The ways we name numbers
From zero through twelve, our number names may seem as if they are totally arbitrary, each number having a totally new name. We might almost expect to keep on the same way -- ...eight, nine, ten, eleven, twelve, francisco, elana, rashid, martha, ryan... -- but a long sequence of such arbitrary names would be impossible to memorize. So, in every language, people give numbers "family" names as well as "personal" names: "the seventies" (like "the Smiths"). Which comes first -- four-and-twenty or twenty-four -- depends on the language. In German, 24 is "four and twenty" (vier und zwanzig), while in Spanish it is "twenty four" (veinte cuatro), as it is in English. Number names change over time even within one language: "four-and-twenty blackbirds baked in a pie" was perfectly natural English when the nursery rhyme was composed.
Numbers, themselves, know nothing about place value. And even though people -- because we have limits on our memory -- spoke the names of numbers in this place-value way, the idea of writing them in what now seems like the only imaginable way was a brilliant insight that vastly simplified calculation. Roman numerals do a fine job of recording the numbers, but are difficult to use for computation.
Using this idea in teaching
The way we name numbers lets us take advantage of children's extraordinary ability with language. (They all acquire a full half their adult vocabularies by the age of five!)
Tchr: Say your whole name.
LG: Laura G——
Tchr: So, what's "Laura G—— minus Laura"?
Tchr: Yes. Say your whole name again.
LG: Laura G——
Tchr: So, what's "Laura G—— minus G——"?
Play for a while, and have a few kids play -- it doesn't come all at once -- and then switch to
Tchr: Pretend your name is "twenty eight"! What's your first name?
Tchr: Yup, and what's your last name?
Tchr: Yes. So say your whole name.
Child: Twenty eight!
Tchr: Same game. What's "twenty eight minus eight”?
And so on... (See addition and subtraction for more detail.)Subtracting 9 from 28 requires some mathematical understanding. But we have named the numbers so conveniently that taking "eight" from "twenty eight" is like taking "panther" from "pink panther." Children in second grade find it funny to get the "hang of it" with their first and last names.
At this point, some children try to switch back to arithmetic and count backwards, not quite trusting that the game can be played safely the same way as with their real name. You may need to switch back to the name game again, but the idea is to use the language clues here. Of course, the parts that are taken away must be there in the first place. For the child who is first learning this naming part of place value, posing "twenty-eight minus five" is like asking for "Laura Gordon minus Dale." "Twenty-eight minus five" requires arithmetic, not just language.
The linguistic "trick" is not a mathematical cheat at all! This is why we named the numbers this way, so that the logic of our language conforms with the logic of our numbers.
Extending the teaching, and building mastery
Once students are thoroughly comfortable with the linguistic idea -- "a hundred thirty nine minus a hundred" is just like "Ricardo Luis Moreno minus Ricardo" -- then you can extend it, still using mental math, like this:
OK, you know what twenty eight minus eight is, right?
So what's twenty eight minus nine?
Here, the child is using language knowledge and mathematical knowledge. Nine is one more than eight (arithmetic knowledge), so we are taking one more away (algebraic knowledge). But it's easy, now, to take eight away and get twenty (language knowledge). Taking one more away (mathematical knowledge) gives 19.
Students who just got the linguistic idea love playing this a lot, mentally (nothing written), developing skill and confidence, really mastering the idea.
When children are getting good at the two-digit "simple" problems, they can start again with three-part names, then three-digit numbers, and then extensions from the three digit numbers, like this:
OK, you know what threehundred twenty four minus threehundred is, right?
So what's threehundred twenty four minus twohundred? (Or, what's threehundred twenty four minus five?)
Grouping and regrouping
Because the idea of grouping and regrouping is so useful, it appears throughout Think Math!. Many kinds of grouping are used: grouping by 2s with liquid measure (2 cups make a pint, two pints make a quart, two quarts make a half-gallon, two half-gallons make a gallon); grouping by 60s with time (60 seconds in a minute, 60 minutes in an hour). Most of these specialized groupings are limited -- we don’t go on to 60-hour units and nobody (now, but we used to!) uses 60ths of a second -- but the principles are identical, no matter how the grouping is done: 12 inches make a foot (but 12 feet don't make anything special, nor do we subdivide inches into 12 parts); 12 months make a year (but again, we don't subdivide months or group years by the same number). In all these contexts, though, students must be able to think in terms of packaging smaller units into larger groups, unpackaging groups, and keeping track.
Aggregating like quantities: When we are adding numbers, we add hundreds to hundreds, tens to tens, and so on. The paper and pencil method "lines up the numbers" to help us keep track, but we'd expect students to aggregate properly even in their heads. If we are adding time, we do the same thing, gathering hours with hours, minutes with minutes, seconds with seconds, and then trading up or down to make sure we have only the right number of units in each group (no more than 60 seconds, no more than 60 minutes, no more than 24 hours, etc.).
Part of this is natural to kids. If we asked what 5 minutes plus 4 inches is, any child who is thinking will feel that something’s wrong. The only times we can combine two quantities simply by adding the numbers is if those numbers refer to the same kind of thing. That is why three thousands and four hundreds is not seven of anything. The “sum” of those two quantities simply restates the problem: three thousand, four hundred. We can add 2 inches and 4 feet, but the answer isn't 6 of anything: the answer is either a restatement of the problem (4'2) or something that doesn't resemble 6 at all (like 50 inches, or two and a sixth feet). The same is true of subtraction: 7 cows minus 3 chickens makes no sense, even though 7 minus 3 is easy to compute.
See Eraser store for more about grouping and regrouping.