Why Word Problems?
Why do we assign word problems in math classes? They aren’t easy. My students usually think that word problems are one of the most difficult and traumatic things I ask them to do, and I doubt I’m alone...
View ArticleAsking the Right Question
I’ve been poking around on math reddit lately, and in particular /r/askmath and /r/learnmath. And one thing I’ve really noticed is that many of the posters are really bad at asking questions. Learning...
View ArticleWorking Backwards
I teach a lot of students who are still learning the basics of proof-writing. My calculus students are seeing their first college math, and often my number theory class is the first really proof-heavy...
View ArticleA Neat Argument For the Uniqueness of $e^x$
In my advanced Calculus 1 class I teach a quick unit on differential equations. We don’t have the tools to solve them since we haven’t done integrals, but I talk about what differential equations are...
View ArticleAn easier approach to partial fractions decomposition
I always found partial fraction decomposition incredibly annoying and tedious. But it turns out there’s a much easier way to compute it. (I learned this a couple years ago from Chris Towse). Suppose we...
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The difference between science and engineering
I wrote this essay a few years back elsewhere on the internet. It still seems relevant, so I’m posting this updated and lightly edited version. I’ve noticed that people regularly get confused, on a...
View ArticleNumerical Semigroups and Delta Sets
In this post I want to outline my main research project, which involves non-unique factorization in numerical semigroups. I’m going to define semigroups and numerical semigroups; explain what...
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Paradigms and Priors
Scott Alexander at Slate Star Codex has been blogging lately about Thomas Kuhn and the idea of paradigm shifts in science. This is a topic near and dear to my heart, so I wanted to take the opportunity...
View ArticleAn Overview of Bayesian Inference
A few weeks ago I wrote about Kuhn’s theory of paradigm shifts and how it relates to Bayesian inference. In this post I want to back up a little bit and explain what Bayesian inference is, and...
View Article2019 Spring Class Reflections: Calculus
Now that the term is over, I want to reflect a bit on the courses I taught, what worked well, and what I might want to do differently next time. (Honestly, it probably would have been more useful to...
View ArticleOnline Teaching in the Time of Coronavirus
I’ve been spending a lot of the past week looking at different options for transitioning my teaching online for the rest of the term. There are certainly people far more expert at online instruction...
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The SIR Model of Epidemics
For some reason, a lot of people have gotten really interested in epidemiology lately. Myself included.I have no idea why.Now, I’m not an epidemiologist. I don’t study infectious diseases. But I do...
View ArticleLockdown Recipes: Red Beans and Rice
Since we’re all stuck at home and cooking more than usual, I wanted to share one of my favorite recipes from my childhood, which is also especially suited to our current stuck-at-home ways.1Red Beans...
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What is the Axiom of Choice?
One of the easiest ways to start a (friendly) fight in a group of mathematicians is to bring up the axiom of choice. This axiom has a really interesting place in the foundations of mathematics, and I...
View ArticleMore Thoughts on the Axiom of Choice
I got a lot of good, interesting comments on my recent post on the axiom of choice (both on the post itself, and in this very good Hacker News thread). I wanted to answer some common questions and...
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Pascal’s Wager, Medicine, and the Limits of Formal Reasoning
Scott Alexander at Astral Codex Ten has a good post recently thinking about what he calls Pascalian Medicine. As always the entire post is worth reading, but here’s an excerpt:Another way of looking at...
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Why Isn’t There a Replication Crisis in Math?
One important thing that I think about a lot, even though I have no formal expertise, is the replication crisis. A shocking fraction of published research in many fields, including medicine and...
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Hypothesis Testing and its Discontents, Part 1: How is it Supposed to Work?
In my last post on the replication crisis, I mentioned the basic ideas of statistical hypothesis testing. There wasn’t room to give a full explanation in that post, but hypothesis testing is worth...
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Hypothesis Testing and its Discontents, Part 2: The Conquest of Decision Theory
This is the second-part of a planned three-part series on hypothesis testing.In part 1 of this series, we looked at the historical origins of hypothesis testing, and described two different approaches...
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Hypothesis Testing and its Discontents, Part 3: What Can We Do?
Hypothesis testing is central to the way we do science, but it has major flaws that have encouraged widespread shoddy research. In part 1 of this series, we looked at the historical origins of...
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