We’ve written specifically about the discipline of mathematics several times — the two specific to concept-based are here and here. To me, it’s the discipline that requires the greatest shift in how it is traditionally taught. Many experts talk about how we need students to be better problem-solvers, allowing them space to figure things out without giving them the formula or only the specific information needed to solve the problem. I agree. At the same time, we need to be sure we are teaching the conceptual relationships behind the “how” of math. Students need to discover the “what” of math.
Like English class with drills about the rules of grammar, procedures are often taught without much thinking behind why we are doing them. But very much unlike English class, there is lots and lots of content behind the procedures. In 2013, Dr. Lois Lanning published her book with the helpful illustration of the Structure of Process which is the complement to Dr. Erickson’s Structure of Knowledge. She describes strategies and skills associated with complex procedures as similar to specific facts associated with complex ideas. Many people see the two images below and think mathematics falls on the Structure of Process side. This is the wrong way to think about it.
sok and sop
Here are some tips that might help:

  • If we find ourselves writing out goals for student understanding that start with “how to…”, we know we are missing the mark for conceptual understanding. Those are skills.
  • If we find ourselves writing goals that sound like things we want kids to do, again missing the mark.
  • We should ask ourselves, what is the content behind this mathematical procedure? How can I write that in a clear, powerful statement of conceptual relationships.

An example:
What is the content behind pi?
The circumference of a circle is roughly 3 times its diameter.
How can I write that in a statement clear, powerful statement of conceptual relationships that doesn’t use the weak verb “is”?
The ratio of circumference to diameter in all circles represents a fixed constant, π.
Once we have this powerful statement of conceptual relationships, we then set to work on writing lesson plans that allow students to discover it.
One thing I’ve noticed from my old school and from the MYP IB maths documents is that when we aim to write inquiry lessons, it’s very easy to veer into “how to” statements rather than statements of conceptual relationships with content or knowledge as the driving force.
Stay tuned for more details on writing those lesson plans. Also, be sure to check out my friend Jennie Wathhall’s book, due out in January!