"Eyes on Progressive Education" Blog

A Green Acres School journal that chronicles progressive education in action, the research that supports and informs our practice, and ways in which we live out our mission each and every day.


But, I Can Do That in My Head! (Or, I Can’t Do That in My Head!)

David, Middle School Math Teacher

“Yeah, I got the wrong answer, but I know how to do this!” Careless calculation errors certainly lead to some big mistakes. Worse, when students struggle to recall six times eight or nine plus three, they are using precious working memory that could otherwise support more complex applications or acquisition of deeper concepts. As they move into algebraic work in 7th grade, cognitive capacity committed to minor computations limits consideration of like terms, coefficients, and constants. Down the road, small calculation errors or chunks of time spent calculating may lead students astray on problems requiring substantial focus on abstract concepts. 

This is not to say that mathematics is all about arithmetic and number crunching. On the contrary, it is about patterns, connections, and concepts. Mental math skills are building blocks. Counting on a number line, using fingers, or finding products with repeated addition are all important strategies as children build numerical understanding. Finding 6 x 5 by counting 5’s is a great stepping stone through the lower grades; in pre-algebra it distracts from deeper concepts. And, we certainly do not want students interrupting their problem-solving flow to do 6 x 5 on a calculator!

We encourage students to work towards automaticity of basic math facts. Knowing multiplication tables, one and two digit addition, common fraction conversions, and basic properties (associative, commutative, and distributive) allows the mind freedom to more easily consume and integrate new knowledge. While rote memorization remains important in mental mastery, it is certainly not sufficient on its own. In fact, the research on memorization of math facts is mixed. Although drilling with flash cards or other tools strengthens automaticity, that does not always transfer into stronger problem-solving skills. Nonetheless, automaticity supports quicker concept acquisition. Contextual activities to enhance numeracy and conceptual understanding supplement the memorization—memorize the multiplication and consider the numerical relationships that make it true. Students then solve problems more successfully when they are not mired in simple computations and can ponder the big picture, instead. 

On the flip side, we often encounter students who want to do all of their work mentally and resist the increasing necessity of showing all of their written work. Many of those students struggle initially in 7th grade when they are required to show increasing written work for every step. Some of them resist out of defiance, planting their flags of superior mental math skills with disdain for the menial, pedestrian pencil-work. Others are genuinely confused about how to properly show each step, stretched by the translation of thought processes into the written language of algebra. Both camps soon encounter the absolute necessity of written skills as they embark on solving complex, multi-step equations and the oft more complex process of checking solutions. 

These two skills, mental math and showing work, are inextricably connected. Efficiently moving through each computation while evaluating an expression or solving an equation leads to each subsequent written step. Each written step supports the next, keeping the student on track and aiding error checking.

You can help at home! Simply play games! Cribbage, backgammon, Mille Bornes, Greed/Farkle, and pyramid solitaire are all excellent choices. On car trips, scoring points by adding or multiplying numbers on license plates is an easy game to play. Have your child mentally find the tip or tax when eating out, calculate mileage when filling up the gas tank, or do your taxes in April (just kidding!) Regularly flexing the mental math muscle will serve them well all of their math classes.

Posted by sarah in Middle School on Monday November, 12, 2018

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