Attend to the verbs in the Mathematical Practices

Editor’s note: This is a guest post by Dev Sinha, a mathematician from the University of Oregon who is working with Illustrative Mathematics. Dev recently attend a meeting of Illustrative Mathematics devoted to elaborating the practice standards at different grade levels, where he made a very important point about verbs in the practice standards, so I invited him to submit this post.

Mathematical objects are key components of content standards.  Practice standards on the other hand describe student actions.  Thus while we usually pay attention to nouns in content standards, for practice standards we must pay attention to verbs.

For  MP7 Look for and make use of structure, “Look for” is a key phrase. Consider the task 2.NBT Making 124, which in brief asks students to decompose 124 into hundreds, tens and ones in all ways they can find (e.g. 6 tens and 64 ones).  In order to be efficient or complete, students will need to use exchanges—of a ten for ones or of a hundred for tens—systematically.  That is, there is a structure of systematic exchanges which students must look for and make use of in order to be highly successful.  We can say that this task implicitly invites students to engage in MP7, through both its “look for” and “make use of” halves.  If the systematic exchanges are suggested by the task or by the teacher before students have had a chance to search themselves, then the practice would not be fully be engaged.

For MP8 Look for and express regularity in repeated reasoning, “express” is an important verb.  Consider this instructional sequence from the progression on Progression on Ratios and Proportional Relationships in which students are to consider equivalent-tasting mixtures of juice.  While students may immediately notice some regularity it is the process of expression, going say from observations about a table to statements like “if we increase the grape juice by 1 cup we must increase the peach juice by 2/5 of a cup to taste the same” and ultimately to writing the equation $y = 2/5 x$, which constitutes the bulk of the mathematical work of the task.  MP8 provides language to discuss this kind of expressive mathematical work.

Development of tasks and lessons involves consideration of the mathematical work students are invited to do.  Content standards provide nouns to be employed in describing this work, while practice standards provide verbs.

6 thoughts on “Attend to the verbs in the Mathematical Practices

  1. I agree it is crucial to attend to the verbs in the practice standards!

    Sometimes it is also important to attend to their other parts of speech. For example, in MP.5, “Use appropriate tools strategically,” the adjective “appropriate” and the adverb “strategically” are both crucial.

    I do think it is true that the nouns in the practice standards differ in kind from the nouns in the content standards. The nouns in the practice standards are these: problems, arguments, reasoning, mathematics, tools, precision, structure, regularity, and repeated reasoning. These are not what we think of as content areas or “topics.” Whereas, the nouns in the content standards generally *are* what we think of as “topics.”

    However, although it is true that “content standards provide nouns,” it is important to observe that the content standards also provide verbs (such as “understand”) and adverbs (such as “fluently”) that are essential to the expectation in question.

    It may be true that people “usually pay attention to the nouns in content standards,” but I wouldn’t want to give a pass to this predilection. Consider 3.NF.3 for example, which says, “Explain equivalence of fractions, and compare fractions by reasoning about their size.” Virtually nothing about this expectation is captured by the noun “fractions,” or by the noun phrase “fraction equivalence.” The verb “explain” is clearly essential. Likewise essential is the adverbial phrase “by reasoning about their size” – without this, one might imagine that students were expected only to use algorithms based on numerators and/or denominators.

    In case helpful or interesting, some related examples can be found here:

  2. Pingback: The Content Standards are the Nouns; the Practice Standards are the Verbs |

  3. Thanks, Jason. I did think of the ubiquity of the verb “understand” in the content standards when I wrote this. (I was also reminded of the importance of conveying how that verb “understand” is intended). Moreover, I paused when writing because in my view mathematics is both a subject area and a way of thinking, and I think the noun/verb dichotomy ultimately divorces things too much. But to begin a conversation, I’m comfortable with talking about these aspects separately, as predominantly (though not exclusively, as you point out) embedded in the standards as content vs. practice.

    In making these distinctions, I think it isn’t so much the parts of speech used in the standards we should attend to as it is the underlying mathematics and student actions. In your 3.NF.3 example, there is a noun there implicitly, namely a preferred proof that fractions are equivalent. (Thinking of proofs as nouns is part of the trouble with this dichotomy; the “verby” language of the standard is perfectly natural.) The verb “explain”, by the way, is essentially being borrowed from MP3.

    [By the way, I happen to be going over this material at the moment with some pre-service teachers, and they came to agree that the choice of argument based on reasoning about sizes (in particular sizes on the number line) in the CCSS is preferable to more formal or algorithmic approaches. Just thought that you’d appreciate the vote of confidence in the choices you made from a room full of undergrads ; ) ]

    • Thanks Dev! Appreciated these additional thoughts. You make a great observation that “explain” in 3.NF.3 is a point of connection between content and practices. And as you say, it helps to realize that explaining is more than just “doing a verb” – explaining also *results in an explanation,* and an explanation is a thing that can be turned over and examined.

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