Teach the Math Behind the Methods

Let me start off with a confession: when I was a student, I had no idea what I was doing in math. Come to think of it, I don’t think many of my peers understood what was going on during class, either. We’d copy the work from the chalkboard (or the overhead projector), and then learn the steps to solve a problem. I hated math because it didn’t make sense to me and made me feel stupid. I don’t blame my teachers, but I do find fault with the prevailing methods of that time.

When I graduated from high school, I had memorized a set ofsteps and equations that helped me solve problems on a chalkboard or in a mathbook, but I had no clue what I was doing and I couldn’t explain how or why themethods and equations worked. I didn’t take math all the way through high school(I stopped after geometry, pulling myself out of algebra 2 when I realized Ihad no clue what was going on).

Now that I’m teaching, I see many students come into my class with that same experience. I don’t blame any of the wonderful educators I work with for this, but I do wonder how we can begin to change the experiences students have in math class. One of the ways I plan on changing math in my classroom is to teach the math behind the methods. Rather than just teach math as a bunch of steps to solve a problem on paper where numbers float around in a nonsensical manner, I want to teach my students to understand what they are doing at a deeper level. To do this, I have decided to incorporate the Standards for Mathematical Practice in my lessons as much as possible.

Here’s a list of the eight practices:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

CCSS Standards for Mathematical Practice
An example of how I might incorporate these practices can be found in a warmup I did with my students earlier this week. I presented a series of photos I took in succession, all captioned with two questions: What do you notice? What do you wonder? I had students silently write down their noticings and wonderings on paper, giving each student time to think before we shared out. Some of the things students noticed were rather interesting (and surprising), and some of their wonders lead us toward making sense of the photos. After each photo was shown and notes were jotted down, students shared out their notes, sometimes having to prove their understanding to other students, and ended up making sense of the structure, finding patterns and using regularity and repeated reasoning (MP1, MP2, MP3, MP4, MP5, MP6 MP7, and MP8!).
What do you notice? What do you wonder?

I wonder why Mr. Seeley is having us do this.

I notice that there are three orange bars and five blue squares.

I notice that there are an even number of orange bars and blue squares on the left side and a odd number on the right.

I notice that the orange bars are as big as ten blue squares.

I notice that the left is worth 22, and the right is worth 13.

I wonder if we’re supposed to add them together, because I notice that if we do, there is 35 altogether.

Student responses

What do you notice? What do you wonder?

I notice that this is the same pieces as before.

I notice that they are laid out differently than the last picture.

I wonder why Mr. Seeley put them this way.

I notice that the 22 is vertical and that 13 is horizontal.

I wonder if we’re not adding them together like I thought before.

I wonder what this has to do with math.

Student responses

What do you notice? What do you wonder?

*Collective bewilderment*

What the heck?!

I notice that there are two hundred blocks and two more tens bars.

I wonder why there are two hundreds and two more tens.

I notice that this can’t be addition any more because 22+13=35, and now we have 35+222, which is 257.

I wonder why there is nothing on the right side.

Student responses

What do you notice? What do you wonder?

*More collective bewilderment*

I notice that now we have six more orange tens sticks and six more blue ones squares.

I notice a pattern. There are tens sticks on the left and the right, and there are tens sticks and ones on the top and the bottom.

This student had to come to the front and point it out for other students who couldn’t see the pattern.

I notice that everything is really organized, like it all has a place to be.

Oh my gosh! That’s an area model for multiplication! This is showing us 22*13! They multiplied the tens together and got 100’s, and the tens by the ones and the ones by the tens and that’s why they are laid out that way!

This student had to come to the front and model his understanding of the structure for other students who weren’t sure.

Student responsesAs more and more of the images were revealed, students began to have Aha! moments, and the excitement in the room built. Although they were supposed to remain quiet during the observation time, one student couldn’t help it. “Oh my gosh! That’s an area model for multiplication! This is showing us 22*13!”

What do you notice? What do you wonder?

I notice that there is only 286 on the screen, which is the product of 22 and 13. [Student] was right! The picture was showing us multiplication!

*Collective cheering and wows*

Student reponsesAfter this warm up, we discussed how modeling can help us demonstrate what is going on with numbers in math, and that they can help us make sense of what we are doing. Students really enjoyed this, and for many, it made multi-digit multiplication come to life. I let my students know (to their delight) that I would start having them demonstrate multiplication with drawings or physical models very soon, letting them prove to me and others that they knew what they were doing.

This who process took around 10 minutes, and it was worth every moment because students were actively engaged and making new connections to the numbers on paper. For those 10 minutes, math came to life for my students in a meaningful way. Students learned the math behind the method because we slowed down and looked at a model, making sense of structure and by critiquing the reasoning of their peers.

I plan on doing the same with multi-digit division, fractions, geometry, and measurement and data in the future this year. My challenge for you is to find new ways to teach students the math behind the methods. Let them see the numbers in a new way and make them make sense of it. The learning takes place in that productive struggle. Give students a challenge, step back, and enjoy the ride.

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About the Author: Shawn Seeley

Father, husband, teacher, retired Marine.

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