For several years, starting shortly after the release of the Common Core State Standards for Mathematics (CCSS-M; Common Core State Standards Initiative [CCSSI], 2010), I spent every day of my working life designing, writing, testing, and editing activities meant to address the K-5 content standards. Often, this work was fun. There were a handful of standards, however, that tended to make the work miserable.
Take this fifth-grade standard from the Operations and Algebraic Thinking strand (5.OA.2):
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. (CCSSI, 2010, p. 35)
True confessions: I came to hate this standard. At first read, it doesn’t seem so bad. However, attempting to develop extended activities on it — a full lesson, and plenty of follow up practice — was a slog. I (and all the members of the team working on this grade level’s materials, from what I remember) quickly ran out of ways to make this work interesting. Grouping symbols were new to the grade level, but the translation and interpretation of expressions from words to symbols was not. The standards in first through fourth grades required students to write equations to represent word problems, a task related to this standard but more interesting because the translation in those cases required more abstraction. Story-to-symbols can be interesting; words-without-context-to-symbols seemed to be a step backward. I just couldn’t understand the point of this standard. Any time I wrote another prompt for students to write about how, for example, (5 + 7) × 2 was twice (5 + 7), I said a silent apology to the future fifth graders of America.
A few weeks ago, though, I read some articles that made me think a bit differently about this standard. The articles focused on the mathematics register, or “meanings that belong to the language of mathematics (the mathematical use of natural language, that is: not mathematics itself), and that a language must express if it is being used for mathematical purposes” (Halliday, 1978, p. 195). The math register does not consist only of vocabulary, but also of language structures and modes of argument that convey meaning (Halliday). The notion of the math register has been discussed and expanded in various ways since Halliday introduced it. Of particular interest to me is the idea that the mathematics register includes symbolic notations and grammatical structures that map onto each other in opaque ways; for example, one way to translate a2 + (a + 2)2 = 340 into spoken or written language is, “The sum of the squares of two consecutive positive even integers is 340” (Schleppegrell, 2007, p. 145).
As I read the Schleppegrell piece, I was struck by the illustration of how complex the translation from mathematical symbols to words and back can be. I thought again about 5.OA.2. The examples given in the standard are trivial (after the first few practice rounds, at least), but in general, there are plenty of examples that are not, particularly when variables start to be used to express general relationships. The text of 5.OA.2 does not allow for the use of algebraic, rather than numeric, expressions, but suddenly I wondered if this standard was intended to support readiness for more complex sorts of translation and interpretation of expressions later.
Based on the information I have at hand, my best answer to that question is “maybe.” According to the draft progression document for the Operations and Algebraic Thinking (OA) strand of the CCSS-M, the intention of 5.OA.2 is to prepare students to work with algebraic expressions in later grades: “[S]tudents in Grade 5 begin to think about numerical expressions in ways that prefigure their later work with variable expressions (e.g., three times an unknown length is 3 × L)” (Common Core Standards Writing Team, 2011, p. 32). They go on to further describe the work in Grade 6:
In Grade 6, students will begin to view expressions not just as calculation recipes but as entities in their own right, which can be described in terms of their parts. For example, students see 8 × (5 + 2) as the product of 8 with the sum 5 + 2. In particular, students must use the conventions for order of operations to interpret expressions, not just to evaluate them. Viewing expressions as entities created from component parts is essential for seeing the structure of expressions in later grades and using structure to reason about expressions and functions. (Common Core Standards Writing Team, 2011, p. 34)
It seems clear that the standards writers were attempting to anticipate an increase in complexity in interpretation of expressions. I’m not entirely convinced they were focused on the communication of meanings as Schleppegrell (2007) was; rather, they seem to be focused specifically on recognizing mathematical structures and reasoning about them, rather than ascribing them with meaning in natural language. Still, there’s some evidence of an intention behind standard 5.OA.2 that I didn’t understand until I read about the mathematics register.
I spent countless hours thinking about 5.OA.2 and many of the other standards. I have thought about them together and separately; as activities in their own right and as learning goals for those activities; in their most literal and most figurative senses. I did not think there were many more ways to think about or interpret them than those I’d already turned over in my head at least once. But this experience proved me wrong. Now I have new way to think about why this standard might matter to students’ understanding of mathematics.
I’m not sure how this new information would affect my curriculum development work if I had a time machine and could go back and try again. Maybe it wouldn’t change anything about the practical implementation or the student exercises. But I suspect it would change the way I wrote about it in the teacher materials.
What do you think? Is building readiness for interpretation of algebraic expressions a worthwhile goal? Is it what 5.OA.2 was meant to do?
References
Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Retrieved 06 October 2017 from http://www.corestandards.org/Math/.
Common Core Standards Writing Team. (2011). Progressions for the Common Core State Standards in Mathematics (draft): K, Counting and Cardinality; K-5, Operations and Algebraic Thinking. Retrieved 06 October 2017 from https://commoncoretools.files.wordpress.com/2011/05/ccss_progression_cc_oa_k5_2011_05_302.pdf
Halliday, M. A. K. (1978). Sociolinguistic Aspects of Mathematical Education. In Language as Social Semiotic: The Social Interpretation of Language and Meaning (pp. 194-204). Baltimore, MD: University Park Press.
Schleppegrell, M. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading and Writing Quarterly, 23(2), 139-159.
Katie the Curious 