Questions About Conceptual Understanding

I’ve been fascinated by the conversations about conceptual understanding, happening on Dan Meyer’s blog and elsewhere. I’ve realized I understand way less about “conceptual understanding” than I thought. Here are some questions that have helped me think about this whole thing:

• Is too much procedural fluency bad for conceptual understanding?
• Is it possible to have lots of procedural fluency without any conceptual understanding? Is it possible to have lots of conceptual understanding without procedural fluency?
• Is conceptual understanding more about what students can do or what they know?
• Does conceptual understanding support student engagement? Does procedural fluency?
• Adding It Up from the National Academies Press defines five strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Are we missing useful complexity by narrowing our focus to conceptual understanding and procedural fluency?
• Here’s a graph. What does an ideal learning trajectory look like?

• Does that trajectory depend on the content?
• If a student can explain how they solved a problem, do they definitely have conceptual understanding? If a student can’t explain it, do they definitely not have conceptual understanding?

My Hot Take 

Here are two tentative ideas that I think might contradict each other, but might also both be true.

1. It’s easy to overcomplicate conceptual understanding, but really it’s just transfer. Can a student take what they learned in one context and apply it in another? And transfer is, or should be, the primary goal of education.
2. Conceptual understanding is actually composed of lots of little pieces, and those pieces depend on the content, the teacher’s goals relative to that content, and the students’ prior knowledge, skills, and dispositions. It’s easy to overgeneralize, but building conceptual understanding is context-specific and there aren’t any one-size-fits-all ways to get there.

Further reading that’s on my mind:

• All of the links in Dan’s blog post
• Kate Nowak on why Illustrative Mathematics’ avoids cross-multiplication
• Dan Willingham on Inflexible Knowledge
• Are Cognitive Skills Context-Bound? by Perkins & Salomon