Common Core is an attempt to teach number sense. Here’s why it’s misguided, where the problem actually came from, and how to fix it.
Number sense is a physical, concrete understanding of numbers and operations. Its understanding that math operations are more than just following steps, but that
The thing about number sense: it generally grows over time. As people use numbers, carry out operations, their number sense continues to improve. When you do problems in your head or on paper, you develop an increased concrete understanding of what’s happening.
Unless, that is, something is done to interrupt that process. For decades, public and private high schools have been dim-wittingly screwing up student’s number sense by encouraging calculator dependency.
When you do arithmetic by hand or on paper, your number sense continually improves. You notice patterns, understand the reasoning behind familiar formulas, etc. But when you do math with a calculator, that number sense goes out the window. Your brain doesn’t need it, so it neither develops nor maintains it.
One part of the solution is to keep calculators out of math education. At Vohra Method, we allow them only for things like trig functions of angles not usually memorized. The rest of the time, students are building their number sense the real way. We don’t need to teach them shortcuts or ways to do problems in their head faster. They have every incentive to discover them on their own, and do.
Another part is to understand that thought processes are developed, not memorized. Common Core attempts to directly teach all parts of the internal thought process. It observes the thought process that people good at math develop by doing math in their head, and then attempts to have students memorize that. It then notices that different people have different processes to do math in their heads, and has students try to memorize all of them. The result is that students are cognitively overwhelmed, and move at a snails pace. Students in 6th grade are often still learning the kind of basic arithmetic that should by mastered by 3rd grade.
Next, educators need to remember that understanding usually comes after mechanics, counterintuitively enough. If you can mechanically do borrowing, it’s easier to then understand why it works. If you try to understand why it works before you can do it, it’s cognitively overwhelming. As an analogy, you and I can discuss word roots because we already know words. But teaching an infant to speak by discussing word roots would be insane. Similarly, you and I can discuss the reasoning behind arithmetical algorithms because we know them. If you try to discuss the rationale before a person knows the formula, try to discuss the “why” before the “what”, you’ll spend more effort to get worse results.
Educators should keep in mind Piaget’s theory of cognitive development, where he shows that formal operations (logic) comes after the concrete operational (mechanics) stage. Young kids can easily learn the mechanics of arithmetic. Once they are a little older, they can discuss logically why those mechanics work.
Educators should also understand that while there are theoretically infinite algorithms for any math operation (e.g. multiplication), there is generally one that is the best. A better algorithm should involve fewer steps, make intuitive sense, and be generally applicable. Common Core algorithms sometimes make intuitive sense (though not more than standard ones), but massively fail the “fewer steps” critera.
Finally, it’s vital to understand what mathematical problem solving is. Problem solving is the ability to figure out unfamiliar math problems. It’s not the ability to just carry out instructions. When you develop number sense through using numbers, you develop the ability to think and create methods that work for you. That creative process builds problem solving ability.
When you just follow elaborate instructions, you are not building that ability. You are trying to memorize the results without going through the creative process. You aren’t preparing yourself to think independently and creatively, but rather just learning how to be programmed. I can see why government schools would want that. But parents and students presumably don’t.
The Equation for Excellence
(Arvin Vohra runs Vohra Method, which repairs the damage done by the counterproductive curricula found at public, private, and parochial high schools, and helps students move years ahead of other students at those schools.)