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Tangible Math

Mathematical Struggle: Experiences of Mathematicians

Even those who are highly skilled in mathematics often consider the subject to be difficult (Devlin, 2000; Hadamard, 1949; Thurston, 1994).  What, then, constitutes this sense of difficulty in those cases when it is almost certainly not due to a lack of mathematical skill?

Research Question

The research question we (Ricardo Nemirovsky and Michael Smith) use to explore this question is thus: What themes and/or notable theoretical constructs regarding the intentionality of experience appear in the mathematical struggles of mathematicians in the process of collaborating?

There are a few background details we should explain in order to contextualize this research question.

Theoretical Framework

We take a phenomenological approach, by which we mean that we examine the nature of how the phenomenon at hand is experienced by our subjects regardless of the mechanisms that cause this experience.  For instance, when a stick goes halfway underwater at an angle, it appears to bend.  We understand why this occurs in terms of refracting light and that the stick does not in fact bend, but our understanding of this mechanism does nothing to change the fact that the stick is perceived as bent.  Similarly, we might understand a reasonable amount about the brain chemistry that is involved in pair bonding, but that kind of description does little to inform us about what it is like to fall in love (although the two accounts are obviously intertwined and should support one another).

In phenomenological circles, the term “intentionality” no longer necessarily refers to any kind of purpose-driven behavior.  Instead, it refers to the “about-ness” of a subject’s experience (Gallagher & Zahavi, 2008; Merleau-Ponty, 1962).  An example of this is affordance: a chair is actually experienced differently when the observer realizes that he or she can use it as a stool rather than as just a seat.  In this case we would say that the chair becomes “about” standing instead of just “about” sitting for this subject, which is to say that the subject’s “intent” changes toward the chair or that the subject “intends” the chair differently.  So when we ask about “the intentionality” of something, we’re asking after the nature of how that something is experienced by a given individual.

In this case study, that “something” is what we term “mathematical struggle”.  By this we mean to refer to the well-known phenomenon of hard mathematical thinking being experienced as a kind of heavy effort (Burton, 2004; Hadamard, 1949; Hiebert & Grouws, 2007).  We find in many cases that this “struggle” seems to be a core component of mathematicians’ research efforts (Burton, 2004; Hadamard, 1949), and it may well be a key component of math students’ ability to develop facility and understanding with mathematics (Hiebert & Grouws, 2007).

We frame this exploration using our own interpretation of embodied cognition.  More specifically, we reject the claim that cognition is done independently of perception or motor activity, instead viewing these three arenas (thinking, perceiving, and physically moving) as inextricable facets of human behavior (which we thus term “perceptuo-motor activity” to reflect this union).  For instance, we maintain that the physical way we move our mouth, throat, diaphragm and so on to form the sound “sharp” is part of our understanding of what that word means.  Even our perception of the sound “sharp” tells us that the sound of the word itself is in some sense sharp.  Similarly, Spanish-speakers apparently hear “afilado” as sounding sharp to them even though it might not sound that way to non-Spanish-speakers.  We suggest that this comes from the fact that perception, meaning-making, and motor activity form an inseparable whole.

This view has methodological implications.  First, we don’t view cognition as a black box whose contents we must infer.  Instead we view cognition as distributed throughout the subjects’ bodily activity including interactions with their environment and with others (Hutchins, 1999).  Thus the way they orient themselves toward another person or a blackboard, the way they hold a pen, their use of inscription, and so on are all seen as part of, rather than merely indicative of, their perception and understanding of the situation.  This means that we can potentially come to an understanding of how a subject experiences a given phenomenon by observing the subject through as many semiotic modalities (Roth & Thom, 2009) as possible, including gesture, speech, eye gaze, use of inscriptions, and body posture.

Furthermore, because perceptuo-motor activity is based on interactions with the environment, we recognize that meaning is co-defined in social and physical interactions rather than existing solely within some “black box” individual mind (Scheler, 2007).  For instance, it’s a matter of debate whether a given individual understands some specific idea.  She might believe that she understands the idea in question, but it’s the belief that is experienced (and expressed!) rather than the truth or falsity of its content.  Whether she “really” understands is something continually defined and redefined in her interactions with others.  This means that understandings, beyond just being incidentally expressed outwardly via bodily activity, are largely constituted by behaviors that others can experience.  In other words, the meaning someone perceives in a given situation cannot be purely private; it must be accessible to others through the multimodal embodied manifestations of perceptuo-motor activity.  This means that subjects’ intentionality is something we can at least partially observe.

The Study

With this background in mind, we observe a graduate student in math working with his or her advisor on his or her dissertation material.  We video-record their interaction and produce a microethnography, which consists of an annotated sequence of still shots that make it fairly easy to see at a glance what happened in the episode in question.  (See below for an example of a microethnography.)  We then iteratively examine the microethnography for insight into the intentionality of the subjects.  Two sorts of patterns are likely to emerge from this:

(1) Some particular segments may come to stand out as particularly significant or offering of insight.  We examine these closely for theoretical constructs, which are descriptions of an individual’s intentionality that can lend insight into potentially similar experiences of others.

(2) Some common tendencies in intentionality appear repeatedly in different contexts.  These are what we term themes.  A theme can reveal one way in which mathematicians can engage in mathematical struggle across scenarios.

Once we’ve noticed and carefully examined and detailed these patterns, we follow up with the subjects in unstructured interviews (Bernard, 1988) in order to gain further insight into the way they were engaging with the mathematics.  Depending on the individual and episode in question, this may include showing particular clips from the original ethnographic observation and asking the mathematician to comment on his or her experience and/or on our interpretation.

Example of Microethnography

The following images come from some pilot data we collected and analyzed.  The pair here is discussing a problem in combinatorics they’re working on together involving separating three cases they’re having difficulty distinguishing.  The advisor (on the left) is trying to develop a strategy based on modifying work they had done previously.  The doctoral student (on the right) is trying to approach the problem by rearranging the way they count the objects they’re interested in.  The missing image in panel 4 is omitted because the advisor did not change position as the student did.  Bolding in the transcript indicates correspondence with the image (with the first bold in panel 5 being for the advisor’s movement and the second being for the student’s), and italicization indicates verbal emphasis.

Panel 1

Panel 2

Panel 3

Panel 4

Panel 5

Panel 6


Bernard, R. (1988).  Research Methods in Cultural Anthropology.  Beverly Hills, CA: Sage Publications.

Burton, L. (2004).  Mathematicians as Enquirers: Learning About Learning Mathematics.  Dordrecht: Kluwer.

Devlin, K. (2000).  The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip.  London: Basic Books.

Gallagher, S. & Zahavi, D. (2008).  The Phenomenological Mind.  New York: Routledge.

Hadamard, J. (1949).  The Psychology of Invention in the Mathematical Field.  Princeton.

Hiebert, J. & Grouws, D. (2007).  The effects of classroom mathematics teaching on students’ learning.  In F. K. Lester Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning, chapter 9, (pp. 371–404). Reston, VA: NCTM.

Hutchins, E. (1999).  Cognition in the Wild.  Cambridge, MA: MIT Press.

Merleau-Ponty, M. (1962).  Phenomenology of Perception.  London: Routledge.

Roth, W-M. & Thom, J. (2009).Bodily experience and mathematical conceptions: from classical views to a phenomenological reconceptualization.  Educational Studies in Mathematics, 70(2), 175-189.

Scheler, M. (2007).  The Nature of Sympathy.  London: Transaction Publishers.

Thurston, W. (1994).  On proof and progress in mathematics.  Bulletin of the American Mathematical Society, 30(2), 161-177.