(A new question of the week) Like many questions we get, this one can be solved in many ways. We like to guide a student to whatever solution will fit what they have learned; along the way, we may find various additional methods, and side trips into other topics of interest.
(An archive question of the week) I’m looking for past questions that led to deep discussions. This week, we have a case where a student realized he was doing algebra by rote, not thinking about what variables really mean. This realization was triggered by a step that many students stumble over, where parameters change their …
(A new question of the week) This week’s question, asked in January on the new site, will take us through some tricky areas of calculus, and also give a glimpse both of the value of quoting the entire problem you are working on when you ask for help, and of the interesting side discussions we …
(An archive question of the week) There are several ways to restrict the range of values you need to test when you are searching for zeros of a polynomial (using the Rational Zero Test or the Intermediate Value Theorem, for example). One of them can be quite useful for difficult problems, but can be hard …
(A new question of the week) It has been a while since I have regularly included recent questions to this site, in part due to my focus on big topics. This summer I will be focusing on individual questions, some large and some small. Today, we’ll have two recent questions that look at the same …
(An archive question of the week) Recently we have had a number of series digging into multiple aspects of an idea like area or percentages. But I am always running into miscellaneous questions that can’t fit into such a series because of their length or uniqueness. I want to spend the summer focusing on random …
We have been looking at some classic “false proofs” or “fallacies”, where a seemingly valid proof shows something clearly false to be true. The goal is to learn from these, how to distinguish a valid proof from an error. In a post from last year, What Role Should a Figure Play in a Proof?, I …
Last time we looked at some false proofs, which are often used to help students understand what does and does not constitute a valid proof, and in particular, to remind them to be careful in algebraic proofs, looking for issues like division by zero and taking square roots. This time, we’ll look at two similar …
One of the ways to show the need for careful proof, which I discussed last time, is the existence of apparent proofs of false “facts”. These are also useful for training us to think carefully. In this and the next couple posts, I will look at a few categories of false proofs, focusing on what …
One aspect of mathematics that students often struggle with, particularly in geometry (which traditionally has been where proof is introduced), is writing proofs. Why do we need to learn about proofs? Why are proofs needed in the first place? Here are a few answers we’ve given to these questions.