Teaching mathematics and especially statistics

February 12, 2020 — February 23, 2023

Figure 1

How do I introduce people ot statistics/data science/analytics? What is the most punchy, most efficient modern curriculum?

1 Pedagogy

I do not know much about pedagogy of mathematics yet.

Here are some links that might help:

See also general pedagogy.

Figure 2

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 visualization 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 tho 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

4 Probability

Esp 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).

5 As computational practice

In courses “for hackers” is 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.

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 behavior 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.

5.1 Philosophical / general

5.2 Bootstrap

See bootstrap.

5.3 Hierarchical models

See hierarchical models.

5.4 Causal inference

See causal inference.

5.5 Hypothesis testing

See statistical tests. My question is: do I need to teach this? Is it ever what my students actually need?

5.6 Regression

6 Bayesian inference

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

See Bayesian inference.

6.1 Teaching R

See R.

7 Tools

7.1 geogebra

geogebra is a neat java web app which creates nice plots pedagogically.

8 Practice-led

Tutorials targetting domain specialists who want to learn data science

9 Incoming

  • 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, targetted 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 instant, 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 organized 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

10 Where to host learning notebooks?

See jupyter notebook hosts. )

11 References

Craiu, Gong, and Meng. 2023. Six Statistical Senses.” Annual Review of Statistics and Its Application.
Gelman, Carlin, Stern, et al. 2013. Bayesian Data Analysis. Chapman & Hall/CRC texts in statistical science.
Gelman, Hill, and Vehtari. 2021. Regression and other stories.
Gelman, and Loken. 2012. “Statisticians: When We Teach, We Don’t Practice What We Preach.” Chance.
Gelman, and Nolan. 2017. Teaching Statistics: A Bag of Tricks.
Gigerenzer, and Hoffrage. 1995. How to Improve Bayesian Reasoning Without Instruction: Frequency Formats.” Psychological Review.
Good, and Good. 1999. Resampling Methods: A Practical Guide to Data Analysis.
Hubbard. 2014. How to Measure Anything: Finding the Value of Intangibles in Business.
Kohavi, Tang, and Xu. 2020. Trustworthy Online Controlled Experiments: A Practical Guide to A/B Testing.
Krishnamurthi, and Fisler. 2020. Data-Centricity: A Challenge and Opportunity for Computing Education.” Communications of the ACM.
Kroese, Botev, Taimre, et al. 2019. Mathematical and Statistical Methods for Data Science and Machine Learning. Chapman & Hall/CRC Machine Learning & Pattern Recognition.
Lovett, Meyer, and Thille. 2008. JIME - The Open Learning Initiative: Measuring the Effectiveness of the OLI Statistics Course in Accelerating Student Learning.” Journal of Interactive Media in Education.
McElreath. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and STAN.
McElreath, and Boyd. 2007. Mathematical Models of Social Evolution: A Guide for the Perplexed.
Miller. 2013. The Chicago Guide to Writing about Multivariate Analysis. Chicago Guides to Writing, Editing, and Publishing.
Nolan, and Stoudt. 2020. Reading to Write.” Significance.
Osborne. 2022. Science Education in an Age of Misinformation.”
Sedlmeier, and Gigerenzer. 2001. Teaching Bayesian reasoning in less than two hours.” Journal of Experimental Psychology. General.
Soch, Proofs, Faulkenberry, et al. 2020. StatProofBook/StatProofBook.github.io: StatProofBook 2020.”