They’re really for me but I plan to share them at some upcoming PL sessions. I’ll take any suggestions and comments of how to make them better…
1. Make sense of problems and persevere in solving them – Teachers of mathematically proficient students understand that students will struggle, however that is not an indicator of teacher effectiveness. They understand that students struggle because they’re engaged in problematic tasks and it is their job to ensure that the struggle remains productive, not destructive. Teachers of mathematically proficient students preserver through their own frustration by questioning relentlessly and continuing to improve their own understanding of student pedagogy. They push relational understanding of mathematics by creating a culture and climate that promotes metacognition and embraces student efficacy.
2. Reason abstractly and quantitatively – Teachers of mathematically proficient students expose students to situations where they are able to recognize patterns and relationships as tools to effectively make sense of problems. Teachers of mathematically proficient students ensure that students are provided multiple opportunities to explore mathematics by flexibly relating numbers and quantities in and out of contextualized situations.
3. Construct viable arguments and critique the reasoning of others –Teachers of mathematically proficient students understand the importance of communication, both verbal and non-verbal. They continually seek for ways to engage students in dialogue and encourage ideas that are wondered, noticed, and questioned. Teachers of mathematically proficient students expect their students to make conjectures through the use of models and diagrams. When conjectures are made, a teacher of mathematically proficient students expect that students are able to explain why it works.
4. Model with mathematics – Teachers of mathematically proficient students understand the importance of teaching in context and the role it plays in the development of relational understanding. They expect that students use mathematics to make sense of the world around them. Contextualized situations create opportunities for students to derive, test, and validate assumptions.
5. Use appropriate tools strategically – Teachers of mathematically proficient students expose and encourage the use of multiple tools to make sense of mathematics. Most importantly, the teacher understands that it is the student who chooses the tool. Teachers of mathematically proficient students recognize that tool selection stems from the student’s understanding of its limitations and restraints.
6. Attend to precision – Teachers of mathematically proficient students model verbal and written precision continuously. They calculate accurately and efficiently through the correct use of numbers, symbols, and mathematical conventions. All of this is accomplished as they simultaneously relate their understanding in context. And because teachers of mathematically proficient students consistently model precision…they expect the same from their students.
7. Look for and make use of structure – Teachers of mathematically proficient students provide meaningful tasks that encourage a relational understanding of patterns, rather than instrumental. The use and sense making of mathematical structure is encouraged to promote efficiency. Teachers of mathematically proficient students understand the importance of a student being able to compose and decompose objects because of their mathematical understanding.
8. Look for and express regularity in repeated reasoning – Teachers of mathematically proficient students encourage generalizations. They are continually asking students “does that work all the time” or “is there a more efficient solution path”. They encourage students to analyze their work, identify shortcuts, and/or derive a rule based on their intermediate results.
I love this idea. My gut tells me though that the beauty of this is more in the process than the product. Perhaps at the PL sessions it could be more about teachers reading a math practice standard and then writing up something about what they believe teachers need to do to make that happen. If time is an issue, it could always be jigsawed so each group gets one of the standards. Great idea though.
Throughout the entire process of writing these up I was thinking “I need to get teachers to do this”. There was a whole different level of understanding that took place and you nailed it…the process far outweighed the product. I just wish I had collaborated when fleshing these out instead of doing it by myself (in retrospect).
I did this very thing with 6th and 7th grade teachers last week and it went great. In previous PL sessions I’ve had teachers create posters for each SMP but the focus was heavily student centered. When I asked teachers to rewrite the standards as if it pertained to them, the process was golden and by the end of it, teachers had created a list of Teacher Math Practices for which they were accountable. Great minds think alike my friend!
There is a need to make crystal clear that the practice standards aren’t just for students or just for teachers. They are for the practice of mathematics – doing it, learning it, teaching it, using it. So explicating just how they apply to any group, subgroup, or individual is likely valuable.
That said, focusing on teachers and the “precision” standard. Yes, teachers should know how to calculate accurately, of course. So should students. But we all make errors, and not all of them are of the “gee, I just realized I don’t actually know how to figure this out” sort. Those are interesting to explore openly with students when they arise (and of course, sometimes it’s useful for teachers to go astray in trying to calculate or how to solve a mathematical problem at all). But let’s not get too hung up on the minor errors – recognize them for what they are and move on. Trying to talk and write at the same time can lead to lots of “board typos” that don’t have much real mathematical significance. And teachers who jump overly hard on trivial errors will be repaid in kind by their students when they goof in front of the class. I’m not sure how productive such things really are.
I have felt since I first saw it that the “precision” standard was put in as at least in part a political move: trying to strike pre-emptively against those who have long opposed the NCTM Process Standards and lots more as “fuzzy” and indifferent or hostile to accurate, “automatic” computation, particularly in K-5. We’ve all heard the accusation that in progressive math classrooms, the right answer doesn’t matter. I find that an enervating discussion, but it’s gone on for at least the last quarter century, shedding much heat and very little light. Unfortunately, from some commentary I’ve read from die-hard “math warriors,” the precision standard has not even slightly convinced them that anyone involved with the Common Core math standards cares about the sort of computational skills that traditionalists value. Frankly, I’d just as soon move on to more fruitful areas of conversation.
So in that light, it would be great to flesh out what you’ve written with specific examples taken from your own practice or that of colleagues. How do those high-sounding principles work specifically? What does it look like in particular contexts for a teacher do any of those eight (or so) sorts of things? Only by putting them in contexts will it be possible, I suspect, to flesh out why each is actually a vital part of mathematical practice.
In any case, you’ve made a useful start with what you’ve done here. I hope it gets shared widely (I will post it to FB with my comment) and that a good conversation grows out of it for teachers, parents, administrators, students, etc.
I really appreciate your comment Michael and thanks for sharing.
You’ve definitely given me something to chew on in terms of precision. Precision does play an important part however I’ve witnessed a lot of instances when a teacher’s over-emphasis on precision has negatively impacted conceptual understanding. There’s a fine line that needs to be totted when seeking precision. The right answer does matter and so does the process.
Effective math instruction can’t be fully achieved until the Mathematical Practices are not only acknowledged but embraced by all stakeholders. I really like the idea of connecting the SMPs to contextual understanding (at all levels). This would help paint a much clearer picture as to what is expected.
Absolutely awesome. I plan to share with my email list if that’s ok with you. Would it be appropriate to add to the precision discussion to speak with precision, using correct grammar and write with precision using correct spelling?
Love what you’re doing!
Robyn
Absolutely Robyn and thanks Robyn! Please share as you see fit.
As much as I agree that speaking, using correct grammar and writing are critical when discussing precision, I’m trying to keep the standards as much in the spirit of the original SMPs as outlined by the CCSS writers.
My hope is that the aforementioned skills will be brought up in the discussion when teachers share these with one another.
Love this! These are so well connected to the teacher keys, it’s about time someone did this. Need to look more closely later but I do have one suggestion. Instead of beginning each with “Mathematically proficient teachers. . .” How about keeping the focus on students: “Teachers of mathematically proficient students. . .” Let’s focus on the aspect of how we teach for these rather than mathematical competency. That’ll be another list!
Love it and thanks Mike! Done!