Hi Andrew,   This is a very interesting and important question.  In fact, it’s related to one of my favorite themes, namely the habit of using language productively and with appropriate precision, what we call “good use of language”.  Yes, we do see that the process of using language productively takes time for just about everyone  “to get into the bones” (students, teachers, and research mathematicians alike).     In her response to Rob (below), Sarah describes beautifully the habits of "offering students concrete experiences before moving to general or abstract definitions or ideas, letting students in on development of ideas and not just finished mathematical products, and modeling clear use of mathematical language.”   Her comments point to the dependence of SMP3 (Construct viable arguments ...) on SMP7 (Look for and make use of structure).   When we started working together on this project, we summed up this process with the acronym SUDS — Seeing, Using, and Describing Structure.   We believe people learn best from experience.  Experience is something we can have alone or within a community.  Our attempt to understand experience brings us naturally to the need for language to describe the structures we’ve seen (i.e.to develop definitions that describe their features, and then to use those features in order to reason about them).   Something that we have noticed repeatedly in our work with teachers (and students — and research mathematicians if we are honest) is that we all struggle mightily to put into words the things that we instinctively know.   So it’s not really surprising that people don’t necessarily feel for themselves the meaning of definitions that were constructed through the (usually hidden) struggles of others.  Without an internal understanding of the meaning of things it’s pretty tough for any of us to construct viable arguments about them.      No doubt there are many ways of addressing these challenges.  The habits Sarah describes are among them.

Hi Rob - the last question feels like the easiest: in general, I don't think any of thinks a habit is ever "cooked" - you get better and better at identifying and using (for example) mathematical structure over the course of your mathematical life, even if you're doing mathematics research. So in some sense, everyone is always working to cultivate habits in themselves. My suspicion is that it's possible to teach a habit of mind if you find that habit interesting, relevant, and useful. Having said that, we see teachers who do have strong habits - and who also believe that students can develop those habits - doing fairly amazing things with their students in terms of fostering habits in their students.

As for pedagogical moves - I'm not sure what counts as a move, but we do see certain approaches associated with mathematical habits - teachers who offer students concrete experiences before moving to general or abstract definitions or ideas, letting students in on development of ideas and not just finished mathematical products, and modeling clear use of mathematical language. 

As for disentangling mathematical knowledge, experience, and other factors... you know how hard that is. We've been immersed in studying the habits and practices of teachers in one very large high school - the teachers work closely together, they share a lot of common practices, and they've shared a lot of common habits-of-mind-based PD. It would be impossible for any one program to "take credit" for the practices that we see in those teachers. But they have had coherent PD experiences as a department, they work closely together, they constantly learn from each other, and it shows. 

 

 

Hi Rob,

Like Sarah, I'm not sure what counts as a pedagogical move, but I have seen teachers with whom we work shift focus and emphasis away from simply using  results  towards a more balanced treatment between using and establishing.

For example, in algebra classes some teachers (not all) are hammering at the habit of abstracting regularity from numerical examples in order to build equations and functions that model situations.  Students practice [sic] this practice in everything from setting up equations to model word problems to finding Cartesian equations for geometric objects, and these folks  explicitly call out this habit when it comes up in student work.  

I think I know the school to which School Sarah refers, and I can vouch for the fact that the incoming  mathematical backgrounds and  dispositions are all over the place.  They have the luxury of getting together once a month to work on problems, and, yes, some are much more facile with mathematical thinking than others.  But they all appreciate the approach of experience before formality (because, well, they experience it)  and cite reasons that range from ``the kids get it'' to ``it saves time in class'' to ``fewer methods to learn.''

Al

It is also notable that in cases where the instrument has been used diagnostically, it can also lead to some really rich discussions among teachers in a district or school. As highlighted in the video, the tasks are designed to lend themselves to multiple approaches. We have heard from teachers that using the instrument as a vehicle for discussion can be instructive, provide opportunities to discuss the different approaches with colleagues, and be fodder for interesting classroom practice discussions.

Hi Jennifer, Thanks for the question. 

While our project was designed to measure the impact of programs that would seek to impact these habits, there are a number of programs that do target these habits. In the realm of in person professional development opportunities, two intensive summer mathematics experiences would be Promys for Teachers (a piece of which is shown in the video) and PCMI’s Teacher Program.

In terms of written resources that focus on these ideas, the PCMI materials have recently been published with facilitator notes, solution sets, and discussions of how the ideas fit into the mathematical landscape. The Developing Mathematical Ideas series also focuses on the big ideas of mathematics at K-8 and can serve to support teachers in developing mathematical ideas and think about their application to the classroom.

 

Thanks, for your reply, Sarah. I am intrigued by the emergent design from teacher response which then cycles back as feedback when debriefing the assessment. To answer your question, I'd serve you better to refer you to my good colleagues Tom McDougal and Akihiko Takahashi at info@LSAlliance.org. I look forward to watching your work inspire algebraic habits of mind!