In my last newsletter, and the one before, I mentioned my blog post about eclecticism, wherein I argued that teachers cannot be beholden to any one pedagogical framework, but instead should be flexible, and pick and choose approaches that make sense to them and their school. This ended up being among my most popular posts in 2016.
A few weeks ago, a math educator who is familiar with my work as a curriculum developer told me she did not believe me when I said I had no pedagogical framework. She pointed out that there are many things I would not dream of doing, and that the materials I develop have a certain consistency. Fair enough, but to describe that, I have to resort to generalities like my "puzzle constructor's sensibility", as I did in this post on "enrichment".
Soon after that conversation, I remembered a diagram I made in 2014. Perhaps it can serve as a basis for a pedagogical framework? Read on.
Here are links to posts on my Math Education Blog that you might find interesting.
If you are so moved, you may comment on the posts, and/or subscribe to the blog.
A Pedagogical Framework?
This diagram describes many ingredients of a hypothetical math program, one aiming to help develop conceptual understanding:
So this might be my pedagogical framework: do all the things! But remember that the bottom of the diagram provides the foundation for the top. Resist the temptation to rely solely on explanations and drills. Those can help, but only if the foundation has been laid. Read more about this model here. (Note: In the diagram above, I added "Problem Solving" to the center of the larger triangle in theversion I created in 2014. And really, it belongs also in the smaller, upper triangle. Thus speaks the puzzle constructor!)
Too complicated, you say? You would like a simpler framework? Alas, I can't help you! Teaching is a complex endeavor, and searching for uncomplicated panaceas is doomed to failure.
Number sense takes many years to develop. Really, most of grades K-12. The development of symbol sense starts somewhat later, but again, it is not acquired quickly. It is a crucial destination of secondary school math, and many ingredients go into it: generalizing numerical relationships, translating graphs into formulas, manipulating expressions, and reading algebra. I wrote about the latter here.
I attended the NCTM meeting in Phoenix last October, and the California Math Council conference in Asilomar in December. Read my notes:
- NCTM President Matt Larson's insightful presentation on the history of the "math wars"
- Dylan Kane on Growth Mindset, James Tanton on Exploding Dots, David Wees and his colleagues on Structured Rehearsals, and me on Equivalent Expressions — all here
- My three Asilomar talks: Transformational Geometry Proofs (with Lew Douglas), Computing Transformations with Complex Numbers and Matrices, and my Ignite presentation: Time Pressure: Bad for Students, Bad for Teachers. That last talk was somewhat self-referential, as the Ignite format is all about time pressure: twenty 15-second slides to say something substantial. (If you have five minutes, you can watch the video here.) The blog post is mostly about this topic — an elaboration in particular on the tyranny of the clock.
New on my Web site.
Functions Home Page
The Common Core State Standards include functions starting in 8th grade, and throughout high school. I support that emphasis. As it turns out, I have developed chunks of curriculum about functions at all those levels. See if any of the links on this page take you to something you can adopt or adapt for your classes.
The Assessment Trap
This is a somewhat extended and edited concatenation of my blog posts on grading and assessment. Many common assessment practices tend to undermine education, in order to satisfy regrettable societal and administrative priorities. I suggest alternatives that prioritize student learning over the ranking of students. This may end up being one of the most countercultural and provocative pages on my site.
Math Education Consulting
I have retired from the classroom in order to be more available to work with adults. As a consultant, I offer these services: professional development workshops, math program review, teacher and department chair coaching. I created a new page on my site about this.
I also offer summer workshops. See below!