Math Practice Standards

The last week of June I attended an excellent course at the EDC via DSAC which is a part of the MA DESE about the math practices in the CCSS.  Does every profession have such ridiculous acronyms?  Crazy jumblings of letters aside, it was a great week.  The course was specifically designed for high school math courses beyond Algebra 1, so it was ideal for me as I continue to teach Geometry and start up with Pre-Calculus next year.  Each day was focused on examples of a particular math practice.  Since I took the course for credit I wrote a reflection each evening about my current and future use of the math practices.  Since life is busy (I'm at PCMI!) I'm sharing excerpts of those reflections here with you.  I really hope that you are thinking deeply about how to incorporate the practice standards in your classroom.  New standards mean nothing unless you, the teachers, make sure students need to use them (more on that in my next post- reflecting on my first week at PCMI this year).

The Standards for Mathematical Practice are:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

I try to encourage students to use all of the standards for mathematical practice, although I do so implicitly.  A goal for the upcoming year is to post the standards and explicitly ask students what methods they are using or which method they could apply to a problem when they are stuck.  This level of metacognition should help students be able to better help themselves.

In Geometry, I try to invoke the ideas of mathematical practice 3 outside of formal proofs and the chapter on conditional statements so students recognize that justification is a part of all mathematics, not specific to two column proofs.  A phrase that I utter in class so often that by the end of the first month students can already anticipate it is: “defend your answer.”  I introduce it the first few times as imagining that they are lawyers and need to present evidence to the jury beyond reasonable doubt to convince the class that their answer is correct.  Asking this of students moves the focus from the answer to the method of solving.  When we are in the middle of discussing a problem and a student yells out an answer I redirect them to the step we are on and refuse to confirm or deny their number.  In the future I would like to make my reasoning for focusing on the process rather than the product clearer to students- every student quickly realizes that I do focus on the process, but they don’t necessarily realize why.

I am not yet sure which of the contexts in Pre-Calculus will lend themselves best to MP3, but at the very least I will make sure to include questions that require interpreting results to justify why the solution makes sense. And throughout the course students will be required to justify their solutions using the laws of algebra. I look forward to finding other opportunities for proof, analysis and critique in my classes. 

One way I plan to support my students in using Mathematical Practice 7 next year is by increasing their familiarity with structures that they should already know.  For example, I will be teaching two sections of Honors Pre-Calculus.  For that class I assigned a summer project where students have to graph a few examples of the families of functions they have studied so far.  After attending class today I wish that I had specified including a table of values for at minimum the parent functions.  Since I neglected to assign it over the summer I will plan to have students work with perfect squares, perfect cubes, powers of two and any other important sets of numbers that students should quickly recognize.  Knowing these numbers by heart makes it so much easier to notice when a set of numbers is one less or one more than a perfect square or seems to be increasing like an exponential function.

Math Practice 8 seems to apply differently in Geometry than in Algebra based courses.  However, we do use this process on a regular basis.  In class I ask students to draw a model and take some measurements, then make a conjecture based on the data everyone produces.  I try to do this in some form for every theorem we introduce, so students have the opportunity to at the very least gain intuition, but in the best cases they get to experience the thrill of discovering a new property. 

Another thing I learned in this class is how very hard it is to sit and work for 8 hours!  Students at least get to change rooms and subjects a few times a day, but it still is draining to be asked to think for long stretches of time (especially the first week of summer!).  Our instructors anticipated this and provided candy and play dough to keep us on a sugar high and give us something to play with other than our phones.