In a class on symbolic logic, students are taught the truth tables that define the “logical connectives” ∧ (and), ∨ (or), ¬ (not), and → (if … then). Everything makes sense until they are told that if p is false, then is true whether or not q is true. How can we say that “If pigs fly, then 2 is …
(New Question of the Week) This week I answered a seemingly simple question that can be solved in several different ways when presented as multiple choice, but is rather difficult as a straightforward algebra problem. Trying to guess what the “patient” had done yielded an invalid method that gave the right answer — or was it really invalid? …
(Archive Question of the Week) Having recently discussed a couple different issues that touched on the relationship of math to reality, I was reminded of this old favorite – a question that is not asked often enough, and reveals some of the “dirty little secrets” behind math. Math is not reality; it is often used …
The other day a student I was helping face to face asked how she can know when to check for extraneous solutions of an equation. I gave her a quick version of my standard answer, and the light went on! Today I want to share these thoughts here, because they are very important in several …
(New Question of the Week) Occasionally we get questions challenging the correctness of a textbook problem, or of a test grade. And sometimes we get questions about mathematics used in sciences like physics or chemistry, which lead us to explore unfamiliar fields. The most interesting question this week is one of these. It is one …
(Archive Question of the Week) Recently I discussed the definition of the median of a data set, pointing out how it needs refinements that are not often discussed. In searching for questions in our archive on that topic, I ran across a discussion of an opposite issue: the breadth of the general term “average”, which …
We frequently get questions about how to round; so many different issues arise that I won’t try to fit them all into one post. Children have trouble learning how to do it, and sometimes their parents are surprised to find that they are being taught a different way than they learned. There are several common …
(New Question of the Week) One of the benefits of being a Math Doctor is interacting with the math of many cultures around the world, as we attract an international following. We have observed variations in terminology and notation from country to country, as well as variations in the content of math education (some better …
(Archive question of the week) Sometimes, we get a question that seems familiar, and can look back at past questions to see if we have already answered it, or to get ideas about what it means. This is the story of a problem that seems to float around as folklore, in varied forms. At first …
From time to time, a student will write to us asking for advice on studying, rather than on math itself. As either successful students, or teachers, or (quite often) as adults who recall overcoming difficulties in the past, we have some good advice to offer. Today, I want to look at three answers, by three …