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mfast: To get to the documents you need to click on the links in the blog post. These are:
Algebra: http://commoncoretools.me/wp-content/uploads/2012/12/ccss_progression_algebra_2012_12_04.pdf
Functions: http://commoncoretools.me/wp-content/uploads/2012/12/ccss_progression_functions_2012_12_04.pdf
You’re replying to the post about Algebra and Functions, which suggests you are thinking of high school. However, the term “math specialist” is often used to describe elementary mathematics specialists.
For information about elementary mathematics specialists (EMS), including examples of programs, see https://sites.google.com/site/emstlonline/. This site acts as a clearinghouse for information about EMS.
For information about preparation and professional development for PreK–12 teachers (not a textbook, sorry, but there are links to examples of programs and other information), see the new Mathematical Education of Teachers report: http://www.cbmsweb.org/MET2/index.htm
Duane, does “these tasks” refer to 4.G.1? Would it help if we put references to this standard after:
“For instance, what series of commands would produce a square?” (p. 14)
“Given a segment on a rectangular grid that is not parallel to a grid line, draw a parallel segment of the same length with a given endpoint.” (p. 15)
Or does your question refer to the point of the tasks such as the two above?
Nancy, I’m not Bill, but I edit the progressions, so I can tell you that not every standard appears in some progression and that’s the case for the ones about money. (There’s a discussion of the issue of which standards appear in progressions somewhere on this site, perhaps before the forum was set up, but I’m not finding it readily.)
It may be helpful to you to look at the pre-forum blog posts and search on “money” here: http://commoncoretools.me. As you can see, there are several discussions under “general comments” and two mentions of money elsewhere. Note that money is also in 4MD2.
Jack, the main purpose of the progressions is described here: ime.math.arizona.edu/progressions/. It is not communication of education research to education researchers, so I think you’re in a minority as regards usefulness of references. From your point of view as a researcher, you might think of the progressions as being somewhat more like a methods book (e.g., Van de Walle) than like a research article.
Verbatim text without citation would, of course, be a violation of copyright. That’s why when something is quoted verbatim in the progressions the reference is given, though in a very brief format as with the Usiskin reference. (The quote about T(E) and T(I) should be completely indented. That will help it look more like a quote.)
Ideas are a different matter. The progressions don’t give references for who invented the box plot (John Tukey, 1977, refining Mary Eleanor Spear’s “range bar,” 1952). Similarly, the van Hiele ideas have been refined since the van Hieles wrote (which was between the 1950s and 1980s). That’s reflected in the geometry progression. Similarly, a lot of ideas such as quotitive and partitive interpretations of division have passed into more common use (although maybe under different names).
Yes, these are correct. We switched the order so that the kindergarten situations would be adjacent to each other in the table (trying to make it easier to read).
In the CCSS, the table didn’t have that problem because details about when various problem types occur are only listed in the cluster headings and the overviews for the grades. The reader needs to do the work of connecting those with the table. The cluster heading for OA in kindergarten says: “Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.” That corresponds to the four dark grey cells in the progressions table 2.
In grade 1, 1OA1 is more explicit and says,
“Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.” It also has a footnote directing the reader to the table.
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).
What standard might that question be testing? My guess would be 5NBT2 except that standard is about explaining patterns and one instance doesn’t make a pattern.
So maybe the question is more about being familiar with the pattern? Maybe that makes the question more about 5NBT7 and knowing about one piece of the calculation.
Could you elaborate about what the question is supposed to tap?
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 . . .
I’m not sure that the analogue with whole number subtraction and the possible meanings of “conversion” are clear. Let’s suppose we’re computing $2 \frac 13 – \frac23$.
There are at least three options (I’ll put fewer steps than a student might and put parentheses to push the analogue that I’m trying to make):
A. $(2 + \frac13) – \frac23 = \frac73 – \frac23 = \frac53$ (converting the entire mixed number to an improper fraction, i.e. decomposing two 1s as six thirds)
B. $(2 + \frac13) – \frac23 = (1 + 1 + \frac13) – \frac23 = (1 + \frac13) + (1 – \frac23) = (1 + \frac13) + \frac13 = 1 + \frac23$ (using properties of operations)
C. $(2 + \frac13) – \frac23 = (1 + 1 + \frac13) – \frac23 = (1 + \frac43) – \frac23 = 1 + \frac23$ (decomposing one 1 as three thirds)
Versions A and C both involve writing an improper fraction (though one could do a variant of C that didn’t), but C is a closer analogue to decomposing a unit as done in the subtraction algorithm for whole numbers. So, when you get to decimal fractions, you can use an analogue of C to mimic what’s happening in the subtraction algorithm:
$(2 + \frac1{10}) – \frac2{10} = (1 + 1 + \frac1{10}) – \frac2{10} = (1 + \frac{11}{10}) – \frac2{10} = 1 + \frac9{10}.$
The advantage of just decomposing a 1 as in C is that it’s the analogue of what students learned in multi-digit subtraction, so the same explanation of decomposing a unit of the minuend (in this case a 1 of the 2 + 1/3) applies for fractions and later to fractions expressed in decimal notation. (In case you’re familiar with Liping Ma’s book Knowing and Teaching Elementary Mathematics, I’m thinking about the discussion on pp. 8–9.)
I checked with one of the progressions writers. The second to last paragraph of the sidenote on p. 3 can be revised to read:
“These different meanings result in different classifications at the abstract level. According to T(E), a parallelogram is not a trapezoid; according to T(I), a parallelogram is a trapezoid. At the analytic level, the question of whether a parallelogram is a trapezoid may arise, just as the question of whether a square is a rectangle may arise. At the visual or descriptive levels, the different definitions are unlikely to affect students or curriculum.”
That does still seem to leave a lot of years when differences in definitions might affect curriculum. However, given that classification only occurs in a few grades, my guess would be that the different definitions would affect curriculum in grades 3 to 5 and statements of theorems in high school (just how many would need to be checked—and it’s possible that students might create their own). Obviously, that’s not good if definitions change during those grades, but it’s better than having fluctuating definitions affecting all of K–12. And certainly students who happen to be curious should have their questions answered in consistent ways, so one would hope to have this issue mentioned in teachers manuals and professional development.
Duane, assessment developers also get guidance from http://illustrativemathematics.org/standards/k8. Check out the Peaches task. Two solutions are given (with and without conversion). Neither is labeled “preferred.”
The standard is written to indicate that the methods are examples: “Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.”
The “e.g.” is there for a reason, it indicates that the method is not prescribed. In the sentence, one reason to put the conversion method before the other method is that the sentence is hard to read if the order is switched (too many “and”s).
The focus of the standard is “can the student add and subtract mixed numbers”? I think that we would all be unhappy if an assessment developer misinterpreted the sentence but I don’t think it’s easy to misinterpret in the way you fear. I realize that the US has a history of poor quality tests (e.g., http://www.educationsector.org/publications/margins-error-testing-industry-no-child-left-behind-era), but some of the conditions of test development have been changed.
In the progression, line 5 of page 7 says: “Students use this method to add mixed numbers with like denominators.” I think the problem is that the sentence can be interpreted as saying “Students MUST use this method” and there’s no example that shows another method. We can put one in.
Duane, thanks for this comment. Note that the example of converting 47/6 isn’t coming from a sum and that the CCSS say “Grade 4 expectations in this domain [NF] are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.” Are you chaining together two examples from the progression to extrapolate a single method for adding mixed numbers that you think the progression is recommending? The standard about adding mixed numbers doesn’t prescribe a single method:
4NF.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
Another reason to mention converting mixed numbers to fractions (and vice versa) is that it is yet another example of decomposing or composing a unit. For fractions with base 10 or 100, this can be used to as preparation to extend ideas of the base-ten system, e.g., 2 and 1/10 is 21 tenths.
Your second-to-last paragraph reminds me of Stanley Erlwanger’s famous case study of Benny, the pseudonym of a student who was learning about fractions and decimals in the 1970s. It’s posted here, together with an introduction: http://www.uky.edu/~mfi223/EDC670OtherReadings_files/ErlwangersBenny.pdf. One of the things that Benny believed was that if you did something (e.g., converted mixed numbers to improper fractions) pictorially and computationally, it didn’t matter if you got different answers from different methods. That sort of belief is countered by MPS1 which says that students can explain correspondences between different representations of the same thing and between different approaches to computing the same thing.
