1.7: Mathematics as a Language

This section might seem like a bit of a sidetrack from the idea of problem solving, but in fact it is not. Mathematics is a social endeavor. We do not just solve problems and then put them aside. We need to know how to communicate our mathematics to others. With that in mind, many prospective elementary teachers may not initially think of mathematics as a language, yet it has its own symbols, structures, and ways of communicating ideas. Like spoken and written languages, math relies on a system of notation and grammar that allows for precise expression and logical reasoning (Wager and Stinson, 2012). Just as letters form words and sentences, mathematical symbols create expressions and equations that convey meaning. When teachers understand math as a language, they can present it more flexibly, helping students see connections rather than isolated rules.This perspective also highlights math as a tool for communication—students should be encouraged to verbalize their reasoning, bridging the gap between informal descriptions and formal mathematical vocabulary (Seda and Brown, 2021)

Viewing math as a language fosters a sense-making approach rather than rote memorization, which is crucial for reducing math anxiety in both teachers and students (Boaler, 2022). It also acknowledges the cultural context of mathematics, as different societies have developed unique ways of representing and thinking about numbers. By recognizing these diverse perspectives, teachers can make math more accessible and meaningful for all students (Seda and Brown, 2021). Ultimately, when educators embrace the linguistic nature of mathematics, they empower students to become fluent, confident mathematical thinkers who see math as a creative and relevant part of their lives (Wager and Stinson, 2012). Mathematics is more than just a subject to be learned in school; it is a language for understanding and interacting with the world (NCTM, 2020)

Mathematical Statements

Definition

A mathematical statement is a complete sentence that is either true or false, but not both at once.

NOTE: In mathematics, we reserve the term mathematical statement for statements that are:

• About mathematical objects (numbers, sets, functions, structures, logical relationships)
• Or used in mathematical reasoning or formal logic systems

"Modern, pure mathematics is primarily concerned with statements about mathematical objects. Mathematical objects are things such as: integers, real numbers, sets, functions, etc." (Devlin, 2012 Math Thinking). Examples of mathematical statements are:

• (1) There are infinitely many prime numbers.
• (2) For every real number a, the equation \(x^2 + a = 0 \) has a real root.
• (3) \( \sqrt{2} \) is irrational.

Not only are mathematicians interested in statements of the above kind, they are, above all, interested in knowing which statements are true and which are false. For instance, in the above examples, (1), and (3) are true whereas (2) is false. The truth or falsity in each case is demonstrated not by observation, measurement, or experiment, as in the sciences, but by a proof. (Devlin, 2012 Math Thinking)

So a “statement” in mathematics cannot be a question, a command, or a matter of opinion. It is a complete, grammatically correct sentence (with a subject, verb, and usually an object). It is important that the statement is either true or false, though you may not know which! (Part of the work of a mathematician is figuring out which sentences are true and which are false.)

Think About and Discuss

For each English sentence below, decide if it is a mathematical statement or not. If it is, is the statement true or false (or are you unsure)? If it is not a mathematical statement, in what way does it fail?

1. Blue is the prettiest color.
2. 60 is an even number.
3. Honolulu is the capital of Hawaii.
4. This sentence is false.
5. 1/2 = 2/4.
6. There are a total of 204 squares on an 8 × 8 chess board.

Now write some mathematical statements and some English sentences that fail to be mathematical statements.

Notice that “1/2 = 2/4” is a perfectly good mathematical statement. It does not look like an English sentence, but read it out loud. The subject is “1/2.” The verb is “equals.” And the object is “2/4.” This is a very good test when you write mathematics: try to read it out loud. Even the equations should read naturally, like English sentences. "Honolulu is the capital of Hawaii" is a factual statement from geography, but it does not concern mathematical objects. Statement (4) is different from the others. It is called a paradox: a statement that is self-contradictory. If it is true, then we conclude that it is false. (Why?) If it is false, then we conclude that it is true. (Why?) Paradoxes are no good as mathematical statements, because it cannot be true and it cannot be false.

Mathematical vs. Everyday Language: Mathematics requires precision, and the use of language in mathematics is often more precise than in everyday settings. When using language, people often rely on context to determine what their words convey. However, in mathematics, the context must be specific (Devlin, 2012).

And / or

Consider this sentence:

After work, I will go to the beach, or I will do my grocery shopping.

In everyday English, that probably means that if I go to the beach, I will not go shopping. I will do one or the other, but not both activities. This is called an “exclusive or.”

We can usually tell from context whether a speaker means “either one or the other or both,” or whether he means “either one or the other but not both.” (Some people use the awkward phrase “and/or” to describe the first option.)

Remember that in mathematical communication, though, we have to be very precise. We cannot rely on context or assumptions about what is implied or understood.

Definition

In mathematics, the word “or” always means “one or the other or both.”

The word “and” always means “both are true.”

Think About and Discuss

For each sentence below:

• Decide if the choice x = 3 makes the statement true or false.
• Choose a different value of x that makes the statement true (or say why that is not possible).
• Choose a different value of x that makes the statement false (or say why that is not possible).

1. x is odd or x is even.
2. x is odd and x is even.
3. x > 5 or x < 5.
4. x > 5 and x < 5
5. x · 1 = x or x · 0 = x.
6. x · 1 = x and x · 0 = x.

Quantifiers

Think About and Discuss

You are handed an envelope filled with money, and you are told “Every bill in this envelope is a $100 bill.”

• What would convince you beyond any doubt that the sentence is true? How could you convince someone else that the sentence is true?
• What would convince you beyond any doubt that the sentence is false? How could you convince someone else that the sentence is false?

Suppose you were given a different sentence: “There is a $100 bill in this envelope.”

• What would convince you beyond any doubt that the sentence is true? How could you convince someone else that the sentence is true?
• What would convince you beyond any doubt that the sentence is false? How could you convince someone else that the sentence is false?

What is the difference between the two sentences? How does that difference affect your method to decide if the statement is true or false?

Some mathematical statements have this form:

• “Every time...”
• “For all numbers. . . ”
• “For every choice. . . ”
• “It’s always true that. . . ”

These are universal statements. Such statements claim that something is always true, no matter what.

• To prove a universal statement is false, you must find an example where it fails. This is called a counterexample to the statement.
• To prove a universal statement is true, you must either check every single case, or you must find a logical reason why it would be true. (Sometimes the first option is impossible, because there might be infinitely many cases to check. You would never finish!)

Some mathematical statements have this form:

• “Sometimes...”
• “There is some number. . . ”
• “For some choice. . . ”
• “At least once...”

These are existential statements. Such statements claim there is some example where the statement is true, but it may not always be true.

• To prove an existential statement is true, you may just find the example where it works.
• To prove an existential statement is false, you must either show it fails in every single case, or you must find a logical reason why it cannot be true. (Sometimes the first option is impossible!)

Think About and Discuss

For each statement below, do the following:

• Decide if it is a universal statement or an existential statement. (This can be tricky because in some statements the quantifier is “hidden” in the meaning of the words.)
• Decide if the statement is true or false, and do your best to justify your decision.

1. Every odd number is prime.
2. For all positive numbers \(x, x^{2} > x\).
3. There is some number \(x\) such that \(x^{3} < x\).
4. Addition (of real numbers) is commutative (a + b = b + a).

Look back over your work. You will probably find that some of your arguments are sound and convincing while others are less so. In some cases you may “know” the answer but be unable to justify it. That is okay for now! Divide your answers into four categories:

1. I am confident that the justification I gave is good.
2. I am not confident in the justification I gave.
3. I am confident that the justification I gave is not good, or I could not give a justification.
4. I could not decide if the statement was true or false.

Conditional Statements

Wonder, Play, Grow

You have a deck of cards where each card has a letter on one side and a number on the other side. Your friend claims: “If a card has a vowel on one side, then it has an even number on the other side.”

These cards are on a table.

Which cards must you flip over to be certain that your friend is telling the truth?

After you have thought about the problem on your own for a while, discuss your ideas with a partner. Do you agree on which cards you must check? Try to come to an agreement on an answer you both believe.

Here is another very similar problem, yet people seem to have an easier time solving this one:

Wonder, Play, Grow

You are in charge of a party where there are young people. Some are drinking alcohol, others soft drinks. Some are old enough to drink alcohol legally, others are under age. You are responsible for ensuring that the drinking laws are not broken, so you have asked each person to put his or her photo ID on the table. At one table, there are four young people:

• One person has a can of beer, another has a bottle of Coke, but their IDs happen to be face down so you cannot see their ages.
• You can, however, see the IDs of the other two people. One is under the drinking age, the other is above it. They both have fizzy clear drinks in glasses, and you are not sure if they are drinking soda water or gin and tonic.

Which IDs and/or drinks do you need to check to make sure that no one is breaking the law?

After you have thought about the problem on your own for a while, discuss your ideas with a partner. Do you agree on which IDs/Drinks you must check? Compare these two problems. Which question is easier and why?

Definition

A conditional statement can be written in the form

If some statement then some statement.

Where the first statement is the hypothesis and the second statement is the conclusion.

These are each conditional statements, though they are not all stated in “if/then” form. Identify the hypothesis of each statement. (You may want to rewrite the sentence as an equivalent “if/then” statement.)

1. If the tomatoes are red, then they are ready to eat.
   The tomatoes are red. / The tomatoes are ready to eat.
2. An integer n is even if it is a multiple of 2.
   n is even. / n is a multiple of 2.
3. If n is odd, then n is prime.
   n is odd. / n is prime.
4. The team wins when JJ plays.
   The team wins. / JJ plays.

Remember that a mathematical statement must have a definite truth value. It is either true or false, with no gray area (even though we may not be sure which is the case). How can you tell if a conditional statement is true or false? Surely, it depends on whether the hypothesis and the conclusion are true or false. But how, exactly, can you decide?

The key is to think of a conditional statement like a promise, and ask yourself: under what condition(s) will I have broken my promise?

Example \(\PageIndex{1}\):

Here is a conditional statement:

If I win the lottery, then I’ll give each of my students $1,000.

There are four things that can happen:

• True hypothesis, true conclusion: I do win the lottery, and I do give everyone in class $1,000. I kept my promise, so the conditional statement is TRUE.
• True hypothesis, false conclusion: I do win the lottery, but I decide not to give everyone in class $1,000. I broke my promise, so the conditional statement is FALSE.
• False hypothesis, true conclusion: I do not win the lottery, but I am exceedingly generous, so I go ahead and give everyone in class $1,000. I did not break my promise! (Do you see why?) So the conditional statement is TRUE.
• False hypothesis, false conclusion: I do not win the lottery, so I do not give everyone in class $1,000. I did not break my promise! (Do you see why?) So the conditional statement is TRUE.

What can we conclude from this? A conditional statement is false only when the hypothesis is true and the conclusion is false. In every other instance, the promise (as it were) has not been broken. If a mathematical statement is not false, it must be true.

Example \(\PageIndex{2}\):

Here is another conditional statement:

If you live in Honolulu, then you live in Hawaii.

Is this statement true or false? It seems like it should depend on who the pronoun “you” refers to, and whether that person lives in Honolulu or not. Let us think it through:

• Sookim lives in Honolulu, so the hypothesis is true. Since Honolulu is in Hawaii, she does live in Hawaii. The statement is true about Sookim, since both the hypothesis and conclusion are true.
• DeeDee lives in Los Angeles. The statement is true about DeeDee since the hypothesis is false.

So in fact it does not matter! The statement is true either way. The right way to understand such a statement is as a universal statement: “Everyone who lives in Honolulu lives in Hawaii.”

This statement is true, and here is how you might justify it: “Pick a random person who lives in Honolulu. That person lives in Hawaii (since Honolulu is in Hawaii), so the statement is true for that person. I do not need to consider people who do not live in Honolulu. The statement is automatically true for those people, because the hypothesis is false!”

Example \(\PageIndex{3}\):

How do we show a (universal) conditional statement is false?

You need to give a specific instance where the hypothesis is true and the conclusion is false. For example:

If you are a good swimmer, then you are a good surfer.

Do you know someone for whom the hypothesis is true (that person is a good swimmer) but the conclusion is false (the person is not a good surfer)? Then the statement is false!

Wonder, Play, Grow

For each conditional statement, decide if it is true or false. Justify your answer.

1. If \(2 \times 2 = 4\) then \(1 + 1 = 3\).
2. If a number is even, then the number has a 4 in the one’s place.
3. If the product of two numbers is 0, then one of the numbers is 0.
4. If \(2 \times 2 = 5\) then \(1 + 1 = 3\).

On your own and with others, come up with two conditional statements that are true and one that is false. Share your three statements with peers, but do not say which are true and which is false. See if they can figure it out!

REAL WORLD ANALOGIES

Weather Promise

“If it rains tomorrow, I will bring my umbrella.”

• If it rains and I don’t bring the umbrella, I broke my promise. (False)
• If it rains and I bring it, I kept my promise. (True)
• If it doesn’t rain, you can’t say I broke my promise, because the promise only applied if it rained. So it counts as true (kept).

Nonsense Promise

“If I am the Queen of England, then I will give you a million dollars.”

Since I’m not the Queen, this never applies. So you can’t say I lied—I just said something that doesn’t trigger.

A conditional only promises something about the world if the hypothesis is true. When the hypothesis is false, it “says nothing,” so by definition it counts as satisfied (true).

Mathematics serves as a universal language, allowing for precise communication of ideas, logical reasoning, and problem-solving (CDE, 2013; NCTM, 2020). Just as spoken and written languages have grammar and structure, math employs symbols and notation to convey complex relationships and patterns. For future elementary school teachers, understanding math as a language is crucial because it transforms the subject from a rigid set of rules into a dynamic tool for communication and reasoning. When educators embrace this perspective, they can help students see mathematical concepts as interconnected rather than isolated, fostering deeper comprehension and reducing math anxiety. By encouraging students to verbalize their thought processes and relate mathematical ideas to real-world contexts, teachers create an inclusive and engaging learning environment where all students can become confident, fluent mathematical thinkers. (Seda and Brown, 2021; Wager and Stinson, 2012).

Retrieval Practice