Ethan Bolker
• Member for 6 years, 9 months
• Last seen this week
• Boston, MA

Lots of good answers here (I've upvoted many). I'm won't try to add to the discussion about why induction is hard, but I can suggest some approaches that have helped some of my students. Many have ...

Clearly there is no historical data that addresses this question I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or ...

Some students will cheat. (So do some grownups.) In my experience it's not a large fraction, despite the studies that seem to say otherwise. I think I caught most of what went on in my classes. I was ...

A good way to lead to the uniqueness of prime factorization and the convention that $1$ is not a prime is to build factor trees (that's common in elementary school these days in fourth grade, ...

There is no generally applicable answer to your question. In my own career I often taught while learning the material as my mathematics department began to offer more and more computer science. I ...

Most of real mathematics is not about numbers at all. If you can solve hardware and software problems that are essentially puzzles then you could probably do better at mathematics if you wanted to and ...

Grade appropriate elementary number theory: Counting even and odd license plates to see if parity is equally distributed. Same for the residue mod 3. using the sum of digits algorithm. Factoring ...

One way to address this students cannot comprehend that it isn't necessary to always work out the arithmetic right away but can keep them as expressions to observe patterns. is to do (or ...

The other (correct) answers explain why the modulus operation doesn't fit with the other ordinary operations. Nonetheless there are very good reasons for introducing it at many places in the K-8 ...

As you realize, this In my brief experience, I’m finding that my written diagrams or notations are not ideal. Admittedly, I will draw operators, numerals, and arrows on-the-fly, and they don’t ...

Perhaps start with Riemann sums to find distance covered when velocity is known but you can't guess an antiderivative - something they might appreciate if they are fond of derivatives. Gives you a ...

tl;dr Does he need to know algebra, or just to pass the course? Rant follows. The kinds of struggles you describe suggest that he won't/can't major in a science. Perhaps college algebra is simply a ...

I think how you read (and write) them depends on how they are being used. $$x^4 - y^4 = (x^2- y^2)(x^2 + y^2) = (x - y )(x+y)(x^2 + y^2)$$ is fine but pedagogically  \begin{align} x^4 - y^4 &...