The Dangers of Mental Math

by John R. Walkup, Ph.D.

The mathematics education community has long stressed the importance of mental math, that is, the ability to perform mathematical manipulations in one's head. In my travels, I have observed many teachers assign mental math activities for students, with the idea that, with practice, skill in mental math will increase.

And this is fine. I think mental math is a good skill to possess. Teaching mental math is also a great way to exercise brain processes related to the recall-level of Bloom's Taxonomy. However, I suggest approaching the teaching of mental math with some caution.

Cognitive Rigor of Mental Math

While a person performing mental math may impress with her ability to juggle numbers in her brain, the Cognitive Rigor of mental math is actually quite low. With rare exception, the algorithm employed to derive an answer is straightforward, so much of mental math resides at the lowest levels of Depth of Knowledge. Also, most mental math relies heavily on the recall and apply levels of Bloom's Taxonomy, and little else. 

This should not discourage teachers from teaching mental math. As I stated previously, the ability to perform mental math is quite beneficial. In fact, I wish I was better at it myself.

Problems with Mental Math

Many students struggle with mental math for a reason that lies outside mathematics, and I am referring specifically to memory. Let me provide an example:

Jim is buying a pizza, which normally costs $15. But the restaurant is offering a 20% discount. Jim, not having a pencil or paper, tries to perform the computation in his head.

He knows how to find the net cost. He first computes 10% of the pizza, which is $1.50. But, since 20% is double 10%, he knows to double his result to get $3.00. Now, he simply needs to subtract $3 from $15 to get the net price, that is, $12.

Sounds good. But, many people will struggle to hold the original $15 price in their head during the computation, so once they compute the $3 discount they get stuck.

I struggle with this problem, myself. (And I think my math is pretty good.) I simply can't hold numerical values in my short-term memory long enough to retrieve them when needed.

Those students who struggle with mental math are often unfairly labeled as mathematically ignorant. We often encounter clerks in stores who seem to struggle with even the most basic mathematics. Give them a pad and pencil, however, and they often do just fine. For these people, the problem wasn't the math; it was the memory.

Teaching Mental Math

I suggest teachers consider the following rules of thumb when teaching mental math.

1. Do not confuse the ability to perform mental math with intelligence or math skill. 
2. Unless you are purposefully trying to exercise memory skills, let students write down intermediary values so that they don't have to hold them in memory. 
3. Use the think-aloud strategy for teaching students mental math. That is, perform some mental math of your own, but verbalize your thought processes so that students can hear for themselves the logical progression you employed to perform the computations.
4. Follow up your own think-aloud with questions that check for understanding and promote critical reflection.

Teacher Use of Mental Math

Picture a middle school teacher walking through the steps needed to carry out a mathematical computation. Periodically, she stops and performs "sidebar" computations in her head before moving on to the next step. "Let's see, we need to multiply 25 X 30 and that's 750 (of course!), so we just need to divide by 750..."

While it may make some of her students think she must be enormously smart, her performing the sidebar math in her head could also destroy her students' confidence. ("The answers just pop into her head!") The last thing some students need is to feel that, as far as math goes, "Some have it and some don't."

As such, I always try to avoid performing mental math when teaching unless I am specifically trying to teach that skill. I think we need to be careful about trivializing mathematical difficulty by doing computations in our heads without verbalizing our thoughts.

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Seeking training at your school or district centered on Cognitive Rigor or Depth of Knowledge?  Call me at (559) 903-4014 or email me at jwalkup@standardsco.com. 

We will discuss ways in which I can help your teachers boost student engagement and deep thinking in their classrooms. I offer workshops, follow-up classroom observation/coaching, and curriculum analysis to anywhere in the country (and even internationally).

Follow me on Twitter at @jwalkup.