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Making precise logical statements is an important part of teaching and learning mathematics. There are many ways to write such statements, and let me divide them into two main types1: writing in symbols ($\forall,\exists,\iff,\dots$) or writing in words (for all, there is, if and only if, …). Are there studies about how emphasizing one of these types affects the way students learn? What harm does it do if only one type is present in teaching? How important is it that students can translate statements between these two languages? What are some key skills having to do with formal logical statements that I should make sure every student learns? Is it easier or faster for students to read statements in words than in symbols?

Many style guides and older colleagues advice against using logical symbols. I am not looking for experience or anecdotal evidence, but actual studies about teaching mathematics. This does not exclude unpublished teaching experiments done within a department, but I would prefer more rigorous studies.

In my limited teaching experience I have encountered numerous students with severe problems in reading, writing and understanding logical statements. (I also fear that some students don't see the connection between symbolic logical expressions and meaningful statements in words.) This is clearly a grave impediment to learning mathematics, and I would like to know what is actually known to work helping students overcome it.

This question was inspired by this question at MSE.

For concreteness, here is an example of a statement in symbols or one in words2:

  1. $\forall y\in\mathbb R \;\exists \delta>0\;\forall x\in\mathbb Q:|y-x|<\delta$.
  2. For any real number $y$ there is such a number $\delta>0$ so that for any $x\in\mathbb Q$ we have $|y-x|<\delta$.

Added clarification: The main focus here is on logical statements and logical symbols. I am mainly interested in whether (or when) it would be good to replace logical symbols (quantifiers, implications, negations) with words. I am quite confident that writing equations out in words would be a bad idea and university level students seem to parse equations much more reliably than logical statements.


1 Some common abbreviations fall between these two classes (iff, w.r.t., s.t., …) and, of course it is possible to mix symbols and words together.

2 There are many conventions for punctuation and parentheses in symbolic statements and many possible wordings of the second example, but I don't want to discuss that (unless it turns out to be important for learning). Neither do I want to discuss whether the two statements above are correct; that is left as an exercise.

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    $\begingroup$ I challenge (what I think is an assertion of yours) the notion "not seeing the connection between s. l. expressions and meaningful statements in words is a grave impediment to learning mathematics". I agree it is challenging but some people don't process symbols as well as words. Everyone eventually needs the mini-logic course that includes distinguishing "For every apple there is a worm..." from "There is a worm so that for every apple...", but those ideas are needed before training them in notational (mis-)use. Gerhard "Thinks Parenthetical Use Is Underappreciated" Paseman, 2015.07.22. $\endgroup$ – Gerhard Paseman Jul 22 '15 at 19:19
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    $\begingroup$ @GerhardPaseman, the impediment I had in mind was difficulties with logical statements, not not seeing the connection. I added some parentheses to make it more clear what the main impediment is. Some students have big problems with simple logical statements after years of mathematical education, but most of these problems occur in the first year or two. $\endgroup$ – Joonas Ilmavirta Jul 22 '15 at 19:36
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    $\begingroup$ Maybe take a quick look at this paper ("Strict Logical Notation Is Not a Part of the Problem but a Part of the Solution for Teaching High-School Mathematics") and see if chasing the paper down in either directions (its references, or those that referenced it) turns out to be useful. (You may find it especially interesting insofar as it concerns research at a Finnish high school...) $\endgroup$ – Benjamin Dickman Jul 23 '15 at 7:19
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    $\begingroup$ Or further, 3. Every real has a positive $\delta$ which bounds its distance to any rational. $\endgroup$ – user173 Jul 23 '15 at 7:28
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    $\begingroup$ @BenjaminDickman: That looks interesting. Thanks! Others: I don't want to discuss the wording of my examples. The point is communicating and teaching logic when teaching mathematics. Please try not to digress. $\endgroup$ – Joonas Ilmavirta Jul 23 '15 at 7:34
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Note: This came up in my "hot questions" feed - I didn't notice how old the question was until after I answered it. Sorry if this isn't required anymore or digs up a dead and buried debate!

Studies (and my own experiences) would suggest that using more words than symbols (to start with) would help everyone.

There's a page Here that deals specifically with guidelines on how to write mathematics (and consequentially, logic). There's an awful lot there, with examples and preferences over many different cases, and there is a list of sources to papers and studies at the bottom.

The general gist of the page is that mathematical language has a tendency to be too succinct when it comes to beginners - professional level communication (which assumes knowledge of the subject) is too often used with beginners. Words are left out, context isn't given and a glut of symbols is used where words could be more clear, if not more descriptive to someone with professional knowledge of the subject.

If you're only interested in literature to back up a point of view, stop here. If you want some more input and specific answers to your questions within your question, feel free to read on, but be aware that the rest is more opinion based on personal experience.


This backs up my own personal experiences on the subject - I studied Maths in university and had a module on Mathematics Education. Our lecturer said she has started the course because there weren't enough good maths educators in the world; either you understood the language - and then found it hard to understand when someone else didn't - or you didn't, and while you could then empathise and help someone else who didn't, you never truly felt comfortable with it. People who both "Got it" AND could see how to explain it to someone who didn't are incredibly rare, at least in the western world - the amount of people out there convinced they "can't do maths" is proof of that.

All this being said, it is still extremely important to learn the symbols properly if you are going to converse in the mathematical world. While it might be best to start with words to encourage understanding, at some point that understanding needs to be extended to the symbolic representations of the same concepts - learning the language of maths is in many ways just as important as learning the concepts themselves.

To address the questions in your question directly (and succinctly - if you want me to expand on any let me know, I'm just way of the length of this answer):

  • Are there studies about how emphasizing one of these types affects the way students learn?

Yes, at least partially - see the link at the start of this answer (This one!)

  • What harm does it do if only one type is present in teaching?

In most (those who don't "find maths easy"), only teaching the symbols will prevent true understanding and in many cases any understanding of what the student is learning. Only teaching the words prevents students from learning the language they need to converse in the mathematical community and will also ostracise those students who are gifted at maths but find language hard (quite common).

  • How important is it that students can translate statements between these two languages?

I would say extremely important, but that's a personal opinion. For one, it shows sufficient comprehension. If the student can convert a symbolic statement into language, it shows they understand the concept correctly, and if they can convert from language to symbols, it shows they understand the symbols correctly.

  • What are some key skills having to do with formal logical statements that I should make sure every student learns?

I don't feel fully qualified to answer this, as I'm not a trained mathematics educator (I've just studied the stuff), and I cannot claim any list I make to be remotely definitive and may even be vastly misleading due to glaring omissions. All I can say is that you're on the right track with this question in the first place - communication and clarity are key in logic and maths. If a student learns to phrase their statements in a way that makes the content as clear and obvious as possible, they've done the hard part.

  • Is it easier or faster for students to read statements in words than in symbols?

for an absolute beginner, words will always be easier as they know what the words mean. As they do more, learn more of the symbols, the ratio of words:symbols that they find "fastest" and "easiest" to read will tend more towards symbols than words, because words take longer to read, but the actual ratio will likely depend on the person. For me, the below is easier to read than either of the examples in your question:

$∀y∈R, ∃δ>0$ such that $∀x∈Q : |y−x| < δ$

That's because I understand all the symbols and the connotations of the symbols that would take a lot of words to properly write out, but I still feel it benefits from a little English in the middle, just to give the "sentence" some direction. Of course, the advantage of pure mathematical/logical symbolism is that it transcends language; if I can write the whole thing without using any English, it will be absolutely clear to a French mathematician, which has its own benefits (and is another reason to learn the symbols!).

In conclusion, I would say that studies support the concept that words are (for the general population) far easier to understand (at least at first) than symbols, and that if your aim is to help spread the love of maths to more than the disappointingly small portion of the population that currently understand it, then advocating a "words, then symbols" approach would probably be vastly beneficial, even to those who would understand it without the wordy approach, because it may help them get their ideas across when communication with symbols breaks down.

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Mathematics is a language, with it's own rules. Symbols play a role, sure. Just like technical terms play a role in any specialized area like engineering or antropology. But in the end, we want to transmit understanding, and that requires explanations. Those are expressed in natural language, not just symbols.

Use symbols where words are too imprecise or long (write $\frac{\mathrm{d}}{\mathrm{d} z}$ for derivatives, don't use the quaint terms e.g. Cardan used to describe his equations); use words as far as possible, as you are talking to people, not some automatic proof checking system. By the way, check out some of the "automatically checked proofs", and ask yourself if you understand something, or if you wanted to learn them that way.

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    $\begingroup$ $-1$. The OP specifically writes: "I am not looking for experience or anecdotal evidence, but actual studies about teaching mathematics." So: I don't think this response answers the question. (Please let me know if I've misunderstood.) $\endgroup$ – Benjamin Dickman Aug 25 '15 at 1:54
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    $\begingroup$ @BenjaminDickman is correct. I am not looking for opinions that are not clearly backed by testing different methods. Moreover, the focus of the question was on logical symbols or lack thereof, not so much on all mathematical notation. (I will edit to emphasize this point.) $\endgroup$ – Joonas Ilmavirta Aug 25 '15 at 4:29

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