In Education Week “Teacher”, a recent article reproduces the responses of four high school math teachers to the question: “What are some best practices for teaching high school mathematics?”
I got as far as the first teacher who starts off by telling us:
“Focus on the processes and connections between different processes rather than just finding the answer.”
Those who believe the big lie told at school board meetings and repeated ad nauseum by math reformers (in publications like Education Week and others of similar ilk) that “traditionally taught math failed thousands of students” likely nodded their heads at this with a big “Yup, that’s what’s wrong with how math is taught.” What goes unnoticed and unacknowledged is that a good textbook does everything claimed to be his, but in a more efficient way and one that focuses on individual success, not feel-good student reactions in class.
To be more specific, in elaborating on this idea about processes and connections, he starts off with: “Teach to big ideas rather than 180 disconnected lessons.”
I don’t know about you, but I haven’t heard of or witnessed any high school teacher who teaches 180 disconnected lessons. And though math books from previous eras like the 50’s and 60’s are held in disdain by today’s math reformers and others who claim the high ground in “best practices in math teaching”, the lessons contained therein were well sequenced, well connected and well scaffolded. Such textbooks taught to big ideas; in fact, my old algebra book was called “Algebra: Big Ideas and Basic Skills”.
But then he gets to the heart of what’s really bugging him:
The answer is important, but the processes that lead to answers are far more important to learn. The connections between processes and between different representations of that process are where the core mathematical ideas you want your children to learn reside. This means that while teaching processes, you should use these processes to teach students mathematical principles they can apply to solve problems, not how to solve problem types.
The connections he thinks are not happening have always been a part of traditional math. We learned big ideas from unit material. Given a governing equation, like distance equals rate times time, we solved straightforward problems with worked examples to get us started. Then, in working the homework problems, there were many problem variations. We learned how to rearrange the equation and solve for the unknown, to draw diagrams, label unknowns, and start writing down known equations to find an equal number of equations as unknowns. There was never one type of problem for any new unit of material.
There was no special process or set of ideas that guaranteed transference other than practice. From the unit material homework variations, we saw patterns of solutions that could be applied to other types of problems. The variations took us well past the initial worked example and required us to stretch our prior knowledge and apply it to new situations within the context of the original problem type. What if you had five people working on a job at different rates? What if they started at different times? If they could only work for 5 hours, how much might be left over for you to do? (For more on the topic of non-routine problems within the context of problem types, see this article. )
The teacher next exhorts us to:
Use instructional routines to support all of your students in having access to the mathematics. Instructional routines shift the cognitive load for students as they focus less on what their role is and what they are supposed to next, since these tasks are delegated to long-term memory, and are therefore more able to focus on learning mathematics.
He provides a link to his blog to provide an example of instructional routines. There, we see the typical “growing tile” type of problem presented as an instructional routine. He states that
[t]he high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem. A critical aspect of this and other instructional routines is that they embody a routine in which one makes teaching decisions, rather than scripting out all of the work a teacher is to do with her students.