Three things I believe:

1. Students need to understand math conceptually. They need to make sense of problems, be able to explain why the processes and procedures they use are correct, solve problems in and out of context and apply concepts in novel situations.

2. Successful inquiry-based instruction (coupled with appropriate practice) leads to greater conceptual understanding and a productive mindset around mathematics.

3. Inquiry-based instruction often fails for many students due to insufficient background knowledge, imperfect facilitation, or a lack of a desire to learn the material.

I believe in direct instruction. I believe that, under certain conditions, telling students . There’s a reason teachers don’t pretend in any other discipline that all necessary knowledge can be discovered if only we structure an ideal inquiry environment and inspire students to want to learn. This is not to say that direct instruction is ideal for every lesson, or every moment of any lesson, and in particular I’m pretty put off by teaching that looks like this, but I don’t shy away from telling my students key information when I feel they are engaged, the material is appropriately motivated, and they will have a chance to apply it independently and in novel contexts right away.

The classroom I believe in (and my classroom is moving in that direction, but is far from there) is one where students are inspired to answer mathematical questions every day through perplexing tasks (whether or not they come from the real world), and then find them the tools they need to answer those questions. This often means motivating a new topic through a rigorous task, leading to direct instruction on the topic, some combination of guided practice, partner work, and whole-class problem solving/analysis, and independent practice for students to stamp their understanding. It also often means using inquiry to allow students to talk about math, explore new topics, and construct their own understanding. One is not inherently more valuable than the other. Either way, I believe that the most valuable mathematics comes from students wanting to answer problems that require them to make sense of both mathematics and the world around them, and to pause and think about the mathematical concepts that they are applying.

Anyway, while I’ve spilled lots of digital ink on the idea of problem design here, and in particular thought a lot about the value of a taxonomy and language of problem design, I think it can be effectively distilled into two key ideas that channel the Standards for Mathematical Practice:

1. Great problems give students reason to pause, be thoughtful about mathematics, and consider their solution path.

2. Great problems provoke solution paths that are divergent and different from conventional procedures applied to the concept.

These are imperfect, but the more I write about problem design the more I see these two ideas at the center of problems that promote high-quality mathematical thinking.