Inflexible Knowledge

Dan Willingham’s excellent article Inflexible Knowledge: The First Step to Expertise (American Educator, Winter 2002) opens with the following question.

“So often, even if I inventively present new material or emphasize applying the new knowledge in various situations, what I get back from my students seems ‘rote’. Why is this? What can I do about it?”

If you’ve asked yourself this question as an educator, I recommend reading the article in full. But here are the highlights!

Willingham defines three types of knowledge in his article: rote knowledge, inflexible knowledge, and flexible knowledge. He begins by distinguishing rote knowledge (parroting without any understanding) from inflexible knowledge (a seeming inability to apply knowledge learned). Willingham uses an example from the book Anguished English to illustrate rote knowledge. A student, asked to define equator, writes “a managerie lion running around the Earth through Africa”. The student has memorized not even the right words, but sounds, such that ‘imaginary line’ has turned into ‘managerie lion’ perhaps subconsciously associating Africa with the king of the beasts.

Much more often, we encounter the situation where the student can correctly produce a newly learned definition but seems unable to use it. According to Willingham, “cognitive science has shown us that when new material is first learned, the mind is biased to remember things in concrete forms that are difficult to apply to new situations.” This is inflexible knowledge and, Willingham argues, an important first step in the learning process. Why does this happen? If something is unfamiliar, we automatically consider surface features first checking if they resemble anything else we already know.

Abstracting the deep features from the surface features is what distinguishes the expert from the novice. This is flexible knowledge – the ability to apply our understanding of the deeper structure to a variety of concrete situations, and not just the situation where the concept was first learned. As educators, we want to move our students from surface-level thinking to deeper thinking, leading them down the rabbit hole. You might think that we can teach deep structure and abstraction directly to novice students, but that turns out to hugely difficult. Willingham writes: “… cognitive scientists have tried to [use it] many times. But, the problem with such direct instruction is that the mind much prefers that new ideas be framed in concrete rather than abstract terms.”

Reflecting on all of this reminded me of an office hour session with one of my students three weeks ago. She told me she was having trouble keeping straight all the different periodic table ‘trends’ we had learned. When I asked her the definition of ‘Ionization Energy’ she struggled to parrot the answer, at which point I asked her to stop and think about the words ‘Ionization’ and ‘Energy’ meant. A moment later she was able to give me the correct definition. But then she seemed to be guessing whether the first ionization energy increased or decreased across a row or down a column, and unable to use the definition to help her. I moved the conversation away from ionization energy back to something she could more easily concretize – atomic size. She could correctly tell me the size trends, and with a bit of delving, eventually able to explain her answers in concrete terms. Then I asked her the definition of ‘Effective Nuclear Charge’ and this time she was able to figure it out and then use the concept to explain the size trends. At this point we returned to ionization energy and she was now able to determine and explain the trends. Success!

Next she asked me about electronegativity. I asked her for the definition and got a garbled response. She looked up the definition and was able to repeat it to me, but then together we parsed the definition carefully to make sure she paid attention to all the attendant parts. I asked her how electronegativity might connect to effective nuclear charge, and with a bit of (mental) prodding, she could now explain the trend. To tie it together, I had her summarize all three trends and connect them to each other. The whole process probably took at least 30 minutes one-on-one. It takes time and energy for those disparate pieces of information to click together as an integrated whole. In a classful of students, I had actively connected the definitions and trends, and had the students think-and-reason through the process. That may have clicked for some students, but clearly it did not for others. In this one particular case, I hope I successfully increased the student’s knowledge flexibility at least for this one concept.

How might one overcome the bias to “remember things in concrete forms that are difficult to apply to new situations”? Willingham suggests it “seems best overcome by the accumulation of a greater store of related knowledge, facts, and examples.” He ends the article with suggestions for teachers.

(1) Use multiple examples and encourage students to see what the examples have in common, preferably leading them to the deeper structure.

(2) Distinguish rote knowledge from inflexible knowledge, not despairing from the latter, but using it as a building block to flexible knowledge.

(3) Appreciate that learning facts can be helpful and encourage students’ growing their knowledge even if it’s inflexible. “Knowing more facts makes many cognitive functions (e.g., comprehension, problem solving) operate more efficiently.”

The last point is important especially with the recent popularity of emphasizing broad skills over facts/content in education circles. While we’d like for students to acquire those deeper ‘critical thinking’ skills, the road to it is paved with facts and content. Thanks to cognitive science research, we now have good evidence that higher order abilities are domain-dependent. The more you know about something, the better you are able to turn that knowledge from inflexible to flexible. Content and critical thinking are not opposed to each other. The more factual knowledge you have, the more able you can think about the area critically.