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Not true. Everyone makes their own understanding, and successful students
Remember the Anna Karenina principle: Every successful
student understands the material more-or-less the same,
whereas each unsuccessful student is confused in his own way.
Yes. Students have been taught ratios, but they don't recognize when things
If a student can't handle proportional reasoning, in all
likelihood there isn't a problem with the proportionality;
there's a problem with the reasoning. Directly teaching
"ratios" per se is generally a waste of time. For further
discussion, see below.