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How our 1,000-year-old math curriculum cheats America's kids

71 points by chrisgd 13y ago

dalke 13y ago
It sounds like a return to the "new math" of the 1960s. Compare this article: > If we are to give students the right tools to navigate an increasingly math-driven world, we must teach them early on that mathematics is not just about numbers and how to solve equations but about concepts and ideas. > It's about things like ... clock arithmetic — in which adding four hours to 10 a.m. does not get you to 14 but to 2 p.m. — which forms the basis of modern cryptography, protects our privacy in the digital world and, as we've learned, can be easily abused by the powers that be. to the Wikipedia entry at http://en.wikipedia.org/wiki/New_Math http://en.wikipedia.org/wiki/New_Math : > Other topics introduced in the New Math include modular arithmetic, algebraic inequalities, matrices, symbolic logic, Boolean algebra, and abstract algebra.
- j2kun 13y ago
  Any attempt to force a particular mathematical subject matter on students will undoubtedly be ruined by a committee.
  - dalke 13y ago
    Such cynicism. What do you have against Feynman's presence in the California committee to select, among other things, math books? Ahh, I see. If you are the person behind "Math ∩ Programming" then I can see what you have against that committee. Feynman wrote, arguing against new math: > It will perhaps surprise most people who have studied these textbooks to discover that the symbol ∪ or ∩ representing union and intersection of sets and the special use of the brackets { } and so forth, all the elaborate notation for sets that is given in these books, almost never appear in any writings in theoretical physics, in engineering, in business arithmetic, computer design, or other places where mathematics is being used. I see no need or reason for this all to be explained or to be taught in school.
    - j2kun 13y ago
      The reason new math failed is because it was organized and administered by committees (comprised of people who don't do math). So your example is humorously evidence to my claim. What does a committee do when they are tasked with designing a curriculum or standards? They identify what should be tested, and then they pick the order that the topics are taught, likely outlining a time constraint on how long should be spent on each topic. This is the problem. If you're looking for things to test your students on then you will invariably add pointless content. Physicists are no exception (they invented the elaborate and pointless notation for vectors, the bra and ket). I don't propose we test students on mathematical notation, not at all. I propose we actually have them do mathematics (craft proofs, make conjectures), and I assert that any committee trying to craft a curriculum would ruin that. They have already ruined proofs in high school geometry.
      - dalke 13y ago
        "evidence to my claim"? But it's an example of a committee rejecting new math, not proposing it. And do you seriously want to argue that Feynman doesn't "do math"? Actually, it looks like you do. Why should only a certain type of math practitioner determine the curriculum? Do we only let people like Ramanujan set the curriculum? 'Cause that won't work. What you're complaining about is called "democracy." Yes, democracy is messy. The current movement towards high-stakes testing in the US is anti-democratic in part because it coalesces more power into fewer people. If you don't want a committee then what do you want? A so-called "education czar" with absolute power to make things work the way you think it should?
dabrowski 13y ago
This reminds me of Lockhart's Lament, which I highly recommend reading http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...
- chrisBob 13y ago
  That was a fun read, but no one can teach that way. You can't become a teacher by studying math, physics, chemistry, biology, literature, french, programming, history, geography or english. You become a teacher in the US by getting a teaching degree. I believe that is the biggest problem we have in the US education system. Teachers only need minimal knowledge of their subject, and if someone discovers that they don't actually like teaching then they just keep doing it anyway because they got a teaching degree, and can't do much else with it.
  - j2kun 13y ago
    Low standards for math teachers is a huge concern, and Lockhart addresses it too, I believe. Some possible ways to help: give teachers less class time and more time to prepare interesting lessons, pay teachers more, raise the standards of training and hiring.
    - teach 13y ago
      And that's the real problem: those things all cost money. Quite a bit, actually.
      - j2kun 13y ago
        I can't think of a more worthy cause, and it would be chump change compared to what we spend on bailouts/wars/petty politics.
  - btilly 13y ago
    Not only do you have to get a teaching degree, but smart people don't want to do that. See http://www.statisticbrain.com/iq-estimates-by-intended-college-major/ http://www.statisticbrain.com/iq-estimates-by-intended-colle... and note that, relative to other college majors, education majors tend to be dumb.
Spooky23 13y ago
I don't think that you can appreciate this stuff until you actually have the mechanical background in math. Unlike most subjects in K-12, you can't BS math. You're right or wrong. There's no subjectivity in evaluation, and no opportunity for lazy students like me to BS a tired teacher with flowery prose that doesn't say much. Why not make math accessible in easier or more practical ways? My trigonometry teacher walked us out into the schoolyard and gave teams of students assignments to measure the heights of various objects. It was a simple thing, but it made the not-so-interesting mechanics of looking up sine/cosine/tangent in some table real.
- hessenwolf 13y ago
  I do maths every day and I do nothing but approximations and fudges to roughly model real situations. It's like recreating a waterfall in lego.
- j2kun 13y ago
  This is the viewpoint of an engineer. The salient difference in thought is that the goal is rarely to be right (initially), but to get insight about mathematical stuff. Mathematicians do (and students should) focus on proofs over computations, and there is an extreme amount of subjectivity there. If you produce a false proof, it can nevertheless be beautiful and yield fantastic insights. Likewise, a correct proof can be unsatisfying and ugly. Your goal then is to revise it (or completely rewrite it) to make it more beautiful and insightful. Here is an example: the 7 Bridges of Konigsberg problem is one of the most famous (solved) problems in all of mathematics. One can easily give a proof by exhaustion, but that is wholly unsatisfying and perhaps the ugliest possible proof. A much better proof involves some insight into graph theory, and a satisfying and elegant proof would give you a characterization of these kinds of trails in graphs. You don't need a mechanical background to start trying to solve such puzzles (I know because I've taught it that way), and you don't need any mathematical background to appreciate the very elementary proof. But as you let students flounder with puzzles like these, you can slowly introduce technical matter. The point is that it has a context they care about (people like working on puzzles!).
  - ghaff 13y ago
sn41 13y ago
As a person who does some math, I disagree with this article. I agree with the author that mathematics is the language of abstraction, but do not think that school mathematics should put abstract before the concrete. One would think that the failure of "New Math" of the 1960s would have been enough: http://www.youtube.com/watch?v=8wHDn8LDks8 http://www.youtube.com/watch?v=8wHDn8LDks8 [Tom Lehrer, "New Math"] Just because something is abstract and more general does not necessarily imply that it is more insightful than concrete calculations. I feel that abstract mathematics should be confined to higher education. A good antidote to such articles are these by legendary mathematicians: http://pauli.uni-muenster.de/~munsteg/arnold.html http://pauli.uni-muenster.de/~munsteg/arnold.html [V. I. Arnold] http://ega-math.narod.ru/LSP/ch1.htm http://ega-math.narod.ru/LSP/ch1.htm [L. S. Pontryagin]
jerf 13y ago
Focusing on the age is probably the wrong approach. Math true then is still true now. The problem is that the curriculum is now held sacred (and I mean that fully as nastily as possible), and it is being held there by people who are now the blind-leading-the-blind unto the sixth or seventh generation. I've gotten around in the math world a lot since primary school, and what I was taught vs. any actual modern mathematical practice, across the entire spectrum from utterly practical to the utterly theoretical, bears surprisingly little resemblance to what I wasted my time with in school. The current curriculum is bizarrely overfocused on real analysis to the near exclusion of all else (a few other disciplines will be introduced, generally poorly motivated either practically or theoretically, fiddled with for a couple of weeks, then dismissed; matrix math, for instance... in primary school nobody ever gave me a clue why we cared about this, and once you're done solving polynomial sets of equations (itself an unmotivated activity), you're pretty much done), and even if we grant that real analysis really is worth crowding out everything else, it still comes with a very 19th century flavor to the whole thing, too, wasting time on some dead-ends and missing some things that would be both mathematically and practically useful, too. (For instance, numerical approximations methods are usually covered as "Newton' Method For Finding Roots", and that's it. There's several things I'd trade away in favor of more of this, and it's easy to motivate why this is useful stuff.) I'm pretty pessimistic in the short term about any efforts to reformulate math curricula, though, due to the aforementioned blind-leading-the-blind unto the seventh generation problem; math educators aren't even aware how bad the problems are, let alone in any position to fix them themselves, or aware enough to ask for help.
- gus_massa 13y ago
  The curriculum changed. For example, logarithm tables were a standard topic. I don't know haw many hours in a year were use to explain it. Now that subject luckily has disappeared because it's not necessary. Note: There are some tables that are very similar, for example the voltage of a thermocouple.
  - j2kun 13y ago
    And yet, we somehow think that being able to draw an accurate picture of an ellipse given its equation is useful, along with knowing the derivative of every inverse trig function, and countless other things.
spenrose 13y ago
Frenkel laid out his perspective in a mathematical memoir, much admired: http://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465050743
AutoCorrect 13y ago
And then there's Common Core, which seems to teach children insanity.
- Shivetya 13y ago
  I thought some if not most of CC in regards to Math was to follow the Singapore model? Is there something inherently wrong with that approach?
reallysmart 13y ago
im so smart I can learn math just by thinking about it
w1ntermute 13y ago
The biggest problem in education isn't the curriculum or method of teaching or the teachers, it's how motivated the student is (which is usually a product of their home environment). Trying to solve this social issue with "technical" fixes is not going to succeed.
- drcube 13y ago
  Try motivating art students to paint fences. And never show them any "art" except different colored fences. That's what this article is about. If you teach real math, you can actually motivate people to want to learn it.
- ronaldx 13y ago
  >The biggest problem in education isn't the curriculum or method of teaching or the teachers, it's how motivated the student is (which is usually a product of their home environment) While I get what you're saying, the motivation of the student is highly influenced by the curriculum and the teacher. Having a meaningful curriculum content with a teacher who presents it in a way which the student can relate to is surely possible (but woefully rare). Good teaching can change things around.
Steuard 13y ago
I'm very much a math person, but I think this author is being rather naive. In the current political climate surrounding education, it seems like madness to say, "Teach math more like we teach art!" After all, art classes in the US are pretty much first on the chopping block for the many, many people who think education should be all about test scores and checklists. I would love to see more kids exposed to "real math" somehow. But there's a valid question of how to make sure they wind up with practical, applicable skills in the process, and I found the arguments here about "the value of abstract thinking" too vague to feel convincing (at least to folks who aren't already immersed in real math). I don't know the answers to those questions, but I certainly hope we can find them.
- j2kun 13y ago
  The problem is that they already don't end up with practical applicable skills, and this method (teaching with an emphasis on discovery over lecture) is known to have results. See, for example, this lengthy technical document describing (in more scientific terms) teaching math like art [1] [1]: http://www.ams.org/notices/201308/rnoti-p1018.pdf http://www.ams.org/notices/201308/rnoti-p1018.pdf
- jcampbell1 13y ago
  Not to mention, "The curriculum is 1000 years old" is a terrible argument from a logical perspective. I have no idea if that means math is timeless or dated. There are other funny gems. "A mathematical statement is either true or false"
  - arbitrage 13y ago
    Gödel might have something to say about that. Then again, he might not.
- acknowledge 13y ago
  I agree, and I don't see the relevance of 1,000 old findings to an bad math curriculum. The beauty of math is that findings 1,000 years ago are still relevant today. I think first steps in improving math education are teaching concept instead of calculation, and focus on understanding math principles instead of memorizing. As a math/CS major, when I wanted to fully understand the basic concepts of pi, pythagorean theorem, I looked back at how they were originally discovered. It gave math a story behind it, and these concepts stuck with me because I understood them. I think we could improve people's understanding of math through telling the story of many of the concepts, rather than throwing concepts at students and hoping that they stick. Stephen Wolfram discusses the concept will on Concept vs. calculation: http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_m... Memorization is the road to failure in math, and many teachers push memorization rather than understanding.
gus_massa 13y ago
Math is like knitting. It's not useful to read about it. You must do it. When reading an advanced math book for the fist time, I can spend 1 hour per page, because I must copy the proof, try a few "improvements" that are dead ends, do some exercise, and understand what is the main idea of that part. It's very difficult to mixing some modern subjects without calculation. Most of the time, the idea is to just hide the technical details so now it's totally unintelligible and the only possible way to learn it is by memorization. It's possible to teach magic and religion in the same way. I like to add some comments about applications in my classes, but they are marked as off topic, and they are short because usually there is no time to spare. Some of the proposed subjects are plausible to teach, for example module arithmetic. I'd like to see a discussion in school about the module 2, 9 and 10 arithmetic. (2 is almost intuitive, 10 is easy to see, and 9 is the base for the "rule of nines" test, that is usually teached without proof) But, for example, Riemannian geometry is difficult. It's possible to pick a sphere surface and show that the geodesic intersects in two points. Explain the sum of the angles in a triangle? And then what? I don't have any intuition about hyperbolic geometry. It's difficult to students to see what is happening with the geometry in a plane in spite it's easy to draw, and we have paper everywhere to tray. > I recently visited students in fourth, fifth and sixth grades at a school in New York to talk about the ideas of modern math, ideas they had never heard of before. [...] I used a Rubik's Cube to explain symmetry groups: Every rotation of the cube is a "symmetry," and these combine into what mathematicians call a group. I saw students' eyes light up when they realized that when they were solving the puzzle, they were simply discerning the structure of this group. It's an interesting topic, but the problem is not how much you can explain but how much the students understand. Did he make any test to measure it? Can the students solve the Rubik's Cube? I'd like to see a test for this idea. Pick 10 classes and divide them in two groups. Give 5 of them the traditional education and 5 of them the "modern topics" proposal. Pick the classes at random. Pick the teachers at random!!! Don't compare one class with a standard underpaid "traditional" teacher with 30 students and a class with a specially selected for the experiment teacher with 10 students. And compare them with multiple choice exercise, not teacher opinion or self report interest.
- kaitai 13y ago
  Since you mentioned knitting (and I agree with your analogy), check out the great intuition you can get about hyperbolic geometry via crochet and knitting. See how "parallel" lines act and the angle sums of triangles end up smaller than 180 degrees: http://theiff.org/oexhibits/oe1e.html http://theiff.org/oexhibits/oe1e.html A TED lecture is linked: http://geknitics.com/2010/11/sunday-knitting-science-hyperbolic-geometry/ http://geknitics.com/2010/11/sunday-knitting-science-hyperbo... Hyperbolic baby pants also convertible into a 2-holed torus (not while on baby I guess), with added note that on 2-holed tori you can make maps that require 8 colors instead of just 4 like maps on the plane or sphere: http://www.ravelry.com/projects/smbelcas/hyperbolic-pants-2 http://www.ravelry.com/projects/smbelcas/hyperbolic-pants-2 Geometry, topology, map-coloring theorems, crafts: all things many kids never get to do anymore.
bhixon 13y ago
This is like learning art history instead of how to draw. How does the "majestic harmony of Platonic solids" help a middle schooler actually do some math? Why should the average sixth grade class learn about Riemann surfaces when most of them haven't even learned what an exponent is yet? An occasional lecture on a 'fun' topic (or even better, an assigned reading and short essay -- expository writing should be practiced in every subject starting in middle school or earlier) could be motivational. But the author wants to spend "20% of class time opening students' eyes to the power and exquisite harmony of modern math," which he expects would "feed their natural curiosity, motivate them to study more and inspire them to engage math beyond the basic requirements." That's fantasy land. The author sounds like what he is, a UC Berkeley mathematics professor. If students are going to succeed in his college-level courses then they need the basics down cold, which means memorization, repetition, and application to concrete problems. That is, the "stale and boring" stuff.
- kaitai 13y ago
  I've taught a lot of enrichment seminars on Saturdays. They involve "clock arithmetic," or Fermat's little theorem and the basics of RSA encryption for older students, or fractals and ways of computing fractal dimension (and exploring polynomials and iteration and Newton's method), or the differences between geometry on a beach ball, a paper towel center tube, and a piece of paper, or even a bit of Mill and Frege's ideas on how to define a number... Kids love this stuff and they get really excited and they start asking questions about infinity and making connections with their other learning. They have fun, but the topics aren't fun -- they are important and useful! Learning about solids has applications (if that's what you value) to proteins and polymers and all sorts of funky stuff. Check out http://phys.org/news/2014-02-years-mathematicians-class-solid.html http://phys.org/news/2014-02-years-mathematicians-class-soli... for some new solids mathematicians have just discovered, inspired by biological research. Seems better than just learning to hate math. Our current system is certainly not effective in teaching young people to add fractions, which I see as the single greatest indicator for competence in college mathematics (sigh). Ok, after another moment's thought your argument seems similar to "Why should kids read stories and books? To succeed in writing dissertations they need grammar. All that extra reading of literature should be confined to the side..."
- 13y ago
drakaal 13y ago
Ok. First I'm not taking lessons from anyone who plagiarizes. This is a re-write of: http://j2kun.svbtle.com/you-never-did-math-in-high-school http://j2kun.svbtle.com/you-never-did-math-in-high-school Which may also be someone's work, but it was one I read recently. Second, I learned some Calculus in school, how about you? That is not a subject a 1000 years old. So we haven't been teaching it that way for 1000 years. Third, did this guy not have story problems? Applied Math is basically what this author is claiming we don't have. Fourth, Graphing is pretty new. We didn't use to visualize plots because we didn't have the resources, I can't tell you when that became part of algebra, but it was in the past 100 years. I can't believe LA Times ran such a poorly informed piece.
geebee 13y ago
I read that article over the weekend. I agree completely, but it's downstream of a bigger problem. Think of it this way - when we file the bug fix, the report will reference a different bug fix that solved this one downstream. We need to draw math teachers from the top tier of math majors (or physics, stats, etc). And if we do that, it's very likely that people with real insight into math will teach it very effectively but less mechanically. One thing I've noticed is that the US system loves a "plan". This might be a result of the career ladder - a teacher is on the bottom rung, whereas someone who sets the curriculum for all teachers has moved beyond that lowly spot. But what we really need is an armada of accomplished math majors in actual teaching positions. Without that, the plan won't really matter. With it, the plan? It matters, sure, but I'd personally be inclined to give quite a bit of latitude and autonomy to people who were in the top tier of their class and majored in math and who are inclined to teach.
- j2kun 13y ago
  Latidude? Autonomy?! If only you were in charge teachers would get something done!
- DerpDerpDerp 13y ago
  > We need to draw math teachers from the top tier of math majors (or physics, stats, etc). And if we do that, it's very likely that people with real insight into math will teach it very effectively but less mechanically. Then don't pay me 25%-33% of what I'd make working in industry.
j2kun 13y ago
> By hiding math's great masterpieces from students' view, we deny them the beauty of the subject. The problem is that math's great masterpieces are problems and proofs. High school math educators are largely the folks who hated doing proofs during their education, and they're just as uninformed about what the interesting open problems are as their students.
- arbitrage 13y ago
  Wow, that's a really bold statement. Do you have a proof of that?
  - Double_Cast 13y ago
    "great" is a superlative. His statement an opinion. But I imagine he knows what he's talking about since he has a PHD in math. [1] [1] http://jeremykun.com/about/ http://jeremykun.com/about/
  - j2kun 13y ago
    This comes from my interactions with colleagues at my own undergraduate institution. So it's biased evidence, but my institution was also one of California's largest math teaching credential programs, so I think it's a reasonable sample. I can't say how many times my 'teaching concentration' colleagues would look at a basic linear algebra problem (which requires a proof) and whine saying things like, "I don't care about this stuff, it's so stupid and pointless, I just want to get my degree so I can go teach high school math." All irony aside, these are the students who actually enjoyed the structure of high school math, and were rudely awakened when they were asked to apply their minds to reason. As a result, 'teaching concentrations' in undergraduate and graduate school usually provide mathematically watered down courses and automatic passing grades, and the teachers-in-training don't learn anything new. They certainly don't learn, for example, any mathematics that has been done since 1900, or any open problems that aren't Millenium Prize Problems. And god forbid they ask and try to answer an original question (as they expect their students to do when they're confused). This is what they mean by the "blind leading the blind."
- SamuelMulder 13y ago
bane 13y ago
I was a terrible math student in school. Looking back, it wasn't because the material was hard, or boring, but because it was completely unmotivated. They may as well have been asking me to organize piles of toothpicks or count ceiling tiles to fill up the class periods. There was simply no sense to it. I think this is partially understood, and the attempted solution is to try to find applications for the math you're learning. Except that once you hit even basic algebra, you quickly run out of applications a young student can relate to. So they go home, ask their parents "what do you use Algebra for?" and get "I don't." and that's that. I think, along with application concepts, the history of these maths...why were they created, about the creators who came up with the concepts, what were they trying to solve, let's solve the same problems with these new techniques, etc. This would have grabbed me and pulled me in. Instead you sometimes get a little box in the reading quickly going over all this (if you are lucky) before a hundred problems are dumped on you to drill through. It's rarely talked about in class, and if it is, it's by a teacher who doesn't know or appreciate the value of this history at all. By doing this you show applicability to entire fields, even if the child doesn't understand what's involved in the field. Those kids that say "when I grow up I want to be a biologist" will encounter stories about people who learned to apply logic or math or fractions or whatever to solve difficult problems in biology or whatever, and then relive those moments as they try to solve the same or similar problem using the tools they were just given.
- wisty 13y ago
  > Except that once you hit even basic algebra, you quickly run out of applications a young student can relate to. Algebra is problem solving. You have a formula describing the problem, and want to re-arrange it to find one of the variables. It has a lot of useful and logical applications. Trigonometry though ... screw that. "Bob the builder has made a truss, and wants to figure out the angle A. He doesn't own a protractor, and can only measure a few of the lengths." Seriously?
  - bane 13y ago
    > Algebra is problem solving. You have a formula describing the problem, and want to re-arrange it to find one of the variables. It has a lot of useful and logical applications. It does, but for average Joe and Carla, who's day job involves making sure the box of sprockets sitting on the loading dock matches the P.O., and who's most pressing math problem is balancing their check book every few days and using turbo tax once a year, they'll never encounter anything more complicated than very basic arithmetic. I think the hardest math problem most people encounter is figuring out percentages so they can fill out the "tip" on the credit card receipt when they eat out. Yeah it's "algebra" but it's a very limited class of problem and people just use a calculator or a tip calculator to do it these days. >Trig Yeah, trig really requires good motivation. The answer to most trig questions really is "buy the appropriate measuring tool" not "do a bunch of calculations and look up the result of the trig functions in a table". I don't ever remember having the calculations for cos(), sin() explained to me, they're just magic functions, number goes in, number comes out.
WalterBright 13y ago
Math curriculum may be rote and boring, but if you don't master arithmetic, fractions, exponents, etc., you're going to have a lifetime of poor decisions regarding investments, financing, etc., which will be very costly.
- sitkack 13y ago
  You can teach exponential growth, compound interest and ratios before mastery of the mechanics. To many, mastery of the mechanics inoculates them not from darkness but from a scientific mind. I know many people with a mastery of arithmetic and fractions yet still make poor mathematical decisions. Conversely, I see people make projections and use algebra w/o even knowing they are doing it. Learning assembly isn't is a prerequisite for making useful higher level software. Knowing all the words doesn't make us a better essayist. Yes the low level stuff can make the result better, the spark still has to exist for the flame to grow.
  - WalterBright 13y ago
    I've talked with people who sell finance, and they make bank off of people with a poor grasp of mathematical fundamentals. And that's just a small bit of what it will cost a person to not master it.
    - sitkack 13y ago
      These people also don't understand concepts like exponential growth, I think a lack of fundamentals is a symptom not a cause.
SixSigma 13y ago
it should be taught as two different subjects: arithmetic and mathematics just like painting and decorating isn't called interior design

71 points by chrisgd 13y ago

dalke 13y ago

It sounds like a return to the "new math" of the 1960s. Compare this article: > If we are to give students the right tools to navigate an increasingly math-driven world, we must teach them early on that mathematics is not just about numbers and how to solve equations but about concepts and ideas. > It's about things like ... clock arithmetic — in which adding four hours to 10 a.m. does not get you to 14 but to 2 p.m. — which forms the basis of modern cryptography, protects our privacy in the digital world and, as we've learned, can be easily abused by the powers that be. to the Wikipedia entry at http://en.wikipedia.org/wiki/New_Math http://en.wikipedia.org/wiki/New_Math : > Other topics introduced in the New Math include modular arithmetic, algebraic inequalities, matrices, symbolic logic, Boolean algebra, and abstract algebra.

j2kun 13y ago
Any attempt to force a particular mathematical subject matter on students will undoubtedly be ruined by a committee.
- dalke 13y ago
  Such cynicism. What do you have against Feynman's presence in the California committee to select, among other things, math books? Ahh, I see. If you are the person behind "Math ∩ Programming" then I can see what you have against that committee. Feynman wrote, arguing against new math: > It will perhaps surprise most people who have studied these textbooks to discover that the symbol ∪ or ∩ representing union and intersection of sets and the special use of the brackets { } and so forth, all the elaborate notation for sets that is given in these books, almost never appear in any writings in theoretical physics, in engineering, in business arithmetic, computer design, or other places where mathematics is being used. I see no need or reason for this all to be explained or to be taught in school.
  - j2kun 13y ago
    The reason new math failed is because it was organized and administered by committees (comprised of people who don't do math). So your example is humorously evidence to my claim. What does a committee do when they are tasked with designing a curriculum or standards? They identify what should be tested, and then they pick the order that the topics are taught, likely outlining a time constraint on how long should be spent on each topic. This is the problem. If you're looking for things to test your students on then you will invariably add pointless content. Physicists are no exception (they invented the elaborate and pointless notation for vectors, the bra and ket). I don't propose we test students on mathematical notation, not at all. I propose we actually have them do mathematics (craft proofs, make conjectures), and I assert that any committee trying to craft a curriculum would ruin that. They have already ruined proofs in high school geometry.
    - dalke 13y ago
      "evidence to my claim"? But it's an example of a committee rejecting new math, not proposing it. And do you seriously want to argue that Feynman doesn't "do math"? Actually, it looks like you do. Why should only a certain type of math practitioner determine the curriculum? Do we only let people like Ramanujan set the curriculum? 'Cause that won't work. What you're complaining about is called "democracy." Yes, democracy is messy. The current movement towards high-stakes testing in the US is anti-democratic in part because it coalesces more power into fewer people. If you don't want a committee then what do you want? A so-called "education czar" with absolute power to make things work the way you think it should?

dabrowski 13y ago

This reminds me of Lockhart's Lament, which I highly recommend reading http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...

chrisBob 13y ago
That was a fun read, but no one can teach that way. You can't become a teacher by studying math, physics, chemistry, biology, literature, french, programming, history, geography or english. You become a teacher in the US by getting a teaching degree. I believe that is the biggest problem we have in the US education system. Teachers only need minimal knowledge of their subject, and if someone discovers that they don't actually like teaching then they just keep doing it anyway because they got a teaching degree, and can't do much else with it.
- j2kun 13y ago
  Low standards for math teachers is a huge concern, and Lockhart addresses it too, I believe. Some possible ways to help: give teachers less class time and more time to prepare interesting lessons, pay teachers more, raise the standards of training and hiring.
  - teach 13y ago
    And that's the real problem: those things all cost money. Quite a bit, actually.
    - j2kun 13y ago
      I can't think of a more worthy cause, and it would be chump change compared to what we spend on bailouts/wars/petty politics.
- btilly 13y ago
  Not only do you have to get a teaching degree, but smart people don't want to do that. See http://www.statisticbrain.com/iq-estimates-by-intended-college-major/ http://www.statisticbrain.com/iq-estimates-by-intended-colle... and note that, relative to other college majors, education majors tend to be dumb.

Spooky23 13y ago

I don't think that you can appreciate this stuff until you actually have the mechanical background in math. Unlike most subjects in K-12, you can't BS math. You're right or wrong. There's no subjectivity in evaluation, and no opportunity for lazy students like me to BS a tired teacher with flowery prose that doesn't say much. Why not make math accessible in easier or more practical ways? My trigonometry teacher walked us out into the schoolyard and gave teams of students assignments to measure the heights of various objects. It was a simple thing, but it made the not-so-interesting mechanics of looking up sine/cosine/tangent in some table real.

hessenwolf 13y ago
I do maths every day and I do nothing but approximations and fudges to roughly model real situations. It's like recreating a waterfall in lego.
j2kun 13y ago
This is the viewpoint of an engineer. The salient difference in thought is that the goal is rarely to be right (initially), but to get insight about mathematical stuff. Mathematicians do (and students should) focus on proofs over computations, and there is an extreme amount of subjectivity there. If you produce a false proof, it can nevertheless be beautiful and yield fantastic insights. Likewise, a correct proof can be unsatisfying and ugly. Your goal then is to revise it (or completely rewrite it) to make it more beautiful and insightful. Here is an example: the 7 Bridges of Konigsberg problem is one of the most famous (solved) problems in all of mathematics. One can easily give a proof by exhaustion, but that is wholly unsatisfying and perhaps the ugliest possible proof. A much better proof involves some insight into graph theory, and a satisfying and elegant proof would give you a characterization of these kinds of trails in graphs. You don't need a mechanical background to start trying to solve such puzzles (I know because I've taught it that way), and you don't need any mathematical background to appreciate the very elementary proof. But as you let students flounder with puzzles like these, you can slowly introduce technical matter. The point is that it has a context they care about (people like working on puzzles!).
- ghaff 13y ago

sn41 13y ago

As a person who does some math, I disagree with this article. I agree with the author that mathematics is the language of abstraction, but do not think that school mathematics should put abstract before the concrete. One would think that the failure of "New Math" of the 1960s would have been enough: http://www.youtube.com/watch?v=8wHDn8LDks8 http://www.youtube.com/watch?v=8wHDn8LDks8 [Tom Lehrer, "New Math"] Just because something is abstract and more general does not necessarily imply that it is more insightful than concrete calculations. I feel that abstract mathematics should be confined to higher education. A good antidote to such articles are these by legendary mathematicians: http://pauli.uni-muenster.de/~munsteg/arnold.html http://pauli.uni-muenster.de/~munsteg/arnold.html [V. I. Arnold] http://ega-math.narod.ru/LSP/ch1.htm http://ega-math.narod.ru/LSP/ch1.htm [L. S. Pontryagin]

jerf 13y ago

Focusing on the age is probably the wrong approach. Math true then is still true now. The problem is that the curriculum is now held sacred (and I mean that fully as nastily as possible), and it is being held there by people who are now the blind-leading-the-blind unto the sixth or seventh generation. I've gotten around in the math world a lot since primary school, and what I was taught vs. any actual modern mathematical practice, across the entire spectrum from utterly practical to the utterly theoretical, bears surprisingly little resemblance to what I wasted my time with in school. The current curriculum is bizarrely overfocused on real analysis to the near exclusion of all else (a few other disciplines will be introduced, generally poorly motivated either practically or theoretically, fiddled with for a couple of weeks, then dismissed; matrix math, for instance... in primary school nobody ever gave me a clue why we cared about this, and once you're done solving polynomial sets of equations (itself an unmotivated activity), you're pretty much done), and even if we grant that real analysis really is worth crowding out everything else, it still comes with a very 19th century flavor to the whole thing, too, wasting time on some dead-ends and missing some things that would be both mathematically and practically useful, too. (For instance, numerical approximations methods are usually covered as "Newton' Method For Finding Roots", and that's it. There's several things I'd trade away in favor of more of this, and it's easy to motivate why this is useful stuff.) I'm pretty pessimistic in the short term about any efforts to reformulate math curricula, though, due to the aforementioned blind-leading-the-blind unto the seventh generation problem; math educators aren't even aware how bad the problems are, let alone in any position to fix them themselves, or aware enough to ask for help.

gus_massa 13y ago
The curriculum changed. For example, logarithm tables were a standard topic. I don't know haw many hours in a year were use to explain it. Now that subject luckily has disappeared because it's not necessary. Note: There are some tables that are very similar, for example the voltage of a thermocouple.
- j2kun 13y ago
  And yet, we somehow think that being able to draw an accurate picture of an ellipse given its equation is useful, along with knowing the derivative of every inverse trig function, and countless other things.

spenrose 13y ago

Frenkel laid out his perspective in a mathematical memoir, much admired: http://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465050743

AutoCorrect 13y ago

And then there's Common Core, which seems to teach children insanity.

Shivetya 13y ago
I thought some if not most of CC in regards to Math was to follow the Singapore model? Is there something inherently wrong with that approach?

reallysmart 13y ago

im so smart I can learn math just by thinking about it

w1ntermute 13y ago

The biggest problem in education isn't the curriculum or method of teaching or the teachers, it's how motivated the student is (which is usually a product of their home environment). Trying to solve this social issue with "technical" fixes is not going to succeed.

drcube 13y ago
Try motivating art students to paint fences. And never show them any "art" except different colored fences. That's what this article is about. If you teach real math, you can actually motivate people to want to learn it.
ronaldx 13y ago
>The biggest problem in education isn't the curriculum or method of teaching or the teachers, it's how motivated the student is (which is usually a product of their home environment) While I get what you're saying, the motivation of the student is highly influenced by the curriculum and the teacher. Having a meaningful curriculum content with a teacher who presents it in a way which the student can relate to is surely possible (but woefully rare). Good teaching can change things around.

Steuard 13y ago

I'm very much a math person, but I think this author is being rather naive. In the current political climate surrounding education, it seems like madness to say, "Teach math more like we teach art!" After all, art classes in the US are pretty much first on the chopping block for the many, many people who think education should be all about test scores and checklists. I would love to see more kids exposed to "real math" somehow. But there's a valid question of how to make sure they wind up with practical, applicable skills in the process, and I found the arguments here about "the value of abstract thinking" too vague to feel convincing (at least to folks who aren't already immersed in real math). I don't know the answers to those questions, but I certainly hope we can find them.

j2kun 13y ago
The problem is that they already don't end up with practical applicable skills, and this method (teaching with an emphasis on discovery over lecture) is known to have results. See, for example, this lengthy technical document describing (in more scientific terms) teaching math like art [1] [1]: http://www.ams.org/notices/201308/rnoti-p1018.pdf http://www.ams.org/notices/201308/rnoti-p1018.pdf
jcampbell1 13y ago
Not to mention, "The curriculum is 1000 years old" is a terrible argument from a logical perspective. I have no idea if that means math is timeless or dated. There are other funny gems. "A mathematical statement is either true or false"
- arbitrage 13y ago
  Gödel might have something to say about that. Then again, he might not.
acknowledge 13y ago
I agree, and I don't see the relevance of 1,000 old findings to an bad math curriculum. The beauty of math is that findings 1,000 years ago are still relevant today. I think first steps in improving math education are teaching concept instead of calculation, and focus on understanding math principles instead of memorizing. As a math/CS major, when I wanted to fully understand the basic concepts of pi, pythagorean theorem, I looked back at how they were originally discovered. It gave math a story behind it, and these concepts stuck with me because I understood them. I think we could improve people's understanding of math through telling the story of many of the concepts, rather than throwing concepts at students and hoping that they stick. Stephen Wolfram discusses the concept will on Concept vs. calculation: http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_m... Memorization is the road to failure in math, and many teachers push memorization rather than understanding.

gus_massa 13y ago

Math is like knitting. It's not useful to read about it. You must do it. When reading an advanced math book for the fist time, I can spend 1 hour per page, because I must copy the proof, try a few "improvements" that are dead ends, do some exercise, and understand what is the main idea of that part. It's very difficult to mixing some modern subjects without calculation. Most of the time, the idea is to just hide the technical details so now it's totally unintelligible and the only possible way to learn it is by memorization. It's possible to teach magic and religion in the same way. I like to add some comments about applications in my classes, but they are marked as off topic, and they are short because usually there is no time to spare. Some of the proposed subjects are plausible to teach, for example module arithmetic. I'd like to see a discussion in school about the module 2, 9 and 10 arithmetic. (2 is almost intuitive, 10 is easy to see, and 9 is the base for the "rule of nines" test, that is usually teached without proof) But, for example, Riemannian geometry is difficult. It's possible to pick a sphere surface and show that the geodesic intersects in two points. Explain the sum of the angles in a triangle? And then what? I don't have any intuition about hyperbolic geometry. It's difficult to students to see what is happening with the geometry in a plane in spite it's easy to draw, and we have paper everywhere to tray. > I recently visited students in fourth, fifth and sixth grades at a school in New York to talk about the ideas of modern math, ideas they had never heard of before. [...] I used a Rubik's Cube to explain symmetry groups: Every rotation of the cube is a "symmetry," and these combine into what mathematicians call a group. I saw students' eyes light up when they realized that when they were solving the puzzle, they were simply discerning the structure of this group. It's an interesting topic, but the problem is not how much you can explain but how much the students understand. Did he make any test to measure it? Can the students solve the Rubik's Cube? I'd like to see a test for this idea. Pick 10 classes and divide them in two groups. Give 5 of them the traditional education and 5 of them the "modern topics" proposal. Pick the classes at random. Pick the teachers at random!!! Don't compare one class with a standard underpaid "traditional" teacher with 30 students and a class with a specially selected for the experiment teacher with 10 students. And compare them with multiple choice exercise, not teacher opinion or self report interest.

kaitai 13y ago
Since you mentioned knitting (and I agree with your analogy), check out the great intuition you can get about hyperbolic geometry via crochet and knitting. See how "parallel" lines act and the angle sums of triangles end up smaller than 180 degrees: http://theiff.org/oexhibits/oe1e.html http://theiff.org/oexhibits/oe1e.html A TED lecture is linked: http://geknitics.com/2010/11/sunday-knitting-science-hyperbolic-geometry/ http://geknitics.com/2010/11/sunday-knitting-science-hyperbo... Hyperbolic baby pants also convertible into a 2-holed torus (not while on baby I guess), with added note that on 2-holed tori you can make maps that require 8 colors instead of just 4 like maps on the plane or sphere: http://www.ravelry.com/projects/smbelcas/hyperbolic-pants-2 http://www.ravelry.com/projects/smbelcas/hyperbolic-pants-2 Geometry, topology, map-coloring theorems, crafts: all things many kids never get to do anymore.

bhixon 13y ago

This is like learning art history instead of how to draw. How does the "majestic harmony of Platonic solids" help a middle schooler actually do some math? Why should the average sixth grade class learn about Riemann surfaces when most of them haven't even learned what an exponent is yet? An occasional lecture on a 'fun' topic (or even better, an assigned reading and short essay -- expository writing should be practiced in every subject starting in middle school or earlier) could be motivational. But the author wants to spend "20% of class time opening students' eyes to the power and exquisite harmony of modern math," which he expects would "feed their natural curiosity, motivate them to study more and inspire them to engage math beyond the basic requirements." That's fantasy land. The author sounds like what he is, a UC Berkeley mathematics professor. If students are going to succeed in his college-level courses then they need the basics down cold, which means memorization, repetition, and application to concrete problems. That is, the "stale and boring" stuff.

kaitai 13y ago
I've taught a lot of enrichment seminars on Saturdays. They involve "clock arithmetic," or Fermat's little theorem and the basics of RSA encryption for older students, or fractals and ways of computing fractal dimension (and exploring polynomials and iteration and Newton's method), or the differences between geometry on a beach ball, a paper towel center tube, and a piece of paper, or even a bit of Mill and Frege's ideas on how to define a number... Kids love this stuff and they get really excited and they start asking questions about infinity and making connections with their other learning. They have fun, but the topics aren't fun -- they are important and useful! Learning about solids has applications (if that's what you value) to proteins and polymers and all sorts of funky stuff. Check out http://phys.org/news/2014-02-years-mathematicians-class-solid.html http://phys.org/news/2014-02-years-mathematicians-class-soli... for some new solids mathematicians have just discovered, inspired by biological research. Seems better than just learning to hate math. Our current system is certainly not effective in teaching young people to add fractions, which I see as the single greatest indicator for competence in college mathematics (sigh). Ok, after another moment's thought your argument seems similar to "Why should kids read stories and books? To succeed in writing dissertations they need grammar. All that extra reading of literature should be confined to the side..."
13y ago

drakaal 13y ago

Ok. First I'm not taking lessons from anyone who plagiarizes. This is a re-write of: http://j2kun.svbtle.com/you-never-did-math-in-high-school http://j2kun.svbtle.com/you-never-did-math-in-high-school Which may also be someone's work, but it was one I read recently. Second, I learned some Calculus in school, how about you? That is not a subject a 1000 years old. So we haven't been teaching it that way for 1000 years. Third, did this guy not have story problems? Applied Math is basically what this author is claiming we don't have. Fourth, Graphing is pretty new. We didn't use to visualize plots because we didn't have the resources, I can't tell you when that became part of algebra, but it was in the past 100 years. I can't believe LA Times ran such a poorly informed piece.

geebee 13y ago

I read that article over the weekend. I agree completely, but it's downstream of a bigger problem. Think of it this way - when we file the bug fix, the report will reference a different bug fix that solved this one downstream. We need to draw math teachers from the top tier of math majors (or physics, stats, etc). And if we do that, it's very likely that people with real insight into math will teach it very effectively but less mechanically. One thing I've noticed is that the US system loves a "plan". This might be a result of the career ladder - a teacher is on the bottom rung, whereas someone who sets the curriculum for all teachers has moved beyond that lowly spot. But what we really need is an armada of accomplished math majors in actual teaching positions. Without that, the plan won't really matter. With it, the plan? It matters, sure, but I'd personally be inclined to give quite a bit of latitude and autonomy to people who were in the top tier of their class and majored in math and who are inclined to teach.

j2kun 13y ago
Latidude? Autonomy?! If only you were in charge teachers would get something done!
DerpDerpDerp 13y ago
> We need to draw math teachers from the top tier of math majors (or physics, stats, etc). And if we do that, it's very likely that people with real insight into math will teach it very effectively but less mechanically. Then don't pay me 25%-33% of what I'd make working in industry.

j2kun 13y ago

> By hiding math's great masterpieces from students' view, we deny them the beauty of the subject. The problem is that math's great masterpieces are problems and proofs. High school math educators are largely the folks who hated doing proofs during their education, and they're just as uninformed about what the interesting open problems are as their students.

arbitrage 13y ago
Wow, that's a really bold statement. Do you have a proof of that?
- Double_Cast 13y ago
  "great" is a superlative. His statement an opinion. But I imagine he knows what he's talking about since he has a PHD in math. [1] [1] http://jeremykun.com/about/ http://jeremykun.com/about/
- j2kun 13y ago
  This comes from my interactions with colleagues at my own undergraduate institution. So it's biased evidence, but my institution was also one of California's largest math teaching credential programs, so I think it's a reasonable sample. I can't say how many times my 'teaching concentration' colleagues would look at a basic linear algebra problem (which requires a proof) and whine saying things like, "I don't care about this stuff, it's so stupid and pointless, I just want to get my degree so I can go teach high school math." All irony aside, these are the students who actually enjoyed the structure of high school math, and were rudely awakened when they were asked to apply their minds to reason. As a result, 'teaching concentrations' in undergraduate and graduate school usually provide mathematically watered down courses and automatic passing grades, and the teachers-in-training don't learn anything new. They certainly don't learn, for example, any mathematics that has been done since 1900, or any open problems that aren't Millenium Prize Problems. And god forbid they ask and try to answer an original question (as they expect their students to do when they're confused). This is what they mean by the "blind leading the blind."
SamuelMulder 13y ago

bane 13y ago

I was a terrible math student in school. Looking back, it wasn't because the material was hard, or boring, but because it was completely unmotivated. They may as well have been asking me to organize piles of toothpicks or count ceiling tiles to fill up the class periods. There was simply no sense to it. I think this is partially understood, and the attempted solution is to try to find applications for the math you're learning. Except that once you hit even basic algebra, you quickly run out of applications a young student can relate to. So they go home, ask their parents "what do you use Algebra for?" and get "I don't." and that's that. I think, along with application concepts, the history of these maths...why were they created, about the creators who came up with the concepts, what were they trying to solve, let's solve the same problems with these new techniques, etc. This would have grabbed me and pulled me in. Instead you sometimes get a little box in the reading quickly going over all this (if you are lucky) before a hundred problems are dumped on you to drill through. It's rarely talked about in class, and if it is, it's by a teacher who doesn't know or appreciate the value of this history at all. By doing this you show applicability to entire fields, even if the child doesn't understand what's involved in the field. Those kids that say "when I grow up I want to be a biologist" will encounter stories about people who learned to apply logic or math or fractions or whatever to solve difficult problems in biology or whatever, and then relive those moments as they try to solve the same or similar problem using the tools they were just given.

wisty 13y ago
> Except that once you hit even basic algebra, you quickly run out of applications a young student can relate to. Algebra is problem solving. You have a formula describing the problem, and want to re-arrange it to find one of the variables. It has a lot of useful and logical applications. Trigonometry though ... screw that. "Bob the builder has made a truss, and wants to figure out the angle A. He doesn't own a protractor, and can only measure a few of the lengths." Seriously?
- bane 13y ago
  > Algebra is problem solving. You have a formula describing the problem, and want to re-arrange it to find one of the variables. It has a lot of useful and logical applications. It does, but for average Joe and Carla, who's day job involves making sure the box of sprockets sitting on the loading dock matches the P.O., and who's most pressing math problem is balancing their check book every few days and using turbo tax once a year, they'll never encounter anything more complicated than very basic arithmetic. I think the hardest math problem most people encounter is figuring out percentages so they can fill out the "tip" on the credit card receipt when they eat out. Yeah it's "algebra" but it's a very limited class of problem and people just use a calculator or a tip calculator to do it these days. >Trig Yeah, trig really requires good motivation. The answer to most trig questions really is "buy the appropriate measuring tool" not "do a bunch of calculations and look up the result of the trig functions in a table". I don't ever remember having the calculations for cos(), sin() explained to me, they're just magic functions, number goes in, number comes out.

WalterBright 13y ago

Math curriculum may be rote and boring, but if you don't master arithmetic, fractions, exponents, etc., you're going to have a lifetime of poor decisions regarding investments, financing, etc., which will be very costly.

sitkack 13y ago
You can teach exponential growth, compound interest and ratios before mastery of the mechanics. To many, mastery of the mechanics inoculates them not from darkness but from a scientific mind. I know many people with a mastery of arithmetic and fractions yet still make poor mathematical decisions. Conversely, I see people make projections and use algebra w/o even knowing they are doing it. Learning assembly isn't is a prerequisite for making useful higher level software. Knowing all the words doesn't make us a better essayist. Yes the low level stuff can make the result better, the spark still has to exist for the flame to grow.
- WalterBright 13y ago
  I've talked with people who sell finance, and they make bank off of people with a poor grasp of mathematical fundamentals. And that's just a small bit of what it will cost a person to not master it.
  - sitkack 13y ago
    These people also don't understand concepts like exponential growth, I think a lack of fundamentals is a symptom not a cause.

SixSigma 13y ago

it should be taught as two different subjects: arithmetic and mathematics just like painting and decorating isn't called interior design