## Discovery Learning with Square Roots

A few years ago I learned about the idea of discovery learning. It was revolutionary to me at the time. Math was taught to me in the most traditional ways for most of my life and I taught it that same way. You know- the teacher gives notes and the students take notes. Then, after seeing a few examples, the students do work and we go over the work together. This framework has been so ingrained in my that it was and still is to some degree hard to let students do the work of discovering.

There are a few reasons for that fear of letting students think. I hear them whenever I first talk with other teachers about discovery learning for students. Usually they’re a variation of this question: “What if they get it wrong?” Well, that’s easy to answer- of course students are going to get it wrong. That’s the whole point. We need to let them make mistakes. Let them play around with numbers. Ultimately, we need to let them create their own understanding of the concept. (Read more about how I got started with discovery labs and discovery learning here)They have a much better chance of remembering the concept that way. When you start with discovery learning experiences for students, by the time you get to teaching the algorithm through note-taking they are much more likely to actually help you give the notes. Or they may fill their notes in before you even say anything.

## Building Background

I like to start off with some background building with each discovery lab. It’s a quick way for me to get a glimpse of the students’ understanding. Our background building activity for this discovery lab with square roots was just to categorize a series of numbers as either having a perfect square root or not. This year I used this particular background building activity for the first time and I found that many of my students couldn’t recognize if a number had a perfect square root or not. I have to admit, that surprised me!

Something that was fascinating is that in all of my classes there were a couple of mistakes that were made a lot. All of my classes put 26 into the category for “has a perfect square root”. That was fascinating to me because it seems so simple. I wasn’t trying to be tricky, and yet that number tricked a lot of kids. It showed me that they clearly don’t have a strong handle on the series of square numbers that we were about to explore.

## Discovering Square Numbers and Square Roots

There’s a pretty natural activity when it comes to squaring numbers and finding square roots. Square numbers literally make a square when they are created visually. The square root of a square number is the length of a side. You could just tell the kids that and guarantee that only a couple of kids will see what you’re saying and remember it, or you could show them some examples and let them discover that pattern on their own.

For my discovery lab I gave them a visual representation of the number 1, 4, 9, 16, 25, and 36 as pictured at the top of the picture above. Students worked to answer questions about what patterns they saw in the visual representations and how they thought that might relate to square numbers and square roots. The core of a discovery activity is to ask questions that create an environment where students can discover something mathematical.

## My students’ experience

We completed this discovery lab on the 3rd day of school. At that point in time this was a new and different kind of activity for them to do. Most of my new students had limited experience with discovery learning, so I could see a lot of them just wanting to get the right answer from me, the teacher. I gave them a bit more guidance on this lab than I will on future labs because I like to ease them into it a little bit. By doing this I wanted to make sure that there was productive struggle, but not to the point of giving up.

My students were very hesitant to show their thinking in this activity. They were cautious when listing what the squares had in common. I expected to see a list of common characteristics and instead they usually had just one characteristic. The process of breaking down their reliance on getting the right answers may take a few discovery labs, but I know it will happen in time.

## Biggest discovery

The biggest discovery that students seemed to make is that the side length of the square is the square root. I was very excited to see so many of them draw this conclusion. That type of discovery can go a long way in preventing students from multiplying a number squared by 2 (which, how many times have we seen that misconception?!). When they draw that conclusion and have a picture of a square number in their head they are so much more likely to remember that when they see a number squared, it actually means that number times itself.

## Reflection writing

Students ended the discovery lab with a reflection activity that showed what they learned or didn’t learn. Some of them simply wrote what we learned about square roots. Others had very well written statements about the relationship between square numbers and their roots. Their first discovery lab of the year is a learning experience. So, when I gave them feedback on their work it was almost exclusively about how well they expressed their thinking. I emphasized this because I hope they will develop their thinking process over time.

## Into your classroom..

I hope you can see how this discovery lesson would benefit your students. Try it in your classroom and watch them make sense of math. I promise you that over time you will see the benefits of having students learn with discovery labs. They will remember things so much better than by memorizing algorithms. Imagine if your students could tell you all about square numbers and their roots in a way that shows they really see the relationship between them. When you use discovery learning that’s the type of experience you’ll have on a regular basis.

For a print and go version of this discovery lab activity, you can get your own copy here. Otherwise, following the steps outlined above is a great way to start off this unit and help students show their thinking, draw conclusions, and make mathematical arguments.

To find even more activity ideas and resources, check out “12 Square Roots and Cube Roots Activities with Big Impact.” Thanks for reading! Until next time.