new method to be more understandable.

As for problems: math should be taught as it's used. Some of the best math classes I ever had were learning to figure out mortgages and fill out tax forms - in grade six. Later, I was all excited about learning calculus so I could figure out how far a cannon ball would fly, but they had dropped the real world examples by then.

posted by jb at 5:58 AM on October 19, 2012 [2 favorites]

IANAM (Mathematician), but now I'm questioning my k-5 math education myself. So I'm hoping that someone can clarify this math equation that I would expect at this level:

Simple version:

I have 12 apples. 5 apples will fit into a basket. How many baskets of apples do I have?

Remainder version: 2 baskets with 2 apples remaining

Decimal version: 2.4 baskets

Complex version with long-ish division:

I have 120 apples. 50 apples will fit into a bushel.

Remainder version: 2 bushels with 20 apples remaining

Decimal version: 2.4 bushels

Each of these answers are useful in difference scenarios, but how are either one of them wrong?

posted by Blue_Villain at 6:00 AM on October 19, 2012

Simple version:

I have 12 apples. 5 apples will fit into a basket. How many baskets of apples do I have?

Remainder version: 2 baskets with 2 apples remaining

Decimal version: 2.4 baskets

Complex version with long-ish division:

I have 120 apples. 50 apples will fit into a bushel.

Remainder version: 2 bushels with 20 apples remaining

Decimal version: 2.4 bushels

Each of these answers are useful in difference scenarios, but how are either one of them wrong?

posted by Blue_Villain at 6:00 AM on October 19, 2012

???? Even outside of my job, I find myself doing division and multiplication from scratch. Some people have the idea that I'm some kind of "math genius" because I went to a fancy school and got some nice degrees, but in fact I just do arithmetic by hand, just like everyone else. Things like my financial planning, measuring things, and every day life involves this kind of thing. For spreadsheets, even before you create the spreadsheet, you do some simple tests and explorations to flesh out your model that involves doing things by hand (or at least having an understanding of the math that comes with doing it by hand).

Milgram and Bishop, I think, suffer from the classic problem of people who were very successful under a certain system and want to defend the system they were raised under. And for the most part, I agree with them in the general case: learning math is hard, and there aren't any shortcuts, and you have to do a lot of memorization, drill-and-kill, and the like to "get it." And this takes time and isn't pleasant and a lot of people simply aren't raised with the kind of discipline and endurance it takes to get there. And I get the impression that teachers don't like to do it-- that kind of curriculum isn't enjoyable

So the question is whether there's a math curriculum that the average teacher can teach that the average student under normal circumstances can learn that will make the students math literate? This is actually a pretty good question... I don't want to hear, "oh, when you train the teachers

posted by deanc at 6:16 AM on October 19, 2012 [7 favorites]

See, you're looking at arithmetic in the wrong way, entirely. Arithmetic is a skill. You don't get better at lifting weights and running by injuring yourself or doing the exercises wrong. You get stronger and faster and better by doing something over and over and over again while concentrating on

For the most part, I do think that our high school math curriculum is a bit over-focused on engineering applications. I think we could teach calculus sooner and give people time to understand more about proofs, probability, and discrete math before they get to college. But there is so much material that has to be learned that you can't just claim that people will figure it all out through project-based learning. At best, you can just figure out a means of keeping students engaged while they grapple with the avalanche of information and skill-building they need to do.

posted by deanc at 6:25 AM on October 19, 2012

posted by Kirth Gerson at 6:28 AM on October 19, 2012

...* it's the students who get remaindered.*

...* it was just this thing left over that you threw away.*

I see what you did there...

posted by sammyo at 6:38 AM on October 19, 2012

...

I see what you did there...

posted by sammyo at 6:38 AM on October 19, 2012

Actually if you calculate a ballistics problem you've done a problem that is essentially calculus, it may not use Euller notation but unless you're using a table it's approaching basic calculus.

posted by sammyo at 6:43 AM on October 19, 2012 [1 favorite]

I have 12 apples. 5 apples will fit into a basket. How many baskets of apples do I have?

Remainder version: 2 baskets with 2 apples remaining

Decimal version: 2.4 baskets

Each of these answers are useful in difference scenarios, but how are either one of them wrong?

Neither of these are wrong. They fall under the "decimals are exact when dividing by a product of powers of 2 and 5" rule. But if you put the 12 apples in baskets of 7 and say that you have 1.714 baskets of apples, you are wrong, because the decimal representation is infinite.

posted by yeolcoatl at 7:01 AM on October 19, 2012 [3 favorites]

Unfortunately for lots of kids, pencil and paper arithmetic is a skill that they need to master in order to score well enough on test that determine whether they will pass to the next grade. Yes, it sucks, but unless someone can come along with a magic wand and make the tests have no meaning, kids need to learn a way to divide that doesn't end with "and there's the remainder, which we do nothing with". If the problem on the Important Standardized Test is 375/12, I guarantee that that 31 (remainder 3) won't be showing up as an answer from about 4th grade on, no matter how accurate that answer may be.

posted by 23skidoo at 7:17 AM on October 19, 2012 [1 favorite]

Thanks for giving me a reminder to re-learn long division so I can tutor my own kids, who are about to start learning it.

posted by davejay at 7:20 AM on October 19, 2012

posted by davejay at 7:20 AM on October 19, 2012

Okay, so, I just re-learned long division. Which is to say, I sat down and tried to remember how I did 12 / 5 as a kid, and wouldn't you know it, I remembered. And got 2.4. And it wasn't hard, just a process. Is this hard for kids, generally?

posted by davejay at 7:23 AM on October 19, 2012 [1 favorite]

posted by davejay at 7:23 AM on October 19, 2012 [1 favorite]

Just did 1400 / 37, and got (on paper) 37.83, looked at it, and realized it would repeat 37.837837837837837837...and thought "whee!" I'd forgotten that little happy feeling of "I understand this." I hope my kids feel that way at least once a day at school.

posted by davejay at 7:27 AM on October 19, 2012

posted by davejay at 7:27 AM on October 19, 2012

As a kid who at first excelled, then struggled, with math, this tallies with my experience. I didn't get anywhere until my brother sat me down and helped me memorize times tables. Rather than being boring, that knowledge opened up the world, in that suddenly I could do simple calculations much faster and larger numbers could now be seen as just extensions of these smaller numbers. In other words, once you know 9 x 5, then 900 x 500 isn't intimidating. The times tables gave me a sense of the underlying structure of math that I hadn't had before, and are among the most useful things I ever learned. I use them every day.

It's hard for me to imagine a way of teaching math that doesn't include carrying around a basic times table knowledge.

posted by emjaybee at 7:51 AM on October 19, 2012 [2 favorites]

Personally, I didn't memorize vast majorities of the times table until I took the GRE at age 24. I was a degreed engineer who had successfully passed the EIT exam. Every kid is a little bit different, and a successful teacher of any subject recognizes this.

posted by muddgirl at 7:59 AM on October 19, 2012 [1 favorite]

posted by muddgirl at 7:59 AM on October 19, 2012 [1 favorite]

I spent some more time looking at the Everyday Mathematics methods, and the Column Addition Method, Trade First Subtraction Method and Partial Products Method are just different ways to write the standard (traditional) method and which may well provide a better bridge between arithmetic on single digits using rote-learned tables to the methods on multi-digit numbers.

Of all the methods presented on that page, it's the lattice method that I had the hardest time relating to. But notice that the diagonals of the lattice are just the same as the columns in the "partial products method" just above it, and the structure of the lattice saves you from taking care to write the right number of zeros in every step.

posted by jepler at 8:03 AM on October 19, 2012

Of all the methods presented on that page, it's the lattice method that I had the hardest time relating to. But notice that the diagonals of the lattice are just the same as the columns in the "partial products method" just above it, and the structure of the lattice saves you from taking care to write the right number of zeros in every step.

posted by jepler at 8:03 AM on October 19, 2012

I thought geometry in ~9th grade was typical, containing a dose of logic and proofs.

Kickball wasn't something I needed for any job. Reading the Bridge of San Luis Rey isn't something I needed for any job. A lot of education is about teaching lower level skills and behaviors that are good for human beings to have. Learning to follow an algorithm correctly to solve something bigger than what you can do in your head, and being precise about what you put on paper, these are good things. Elementary education is not vocational training.

posted by fleacircus at 8:05 AM on October 19, 2012 [1 favorite]

jb, you don't need calculus to figure out ballistics.

posted by sammyo at 8:43 on October 19 [+] [!]

Ballistics may be described using kinematic equations that are to be taken as a given. But having taught as a TA in both calculus and non-calculus based physics courses at the university level, I can tell you that the odds of perfectly implementing a procedure that one does not actually understand to generate an answer are not as high as the chances a student has approaching the problem using calculus.

Of course, this is anecdotal evidence, and one of the problems with the discussion of these problems is that so many of us see our own anecdotes as sensible and intuitive, and discount the experiences of others. I can tell you that I scoffed at a number of comments in this thread, until I realized I was doing so, because for me the traditionally taught mathematics was easy.

Still. In my own little world, I have a lot of skepticism of reform in math education. Any time someone says that we're crippling students by forcing them to master skills with little practical use, I smile and nod and back out of the room. The purpose of learning complicated, rigorous methods of thought isn't so that you can perform proofs or optimize some linear system. It's so that you get smarter and don't swallow bullshit. Math can make you smarter. I fully submit that this opinion again reflects my own biases, etc etc.

posted by samofidelis at 8:06 AM on October 19, 2012 [1 favorite]

Well, technically it would still be 'peer-reviewed'. Your peers reviewed it, and declared that it sucked.

Sorry, this is bullshit, and gets to the heart of why Milgram and Bishop's critiques are impossible to see as anything but small-hearted sniping at someone whom they disagree with, but don't have the facts or analysis to actually engage with.

Failing peer-review does not make their paper peed-reviewed, anymore than failing to qualify for the Olympics makes one an Olympian. Peer review includes as part of the process a chance for the authors to correct their manuscript, even perform new work and reanalyze their conclusions. There is a fair bit of flexibility in the back and forth between author reviewer and editor to refine and develop a reasonable quality paper. Milgram and Bishop have avoided that process and have published something that is little more than a blog post.

There are many right ways to do critiques: letters to the editor, write a peer-reviewed paper themselves with new data to support their own position, do a comprehensive review of the literature placing Boaler's work in the context of other data. They apparently did none of these things, preferring to shoot from the sidelines. That's what makes this academic harassment.

posted by bonehead at 8:07 AM on October 19, 2012 [4 favorites]

Actual artillery officers didn't do calculus -- since each cannon was itself a little non-standard, they tested it at different angles and wrote down the distance it would shoot. I just wanted to figure it out mathematically - and then I realized that I didn't know the force or a whole bunch of other variables.

My overall point was that real-world applications of mathematics - like destroying your enemies' fortifications - was more exciting to me than pure maths.

posted by jb at 8:12 AM on October 19, 2012

Do people not do "back of the envelope" calculations anymore? Do landlords not need to use math to figure out whether rental income will cover the cost of their mortgages? Do people no longer need to divide the carried balance on their credit cards by a number of months to estimate how much extra they will have to pay each month to pay off their balance? Do people who own land not need to figure out how much each plot might be worth if sold separately? How about calculating $/sq.ft. of a condo you're thinking of buying?

The claim that we no longer need to learn basic arithmetic and drill it into our heads so that it becomes 2nd nature is bizarre. I have a feeling that all of you simply learned math so well and have such an intuitive grasp on it that you no longer even realize that you use it all the time.

posted by deanc at 8:14 AM on October 19, 2012 [3 favorites]

The claim that we no longer need to learn basic arithmetic and drill it into our heads so that it becomes 2nd nature is bizarre. I have a feeling that all of you simply learned math so well and have such an intuitive grasp on it that you no longer even realize that you use it all the time.

posted by deanc at 8:14 AM on October 19, 2012 [3 favorites]

Geometry is most common in 11th grade (10th grade for college prep), and in my class most of the proofs are of the concrete kind (lots of cutting and pasting paper). Any logic is... intuitive, I guess I would say. At least when I took geometry, we didn't learn any sort of formal, abstract logic structure. We were just supposed to understand that if A = B, and B = C, then A = C. (Also, a lot of kids struggled with geometry, more so than other math classes.)

posted by muddgirl at 8:14 AM on October 19, 2012

(Oh - but the artillery officers did have to use trigonometry to figure out how far away that thing was that they wanted to hit. But they had tables for the trigonometry, just like we have functions on a calculator. Have you ever tried to do trigonometry without tables or a calculator? it's really hard.)

posted by jb at 8:15 AM on October 19, 2012

posted by jb at 8:15 AM on October 19, 2012

I don't think anyone is saying, "We don't need to ever do arithmetic, ever."

posted by muddgirl at 8:19 AM on October 19, 2012

Well, someone did:

The reality is, pencil and paper arithmetic is not a skill anyone needs for any job, or any other aspect of life, as far as I know.As an aside, my experience in geometry was that it was taught in 9th or 10th grade, depending on what math track you were on, and that it was your first introduction to the concept of "rigorous proofs."

posted by deanc at 8:22 AM on October 19, 2012

Considering nearly every person carries a powerful computer in their pocket that has a calculator, I do agree that to a large extent the need to accurately calculate numbers with a pencil and paper is all but eliminated. Estimating arithmetic calculations in ones head does not require memorizing times tables or using a pencil and paper.

If I am trying to figure out what product is the best value, I'm using a spreadsheet anyway, because if I have to record and compare I might as well let it do the math as well. If I need to quickly determine price per square foot, I'm rounding to the nearest tens place.

posted by muddgirl at 8:26 AM on October 19, 2012

I admit to getting irritated when asking new students to figure out volumes or concentrations in the chem lab, and having them look for a calculator to do simple arithmetic. For 4+12 you need a calculator? Really?

posted by bonehead at 8:29 AM on October 19, 2012 [2 favorites]

posted by bonehead at 8:29 AM on October 19, 2012 [2 favorites]

I see this all the time. People will swear to me that algebra was a waste of time and that they never

posted by eabomo at 8:30 AM on October 19, 2012 [2 favorites]

Fascinating post. Thanks!

posted by brundlefly at 8:57 AM on October 19, 2012

posted by brundlefly at 8:57 AM on October 19, 2012

This took a little longer to write than I intended, so it does not quite address everything already here. Pitfalls of writing huge comments at work.

I was recently (3 years ago) in a masters program to get a degree in Math Education. This was definitely a topic. I was looking more at teaching middle and high schoolers than elementary school, but here's my $.02.

I flourished under traditional math. I rocketed through it, to the extent of being at a nerd camp between seventh and eighth grade and completing Algebra I from a textbook in a week.

So here's my take, from what I read in the literature, from teaching 2 years at a private boarding school, and from what I saw with friends who had to struggle with math to get through their majors (I still feel bad about almost laughing at a friend who had forgotten how to do any arithmetic with fractions).

Traditional math is something of a gatekeeper. You need to either be well enough off to get tutors and have concerned parents, be incredibly talented or be naturally decent at it and have an enthusiastic teacher. One of the few things that NCLB did that I like is that it give kids at "failing" schools free tutoring if they sign up for it. The record keeping is onerous and the pay is lousy, but it's a step.

In most traditional classes, students do not have to collaborate. Math is cast as a solitary enterprise, done by a single person sitting alone with pencil and paper. Perhaps they get to use a calculator, but on the whole, they should not need it. Learning math must be rigorous and difficult for most students because that is how it has always been and dmanit, if the education I received back in the day was enough to make me hate and misunderstand math, it should be sufficient for my kids too. From the side of the professional mathematicians, if traditional math education was sufficient to elevate them to the position they are in and allowed them to master their subject, changing it is dangerous to the continuation of math as whole.

The reform side seems to be trying to impart a basic understanding of mathematics into every student. It feels less likely to propel students rapidly into advanced levels, but basic competency should be achieved by more students. Students should use modern tools because there is no reason to pretend they do not exist and after all, students will always have access to a calculator or computer, won't they? (A decade ago this assertion seemed ridiculous, now, well, it's becoming truer with the ubiquity of cell phones.)

Curriculum wars, are quite honestly not all that important in the competency that students achieve in math. They help, but bad teachers can ruin even the most formulaic Saxon curriculum. I was lucky and head decent to excellent teacher up until my senior year of high school, by which time I knew I loved math and had already taken the calculus AP.

More than simply altering the curriculum, teaching training is needed. The data on what pedagogical methods are most effective is sparse as compared to most other disciplines and generally very recent. Pedagogical research tends to lag behind developmental psychology (a friend on mine in a psychology PhD program said it tends to be about 10 years behind).

What I think people miss is that there is no one curriculum that will work for every child. EDM may reach more children than traditional systems, but there will be children in EDM classes who will not get it. I know that EDM would not have worked well for me. I had trouble with math as a first grader until I was allowed to approach it algorithmically with memorization. I also saw a great number of my peers, some of who were (and are) very bright struggle with concepts that came to me naturally.

The fear with EDM is that we will eventually end up in a situation like The Feeling of Power by Asimov. (If you haven't read it, please do so. It takes moments and is a wonderful story.) However, if it can be used to help people become more comfortable with math, to have less fear of math, less inclined to simply accept numbers when faced with them, it will be a success. From what I've seen of programs like this, they try to convey comfort with number and with estimations. And honestly, in circumstances where someone does not have access to computational devices, estimation is generally what is needed.

Final brain teaser: given a dump truck loading every 15 seconds, how long with it take for dump trucks to cart away Mt. Rainier (ignoring the time to dig, just 15 seconds to load)?

posted by Hactar at 9:01 AM on October 19, 2012 [3 favorites]

I was recently (3 years ago) in a masters program to get a degree in Math Education. This was definitely a topic. I was looking more at teaching middle and high schoolers than elementary school, but here's my $.02.

I flourished under traditional math. I rocketed through it, to the extent of being at a nerd camp between seventh and eighth grade and completing Algebra I from a textbook in a week.

So here's my take, from what I read in the literature, from teaching 2 years at a private boarding school, and from what I saw with friends who had to struggle with math to get through their majors (I still feel bad about almost laughing at a friend who had forgotten how to do any arithmetic with fractions).

Traditional math is something of a gatekeeper. You need to either be well enough off to get tutors and have concerned parents, be incredibly talented or be naturally decent at it and have an enthusiastic teacher. One of the few things that NCLB did that I like is that it give kids at "failing" schools free tutoring if they sign up for it. The record keeping is onerous and the pay is lousy, but it's a step.

In most traditional classes, students do not have to collaborate. Math is cast as a solitary enterprise, done by a single person sitting alone with pencil and paper. Perhaps they get to use a calculator, but on the whole, they should not need it. Learning math must be rigorous and difficult for most students because that is how it has always been and dmanit, if the education I received back in the day was enough to make me hate and misunderstand math, it should be sufficient for my kids too. From the side of the professional mathematicians, if traditional math education was sufficient to elevate them to the position they are in and allowed them to master their subject, changing it is dangerous to the continuation of math as whole.

The reform side seems to be trying to impart a basic understanding of mathematics into every student. It feels less likely to propel students rapidly into advanced levels, but basic competency should be achieved by more students. Students should use modern tools because there is no reason to pretend they do not exist and after all, students will always have access to a calculator or computer, won't they? (A decade ago this assertion seemed ridiculous, now, well, it's becoming truer with the ubiquity of cell phones.)

Curriculum wars, are quite honestly not all that important in the competency that students achieve in math. They help, but bad teachers can ruin even the most formulaic Saxon curriculum. I was lucky and head decent to excellent teacher up until my senior year of high school, by which time I knew I loved math and had already taken the calculus AP.

More than simply altering the curriculum, teaching training is needed. The data on what pedagogical methods are most effective is sparse as compared to most other disciplines and generally very recent. Pedagogical research tends to lag behind developmental psychology (a friend on mine in a psychology PhD program said it tends to be about 10 years behind).

What I think people miss is that there is no one curriculum that will work for every child. EDM may reach more children than traditional systems, but there will be children in EDM classes who will not get it. I know that EDM would not have worked well for me. I had trouble with math as a first grader until I was allowed to approach it algorithmically with memorization. I also saw a great number of my peers, some of who were (and are) very bright struggle with concepts that came to me naturally.

The fear with EDM is that we will eventually end up in a situation like The Feeling of Power by Asimov. (If you haven't read it, please do so. It takes moments and is a wonderful story.) However, if it can be used to help people become more comfortable with math, to have less fear of math, less inclined to simply accept numbers when faced with them, it will be a success. From what I've seen of programs like this, they try to convey comfort with number and with estimations. And honestly, in circumstances where someone does not have access to computational devices, estimation is generally what is needed.

Final brain teaser: given a dump truck loading every 15 seconds, how long with it take for dump trucks to cart away Mt. Rainier (ignoring the time to dig, just 15 seconds to load)?

posted by Hactar at 9:01 AM on October 19, 2012 [3 favorites]

A pox on both their houses.

Milgram and Bishop should be ashamed for making unsubstantiated claims of misconduct.

The sad truth is that in the social sciences it is so easy to produce work that is driven by ideology and subject to bad methodology, that there is no need to engage in misconduct.

And Boaler should be ashamed for being a good example of that point.

posted by ocschwar at 9:06 AM on October 19, 2012

Milgram and Bishop should be ashamed for making unsubstantiated claims of misconduct.

The sad truth is that in the social sciences it is so easy to produce work that is driven by ideology and subject to bad methodology, that there is no need to engage in misconduct.

And Boaler should be ashamed for being a good example of that point.

posted by ocschwar at 9:06 AM on October 19, 2012

And because no one has yet posted this:

New Math by Tom Lehrer.

(Watching this with the sound off because I'm at work, I realized something. I have no idea how the hell the initial example works. Once he starts talking about 10s places, that makes sense. But in the beginning, how is he changing the lower number?)

posted by Hactar at 9:09 AM on October 19, 2012 [1 favorite]

New Math by Tom Lehrer.

(Watching this with the sound off because I'm at work, I realized something. I have no idea how the hell the initial example works. Once he starts talking about 10s places, that makes sense. But in the beginning, how is he changing the lower number?)

posted by Hactar at 9:09 AM on October 19, 2012 [1 favorite]

As well it should. The question is whether

posted by pwnguin at 9:10 AM on October 19, 2012

What'd Boaler do wrong?

posted by fleacircus at 9:10 AM on October 19, 2012

posted by fleacircus at 9:10 AM on October 19, 2012

bonehead:Years ago I posted a blog entry entitled "Why Johnny Uses a Calculator to Solve 6 x 4" and linked to this extremely skeptical video concerningI admit to getting irritated when asking new students to figure out volumes or concentrations in the chem lab, and having them look for a calculator to do simple arithmetic. For 4+12 you need a calculator? Really?

I don't really have a dog in this hunt. My layman's opinion is that traditional arithmetic is superior precisely because it doesn't require you to understand mathematics. They can be used as strictly mechanical methods that always work and don't require anything more of the student than knowing basic math facts,

posted by ob1quixote at 9:13 AM on October 19, 2012 [1 favorite]

This is actually pretty true for everything significant. The only way to learn to be a good writer, for example, is do a lot of writing and study writing and practice writing and a lot of people don't want to take the time. The difference with mathematics is that, culturally, we are way more likely to give people a "pass" and let them go "I don't get math" instead of pushing past that (and I know I've been there) into learning. It's not that the subject is so much harder, but we let students get discouraged and quit.

I have a feeling that all of you simply learned math so well and have such an intuitive grasp on it that you no longer even realize that you use it all the time.

I tutored a woman working on her GED one summer, and she was hung up on fractions and decimals. After a few days of drills, she was all "I can't do this." And I said "Yes, you can, you just don;'t realize that you already know it." She said "No way." In desperation, I threw some change on table and said "what is that?" She said "65 cents." I said "See? You turned 1/4, 1/4, 1/10, and 1/20 into .65.

Or so I like to tell myself.

posted by GenjiandProust at 9:35 AM on October 19, 2012

You know, I finally saw Wall-E, and I really don't think that this sort of attitude leads to a very good place.

Seriously: we don't want to be reliant on computers for simple calculations. For SIN functions, sure (seriously, don't try to do trigonometry without a calculator -- I passed, but barely). But we need to exercise our brains and practice the skills of simple arithmetic -- so that we can go "hey, that's not right" when our calculators/spreadsheets/cash tills spit out wrong numbers due to typos or miswritten formulae.

Estimating arithmetic does require memorizing times tables - if I told you 6 times 7, you have no idea what it is if you haven't memorized a times table. But if someone is buying 6 things for $7 each and the till gives you a total of $36 before tax, you know that you didn't scan one of them. Recently, I've found myself correcting people's arithmetic when they've given me back

But I find that divisible food works well with fractions - pie, pizza, etc - because it gets to the heart of the idea that a fraction is part of a whole.

posted by jb at 9:40 AM on October 19, 2012 [2 favorites]

deanc:[…]learning math is hard, and there aren't any shortcuts, and you have to do a lot of memorization, drill-and-kill, and the like to "get it." And this takes time and isn't pleasant and a lot of people simply aren't raised with the kind of discipline and endurance it takes to get there. And I get the impression that teachers don't like to do it-- that kind of curriculum isn't enjoyable for them.

GenjiandProust:Exactly. I liken arithmetic to playing an instrument. It requires every day practice to get even "passably" good at it. Not everyone is going to be aThis is actually pretty true for everything significant. The only way to learn to be a good writer, for example, is do a lot of writing and study writing and practice writing and a lot of people don't want to take the time.

If people are clamoring to introduce a non-standard text to give students more of an analytic approach to math, I can't understand how no math educator has brought up Pólya's

posted by ob1quixote at 9:44 AM on October 19, 2012 [2 favorites]

People, please remember that the singular of data is datum, not anecdote. From what I have read, Boaler has empirical evidence that with proper training, this method works better than the method that Milgram and Bishop are advocating. I will happily accept studies (preferably peer reviewed, but published by school districts or departments of educations are acceptable) that show this does not work. An expose by a meteorologist does not disprove empirical data. Please don't make me break out the global warming analogies. Please.

posted by Hactar at 10:17 AM on October 19, 2012 [3 favorites]

posted by Hactar at 10:17 AM on October 19, 2012 [3 favorites]

Part of the reason people go for the calculator even when confronted with 2+6 is because OMG here comes the math, I'm in a fuckin' math context now, who knows what will come next, better get a calculator and be prepared.

After all, it's also common for people who are*good* at arithmetic to start off doing it in their head or by hand, get to the point where they need a calculator and maybe even wind up doing it all over from scratch on the calculator.

*I have no idea how the hell the initial example works. Once he starts talking about 10s places, that makes sense. But in the beginning, how is he changing the lower number?*

Just a different place to borrow from I guess. Subtracting one from what you're starting with is the same as adding one to the other number you are about to subtract. (4-1) - 7 = 4 - (7+1)

The phrase "three from two is nine" hurts my brain a little bit. It's not! Carry the 1.. but it's a -1 dammit. Or they talk of borrowing, but they never give it back... Maybe this is why I had to think of my own way of doing it when I was a kid.

I suppose they can't teach them 3 from 2 is -1. So write out 342 - 173 = 2|-3|-1 then make them cascade down until there are no negatives to get a proper number: 2|-3|-1 = 1|10-3|-1 = 1|7|-1 = 1|6|(10-1) = 1|6|9 = 169. Okay fine, dumb way to teach it but it amuses me.

Or "ten's compliment" that shit up, 342-173 => 342 + ( 8|2|6 + 1 ) => 342 + 827 => 1169 => 169.

I'll be in my bunk.

posted by fleacircus at 10:18 AM on October 19, 2012 [1 favorite]

After all, it's also common for people who are

Just a different place to borrow from I guess. Subtracting one from what you're starting with is the same as adding one to the other number you are about to subtract. (4-1) - 7 = 4 - (7+1)

The phrase "three from two is nine" hurts my brain a little bit. It's not! Carry the 1.. but it's a -1 dammit. Or they talk of borrowing, but they never give it back... Maybe this is why I had to think of my own way of doing it when I was a kid.

I suppose they can't teach them 3 from 2 is -1. So write out 342 - 173 = 2|-3|-1 then make them cascade down until there are no negatives to get a proper number: 2|-3|-1 = 1|10-3|-1 = 1|7|-1 = 1|6|(10-1) = 1|6|9 = 169. Okay fine, dumb way to teach it but it amuses me.

Or "ten's compliment" that shit up, 342-173 => 342 + ( 8|2|6 + 1 ) => 342 + 827 => 1169 => 169.

I'll be in my bunk.

posted by fleacircus at 10:18 AM on October 19, 2012 [1 favorite]

I want to make something clear here, and that's that no one, as far as I know, are suggesting that kids *not* learn their times tables.

Some aspect of computation must be done by rote - the estimation method requires you to know your times tables by heart, even!

*My layman's opinion is that traditional arithmetic is superior precisely because it doesn't require you to understand mathematics.*

I completely disagree. There's a famous example in a book by W. W. Sawyer (probably "Vision in Elementary Mathematics") where an examiner came to a top school in the British Isles and asked them, "You have 60 sheep and 15 die, how many are left?" A quarter of the class subtracted 15 from 60, but a quarter added, a quarter multiplied and a quarter divided...

For both professionals and non-professionals, there are only two tasks you ever have. Either you need to quickly get a good estimate of a calculation, or you need to get a completely accurate result.

If you don't understand the mathematics, you simply aren't going to get good results either way. You will misapply the operations, or you'll move the decimal place, and you simply won't know that you've gone wrong because you're performing operations that you don't understand. And in all the cases you need a completely accurate result, you have a calculator.

From probably the same W. W. Sawyer book, he compares rote teaching of mathematics to teaching a deaf child the piano. You could do it with enough punishments and rewards, show the child where to put the fingers and have a visual metronome, and the kid could no doubt learn to play a few simple tunes - but the child would never really understand what was going on and it would be relegated to being an obscure torture foisted on them by adults - just like mathematics is for most people.

Again, you need some sort of rote learning of the absolute basics. Even then you need to explain what's going on, so that if the student doesn't remember what 7x8 is, they can work it out - but times tables and addition need to be memorized.

Beyond that, no. Teach them what's going on, show them how to use a calculator properly, you're golden.

One more example from history - they used to teach kids (including me!) how to perform arithmetic in Roman numerals, because it was "good for you". The long division algorithm is a little more relevant than that - but only a little.

posted by lupus_yonderboy at 10:32 AM on October 19, 2012

Some aspect of computation must be done by rote - the estimation method requires you to know your times tables by heart, even!

I completely disagree. There's a famous example in a book by W. W. Sawyer (probably "Vision in Elementary Mathematics") where an examiner came to a top school in the British Isles and asked them, "You have 60 sheep and 15 die, how many are left?" A quarter of the class subtracted 15 from 60, but a quarter added, a quarter multiplied and a quarter divided...

For both professionals and non-professionals, there are only two tasks you ever have. Either you need to quickly get a good estimate of a calculation, or you need to get a completely accurate result.

If you don't understand the mathematics, you simply aren't going to get good results either way. You will misapply the operations, or you'll move the decimal place, and you simply won't know that you've gone wrong because you're performing operations that you don't understand. And in all the cases you need a completely accurate result, you have a calculator.

From probably the same W. W. Sawyer book, he compares rote teaching of mathematics to teaching a deaf child the piano. You could do it with enough punishments and rewards, show the child where to put the fingers and have a visual metronome, and the kid could no doubt learn to play a few simple tunes - but the child would never really understand what was going on and it would be relegated to being an obscure torture foisted on them by adults - just like mathematics is for most people.

Again, you need some sort of rote learning of the absolute basics. Even then you need to explain what's going on, so that if the student doesn't remember what 7x8 is, they can work it out - but times tables and addition need to be memorized.

Beyond that, no. Teach them what's going on, show them how to use a calculator properly, you're golden.

One more example from history - they used to teach kids (including me!) how to perform arithmetic in Roman numerals, because it was "good for you". The long division algorithm is a little more relevant than that - but only a little.

posted by lupus_yonderboy at 10:32 AM on October 19, 2012

I am reminded of an earlier MeFi post on a community college writing class where the professor complained about the poor talent of his students. The problem was that the class he

It's not merely that elementary school teachers didn't like math growing up. I suspect that for them, their vision for what they saw the careers being did not involve teaching children to chant out times tables on demand or being asking kids to convert fractions to decimals every day until it became instinct.

I disagree entirely. My ability to master mathematics (the highest level I studied was things like discrete math and linear algebra, so I didn't do any abstract, theoretical work) did not come from any inborn genetic "math" talents (it was freaking hard for me all through high school and college). Rather, it came from my ability to sit still and study in the basement of the library for hours at a time to figured it out.

It may well be that the need to concentrate for several hours a week on basic math skills is not something that can scale up to an audience of tens of millions of students, and I'm open to hearing what the alternatives are. But math "talent" is totally meaningless below graduate-school-level mathematics: your "talent" will have little effect on your grades and ability to absorb the information because people simply willing to practice and master the material will be able to compete on an equal level with the "talented."

There's the rub, now, though. I might add, though, that Boaler claims to get "better" results by redefining "better." In the introduction, she

What it seems is that Boaler's methods are essentially a self-esteem-building exercise for the students. Focused less on learning math, it's more about making sure that kids are less afraid of math. The only tangible outcome of her methods is that more people agree “Anyone can be really good at math if they try” (84%) than in traditional schools (52%). This is a good thing, and leads to the next tangible result, which is that more students take calculus when they're older (how well do they perform? We don't know, and I don't think Boaler cares because she doesn't think that "executing procedures correctly and quickly" is of primary importance in math).

posted by deanc at 10:38 AM on October 19, 2012 [2 favorites]

In my experience, a very large chunk of students didn't actually "figure it out" - they simply learned enough of the procedures by rote to satisfy an examiner. Hardly the same thing!

There's a huge difference between "doing a lot of work" and "doing a lot of meaningless, rote work".

There is an argument to be made that school is deliberately boring and meaningless to prepare kids for lifetimes of meaningless, boring work. I would hope that no compassionate, rational person would buy into it...

I've had a lot of experience with people who are afraid of math and I'd say that's the number one reason that people are incapable of doing math, so I'm all for attacking it as our number one pedagogical problem. If the kids aren't being given enough challenging homework that they actually gain mastery, that's a problem - in

posted by lupus_yonderboy at 10:51 AM on October 19, 2012

I see it very differently. Because they're our only hope for a better society. If we let traditional education continue its reign, you will get your doctors and engineers and specialists and analysts. But these generations would be rotten and unhappy and screwed

posted by polymodus at 11:08 AM on October 19, 2012 [2 favorites]

But we're discussing competence in specific skills, like the long division algorithm, that have essentially no use whatsoever in today's society. The argument that we are teaching these useless skills because it makes you a better person would have some resonance with me - if you weren't teaching people arbitrary mechanical skills that bring people further away from actually understanding what they are doing.

It might be that at the end of this new program, kids aren't as good at doing the real world arithmetic that they will need to do in the rest of their lives. If so, this is a problem - but nothing above actually indicates that this is the case.

Frankly, if I needed to get some people to do arithmetic, it'd be far far easier for me to start with a group of people who were positive and enthusiastic about math but didn't have the specific problem solving skills, than it would be to start with someone who knew the division algorithm perfectly but thought of math as a meaningless and hateful occupation.

(And just a suggestion that using the word "repulsive" to refer to people in discussions whose opinion differs from yours might not be the nicest tactic...)

posted by lupus_yonderboy at 11:18 AM on October 19, 2012

Here's what's unacceptable: the treatment of children and parents as pawns and resources to be exploited. It comes out through the idea that the job of the high-performing students is to tutor the lower-performing students and that the ability of parents to raise academically minded children is actually a threat to the new society you're trying to create.

You can't sacrifice kids on the altar of your religion about how you "think" things should work. It might not be very pleasant to realize that some things only come about through intense practice and concentration, but that's the reality of the situation.

Boaler says explicitly that competence or improvement at real world arithmetic skills is not the goal of her teaching methods but rather about comfort with math (which is a valuable thing, but it is just a thing).

posted by deanc at 11:25 AM on October 19, 2012 [1 favorite]

Do you realize your sort are the management consultants of the education world? For some reason, school districts always have 6 figure sums to bring in new teaching paradigms and consultants to "train" teachers to use them. Those consultants always talk about the "research," and how it shows the new program is totally great, the studies all say.... until the next new program and the next round of consultants.

posted by ennui.bz at 11:40 AM on October 19, 2012 [1 favorite]

"31 remainder 3" can be trivially converted into a whole number and fraction. It's 31 3/12, or 31 1/4.

Also, if the student goes on to STEM they will find this a breeze if they learned what a remainder is early on.

posted by Pruitt-Igoe at 11:53 AM on October 19, 2012 [1 favorite]

It might even come up in high school if they take AP computer science.

posted by Pruitt-Igoe at 11:58 AM on October 19, 2012

posted by Pruitt-Igoe at 11:58 AM on October 19, 2012

Three, but one of them still has some room if you want to stick a couple of oranges in there. Or maybe even a grapefruit; that'd probably fit.

posted by infinitywaltz at 12:06 PM on October 19, 2012 [5 favorites]

The following questions appeared in my kid's homework recently and left me scratching my head as to their value.

1- Write a sentence explaining how to add 3 + 4?

2- There are three ones and three tens how many threes and tens are there.

a) 33

b) 6

c) 101010111

d) 7

posted by humanfont at 12:10 PM on October 19, 2012

1- Write a sentence explaining how to add 3 + 4?

2- There are three ones and three tens how many threes and tens are there.

a) 33

b) 6

c) 101010111

d) 7

posted by humanfont at 12:10 PM on October 19, 2012

> Boaler says explicitly that competence or improvement at real world arithmetic skills is not the goal of her teaching methods but rather about comfort with math (which is a valuable thing, but it is just a thing).

Comfort with math results in better performance in math - but more, it results with people actually using math when they need to.

The evidence seems to show that students taught with Boaler's method also display improved competence in math. If that is true, it is surely irrelevant that this competence isn't the number one goal of her method, yes? Or if you have evidence that this isn't so, lay it on us...?

Hactar wrote:

> > I will happily accept studies (preferably peer reviewed, but published by school districts or departments of educations are acceptable) that show this does not work...

and ennui.bz replied:

> Do you realize your sort are the management consultants of the education world?

Or in short: "I want evidence!" "[series of generic insults, no evidence]"

You're acting as if mathematics was taught well in the past. But there's no evidence it ever was, and every bit of evidence that a huge portion of kids have always been getting out of school with math anxiety and math incompetence.

Hactar is absolutely right - this argument should be made based on solid evidence, not people's hostility toward "new math" or feeling that, "I learned this now-meaningless skill, so my kids should."

posted by lupus_yonderboy at 12:18 PM on October 19, 2012

Comfort with math results in better performance in math - but more, it results with people actually using math when they need to.

The evidence seems to show that students taught with Boaler's method also display improved competence in math. If that is true, it is surely irrelevant that this competence isn't the number one goal of her method, yes? Or if you have evidence that this isn't so, lay it on us...?

Hactar wrote:

> > I will happily accept studies (preferably peer reviewed, but published by school districts or departments of educations are acceptable) that show this does not work...

and ennui.bz replied:

> Do you realize your sort are the management consultants of the education world?

Or in short: "I want evidence!" "[series of generic insults, no evidence]"

You're acting as if mathematics was taught well in the past. But there's no evidence it ever was, and every bit of evidence that a huge portion of kids have always been getting out of school with math anxiety and math incompetence.

Hactar is absolutely right - this argument should be made based on solid evidence, not people's hostility toward "new math" or feeling that, "I learned this now-meaningless skill, so my kids should."

posted by lupus_yonderboy at 12:18 PM on October 19, 2012

lupus, "executing procedures correctly and quickly" and measurement of students' ability to do so was not considered a priority. So we don't know how well they do in math. If anything, Boaler merely demonstrates a method of inculcating a, important cultural value system to the students-- but one that has to be followed up with a drill-and-kill methodology.

The linked-to paper*has no quantifiable results about performance*-- only that in a highly structured, trained environment (which may well have had nothing to do with the cooperative methods themselves), the students gave more positive answers about their experience than other students at other schools. More students took calculus by year 4 (great!) but we don't know anything about their relative performance. There's no comparison of test scores. (because Boaler herself does not consider mathematical skill to be an important evaluation factor). No evidence that the students could learn the same lessons through traditional methods.

There is one important lesson to be learned: students overwhelmingly agree with the statement “Anyone can be really good at math if they try." That's actually extraordinarily important, but the thing is that it is a lesson that lends itself towards drill-and-kill methods of instruction. What it may be is that it turns out that the concentrated individual effort required to practice math and get good at it is not scalable, so possibly Boaler's method is a means of spreading this practice to the masses.

posted by deanc at 12:32 PM on October 19, 2012

The linked-to paper

There is one important lesson to be learned: students overwhelmingly agree with the statement “Anyone can be really good at math if they try." That's actually extraordinarily important, but the thing is that it is a lesson that lends itself towards drill-and-kill methods of instruction. What it may be is that it turns out that the concentrated individual effort required to practice math and get good at it is not scalable, so possibly Boaler's method is a means of spreading this practice to the masses.

posted by deanc at 12:32 PM on October 19, 2012

And where do we get the idea that long division is a meaningless skill? This claim blows my mind. Am I the only person who uses a pen and paper for taking notes and writing down numerical thoughts when solving problems?

Even "EDM" teaches long division, but it's a variant of long division that it an iterative process of homing in on the answer until you get there, rather than traditional long division's method of getting the final answer with a fixed number of iterations.

posted by deanc at 12:37 PM on October 19, 2012 [1 favorite]

Even "EDM" teaches long division, but it's a variant of long division that it an iterative process of homing in on the answer until you get there, rather than traditional long division's method of getting the final answer with a fixed number of iterations.

posted by deanc at 12:37 PM on October 19, 2012 [1 favorite]

I find myself doing long division in the steam on the inside of my shower door at least once a week. And I'm not in STEM, I'm a SAHM. Last week it was "If I want to walk 100 miles in October, how far do I need to walk each day? OK, and I started on 10/3, and I'm probably never going to walk on Sundays, so NOW how far do I need to walk each day?" I've figured out calorie counts per meatball, I've figured out fabric purchases for a quilt, I've figured out how I'm going to get $250 to buy food for four people for the next three weeks. I do arithmetic in my head all. the. time, often when I don't have a calculator available.

posted by KathrynT at 1:12 PM on October 19, 2012 [1 favorite]

I use pen and paper all the time for doing math, but I don't think I've had to do long division in probably close to a decade. Once you reach a certain level of abstraction, you really kind of stop needed to ever explicitly divide actual numbers, because you can just leave everything as improper fractions or fold them into constants.

posted by Pyry at 1:19 PM on October 19, 2012 [1 favorite]

posted by Pyry at 1:19 PM on October 19, 2012 [1 favorite]

I took four years of college math courses and the only ones that were fulfilling were the ones that empowered me to solve real-world problems.

Admittedly I'm blind in the eye that admires the beauty of mathematics, but most of us are in the same boat. I see no reason why pure abstraction should take precedence over relevance - particularly in school systems which coerce student attendance, or fashion channels through which the more valued and less valued students must flow, or always sacrifice art and music classes first.

posted by Twang at 1:23 PM on October 19, 2012

Admittedly I'm blind in the eye that admires the beauty of mathematics, but most of us are in the same boat. I see no reason why pure abstraction should take precedence over relevance - particularly in school systems which coerce student attendance, or fashion channels through which the more valued and less valued students must flow, or always sacrifice art and music classes first.

posted by Twang at 1:23 PM on October 19, 2012

I learned something close to EDM's division method in 1980s Quebec; we didn't really learn to convert the stuff do decimals until 6th grade, I think. Some kids hated math and found it hard, others loved it and found it easy.

It should be noted that K-6's role isn't really to make children learn anything, at least not intellectually; it's mostly there so they can learn to sit at their place for $x hours, and speak when it's their turn to speak. What they do while they're sitting on that chair is mostly an afterthought.

Might be useful to calculate stuff involving money, since they might start earning some once they're in their teens, but otherwise they won't be equipped to tackle stuff like logic, calculus or differential equations until they're a bit older.

Maybe we could teach a bit of the log tables so people can use it to multiply stuff fast in their heads, like...

+ 3 dB ~ × 2

+ 5 dB = ×sqrt(10) ~ 3.16

and so on... you convert your multiplicands to dBs, do the addition, and re-convert to get your answer.

posted by Monday, stony Monday at 2:57 PM on October 19, 2012

It should be noted that K-6's role isn't really to make children learn anything, at least not intellectually; it's mostly there so they can learn to sit at their place for $x hours, and speak when it's their turn to speak. What they do while they're sitting on that chair is mostly an afterthought.

Might be useful to calculate stuff involving money, since they might start earning some once they're in their teens, but otherwise they won't be equipped to tackle stuff like logic, calculus or differential equations until they're a bit older.

Maybe we could teach a bit of the log tables so people can use it to multiply stuff fast in their heads, like...

+ 3 dB ~ × 2

+ 5 dB = ×sqrt(10) ~ 3.16

and so on... you convert your multiplicands to dBs, do the addition, and re-convert to get your answer.

posted by Monday, stony Monday at 2:57 PM on October 19, 2012

sammyo: "*Actually if you calculate a ballistics problem you've done a problem that is essentially calculus, it may not use Euller notation but unless you're using a table it's approaching basic calculus.*"

Wait, that doesn't seem right.

Ok, I've got a cannon, pointed 33% up. It has a muzzle velocity of 300m/s. Using vectors, that means it's going horizontally at 200m/s, and vertically at 100m/s. Gravity decelerates a rising object at 9.8m/s/s, so, given that v=at, it will decelerate from 100m/s to 0m/s in (100m/s=9.8m/s/s * t) t=10.204 seconds. Since it's going sideways at 200m/s, and it will be in the air for 20.408 seconds (it takes as long to go up as it does to go down), the cannon ball will hit the ground 4,081 meters away.

What part of that was calculus?

humanfont: "*The following questions appeared in my kid's homework recently and left me scratching my head as to their value.*

1- Write a sentence explaining how to add 3 + 4?

2- There are three ones and three tens how many threes and tens are there.

a) 33

b) 6

c) 101010111

d) 7"

Not only do I not understand their value, I can't answer them.

You add three to four by counting to four, and then keeping on counting three more times?

And, if there are three ones and three tens, then there is one three and three tens. So that would be 4...which is not one of the multiple choices.

posted by Bugbread at 4:14 PM on October 19, 2012

Wait, that doesn't seem right.

Ok, I've got a cannon, pointed 33% up. It has a muzzle velocity of 300m/s. Using vectors, that means it's going horizontally at 200m/s, and vertically at 100m/s. Gravity decelerates a rising object at 9.8m/s/s, so, given that v=at, it will decelerate from 100m/s to 0m/s in (100m/s=9.8m/s/s * t) t=10.204 seconds. Since it's going sideways at 200m/s, and it will be in the air for 20.408 seconds (it takes as long to go up as it does to go down), the cannon ball will hit the ground 4,081 meters away.

What part of that was calculus?

humanfont: "

1- Write a sentence explaining how to add 3 + 4?

2- There are three ones and three tens how many threes and tens are there.

a) 33

b) 6

c) 101010111

d) 7

Not only do I not understand their value, I can't answer them.

You add three to four by counting to four, and then keeping on counting three more times?

And, if there are three ones and three tens, then there is one three and three tens. So that would be 4...which is not one of the multiple choices.

posted by Bugbread at 4:14 PM on October 19, 2012

The part where you solve that equation numerically? Well that's numerical analysis, or computational fluid dynamics, really; but without calculus, it's going to be quite hard to solve.

posted by Monday, stony Monday at 4:26 PM on October 19, 2012

Shouldn't we be resolving this scientifically? Do a country wide AB test, east of the Mississippi method A, west B, check back in a couple generations.

It will all be soon moot when those lollygagers at the GoogleImplant project get around to releasing the K-12 higher math module.

posted by sammyo at 4:31 PM on October 19, 2012

It will all be soon moot when those lollygagers at the GoogleImplant project get around to releasing the K-12 higher math module.

posted by sammyo at 4:31 PM on October 19, 2012

Ah but that is not what this is about, all of us are derailing. Look at this video: Jo teaching smart beautiful kids sitting on white beanbag chairs. With a great geometric model, "how can you tell how many cubes in this shape without counting?" Well the kids used trig, where did they learn that? I want to learn math in that wonderful room, but I can see where other academics would feel attacked and slip into reactionary badness when Jo starts getting all the grants.

It bothers me a bit that she uses a potentially out of context quote of a private email. Other points do sound like Bishop stepped over the line repeatedly. From the posts here it's easy to see that math education is surprisingly inflammatory.

Actually ocschwar makes the point well.

posted by sammyo at 5:06 PM on October 19, 2012

It bothers me a bit that she uses a potentially out of context quote of a private email. Other points do sound like Bishop stepped over the line repeatedly. From the posts here it's easy to see that math education is surprisingly inflammatory.

Actually ocschwar makes the point well.

posted by sammyo at 5:06 PM on October 19, 2012

The answer they were looking for was cosine theta.

posted by humanfont at 5:13 PM on October 19, 2012

posted by humanfont at 5:13 PM on October 19, 2012

Monday, stony Monday: "*The part where you solve that equation numerically? Well that's numerical analysis, or computational fluid dynamics, really; but without calculus, it's going to be quite hard to solve.*"

I'm sorry, I don't understand. I didn't use the Navier-Stokes equation at all. I assume what you're meaning-but-not-saying is that I should have, but didn't. Which I can totally accept, but looking at that page, I don't see how it applies to cannonballs. Is it because of air resistance?

posted by Bugbread at 5:21 PM on October 19, 2012

I'm sorry, I don't understand. I didn't use the Navier-Stokes equation at all. I assume what you're meaning-but-not-saying is that I should have, but didn't. Which I can totally accept, but looking at that page, I don't see how it applies to cannonballs. Is it because of air resistance?

posted by Bugbread at 5:21 PM on October 19, 2012

Like Barbie says, "Math is Hard"... for some people (like me) who see a bunch of numbers, and then the numbers all get up and start to dance, and then blur together. (But I was good at Geometry, which I could visualize).

Teaching Math is hard. I had some pretty good teachers at the elementary level who patiently tried to gently shove concepts into my head. My High School Instructors were often (but not all) of the harsher: "(here's a bunch of numbers) + (if you don't get it) = (you fail)" philosophy.

Teaching Math for everyone in a single classroom is hard, which is why Educational Theories and trends go through such odd contortions at times (oh, that evil primary school late 1970's new wave "New Math", glad I missed that).

Personally, I do not think that there is any single perfect method of learning Mathematics which can apply to all young people evenly... abstraction with numbers can be complex at times.

posted by ovvl at 5:53 PM on October 19, 2012

Teaching Math is hard. I had some pretty good teachers at the elementary level who patiently tried to gently shove concepts into my head. My High School Instructors were often (but not all) of the harsher: "(here's a bunch of numbers) + (if you don't get it) = (you fail)" philosophy.

Teaching Math for everyone in a single classroom is hard, which is why Educational Theories and trends go through such odd contortions at times (oh, that evil primary school late 1970's new wave "New Math", glad I missed that).

Personally, I do not think that there is any single perfect method of learning Mathematics which can apply to all young people evenly... abstraction with numbers can be complex at times.

posted by ovvl at 5:53 PM on October 19, 2012

I must admit that, like jb, I immediately thought of WALL*E in parts of this thread. Where does this idea that because a computer can do something we shouldn't learn how to do it ourselves come from? I see it all the time. With math, with handwriting, with all sorts of basic tasks.

If you can't do basic math, or write basic English (insert your language here) by hand, or make short logical arguments in paragraph form, you are lacking basic skills of a functional, educated adult. We're not talking about learning how to do calligraphy or triple integrals or whipping out Nabokov quality writing. We're talking about being able to do simple arithmetic or not sound like an illiterate when doing basic composition. Isn't that sort of thing a good in and of itself?

What's next, deciding there's no point in learning to read because kids can point their smartphones at text and have it read to them?

posted by Justinian at 6:34 PM on October 19, 2012 [2 favorites]

If you can't do basic math, or write basic English (insert your language here) by hand, or make short logical arguments in paragraph form, you are lacking basic skills of a functional, educated adult. We're not talking about learning how to do calligraphy or triple integrals or whipping out Nabokov quality writing. We're talking about being able to do simple arithmetic or not sound like an illiterate when doing basic composition. Isn't that sort of thing a good in and of itself?

What's next, deciding there's no point in learning to read because kids can point their smartphones at text and have it read to them?

posted by Justinian at 6:34 PM on October 19, 2012 [2 favorites]

I find myself in a different camp, in that I was pretty good at math, but I never really grokked higher math concepts. I could use calculus to solve problems, even applied problems where there were no indications of which formulas to use, but when the teacher said, " so, you see, the slope at a point indicates the rate of change", all of the other kids in class who were good at math would make be like, "oh, yeah, that makes sense", while I would be like, "ok, if you say so."

So, yeah, there's more than just people who like, understand, and are good at math, and those who dislike, don't understand, and are bad at math.

posted by Bugbread at 6:42 PM on October 19, 2012

So, yeah, there's more than just people who like, understand, and are good at math, and those who dislike, don't understand, and are bad at math.

posted by Bugbread at 6:42 PM on October 19, 2012

Bugbread: I'm very good at arithmetic (including fractions), but not very good at higher maths. I peaked in grade 6 - had 100% on the standardised test - and I seem to just feel quantities (in terms of adding, subtracting, estimating). But basic geometry or statistics breaks my brain.

Maths isn't one skill. My SO has learning disabilities such that he can't really do arithmetic, but he was good at physics (unlike me).

posted by jb at 6:53 PM on October 19, 2012

Maths isn't one skill. My SO has learning disabilities such that he can't really do arithmetic, but he was good at physics (unlike me).

posted by jb at 6:53 PM on October 19, 2012

NO WAY HOW DID YOU DO THAT

I wasn't saying it was HARD to convert a remainder into a decimal or fraction, I said it was a skill that should be taught, because that skill will show up on standardized tests.

posted by 23skidoo at 8:06 AM on October 21, 2012

So I read through this whole post without knowing what "partial quotients" means, and assuming it was some sort of horrid approximation, and I think it's worth actually pointing out that it's a pretty obvious way to make the mental arithmetic in long (/short) division easier.

Suppose (á la Wikipedia) that you want to divide 132 by 8. The usual division algorithm says

Suppose (á la Wikipedia) that you want to divide 132 by 8. The usual division algorithm says

8 goes into 13 once, with 5 left over; 8 goes into 52 six times, with four left over, so 132/8 is 106=16 with four left overTo do it by partial quotients just says

Ten eights is eighty; 132 less 80 is 52. Five eights is forty; 52 less 40 is twelve. One eight is eight; 12 less 8 is four. So 132/8 is ten plus five plus one=16 with four left over.posted by katrielalex at 1:45 AM on October 22, 2012

Bugbread: yes, air resistance is super important in real-life ballistics. And the Navier-Stokes equations are famously impossible to solve analytically except for extremely simplified models.

posted by Monday, stony Monday at 6:59 PM on October 24, 2012

posted by Monday, stony Monday at 6:59 PM on October 24, 2012

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