Oddbean new post about login | logout Thousands of students hit a wall at algebra and calculus because of this aspect of notation. The tendency of more basic courses and examples to omit ratio representation doesn’t help. I have a student who is very process focused (the procedures are safe! reliable) but I’m worried she thinks each of these is a totally different function. Does this mean that ratios never clicked for her? Or is this just about “experience” and with time you are confident that these are all the same beast? https://cdn.masto.host/sauropodswin/media_attachments/files/111/183/613/625/624/072/original/c4bd88613d636d07.png
Most problem sets avoid this stuff and if a student isn’t playing with notation on their own I can understand the confusion.
@134318c2 I feel like some of this I had explained and some I had to reason out myself. I definitely had to sit there with a number line and convince myself that "a negative times a positive makes a negative" "works" by going "ok, multiplying is adding repeatedly, when you add a negative, you subtract, so 2x-3… start at zero, subtract 3, subtract 3 again…and you get -6. ok" and then the joy of negative times negative equals positive after that
@134318c2 two of my math books (I think pre-algebra and algebra I) each had about a chapter's worth of exercise sets devoted to modest rearrangements of this sort. At the time I lived in a chaotic and abusive home, and so did not have any reasonable time or place to do homework, and was thus not fond of any kind of homework that seemed overly obvious or too time-consuming, but I could see the necessity of learning to easily do such modest rearrangements.
@134318c2 I think teaching canonicalization helps with this sort of thing. I think the first example I learned was simplifying fractions, but I don’t remember how I got from learning that to habitually rewriting everything before my 2nd attempt at college
