Have you ever been frustrated that your students just don’t seem to remember what you taught them? I think all educators have run into this challenge- you’ve taught it, students have practiced it, heck, they even did pretty well on the test you gave at the end of the unit. And then… BOOM! Their scores tank on the semester final/district benchmark/state end of the year test. And you are looking incredulously at the answers they got wrong wondering why they don’t know now what they seemed to know before. Anyone else been there?

After having this experience more than once, I spent some time reflecting and trying to figure out a different approach. One that would get the students doing more of the mathematical thinking. One that would cement conceptual understanding into students’ brains in a more permanent way. What I found was a discovery math approach that got me fired up about the learning going on in my math classroom. Here I’ll share with you the story of the very first discovery lab I used with students, why they work, and how to implement discovery labs in your own classroom.

#### Experimenting with discovery math

I started thinking about the idea of “discovery learning.” Learning through inquiry always intrigued me. But typically the examples I saw were extensive inquiry-based projects. While those certainly have a place, I wasn’t looking for such a time intensive approach.

I wondered if there was a way for students to do the same level of thinking in smaller chunks of time. How could they get curious about a topic, draw on their previous knowledge, and try to solve a novel problem without setting aside 3 weeks of class time?

With these questions swirling around my brain, I was introduced to the idea of IEEI lessons. IEEI lessons flip the order of the traditional EEI (Madeline Hunter’s Essential Elements of Instruction) lesson sequence. Instead of the traditional lesson approach, the IEEI lessons invert the lesson sequence:

Traditional lesson sequence:

- Teacher delivers information and models
- Students and teachers practice applying information together
- Students apply information independently

Inverted lesson sequence:

- Students work with concept,experimenting and looking for patterns
- Teacher models and explains concept, connecting to the students’ previous experience.
- Students apply understanding with support and then independently.

I started thinking about what switching the order of the first and second step would look like in my class. I started to really see this as an opportunity to get my students to own their math understanding at a deeper level. But, would it work? Only one way to find out- jump right in and try it out. So, I designed a lesson on identifying linear and non-linear functions with this inverted approach.

#### Inquiry math learning in action: linear and non-linear functions discovery lab

As I thought about what this inquiry activity could look like, I thought about the scientific method. I realized that I wanted students to go through this same process of thinking like a scientist. This activity would invite students to tinker with math, make observations, and draw conclusions.

##### Step 1- Building background & setting expectations

To get started I gave students specific directions on how they would work with their partners.

Then we built a bit of background. I quickly explained the vocabulary terms of linear and non-linear. We didn’t spend a ton of time on these (yet) but I wanted them to have these words in mind as we went through the activity.

##### Step 2- Trails and observations

I distributed to students a list I made of functions, both linear and non-linear. Then, I asked them to go to an online graphing calculator (hooray for Desmos!) and input the functions. After graphing the function, students recorded what the function looked like on a graph and made predictions based on what they were seeing. Throughout the lesson they shared, reviewed, and revised their predictions.

##### Step 3- Drawing conclusions

At the end of the activity, I asked students to write a rule about linear and nonlinear functions based on what they had done. I gave them several questions to respond to using their observations from the trials. After this activity, we moved into the traditional instruction and students added notes to their interactive math notebooks. Note-taking was different this time, though. Students kept referring back to the experience they had and the patterns they saw. Rather than just copying notes, students were actively connecting them with something.

#### Reflections on the first discovery lab

After watching my students get their hands dirty *doing * the math, not just hearing about it, the process reminded me of hands-on labs in the science classroom. It was so exciting- I saw students actually experimenting with math, formulating hypotheses and then testing those hypotheses. I saw them drawing and sharing their conclusions. I couldn’t wait to have another lesson like this one. And thus was born what I’ve called the Math Discovery Lab.

I wasn’t the only fan of how things went that day- the students absolutely loved this activity. They were engaged and thinking about math in a way that I haven’t often seen before. Students who were typically more “checked out” found that they couldn’t wait for information to be spoon fed to them. As that realization hit them, they began to be more and more willing to participate and take some risks. I especially saw my students with IEPs (special education) and other students who typically struggled at math able to work with limited support and experience success.

But what did this change mean for test scores? The class did very well on the unit test! Then, when the district benchmark assessment rolled around a few months later, they outscored other classrooms on that topic- they remembered what they learned! Clearly something good happened here.

#### The Birth of Discovery Math Labs

I began anxiously looking for ways to replicate this experience for students with other topics. I developed Math Discovery Labs for other topics: combining like terms, transversals, powers of exponents, and others. With each passing discovery lab, I get more and more convinced of the power of the approach. While units and lessons take a smidge more in class time than me simply giving notes for 20 minutes, the students show better retention of math concepts than ever before.

After some fine tuning, here is the basic recipe for a successful discovery lab:

Some topics lend themselves to slightly different components, but overall these are the steps students go through. After this discovery activity, students are primed and ready for formal instruction. And they get great practice solving problems, drawing on their math understanding, justifying a position, and seeking patterns. So many of the math practices are naturally embedded within these discovery labs. This is a great, low-stress way to get students to take chances in math. It ignites their curiously and gives them context for their future learning.

#### Your Turn and a Freebie

I hope I’ve convinced you to give discovery labs a shot. They are a great way to break up the pattern of “I teach- you practice” and encourages productive struggle.

If you haven’t dipped your toe into the inquiry-based learning structure, try it out! The lesson I’ve talked about in this post is **available for free in my Teachers Pay Teachers store: **

#### More Discovery Labs for Math Concepts:

Want to see what the Discovery Lab approach looks like for other topics? Check out these posts for more discovery math goodness:

- Teaching Systems of Equations with Discovery Labs
- Teaching Area and Circumference of Circles Through Discovery
- Exploring the Volume of Cone, Cylinders, and Spheres

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