Math Reimagined

Guest Post by Susan Smith, Elementary Math Teacher.

This post explores a topic that we also feature in our upcoming book, Developing Mathematical Reasoning: The Strategies, Models, and Lessons to Teach the Big Ideas in 3–5. Want to dive deeper into our work? Browse our collection of books here.

To celebrate the launch, join us for a live webinar with Pam Harris on March 25 where we’ll dig into the big ideas, models, and strategies that help students truly reason. (Details and registration below).

I Thought I Was Teaching Math Well

October 2022: “It’s Halloween! Do you remember what I told you on the first day of school?” I asked, smiling expectedly at my fourth-grade students. “I said that by the end of October you would be able to solve problems like 49 x 86 and 1,937 ÷ 4. Many of you doubted that would be true. Can you solve these now?”

All the students dutifully nodded their heads and immediately got to work on the first problem. Some drew area models divided into 4 boxes for 40 x 80, 40 x 6, 9 x 80, and 9 x 6. Others listed these exact same 4 partial products vertically, computed each, and added to find the total product. “Wow!” I exclaimed, feeling quite proud and satisfied as a teacher, “ Look at the math skills you have learned this year.” Every child had computed the correct answer by accurately executing a process I had taught them step by step. Mission accomplished. That was the goal of math instruction, wasn’t it, to get correct answers to problems? At least that is what I thought at the time. Later that year, I would embark on a journey that would lead me to discover I only thought I knew how to teach math well. What I really knew was how to teach “fake math” well.

Discovering “Real Math”

My journey began by sheer chance, or fate, when I discovered Pam Harris and the Math is Figure-out-able movement over Spring Break 2023. I was intrigued by Pam’s message of the need to teach more “real math.” I was not exactly sure what that meant or looked like. I took the free Development of Mathematical Reasoning course and listened to multiple podcast episodes. I began reevaluating my teaching practices and long-standing beliefs about the purpose of math instruction. What does it mean to teach math well? I asked myself repeatedly. Could my fourth graders still be in additive reasoning even after all the multi-digit multiplication and division problems they had learned steps to solve? Even worse, could they be stuck in counting strategies? Are my students even able to reason additively? I was curious.

The Moment Everything Changed

I will never forget the first day after that spring break, I wrote the problem 78 + 99 on the board and asked the class to solve it. I watched in horror as a student in the front row proceeded to stack the numbers, use her fingers to count the total of 8 + 9, write the 7, “carry the 1” and then add the tens column. She smiled up proudly with the correct answer. That is when it struck me, Pam Harris is right, my students have been doing “fake math.” “How about 500 – 103?” I asked. And again, several students stacked and used the algorithm. Where was their ability to reason and think? I felt like someone who was unaware their vision was blurry until they put on glasses and saw the world with clarity for the first time. I could see now that my very own students, who scored amazingly well on the state tests and appeared to be successful math students, were not developing their mathematical reasoning abilities. I had 2 months left in the school year. I needed to empower my students by giving them experiences of what it really means to mathematize like mathematicians.

Over the following weeks I did problem strings daily, pushing my students to try new strategies and discover relationships. Of course I was learning right along with them, stumbling my way through the strings that I found on the website, in podcasts, and through the workshop. I experimented. I observed. I learned. And then I experimented some more.

Problem Strings and Powerful Shifts

One morning I began our daily string, “My daughter loves gum. Her favorite kind of gum has 17 sticks in a pack.” I recorded on a ratio table 1 pack/17 sticks. (I had never done a ratio table with students in my life!) “What if she had 2 packs, how many sticks of gum?” Students shared their reasoning, and the string continued. In the middle of the string, Melody, who had struggled all year with memorizing and mimicking the steps of math, shared what she had been thinking. “I knew 5 packs would be 85 sticks. 17 x 10 is 170. I took half of 170 since 5 is half of 10.” When I asked her how she thought about half of 170, she responded confidently, “Half of 100 is 50 and half of 70 is 35, so that is 85.”

I was astounded. “Sit up tall,” I said, “You just mathematized using relationships you own.” Every problem string I presented to my students was like a treasure. I was excited to see how my students would use relationships to reason and solve problems. Now I understood the limitations of algorithms and how they restricted my students’ abilities to develop their reasoning and grow denser mathematical connections. I felt transformed. I would never be the same math teacher again.

When It Hit Close to Home

When my 22-year-old daughter came home from college, I asked her curiously, “How would you solve 35 x 18?”

“I don’t know,” she said. “I have not done that kind of math in years. Is that the “one” where you multiply across and then you must put the zero in the space on the next line or something like that?”

WHAT?! I looked at her dumbfounded. Oh my, I thought, she truly views math as a series of rules and steps you must memorize and follow. This is an intelligent, college educated woman. Can she not reason about this problem? “Just think about what you know. How could you solve it?” I prompted. She stared blankly at me. “Do you know any relationships that could help you?” I continued.

“I guess I know 35 x 10 is 350.”

“Okay, how could that help you?”

“Well, I need 8 more 35s… which I don’t know.”

“Could you figure it out?” I suggested. “Start with what you do know and go from there.”

“Okay, let’s see 2 x 35 is 70,” she said,” so 4 x 35 would be 140 and 8 x 35 would be 280. Then add the 350, and I get… 630. Hey, look at me. I just figured that out! I thought you always had to do those steps I could never remember.”

Then I shared my strategy of 35 x 18 being equivalent to 70 x 9, which is 630.

Now she was dumbfounded. “Why didn’t anyone ever teach me strategies like that? Why didn’t anyone ever tell me math was figureoutable. I think I might have liked math if I had known that! You know the reason math was so hard for me is I always confused all those rules and steps. You MUST teach your students this way, mom! You HAVE to!”

That was it!

Changing My Teaching (and Myself)

There was no going back now. I was on a new mission. I felt empowered as a math learner and as a math educator to change how I thought about math and how I taught math. Now that I knew better, I had to do better. And I could not wait. I took my first Deep Dive Workshop on Multiplication. Within a year, I completed workshops on subtraction, fractions, and division. I joined Journey and Journey Leader. I soon realized my own lack of numeracy. I too was a “math mimicker.” I was excellent at following rules and learning the steps to solving problems. As I implemented more strategies, problem strings, and rich tasks, my numeracy developed right alongside my students.

That next fall, I was inspired to not only develop mathematical reasoning in my students, but to share this vision with other teachers and help them teach more “real math.” As I learned more, I shared my learning and passion with colleagues, inviting them into my room, modeling lessons, and providing resources and support. I too wanted to spread the message: “Math is Figureoutable and teachers and students can ALL learn and do more real math!”

What Mathematical Reasoning Looks Like Now

Now when I ask students to solve 49 x 86, I am intrigued to see the clever and efficient ways they approach this problem using thinking and reasoning. It never ceases to thrill me. Michael: “I found 50 x 86, which is 4300 because 100 x 86 is 8600, and I divided it in half. Then I took away one group of 86, which left 4214.” Or Brielle: “Next time I want my brain to do what Michael did and use 5 is half of 10 strategy. But instead, I did 50 x 80, which is 4000 and 50 x 6 which is 300. That gave me 4300. Then I subtracted the extra 86, which I knew was 14 because the partner of 100 for 86 is 14.” BAM!

Ready to Develop Mathematical Reasoning?

“Do the best you can until you know better. Then, when you know better, do better.” Maya Angelou

If this story resonates with you, you’ll want to explore Developing Mathematical Reasoning: The Strategies, Models, and Lessons to Teach the Big Ideas in Grades 3-5, releasing March 25. In it, Pam lays out the strategies, models, and lesson structures that help students build the big ideas in Grades 3–5, so they’re not just getting answers, they’re developing reasoning.

📘 Order the book now.
🎥 Join us March 25 at 7:00 PM for a free live webinar with Pam Harris to celebrate the launch and dig into what it really takes to teach the big ideas through reasoning.

Ready to move beyond steps and rules? Let’s help students—and teachers—do better math.

Susan is a retired math teacher with 34 years of classroom experience. Passionate about mathematizing, she shares practical tips and strategies to make math meaningful for students.