After Joanne’s introduction and Diana’s open cheerfulness at being a co-guest blogger here, you might think I’d offer up more intellectually stimulating fare than funny pictures of Starbucks references. For those of you who were disappointed, I’ll endeavor to make it up to you in this post.
In a recent Wall Street Journal piece, Barbara Oakley posits that the Common Core standards, and the pedagogy that is often pushed with those standards, prioritizes “conceptual understanding” at the expense of slighting repeated and varied practice that leads to computational mastery:
Conceptual understanding has become the mother lode of today’s [Common Core] approach to education in science, technology, engineering and mathematics—known as the STEM disciplines. However, an “understanding-centric approach” by educators can create problems….
True experts have a profound conceptual understanding of their field. But the expertise built the profound conceptual understanding, not the other way around. There’s a big difference between the “ah-ha” light bulb, as understanding begins to glimmer, and real mastery.
As research by Alessandro Guida, Fernand Gobet, K. Anders Ericsson and others has also shown, the development of true expertise involves extensive practice so that the fundamental neural architectures that underpin true expertise have time to grow and deepen. This involves plenty of repetition in a flexible variety of circumstances. In the hands of poor teachers, this repetition becomes rote—droning reiteration of easy material. With gifted teachers, however, this subtly shifting and expanding repetition mixed with new material becomes a form of deliberate practice and mastery learning….
True mastery doesn’t mean you use crutches like laying out 25 beans in 5-by-5 rows to demonstrate that 5 × 5 = 25. It means that when you see 5 × 5, in a flash, you know it’s 25—it’s a single neural chunk that’s as easy to pull up as a ribbon. Having students stop to continually check and prove their understanding can actually impede their understanding, in the same way that continually focusing on every aspect of a golf swing can impede the development of the swing.
Understanding is key. But not superficial, light-bulb moment of understanding. In STEM, true and deep understanding comes with the mastery gained through practice.
I recently wrote on a similar theme, quoting from a newspaper article about a Stanford paper that demonstrated, among other things, that children should memorize multiplication tables and addition tables. Quoting:
Next, Menon’s team put 20 adolescents and 20 adults into the MRI machines and gave them the same simple addition problems. It turns out that adults don’t use their memory-crunching hippocampus in the same way. Instead of using a lot of effort, retrieving six plus four equals 10 from long-term storage was almost automatic, Menon said.
In other words, over time the brain became increasingly efficient at retrieving facts. Think of it like a bumpy, grassy field, NIH’s Mann Koepke explained.
Walk over the same spot enough and a smooth, grass-free path forms, making it easier to get from start to end.
If your brain doesn’t have to work as hard on simple maths, it has more working memory free to process the teacher’s brand-new lesson on more complex math.
‘The study provides new evidence that this experience with math actually changes the hippocampal patterns, or the connections. They become more stable with skill development,’ she said.
‘So learning your addition and multiplication tables and having them in rote memory helps.’
As a math teacher I explain it this way: to truly understand algebra a student must already have mastered operations with fractions, decimals, and negative numbers. The simple calculation of calculating the slope of a line between two given points could include all three of those, and if one expends all his/her brain power on that simple calculation, there won’t be as much brain power “left over” to understand what the answer, the slope, actually means or represents.
Some things must be memorized–not for their own sakes, but because they are useful tools, they are means to an end.
In professional development sessions I’m often told, as if it’s an obvious fact that cannot possibly be doubted, that if you cannot explain how something works, then you truly don’t have a “deep” enough understanding of it. You have rote memorization, nothing more, and rote memorization is useless. Sometimes I’m even told this by math teachers, who will at lease concede that memorizing the multiplication tables is a valuable exercise. I put up a division problem, usually something simple like 515/3, and ask “Who can perform this calculation?” Everyone can and all hands are raised. Then I ask, “Who can explain why the standard algorithm (which everyone our age knows and uses) works, and why?” Even most math teachers cannot, but everyone recognizes why that standard algorithm is important, useful, and efficient–everyone, that is, except for those who think that some Indian lattice method leads to “deeper understanding”. Beyond knowing that division is akin to finding out how many “groupings” of a certain size can be made from a certain number, how “deep” does one need to understand division? It’s useful only as a tool to get to bigger and better things, IMNSHO.
So repeat and repeat and repeat until the repetition begets memorization. That’s what Mrs. Barton did until every one of her students knew the multiplication tables. Don’t allow a pet pedagogical theory to harm students’ ability to calculate. Teach them what works. Give them the most efficient tools out there.