Beyond memorization: when numbers make sense

“What does it mean to know mathematics?”

This question, posed by former National Council of Teachers of Mathematics (NCTM) President Cathy Seeley,is really what we’re asking when we discuss the best way to make sure kids are proficiently numerate by graduation.

Seller’s answer is that to know mathematics is, ultimately, to be capable of working out the relationships between numbers in your head. She explains:

"Problem solving continues to be a high priority in school mathematics. Some argue that it is the most important mathematical goal for our students. Mental math provides both tools for solving problems and filters for evaluating answers. When a student has strong mental math skills, he or she can quickly test different approaches to a problem and determine whether the resulting path will lead toward a viable solution. Estimation skills require both a sense of number and facility with mental computation and can provide a ballpark answer to a problem before the student attempts to solve it. They also offer a comparison point by which to judge whether a result is reasonable for the given situation. Estimation is an important skill for inclusion in students’ tool kits, whether they perform calculations with a pencil and paper or on a calculator."

Research suggests that teaching mental math builds numerical reasoning and supports computational accuracy, efficiency, and overall student confidence.

I was able to watch this in action in my colleague Rita Lahiri’s middle school math class (sixth and seventh graders). Her lesson invited students not only to master content, but to reason as mathematicians. They would multiply numbers by 11 to gather data on the pattern, and then come up with a theory explaining why that pattern happens. Her directions were concise, clear, and inspiring:

"Whatever rule or strategy you come up with has to apply to every problem. Write it down, because there are some theories that when you write it down on paper, you own it. And if you own it, you can contribute to why it’s happening."

I was so struck by Ms. Lahiri’s words. You, quiet sixth grade girl in the back row, can contribute your theory to the conversation. That is an incredibly powerful invitation to offer to a young learner. You are part of the collaborative construction of knowledge, and you have something to say. It was also an immensely convincing argument for taking notes!

Ms. Lahiri began by grounding the lesson in real-world contexts. In what direction do you typically read words and numbers? (Left to right). In what direction do you typically calculate? (Right to left). Why is that? (Engaged silence!). What are some places where you or your parents might use mental math?

Then kids worked individually, computing and observing the data. Ms. Lahiri moved through the room, peppering kids with challenging questions to move them forward.

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When she noticed one of the students had figured it out before everyone else, she honored the student’s abilities while still managing to keep her engaged:

"We’re going to do our best not to say anything for five minutes. The goal is for all of us to come up with theories for why this happens. Another thing to consider while you wait is: how does this change when we multiply by three-digit numbers? If you’re extremely confident in what you are doing now, and you have written it all out, you can work on that next."

Those five minutes were difficult. Kids were brimming with excitement to share their theories.

Evalynn (6th) can barely contain her excitement to share her research, but she knows she must wait.

Evalynn (6th) can barely contain her excitement to share her research, but she knows she must wait.

By the last twenty minutes of the period, Ms. Lahiri placed students in the teacher’s role. Each student contributed a vital part to the theory, and each was responsible for holding the group’s ideas accountable to the standard Ms. Lahiri had set (does it work for every problem or just a few conveniently selected ones?)

Taz (7th) shares his research on multiplication by 11.

Taz (7th) shares his research on multiplication by 11.

While the work was rigorous and worthwhile, the biggest takeaway I had from sitting in this classroom was that every student was free to engage the content at his or her level of ability and interest, and Ms. Lahiri’s fundamental role was to methodically, humanely, and uncompromisingly push each kid a bit past where they were at the beginning of the period. No one was off the hook from having to contribute, and no one was left behind because of boredom or inability. There was a place for everyone in this work.

Ms. Lahiri’s parting words to the kids were:

"Thank you to everyone for bringing enlightenment to our research on multiplication by 11."

That really sums up her work with the students that day. They were not doing worksheets or rote memorization; they were doing research. They were not checking the box for participation points; they were bringing enlightenment. And the knowledge is owned and produced collectively, just as it is in a research university or laboratory. This is a beautiful education