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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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