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I was wondering, how would you explain to young high school students or grade school kids the relationship between math and English?

For me, both are systems that follow certain rules. Math is often referred to as a language and you learn it by using it, just as you would any other language. Furthermore, one can help in decoding the other. For example, if you have an equation, it doesn't really mean anything until we give it some sort of context for your problem. Without a story behind the equation, it's just math for math's sake and it seems pointless.

I'm looking for something along those lines. I will be teaching a summer school class soon for the first time and I have to come up with something on the fly, since last year's materials were lost.

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    $\begingroup$ to clarify what you mean by "the relationship between math and English": are you trying to compare the grammar and syntax of English and the rules / logic of mathematics directly, or are you just trying to have students be able to articulate the information from an English paragraph by writing one or more mathematical expressions? $\endgroup$ – Marian Minar Jun 16 '16 at 21:00
  • $\begingroup$ Certainly, there are some relations, but please, be careful. You might find this answer and this question related. $\endgroup$ – dtldarek Jun 28 '16 at 9:12
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Assign exercises explicitly translating natural-language expressions/sentences into algebraic expressions/equations. Assign exercises like number puzzles requiring that the student "translate to an equation" and then solve and check/interpret. These are fairly standard, useful, but perhaps not sufficiently emphasized.

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I am not sure it answers your question, but I strongly advocate to learn as soon as possible that mathematic is expressed in a language. You can phrase mathematics in English, in French, but there is no more way to do mathematics without a language to express it than there is, say, to do politics. Too often, people have the idea that mathematics should look like Russel and Whitehead's Principia: they don't, and pupils should know that.

To take an example, compare

$n$ odd $n=2k+1$. $n^2=(2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1=2k'+1$. $n^2$ odd.

and

If $n$ is an odd integer, then there is an integer $k$ such that $n=2k+1$. Then it holds: $$n^2=(2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1=2k'+1$$ where $k'=2k^2+2k$ is an integer. It follows that $n^2$ is odd.

The first writing is commonly found in students' papers, and would be inadmissible from a teacher. I think it should be inadmissible from students too, because it won't enable them to articulate their reasonings clearly. This may be a slightly too complicated example, but it should be clear enough.

Then, in the context of mathematics, there are some additional conventions to be followed, in particular some specific vocabulary and a more rigid use of some terms. But even this last point is overrated. Let me take the example of the word "and". Many would say that in mathematics, "and" is used only in the very precisely defined sense of logical conjunction. This is not true, as witnessed by the following example:

The solutions of the equation $x^2-1=0$ in the unknown $x$ are $1$ and $-1$.

Here, it is important to point out that "and" is not a logical conjunction, but a mere enumeration. So we use that word in a common way, not specific to maths.

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  • $\begingroup$ I actually don't allow students to write "the solutions are 1 and -1" in that way. Proper phrasings, IMO, are "the solution is 1 or -1" or "the solution set is {1, -1}". $\endgroup$ – Daniel R. Collins Jun 17 '16 at 12:46
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    $\begingroup$ @DanielR.Collins: it might be because I am not a native speaker, but I fail to see what is wrong with "1 and -1 are solutions" (or, for example, let A and B be points in the plane"), while I strongly feel "the solution is 1 or -1" is very wrong: one should not speak about "the" solution unless it is known to be unique (this is what the definite article precisely means!). Using the set of solution is ok in this example, but not an option in others (such as the one above). $\endgroup$ – Benoît Kloeckner Jun 17 '16 at 15:14
  • $\begingroup$ Perhaps to be perfectly precise I should have not used the word "solution". Granted that my students aren't writing full sentences, I would prefer to see "$x^2 - 1 = 0 \text{... } x = 1 \text{ or } x = -1$" rather than "$x^2 - 1 = 0 \text{... } x = 1 \text{ and } x = -1$", following the statement of the Zero Product Property (which certainly uses "or", not "and"). $\endgroup$ – Daniel R. Collins Jun 18 '16 at 2:46
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    $\begingroup$ @DanielR.Collins: ok, but then this substantiates my first claim, that if we expect students to understand anything at all rather than trying to replicate meaningless recipes, they should write full sentences. Saying "Since $x^2-1=0$, we must have $x=1$ or $x=-1$" is perfectly fine, but different from "the solutions is 1 or -1". And the difference can only be seen when one writes sentences. $\endgroup$ – Benoît Kloeckner Jun 18 '16 at 11:41
  • $\begingroup$ That would be nice, but for my students writing in sentences is definitely a non-option. I've committed to at least assessing proper use of the equals sign (=), which the majority of my students cannot do even if I tell them every day; even while they take the tests. I'm already infamous at my U.S. community college even for that. $\endgroup$ – Daniel R. Collins Jun 18 '16 at 13:38
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I'll break this down into three parts with respect to "relationship between basic math and English".

Translation and Transliteration

I've found that it is a useful exercise to have students simply take a mathematical expression and say / write it out with as little mathematical symbolism as possible. In my introductory Set Theory class, we will take a statement like $\{x : x > 0, x \in \mathbb{N}\}$ and read it as "The set of $x$'s such that $x$ is greater than zero and $x$ is in the set of natural numbers" (or other equivalent readings).

I further make the point that while the mathematical statement written in mathematical symbolism is short, it is not a short statement. In fact, it is a lot of information, compactly written. As an aside, I often tell my students that while the word "cat" is a three character word which we can read at a glance, it is usually not the case with mathematical statements. For example, $x = 5$ is also three non-whitespace characters, but it reads as "$x$ and five are equal to each other" (or the more common "$x$ equals five").

Similarly it is helpful to go the other way. Start with a statement with as little mathematical symbolism as possible and try to write it in the most condensed way --- just as an exercise. Somewhere in between is something that is readable.

Abuse of Vocabulary and Symbolism

I also discuss multiple definitions of words with my students. In a Calculus class, I will first ask for my students' understanding of the word "continuous" and show that there is a difference between colloquial use and the mathematical use.

In an Algebra class, you may want to try this with words like "infinity", "and" vs "or", etc.

Then, there is the abuse of vocabulary within mathematics itself. There is the word "graph", for example which has a formal and general usage within Graph Theory as an object and the slightly more colloquial and different usage in an Algebra course: "Graph the function ...".

Similarly, there is the abuse of symbolism and the biased use of symbolism. Break your students out of the habit that "$x$ is the independent variable and $y$ is the dependent variable". Use different letters and change the context. Make $x$ the dependent variable; let the vertical axis be associated with $x$, as for instances.

Left to Right Reading

English is read from left to right and then from top to bottom, like this response. Other written languages are read from right to left and from to bottom. And still others are read top to bottom and then left to right. And there may be other cases.

Mathematical statements aren't necessarily read in any of these ways. Consider $\Big(\sum_{k=1}^{n}k^{2}\Big)^{\frac{1}{2}}$. Ack! This is "the square root of the sum of the squares of $k$ from $k = 1$ to $n$". "Diagramming" this would be not be so "linear" [purposeful use of mathematical term in a technically, non-mathematical way].

It is helpful to get students to understand this notion and to see and read the mathematical phrase differently than they would English. This is why I say "$x$ and five are equal to each other" rather than "$x$ equals 5". The former talks about the two objects as equal, the latter has a subtle bias of causation [case in point is that when one write "5 = $x$" there is typically an open revolt from students since they reject the idea that "five can equal $x$"].

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