I often justify my existence as a math teacher by arguing that math is worth learning because it teaches humans to think clearly and reason abstractly. Or, in the words of Underwood Dudley:
What mathematics education is for is not for jobs. It is to teach the race to reason. It does not, heaven knows, always succeed, but it is the best method that we have. It is not the only road to the goal, but there is none better.
Is this true? Does a mathematical education teach transferable skills that can be applied beyond the classroom?
Thanks to Michael Pershan for sharing an informative research paper on this topic that gave me a new perspective on teaching for transfer: Are Cognitive Skills Context-Bound?, by D. N. Perkins and Gavriel Salomon.
Half a century ago, many psychologists would argue that effective thinking is a function of intelligence and general strategies for problem solving and critical thinking. Polya’s work in problem solving was particularly influential, as he identified a number of heuristics such as breaking a problem into subproblems or examining extreme cases that could be applied to a wide variety of problems in different domains. Many thought that expertise consisted largely of general strategies like these that helped humans to reason across a range of contexts.
In the following decades, a growing body of research provided evidence that seemed to contradict this hypothesis. Studies of experts in various fields showed that their knowledge did not transfer readily outside of the domain in which it had been learnt, suggesting they had developed a specialized skill rather than a broad array of general reasoning strategies. Chess players could excel at chess, but not broader strategic thinking. Doctors who were expert diagnosticians in one field were no better than chance in another. Research looking specifically for transfer found that it rarely happened spontaneously and seemed much more elusive than had been previously thought.
For many today, this is a dominant paradigm of psychology: knowledge gained in one domain is unlikely to support thinking in another domain. However, there are plenty of contradictory results suggesting there is more to the story. That transfer exists is self-evident; under some conditions humans are capable of solving problems they haven’t seen before. This has been replicated in some studies but not in others; the question is, what are the conditions to make transfer more likely?
The Low Road and the High Road to Transfer
The authors describe two conditions under which transfer seems more likely. The “low road” to transfer involves “much practice, in a large variety of situations, leading to a high level of mastery and near-automaticity” (22). Practice and fluency in a domain makes it more likely that those skills will be drawn upon in a novel situation. The “high road” to transfer “depends on learners’ deliberate mindful abstraction of a principle” (22). Knowledge that is contextualized and connected with other ideas or broader principles is better primed to apply to a new situation. In short, there are two ways I can teach students to increase the odds they will be able to flexibly apply their knowledge in the future: effective practice and mindful abstraction.
Neither of these “roads” is certain, but they also provide a blueprint for learning that is unlikely to transfer: if there is insufficient practice and learning only takes place in one context without explicit abstraction, transfer seems all but impossible.
Where To From Here?
I often get frustrated with the arguments between the inquiry-oriented “progressive” folks and the explicit instruction “traditionalist” crowd. From my perspective, they’re both right. Learning needs to focus on connections to different elements of prior knowledge, to examine the ways that mathematical content can apply in other domains, and to focus on depth and flexibility of knowledge, all arguments of progressive educators. At the same time, purposeful practice spaced over time leads to fluency and automaticity with key ideas, making it more likely that they can be applied and synthesized with other knowledge in the future, as emphasized by traditionalists.
I think both sides have a point. I’ve seen too many students struggle with a challenging problem because they lack prior knowledge I wish they had — whether that’s addition, fraction operations, integers, or reasoning about the structure of functions. With more practice and better fluency and automaticity, they could be more successful. And I’ve seen too many students struggle with a challenging problem because their prior knowledge is totally context-bound — they’ve only solved problems from one perspective, and are unable to see how their knowledge applies to a novel problem at hand. Prior mindful abstraction of the principles they need would support this thinking by making explicit connections they may be able to make use of in the future.
It’s not an either-or, it’s a both-and. I need to be teaching students so that they have access to both the low road and the high road to transfer. And doing that depends on the content, the students in the room, and where we are in the broader curriculum. There are no easy answers. But I think that considering these two pathways to transfer is a useful touch point for my pedagogical priorities in the classroom.