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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
- 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?