4 ms·

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

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.
mjmahone17 13y ago
A fun one is to fill out a multiplication table of different sizes and bases, especially once kids reach about 5th or 6th grade. I remember reading about how Sumer did math in base 12, and so I tried to replicate a base-12 table, and noticed the way numbers lined up was really different.

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.

mjmahone17 13y ago

A fun one is to fill out a multiplication table of different sizes and bases, especially once kids reach about 5th or 6th grade. I remember reading about how Sumer did math in base 12, and so I tried to replicate a base-12 table, and noticed the way numbers lined up was really different.