Cf. the binomial cube (http://homepage.mac.com/montessoriworld/mwei/sensory/sbinoml...), which has been part of Montessori preschool classrooms since, umm, probably since Maria Montessori herself was teaching them a hundred years ago. I think the Montessori cube was based on a similar object designed by Freidrich Fröbel, the man who invented kindergarten.
When I saw something like this at my kids' Montessori school, I thought, they could do that in a textbook. (Most Montessori materials involve manipulation that can't be done on paper.) I wonder if modern textbooks have this illustration.
Thanks to one of the comments on that post I found http://www.betterexplained.com! This is a great resource that I plan on sharing with anyone who asks the "but WHY?" question (whichis a good thing).
I keep meaning to write more of these too. My blog is (ostensibly) about programming, but I find that I keep drifting further and further into mathematical topics without tying them back into programming somehow. I should probably commit to one or the other, I just can't decide.
If you want to read about Zippers and don't mind using your brain, then the presentation in "he Derivative of a Regular Type is its Type of One-Hole Contexts" (http://www.cs.nott.ac.uk/~ctm/diff.pdf) is right for you. He talks about the connection between math and computer science:
"Apart from the implementation of this technology, and
the development of a library of related generic utilities, this work opens up a host of fascinating theoretical possibilities - one only has to open one’s old school textbooks almost at random and ask ‘what does this mean for datatypes?’."
Permutations and combinations.
Taking 3 (visualized as dimensions) from 7 (length of side).
License: Public domain. Attribution requested.
KEY:
Permutation: (order matters).
Repeating (Sequence): All (little cubies). Eg: Passwords.
Non-repeating (Arrangement): All except the reds. (ie transparent + blues). Eg: Number of ways of podium finishes in a race.
Combination: Selection (order doesn't matter).
Repeating: Cubies with white outline. Eg: Number of 3 scoop icecream serving from a set of flavours.
Non-repeating (Subset combination): The blues. Eg: Number of ways to choose 3 member committies from a group.
I use to wonder why there was an outside and inside part when doing FOIL. It was after that I saw the visual proof, and doing other types of multiplication with multiple 'parts/terms', that it slowly dawned on me that there's cross-components. It's the same sort of thing when you multiple matricies or covariances. You need the contribution from every combination of terms.
Yeah...I was in a program called Kumon and learned most of my math there, more than I learned in school. There was a lot of multiplying together polynomials, so I got used to the idea of handling each combination of terms.
There was also a fair amount of factoring. Let me rephrase that: an unreasonable amount of factoring. Factoring down 4th and 5th and I think 6th degree polynomials by hand, for instance, in the equivalent of 10th grade algebra.
By the way, there are analogous patterns between even degreed polynomials, 4th degree polynomials look kind of like quadratics and 6ths do as well, with a kind of rippling pattern in the intermediate terms.
By the time anyone told me about FOIL, I just squinted and shrugged it off. I was so used to taking these polynomials apart that enumerating through all possible combinations was trivial to me.
Incidentally, I always considered it more artful to factor quadratic equations to solve them rather than applying the quadratic formula.
Some people might be surprised to hear this but such visualizations are part of the curriculum for the CBSE of 10th grade (maybe 9th also) students in India. They have to do such things for lots of equations and some geometrical problems. This is called practical Maths, and comprises about 20% of our marks. We have to solve puzzles and do other fun stuff with math visualization.
This is how the EPGY software (I think? It might have been Hopkins CTY, this was over a decade ago) taught factorization in algebra: it was incredibly effective and simply made sense, and was IMO far more effective than what I saw being taught in the classrooms.
