Alison Blank makes good points in her interesting presentation: “Math is not linear“, where she encourages us to make connections, go on tangents, preview future topics and review past ones. In short, we should not be trapped in the inflexible sequence suggested by textbooks and school culture. In a recent blog post, Jim Tanton makes the related point that presenting the content of a course to students on a “mind map” should help students see that math content is rich with interconnections, and should not be seen as a straightforward linear march through topics as suggested by a table of contents, or a syllabus. I very much agree with both of them.
However, as a teacher and curriculum developer, I cannot limit myself to think only about content. I prefer a more complicated framework, based on “themes, tools, concepts” (TTC) which I summarize in this figure:
Traditional pedagogy is near the top. It need not be thrown away, but it can only be effective if it is based on a foundation of themes (contexts), multiple representations, and learning tools. I explained this in more detail in a 2014 post and in various articles over the years. Later in 2014, I sketched a mind map using the TTC framework for the concept “proportional relationships“. Check it out. And way back in the 1990’s, I made a TTC mind map for the theme “area”. Here it is:
(If the type is too small, see the full-scale map here —scroll down to page 23)
Compare with the corresponding map in too many math programs:
For anyone planning a course, or preparing a unit, I strongly recommend starting with brainstorming a TTC mind-map for what needs to be taught. This can be done alone, but it is sure to be more productive as a departmental project. The teacher’s familiarity with the TTC landscape and connections for that course or unit is sure to provide a stronger foundation for teaching or writing curriculum. (See related ideas about “forward design” in my article on The Assessment Trap.)
That said, Bill McCallum points out that while math is indeed not linear, time is linear, and curricula and lesson plans must be sequenced. Organizing things in a strictly logical sequence is almost always a mistake: such sequencing may make sense to a mathematically sophisticated person, but it is not necessarily best from a pedagogical standpoint. Doing things in a certain order because tradition requires it is another loser, as traditional sequencing is often terrible. And looking at lists of standards is not particularly helpful.
Effective sequencing, like everything else in teaching, requires paying attention to what happens with actual students. I did this for 30+ years, and came up with some ideas you may find useful. See my article on the Common Core for a discussion of realistic sequencing across the four years of high school. For sequencing within a course and within a unit, see my article “Nothing Works“. But whatever sequencing you end up with, consider it provisional, and don’t lose sight of the reality that if it doesn’t work, you can and should change it. There is no God-given sequence for teaching math. Still, I suggest sequencing guidelines in my next post.