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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:


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 $$|(x_n+y_n)-(x+y)|\leq |x_n-x| + |y_n-y|<\frac\epsilon2+\frac\epsilon2=\epsilon. \quad \blacksquare$$


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


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


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.:


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 $\max\{M,1+|a|\}. \quad \blacksquare$


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


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 $n. \qquad \blacksquare$


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.

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    $\begingroup$ Writing this as a comment because this is totally just my opinion: I do not think there is much value in having the instructor write proofs on the board at all. Much better to have students write self explanations of the proof, get the students to draw pictures illustrating the proof, ask students to modify the proof, etc. They have it in their textbook already. However, if I did write a proof, I would certainly use full sentences. Abbreviations (which may be standard for you but not the student) just make it even harder to read. $\endgroup$ – Steven Gubkin Jan 24 '18 at 14:18
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    $\begingroup$ What you're doing is fine. Some abbreviations are wise to save time and space in class. A slightly different perspective is valuable. $\endgroup$ – Daniel R. Collins Jan 24 '18 at 14:21
  • $\begingroup$ I appreciate the comments. I also appreciate that both comments present different views. This is helpful. $\endgroup$ – jdods Jan 24 '18 at 21:02
  • $\begingroup$ Do not use different notation from the textbook. $\endgroup$ – Gerald Edgar Jan 24 '18 at 23:59
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    $\begingroup$ Besides what you and others are saying, I suggest also trying to include suggestions to students for HOW the proof could be discovered. For example, in the case of $x_n+y_n\rightarrow x+y,$ off to the side somewhere you could start with this, then write $(x_n+y_n)- (x+y)\rightarrow 0,$ and then write $(x_n−x)+(y_n-y)\rightarrow 0,$ and then rhetorically ask (or explicitly ask students) how this might be done, such as more specifically asking how to show that a sum approaches zero (one way is to show that each summand approaches zero), and then asking how one might go about showing this, etc. $\endgroup$ – Dave L Renfro Jan 25 '18 at 14:22
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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.

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  • $\begingroup$ This is a great answer. Can you clarify on how you might get through everything in the class? Do you skip theorems and proofs and have the students read through them on their own and only show a select few in class? Or do you write very fast? Or do you cover less over the course? I'm trying to not sacrifice the content of the overall course, but also to give the students less to copy in notes so they can pay attention to my words. I'm not sure if these students will have had rigorous formal logic besides a few lectures in a previous intro to proofs course. $\endgroup$ – jdods Jan 25 '18 at 13:12
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    $\begingroup$ @jdods: here I will give the same advice I give about research talks. Define the right pace, and then see what fits in. Trying to fit more in a given time will only lower the level of understanding, increase superficial learning, and you will loose everything you try to achieve in the long run. Then, several options can be used: have student work more outside class (very good if they actually do it, which is not easy to achieve), skip some of the less relevant parts (depends on your control over the curriculum), skip some proofs (perfectly acceptable for most service courses). $\endgroup$ – Benoît Kloeckner Jan 25 '18 at 15:25
  • $\begingroup$ Thank you for the advice. I think skipping some of the proofs would be my choice. I could offer brief sketches/explanations of them instead of every detail. That way I could concentrate more on how to use the theorem in example exercises. $\endgroup$ – jdods Jan 25 '18 at 16:39

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