Category Archives: Language

More on the language of place value

Read aloud the following number:


Now mentally answer this question: What is the value of the 8 in this number?

I see two correct ways of stating this:

Eighty thousand, and

Eight ten-thousands

And I’m trying to decide whether I care about the difference between these. I’m not sure that I do.

So now I do what I always do to test ideas. I ask, What if? Specifically, What if we looked to the right of the decimal point? What would this question look like there?

So consider the number 0.0008.

If you said Zero-point-zero-zero-zero-eight, then I’ve got a lot more work to do with you.

Place value language matters in math classrooms.

And the sentence in the picture below is confusing.

No, I’m guessing we would all agree that this is eight ten-thousandths, which is not at all the same as eighty-thousandths (although it is the same as eighty hundred-thousandths).

So now I see that my What if? question has muddied the waters, rather than clarified them.

What can we conclude?

I suppose that the major conclusion is this:

We need to stop pretending that decimal place value (i.e. to the right of the decimal point) behaves exactly like whole-number place value (i.e. to the left of the decimal point).

In the abstract, this is certainly true. But composing units is not conceptually (or linguistically) equivalent to partitioning them.

Like I was saying here:



How many dalmatians were in that movie again?

It was one-hundred-and-one, right?

Have I angered you yet? If you teach math, I probably have.

See, we math teachers are precise people. We like things to be just right. Part of being just right is using language correctly. In English number language, and is reserved for separating the whole number part of a number from the fractional part.

One and one-half

One and seven tenths

That sort of thing.

So when we hear one hundred and one, we freak out.

But separating the whole number part from the fractional part? That’s just one perspective. Sure, it’s the grammatically correct one.

But here’s the thing about grammar rules…they are arbitrary. Totally and completely arbitrary. Name a grammar rule and there’s a language somewhere that violates it completely.

Yet we English-speaking math teachers act as though the use of and were a signifier of mathematical understanding or competence. Which it is not.

Here’s another interpretation of and in English number language. Maybe and signifies a change of unit. In one and seven tenths, one is counting the original unit, seven is counting tenths. The and helps the listener to follow along; it signifies this shift.

In that case, one hundred and one is the same way. The first one counts a unit-hundreds. The second one (following the and) counts a different unit (which, awkwardly, we call either ones or units).

If I’m right about this, then you will have heard native speakers of English say something such as,

Three hundred and four thousand and twelve.

Three hundred and four counts thousands. Twelve counts ones. Then within the three hundred and four part, there are two different units also. Hence the and.

If I’m right, then you will probably not have heard native speakers of English say something such as,

Thirty and four.

Both of those words count the same units.

So I say let’s give up on this little obsession we have about and. Let’s not let it get in the way of effective, efficient communication in mathematics classrooms.

Let’s save our wrath for this:

Three point twelve.

What is ten?

Consider the seemingly simple question What is ten?

Quantity. This refers to how many things there are. If ten is a quantity, then it refers to this many things: ***** *****

Numeration. This refers to how we write how many things there are. If ten is a set of symbols, then it refers to this: 10.

Number language. This refers to how we say how many things there are. If ten is a word, then it refers to this word: ten.

To illustrate the difference, ask a French person to read this number: 10. Then ask a fifth grader what this Roman numeral stands for: X. Finally ask a computer programmer what number this refers to in binary: 10.

In order, the French person’s dix illustrates that we can use different number language for the same numeration and quantity. The fifth grader’s ten illustrates that we can use different numeration for the same number language and quantity. And the programmer’s two illustrates that we can use the same symbols to represent different quantities.

If I weren’t so lazy, I’d link to a Karen Fuson reference for the research details. Maybe I’ll get to that sometime. But she’s the go-to person on this.


Ratcheting back the rhetoric on Common Core

Bill McCallum writes in the comments here at OMT:

 I don’t think that effort [to define terms such “ratio” and “rate” that CCSS leaves undefined] deserves quite the ridicule it is receiving here, but never mind, the criticism will be taken into consideration nonetheless and inform the final draft. I’ll only say that if I had a dollar for every time someone told me the answers to all these questions were obvious, I’d be a rich man. Of course, the “obvious” answers are mutually self-contradictory. This seems to be an area where it is very difficult indeed to find common language, and where emotions run high.

Fair enough. I’m happy to tone things down a bit.

I do need to observe that no one here at OMT (least of all me) has suggested that the answers to the questions at hand are “obvious“. I agree that they are not obvious at all, and I agree that it is very difficult to find common language with respect to these ideas. (Although this last bit is tricky; if we all use the same words but mean different things by them, are we speaking a common language?)

No, my critique is not at all that Common Core has failed to state the obvious definitions.

My critique is that I see no evidence that Common Core-either the Standards or the Progression on rational number-take into account research on how children learn this content, nor do they seem to coincide with everyday uses of these terms. In the era of No Child Left Behind and “evidence-based practice”, I find it troublesome that results of important research work such as that in the Rational Number Project or Cognitively Guided Instruction don’t seem to form a basis for either document.

I find it surprising that there are no research references in the Progression.

In my work with Connected Mathematics, I have many times had teachers ask for definitions of rate, ratio, fraction and rational number. As writers, we have hashed out these ideas many times as well. Answers are not obvious and reasonable people can disagree.

But we sort of agree that answers to these questions ought to be consistent with, and explain relationships to, uses of these terms in mathematics and the world. I don’t understand why this isn’t the starting place for the Progression.

So how about this for a first attempt at the relationships involved here?

ratio is a multiplicative comparison of two quantities (usually both are non-zero). Conventionally, we use the term “ratio” to apply to part-part comparisons, but this need not be the case.

We can express ratios in several forms. If there are 5 girls for every 3 boys in a certain class, we say that (1) the ratio of girls to boys is 5 to 3, (2) the ratio of girls to boys is 5:3, (3) the ratio of girls to boys is \frac{5}{3}, (4) there are 5 girls for every 3 boys.

The fraction notation \frac{5}{3} is problematic in early ratio instruction because children may confuse it to mean that \frac{5}{3} of the students are girls. Children are accustomed to fraction notation being reserved for part-whole relationships; for this reason the notation should be saved for later instruction.

The term rate suggests change. We tend to talk about a “ratio” in static situations where the values remain constant but a “rate” in a situation where the quantities are changing. In the girls and boys situation above, it would be correct to say that there is a rate of 5 girls for every 3 boys, but this feels awkward. If students were enrolling in a school and there were 5 girls enrolling for every 3 boys, the term “rate” is a more natural fit.

unit rate is a rate where one of the quantities being compared is 1 unit. If we enroll five girls for every three boys, this is not a unit rate. We could say that there are \frac{5}{3} girls per boy enrolling at the school, or \frac{5}{3} girls for every boy. For every (non-zero) unit rate, there is a reciprocal unit rate. So we can also say that there are \frac{3}{5} boys per girl.

What counts as a unit varies. When computing a “unit rate” for buying pop, we could compute the cost per ounce, the cost per can, the cost per six-pack or the cost per case of 24. Which of these is considered a unit rate depends on our choice of unit.

To summarize the discussion, ratios and rates are different mainly in connotation. Each expresses a multiplicative relationship between two numbers (as opposed to an additive relationship, for which we use the term “difference”). Unit rates are important forms of rates because of their intimate connections to algebraic and calculus ideas such as slope and rate of change.

It’s just a first stab at discussing these terms in this context, but is consistent with common usage and focuses the discussion on the main idea that is important at this level-rates and ratios are about multiplication relationships, which are the heart of proportional reasoning.

Going back to the lawn example that has been the focus of discussion here as well as over at the Common Core Tools website, this would suggest that “7 lawns in 4 hours” is a rate (there is change involved, and it’s not a part-to-part relationship), and that there are two unit rates: \frac{7}{4} lawns per hour and \frac{4}{7} hours per lawn.

Again, I am not claiming that these relationships are obvious. But if a couple of important goals for the Progressions work are (1) clarity and (2) usefulness for teachers, professional development and curriculum development, I think my proposal above is an improvement over the present document.

What is a rate? Common Core revisited

A commenter (not me) asks over on the CCSS Progressions blog:

Are rate and unit rate interchangable? Or should a teacher define them for a middle school students as… Rate: a quantity derived from the ratio of two quantities that describes how many units of the first quantity corresponds to one unit of the second quantity. Unit rate: the numerical part of a rate (e.g. For the rate 8 feet per second, the unit rate is 8.) If these are correct, I would then ask for clarity on the phrase “at that rate” in this example from 6.RP.3b. “For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?” Does “at that rate” here really mean “at the rate implied by the ratio of 7 hours to 4 lawns”? You aren’t suggesting that “7 hours to mow 4 lawns” is a rate? The rate, which you ask for in the last question, is “7/4 hours per lawn”?

The answer to this last question is going to be “yes”.

Whether it matches the meaning of these terms in real life or not, the answer will be “yes”.

Whether it matches the grammatical structure of the English language, in which unit would be seen to modify rate, the answer will be “yes”.

A unit rate, in the Looking Glass world of Common Core is not a kind of rate; it’s a different thing altogether. A rate is a numerical/linguistic construction. A unit rate is a number. Each is associated with a ratio.

But why?

The best sense I can make of this is that CCSS wants these terms to be precisely enough defined to admit a sort of mathematical clarity. No such definitions previously existed. So CCSS made them up.