Teaching mathematics and especially statistics

1 Pedagogy

I do not know much about the pedagogy of mathematics yet.¹

Here are some links that might help:

• Why aren’t math textbooks more straightforward?
• No such thing as a bad math student?

See also general pedagogy.

2 Foundational statistics

I do not mean “measure theoretic probability” but rather “intuition-building introductions to the project of learning about the world systematically from evidence.”

One incredible project is Hubbard (2014), the book by Douglas Hubbard which reframes all the traditional statistics in terms of measuring things. He then compresses an incredible amount of medium-to-advanced methodology into some Excel spreadsheets. The art is he gets lots of mileage out of statistical tricks that are usually emphasised for not being mathematically lavish enough to still make good exam questions.

The Curious Journalist’s Guide to Data By Jonathan Stray.

This is a book about the principles behind data journalism. Not what visualisation software to use and how to scrape a website, but the fundamental ideas that underlie the human use of data. This isn’t “how to use data” but “how data works.”

This gets into some of the mathy parts of statistics, but also the difficulty of taking a census of race and the cognitive psychology of probabilities. It traces where data comes from, what journalists do with it, and where it goes after—and tries to understand the possibilities and limitations. Data journalism is as interdisciplinary as it gets, which can make it difficult to assemble all the pieces you need. This is one attempt. This is a technical book, and uses standard technical language, but all mathematical concepts are explained through pictures and examples rather than formulas.

The life of data has three parts: quantification, analysis, and communication. Quantification is the process that creates data. Analysis involves rearranging the data or combining it with other information to produce new knowledge. And none of this is useful without communicating the result.

Carl T. Bergstrom and Jevin West, in Calling bullshit: Data Reasoning in a Digital World have excellent framing and a wide syllabus of different types of bullshit curation.

Miller (2013) attempts to systematize data analysis for non-specialists who need to analyse and present it to others. The table of contents looks incredible — kind of a minimum viable statistician program — but I have not read it.

3 Memes, puns and cartoons

• 🔥Kareem Carr🔥 (@kareem_carr)
• ML Hipster (@ML_Hipster)
• AI Memes for Artificially Intelligent Teens (@ai_memes)

4 Probability

Especially Bayes’ Theorem for discrete events. We can start from axiomatic probability theory but there is evidence that the best practice is by teaching frequencies (Gigerenzer and Hoffrage 1995; Sedlmeier and Gigerenzer 2001).

• David Spiegelhalter, Using expected frequencies when teaching probability.
• Great Expectations: Probability Through Problems
• DRMacIver’s Notebook: Probably enough probability for you

5 As computational practice

In courses “for hackers” we attempt to give coders stats skills by leveraging their coding skills. I think there is some interesting stuff to be done there, because coding can get you to lots of the same place as mathematics. Cameron Davidson-Pilon, Probabilistic Programming & Bayesian Methods for Hackers (source) is worth trying.

There are some more classical approaches, of course. Here are some freely available online.

• Mine Çetinkaya-Rundel and Johanna Hardin, Introduction to Modern Statistics

• Intro to Statistics In One Hour

• Applied Statistics with R

• Project Mosaic

  publishes university-level texts in statistics, data science, modeling, and scientific computing.

  Handsome lookin’ statistics options include Daniel T. Kaplan’s Statistical Modeling: A Fresh Approach, and his guide to computational calculus.

Going even deeper down this hole, A Data-Centric Introduction to Computing:

we propose a new perspective on structuring computing curricula, which we call data centricity. We view a data-centric curriculum as

data centric = data science + data structures

in that order: we begin with ideas from data science, before shifting to classical ideas from data structures and the rest of computer science. This book lays out this vision concretely and in detail.

Second, computing education talks a great deal about notional machines—abstractions of program behaviour meant to help students understand how programs work—but few curricula actually use one. We take notional machines seriously, developing a sequence of them and weaving them through the curriculum. This ties to our belief that programs are not only objects that run, but also objects that we reason about.

Third, we weave content on socially-responsible computing into the text. Unlike other efforts that focus on exposing students to ethics or the pitfalls of technology in general, we aim to show students how the constructs and concepts that they are turning into code right now can lead to adverse impacts unless used with care. In keeping with our focus on testing and concrete examples, we introduce several topics by getting students to think about assumptions at the level of concrete data. This material is called out explicitly throughout the book.

6 Bayesian inference

For various reasons, probably best done at textbook length. Here are some that look fun.

See Bayesian inference.

7 Tools

• probabilistic spreadsheets
• manim is a tool for pedagogic mathematical animations
• quarto is designed to integrate plots and slides etc

8 Practice-led

Tutorials targeting domain specialists who want to learn data science

• Kaggle Tutorials on Python, Data Viz, Pandas & More
• carpentries incubator: ml-python-supervised-learning: Supervised Learning with Python
• carpentries incubator: Introduction to Machine Learning with Scikit Learn
• fast.ai’s Practical Deep Learning for Coders

9 Moore method

The Moore method sounds delightful and terrifying (Chalice 1995; Cohen 1982; Zitarelli 2004). It’s a pity the creator of the was pretty racist and would have refused to teach some of my colleagues.

The way the course is conducted varies from instructor to instructor, but the content of the course is usually presented in whole or in part by the students themselves. Instead of using a textbook, the students are given a list of definitions and, based on these, theorems which they are to prove and present in class, leading them through the subject material. The Moore method typically limits the amount of material that a class is able to cover, but its advocates claim that it induces a depth of understanding that listening to lectures cannot give.

Anyway, for all that this method sounds fun, I wonder how supported it is by evidence? Should check that.

See R.L. Moore and the Moore Method.

10 Incoming

• The Math Academy Way: Using the Power of Science to Supercharge Student Learning

• Arbital Bayes’ rule: Guide

• rmcelreath/stat_rethinking_2023: Statistical Rethinking Course for Jan-Mar 2023

• Larry Wasserman’s stats course

• Shalizi’s regression lectures

• Moritz Hardt, Benjamin Recht Patterns, predictions, and actions: A story about machine learning

• May I draw your attention especially to Kroese et al. (2019), which I proof-read for my PhD supervisor Zdravko Botev, and enjoyed greatly? It smoothly bridges non-statistics mathematicians into applied statistics, without being excruciating, unlike layperson introductions. It is now freely available online.

• Cosma’s links, targeted more to students committed to being statisticians.

• Intro to Statistics In One Hour

• Live Free or Dichotomize is full of examples.

• There are also statistics podcasts.

• Data Science: A First Introduction

• The Book of Statistical Proofs (Soch et al. 2020) is really good! Simple proof of basic probability results shorn of all fluff, oriented to practical application.

  The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

• Thinking and Explaining

  I’m under the impression that mathematicians often have unspoken thought processes guiding their work which may be difficult to explain, or they feel too inhibited to try. One prototypical situation is this: there’s a mathematical object that’s obviously (to you) invariant under a certain transformation. For instance, a linear map might conserve volume for an ‘obvious’ reason. But you don’t have good language to explain your reason—so instead of explaining, or perhaps after trying to explain and failing, you fall back on computation. You turn the crank and without undue effort, demonstrate that the object is indeed invariant.

  Here’s a specific example. Once I mentioned this phenomenon to Andy Gleason; he immediately responded that when he taught algebra courses, if he was discussing cyclic subgroups of a group, he had a mental image of group elements breaking into a formation organised into circular groups. He said that ‘we’ never would say anything like that to the students. His words made a vivid picture in my head, because it fit with how I thought about groups. I was reminded of my long struggle as a student, trying to attach meaning to ‘group’, rather than just a collection of symbols, words, definitions, theorems and proofs that I read in a textbook.

Numbas

Numbas is an online assessment system designed for mathematical subjects.

Developed by mathematicians at Newcastle University, Numbas is free to use and open-source.

Create a test in the online editor.

Share a link with your students, or upload it to your learning environment.

Students get randomised questions in their browser.

Answers are marked automatically and feedback is instant.

11 References