Amid the controversy of how Common Core Standards are changing American K-12 math education, engineering professor Barbara Oakley argues that the value of memorization and practice continues to be downplayed.
Common Core Standards “propose that in mathematics, students should gain equal facility in conceptual understanding, procedural skills and fluency, and application”. But implementation of CCS does not always follow those guidelines.
The devil, of course, lies in the details of implementation. In the current educational climate, memorization and repetition in the STEM disciplines (as opposed to in the study of language or music), are often seen as demeaning and a waste of time for students and teachers alike. Many teachers have long been taught that conceptual understanding in STEM trumps everything else. And indeed, it’s easier for teachers to induce students to discuss a mathematical subject (which, if done properly, can do much to help promote understanding) than it is for that teacher to tediously grade math homework. What this all means is that, despite the fact that procedural skills and fluency, along with application, are supposed to be given equal emphasis with conceptual understanding, all too often it doesn’t happen. Imparting a conceptual understanding reigns supreme—especially during precious class time.
The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. Worse, students often believe they understand something when, in fact, they don’t. By championing the importance of understanding, teachers can inadvertently set their students up for failure as those students blunder in illusions of competence. As one (failing) engineering student recently told me: “I just don’t see how I could have done so poorly. I understood it when you taught it in class.” My student may have thought he’d understood it at the time, and perhaps he did, but he’d never practiced using the concept to truly internalize it. He had not developed any kind of procedural fluency or ability to apply what he thought he understood.
I’ve read similar reports about how CCS continues to prioritize understanding at the expense of fluency. My experience several years ago was that teachers used to focus heavily on understanding, and I was told not to be concerned when my child had not mastered basic math facts because it was more important that she understand the concepts. Meanwhile she was falling behind in all aspects of learning. Is this still common?
Explaining and discussing alone do not lead to full understanding and mastery.
In the years since I received my doctorate, thousands of students have swept through my classrooms—students who have been reared in elementary school and high school to believe that understanding math through active discussion is the talisman of learning. If you can explain what you’ve learned to others, perhaps drawing them a picture, the thinking goes, you must
Chunking is important in analyzing and reacting to new learning situations.
Chunking was originally conceptualized in the groundbreaking work of Herbert Simon in his analysis of chess—chunks were envisioned as the varying neural counterparts of different chess patterns. Gradually, neuroscientists came to realize that experts such as chess grand masters are experts because they have stored thousands of chunks of knowledge about their area of expertise in their long-term memory. Chess masters, for example, can recall tens of thousands of different chess patterns. Whatever the discipline, experts can call up to consciousness one or several of these well-knit-together, chunked neural subroutines to analyze and react to a new learning situation. This level of true understanding, and ability to use that understanding in new situations, comes only with the kind of rigor and familiarity that repetition, memorization, and practice can foster.
Gifted students and experts are often able to quickly reason their way into algorithms when solving math problems, arguably because of their deep understanding. But I don’t think most students find that to be a good approach in the typical learning process, as it slows them down as they are practicing to gain both fluency and understanding. On the other hand, I’m sure many students simply memorize with no understanding, thereby failing to build the foundation for later math success.
Of course both understanding and fluency are important, and in the real world the critical questions are usually: Can you do it? Can you do it quickly?
Barabara Oakley, “How I Rewired My Brain to Become Fluent in Math”, Nautilus, October 2, 2014.