How to write proofs on the board in the classroom

Question

I'm teaching an introductory analysis course, and I am seeking some feedback on how proofs should be written on the board in class in order to maximize learning. I realize that there is an opinion-based component to this question, but would appreciate any citations of research on the subject (not necessary) in addition to arguments on how student learning might be impacted.

My question is regarding the actual style that the proof is written on the board including but not limited to the following points :

• Should theorems and proofs always be written on the board in complete English sentences or is it ok to use abbreviations and symbolic shorthand?
• Is it ok to use slightly different notation than in the textbook?
• Is it ok to use slightly (or even completely) different arguments than what is used in the textbook?

I'm including a couple theorems from Wade's introduction to analysis as examples. I'll type up the theorem statement and proof exactly as in the text and then how I might write it on the board in class.

Here is a nearly word for word as the theorem and proof appear in the book:

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2.12 Theorem. Suppose that $\{x_n\}$ and $\{y_n\}$ are real sequences and are convergent, then $$\lim_{n\rightarrow\infty}(x_n+y_n)=\lim_{n\rightarrow\infty}x_n+\lim_{n\rightarrow\infty}y_n.$$ Proof. Suppose $x_n\rightarrow x$ and $y_n\rightarrow y$ as $n\rightarrow\infty.$ Let $\epsilon>0$ and choose $N\in\mathbb N$ such that $n\geq N$ implies $|x_n-x|<\epsilon/2$ and $|y_n-y|<\epsilon/2.$ Thus $n\geq N$ implies $$|(x_n+y_n)-(x+y)|\leq |x_n-x| + |y_n-y|<\frac\epsilon2+\frac\epsilon2=\epsilon. \quad \blacksquare$$

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Now here is how I might actually write it on the board in class:

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Thm 2.12 $\quad x_n\rightarrow x, y_n\rightarrow y \quad \Longrightarrow \quad x_n+y_n\rightarrow x+y.$

Proof. Given $\epsilon>0$, choose $N$ s.t. $|x_n-x|<\epsilon/2$, $|y_n-y|<\epsilon/2$ $\ \forall n>N.$ $$ \begin{aligned} |(x_n+y_n)-(x+y)|&\leq |x_n-x| + |y_n-y|\\ &<\frac\epsilon2+\frac\epsilon2=\epsilon. \qquad\qquad \blacksquare \end{aligned} $$

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The main differences is that the text is more "wordy", and I will write everything out more "symbolically". Note that I do stay consistent with the numerical labeling of theorems (e.g. 2.12) in the book so that students can more easily reference the text to compare/study.

I also sometimes use slightly different arguments than in the text, add in details that are left out of the book or leave out details that are explained in the book.

E.g.:

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2.8 Theorem. Every convergent sequence is bounded.

Proof. Assume $x_n\rightarrow a.$ Given $\epsilon=1,$ there is an $N\in\mathbb N$ such that $n\geq N$ implies $|x_n-a|<1.$ Hence by the triangle inequality, $|x_n|<1+|a|.$ On the other hand, if $1\leq n \leq N,$ then $$|x_n|\leq M:=\max\{|x_1|,|x_2|,\ldots,|x_N|\}.$$ Therefore, $\{x_n\}$ is dominated by $\max\{M,1+|a|\}. \quad \blacksquare$

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Now here is how I might actually write it on the board in class:

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Thm 2.12 $\quad x_n\rightarrow x \quad \Longrightarrow \quad x_n$ bdd.

Proof. Given $\epsilon>0$ we can find $N$ s.t. $\forall n>N,$ $|x_n-x|<\epsilon.$

So $|x_n|\leq |x|+\epsilon$ when $n>N.$

Let $M=\max\{|x_k|; k\leq N\}.$

Therefore $|x_n|\leq\max\{M,|x|+\epsilon\}$ for any $n. \qquad \blacksquare$

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Of course, I also verbally explain each step of the work and will sometimes draw diagrams to help them informally/intuitively understand the argument, etc.

My justification is that there is no need to copy it exactly as it is in the text as the students can simply read that, and that it might be beneficial for them to see it written up slightly differently so that they can become accustomed to different styles of mathematical writing.

Answer 1

I very strongly think that theorems and proofs should be written on the board in complete sentences. It is OK to use symbols such as $\implies$ or $\forall$, but only inside a formal assertion. I could for example write

Let $(x_n)_{n\in\mathbb{N}}$ be a sequence of real numbers. It holds : $$ (x_n)_{n\in\mathbb{N}} \text{ converges} \implies (x_n)_{n\in\mathbb{N}}\text{ is bounded}.$$

even if that would not be my preferred way to state the result. But it is in my opinion very important to be a model for student's expected answers to test, which should include

• to write in full sentences and always introduce all variables (too often, one reads statements $x=\dots$ with no way to determine whether the statement is claimed, is to be proved, or is to be refuted; with no way to determine whether there is an $x$ with the claimed property or the claim is for all $x$),
• to correctly type all variables; in particular you need not to write $x_n$ for the sequence, because $x_n$ represents a number and overloading it can cause great deals of confusion. There is a perfectly fine way to write the sequence, namely $(x_n)_{n\in\mathbb{N}}$ or even $(x_n)$ if you want to shorten a bit, and this writing is pretty clear, very directly related to the sequence being a collection of numbers.

Confusion about the type of mathematical object, the status of variables are a primary cause of difficulties among students. As teachers, we need to pave the way as clearly as possible, and that measn be as explicit and precise as we can. It will not be sufficient, but it will be necessary for many students.

To illustrate this claim of mine, which I fear might seem pedantic to some, let me present an example which I guess every experienced math teacher has met in one form or another.

Question Let $k$ be a positive integer and let $f$ be the function defined for each $x\in \mathbb{R}$ by $f(x) = k \sin(x/k)$.

1. Show that $f(0)=0$.
2. Compute $f'(0)$

...

Student's answer

1. $f(0) = k \sin(0/k)= k\sin(0) = 0$
2. $f'(0) = 0' = 0$

This comes from a wrong typing: there is a confusion between the function and its value at some point. There is no way to debug this kind of problem if one is not very rigorous in its writing.