In our article in the Summer 2022 issue of Educational Leadership, my colleague Ann C. Holm and I…

# The Right Kind of Lazy for Math

What if students approached math problem sets as if they were fascinating puzzles rather than boring worksheets? And did great work while being lazy? Sound crazy? It’s not–and there’s a neuroscience connection!

Quick–multiply 19 x 24. Are you reaching for paper and pencil? Or a calculator? That’s what students (and adults) trained in algorithms do. My colleague, Dr. Dario Nardi, has gathered EEG images of dozens of college students solving such problems. Those trained in algorithms use just a couple areas in their brains for calculations. The good news is that they solve these problems quickly and accurately. The bad news? *They don’t access the part of the brain used for skillful working of unusual problems.*

Instead of reaching for paper or calculator, did you say to yourself, *Well, I can make this easier and do it in my head…20 x 24 – 24. *If so, you used more parts of your brain, connected with understanding the steps in a process, rotating images, and viewing things holistically. And, if you learned to use strategies such as “friendly numbers” in collaborative groups, you also activated the speaking and listening areas of your brain. You did less work and fired more areas of the brain, creating new synapses for later, even more complex mathematical thinking!

This is the right kind of “lazy” for math. Why would you use the standard algoritm when a better strategy lets you calculate mentally without having to keep track of 9 X 4 + 10 x 4 and so on? Mathematical thinking involves efficiency. **And it turns these kinds of problems into interesting puzzles rather than repetitive drills.**

Sometimes the standard algorithm turns out to be the best method–numbers can be messy and it adds order! However, try another: 16 x 35. Can you find a more efficient method? I’d use factoring and change it to 8 x 70. Pretty slick, right?

These aren’t tricks–these strategies require not just knowledge, but **application of foundational mathematical concepts** such as

- The distributive, associative and commutative properties (how many students and adults
*really*know what those mean, even after years of worksheets? Think if they were using those properties to be “lazy” about calculations!) - Hierarchical Inclusion
- Compensation
- And many more. Here’s a partial list, Common Definitions of Big Math Ideas, with ways to assess student understanding and some activities that help reinforce each concept.

Teaching just the algorithm actually blocks efficiency, but not teaching it does the same thing. For mathematical thinking, students need algorithms, strategies, and concepts. With that knowledge, they apply the right kind of laziness–they actually use more of their brains to do less work. Pretty slick, right?

How’s your mathematical knowledge? Does this increase your interest in mastering mental math?

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Thanks for the post! I like this idea of thinking through different approaches to problems being coupled with knowledge of appropriate algorithms and processes. I learned what I think is a better approach to division of large numbers than the standard algorithm, but there would be many cases when factoring or using addition/subtraction to change the problem to a much simpler one would be better still then the more open division algorithm I like.

ps. I’m incredibly late on the Explore the #MTBOS but am glad I came across this post!

Thanks–I hope to help reluctant parents, educators and students see the value in different approaches.

I totally agree with your post. I’m so tired of parents attacking the common core because suddenly their students aren’t using the same old algorithms. I was able to explain this to the parents who came to back to school night, but, there are so many other parents that haven’t had the explanation. I wish someone could educate all the parents at once!

Kristen, I do think schools can work toward educating parents. The best arguments I’ve seen for moving beyond algorithms are in Boaler’s

What’s Math Got To Do With It?For a problem like the one posed in this blog, no self-respecting engineer would use the standard algorithm. It’d be a waste of time. Understanding why all the different methods work is key, but having different methods available is essential for anyone who is actually going to use math in real life.