Part 2: Three Aspects of Rigor (Why is it necessary?)

The California Mathematics Framework calls on teachers to teach conceptual understanding, procedural fluency, and application with “equal intensity”. The way I think of “equal intensity” is with a Venn diagram. I first saw a Venn used to describe rigor here.

The ambitious instruction we are seeking – rigor – is the sweet spot in the middle. Why are all three components necessary?

I created this Venn diagram to explain how I see mathematics that is missing one or more of the three components of rigor.

Conceptual + Procedural without Application: Students will ask, “When am I ever going to use this?”

Procedural + Application without Conceptual: Students will ask, “Why does this work?”

Conceptual + Application without Procedural: Students will ask, “Can this be done more efficiently?”

Focusing exclusively on Conceptual Understanding makes math aimless. Mathematics is severely limited in its usefulness if we focus only on drill-and-kill of procedural fluency. Students who are subjected to real-life applications without the additional support of conceptual understanding and procedural fluency become frustrated.


A friend of mine, Graham Fletcher, doesn’t particularly enjoy the word RIGOR.

I don’t blame him. If you were to look up RIGOR in the dictionary, none of the definitions are particularly inspiring. Perhaps a better word is THOROUGH. Or the phrase DEEP LEARNING.

Until I figure out a better word, I’ll be sticking with RIGOR.

So, now we understand that we need all three components of rigor in order for students to experience the rich mathematics we envision. But, how do we achieve “equal intensity”? What does it look like?

That will be next week’s blog post.