“There are no coincidences in math.” -A 12-year-old student, reacting to an exercise from James Tanton’s “Guide to Everything Fractions“
This is deep, right? I mean, are there? Or aren’t there?
If you believe that mathematics is about looking for patterns, then there shouldn’t be any coincidences.
Not sure I actually believe this, but it’s one possible answer….
Richard Guy’s “Strong Law of Small Numbers” (http://www.math.sjsu.edu/~hsu/courses/126/Law-of-Small-Numbers.pdf) pretty much says the opposite, that math is full of coincidences … I suppose it depends on how you interpret “coincidence”, though.
Given the amount of my students who by some miracle get the right answer by using the wrong method – one that doesn’t translate to other instances – I will disagree with this 12 year old’s analysis.
I teach elementary children who don’t necessarily know what conjecture means, but when someone makes an observation, I often ask “Is that a coincidence or will that always happen?” and I’ve found this question helps younger kids think about generalizations.
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