Challenges of particularity and generality in depicting and discussing teaching

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For the Learning of Mathematics 31, 1 (March, 2011)
FLM Publishing Association, Edmonton, Alberta, Canada
This essay focuses on representing and discussing the
ephemeral, time-bound activity of teaching which, for the
sake of this essay, we define as the interaction that happens
in classrooms among teachers, students, and the subject mat-
ter that is taught [1, 2]. How might one depict the activity of
teaching so that the teaching can be discussed among prac-
titioners across differences of context (e.g., school, students,
curricula, …)? To reflect on the deep challenges of such
depiction and discussion, we consider non-fictional video-
tapes of actual classroom interaction and fictional
animations depicting scenes from classrooms. Such repre-
sentations of practice raise questions about the degree to
which practitioners in different circumstances can discuss
their circumstances by projecting them onto a common text.
To understand discussions of teaching mediated through
these two sorts of depictions of classroom interaction, we
look to mathematics where much effort is devoted to repre-
senting abstract aspects of human experience [3] in the
service of making general claims. We examine representa-
tional systems mathematics has developed for representing
the particular and the general when discussing the objects
of Euclidean geometry and of algebra. These representa-
tional systems were once cutting edge technologies in
mathematics that are now taken for granted. Over time,
norms for talking about these representational systems have
been developed. And, since they are taught in school, math-
ematics educators have had the chance to see how novices to
these representational systems have difficulty learning these
norms. Reference to these representational systems will aid
us in contemplating discussion of teaching around depic-
tions of classroom interaction.
Videotape of real classroom interaction to
depict teaching and support discussions
about teaching
Video and film of actual classrooms have long been used to
represent teaching (consider, for example, the films that
Robert Davis produced in the 60’s; Davis, 1964). Much cur-
rent professional development is based on videotapes of
classroom interaction (see Seago, Mumme & Branca, 2004).
Video records of practice are arguably a means to bring
novice teachers in contact with practice (Lampert & Ball,
1998) and can stimulate reflection by experienced practi-
tioners as well. Van Es and Sherin (2008) note:
Video has been used for decades in teacher learning
and it appears to show promise in supporting teachers
in learning to notice. Video is able to capture much of
the richness of classroom interactions, and it can be
used in contexts that allow teachers time to reflect on
these interactions. (p. 244)
Video has been used to showcase and transmit desirable
techniques and strategies (e.g., Burns, 1996, and the related
Mathematics with Manipulatives videotapes [4]) as well as
to record and research intact instruction (e.g., the videos col-
lected by TIMSS; Stigler & Hiebert, 1997). Video has also
been used to disseminate the findings of comparative studies
of instruction (e.g., Stigler & Hiebert, 1997, and the related
TIMSS videotapes [5]).
Video is an appealing technology with which to repre-
sent teaching, possibly because it combines some of the
immediacy of the experience with the opportunity to relive
the experience multiple times. The capacity to identify spe-
cific moments and replay them makes it possible to share the
same visual and auditory experience repeatedly, though the
meaning that different constituencies may make of it can
vary (see Jacobs & Morita, 2002). As opposed to narrative
cases, video allows one to point to an example rather than
have to rely on language for describing it (see Chazan et
al., 1998, for a related point). To the extent that our language
for describing teaching is still under-developed, one cannot
underestimate these advantages of video technologies: if our
goal is to achieve some deep understanding of teaching
through the consideration of the particulars of a teaching
instance, making use of a video record of that instance is a
reasonable choice. Still, while widely used and extremely
useful, video records have their limitations.
Video as the diagram in a two-column proof
To flesh out the limitations of video, we draw on an anal-
ogy with the diagrams that accompany a proof in the two–
column proof format (see Herbst, 2002, for the development
of this practice over time), as well as challenges students
sometimes experience when interacting with two–column
proofs. Netz (1999) argues that Greek mathematicians
developed norms for using letters to allow diagrams to com-
municate the way in which geometric arguments were
constructed. But, this practice has limitations. Regardless
of whether geometric objects are sketched or constructed,
the resulting sketches or diagrams are always particular;
they have particular angle measurements, particular lengths,
CHALLENGES OF PARTICULARITY
AND GENERALITY IN DEPICTING AND
DISCUSSING TEACHING
DANIEL CHAZAN, PATRICIO HERBST

and particular areas; any diagram has specific properties
(Laborde, 2005, refers to them as spatio-graphical proper-
ties) that go beyond the properties necessitated by the figure
or geometric object that the diagram purports to represent.
Nineteenth century French textbook writers became con-
cerned about the intrusion of particularity into a discourse of
generality and therefore banned geometric diagrams from
Euclidean geometry texts (Haggarty & Pepin, 2002).
In US mathematics education, as Herbst (2002) outlines,
developments in the ways of representing geometric objects
took place around the turn of the 20th century, as mathemat-
ics educators faced the challenge of helping a larger
percentage of the high school age cohort to learn to produce
mathematical proofs in geometry classrooms. Cognizant of
both the strengths of diagrams and their limitations, these
educators developed a format for the communication of
deductions regarding general if-then statements in Euclidean
geometry. Following in the Greek, rather than French, tradi-
tion, in the two–column proof format, the “givens” and “to
prove” are accompanied by a lettered diagram. As a particular
representative of the class of figures being discussed (the geo-
metrical object), the diagram creates a register that allows
for the discussion of the general through the particular
(requiring of the reader what Mason, 1989, calls a delicate
shift of attention). The two–column proof states the “givens”
and “to prove” (the characteristics that all figures in this class
would be shown to have) in relationship to a particular dia-
gram. Subsequent numbered statements about the general
class of figures described by the “givens” are articulated with
reference to the lettering in that diagram. Each of these num-
bered statements is supported by a previously proved
theorem, an axiom, a definition, or the givens. These “rea-
sons” justify that, given what is already known about this
class of figures, the content of the statement must also be true.
Thus, if given three non-collinear points in the plane and
the construction outlined by the givens and diagram, the
argument in Figure 1 is an attempt to prove that in all such
arrangements the original median divides the triangle into
two triangles of equal area (in the more specific, diagram-
matic register, that the areas of triangles ADB and ADC in
the diagram in Figure 1 are equal).
Yet, research on students’ understandings of this format as
it appears in their instruction indicates that the goals of
instruction are often unclear to students (e.g., Fischbein,
1982); for example, for some students the two column proof
format does not convey the intended generality. Chazan
(1993) demonstrates that because there are no explicit mark-
ers of generality in a two-column proof, some students in his
study felt that they had proven a result for the particular tri-
angle pictured in the diagram accompanying the proof, the
triangle with these angles and these lengths. In order to
establish this result for a new triangle with its medians
drawn in, particularly one that was visually quite different,
they would need to repeat the steps of the proof. Thus, the
norms developed by mathematicians and mathematics edu-
cators are not transparent to students. Students cannot
automatically see through the particular to the general.
Rather, in their geometry classrooms, they are being encul-
turated into a way of gaining the affordances of the
particular to discuss the general, while minimizing the limi-
tations that the particular has for discussing the general (see
Chazan, 1990, for the work of a group of teachers on helping
students with issues of particularity and generality exempli-
fied with the construction implicit in Figure 1).
Learning from the analogy
Videotapes of actual classroom teaching practice, while
much more complex than the diagrams in a two–column
proof, are analogous to the diagrams of the two–column
proof format in geometry in the way they can enable access
to the general through a particular. A video record from a
particular class can play the same role in a discussion about
teaching that a particular diagram plays in a discussion about
a class of figures. For whomever uses the video to make a
point, a video record may stand as a sign for a general class
of events in the same way that the diagram may stand as a
representation of theoretical properties for the geometer.
But the particularity of the video can be a source of chal-
lenges, in addition to being a scaffold between a discourse of
generality about teaching and the particular instances captures
in a clip; viewers of the video, like the geometry students
interviewed by Chazan (1993), may instead see it as a depic-
tion of particular events. Thus, a video record of a classroom
event may be meant as an illustration of a teaching strategy;
yet the viewer may have difficulty teasing apart the strategy
from other, inessential details of the classroom events in
which the strategy was embedded. A similar challenge exists
when using records to communicate how teaching is done in
a particular culture: the viewer can have difficulty separating
what is characteristic of the culture and what is incidental to
the episodes recorded. These challenges also exist for teacher
educators who broker conversations about teaching: to
explore the boundaries between the typical and the atypical in
teaching, facilitators may be challenged in distinguishing
between what appears in an instance of practice and what the
instance is meant to be a case of.
The analogy we are pursuing breaks down when it comes
to the register for expressing generality; educators who use
video do not have a register of generality analogous to the
one available to geometers when they articulate a theorem
without recourse to a diagram; nor do they have a mecha-
nism to translate this generality into a specification of the
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Figure 1. A two-column proof.

givens and the to prove, which in geometry set the granu-
larity with which one is to look at the diagram. Such
coordination of the particular and the general in discussing
teaching does exist, but is made possible through the artistry
of facilitation rather than through shared conventional
practices. In the absence of such conventions, some conver-
sations about teaching are hard to sustain and benefit from:
they may lose their focus on target facets of teaching and
instead jump from one of the multitude of incidents and
peculiarities shown in the video record to another, giving a
sense of excessive richness in the video.
In terms of particularity and generality, sometimes video
records are perceived as too particular and that prevents dis-
cussion: viewers may want to discuss the teaching of lower
track secondary mathematics and feel that a video that
shows the teaching of fifth grade mathematics has nothing to
say to them, even if the facilitator meant it as illustrating a
general issue about teaching. On the other hand, sometimes
video records do not seem to include enough particular
information; they do not have what people seem to need in
order to engage in a conversation about teaching: for exam-
ple, they may lack information about what happened before
and after the video clip.
In general, it seems that some of the same characteristics
that recommend video as a useful tool for supporting con-
versations about teaching are also the sources of challenge.
The richness and particularity of a video are what allow peo-
ple to feel that they are in the presence of a lesson, yet the
particularity of video can be such that it does not allow
viewers to project their circumstances onto the provided rep-
resentation of teaching and instead to focus too much on
the circumstances of a different teacher in some other place.
Depicting with the unreal rather than the real
To return for the moment to the issue of representation of
particularity and generality in geometry, the advent of com-
puter technology brought a new wrinkle, the capturing of
geometrical procedures by the computer and the capacity
quickly to create multiple diagrams from a single procedure.
Thus, in a software application that predated the computer
mouse, one could do a construction on three particular start-
ing points and then repeat this construction starting with
other points (Schwartz & Yerushalmy, 1987). Or, in current
software applications (as described by Goldenberg & Cuoco,
1998), one can create a diagram according to a construction
and then drag base points to create new instances. The speed
of the recreation of the images creates the sensation that one
has a diagram with stretchy segments that can be dragged
about, rather than a collection of diagrams.
While the static images that one sees on the computer
screen are still particular, the ease with which they are
changed supports the asking of all sorts of “what if” ques-
tions. What if we drag the points and make all three points
collinear, what happens? Building on the earlier analogy
between diagrams and video, the question is: what sorts of
representations of teaching might be particular, but easily
changed; what sorts of representations of teaching might
support the asking of “what if” questions. What would be the
equivalent of stretchy geometrical lengths in a representa-
tion of teaching?
In the Thought Experiments in Mathematics Teaching
(ThEMaT) Project, we have explored two-dimensional, ani-
mated depictions of classroom interactions [6]. While, as
will be explicated later, these animations are influenced by
an idea of creating sketches of classroom interaction, in
making these animations, we chose consciously to repre-
sent classrooms with characters that are patently unreal. An
analogy with the use of literal symbols in algebra will help
provide a rationale for this choice and will bring out other
aspects of these depictions of classroom interaction.
An alternative mathematical strategy: gener-
alizing with x
In mathematics, abstractions tend to be built out of other
abstractions (using a process that Sfard and others call “reifi-
cation”, e.g., Sfard and Linchevski, 1994). For example,
people have abstracted counting numbers as a characteristic
of collections of objects (Frege, 1884/1980). Once this aspect
of experience was identified it was represented in tallies,
using alphabets, and with the Hindu-Arabic numerals.
But, numbers, as characteristics of collections, were not just
represented; with arithmetic, they were also acted upon and
used to calculate. In what some (e.g., Cajori, 1919) call rhetor-
ical algebra, unknown quantities are represented with words
and diagrams. But, such representations had their limits. Start-
ing in the 17th century, the x’s and y’s that we are familiar
with from school algebra were used to represent particular
unknown numbers, representatives of a class of numbers, or
the totality of the class all at once. While the ambiguity of
this notational system is one of the key challenges in learn-
ing school algebra (as Yerushalmy & Chazan, 2002, argue),
the power of this notation was soon evident. One can write
equations involving numbers, but with a letter representing
one of the numbers, and then reason about the letter as if it is
a number. Viète (1983) noted that this strategy allows a math-
ematician to work analytically (as opposed to synthetically,
in the Greek oppositional sense of these terms): one can
assume that an equation has an answer, name that answer x,
reason assuming that x represents an as–of–yet unknown
number, and then in the end unmask the numerical identity
of x. If one’s assumption that there was an answer was incor-
rect, then one’s reasoning will lead to a contradiction.
A key aspect of this notational advance is to use one kind
of thing to stand for another, to use a letter to stand for a
number (in this use, the letter is sometimes referred to as a
literal number). Precisely, because a letter does not usually
denote a number, it has few of the characteristics associated
with numbers. One cannot look at it and determine if it is
even or odd. It does not telegraph whether or not it is prime.
Alternatively, because the letter is not a number, it can be
conceived of as representing any possible number, or even
all numbers at once. While mathematics educators have
developed clever ways to have young students see through
the specificity of numbers to the role that they play in arith-
metical equations (e.g., Fujii, 2003), the affordance of
literals (x’s and y’s), as opposed to numbers or geometrical
diagrams, is that they aid in seeing generality without having
to ignore some particularity.
Of course, use of this technology also requires encultura-
tion. Algebra students are known to wonder what the x’s and
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y’s are all about (see Usiskin, 1995, for a response). As we
have suggested earlier, some of this wondering is due to the
different uses to which these symbols are put (see, for exam-
ple, Usiskin, 1988). And, then there are important notational
ambiguities to address: for example, to learn to use this tech-
nology students must learn that 2n does not imply a number
in the twenties, but rather a number multiplied by 2; that n
is not representing a digit, but rather a number, however
many digits it may take to write that number.
Learning from this metaphor
As with the earlier mention of geometrical constructions that
can be “dragged” with a mouse, the technology of literal
symbols in algebra raises questions about depiction of teach-
ing. For the purpose of depicting teaching, is there an
analogue to the literal? Is there some way of depicting inter-
action among teachers and students that would suggest
either an unknown particular or a more general class? Would
there be discussions about teaching for which such a depic-
tion of teaching would be useful?
ThEMaT has created animations peopled by cartoon char-
acters. In an analogue to the literal number, inspired by
McCloud (1994), our animations are built around non-
descript cartoon characters. These characters are clearly not
people, though the interactions between them model some
aspects to be found in classrooms. In the same way that
mathematical norms ask us to treat literals by the rules gov-
erning numbers, we ask participants in our study groups to
discuss these characters as if they were teachers and stu-
dents. While our teacher characters do not have all the
characteristics that a particular teacher has (e.g., they do
not have hair, knees, etc.), by virtue of the role assigned to
the cartoon characters in the talk depicted in the animation,
they have some key characteristics, and a kind of indeter-
minacy, that potentially allow a wide range of teachers to
identify with them as teachers.
One important side benefit of such indeterminacy is that
one is no longer watching the classroom of some particular,
real teacher. Instead, one is watching the classroom of a fic-
tional teacher. And, alternative scripts can be represented
by the same set of characters without one alternative having
a privileged status over others. In the same way that x can be
an even number in one problem and an odd number in
another, a teacher character can be a “traditional” teacher in
one alternative enactment of a story and a “reform-based”
teacher in another. Finally, given the difficulties often noted
around critiquing instances of actual practice represented in
video, one can criticize the actions of this teacher character,
and implicitly of the person who created the animation,
without criticizing the teaching of any particular individual.
Thus, in the animations we have produced, one thing is
substituting for another; a cartoon character is representing a
real person. At the same time, they have the feeling of a
sketch. Some essence is being conveyed, without much
specificity about other aspects.
It is important to note, however, that in creating this sort
of indeterminacy, as designers of the representation, we are
not claiming that the aspects of teaching we represent specif-
ically in the animation are the important ones, and therefore
are to be represented and to be discussed, and that other
aspects of the interaction are not important, and therefore are
either left vague or not represented and should not be dis-
cussed. Instead, we are interested in identifying the
characteristics that hold together the class of events that we
seek to discuss and make these characteristics specific, while
leaving other characteristics of the interaction indeterminate.
To return to the analogy with diagrams in a two-column
proof, we seek to make the givens salient, without providing
too much additional specificity (in the same way that the
diagram may also convey some information that goes
beyond what is in the “givens”). This sketchiness allows par-
ticipants in a conversation to project their own
circumstances onto the parts of the interaction that we have
left vague or have omitted.
Unlike the conventions around the discussion of geomet-
ric figures, conversations around an animation might thus
productively focus on aspects of the classroom interaction
that are left vague in the animation, or even omitted, not
only those that are represented specifically. In addition, par-
ticipants in the conversation might discuss whether
something we have represented is a part of the “givens,” or
is a corollary of the “givens,” or whether it is an artifactual,
particular aspect of the interaction that could have been
changed, or even left unrepresented.
Conclusion
In recent years, video recordings of classroom interaction
have been a mainstay of our repertoire for supporting con-
versations about teaching. At the heart of how videotape
supports conversations about teaching is the particularity of
this representation and the multitude of characteristics it pre-
sents the viewer for interpretation. Video seems especially
well suited to conversations in which teachers or prospective
teachers are being taught to use evidence of the sort that
videotapes can capture to make arguments about student
thinking, for example.
In this essay, we have suggested that the very strengths
of video for supporting some conversations about teaching
may be drawbacks in allowing other conversations. Because
of its specificity, teachers may feel that a video does not
allow them to project their own circumstances onto an inter-
action and thus rules out possibilities that might be
important for conversation. Stimulated by analogies with
diagrams and literal symbols as tools for representing and
discussing general claims in mathematics, we have argued
for projective representations of teaching, representations
which viewers can use to share their own perspectives.
Notes
1. The research reported in this article is supported by NSF grant ESI-
0353285, to the authors. Opinions expressed here are the sole responsibility
of the authors and do not reflect the views of the Foundation.
2. Of course, by doing so, we do not intend to suggest that classroom inter-
action represents all of the work of teaching.
3. Though, of course, there are important disagreements among philoso-
phers of mathematics about the relationship between mathematics and
human experience.
4. http://www.mathsolutions.com.
5. http://nces.ed.gov/timss/video.asp.
6. For a short excerpt see http://www.youtube.com/watch?v=
lyTNP3IXwqk, more of these animations can be seen in ThEMaT Online,
http://grip.umich.edu/themat.
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