Topics: Mathematical Language

Children are extraordinary linguists! They acquire half their adult vocabulary and nearly all their adult grammar by the time they are 5 years old, mostly informally and outside of school. That language-learning capacity can help them learn mathematics, too, *especially* when the mathematical idea they are learning depends on the way we name numbers. (See Language PowerPoint and also Early algebra for more.)

This natural facility with language is especially helpful in learning to add and subtract 10s. Why? Numbers aren’t born with names; *people* invented names for them, names that help us compute. The numbers from zero to twelve have names that feel so arbitrary that they might just as well be called Carole, Charles, Carla, Cyril, and so on. But pretty soon, the number names settle down to a pattern. That pattern is not in the *numbers*, but in the way we *name* (and write) them. Different languages make different choices about that naming, but *every* language that has names for numbers beyond 20 has evolved a *patterned* way of naming them so that the *names*, themselves, support mental computation.

Some languages, like Hebrew and Japanese, name numbers under 100 explicitly as so many tens and so many ones. For example, 58 in both languages is literally “five tens and eight.”

The Indo-European languages — the large family of languages that includes English and its Germanic relatives (e.g., Dutch, Danish, Swedish, German), Latin and its descendants (e.g., Spanish, Portuguese, French, Italian), the Slavic languages (e.g., Russian, Polish, Czech), Greek, and some of the languages of India — are not, in general, as regular. Most of these languages are, like English, a bit disorderly up to 20. Spanish makes clear that 16 through 19 are “ten and six,” “ten and seven,” and so on, but gives numbers unique names through 15. The number names for 11 through 15 are *related* to the words for 1 through 5, but don’t make their meaning explicit. English uses special words through 12 (and, if one digs deeply enough, one can understand what the names for eleven and twelve have to do with one and two). English becomes somewhat more regular after that, but names like “thirteen” don’t generally feel to children like “three and ten.”

The irregularity under 20 is a remnant of using 20, rather than 10, as the primary organizer of number names in the ancestor of these languages! Lincoln’s “Four score and seven years ago” was the now archaic English equivalent of what French still uses as a name for 87: *quatre-vingt sept* ‘four-twenty seven.’ French 73 and 93 are “sixty thirteen” and “four-twenty thirteen,” respectively. Danish shows even more of the old organization by 20 in the larger numbers under 100. The numbers 3, 4, and 5, in Danish are *tre, fire, fem.* Sixty and eighty are *tres* and *firs*, suggesting ‘three twenties’ and ‘four twenties.’ And fifty, seventy, and ninety are, respectively, *halvtres*, *halvfirs* and *halvfems*, where the *halv* means “half” but signifies “half way to” the next group of twenty. So, *halvfirs* is ‘half way (from three twenties) to four (twenties).’

The regularity of the names of numbers means that it is easier for children to use linguistic cues to add 10 to 43 than to add 10 to 3 or 13.

**Learning the litany of numbers.**

- Counting is a matter of matching words, learned in a particular order, to objects. Learning the words is like learning any poem or song; it is a linguistic act, not a mathematical one, exactly like learning the alphabet in order (which usually
*is*taught as a song). *Matching*those words to objects — saying one and only one word as each object is touched, and saying those words in order — is a cognitive skill that requires attention and control.- Using the last-named word as the name for the “number of objects” completes the counting process.

Number names in English are not entirely regular before 20, leaving children a longer “poem of numbers” to learn before they can start combining familiar names of small numbers to make the names of larger numbers, but the names “one” through “nine” repeat themselves (with the exception of “ten” and “twenty”) fairly recognizably in “thirty,” “forty,” “fifty,” “sixty,” through “ninety.” Games that treat fingers as dimes connect counting by 10 with counting by 1.

- It is
*very*important for children to learn to count backwards as well as forwards. As soon as children are comfortable counting to 30, a nice “cooling-down” exercise after coming back from recess or gym or lunch is to count down from 30 to zero.

Children often enough learn to count by learning *linguistically* how the repetitive sequence of words works. They learn that the words “…seven, eight, nine…” follow each other in sequence, so the words “…thirty-seven, thirty-eight, thirty-nine…” would also follow each other in sequence. At that stage, even though a word like *thirty-eight* has parts that are used in other number words — the *thirty* that is heard in the numbers just before and after it, and the *eight* part that always comes after a word that has a *seven* part in it — the word parts don’t necessarily have *mathematical* significance to the child.

Two experiences help children be aware of the “etymology” of those words and use it mathematically. One is to hear the “poetry” of counting-by-tens when the starting number is not 10 (or zero) and to connect that with *adding ten*.

**Adding 10** to any number involves a mathematical idea, but *naming* the answer is mostly a *linguistic* idea. We named the numbers to make adding a power of 10 easy. Children often learn the sequence “…twenty, thirty, forty, fifty…” without ever hearing the corresponding (and rhyming) “…twenty-three, thirty-three, forty-three, fifty-three….” But the first connection to make is that these numbers are actually 10 apart. Children initially need to experiment, by counting the additional 10, to see that “twenty-three” plus 10 more is “thirty-three”, and that another 10 gives “forty-three”, and another 10 more gives “fifty-three”. But usually, children then begin to “hear” the counting-by-tens pattern and, when asked to add “another 10,” say “sixty-three” spontaneously. Making this solid, like any other skill, takes some practice, but the foundation is easy to build very quickly. It is essentially a linguistic act, not a mathematical one, and, when done out loud, *verbally* — *not as a written exercise* — it draws upon children’s formidable linguistic skill. Knowing how to *write* those numbers is a slightly different act, though also more linguistic than mathematical. (For ideas about how to help children see how the numbers appear in written form, after they are comfortable with the names, see “hundreds chart.”)

Another experience that helps children be aware of the “etymology” of those words and use it mathematically is explicitly pulling apart a number name like twenty-eight as if it were the first and last names of a number.

**The first, middle, and last names of a number.** Toward the beginning of second grade, Laura was trying to figure out 28 − 8. To her, at that point, 28 − 8 was just as arbitrary a problem as 24 − 7. The approach that she took was to count backwards. The teacher wanted her to see how the number *names* could help. Here is an excerpt from their conversation.

Tchr: *playfully* “Hi there, Laura G——! What’s *your* name?”

LG: *amused* “Laura!”

Tchr: “Your first and last name.”

LG: “Laura G——”

Tchr: “Hmm, Laura G——… What if I take away the Laura? What’s left?”

LG: *hesitantly* “G——?”

Tchr: “Sure! Would you say your name again?”

LG: “Laura G——”

Tchr: “So what was ‘Laura G——’ minus ‘Laura’?

LG: *more confidently* “G——.”

Tchr: “Yup! Say your name again!”

LG: *amused again* “Laura G——!”

Tchr: “What’s ‘Laura G——’ minus ‘G——’?”

LG: “Laura?”

Tchr: “And what’s ‘Laura G——’ minus ‘Laura’?”

LG: “G——?”

Tchr: “Yay. Ok, we’re going to pretend your name is twenty-eight! *playfully* So, what’s your name?”

LG: “Twenty-eight.”

Tchr: “Hi! Nice to meet you, Twenty Eight! What’s your first name?”

LG: *hesitantly* “Twenty?”

Tchr: “Hi, twenty! And what’s your last name?”

LG: *less hesitantly, but still with the questioning tone* “Eight?”

Tchr: “And… your whole name is…?”

LG: *confidently* “Twenty-eight!”

Tchr: “Ah, and what if I take away the Twenty? What’s left?”

LG: “Eight?”

Tchr: “Yes! Say your name again?”

LG: “Twenty eight!”

Tchr: “And what’s ‘Twenty Eight’ minus ‘Eight’?”

LG: *with real confidence* “Twenty!”

This is *not* a mathematical idea, but it is also not a trick. It is a linguistic idea. The name ‘twenty-eight’ evolved to make this computation easy! Many languages insert “and” in the number name (for example, “eight and twenty” rather than “twenty eight”), making it even clearer that “twenty-eight” is “twenty” and “eight.” We could have called that number fourweek (suggesting multiplication/division), but instead chose a name that suggests addition/subtraction and lets us understand 28 − 8 through *language alone*. Of course, language alone does not solve all problems — for problems like 24 − 7 we need mathematical ideas, too — but even for 24 − 7 the linguistic idea is valuable, because we want *24 − 4* to feel so trivial that we can *use* it in the solution of 24 − 7.

Base 10 blocks help, of course, but take advantage of the language, too!

Number names make lots of other computations easy, too. To a first grader, the *spoken* question (not the written one) “how much is two hundred plus two hundred” is just as easy as “how much is two plus two.” The reason is that “how much is two hundred plus two hundred” is like “how much is two fish plus two fish” or “how much is two sheep plus two sheep”: the linguistic form draws attention to adding 2 + 2 and doesn’t depend on the child having any notion of what “a hundred” is. Interestingly, to the question “how much is a hundred plus a hundred” (with no audible “small” numbers), a young enough child is asked might give an answer like “a million.” Why? Precisely because they *don’t* really know what “a hundred” is, except that it is “big”: when their attention is not drawn away from that one thing they know about “hundred,” they may just name a number that they know is even bigger, like a million! In a similar way, walk into any first grade class and ask “what is two fifths plus four fifths?” They haven’t a clue what “fifths” are, but don’t hesitate to say “six fifths.”

The linguistic help, mental addition of three hundred and five hundred is easy for children (if they can add 3 and 5). In fact, adding these large numbers (but spoken, not written) is a great way to build facility with adding the small numbers, because the fact that the numbers are huge makes kids feel smart, and *that* is very useful. Just as no corporation invests effort in a venture it does not think will succeed, neither does a child. I will work harder at attaining a skill if I believe I’m capable of attaining it, so *feeling* smart at math helps me *become* smart at math.

**Adding things that are not alike:** Being able to add “two hundred plus three hundred” (spoken, not written) does *not* help them with “two thousand plus three hundred.” If posed the question, children are likely to feel that they are being asked to “do” something — after all, the correct answer, “two thousand three hundred,” feels too much like just repeating the numbers back “without doing anything.” Even though they’d instantly recognize “two inches plus three cows” as absurd, “two thousand three hundred,” may feel “too simple,” and so they can be tempted into nonsense answers that somehow involve adding “two” and “three.” So they need to relearn what they already know about numbers under 100. Just as “four and twenty” make “twenty-four,” “three hundred plus two thousand” make “two thousand three hundred” — in both cases, we simply *list* the added parts, but list them in a particular order with the bigger quantities first (i.e., thousands before hundreds before tens before ones before any fractional parts).

So strong is children’s sense that unalike things can’t be added (just by adding the numbers of them) that they apply it naturally when they report what number an unsorted collection of base-10 blocks represents, or what number they’ve “heard” when it is presented as Thunk! Swoosh! Pop!. They aggregate the blocks or sound by their values, count each, and report the largest values (e.g., hundreds) first. Children might mix up which block or sound stands for what value, but almost never does a child add, say, the number of rods (say, two 10s or swooshes) and the number of flats (say, three 100s or thunks) and report them together (as, say, five of something).

**Written arithmetic is sometimes harder than mental arithmetic:** Though a first grader might be happy *saying* “five hundred” in response to the *spoken* question “what’s two hundred plus three hundred,” dealing with the written expression 200 + 300 may be very different. The 5 part is still tempting, but what do we do with all the zeros? 500? 50000? 200300? The *notation* is extremely convenient for complex computations like 487 + 956 *once one understands how to perform the algorithms* but, at least at the very beginning and for “round” numbers can be harder than spoken computations performed mentally. The situation is even more striking when children are working with fractions. (See below.)

Try asking any first grader (speaking the question without writing anything) “What is two eighths plus three eighths?” The chances are very good that the child will answer “five eighths” as if you had asked “what is two cats plus three cats?” And the kindergartener who can add 2 and 3 will do the same thing. This does not, of course, mean that they “understand fractions.” But they do understand numerating (counting) things. (The word “numerator” comes from its role in numerating, counting, the number of parts of some particular denomination.)

When + is first encountered *in writing*, a child who doesn’t already have a very strong sense of the meaning (and, perhaps, even the sound) of it will see a plus sign, and be tempted add everything in sight, producing the common wrong answer: . This is one reason why it is so important to establish the *meaning* first (e.g., with shaded bars, or on the number line, or with Pattern Blocks or Cuisenaire Rods) and do enough problems “mentally” (spoken question and spoken answer, with no writing, to help them to keep relying on their language intuitions) *before* being asked to calculate from the written symbols.

Just as we want children to recognize 23 as *one number* “spelled” with two digits (like *at* is one word spelled with two letters), we want them to recognize as *one number* spelled with two digits. In the case of 23, the left-hand digit counts 10s and the right-hand digit counts 1s; in the case of the fraction, the bottom number tells how we’ve subdivided the space between two whole numbers (in this case, into eighths) and the top number tells how many of those subdivisions we have.

**Teacher story:** Here’s how I do it. I begin a mini lesson — all parts of which are pretty short — something like this:

- Tchr: I just want to check what 3 apples + 2 apples equals
- Child:
*(probably protesting how easy this is!)*Five apples. - Tchr:
*(with a serious expression)*What about 3 pens + 2 pens? - Child:
*(probably more protests of extreme ease)*Five pens.

…The same, another few times, always with the same amounts with different concrete objects in each question. Then, without any hint that this problem is at all different

- Tchr: What about 3 eighths + 2 eighths?

Children usually answer correctly: “five eighths.” On the rare occasions when they don’t, remind them that this is no different from 3 apples + 2 apples, and repeat the fraction problem.

- Tchr: Perfect! Now let’s see how we can write what you just said as a sentence! Who can write “three eighths” on the board?

*(a child writes **)*

- Tchr: Great! Now someone else write “plus.” Good. Now someone else write “two eighths.”

The board now contains +

- Tchr: Excellent! And what did we say 3 eighths + 2 eighths was?
- Children’s chorus: Five eighths!
- Tchr: So, who wants to write that answer?

Before having the kids come up and record, I may ask many such fraction-with-common-denominator problems and *I* record the problems and the kids’ answers on the board, so that the kids are concentrating on doing the activity verbally and strengthen the logical language connection before they are made responsible for handling the written form. But I always have them *write* what they know — what they’ve just answered correctly verbally — before I have them *read* a problem that I present (or that the book presents) in fraction notation.

In this way, the understanding comes first, verbally, and the writing follows from it. By the way, that is how children first learned writing: the words come first and only later do we learn to write them down!