Disintegration by design.
You should collect all your wisdoms in a book, Dan. Longing to read it!
And I loved watching the video of you teaching. So much to discuss! Something that I was thinking about afterwards was the habit we teachers have of asking students the answer to simple arithmetic questions, i.e. "What is 10 divided by 5?", while solving a problem (an equation/an inequality). I wonder if that runs the risk of actually overflowing the students' working memory (some of the students in the video seemed to struggle with even the simpler calculations), so that they don't have enough space in working memory to actually follow the argument. We're so used to asking those questions, as a means of checking if the students are with us. But for the students who do those calculations effortlessly, the question probably seems "stupid". And for the students who don't do them effortlessly, the question might deprive them of following the main ideas. What are your thoughts on this?
There are instructional videos that demonstrate “best practices” in teaching math, but the reason you aren’t finding them easily on YouTube is that researchers who are known as leaders in this area only make them available by subscription. Some subscriptions are only available to higher ed institutions. If you look up Deborah Lowenstein Ball you’ll find some material on YouTube but the really good stuff is available only to subscribers or educators who attend her Teaching Works PD sessions. Folks gotta make a living!
I once read a thesis by Wilenski (sp?) many years ago that proposed that there are no abstract (or concrete) mathematical concepts. But what is concrete or abstract is the learner’s relationship to the concept. Math professionals have many more ways to relate to and representations for a mathematical concept and so their understanding is more concrete. Learners that don’t have as many connections to the concept have a relationship that’s more abstract.
We recorded different teachers in one high school using Vertical Spaces for students to make thinking visible. Our PD time in Summer Math Institute this Wednesday will be 19 teachers analyzing those videos, among other things like 3 ACT Tasks used by our teachers, Rigor as the Precison of Language, Desmos Activities, Vertical Alignment and a book study. Thank you for sharing all of your ideas! The practice of videotaping teaching and learning AND reflecting on what you see was one of the best practices I got from National Board Certification.
Neat! And I love that quote: "The interesting writer, the informative speaker, the accurate thinker, [the effective math teacher? -dm] and the sane individual operate on all levels of the abstraction ladder, moving quickly and gracefully and in orderly fashion from higher to lower, from lower to higher, with minds as lithe and deft and beautiful as monkeys in a tree."
That video reminds me of the ManningCast on Monday Night Football. Real-time analysis gives insight into the "why" as well as the "what" the teacher (dan) is doing and possibly thinking...best observation model, ever...imho.