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There are a few standards that use “and/or” when describing ways that students are meant to perform operations. For example, 1.NBT.4, 1.NBT.6, 2.NBT.5, 2.NBT.7, 3.NBT.2, …
Are teachers expected to demonstrate all methods surrounding the “and/or” over the course of a year? I know that students do not have to use every single method all at once, but if a teacher taught only one method, or perhaps two out of three, is this acceptable? If they use none of these methods is it acceptable?
One particular case I’m thinking of is 4.NBT.6 where, in reference to division, students are to “illustrate and explain the calculation by using equations, rectangular arrays, and/or area models”. Only one example in the NBT Progressions on page 15 really fits this description: the model that sets division as finding an unknown side length. It is clearly an area model that links strongly with the area model for multiplication, though it would be an onerous task to show concretely or pictorially with arrays. The other model shown, finding group size, does not rely in any large part on equations and has nothing to do with arrays or area models. This model could be demonstrated concretely by sharing base-ten blocks between students and the process literally “illustrated” using pictures.
The second method is a valid approach but is not listed in that final sentence of 4.NBT.6 (nor 5.NBT.6). It is obviously acceptable according to the first part of the standard because it relies on place value. Because I can see a direct pathway from that method to the standard algorithm expected in Grade 6 my preference is to only teach this method to avoid any student confusion over methods and confusion over models.
So, to my original questions, do any and all methods listed in the Standards have to be taught over the course of a year and can only those methods be taught?
Quoting from the standards: “These Standards do not dictate curriculum or teaching methods” (p. 5).
Duane, you seem to be identifying “method” for operations with the use of a particular representation and interpretation of the operation. Certainly some representations lend themselves better to different interpretations and computation methods than others, but the association isn’t necessarily always uniform. And anyway, the OA standards ask that students understand different interpretations of operations. So at some point students need to use interpretations of division that involve measurement (e.g., side length of a rectangle) as well as counting (group size, number of groups, number of objects in a row, number of rows in array).
One thing that you are getting at is the difference between a count model (e.g., number path, group of objects, array) and a measurement model (e.g., number line diagram, area model). For multiplication and division, the count models are associated with interpretations like equal groups of objects or arrays of objects (the first two sections of Table 3 in the OA progression which are similar to the upper rows of the first two sections of Table 2 in the CCSS). The measurement models tend to be associated with interpretations like measurement and area, which might be represented by such models or just stated in ways that indicate measurement, e.g., “You need 3 lengths of string, each 6 inches long. How much string will you need altogether?” At some point, these associations might get blurred because units in a measurement model can act like objects in a count model, e.g., a length of string 6 inches long might be thought of as 6 inch-units. And, a 7 x 100 rectangle can be thought of as 700 square units arranged in 7 rows of 100.
There’s a correspondence between the two types of representations on p. 15 that we might consider illustrating, at least for an example with small numbers, say, 255 divided by 3. That can be represented by the base-ten blocks (or connecting cubes) arranged in 3 equal groups of ones and tens or arranged as a rectangle with side lengths of 3 and 80 + 5. Independent of whether the calculation is interpreted as being about an array or a rectangle, there’s a connection with 255 = 3 x (80 + 5) and other equations (e.g., 255 = 3 x 80 + 3 x 5. Correspondences between the numbers and operations in those equations can be shown for the rectangle and for the base-ten block groups (in keeping with MPS 1: identify correspondences between different approaches).
You may be worrying about how adept students are with length and area. My sense from reading research is that measurement is often neglected in elementary grades, so this is an understandable worry. But, there’s also the possibility of a vicious circle . . . if measurement is neglected, then students don’t do well with measurement, so measurement is neglected, then students don’t do well with measurement . . .
It’s certainly food for thought and I can see the connections you’re talking about. I’m not concerned that students won’t understand the measurement model of division. It’s more that the “finding group size” model on page 15 of the Progressions seems to fit easily with the standard algorithm that students need to be fluent with in Grade 6, yet it isn’t clearly mentioned in the last sentence of 4.NBT.6:
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Is there a link there that I’m missing? You seem to suggest that if students use blocks and a sharing model then it is symbolically equivalent to any other model, and that this demonstrated via equations. This is true, but arrays and area are probably closer to what teachers think of as models, and equations as ways of representing those models (models of models?). Not including “equal groups” as a model in the last sentence of 4.NBT.6 has probably muddied the waters unintentionally.
Also, speaking more generally, despite the statement that the Standards do not dictate curriculum or teaching methods, to my eyes 4.NBT.6 is a counter-example (among others). When phrases such as “using strategies based on” and “illustrate and explain the calculation by using” are used it directs teachers towards certain methods. To make 4.NBT.6 devoid of direction it would simply say “Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors. Illustrate and explain the calculation.” When lists of strategies and models are presented as well, then the choice of teaching methods becomes skewed towards those strategies and methods. This isn’t necessarily bad, depending on the goals, but the use of “and/or” makes it more difficult.
Just one single method may not be dictated but, in the specific case of 4.NBT.6, three methods are presented instead. To the first part of my original question, is it the intention that all three models be presented over the course of a year? If they use none of them is that okay too? The models in the last sentence of 4.NBT.6 are not given as examples – they read as requirements – so at least one of them must be used. So I’m not asking for which one, but how many?
Duane, there are a few words that you’re using that have multiple meanings in this context: demonstrate, model, and method.
To some people, “demonstrate” means “show” and to some it means “prove.” (To add to the confusion, “show” and “prove” are sometimes used to mean the same thing.) “Model” is sometimes used to mean “exemplar.” This suggests to me that when you say “demonstrate a model” it means “illustrate how to use a type of diagram” or maybe “illustrate how to use a given procedure. This procedure includes writing a certain type of diagram.” Could you perhaps use a word other than “demonstrate”?
I think that one thing that you’re missing is the calculation might come first, followed by the equations or diagram. When you say “method” you seem to mean “calculation plus equations or rectangular array or area model.” But, an illustration is not necessarily part of a calculation. So, I don’t see 4NBT6 as asking for three methods. I see it as asking the student to be able to calculate using one of the strategies listed, then to illustrate with one of three options.
Model is also used to mean “representation” (e.g., area model). Because you are listing “equal groups” with area model, I think that you mean that “equal groups” is a type of representation. (Note that the “equal groups language” described in OA Table 3 is not necessarily associated with an “equal groups representation.”) Or do you mean the “equal groups model” is the example “division as finding group size” on the lower right of p. 15? That is not the same as one of the “equal groups” representations that I saw on the web.
“Model” is also used to mean “interpretation of an operation,” e.g., quotitive or partitive model of division.
Quotitive corresponds to “how many groups”? It is, unfortunately, also called “measurement model.” An example of a quotitive interpretation of division in a measurement context is “How many 1/2-foot lengths in a board that is 1.75 feet long?”
Partitive is about “how many in a group?” An example of a partitive interpretation of division in a measurement context is “If a 1.75-foot board is partitioned into 3 equal pieces, how long is each piece?” According to my Google search results, this is also called the “sharing model of division.”
To avoid confusion with quotitive vs partitive, instead of talking about representations that are “count models” (e.g., number path and array) vs “measurement models” (e.g., number line diagram and area model), one might talk about representations that depict discrete vs continuous objects.
Another interpretation of division that isn’t always obviously partitive or quotitive is sometimes called “product and factors.” (For more discussion and examples, see Liping Ma’s Knowing and Teaching Elementary Mathematics, p. 72.) That fits some division situations which can be represented as rectangle area and side length but also fits some division situations which can be represented with rectangular arrays (see the row and column language section of OA Table 3).
Cathy, I can see the distinction you’ve made between how the students calculate and what they use to illustrate their thinking. The “equal groups” reference I made was in relation to the bottom margin diagram on p.15 of the Progressions – this was a sloppy mistake on my part.
Essentially though, the trouble is with students calculating using the sharing process shown in that diagram, then not having an obvious means to illustrate their calculation from the options of equations, arrays, or area models. I’m guessing the closest one is equations. Is this correct? If so, could you please provide an example of what the matching equations would look like for the bottom margin diagram on p.15?
“By allowing students to start with a more elaborated method, such as Area Method 1, and then progress to Method 2 or 3 as they no longer need the support of a drawing, students can use methods that allow them to make sense of the algorithm while also working towards fluency.”
This comment was made in the context of multiplication, but is relevant for thinking about division as well. The standards are benchmarks that students reach at the end of a grade. The Progressions sometimes illustrate things that students might do before reaching those benchmarks.
I’ve put some different ways to illustrate the figure from p. 15 of the NBT Progression with equations here: http://wp.me/aJHdC-6K.
Because there is a remainder, using a rectangle might seem a bit strange but it could be done with the inclusion of a unit square. For this example, using an array seems pretty time-consuming.
This is also an opportunity for identifying correspondences between equations and diagrams (MP1), which I didn’t do. In particular, students might be asked about what the 6 corresponds to.
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