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.

## 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 may 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.

There are many benefits to this. It may reveal strategies that make more sense to the students than the ones you come up with. It gives students the confidence they can recreate the formula if they don’t remember it. It makes “base” and “height” concrete and visible rather than numbers on a figure. A discussion of student solutions can lead to a discussion of how to generalize what was learned, leading up to a formula. And of course, starting this way does not prevent you from providing your own explanation later.

## Reverse traditional questions

Many traditional math class questions are, well, boring. What is 7 + 3? What is the greatest common factor of 12 and 15? Graph y = 6x + 4. Solve 6x + 4 = 3x + 10 for x. And so on. Reversing the traditional question often yields great engagement and powerful discussions. Here are some examples.

**Find pairs of numbers that add up to 10**. Even answering this with positive whole numbers gives more students a chance to contribute answers. But this can expand in many directions, depending on the class. You can ask for *sets* of numbers that add up to 10, such as 3, 3, and 4. If you are getting started with integer arithmetic, great patterns will emerge if you allow negative numbers in the pairs, and your students will be practicing their new skills as part of an interesting quest. Likewise, in Algebra 2, if you ask that same question about complex numbers.

**Find pairs of numbers whose greatest common factor is 3. **This is sure to reveal what understanding your students have about common factors. If some students don’t understand the underlying concepts well enough to do this, they can get help from their classmates, or from you. If even that is not sufficient, some review is in order!

**Find the equations that will yield these graphs. **

This is both more fun, and more educational than “graph this, graph that, what do you notice?” (More about this.)

**Find equations whose solution is x = 2**. Instead of the standard “simplify” process, students get to “complicate” the equation. This gives them an opportunity to practice “doing the same thing to both sides” as part of a creative challenge.

All these examples show the power of reversing traditional questions. On the one hand, it makes it possible for stronger students to show off their fancy solutions, and to remain interested and engaged. At the same time, all students can find their own level of challenge, and you can support them as they take that next step. This is a sort of differentiation with no artificial ceiling: everyone gains.

## Aim high

When I first started teaching in a high school, I was given a terrible piece of advice: “Aim for the middle”. Our program was not tracked, and we had a wide range of talents and backgrounds in each class. Since that is difficult to manage, the idea of aiming for the middle seemed like common sense: if you aim too low, you are betraying your stronger students; if you aim to high, too many kids will be frustrated. As is often the case, common sense was not a good guide.

Since then, I have developed some strategies to handle a wide range of students in the same class, which I share in Reaching the Full Range. One key component of these strategies is to *aim high*. In everyday class work and homework, by all means include some material that is too easy, and some material that is too difficult. But for the activities and challenges that anchor an important concept, it is best to be ambitious. If it turns out the question is too difficult, it is always possible to support students with hints, or some other kind of scaffolding. If it is too easy, it is not moving the class forward, and it is giving the wrong message. Here are some examples.

**Rich anchor problems. **Start a new unit with a Big Question that encompasses the concepts you are about to teach. For example, to prepare students for sequences, explore what happens when you iterate the function *y* = *mx* + *b*. In other words, if you start with a certain value for *x*, and it yields a *y*, use that *y* as the next *x. *Repeat and see what happens. This is especially interesting when *m *is between -1 and 1, but all cases are worth exploring. A full discussion provides an opportunity to introduce subscripts, to think about limits, and eventually to zero in on the usual arithmetic and geometric sequences.

**Add another representation**. Do not limit yourself to a single approach to your topic. Look for other ways to think about it. Can manipulatives provide a productive environment to explore these ideas? Can technology help? Is there a visual representation that could be helpful? For example, use well-chosen graph paper rectangles to illustrate the addition of fractions:

In this example, a 3 by 5 rectangle stands for the number 1. That choice makes it easy to represent the fractions we are adding. By examining the figure, we see that the sum is 13/15. Doing this once, of course, would not be very helpful, but discussing the dimensions of useful rectangles, leads to a powerful strategy which can be applied to comparing, adding, or subtracting fractions.

**Have students make things.** In some cases, a *constructionist* approach can yield much learning. Constructionism is a theory of learning espoused by the Logo movement in the 1980’s, and it is still going strong with Logo descendants such as Scratch and Snap, or with the current popularity of maker spaces and STEM. To enhance math education, what students make needs to have a curricular payoff. It can be a turtle geometry computer program, a GeoGebra construction, a tiling of the plane, etc.

For an example of the latter, using a plastic template and a pencil (or a computer), students can create tessellations using triangle or quadrilateral tiles. Here is one based on a scalene triangle:

This figure offers a context to discover and discuss many basic geometry theorems: sum of the angles in a triangle, exterior angle, parallels and transversals, translations and rotations, …

## Conclusion

Pursuing any of these suggestions should enhance discourse in a math class: who talks? who listens? who does the thinking? The key is to keep the students at the center. We, their teachers, can help, but they are the ones who need to do the learning.

Thank you for your share. It’s understanding. What a child or teen understands is what remains with them. Yes, with time and lack of usage, they may forget, but the understanding remains with them, and they can recall quicker because they understood it before. Understanding is like “living” knowledge. It’s real. It sees to the core. It’s alive. Memorizing or giving academic answers does not stay as long for it doesn’t bring the person themselves into the equation. I will always know my times table, for I drilled myself and practiced, it now imprinted on my brain. But things I understand, though I might forget the factual information surrounding, remain through all the years.

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