Turning Complex Knowledge into Basic Knowledge
It’s no secret that math builds on itself. You need to understand pre-algebra before algebra and pre-calculus before calculus. The idea is literally within the names of our subjects. How then, as teachers, do we facilitate the transition between the once difficult topics of pre-algebra to the new set of difficult topics in algebra? Students must understand the prerequisite topics well enough use them as the basic foundation of the new topics. How do we help students turn complex knowledge into basic knowledge?
Let’s take a look at a recent incident in my classroom where students worked on solving three-variable systems. We started by reviewing two-variable systems and some of the methods used to solve them (equal values, substitution, and elimination). We went through a few three-variable problems, and at the end of the lesson I asked students to take some notes on the process. A few days later, I asked some struggling students to read through their notes to help them get started, but, to my surprise, they did not understand the notes they had taken.
They had notes with familiar words and familiar processes. I have seen them use these processes successfully on tests in the past, but they could not put all of these steps together into a new sensible process. I initially thought this was an issue with their ability to take good notes (and that may still be concern to look into), but I realized after reading an article shared by Mark Childs that there was a bigger issue at hand. My students did not understanding the basic steps well enough to string them together or write about them in a simple manner. This is what the article “Inflexible Knowledge: The First Step to Expertise” by Daniel T. Willingham called inflexible knowledge. Our job as teachers is to turn inflexible knowledge, where students understand concepts based on their surface structure (superficial look and context), into flexible knowledge, where students understand and apply content based on its deep structure (true purpose).
At this point my students had a strong surface level understanding, but their knowledge was still inflexible. They recognized two-variable systems and the processes involved, but didn’t understand the deeper structure of solving systems. They were not yet able to transfer the eliminating variables processes to three variables, and did not see the purpose behind eliminating a variable.
This difficulty with applying knowledge to a new context is common in many classrooms, so, naturally, I wondered how to help students with this obstacle. I remember learning that the brain works a bit like a computer’s data storage. As the brain is learning a new topic, it searches through many related folders to determine which ones are needed and in what order. As the brain masters a topic, it compresses the knowledge into an organized ZIP file for quick access later on. Freeing our brain space allows us to focus on other details and be creative with the task at hand. According to Practice Perfect by Doug Lemov, Erica Woolway, and Katie Yezzi, the key to freeing your mind and unlocking creativity is repetition. At first, this made me a bit uneasy, as it sounded like they were suggesting a drill and kill approach which I am generally not a fan of. They went on to explain that drill and kill has its place and, similarly to Willingham’s article, emphasized that it should not be the exact same exercise repeatedly. Practicing should evolve to help increase flexible knowledge and discourage misconceptions.
“Cognitive leaps, intuition, inspiration — the stuff of vision — are facilitated by expending the smallest possible amount of processing capacity on lower-order aspects of a problem and reapplying it at higher levels. You leap over the more basic work by being able to do it without thinking much about it, not by ignoring it.” — Lemov, Practice Perfect
This quote, this entire chapter, spoke to me. Not only did it parallel Willingham’s article, but also my experience as a violinist. I didn’t learn how to practice effectively until college. I used to repeat the piece over and over, and occasionally focused on repeating a difficult passage. In college, I learned how to break apart a passage and practice it effectively by playing it with different rhythms, chunking, bow strokes, and fingerings. After learning how to master a passage, I realized I was free to work on emoting through phrasing and bowing during rehearsal. I also noticed that my mind and ears were more attuned to the rest of the orchestra and I found sections with similar or juxtaposing passages in real time.
Similarly, my students needed more repetition, repetition that stretched their understanding of solving systems. This sounds like a simple solution, but I think the complexity lies in how to make the practice effective enough to deepen their surface understanding of the topic. For example, the answer is not to give my students a page full of two and three variable systems to solve. The answer is to provide students with examples of systems in various forms to encourage a variety of solving methods. Students should also engage in deeper discussions about the purpose of eliminating variables. Why does a system of equations require eliminating variables? How many equations and sets of eliminations would be needed in four-variable systems? Why would you need to eliminate the same variable in each equation?
By pulling together the idea of effective practice and flexible knowledge, I can try to provide more purposeful practice for my students. Hopefully after broadening students’ definitions and usage of systems we can also go back to their notes to make some improvements. I don’t anticipate this practice to come easily, and I’m sure I will need to tweak it as students struggle and progress. We as teachers can use these ideas about practice to help students transform and compress their knowledge not only in math but in other subjects as well.
