Reading math and history effectively is not the easiest thing to do, and likely something you have done little of previously. One of the main difficulties students have found in this class is keeping up with this, and doing it well. To help you keep up with reading actively, there will be weekly reading assignments, almost a reading journal. You will make individual entries each week, and a bit later in the week, in.-class and online discussions based on a few of the material and questions that come up.

Why is it hard?

1. Not likely aligned with other interests. Most people don’t go out of their way to do math or learn its history. To succeed, you need to find a way in, a reason to care about this stuff, at least a bit.
2. It’s not linear. Reading math is more like reading comics than novels. Everyone skims, gets lost, and back tracks, a few times even. Many people start doing this, worry that it’s because they are dumb, and give up. Don’t. This is how it is for everyone.
3. Equations and such are slippery. It is soooo easy to gloss over the writing as it turns to calculations. “Yeah, yeah, it makes sense” we say to ourselves even as we don’t really read what’s there. Again, this is universal. We only really read mathematical detail when we get out pencils and try to work through it ourselves as we read.
4. It’s hard to know what to do with it. The trick here is to learn how what’s on the written page might connect to something else.

Topics for Reflection

To make it easier to read purposefully and actively, the key is to do something active. Make notes, ask questions, look for connections, follow-up with research and problem solving. Here are several categories relevant to this book:

• Historical Facts, and their hedging – This book involves history, but is the history any good and what makes the history worth considering? Note historical statements and rhetorical devices used to gloss over specifics.
• Mathematical facts, ideas, arguments – It is easy to have your eyes glaze over when reading math, but you have to fight back. Note the content and how it is developed. All of it, not just the great theorems. Try to write down arguments in your own words.
• Value judgements, evidence – It is hard to separate historical facts from value judgements. Dunham is not careful. See if you can locate his uncritical biases.
• Vocabulary – There are bound to be new words or words used in a technical rather than everyday manner (e.g. unbounded). Pay attention to words that feel new or storage and find more out about them.
• Further reading – What other books, articles, and authors are mentioned? Why are they related and why should we read them? Are there other books/videos/etc. you know of that are related and would be interesting for someone to follow up with?
• Questions – Most of this list pretends you know everything, but the real idea is to find what you want to know more about. This section is specifically to remind to to ask questions based on the text and to consider where answers might lie and what they could look like.
• Philosophical Connections – A lot of philosophical questions come up in JTG. They are important, but something we should all learn to do is note when we are verging on philosophical (as opposed to mathematical or historical) inquiry. e.g. “Is there really such a thing as 2‾√2?” is usually asked in a philosophical sense. We have to discuss what it means for something to exist.
• Psychology and Sociology of Math – History and Biography, topics of our book, lead us into thinking about how math fits into our world: the stereotypes of math and mathematicians, what mathematical discovery feels like, how mathematical proof compares to verifiability in other areas. What in our reading brings up these sorts of issues and where do we want to go with them?
• One thing for us to work out/on – There are generally more questions than answers, more good ideas than time to track them down, and really not every interesting fact or question has good answers or can be worked on productively by us together. But some can. And the value of studying together and meeting in-person is that some of these ideas can be worked on. So for each chapter, make sure you pick one thing that would be a good topic for class discussion. This can be a confusing part of a proof, background knowledge that Dunham leaves out, following up on a cool idea, or sorting through a tricky issue in history, philosophy, education, psychology, or sociology.

A self-imposed program to keep reading

Reminder – The schedule is here

Pretty much every week we will read a chapter in JTG and use it as the starting point to do some math. We will look closely at the Great Theorem in each chapter, and in fact, proving this theorem clearly is the backbone of what each group needs to do. In addition to this bit of content, there is often a good deal more math done by Dunham in each chapter and this may be worth considering in greater detail.

But the great theorem and the math that Dunham shows us in the book is not the sum total of the content we will cover. The chapters bring up more than they finish, and there is a fair amount of room for us to explore using the chapter’s content as a starting place. Going beyond the book is one of the main benefits of learning this stuff with an experienced guide and in a small group together. The real value is not in digesting what Dunham wrote, but in pursuing our own ideas based on his provocation.

This only works if each of us is curious, committed, and persistent. You need to read the chapter and think about what’s in it, not just accomplish the act of passing your eyes over it. You need to take responsibility for what interests you by writing it down, pursuing it further, doing independent research, and refining vague, initial ideas to the point where they are worth sharing in larger company.

This has been a difficult discipline for students to develop and maintain in past semesters. So each semester I look for low-impact ways to help students remember to stay involved adn to keep going when the natural inclination may be to fall asleep or totally skim past the details. Quizzes miss the point too far, so it’s always something a bit more interesting.

How – Spaced Repetition

This semester we’d like to try building a habit using spaced repetition and either index cards or Anki.

At its most basic, the idea is to use the reading to develop a set of flashcards and questions that do not yet have answers. A good starting point is 5 of them. We’ll give it a try when we start JTG in earnest. If you’d like to read about it ahead of time, we have a couple resources.

• How to remember anything foreverish an interactive website by Nicky Case
• Using spaced repetition systems to see through a piece of mathematics a blog post by Michael Nielsen (author and Quantum Information Physicist) about learning math deeply with spaced repetition

This last one is maybe the most convincing. Nielsen isn’t trying to get you as students to make it through school, and he doesn’t care about short term recall. He is talking about his own learning, deep learning, as a professional.

Why Bother?

Practically speaking, it is difficult to get yourself to actually read mathematics, let alone to develop the habit to keep it up through the entire semester. Yet, reading is our starting point for understanding. We want to give you a regular opportunity to make sure you are reading with your hands and mind and not just your eyes. The question format is additionally a reminder that the point of engaging with math is not to know but to think. It is an unrealistic expectation to imagine that reading through this stuff once would give you all the answers, and forgets that the point isn’t often the answers we know but how knowing a few answers allows us to learn to ask questions.

Flashcards

To make one of these, all you need is a simple piece of information you hope to remember or eventually learn. The piece of information needs to carry meaning for you, and it needs to be simple enough for you to provide an answer. Complex topics will likely result in many cards. You don’t need to memorize everything. And maybe some of your cards will end up in the trash.

What makes a good question?

There are many kinds of good questions. A good start is to ask questions whose answer you are interested in. Yes/no questions seldom work. Save the simple ones for the flashcards. Ask questions about things that are difficult for you to understand. The trick is being able to explain where you get lost. If a math problem is talked about in the text, other related problems might be a good source for questions. JTG is not just about math itself but also the human contexts in which this math arose. It is a short book so of course these contexts are hardly described. They can be a source of many good questions.

One final tip: Questions are better when the asker does some of the legwork. If you’re asking questions without doing a little work with pencil and paper or 5 minutes of Googling, that’s lazy. No one will want to help satisfy your curiosity. Your question can and often should contain the information or resources you found while thinking about the question yourself.