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?
A 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 , (4) there are 5 girls for every 3 boys.
The fraction notation is problematic in early ratio instruction because children may confuse it to mean that 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.
A 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 girls per boy enrolling at the school, or girls for every boy. For every (non-zero) unit rate, there is a reciprocal unit rate. So we can also say that there are 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: lawns per hour and 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.
I could read this. It made sense. Apparently, that’s a hard thing to achieve with this sort of writing. Well done!
You might want to move the parenthetical “(as opposed to an additive relationship, for which we use the term “difference”)” to the first mention of ratio, “A ratio is a multiplicative comparison of two quantities”. That felt weird to me, and it might be much more disorienting to an education student, who hasn’t really grasped yet that division can be thought of as a multiplicative relationship.
Christopher, thanks for this. I like your paragraphs about ratio, rate, and unit rate. Eventually (in high school, say) we want students to have a central conception of rate as a quantity describing a proportional relationship between two varying quantities, expressed in derived units. By that point the distinction between ratio, rate, and unit rate almost disappears; they are all tied together, and people with a solid understanding have many ways of describing them and use ratio and rate language interchangeably. Still, kids in Grade 6 can’t start out with the whole package; the tricky part is to figure out when to introduce which threads and how to pull them together.
Your question about the research base deserves a longer response than I can give here, but I’ll make two quick points in regard to CGI and RNP. First, the K–5 OA and NBT progressions made extensive use of the 2009 National Research Council Report on early childhood learning. (The standards on Counting and Cardinality in particular follow that report quite closely.) In particular the table of types of word problems comes from that report, and there is a corresponding stream of standards that moves through the levels of difficulty in word problems. This all seems compatible with CGI.
As for the approach of the fraction progression, building up fractions out of unit fractions and moving over the grade levels to the number line as a central approach, it is strongly influenced by East Asian curricula, where this approach is common. The Rational Number Project submitted an extensive critique of the public draft of the standards, and although we probably didn’t do everything they wanted, we did make changes in response to that critique, softening the number line approach and making room for visual fraction models.
In general, many of the decisions we made involved looking at contradictory sources: we looked carefully at these sources, including education research where clear conclusions were available, and we listened carefully to what people had to say. We tried in each case to make reasoned decisions that retained the focus and coherence of the standards.
Well, I guess that was a pretty long response after all.
Bill writes: [t]he tricky part is to figure out when to introduce which threads and how to pull them together.
True enough. Traditionally in this country, these tricky bits have been dealt with at the level of curriculum materials. Consider the very real and very strong differences between, say, Saxon, UCSMP and Connected Math on these issues.
My understanding is that our Canadian neighbo(u)rs think differently about the term curriculum. They use that term to describe the agreed-upon learning trajectory but not the materials themselves. Thus, Connected Math would be “support materials” or some such, while CCSS would be the “curriculum”. And further, the curriculum is more prescriptive about learning trajectories than U.S. standards have been-either in terms of NCTM-type documents or state-level NCLB-inspired ones.
I wonder if it is a goal of Common Core to end up with something more like a curriculum and less like a set of U.S. standards.
Well put sir. Well put.
Common Core is indeed a set of standards, that is, it is a description of understandings and skills we want our children to have. A curriculum is a sequence of experiences intended to help children acquire those understandings and skills. The language of the Common Core is not the language of the classroom. For example, the cluster heading “Use place value understanding and properties of operations to add and subtract” does not mean that students should use the term “properties of operations”. The same goes for “unit rate”. It’s a name for an idea (the number of units of one quantity per unit of another quantity). A curriculum provides students with experiences that develop that idea. Whether or not the curriculum uses the term, or expects students to, is entirely up to the curriculum designers, and is not specified by the Standards.