I. Mental frameworks
The brilliant French mathematician and Einstein’s contemporary, Henri Poincaré, was mid-vacation in the town of Coutances, mid-conversation, and mid-stride — one foot stretched to step into his bus — when the solution to his problem suddenly appeared in his consciousness and seared itself in his mind. He had no time to write down and verify the mathematics, but also no need, so he continued his conversation. Without working on his problem or even actively thinking about it, his subconscious mind had presented him with the solution in a single moment.1 Controversial and perennially misunderstood, occurrences like these fascinated the earliest students of human academic thought, and help motivate the broader goal of this piece: to take a step back and ask “What can we learn about effective teaching by thinking about the minds of students?”
And so I want to briefly explore: what constitutes a framework of knowledge, how is it built, and how can we be more effective teachers by thinking about these things?
Inspired by a series of lectures by Poincaré in 1937, Jacques Hadamard (also a renowned French mathematician) wrestles issues like the above in his short work The Psychology of Invention in the Mathematical Field. While a powerfully thought-provoking piece in its own right, I am particularly interested in his questions to prominent scientists of the day, including Albert Einstein, as to how they “did their thing.” Did they think in words? Imagine mathematical symbols? Interestingly enough, Hadamard notes that “practically all of them… avoid not only the use of mental words but also, just as I do, the mental use of algebraic or any other precise signs; also as in my case, they use vague images.” [i] Einstein reports the following famously fascinating observations:
The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced or combined… The above mentioned elements are, in my case, of visual and some of muscular type.” [i]
The “muscular type”? Yes, Einstein could “feel the abstract spaces he was dealing with, in the muscles of his arms and fingers.” [ii][iii]
I love reading things like this, both because I myself rarely think (scientifically) in words or symbols, and because I wonder at the possibilities— how many ways are there to think?2 How can I improve my understanding of the world around me by exploring modes of thought? But back to the point: what implication does this have for our students?
II. Synthesis as a context for learning
If we agree that our students mentally work in unpredictably unique ways, how can we introduce them to a subject in a way that they can build a useful framework in their own minds, according to their own way of thinking? An answer to this, in many regards, is what Hadamard calls “synthesis,” and by which I mean the active process of discovery, or of actively incorporating new ideas into one’s own mental model. In the classroom, by actively working to understand or apply a new concept, the student builds a mental framework organically consistent with his or her own mode of thinking.3
How do we teach in a way that promotes this active synthesis for our students? This can be especially difficult since in mathematics (and possibly in other fields) we often have to communicate in a language foreign to our own — or our students’— actual mode of operational understanding. Here’s an example: a well-known law in physics and mathematics is called “Gauss’s Law.” I know this law well, but like the scientists mentioned above, it exists in my mind only as a sort of image or impression. Even though the law is elegant and simple, I cannot convey it as such. Instead I have to conjure my mental image, and interpret it into a series of mathematical relationships, which is a common language between my students and myself. Is it then sufficient to derive a mathematical law by simply presenting a series of individually consistent steps?
According to Poincaré (who we started with), this is necessary, but is not usually sufficient. As he puts it:
Context is everything. [viii]
To understand the demonstration of a theorem, is that to examine successively each syllogism composing it and ascertain its correctness, its conformity to the rules of the game? … For some, yes; when they have done this, they will say, I understand. For the majority, no. Almost all are much more exacting; they wish to know not merely whether all the syllogisms of a demonstration are correct, but why they link together in this order rather than another…Doubtless they are not themselves just conscious of what they crave…but if they do not get satisfaction, they vaguely feel that something is lacking. [i]
Teaching by simple progression, one abstract idea to the next, leaves a student missing something; Hadamard more clearly states exactly what’s missing:
In this way of working, which seems to be the best one of getting a rigorous and clear presentation for the beginner, nothing remains, however, of the synthesis… But that synthesis gives the leading thread, without which one would be like the blind man who can walk but would never know in what direction to go. [i]
The missing piece is synthesis, the original context by which the idea came to fruition. Of course, this is commonplace. We discover something through a particular thought-process, then rewrite and rework our logic a dozen times before presenting our work anywhere else. The advantages are concise and logical publications; the disadvantages are the production of learning materials that can leave us feeling led, but blind. This is a pitfall for textbooks especially, which are as rigorous as reference manuals, but are often lacking in synthesis, context, or motivation.
A further example to conclude: suppose this week you become fascinated by a close relative of “Gauss’ law”, called “Stokes theorem” (we use this in introductory Physics, so naturally you are enthralled). You look it up on Wikipedia only to read:
“In vector calculus, and more generally differential geometry, Stokes’ theorem is a statement about the integration of differential forms on manifolds… Stokes’ theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω.” [iv]
Well good luck sipping knowledge from that fire hose. It is logical, concise, defensible, and largely useless, links and all (exactly 6 clicks deep into the first links will lead you to the “philosophy” page anyways—again, good luck) [v][vi][vii]. But Hadamard’s point is deeper than this— he claims that even if you “understood” the above statement, meaning you could verify each part yourself, you are still lacking the synthesis, the ideas and thoughts and problems that could lead a person to such a formulation as a whole.
So both Poincaré and Hadamard claim that synthesis is an important ingredient in the teaching process: by presenting the topic in the fullness of its historical or logical context as much as possible (the context in which the idea was synthesized in the first place), the mind of the student is automatically set to contextualizing and reframing that problem according to their own mental processes as they search for a solution.4
III. Application: lessons from my own teaching
To incorporate synthesis in my teaching, I have borrowed from the examples of my own teachers, who were an unbroken chain of great mentors, especially my undergraduate advisor and friend at the University of Dallas, Dr. Richard Olenick. With them in mind, I try to make synthesis a habitual practice in my teaching, using the following two methods:
- Provide the context for synthesis. To do this, I always try to connect, contextualize, then motivate the material. Connect: Each and every student sits down in class with something different on their mind: food, sleep, relationships, intimidation, excitement, and more. I’ve found it highly effective to allow the first sixty seconds or so be content-free. Discussing the course, upcoming assignments, or their current course load gives each mind time to adapt and settle in to their surroundings, and build attention towards you and the class. Only a few seconds are needed and provide a very high return on investment. Contextualize: I can then easily discuss a background to the topic and properly motivate it. In the sciences particularly, history has been my friend; a short background to your topic is a powerful incentive, human and logical. Motivate: With some background, however brief, the material is ready to be motivated: why is this interesting? How does it impact your life? Why did it fascinate the people who came up with it? Reasons and goals are great allies.
- Provide tools for synthesis. Even with the proper context and motivation, you can’t actively search for answers (synthesize) if you don’t understand the question. With complicated material, I’ve found it helpful to first provide simple overviews of the topic, verbally and visually. This reduces intimidation and clarifies the topic, allowing students to be more receptive to the material itself. With a basic overview and understanding of the relationship among different aspects of a topic, students have a mental “scaffold” on which to place the more complicated material as it arises, providing mental space and the prerequisite knowledge to begin active problem-solving and synthesis.
To frame this according to modern research: “To develop competence…students must: (a) have a deep foundation of factual knowledge, (b) understand facts and ideas in the context of a conceptual framework, and (c) organize knowledge in ways that facilitate retrieval and application” [xii]. That is, competence requires knowledge, in a framework and organized. So, to restate: with complicated material, I find that first exposing students to a clarifying framework5 of a topic can help them retain the knowledge itself. Students are then better equipped to synthesize and create mental organization as they internalize the knowledge. To accomplish this, I first discuss the larger framework (“how are all these ideas connected?”) using simple versions of the ideas, only then to follow through with more detailed explanations.
Innovative thoughts about metacognition from over fifty years ago provide insights into thought and pedagogy that remain highly relevant today. The sources referenced here gave me important mental models for understanding active learning and backward design as a teacher, but student metacognition has also been shown to produce learning gains [xiii]—so as your students learn and synthesize, have them think about how ideas fit into their own broader understanding. Everyone benefits from being mindful of the mind!
By looking into the foundations of education research, we can continue to find innovative ways to relate current research to our own teaching. This innovation “is highly important for the further development of educational professions… and for our development as a knowledge society.”6 In our rapidly accelerating information era, we can foster effective pedagogy by applying research old and new. I hope you find ways to apply mindful teaching in your own discipline, and I hope you tell me how you do it! I’d love to hear.
1 “At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the result at my leisure.”
2 Some would say there is only one way to think. I would refer them to the broader discussion on [i] and documentation on alternative mental processes such as synesthesia.
3 For more on active learning, a good starting place is the CTL’s overview [ix].
4 Ever had a student reach the “a-ha” moment, and then explain the concept back to you in a strange way? “OH! So it’s just like [insert unanticipated or confusing analogy here].” But you realize their analogy does make some sort of sense. That’s the concept being adopted and adapted into their particular mental framework, and being re-expressed.
5 While the full meaning of a “knowledge framework” is the subject of much discussion, here I simply use “framework” in the very specific sense discussed in the prior paragraph: a basic overview of a topic with stated relationships among its various components.
6 This importance was recently emphasized in Review of Educational Research [x].
[i] Jacques Hadamard, The Mathematician’s Mind: The Psychology of Invention in the Mathematical Field. Princeton University Press, Princeton NJ. 1945. Pages cited: 84, 143, 104-106.
[ii] L’Enseignement Mathematique, Volumes 4 and 6. International Committee on the Teaching of Mathematics.
[iv] https://en.wikipedia.org/wiki/Stokes%27_theorem (subject to change without notice, especially after publication of this article)
[v] https://en.wikipedia.org/wiki/Wikipedia:Getting_to_Philosophy (subject to change without notice, but less likely)
[x] M. Thurlings, A. Evers, M. Vermeulen, “Towards a Model of Explaining Teachers’ Innovative Behavior: A Literature Review.” 2015. Review of Educational Research, Vol. 85, No. 3, pp. 430-471. DOI: 10.3102/0034654314557949.
[xi] For further thoughtful commentary about Hadamard’s book (and other topics), see https://www.brainpickings.org/2015/02/24/mozart-on-creativity/
[xii] M. Donovan, J. Bransford, and J. Pellegrino, How People Learn: Bridging Research and Practice. Committee on Learning Research and Educational Practice, National Research Council. 1999. Page cited: 12.
[xiii] K. Tanner, “Promoting Student Metacognition.” 2012. Life Sciences Education, Vol. 11, pp. 113-120. DOI: 10.1187/cbe.12-03-0033.