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.

Once every few months or so I receive a message about the following standard:

6.G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas $V=lwh$ and $V=bh$ to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

See if you can guess what people think the problem is before reading on. Continue reading →