## But I Thought I Was Teaching Conceptually! (An Unfinished Journey)

by Sarah Lonberg-Lew

What does it mean to teach math conceptually? For me, it seems to be a moving target – each time I think I have it figured out, I discover that there are new understandings for me just around the bend. So I keep traveling down the road toward conceptual teaching, not knowing if there will ever be an end in sight. Here’s a peek into my journey along this road. What does your journey look like?

### Chapter 1 – I Go to School

I mostly enjoyed math in school, but it largely depended on the teacher. Some math classes I looked forward to and others I dreaded. In 11th grade, Mr. K. stoked my intellectual curiosity. He was excited when students asked thoughtful questions and took those questions seriously, encouraging us to investigate things we were interested in. By contrast, my 10th grade teacher, Mr. B., didn’t seem to want us to think at all. He wanted us to practice procedures until they became automatic. His refrain was, “Keep your pencil moving. Don’t think!” I was miserable in his class. There was this one procedure (called “completing the square”) that I just couldn’t remember and having him stalking the aisles telling us to move faster and not to think didn’t clear it up for me at all. I concluded that it was just impossible for me to learn it.

### Chapter 2 – Understanding Why

In college I learned from professors who were much more in the mold of Mr. K. and I spent a thrilling four years taking exciting math classes and loving learning. After college I got a job teaching high school math – algebra I, geometry, and algebra II. I was determined that my students would never get the message that I had gotten from Mr. B – that they should just do and not think. So, when I taught them a procedure, I also explained to them why the procedure worked. I even finally made sense of completing the square and explained my new understanding to my students. This, I thought, was much better teaching. It was conceptual teaching because I was helping my students understand the concepts behind the procedures. My goal was to help students remember the procedures, and I figured they would be better able to do that if they knew why they were doing what they were doing. Ultimately, though, my students didn’t seem very interested in the explanations I offered. They seemed to just want to memorize the steps so they could do the exercises on their homework and on their tests. While I was excited to understand the why behind the procedures, they were not.

### Chapter 3 – Some Things You Can Discover, and Some You Just Have to Memorize

Eventually I found my way to teaching math to adults. Here were people whose school math experience had often reflected the worst of mine or much worse. They were wounded from their math education and needed healing as much as they needed math learning. Having learned something from my experience in K-12, my new goal was to help students discover the procedures themselves so they would be invested in them, so they would feel the procedures belonged to them. Well … I did that with some procedures anyway. Some were just too complicated to explain and it was easier just to teach a trick.

Here’s an example of what that looked like in my class:

I created a PowerPoint (of which I was very proud) that used area models to extend student understanding from multiplying whole numbers to multiplying fractions, guiding them step-by-step to discovering the procedure I wanted them to learn. Once my students felt comfortable and confident with the procedure for multiplying fractions, I told them the good news that they wouldn’t have to go through all that stuff with area models to learn how to divide fractions – all they had to do was flip over the second fraction and then use what they already knew about multiplying fractions. I even gave them a handy poem to help them remember:

When you must divide a fraction,

Do this very simple action:

Flip what you’re dividing by…

Then it’s easy – multiply!

Even though I knew it was possible to give an algebraic explanation for the why behind the procedure for dividing fractions, it didn’t seem worth the classroom time or the struggle for students to understand it. Some things you just have to memorize, right?

### Chapter 4 – There Isn’t Just One Way

I’ve entered a new chapter in my journey. My new goal now is to help students grow in their ability to make sense of problems and construct their own strategies for solving them. I am no longer trying to guide students to discovering the procedure they are supposed to know. Now I want them to create something that really belongs to them – something that works but also that truly makes sense to them. The standard procedure for dividing fractions is not the only way to do it. (That was news to me – I believed for a long time that it was the one and only way.) By providing students with thought-provoking tasks, tools (like concrete and visual models), and appropriate support, I can position them to be creators of mathematics and to confidently bring their mathematics with them into the world.

### Chapter 5 – The Future

Have I finally got it figured out? I’ve come a long way in my understanding of what it means to teach math conceptually, but if the previous chapters are any indication, there’s no reason for me to think I won’t find something even better around the next corner! Maybe you will, too.

Sarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency.

Sarah’s work with the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center at TERC includes developing and facilitating trainings and assisting programs with curriculum development. She is the treasurer for the Adult Numeracy Network.