When I was much younger, I was under the impression that anything students “discover”, they will remember. Over time, I realized that this is not really true. First of all, what I hope they discovered may not be what they actually understand. But also, it’s not clear to them what is important about their discovery, what is worth remembering, how it connects with other concepts, and so on.

On the other hand, merely providing good explanations does not turn out to be any more effective, as students don’t necessarily listen to those, and even if they do, they may not understand them. Thus, in order to remember things, they are forced to memorize poorly understood techniques and ideas. Because there is so much math to be learned, memorization often only works until the quiz. A few weeks later, it’s mostly gone.

As a teacher, I had to learn how to combine student inquiry with teacher guidance. I reject the hardcore “never give a hint” position of people who overestimate inquiry. I also reject the belief that excellent teacher explanations suffice. Effective teaching requires us to navigate between many strategies which appear to be mutually exclusive, but are in fact complementary. Well-chosen problems help students engage with the question at hand, and prime them to listen to and understand teacher explanations. How to choreograph this back-and-forth dance is learned through practice, and through teacher observation and collaboration.

Unfortunately, a culture of “I explain it, then you practice in silence” still dominates too many math classes, leaving no room for student intellectual engagement. In this post, I will share pedagogical bits and pieces which I used as part of a *guided inquiry*approach. I do not claim these add up to a full scheme. Rather, they are components that can be added to any teacher’s repertoire. I hope they can help you move from the lecture + practice paradigm towards active student-centered learning. And if your class is already discovery-based, you may find here some techniques to add to your toolbox.

**1. ASK THE QUESTION BEFORE TEACHING A WAY TO THE ANSWER**

Students cannot always hear the answer to a question they don’t have. A consequence is that even after a great explanation from the teacher, they did not absorb what they have been told. If instead of starting with an explanation, you start with a question to get students to think about the topic, they are much more likely to understand your explanation.

For example, ask students to find the area of graph paper triangles (or parallelograms, trapezoids, etc.) before mentioning that there is a formula for that.