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Algebraic and calculus concepts may be better way to introduce children to math

161 points by tokenadult 13y ago

fdej 13y ago
Obligatory Star Trek reference: http://www.youtube.com/watch?v=ETt8GJRbqLc http://www.youtube.com/watch?v=ETt8GJRbqLc
sudhirj 13y ago
http://siterecruit.comscore.com/sr/atlantic/broker.js http://siterecruit.comscore.com/sr/atlantic/broker.js is holding up the entire site for me. Async, anyone?
mathattack 13y ago
The article headline is a little sensationalist. It's very rudimentary, and it's more about learning limits than equations. That said, games like DragonBox highlight that kids are capable of being introduced to these concepts well before schools get around to it.
- MichaelGG 13y ago
  My 5-yr-old played DragonBox a bit - first for the colourful critters and powering up her monster, but then for the problems. It was pretty neat when I wrote down 2 + x = 7 and she was able to rearrange the equation to solve for x. (The later stages get considerably more complicated.) It's fantastic that kids think in terms of symbolic manipulation. Most adults still don't get that, and think maths is just about coming up with numbers.
  - mathattack 13y ago
    I haven't figure out how to convert the on-line intuition into pen and paper. I saw a paper on their website. It still seems like a leap, when the kids haven't formally been taught multiplication, division and fractions. I'm glad that it worked in your case.
    - MichaelGG 13y ago
      Hmm, have you tried having them do the actual problems on paper? No numbers or letters, just copy the screen onto paper: Like write out: Side 1 | Side 2 Fly Box | Mushroom And have them solve into: Side 1 | Side 2 Box | Mushroom Inverted Color Fly ("Dark Fly") From there it's just a visual replacement to turn Box into x, Mushroom into a 7, and Fly into a 2. The game introduces "inverse color = -", so going from 2 to -2 should be a visual translation that doesn't actually require them to understand subtraction or inverse. After they're all done, they might notice "hey, wait, I recognize 7 -2 and it can also be written as 5!" But that's not relevant to manipulating the equation, is it? Sure, without arithmetic they won't be able to calculate the most simple form (7 + -2 -> 5 or 2/4 + 3/6 -> 1) -- but that's a different skill. And people seem to have more issues with handling the equation than actually doing the arithmetic part.
michaelfeathers 13y ago
I know that when I learned calculus the hardest part was reconciling the notion of infinitesimals with algebra. The delta/epsilon definition of a limit was unsettling. It's a shame. Much of calculus is easy to grasp intuitively - Riemann integration is easy to see and explain.
- Pxtl 13y ago
  Yes, the concepts of limits and derivatives are simple and intuitive, but the concept of the epsilon/delta proof just completely soars over students heads. I didn't learn the epsilon/delta stuff until after all that other stuff, and I don't feel like it was the wrong order of things. That and the prof I had never just really explained that epsilon just means "really insanely small" which probably would've helped.
  - ChristianMarks 13y ago
    I had to learn the epsilon-delta definition before I really understood limits. As far as insanely small goes, as you must know, the definition is equivalent to "for every epsilon < M, there exists a delta >0, ..." where M is some positive constant (it could be 1 or 1/10^100...). Sometimes it helps to play with the quantifiers...
- gus_massa 13y ago
  In my university, the first calculus course for most of the people uses only an intuitive definition of limit (without epsilon/delta or epsilon/n_0). Also the definition of derivative and integral is intuitive. The idea is that the students can do the calculations following some rules. And then see some applications, like the tangent line of a function or the area below a function. The first calculus course for engineers and exact science students is more difficult, but the discussion of the technical details of the limit is very small. Perhaps calculate the limit by definition only for 2 or 3 easy functions, and then just use the algebraic properties of the limit. The second course for Math and Physics students has more time assigned to the calculation of limits by definition.
- Shorel 13y ago
cabinpark 13y ago
A big secret in mathematics is that many ideas in mathematics are rather simple and straight forward. Formulating and proving these ideas in a rigorous and logical manner, however, is the difficult part.
- yardie 13y ago
  Explains my joy in puzzles and sequences. Sometimes I don't know why the next number in a series is. I used my intuition, arrived at the correct answer, and now the hard part is explaining myself.
3rd3 13y ago
Discussion on reddit: http://www.reddit.com/r/math/comments/1zfizg/5yearolds_can_learn_calculus_why_playing_with/ http://www.reddit.com/r/math/comments/1zfizg/5yearolds_can_l...
tokenadult 13y ago
I see the article title (which I put on the submission, and one comment commented on) was replaced by an HN moderator with a rewording of the article subtitle. I guess that's okay. The article was especially eye-catching for me, because it was first posted among our mutual Facebook friends by my mentor in homeschooling mathematics, the mother of the first United States woman to win a gold medal in the International Mathematical Olympiad. The teacher featured in the article as the developer of innovative teaching methods for young learners is also a Facebook friend of mine, and we are part of a network of parents and teachers who are curious about how to stimulate interest in mathematics among young learners by doing things differently from the United States typical school curriculum in mathematics. I'm grateful that my children have been exposed to approaches like that (which I follow imperfectly in my homeschooling my children and in teaching other learners in classroom courses for my occupation), as that has helped my oldest son launch into the adult world with good problem-solving skills for hacking at the startup where he works. There is a continual tension in mathematics education between teaching topics in their logical order as we understand mathematical topics in light of modern higher mathematics and teaching topics in their historical order of development (which is largely what happens in school mathematics courses). Children need some concrete, tangible experience with counting and with shapes to understand much about mathematics. The cool abstractions that motivate higher mathematics may be inaccessible to children who have no experience with tangible examples of those absractions. (See http://www.latimes.com/opinion/commentary/la-oe-adv-frenkel-why-study-math-20140302,0,5177338.story http://www.latimes.com/opinion/commentary/la-oe-adv-frenkel-... an article submitted to Hacker News yesterday, for more about this.) But it looks like we can gain in mathematics instruction by letting children play with more sophisticated representations of mathematics than most children get to play with. Games are great learning tools. One of favorite games for teaching mathematics I learned about from John Holt's book How Children Fail (originally published 1964, which I read in 1971 on the advice of my school's assistant principal). The twenty questions game asks children to find a number from 1 to 1,000,000 by asking twenty yes/no questions, which the person who chose the unknown number must answer truthfully. It is an interesting mathematical exercise to show that twenty questions is (barely) enough for finding one number in one million if the twenty questions are used optimally. I have played this game many times with the children in my classes, and the opportunity to play this game again at the end of class is one of my strongest incentives for the children to stay focused during a lesson on a Saturday morning to get through the lesson content efficiently. Playing with challenging problems appears to be the royal road for learning mathematics. http://www.epsiloncamp.org/ProblemsversusExercises.php http://www.epsiloncamp.org/ProblemsversusExercises.php I hope ideas like this spread through many communities in the United States and the English-speaking world, so that more young people gain more opportunity to learn to enjoy challenging mathematics early and often.
nikhizzle 13y ago
I love it. This really appeals to me because of the all the bad ideas I had to unlearn in my 20s: - math is about numbers and arithmetic operations on them - being good at math meant you were good at arithmetic - some people (meaning me) just didn't have what it took to be "good" at math. Reinforced by my high school math and physics teachers. I hated math because I didn't understand that mathematics is a system for representing abstract concepts and manipulating them. Eventually on my 4th try to get calculus, I took a class from nick fiori(http://www.yale.edu/education/about/faculty/fiori.elw062111 http://www.yale.edu/education/about/faculty/fiori.elw062111). His teaching methods opened my mind, and I've gone on since then to become an ardent amateur mathematician with minor publications and career tangents in machine learning and data science. I can only imagine what would have happened if I had been taught math well from an early age.
- Zolomon 13y ago
  This is really great and inspiring to hear! I also had a very bad experience to mathematics during my earlier youth. Relentless studying, book reading and search for motivational information have helped me get over the bullshit I was introduced to. I love math, and it is ing beautiful.
- roasty 13y ago
  Good to hear, any math books you'd recommend to someone who's not there yet, or remember from that class?
  - nikhizzle 13y ago
    Sorry, did not learn much from books. The only one of note is the Tom Mitchell Machine Learning Book, which already requires a basic advanced mathematics fluency. I also believe it is a little out of date. I can recommend a few professors who really opened my mind to how to use math, all at UCSC (CS grad school for me): - Dave Helmbold (Machine Learning) - Kevin Ross (Operations Research) - Martin Abadi (Security)
yardie 13y ago
Does anyone know of any resources to introduce math to a 6yo? I have apps like Dragonbox to introduce algebra to my son. He played with it, completed it, but now is back to the crap IAP games typical on tablets these days. I wouldn't mind a few more like Dragonbox. I also use flashcards to help with the rote math homework.
- tokenadult 13y ago
  I don't ever use flashcards for mathematics learning. Not once, not ever. (I hardly use flashcards for foreign language vocabulary either, not even for Chinese characters, except for recognition of single characters in isolation, because there isn't any one-to-one correspondence of words between languages.) Working a lot of problems[1] eventually makes the math facts second-nature, without flash cards. I do recommend to my children and my students that they consider doing one thing I did as a kid, namely fill out a multiplication table (the one I did in childhood was 30 × 30) by hand by hand calculation. I did that in odd moments during the school day and kept the table stashed in my desk between moments of working on it. Filling in a multiplication table helps learners notice number patterns (it made me very conscious of perfect squares, for example) and produces a tangible accomplishment at the end. For a child the age of your child, I enthusiastically recommend the Miquon Math materials,[2] which are inexpensive and take a thoughtful, playful approach informed by higher mathematics for learning much of the content of the first several years of elementary mathematics. All you need to use with the Miquon Math books are Cuisenaire rods,[3] which are available from various sellers and are a fun plaything in themselves. [1] http://www.epsiloncamp.org/ProblemsversusExercises.php http://www.epsiloncamp.org/ProblemsversusExercises.php [2] http://miquonmath.com/ http://miquonmath.com/ [3] https://en.wikipedia.org/wiki/Cuisenaire_rods https://en.wikipedia.org/wiki/Cuisenaire_rods
  - yardie 13y ago
    Brilliant! thanks. I did the multiplication as a child myself, only 20x20. I've also tasked him with the table, 10x10.
  - ChristianMarks 13y ago
    I've used flash cards (SuperMemo and Anki) for mathematics and do not see anything wrong with using flash cards, memorizing the solutions to 500 complex analysis qualifier exam questions and using spaced repetition algorithms. I would prefer to have a perfect memory and not to (have to) rely on such techniques, but life is short. J.F. Jardine once told me that "there is no right way to do mathematics." But I enjoy solving problems, e.g., http://publicsphere.org/2014/02/07/exercise/ http://publicsphere.org/2014/02/07/exercise/ http://publicsphere.org/2014/01/04/fibrations/ http://publicsphere.org/2014/01/04/fibrations/ http://publicsphere.org/2013/12/23/markets-and-market-games/ http://publicsphere.org/2013/12/23/markets-and-market-games/ http://publicsphere.org/2013/12/14/elementary-equivalence/ http://publicsphere.org/2013/12/14/elementary-equivalence/ http://publicsphere.org/2013/12/12/another-problem/ http://publicsphere.org/2013/12/12/another-problem/ http://publicsphere.org/2013/12/08/problems/ http://publicsphere.org/2013/12/08/problems/ I have many more in password-protected blogs, notes, etc. This morning a friend informed me about Oppia https://www.oppia.org/ https://www.oppia.org/. I took a few minutes to solve their introductory example, Euler Project Problem 1 (Question: What is the sum of all the multiples of 3 or 5 below 1000?). I did this in Mathematica by combining inclusion-exclusion (expressed via indicator functions if you like) with little Gauss's formula for summing the integers from 1 to n. SumN35[n_] := 3SumN[Floor[n/3]] + 5SumN[Floor[n/5]] - 15*SumN[Floor[n/15]] where SumN[n_] := Binomial[n + 1, 2] I got SumN35[999] = 233168. [OT lamentation ahead] Unfortunately, in my experience, employers don't seem to value this activity--at least I tend not to impress them as a confident person who will passionately defend his answers until shown wrong. I don't have the personality, patience and energy for this kind of bravado. (I have an acquaintance who has the requisite unaesthetic attitude, who is not more intelligent that I am and who landed a job at Google.) I tend to have inspirations that come from nowhere that I have to go back and reason about. The act of convincing yourself that a correct answer is really correct can sound like a lack of confidence to some individuals. To me, projecting confidence requires extraneous cognitive load, but I can be confident for an additional charge.] If you tell me that using flash cards is morally wrong, or if you have a theory of learning that invalidates their use, I won't argue.
bane 13y ago
Yes yes yes. The mechanics of basic calculus are remarkably simple. After a while it becomes a game of sorts with a pretty simple set of rules. Once you learn to "see" parts of the equation as symbol blocks (instead of numbers needing computation), you can move them around freely so long as you follow the rules. Algebra is simply a subset of the rules. Teaching kids to think symbolically will help them in so many other fields. I'm pretty convinced that with a little thought, you can teach basic derivation and integration to pretty young kids. Carefully craft the problems to avoid difficult division problems, avoid trig, let them use lookup tables for multiplication and you might be able to get kids under 10 to even do some of this. Then "vertically integrate" other algebraic and trigonometric concepts into this framework, like adding new pieces to the game.
- sp332 13y ago
  Hm, you're talking about teaching symbolic manipulation using calculus, but for me the real ideas of calculus are integration and derivation. I think those ideas are very valuable, and symbolic manipulation is just one way to solve those problems.
  - bane 13y ago
    Can you provide some examples? To me, the basic mechanics I remember (for simple equations like x^3 d/dx) were almost entirely "solvable" by moving bits of the equation around... 1. move the 3 in front of the x 2. where the 3 was, subtract 1 from it and put that back where the 3 was but if you think of it as symbols you get more like 1. move the exponent symbol in front of the x symbol 2. subtract 1 from that exponent and put it back where the original exponent symbol was For me at least, bridging the gap from the specifics of the first example to the more generalized second approach meant that when I was given something like 3x^(y^4) d/dx my mind was absolutely blown because I didn't know if there was some rule I needed to know if the exponent had an exponent or something. Going from the years of the arithmetic approach to really groking the symbolic manipulation approach was really hard for me, and looking at lots of young kids suffering through algebra it was the same. Learning how to recognize the symbols through the specificity of the numbers was something I don't think I really got at all until calculus and it's because the mechanics of the rules of simple derivatives and integration really force you to recognize the underlying symbols...more than algebra did (which for me just seems like a really complicated arithmetic rather than what's actually a pretty simple symbol manipulation exercise). It's odd, because like most technically minded kids, I was already great working with symbols and breaking things down and building them up. I could build huge structures out of various small assemblies with legos, write simple software, crank out papers for English class that would guarantee an A and so on, but understanding that the teachers no longer wanted me to be a human tabulation device but to be a manipulator was something I never really cottoned on to. I know from spending a little time with nieces and nephews younger then 10 that you can teach basic derivatives in about an hour or two and have them fairly reliably doing simple work like the above in an afternoon if you don't worry them too much about simplification of the result or all the arithmetic. My gut says that if you start adding more rules, like what do you do when you have something like (x^3)/(x^2) and how exponents should be subtracted, and with a little care in the examples you show them of this rule in action, you end up showing them all of algebra, fractions, exponents and later trig, logs, etc. while building up and hanging all of these concepts off of calculus. TBH I don't know enough about early childhood education to know if this should be used instead of basic arithmetic or not, but I bet if you use shapes and colors instead of letters and numbers and start to teach basic rules, you can just slowly introduce numeracy later anyways when their brains are more developed.
ef4 13y ago
If you're interested in this kind of work, and especially if you're also interested in using computers to open up new kinds of mathematical play and learning, you should read Seymour Papert's "Mindstorms" book. The ideas are powerful and radical. Radical enough to explain why run-of-the-mill schools never successfully integrated awesome tools like LOGO and its descendants. They can't deal with learning that's so child-driven and free.
chrisBob 13y ago
Am I the only one that enjoyed math more earlier than now? I loved calculus and geometry in high school and undergrad, but now in graduate school I can't follow the physicists at all, and my eyes just glaze over.
Myrmornis 13y ago
"Revolutionizing the way math is taught" and not "memorizing multiplication tables as individual facts rather than patterns" sound like very worthy and important goals. At the same time, it's hard to judge this particular initiative based on the article; it might be Hippie Math for Rich Kids Who Will Study Humanities Subjects and Become Trustafarians by a Berkelely Yoga Instructor"
beat 13y ago
I've thought for a long time that an abstract approach might work better than a concrete approach for introducing children to math. I remember when my kids were in kindergarten, and their math education started with estimation. My first thought was, "Great. Now they've turned math into a touchy-feely all-answers-are-right-answers nonsense subject!"
- gshubert17 13y ago
  I don't want to turn math into a touchy-feely all-answers-are-right-answers nonsense subject, but estimation is a good skill in math, engineering, and computing. Two examples: Fermi problems[1] and Jon Bentley's "back of the envelope"[2] estimates. [1] http://en.wikipedia.org/wiki/Fermi_problem http://en.wikipedia.org/wiki/Fermi_problem [2] http://www.csie.fju.edu.tw/~yeh/research/papers/os-reading-list/bentley-cacm84-envelope.pdf http://www.csie.fju.edu.tw/~yeh/research/papers/os-reading-l... and http://www.eecs.harvard.edu/cs261/background/p176-bentley.pdf http://www.eecs.harvard.edu/cs261/background/p176-bentley.pd...
  - beat 13y ago
    I agree that estimation is important. But so important that we start children with it? That smells wrong to me. Estimation is about intuitive thinking. Mathematics is usually a child's first introduction to formal, rigorous thought. Don't go soft on it.
- pbhjpbhj 13y ago
  I think estimation is an important early concept - often I'll give my lad a question and he'll respond with a first answer that's ludicrous. "Now think about it," I'll say. "If you've only got 30 sweets, how can all 7 people get more than 10, does that sound right? Everyone has to get less than 30, surely?". It's about getting a feel for quantities and units of measure.
NAFV_P 13y ago
> But they also need to see meaningful (to them) people doing meaningful things with math and enjoying the experience. I was wondering if this was essential for the learning process of five or six year olds. Making maths enjoyable for young children would be the primary concern, but showing something meaningful (not necessarily to them) being done with maths by someone meaningful (to them)? It would work better if: The kids require to see something that is relevant to their interests or needs being solved with maths by any old codger, as long as said codger can adequately manipulate algebra. By the way, who is going to teach these children algebra? A fair number of teachers won't be up to the task. Reminds me of the learn to code buzz-phrase being passed around - first off you need to spend money and time on teaching teachers how to code. EDIT: bloomin' asterisks.
kalid 13y ago
(Disclosure: I'll be working with Maria on the calculus series for kids.) I'm a big fan of the conceptual approach. One of the largest problems I see with math education is that we don't check if things are really clicking. I graduated with an engineering degree from a great school, and still didn't have an intuitive understanding for i (the imaginary number) until I was about 26. Go find your favorite tutorial introducing imaginary numbers. Got it? Ok. It probably defines i, talks about its properties (i = sqrt(-1)) and then gets you cranking on polynomials. It's the equivalent of teaching someone to read and then having them solve crossword puzzles. It's such a contrived example! (N.B., this anguish forced me to write a tutorial on imaginary numbers with actual, non-polynomial applications, like rotating a shape without needing trig. See https://news.ycombinator.com/item?id=2712575 https://news.ycombinator.com/item?id=2712575) Calculus needs these everyday applications and intuitions beyond "Oh, let's pretend we're trying to calculate the trajectory of a moving particle." They're out there: my intuition is that algebra gives a static description (here's the cookie), while calculus describes the process that made it: here's the steps that built the cookie. Calculus is the language of science because we want to know how the outcome was produced, not just the final result. d/dx velocity = acceleration means your speed is built up from a sequence of accelerations.
- lifeisstillgood 13y ago
  Yes, yes, yes. I have tried (fitfully) three times to sit down and learn what calculus actually means. I still don't get it. And I suspect that clients and families will push it ever further away. Yes, give me real world puzzles and applications for learning new mental tools and I will probably love them. Make it abstract enough and I cannot see the value in learning it.
  - kalid 13y ago
    I was the same way :). I have an intuition-first guide to calculus, here: http://betterexplained.com/calculus/lesson-1 http://betterexplained.com/calculus/lesson-1 I think the essentials on how to think with calculus can be conveyed in 1 minute. Let me know if the above helps :).
vasundhar 13y ago
There was a study done in a place quite unexpected to me now, and interesting back then when I was in school myself. My Math Teacher told there were some schools where a pilot was conducted teaching calculus concepts from 5th,6th std students and were monitored till they did their graduations and career paths. Interestingly the people who were chosen randomly, performed equally well and their decisions in addition to math were more logical. This study was done in India :)
apalmer 13y ago
I definitely appreciate that math is much more than numbers and equations... On the other hand teaching a kid math has to at some level at some point revolve around taking some numbers and getting other numbers out... I cant really see how one can do algebra without knowing arithematic. algebra is fundamentally a different beast but you just cant do it without some kinda arithmatic
- lutusp 13y ago
  > On the other hand teaching a kid math has to at some level at some point revolve around taking some numbers and getting other numbers out... Yes, but perhaps not the very first thing. To me, the first thing should be interesting, aesthetically pleasing patterns -- images -- that have mathematical meaning (http://arachnoid.com/mandelbrot_set/graphics/title_image.jpg http://arachnoid.com/mandelbrot_set/graphics/title_image.jpg). Then, when the student's curiosity has been aroused, we can "open the machine" and show its inner workings. > I cant really see how one can do algebra without knowing arithematic. But that's easy. I'm not necessarily recommending it, but one can easily have algebra without numbers. Remember that the most useful algebraic equations are those in which the numerical values are irrelevant.
craigching 13y ago
This looks extremely interesting, I have a 5 year old daughter that I want to see the playful side of mathematics before she hits the typical US Maths curriculum. One of my guiding documents has been "Lockhart's Lament" [1], but trying to figure out how to make that a reality for her has been difficult for someone like me with so little time! I am going to check this out tonight! [1] http://en.wikipedia.org/wiki/A_Mathematician%27s_Lament http://en.wikipedia.org/wiki/A_Mathematician%27s_Lament
alexvr 13y ago
Does anyone else find math like trig and calculus dramatically easier to comprehend than stuff like combinatorics?
- nilkn 13y ago
  Combinatorics has always been easier to comprehend for me, but the problems have always been harder. That's because calculus depends on higher-level concepts but is otherwise rote, whereas combinatorics depends on only the most elementary concepts but the problems quickly become very complex. I think this is why combinatorics features so much more strongly in math competitions than does calculus.
nilkn 13y ago
This has, to some extent, been known for a long time. How many stories have you heard about math prodigies? I think the problem, though, is in that very sentence. Rather than considering whether this material might be more accessible than traditionally thought, we just assert outright that the child is a prodigy for learning calculus at a young age and then learn nothing from the observation. I learned calculus in sixth grade and was and am very bad at arithmetic. It was easier for me to understand the fundamental theorem of calculus than it was for me to do long division in my head. I don't think I'm alone. I think I'm just among the few who, for one reason or another, actually made a serious attempt to learn the subject at the time.
- nzp 13y ago
  > It was easier for me to understand the fundamental theorem of calculus than it was for me to do long division in my head. I don't think I'm alone. You're not. I had the same experience. I have severe problems with mundane arithmetic, my mind just locks up when faced with concrete numbers and even simple numerical calculations. OTOH, no problem whatsoever with geometry, (abstract) algebra, calculus etc. I remember when I first got calculus in high school, how natural and easy it felt, how it simply made sense and how all around pleasant and interesting it was, and it struck me how different this experience was to the other more "basic and mundane" areas I struggled with during school. A few years ago I was amused when I saw an interview with an astrophysics PhD who struggled with doing basic arithmetic. It's a source of amusement for her now, but you can imagine the dread she had to endure in "school". So, no, we're not alone. What the article says when it compares what children face in school under the label of "mathematics" to torture akin to digging a trench with a spoon sounds very true to me.
aaronsnoswell 13y ago
Relevant anecdote: For a high school ICT project my class had to make an educational game in Visual Basic. I made a 'math invaders' clone where you answer simple algebra sums (e.g. 5 + x = 12) to defeat the relentless hordes of aliens. At home my 8yo brother watched me play for about 5 minutes, then exclaimed 'Oh I get it, the letters are like numbers that you have to find!', then went on to play the game himself for hours. He learnt algebra about 6 years before any of his classmates through gamification, and years down the track enjoys math more than many of his friends.
kps 13y ago
“What is learned without play is qualitatively different. It helps with test taking and mundane exercises, but it does nothing for logical thinking and problem solving. These things are separate, and you can’t get here from there.” This is the core of what people mean when they — legitimately — talk about ‘passion’. Did you learn computing (or any field) for fun, or alternatively because you thought it paid well, or your parents wanted you to do it, or whatever. (Not to be confused with ‘passion’ as mbaspeak for ‘naïvité enough to work ridiculous hours for peanuts’.)
nroose 13y ago
Ever since I took Discrete Math in College, I have thought it would be a great way to introduce school kids to Math.
niio 13y ago
eitally 13y ago
Maria co-authored a great book aimed at making higher level math concepts fun for schoolchildren titled Moebius Noodles. The site appears to be down now, but here's the FB page: https://www.facebook.com/moebiusnoodles https://www.facebook.com/moebiusnoodles I highly recommend it, as well as signing up to receive periodic emails from Yelena/Maria with additional activities, tips and information.

161 points by tokenadult 13y ago

fdej 13y ago

Obligatory Star Trek reference: http://www.youtube.com/watch?v=ETt8GJRbqLc http://www.youtube.com/watch?v=ETt8GJRbqLc

http://siterecruit.comscore.com/sr/atlantic/broker.js http://siterecruit.comscore.com/sr/atlantic/broker.js is holding up the entire site for me. Async, anyone?

mathattack 13y ago

The article headline is a little sensationalist. It's very rudimentary, and it's more about learning limits than equations. That said, games like DragonBox highlight that kids are capable of being introduced to these concepts well before schools get around to it.

MichaelGG 13y ago
My 5-yr-old played DragonBox a bit - first for the colourful critters and powering up her monster, but then for the problems. It was pretty neat when I wrote down 2 + x = 7 and she was able to rearrange the equation to solve for x. (The later stages get considerably more complicated.) It's fantastic that kids think in terms of symbolic manipulation. Most adults still don't get that, and think maths is just about coming up with numbers.
- mathattack 13y ago
  I haven't figure out how to convert the on-line intuition into pen and paper. I saw a paper on their website. It still seems like a leap, when the kids haven't formally been taught multiplication, division and fractions. I'm glad that it worked in your case.
  - MichaelGG 13y ago
    Hmm, have you tried having them do the actual problems on paper? No numbers or letters, just copy the screen onto paper: Like write out: Side 1 | Side 2 Fly Box | Mushroom And have them solve into: Side 1 | Side 2 Box | Mushroom Inverted Color Fly ("Dark Fly") From there it's just a visual replacement to turn Box into x, Mushroom into a 7, and Fly into a 2. The game introduces "inverse color = -", so going from 2 to -2 should be a visual translation that doesn't actually require them to understand subtraction or inverse. After they're all done, they might notice "hey, wait, I recognize 7 -2 and it can also be written as 5!" But that's not relevant to manipulating the equation, is it? Sure, without arithmetic they won't be able to calculate the most simple form (7 + -2 -> 5 or 2/4 + 3/6 -> 1) -- but that's a different skill. And people seem to have more issues with handling the equation than actually doing the arithmetic part.

michaelfeathers 13y ago

I know that when I learned calculus the hardest part was reconciling the notion of infinitesimals with algebra. The delta/epsilon definition of a limit was unsettling. It's a shame. Much of calculus is easy to grasp intuitively - Riemann integration is easy to see and explain.

Pxtl 13y ago
Yes, the concepts of limits and derivatives are simple and intuitive, but the concept of the epsilon/delta proof just completely soars over students heads. I didn't learn the epsilon/delta stuff until after all that other stuff, and I don't feel like it was the wrong order of things. That and the prof I had never just really explained that epsilon just means "really insanely small" which probably would've helped.
- ChristianMarks 13y ago
  I had to learn the epsilon-delta definition before I really understood limits. As far as insanely small goes, as you must know, the definition is equivalent to "for every epsilon < M, there exists a delta >0, ..." where M is some positive constant (it could be 1 or 1/10^100...). Sometimes it helps to play with the quantifiers...
gus_massa 13y ago
In my university, the first calculus course for most of the people uses only an intuitive definition of limit (without epsilon/delta or epsilon/n_0). Also the definition of derivative and integral is intuitive. The idea is that the students can do the calculations following some rules. And then see some applications, like the tangent line of a function or the area below a function. The first calculus course for engineers and exact science students is more difficult, but the discussion of the technical details of the limit is very small. Perhaps calculate the limit by definition only for 2 or 3 easy functions, and then just use the algebraic properties of the limit. The second course for Math and Physics students has more time assigned to the calculation of limits by definition.
Shorel 13y ago

cabinpark 13y ago

A big secret in mathematics is that many ideas in mathematics are rather simple and straight forward. Formulating and proving these ideas in a rigorous and logical manner, however, is the difficult part.

yardie 13y ago
Explains my joy in puzzles and sequences. Sometimes I don't know why the next number in a series is. I used my intuition, arrived at the correct answer, and now the hard part is explaining myself.

3rd3 13y ago

Discussion on reddit: http://www.reddit.com/r/math/comments/1zfizg/5yearolds_can_learn_calculus_why_playing_with/ http://www.reddit.com/r/math/comments/1zfizg/5yearolds_can_l...

tokenadult 13y ago

I see the article title (which I put on the submission, and one comment commented on) was replaced by an HN moderator with a rewording of the article subtitle. I guess that's okay. The article was especially eye-catching for me, because it was first posted among our mutual Facebook friends by my mentor in homeschooling mathematics, the mother of the first United States woman to win a gold medal in the International Mathematical Olympiad. The teacher featured in the article as the developer of innovative teaching methods for young learners is also a Facebook friend of mine, and we are part of a network of parents and teachers who are curious about how to stimulate interest in mathematics among young learners by doing things differently from the United States typical school curriculum in mathematics. I'm grateful that my children have been exposed to approaches like that (which I follow imperfectly in my homeschooling my children and in teaching other learners in classroom courses for my occupation), as that has helped my oldest son launch into the adult world with good problem-solving skills for hacking at the startup where he works. There is a continual tension in mathematics education between teaching topics in their logical order as we understand mathematical topics in light of modern higher mathematics and teaching topics in their historical order of development (which is largely what happens in school mathematics courses). Children need some concrete, tangible experience with counting and with shapes to understand much about mathematics. The cool abstractions that motivate higher mathematics may be inaccessible to children who have no experience with tangible examples of those absractions. (See http://www.latimes.com/opinion/commentary/la-oe-adv-frenkel-why-study-math-20140302,0,5177338.story http://www.latimes.com/opinion/commentary/la-oe-adv-frenkel-... an article submitted to Hacker News yesterday, for more about this.) But it looks like we can gain in mathematics instruction by letting children play with more sophisticated representations of mathematics than most children get to play with. Games are great learning tools. One of favorite games for teaching mathematics I learned about from John Holt's book How Children Fail (originally published 1964, which I read in 1971 on the advice of my school's assistant principal). The twenty questions game asks children to find a number from 1 to 1,000,000 by asking twenty yes/no questions, which the person who chose the unknown number must answer truthfully. It is an interesting mathematical exercise to show that twenty questions is (barely) enough for finding one number in one million if the twenty questions are used optimally. I have played this game many times with the children in my classes, and the opportunity to play this game again at the end of class is one of my strongest incentives for the children to stay focused during a lesson on a Saturday morning to get through the lesson content efficiently. Playing with challenging problems appears to be the royal road for learning mathematics. http://www.epsiloncamp.org/ProblemsversusExercises.php http://www.epsiloncamp.org/ProblemsversusExercises.php I hope ideas like this spread through many communities in the United States and the English-speaking world, so that more young people gain more opportunity to learn to enjoy challenging mathematics early and often.

nikhizzle 13y ago

I love it. This really appeals to me because of the all the bad ideas I had to unlearn in my 20s: - math is about numbers and arithmetic operations on them - being good at math meant you were good at arithmetic - some people (meaning me) just didn't have what it took to be "good" at math. Reinforced by my high school math and physics teachers. I hated math because I didn't understand that mathematics is a system for representing abstract concepts and manipulating them. Eventually on my 4th try to get calculus, I took a class from nick fiori(http://www.yale.edu/education/about/faculty/fiori.elw062111 http://www.yale.edu/education/about/faculty/fiori.elw062111). His teaching methods opened my mind, and I've gone on since then to become an ardent amateur mathematician with minor publications and career tangents in machine learning and data science. I can only imagine what would have happened if I had been taught math well from an early age.

Zolomon 13y ago
This is really great and inspiring to hear! I also had a very bad experience to mathematics during my earlier youth. Relentless studying, book reading and search for motivational information have helped me get over the bullshit I was introduced to. I love math, and it is ing beautiful.
roasty 13y ago
Good to hear, any math books you'd recommend to someone who's not there yet, or remember from that class?
- nikhizzle 13y ago
  Sorry, did not learn much from books. The only one of note is the Tom Mitchell Machine Learning Book, which already requires a basic advanced mathematics fluency. I also believe it is a little out of date. I can recommend a few professors who really opened my mind to how to use math, all at UCSC (CS grad school for me): - Dave Helmbold (Machine Learning) - Kevin Ross (Operations Research) - Martin Abadi (Security)

yardie 13y ago

Does anyone know of any resources to introduce math to a 6yo? I have apps like Dragonbox to introduce algebra to my son. He played with it, completed it, but now is back to the crap IAP games typical on tablets these days. I wouldn't mind a few more like Dragonbox. I also use flashcards to help with the rote math homework.

tokenadult 13y ago
I don't ever use flashcards for mathematics learning. Not once, not ever. (I hardly use flashcards for foreign language vocabulary either, not even for Chinese characters, except for recognition of single characters in isolation, because there isn't any one-to-one correspondence of words between languages.) Working a lot of problems[1] eventually makes the math facts second-nature, without flash cards. I do recommend to my children and my students that they consider doing one thing I did as a kid, namely fill out a multiplication table (the one I did in childhood was 30 × 30) by hand by hand calculation. I did that in odd moments during the school day and kept the table stashed in my desk between moments of working on it. Filling in a multiplication table helps learners notice number patterns (it made me very conscious of perfect squares, for example) and produces a tangible accomplishment at the end. For a child the age of your child, I enthusiastically recommend the Miquon Math materials,[2] which are inexpensive and take a thoughtful, playful approach informed by higher mathematics for learning much of the content of the first several years of elementary mathematics. All you need to use with the Miquon Math books are Cuisenaire rods,[3] which are available from various sellers and are a fun plaything in themselves. [1] http://www.epsiloncamp.org/ProblemsversusExercises.php http://www.epsiloncamp.org/ProblemsversusExercises.php [2] http://miquonmath.com/ http://miquonmath.com/ [3] https://en.wikipedia.org/wiki/Cuisenaire_rods https://en.wikipedia.org/wiki/Cuisenaire_rods
- yardie 13y ago
  Brilliant! thanks. I did the multiplication as a child myself, only 20x20. I've also tasked him with the table, 10x10.
- ChristianMarks 13y ago
  I've used flash cards (SuperMemo and Anki) for mathematics and do not see anything wrong with using flash cards, memorizing the solutions to 500 complex analysis qualifier exam questions and using spaced repetition algorithms. I would prefer to have a perfect memory and not to (have to) rely on such techniques, but life is short. J.F. Jardine once told me that "there is no right way to do mathematics." But I enjoy solving problems, e.g., http://publicsphere.org/2014/02/07/exercise/ http://publicsphere.org/2014/02/07/exercise/ http://publicsphere.org/2014/01/04/fibrations/ http://publicsphere.org/2014/01/04/fibrations/ http://publicsphere.org/2013/12/23/markets-and-market-games/ http://publicsphere.org/2013/12/23/markets-and-market-games/ http://publicsphere.org/2013/12/14/elementary-equivalence/ http://publicsphere.org/2013/12/14/elementary-equivalence/ http://publicsphere.org/2013/12/12/another-problem/ http://publicsphere.org/2013/12/12/another-problem/ http://publicsphere.org/2013/12/08/problems/ http://publicsphere.org/2013/12/08/problems/ I have many more in password-protected blogs, notes, etc. This morning a friend informed me about Oppia https://www.oppia.org/ https://www.oppia.org/. I took a few minutes to solve their introductory example, Euler Project Problem 1 (Question: What is the sum of all the multiples of 3 or 5 below 1000?). I did this in Mathematica by combining inclusion-exclusion (expressed via indicator functions if you like) with little Gauss's formula for summing the integers from 1 to n. SumN35[n_] := 3SumN[Floor[n/3]] + 5SumN[Floor[n/5]] - 15*SumN[Floor[n/15]] where SumN[n_] := Binomial[n + 1, 2] I got SumN35[999] = 233168. [OT lamentation ahead] Unfortunately, in my experience, employers don't seem to value this activity--at least I tend not to impress them as a confident person who will passionately defend his answers until shown wrong. I don't have the personality, patience and energy for this kind of bravado. (I have an acquaintance who has the requisite unaesthetic attitude, who is not more intelligent that I am and who landed a job at Google.) I tend to have inspirations that come from nowhere that I have to go back and reason about. The act of convincing yourself that a correct answer is really correct can sound like a lack of confidence to some individuals. To me, projecting confidence requires extraneous cognitive load, but I can be confident for an additional charge.] If you tell me that using flash cards is morally wrong, or if you have a theory of learning that invalidates their use, I won't argue.

bane 13y ago

Yes yes yes. The mechanics of basic calculus are remarkably simple. After a while it becomes a game of sorts with a pretty simple set of rules. Once you learn to "see" parts of the equation as symbol blocks (instead of numbers needing computation), you can move them around freely so long as you follow the rules. Algebra is simply a subset of the rules. Teaching kids to think symbolically will help them in so many other fields. I'm pretty convinced that with a little thought, you can teach basic derivation and integration to pretty young kids. Carefully craft the problems to avoid difficult division problems, avoid trig, let them use lookup tables for multiplication and you might be able to get kids under 10 to even do some of this. Then "vertically integrate" other algebraic and trigonometric concepts into this framework, like adding new pieces to the game.

sp332 13y ago
Hm, you're talking about teaching symbolic manipulation using calculus, but for me the real ideas of calculus are integration and derivation. I think those ideas are very valuable, and symbolic manipulation is just one way to solve those problems.
- bane 13y ago
  Can you provide some examples? To me, the basic mechanics I remember (for simple equations like x^3 d/dx) were almost entirely "solvable" by moving bits of the equation around... 1. move the 3 in front of the x 2. where the 3 was, subtract 1 from it and put that back where the 3 was but if you think of it as symbols you get more like 1. move the exponent symbol in front of the x symbol 2. subtract 1 from that exponent and put it back where the original exponent symbol was For me at least, bridging the gap from the specifics of the first example to the more generalized second approach meant that when I was given something like 3x^(y^4) d/dx my mind was absolutely blown because I didn't know if there was some rule I needed to know if the exponent had an exponent or something. Going from the years of the arithmetic approach to really groking the symbolic manipulation approach was really hard for me, and looking at lots of young kids suffering through algebra it was the same. Learning how to recognize the symbols through the specificity of the numbers was something I don't think I really got at all until calculus and it's because the mechanics of the rules of simple derivatives and integration really force you to recognize the underlying symbols...more than algebra did (which for me just seems like a really complicated arithmetic rather than what's actually a pretty simple symbol manipulation exercise). It's odd, because like most technically minded kids, I was already great working with symbols and breaking things down and building them up. I could build huge structures out of various small assemblies with legos, write simple software, crank out papers for English class that would guarantee an A and so on, but understanding that the teachers no longer wanted me to be a human tabulation device but to be a manipulator was something I never really cottoned on to. I know from spending a little time with nieces and nephews younger then 10 that you can teach basic derivatives in about an hour or two and have them fairly reliably doing simple work like the above in an afternoon if you don't worry them too much about simplification of the result or all the arithmetic. My gut says that if you start adding more rules, like what do you do when you have something like (x^3)/(x^2) and how exponents should be subtracted, and with a little care in the examples you show them of this rule in action, you end up showing them all of algebra, fractions, exponents and later trig, logs, etc. while building up and hanging all of these concepts off of calculus. TBH I don't know enough about early childhood education to know if this should be used instead of basic arithmetic or not, but I bet if you use shapes and colors instead of letters and numbers and start to teach basic rules, you can just slowly introduce numeracy later anyways when their brains are more developed.

ef4 13y ago

If you're interested in this kind of work, and especially if you're also interested in using computers to open up new kinds of mathematical play and learning, you should read Seymour Papert's "Mindstorms" book. The ideas are powerful and radical. Radical enough to explain why run-of-the-mill schools never successfully integrated awesome tools like LOGO and its descendants. They can't deal with learning that's so child-driven and free.

chrisBob 13y ago

Am I the only one that enjoyed math more earlier than now? I loved calculus and geometry in high school and undergrad, but now in graduate school I can't follow the physicists at all, and my eyes just glaze over.

Myrmornis 13y ago

"Revolutionizing the way math is taught" and not "memorizing multiplication tables as individual facts rather than patterns" sound like very worthy and important goals. At the same time, it's hard to judge this particular initiative based on the article; it might be Hippie Math for Rich Kids Who Will Study Humanities Subjects and Become Trustafarians by a Berkelely Yoga Instructor"

beat 13y ago

I've thought for a long time that an abstract approach might work better than a concrete approach for introducing children to math. I remember when my kids were in kindergarten, and their math education started with estimation. My first thought was, "Great. Now they've turned math into a touchy-feely all-answers-are-right-answers nonsense subject!"

gshubert17 13y ago
I don't want to turn math into a touchy-feely all-answers-are-right-answers nonsense subject, but estimation is a good skill in math, engineering, and computing. Two examples: Fermi problems[1] and Jon Bentley's "back of the envelope"[2] estimates. [1] http://en.wikipedia.org/wiki/Fermi_problem http://en.wikipedia.org/wiki/Fermi_problem [2] http://www.csie.fju.edu.tw/~yeh/research/papers/os-reading-list/bentley-cacm84-envelope.pdf http://www.csie.fju.edu.tw/~yeh/research/papers/os-reading-l... and http://www.eecs.harvard.edu/cs261/background/p176-bentley.pdf http://www.eecs.harvard.edu/cs261/background/p176-bentley.pd...
- beat 13y ago
  I agree that estimation is important. But so important that we start children with it? That smells wrong to me. Estimation is about intuitive thinking. Mathematics is usually a child's first introduction to formal, rigorous thought. Don't go soft on it.
pbhjpbhj 13y ago
I think estimation is an important early concept - often I'll give my lad a question and he'll respond with a first answer that's ludicrous. "Now think about it," I'll say. "If you've only got 30 sweets, how can all 7 people get more than 10, does that sound right? Everyone has to get less than 30, surely?". It's about getting a feel for quantities and units of measure.

NAFV_P 13y ago

> But they also need to see meaningful (to them) people doing meaningful things with math and enjoying the experience. I was wondering if this was essential for the learning process of five or six year olds. Making maths enjoyable for young children would be the primary concern, but showing something meaningful (not necessarily to them) being done with maths by someone meaningful (to them)? It would work better if: The kids require to see something that is relevant to their interests or needs being solved with maths by any old codger, as long as said codger can adequately manipulate algebra. By the way, who is going to teach these children algebra? A fair number of teachers won't be up to the task. Reminds me of the learn to code buzz-phrase being passed around - first off you need to spend money and time on teaching teachers how to code. EDIT: bloomin' asterisks.

kalid 13y ago

(Disclosure: I'll be working with Maria on the calculus series for kids.) I'm a big fan of the conceptual approach. One of the largest problems I see with math education is that we don't check if things are really clicking. I graduated with an engineering degree from a great school, and still didn't have an intuitive understanding for i (the imaginary number) until I was about 26. Go find your favorite tutorial introducing imaginary numbers. Got it? Ok. It probably defines i, talks about its properties (i = sqrt(-1)) and then gets you cranking on polynomials. It's the equivalent of teaching someone to read and then having them solve crossword puzzles. It's such a contrived example! (N.B., this anguish forced me to write a tutorial on imaginary numbers with actual, non-polynomial applications, like rotating a shape without needing trig. See https://news.ycombinator.com/item?id=2712575 https://news.ycombinator.com/item?id=2712575) Calculus needs these everyday applications and intuitions beyond "Oh, let's pretend we're trying to calculate the trajectory of a moving particle." They're out there: my intuition is that algebra gives a static description (here's the cookie), while calculus describes the process that made it: here's the steps that built the cookie. Calculus is the language of science because we want to know how the outcome was produced, not just the final result. d/dx velocity = acceleration means your speed is built up from a sequence of accelerations.

lifeisstillgood 13y ago
Yes, yes, yes. I have tried (fitfully) three times to sit down and learn what calculus actually means. I still don't get it. And I suspect that clients and families will push it ever further away. Yes, give me real world puzzles and applications for learning new mental tools and I will probably love them. Make it abstract enough and I cannot see the value in learning it.
- kalid 13y ago
  I was the same way :). I have an intuition-first guide to calculus, here: http://betterexplained.com/calculus/lesson-1 http://betterexplained.com/calculus/lesson-1 I think the essentials on how to think with calculus can be conveyed in 1 minute. Let me know if the above helps :).

vasundhar 13y ago

There was a study done in a place quite unexpected to me now, and interesting back then when I was in school myself. My Math Teacher told there were some schools where a pilot was conducted teaching calculus concepts from 5th,6th std students and were monitored till they did their graduations and career paths. Interestingly the people who were chosen randomly, performed equally well and their decisions in addition to math were more logical. This study was done in India :)

apalmer 13y ago

I definitely appreciate that math is much more than numbers and equations... On the other hand teaching a kid math has to at some level at some point revolve around taking some numbers and getting other numbers out... I cant really see how one can do algebra without knowing arithematic. algebra is fundamentally a different beast but you just cant do it without some kinda arithmatic

lutusp 13y ago
> On the other hand teaching a kid math has to at some level at some point revolve around taking some numbers and getting other numbers out... Yes, but perhaps not the very first thing. To me, the first thing should be interesting, aesthetically pleasing patterns -- images -- that have mathematical meaning (http://arachnoid.com/mandelbrot_set/graphics/title_image.jpg http://arachnoid.com/mandelbrot_set/graphics/title_image.jpg). Then, when the student's curiosity has been aroused, we can "open the machine" and show its inner workings. > I cant really see how one can do algebra without knowing arithematic. But that's easy. I'm not necessarily recommending it, but one can easily have algebra without numbers. Remember that the most useful algebraic equations are those in which the numerical values are irrelevant.

craigching 13y ago

This looks extremely interesting, I have a 5 year old daughter that I want to see the playful side of mathematics before she hits the typical US Maths curriculum. One of my guiding documents has been "Lockhart's Lament" [1], but trying to figure out how to make that a reality for her has been difficult for someone like me with so little time! I am going to check this out tonight! [1] http://en.wikipedia.org/wiki/A_Mathematician%27s_Lament http://en.wikipedia.org/wiki/A_Mathematician%27s_Lament

alexvr 13y ago

Does anyone else find math like trig and calculus dramatically easier to comprehend than stuff like combinatorics?

nilkn 13y ago
Combinatorics has always been easier to comprehend for me, but the problems have always been harder. That's because calculus depends on higher-level concepts but is otherwise rote, whereas combinatorics depends on only the most elementary concepts but the problems quickly become very complex. I think this is why combinatorics features so much more strongly in math competitions than does calculus.

nilkn 13y ago

This has, to some extent, been known for a long time. How many stories have you heard about math prodigies? I think the problem, though, is in that very sentence. Rather than considering whether this material might be more accessible than traditionally thought, we just assert outright that the child is a prodigy for learning calculus at a young age and then learn nothing from the observation. I learned calculus in sixth grade and was and am very bad at arithmetic. It was easier for me to understand the fundamental theorem of calculus than it was for me to do long division in my head. I don't think I'm alone. I think I'm just among the few who, for one reason or another, actually made a serious attempt to learn the subject at the time.

nzp 13y ago
> It was easier for me to understand the fundamental theorem of calculus than it was for me to do long division in my head. I don't think I'm alone. You're not. I had the same experience. I have severe problems with mundane arithmetic, my mind just locks up when faced with concrete numbers and even simple numerical calculations. OTOH, no problem whatsoever with geometry, (abstract) algebra, calculus etc. I remember when I first got calculus in high school, how natural and easy it felt, how it simply made sense and how all around pleasant and interesting it was, and it struck me how different this experience was to the other more "basic and mundane" areas I struggled with during school. A few years ago I was amused when I saw an interview with an astrophysics PhD who struggled with doing basic arithmetic. It's a source of amusement for her now, but you can imagine the dread she had to endure in "school". So, no, we're not alone. What the article says when it compares what children face in school under the label of "mathematics" to torture akin to digging a trench with a spoon sounds very true to me.

aaronsnoswell 13y ago

Relevant anecdote: For a high school ICT project my class had to make an educational game in Visual Basic. I made a 'math invaders' clone where you answer simple algebra sums (e.g. 5 + x = 12) to defeat the relentless hordes of aliens. At home my 8yo brother watched me play for about 5 minutes, then exclaimed 'Oh I get it, the letters are like numbers that you have to find!', then went on to play the game himself for hours. He learnt algebra about 6 years before any of his classmates through gamification, and years down the track enjoys math more than many of his friends.

kps 13y ago

“What is learned without play is qualitatively different. It helps with test taking and mundane exercises, but it does nothing for logical thinking and problem solving. These things are separate, and you can’t get here from there.” This is the core of what people mean when they — legitimately — talk about ‘passion’. Did you learn computing (or any field) for fun, or alternatively because you thought it paid well, or your parents wanted you to do it, or whatever. (Not to be confused with ‘passion’ as mbaspeak for ‘naïvité enough to work ridiculous hours for peanuts’.)

nroose 13y ago

Ever since I took Discrete Math in College, I have thought it would be a great way to introduce school kids to Math.

niio 13y ago

eitally 13y ago

Maria co-authored a great book aimed at making higher level math concepts fun for schoolchildren titled Moebius Noodles. The site appears to be down now, but here's the FB page: https://www.facebook.com/moebiusnoodles https://www.facebook.com/moebiusnoodles I highly recommend it, as well as signing up to receive periodic emails from Yelena/Maria with additional activities, tips and information.