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And a self-paced course through edX:
There was a great MITx course called Street-Fighting Math. Most of the techniques and ideas taught apply well to mental math. The course materials are freely readable after logging into edX and enrolling. https://www.edx.org/course/street-fighting-math-mitx-6-sfmx
EDIT: Apparently the similarly-titled MIT Press book is freely available as PDF. Now that the course isn't currently running, I'd probably prefer the book. https://mitpress.mit.edu/books/street-fighting-mathematics
⬐ imranqthis is great - thanks!
In fact he has a course on edx for this: https://www.edx.org/course/street-fighting-math-mitx-6-sfmx
⬐ iamcreasyThank you very much for posting this.
The course was also presented on edX: https://www.edx.org/course/street-fighting-math-mitx-6-sfmx
you can enroll and view the archives.
Link to Archived Course on EdX.
For the opposing view see Street Fighting Mathematics which encourages not only using tricks but using them well and often.
The problem is if you only learn the trick without the reasoning behind it. The solution isn't to not learn the trick, it's to learn why it works.
I think it's a disservice to kids who will go into science and engineering if they've never been allowed to use heuristics before. Too often you learn a long-form solution method and then the "trick" to solve it quickly and you need to be able to do both.
⬐ dhimesThe linked website seems to confuse mnemonics (like FOIL) with 'tricks' like cross-multiply (which I thought was for ratios: x/5 = 3/15- I learned 'invert and multiply' for dividing fractions).
I think that tricks that allow you to do things quickly in your head are great- they help build numeracy.⬐ dsharletFOIL is the worst offender in my experience tutoring people in math. I recall several instances of people being completely stumped when trying to multiply a trinomial and a binomial, because they couldn't figure out how to apply FOIL.⬐ dhimesI both tutored and taught math. Never had a problem with FOIL.⬐ nerd_stuffThen come up with a better mnemonic or trick that captures the essence of the general case.
Like Fanana: First times all of them, next times all of them, next times all of them... it's not 100% clear but it took 10 seconds to come up with and it works with trinomials and beyond.
Or FettuchENE: First times each of them, plus next times each of them.... Ok, that's kind of bad, but you get the point. FOIL sticks around not becuase it's the best but because it's the most memorable, it's a meme. We should introduce better memes to compete with it.⬐ julesWhat's wrong with the sum of all ways of picking a term from each factor?⬐ dsharletWhy do we need a mnemonic or a meme for this? Is understanding distribution really that much harder than memorizing FOIL?
At the same time, understanding distribution is so much more powerful because it generalizes to many other uses and later concepts.
I think FOIL sticks around primarily because it is what people are taught. Even when people actually do understand distribution, they may not realize that FOIL is just a special case of distributing.⬐ nerd_stuffIs it harder to understand distribution than memorize FOIL? Probably. It's memorizing a process vs. understanding a concept. FOILing won't help you master algebra but if you're in a science class and your teacher needs to get you up to speed on multiplying pairs of binomials then they'll likely just show you FOIL.
I agree with you that FOILing is stupid and might make the student worse off in the long run, but I think depriving them of getting practive with heuristics might be worse. Ideally a student would notice that FOIL isn't a good trick and stop using it, but there are other "tricks" that are extremely useful.⬐ dhimesIf you're using FOIL to replace the distributive property then you're doing it wrong. It's simply a bookkeeping device, like the little ditties that people have to "memorize" the names of the planets or the Great Lakes.