An Other
Mathematics: Object Relations and the Clinical Interview
Journal of Curriculum Theorizing 14, 2: 35-42, 1998.
Peter Appelbaum, Arcadia University & Rochelle Kaplan,
The William Paterson University of New Jersey
Pre-publication draft -- check the published version for accurate presentation.
appelbaum@arcadia.edu
kaplanr@wpunj.edu
As our unconscious ego processes are released into objects chosen for the dream to evoke a dreaming self by object choice, and as those objects are changed in the encounter, so too in the waking dream might we choose our objects based on unconscious ego processes and object relations so that a self is evoked. From that encounter, subjectivity may develop. I would that curriculum be understood in this fashion; then what an education that would be! (Block 1997: 34)
The link between psychoanalytic theory and mathematics may not be immediately
apparent, but this is perhaps only because mathematics is viewed as an academic
subject field rather than as the object of a highly interactive and affective
relationship. In fact, the quality of this relationship between the self and
mathematics as its object is at the heart of the educational experience of both
pupils and their teachers. Yet this relationship is ignored in the typical
pedagogical interaction of the classroom just as it is undertheorized in curriculum
development and policy debates. Despite some earlier interest in psychoanalytic
approaches to mathematics education (Winnicott 1986, Early 1992,
Blanchard-Laville 1992) current discourse on mathematics and curriculum
continues to circumvent issues of self and its relations with objects,
Mathematics as an object of self rather than merely an academic discipline
or a content to be learned, is expressed more specifically by Stephen Brown
(1984, 1993) who envisions the potential for mathematics to be redefined as an
object through which people can therapeutically reclaim the sense of oneself as
a moral acting being. For Brown, mathematics is transformed from a technique
that links means and ends (a tool for "solving it") into an activity
through which one understands oneself and mathematics in new ways. In this
view, the standard pole established by skill drill versus meaningful conceptual
knowledge is reframed as a persistent bypassing of activity that incorporates
abstractions "out there in such a way that we can begin to gain power over
it and feel that we possess it in some important sense " (Brown, 1993).
Later Brown goes on to say that "If we persist in by-passing this
activity, we desensitize ourselves to the point that we no longer taste the
uniqueness among the phenomena, and though [students] may be able to gain
answers to questions, they become very much insensitive to what it means for
something to be a problem and have even less of an understanding of what it
means to have solved something " (Brown 1993 p. 271). This insensitivity
can be seen as a symptom of a "splitting crisis" or the "split
off mind" referred to by D.W. Winnicott who earlier framed the issue of
mathematical understanding in terms of object relations.
In a talk for teachers in 1968, "Sum I Am,", Winnicott speculated
on the interesting link between the establishment of "the unit self,"
a personal sense of unity or oneness emerging through the infantile
relationship between child and primary caregiver, and the development of the
mathematical concept of "one" or the "unit." In his talk he
noted three types of object relations with mathematics that are characterized
by the extent to which the child or adult has come to experience a sense of
personal unity ("I am"). One type is referred to as not having
achieved unit status or an integrated sense of self that can be distinguished
as "me" and "not me." For this type the concept of
"one" means nothing and we can infer from Winnicott's brief sketch
that this type does not make much progress in mathematical tasks. Another type
may also have failed to develop a personal sense of oneness, yet forges ahead
to manipulate mathematical concepts despite being limited by trivial
considerations of the unit concept. This type may engage in higher mathematics
procedures, yet remain disconnected from understanding basic unit concepts.
Such a condition is fairly common and typifies the person who has not achieved
unit status because as a child the environment required an application of
intellect "too early." This person may function brilliantly without
reference to the human being, but develops a false self in terms of living with
a split-off mind "so that while higher mathematics gets a boost, the child
fails to know what to do with one penny." (Winnicott 1986, p. 59) The
third type described by Winnicott, however, relates to mathematics with an easy
conception of "oneness." This is because the individual here has a
sense of personal unity that has been derived from experience with the "good-enough"
behavior of a mother-figure. This child's feeling of "I am," is then
available to be invested in a wider concept of wholeness and the building of
personally relevant object relationships with mathematics.
If we examine the child's object relations with mathematics in terms of the
educational encounter of teacher and student, we find that the resulting
mathematical conceptions are really a function of the negotiation between the
participants' personal relationships with mathematics, perhaps originating in their
concepts of unity. Framed in this way as a truly psychoanalytical encounter,
mathematical knowledge then includes a meta-knowledge of how one
"does" mathematics as well as how one establishes relationships with
various objects of mathematics. These relationships to mathematics and one's
understanding of how this influences the mathematical conclusions that are
drawn become important considerations in and out of school (Appelbaum 1995).
Current school reform approaches, particularly as described in the National
Council of Teachers of Mathematics Assessment Standards for School
Mathematics (1995), provide an opportunity to examine the true self and the
split off mathematical self by supporting teachers' efforts to introduce
non-test-based forms of assessment into their classrooms. The underlying
constructivist tone of this thrust encourages teachers to understand how their
students "are thinking" as a legitimate component of their classroom
practices. Such approaches allow a teacher to provide an environment in which
children can recreate for themselves mathematical objects that can then be used
in intentional ways. As Winnicott (1986) points out in discussing the teacher's
role as a psychotherapist, "Teachers of all kinds do need to know that they
are concerned not with teaching their subject, but with.....completing
uncompleted tasks that represent parental failure or relative failure."
(p. 63) This implies that although students come to the educational forum with
their own set of relationships to the objects of mathematics, these
relationships are subject to modification and interpretation by teachers who
bring their own set of mathematical object relations to the interaction. At the
same time, the teacher is in the position to "catch on to the creative
impulse," and use this and the child's reaching out to provide a stable
environment enabling some degree of personal integration to take place in the
child (p. 64).
Brent Davis (1996) notes, "...the listening teacher works with the
contingencies of the particular classroom setting. It is founded on the
realizations that no learning outcome can be prescribed, no active setting can
be controlled. But neither must we forego attempts to influence (or fail to
acknowledge our influence upon) what might come about. The key to teaching, in
this conception, is to present a space for action and then to participate in
and through this participation, to shape the joint project that emerges"
(p. 271). In our view, an ideal opportunity to understand how teachers utilize
their "space for action" during teacher-pupil interactions and of the
resulting joint project can be found in the clinical interview. For us, the
clinical interview is a narrowly focused way of viewing teachers' object
relations with mathematics as they would occur in the classroom and how they
impact on children's object relations with mathematics. In fact, we claim that
the clinical interview represents "a little piece of pedagogical
practice" that reveals as much about the teacher's attachment to mathematics
as it reveals something about what the child knows or thinks. It is the
teacher's structuring of the interview that binds and limits - or enables in
particular ways - children's space for action and participation in the
interview process. Moreover, the clinical interview as described in this essay,
presents us with a process for examining psychoanalytic approaches to notions
of self, authority, curriculum, and teaching/learning practices.
Through the vehicle of the clinical interview, this essay describes a
framework for discussing how the self encounters mathematics as an object in
the world and by conscious and unconscious processes transforms the object into
something that exists internally in a structure that may bear little
resemblance to its external content. In object relations, certain alterations
in self take place, of a kind that Winnicott once called "cathexis"
(See Winnicott, 1971, 1996). The object has become meaningful; what has been
internalized, however, is not the object per se but the process through which
the child has internalized the relationship with the object.
Our model and examples are gleaned from several years of experience in
introducing clinical interviewing to pre-service and advanced elementary
mathematics teachers (Appelbaum, 1998; Kaplan & Harris, 1991). This work
included analyses of teachers' reactions to short videotape vignettes of
interviews with single children (Ginsburg, Kaplan, & Baroody, 1992; Kaplan,
1994). All of these interviews afforded us the opportunity to enter both the
children's conceptions of what mathematics means to them and to understand the
interviewers' own relationships with mathematics. From both perspectives the
language of object relations helps us think about the interview as an artifact
of developing notions of mathematics "as an object" through which and
with which the self is constructed.
For a parallel perspective drawing on the work of Lacan in comparing the
conversations of children in home and school, we draw the reader to the seminal
work of Valerie Walkerdine (1988) who persuasively links parenting and school
mathematics practices with the production of rationality and "mathematical
reason." The fantasy of control turns out to be essentially inculcated at
an early age as part of a necessary component of a social order that requires
"reasonable" people in order to govern them. For Jungian and
Balintian approaches, the work of Early (1992) and Balnchard-Laville (1992) are
good introductions. We note that in terms of the child's spontaneous choice of
objects and use of objects, it is possible to consider the interactions of
emotion and the child's sense of quantitative and spatial concepts with the
child's attitude toward itself as a mathematical being. Object relations helps us
to work through the child and teacher as active agents in the processes of
social control and social change. All of this is crucial grounding for
curriculum theorizing.
LISTENING
One particular feature of a clinical interview raised in the mathematics
education literature is the importance of "listening," as in the work
of Julian Weisglass (1990, 1994) and Brent Davis (1996, 1997). For these
authors, listening is a form of "embodied action" as opposed to a
technique of hearing. Weisglass presents a taxonomy of listening forms that he
designates as partially pedagogic; his alternative, dubbed
"constructivist," encourages the talker to reflect on the meaning of
events and ideas, to express and work through feelings to construct new meanings,
and to make decisions. Davis similarly constructs three comparative modes of
listening differentiated by their features of attending to the one listened to;
beyond evaluative and interpretive listening one finds "hermeneutic
listening," which requires a teacher to reach out rather than take in. In
hermeneutic listening, listening becomes the development of compassion,
increasing the capacity of the listener to be aware of and responsive to the
one "listened to;" participants are involved in a project of
interrogating taken-for-granted assumptions and prejudices that frame
perceptions and actions. For clinical interviewing, such a conception of
listening emphasizes the importance of one's own structural dynamic in the
evolution of outcomes in interaction with another person, as opposed to
functional responses to the other person's actions (as in transmission models
of communication and teaching). Interaction is not "instruction" --
its effects are not determined by the interaction; rather, changes result from
the interaction, determined by the structure of the disturbed system (Davis,
1996). Constructivist or hermeneutic listening promotes participation in the
unfolding of possibilities through collective action.
Thus, clinical interviewing by teachers in the classroom requires
substantial reframing of the teaching/learning encounter. On a surface level,
the interview is difficult for both the teacher and the student who are
unaccustomed to relating "therapeutically." Teachers want to teach
and correct mistakes. Students want to be told if they are correct and if not,
how to make themselves correct. The clinical interview, on the other hand,
presupposes an open-ended, accepting attitude toward whatever surfaces. Its
purpose is not to instruct, but to reveal more than either the interviewer or
the interviewee knew about the object of mathematics before the interview. It
looks deeply into the student's thinking and at the same time tells the
interviewer how what he or she projects about mathematics affects what the child
communicates about what has been learned.
As in the psychoanalytic process, the interview does not define a clear and
objective reality. Rather, just as the psychoanalyst must examine his or her
own fears and motivations in the context of the process, so too must a clinical
interviewer carry out this task. Therefore, the metacognitive act of a teacher
creating his or her own interviewing behavior evokes an encounter with one's
own motivations and fears regarding mathematics, children, learning, and teaching.
It stirs up issues of one's level of confidence in doing mathematics, what the
teacher thinks of his or her own relationship with mathematics, and the extent
to which mathematics is seen as egosyntonic with one's own self image apart
from being a teacher. The discrepancy between a conscious view of oneself as a
mathematical being and the "deeper" unconscious or subconscious
feelings that drive behavior is a potentially volatile realm of
self-confrontation.
Over the course of time during which we have studied the clinical interview
process, several interesting patterns revealing teachers' and students'
relationships to mathematics have emerged. We note that during the clinical
interview, the teacher's questions, responses, his or her every reaction to the
child as well as his or her initial selection of an interview task reveal as
much about the interviewer's motivations and object relations as they do about
the child's. In essence, what we have found is that each level of object
attachment that children display toward mathematics is paralleled by a
comparable level of active engagement or lack of active engagement with
mathematics manifested by teachers. Although these parallel levels do not
necessarily imply cause and effect, it is evident to us that teachers certainly
filter children's attachments through their own unconscious investments in
mathematics. These investments are defined by the extent to which teachers
regard mathematics as essentially egosyntonic or egodystonic (i.e., as part or
not as part of a self concept). Observations of these interviews suggest that
teachers selectively listen to those elements of a child's encounter with the
world of mathematics that can be recognized and acknowledged as consistent with
their own perceptions and feelings about mathematics. As pointed out in an
earlier article, "...the constructivist program is filtered through an
incompatible lens and what comes out is a distorted version of some seemingly
objective curriculum....On a deeper level,...her originally stated belief...is
embedded in a system of other beliefs that defines who [a teacher] is and
colors her perception of reality...The way in which she communicates to the
student(s) defines her real instructional goal" (Kaplan, 1991, p 16-17).
MATHEMATICS AS OBJECTS
Children of course do not come to the interview without their own emerging
relationships with mathematics (Kaplan, 1987). In fact, these relationships are
quite obvious to the observer who looks beyond the particular content of responses
and focuses on the intention of those responses. While Winnicott noted that
mathematical ideas begin with the concept of one and that this concept derives
"in every developing child from the unit self," (Winnicott, 1986, p.
58), it is also true that early object relations involving mathematics come
from children's informal senses of quantitative, logical, and spatial qualities
attributed to objects in the environment (Piaget 1952). At the early preschool
stage, at about ages 2-5, the child sees mathematics as an object that belongs
to the self. All mathematical observations are seen as egosyntonic, a part of
the self system. Similar to other objects in the world, mathematics
relationships originate in spontaneous concepts rather than imposed concepts from
others (Vygotsky 1986). The child at this time can be said, in a sense, to be
"in love" with mathematics. Number, space and pattern are playful
things, things to do something with just to see what happens. It is also a very
personal relationship, transforming objective realities to fit the needs and
sensibilities of the learner.
Mathematics as a Child's Personally Relevant Object
For example, when 3-year old Sal (Ginsburg, et al, 1992) listens to the
interviewer's counting to determine if any mistakes are being made, he can say
with complete glee, "You made a mistake!" when the interviewer
recites "1, 2, 3, 10." Or he can say, "Another mistake. You have
to start with one!" when the interviewer starts counting with "9,
10," instead of 1,2,3. Finally, when the interviewer counts from 1-15
followed by, " 17, 19, 30, 31, 32, 100, 200 and 46," the child
responds with a smile and tells the interviewer, "That's good." These
are all signs of Sal's ownership of the mathematics. The numbers he knows are
part of his self concept and may even define that concept to some extent.
Similarly, we have the example of Bruce who stops after counting from 1-29
because he does not know what comes next. Yet when pressed by the interviewer
to take a guess, he responds, "After 29 comes 10." Subsequently, when
the interviewer counts aloud from 1-30, Ben first stops her at 25 to exclaim,
"You're copying me!" and then when the interview gets to 30, she is
told by Bruce that "No, after 9 comes 10." This child, too, indicates
complete ownership of the mathematics and a sense that he has invented it and
can tell others how it works.
A final example of the preschool child's personal connection to mathematics
comes from a 4-year old, Stacey, who is asked to compare the amounts of clay in
two balls of the substance. After agreeing that the balls have the same amount,
the interviewer breaks one of the balls into small pieces, leaving the other
intact. When asked if there was still the same amount of clay in the two
samples, the child proceeds to count the little pieces accurately to nine and
then points and "counts" the ball of clay as "1, 2, 3, 4."
She then concludes that the little pieces "are more because I have
nine." We see here that the mathematics used by the child is that of an
object internalized to conform to the needs of the learner. Her intuitive
mathematical self tells her that the little pieces look like more than the
ball. Still she wants to justify that impression and so creates a counting
system that works for pieces and wholes and uses her sense of number relations
to support her judgment. This child is deeply wedded to her brand of
mathematics. Stacey takes it with her and applies it as needed within her
personal standards of consistency and as mediated by her senses.
A Child's Attachment to Mathematical Rules
Will this personal attachment and self definition of mathematics endure? Not
likely. This is because the preschool mathematics that is internalized is not the
mathematics the child will be finding later in school. That mathematics has
rules and standards about how it can be interpreted and what it means. No
matter how accepting a teacher may try to be about "does anyone have
another way to solve this problem?" or however many multiple approaches
are entertained, the logic of mathematics cannot be violated in schools. Thus,
the child who seriously suggests that the answer to 43 - 17 can be 36 or 24 or
46 depending upon how you do it, will not be allowed to "love" all
those answers without bias. Rather, the child will be encouraged to explain all
the ways the answers were derived and then, at best, be gently guided to accept
one method as superior to the others. Brown's (1983) work provides a stark
contrast in encouraging new interpretations of mathematical operations or
searching for cases in which seemingly absurd procedures actually work out
sensibly; Marion Walter (1996) also suggests such possibilities. Together,
Brown and Walter (1983, 1993) have put forth "problem-posing" as the
context of such "play" for all ages. Nevertheless, the playfulness of
the early childhood period is not part of school mathematics as otherwise
conceived, and so the object of mathematics often becomes removed from the
self. The self does not internalize the rationale imposed, but just comes to
accept the fact that some ways of thinking are considered acceptable and some
are not. Depending upon the child's general nature and capacity to remember
external facts, this kind of learning about mathematics may proceed toward a
successful performance record or to one of increasing failure and misconception
as defined by "the rules of mathematics." In school some children
learn that they can now treat mathematics as some object out there, which has
nothing to do with me and which makes no sense. Conversely, a more positive
outcome might be that the child still treats mathematics as some object out
there, which is not like that mathematics which is mine and that I love, but
which I can and must master to survive in school.
In a very real sense, school forms the child in the denial of imagination
(Block, 1997), in splitting the subject and object and denying the creativity
that makes the self possible. As Block writes, "to deny imagination is to
deny the very creativity that makes self possible; it is to perpetuate the hate
that results from the inescapable discrepancy between subjective and objective,
between the unlimited possibilities of one's dream and what the real world
actually offers us." (Block, 1997, p. 171) In other words, because
schooling is structured by and about boundaries, it denies creativity and makes
us hate ourselves for our thought-dreams; we are taught to split and submerge
our thought-dreams -- fantasies and ideas. The child as a result may, according
to Block, become a "dictatorial egoist" who actively denies the
wishes and needs of the other, and tries to make his or her own wishes alone
determine what happens; he or she may become a "passive egoist,"
retreating from public reality and taking refuge in a world of unexpressed
dreams, becoming remote and inaccessible. Or, writes Block, he or she may
search to avoid conflict altogether, permitting the outside world to become a
dictator, fitting him or herself into that external world and its demands,
doing what others want and betraying his or her own wishes and dreams (p.172).
These outcomes are illustrated by two children who were interviewed during
their elementary school years, one who sees mathematics as an object to be
avoided and the other, one who sees mathematics as a fixed, unchanging object.
Mathematics as a Child's Object to Be Avoided
Jenn (Ginsburg, et al, 1992), represents the kind of blocked approach in a
child for whom mathematics is only an object to be avoided. This third grader
was asked to recall a basic multiplication number fact, 8 x 4, which she had
mistakenly rote memorized as being equal to 35 on one occasion and 28 on
another. When asked how she could figure out which answer was correct, she told
the interviewer, "Ask the teacher." The interviewer then encouraged
her to represent the combination graphically. When the child could not produce
anything on her own, the interviewer suggested that she use tally marks and
demonstrated how to get started. After a long and laborious process in which
Jenn neatly lined up four rows of eight lines, she then proceeded to add the
tallies as though they were "ones," coming up with an answer in the
millions. When probed further about the correctness of her answers, she
indicated that it was definitely not the larger number, but still did not know
how to find out the correct answer. We have here an example of a child whose is
actively engaged, not in the process of connecting to and internalizing some
mathematics, but in the process of creating a barrier between herself and the
object known as mathematics. Her energies are directed toward preventing the
intrusion of mathematical ideas, or the object of mathematics, upon her sense
of self. For this child, mathematics is egodystonic and not something to be
manipulated, personalized, adapted, or cared for. The only emotion evoked by
the object of mathematics is anxiety and the child avoids the mathematics at
all costs in order to diminish the anxiety. The process she uses involves
providing some answer framed in the language of mathematics in order to
"make the teacher leave her alone." Her goal is to rid herself of the
mathematics rather than to internalize it.
Mathematics as a Child's Fixed Object
From the practical perspective of school achievement, a more positive
resolution of the failure to internalize the object of mathematics as part of
the self image is for the child to be successful at rote memorization. In this
case, even though mathematics remains external to the child, the rules of the
system and its objective content are competently manipulated in terms of school
demands. This child is typically an average achiever, sometimes making silly
mistakes because of a faulty memory, but generally able to perform at grade
level in a rote manner. In response to a question about why some procedure is
performed, this child tends to provide an explanation that does not go beyond
reciting rules and describing actions. The relationships within the mathematics
are not dealt with nor is the child able to envision a problem in more than one
way. In reality, this type of child is also math avoiding and actively seeking
ways to prevent becoming attached to mathematics and mathematical
relationships. Rather she is engaged in an interaction with rules, authority,
and appearances. For this child, both relevant and irrelevant variables in a
problem are treated as equally important with as much emphasis placed on form
as on content. Third grader Nancy's (Ginsburg, et al, 1992) interview responses
are typical of this kind of surface engagement.
After demonstrating enthusiasm, speedy and accurate recall of multiplication
facts, prowess in producing essentially accurate answers to a series of mental
arithmetic combinations, and competence with executing the written procedures
for multidigit addition and multiplication procedures, Nancy was asked to solve
a simple one-step word problem. The child read and made these comments about
the problem: "There were 23 people at each bus stop. So there's 23 people.
The bus driver made 3 stops. How many people in all?" When asked what she
would do with a problem like that, after some thought Nancy indicated,
"You go times...23 three times." When asked how she knew how to do
that, the child's response was to restate the facts of the problem before
saying, "You read it. You can't add it because they said there was 23
people at each bus stop. You can't subtract it. Like forget it.....because it
says how many people in all...." When pressed further about why it was not
possible to use addition in some other way to get the same answer as obtained
with multiplication, Nancy responded with, "I don't understand...(now
showing exaggerated puzzlement and disgust in her facial expression)...
(shrugs). I don't actually know....uh, uh....it's too hard."
This snippet from an interview typifies the way in which a child who appears
to grasp school mathematics, but does not really have a relationship with
mathematics, uses standard procedures to automatically apply a known technique
to a familiar context. There is no room here for playfulness with the facts nor
interest in the mathematics in the problem. Rather there is a clearly motivated
urge to provide answers in order to accrue points or credit for being correct
despite, not because of, the mathematics in the situation.
Mathematics as a Child's Flexible Object of Self
At another level, somewhere between the preschooler's complete incorporation
and transformation of mathematics as an object to part of the self and children
who avoid active engagement with the mathematics, we have the child who
develops a clear image of the self as a mathematical thinker, but does not
necessarily make sense of mathematics in conventional ways. This type of child
is actively engaged in an interaction with the mathematical world, forcing
numbers to make sense and creating his or her on rationale for using
mathematics. Whether the conceptions are accurate or inaccurate, this child
will always approach mathematics with a positive outlook and an expectation
that mathematics can be understood. An illustration of this type of child is a
fifth grader, Viola.
Viola was interviewed about her basic conceptions of fractions represented
in standard written notational format. When asked how many fifths were in a
whole, after some serious thought, she questioningly replied, "Five?"
Then she was asked why there were five parts and she responded, "Probably
because 5 and 5 is 10 and 10 is an equal number...and if it would be anything
else, even if it was an equal number like 8, 5 and 3, (smiling now) it doesn't
sound right for some reason. I think 10 sounds better." Subsequently the
child was asked how many eighths were in a whole. This time Viola responded
more quickly and said, "Two?" Then with more confidence she said,
"Because 8 plus 2 is 10...That's what you're trying to get at [the
10]." We can see here thatViola takes a different kind of approach from
Jenn, who worked with the tallies. Viola ponders the questions and appears to
reason them out. Although her responses do not make sense in conventional
mathematics terms, she is engaged in trying to force the mathematical object to
her perspective. She makes the math conform to her sense of reasonableness and
expands upon her judgments by making associations between events as she
grapples with them. For this child, mathematics as an object is part of her
self image, something that belongs to her, and, therefore, egosyntonic. Viola
is a problem solver rather than a number cruncher. For this type of child,
mathematical objects are continually evolving and being adapted as well as
adapting the internalized sense of self as a mathematical thinker.
TEACHERS' OBJECT RELATIONS
Attending to the teachers in these interviews we find that their
relationships with mathematics parallel those of the children. Some teachers
seem to regard mathematics as a collection of facts and prescribed procedurally
accurate answers. For these teachers, mathematics is something of an alien
content or body of knowledge to which they are minimally connected and which,
in general, has nothing to do with definitions of self. Like Jenn, they are
essentially engaged with mathematics as a rote activity, the rules of which
have been determined by others who have a kind of capacity which they lack.
During the clinical interview, such teachers can react in one of two ways.
Mathematics as a Teacher's Object to Be Avoided
On the one hand, some teachers tend to approach the interviewing task with
awe and an openness to accepting almost anything that the child suggests as
indicative of interesting mathematical thinking. For example, we have the
teacher-interviewer who asked a fourth grade student, Thomas, to help her
figure out how many times she would need to fill an empty quart milk container
with water in order to fill up a gallon container. Because the teacher herself
actually did not know how to answer the question, she did not bring any
preconceived notions about how best to approach the problem other than to just
fill the containers. Therefore, she was open to accepting any type of more
mathematical answer that the student could offer. During the course of the
interview as Thomas experimented with different ways of combining linear
measurements of the two containers, the interviewer expressed repeated approval
of his techniques although the efforts did not lead to any clear answer to the
original question. In this case, though, the interviewer's own absence of a
connection to the task allowed her to "discover" along with the child
that they could agree on a variety of interesting observations about volume and
dimensions of containers, as well as the use of rulers. Although none of these
observations actually made much mathematical sense, this did not necessarily
mean that Thomas' level of problem solving for this task proved to be haphazard
and fruitless. Nor did it mean that he was as detached from the mathematics as
the interviewer. Rather it suggested that his engagement was at a different
level from that of the interviewer, a level somewhere between a procedural
rule-based approach and the playful approach of the preschool child who owns
that mathematics. Perhaps, though, the teacher-interviewer's own distance from
mathematics forced Thomas to shift his own level of object relations vis-à-vis
mathematics downward because the interviewer was unable to respond with sharper
more critical questions about his procedural efforts. The question remains open
as to whether the interviewer's lack of mathematical engagement actually
controlled and diminished the level of his object relations with mathematics
during the course of the interview or whether it fostered a more playful
problem solving approach for both her and Thomas.
Mathematics as a Teacher's Fixed Object
One the other hand, as interviewers, the same type of mathematically
disassociated teachers may listen for accurate recall and statements of fact to
questions they pose, questions that for them have only two kinds of answers -
correct or incorrect. They do not go beyond the answer given, but tend to greet
each response the child makes with a new question, one that is predetermined
and not based on the child's response. For example, if such an interviewer
heard Jenn say that 8 x 4 is 28, the follow-up to that response might be,
"and how could you check your answer to see if you are right,"
looking for some kind of procedural explanation. When the child responds with
"do it the other way around," the interviewer would be satisfied that
the child understands the checking procedure and go no further on this task.
Instead he or she would ask the child the answer to a new combination.
To some extent this type of relationship is more a statement of attachment
to the interview process than to the object of mathematics. For this teacher,
the point of the interview (as of work in school generally) is "task
oriented" and a serious business. The teacher is professional about her or
his job, serving the student and the system, under the presumption that
schooling and learning mathematics is the most serious and important event in
students' social lives. This teacher might emphasize accumulation (or
consumption) of skills and knowledge. That is, the more one knows and
accumulates skills and knowledge, the "better" one becomes; continuous
accumulation becomes the end in itself. This teacher might also
compartmentalize time, behavior or tasks, in the sense that she or he manages
the content and ideas of the interview, the tasks, purposes and organization of
ensuing discussion.
Despite the "seriousness" of this approach, there is little
evidence in the experience of an awareness of the worthwhileness of the tasks
imposed. If the seriousness is combined with seeming irrelevance or triviality,
then the nature of school mathematics is debased. In James MacDonald's (1995)
terms it clouds the development of values in the productive activity of the
people involved since all work must be taken seriously whether justified or
not. Participants in an interview characterized by this image become alienated
from their work (the content of the interview) because the pleasure of
worthwhile activity is reduced to satisfaction in the external rewards offered
as a substitute for justifiable standards. Inherent is the technically rational
planning and organization of work tasks and pupil activity which in the
interests of others destroys the spontaneity, creativity, playfulness and
essential risk-taking potentials of everyday living experiences (MacDonald, p.
122).
A Teacher's Attachment to Mathematical Rules
Other teachers who view mathematics as a set of rules and prescribed procedures also see the logic of these components as well as the contexts in which they can be appropriately applied. For these teachers, mathematics is engaging, but like the case of the student, Nancy, the attachment to mathematics is to its authority and predictability rather than to its more subtle, intellectually challenging, and even aesthetic characteristics. These procedurally engaged teachers tend to present interesting yet relatively routine problems as the context for the clinical interview. They might ask a child to solve a word problem, pursue a line of questioning intended to reveal the approach(es) used by the child to come to a solution, and even challenge the child's answer in an effort to reveal deeper lays of procedural knowledge. Still, though, this type of teacher- interviewer predominantly notes how students follow or interpret rules and when they apply procedures. The exploration and attachment in this case is confined to mathematical rules and their finite variations rather than to nuances and highly personalized conceptions of mathematical relationships that emerge in more playful and spontaneous interactions with mathematical objects.
These interviewers do not emphasize the work of the interview but instead
emphasize power or language. In an interview attending to power, the
interviewer disciplines or controls the interview so that it runs smoothly,
efficiently, and accountably. A hierarchic relationship is the dominant feature
as the interviewer guarantees that his or her goals are met. The student in
such an interview cannot be expected to be responsible for the success of the
interview in any way since they are constructed as "immature." As
MacDonald writes, "The consequences of behavior are sheltered from reality
because students learn they are immature, and the imposition of activity is
legitimated because adults are "mature." (p. 123). The interviewer
teaches a need for rewards that are socially satisfying (thereby constructing a
need for the teacher) rather than personal pleasure. The student pleases the
teacher for future value, losing their own sense of value in activity and
substituting a social satisfaction which makes them dependent on abstract
rewards for their sense of worth. This is very much the situation when the
interviewer, as described above, focuses on obtaining procedurally correct
responses, and the student is left waiting for direction from the interviewer
as the power-broker. Yet, if approval or disapproval is provided, the student
at least knows where he or she stands and does not have to decide on the value
or meaning of the response.
Mathematics as a Teacher's Flexible Object of Self
Teacher-interviewers who are searching for a playful manipulation and the
creation of mathematical objects are themselves likely to spontaneously engage
in exploratory activities on their own. These teachers regard mathematics as
part of the soul and view the world quite naturally in mathematical terms. When
this kind of interviewer interacts with a child in some mathematical context,
if there is a positive outcome there is a kind of synergy that takes place in
which both the interviewer and the child travel to a new dimension of knowing
beyond that which either might engage alone. In this circumstance, the
interviewer becomes part of the child's process and connection to the
mathematics. Even when the child is not a spontaneously mathematically
connected individual, this kind of interviewer may be able to break down
barriers of procedural ritual or illogical random guessing so that in the
process of the interview the child is able to see new mathematical meaning in
the tasks at hand.
For example, Rasheen was asked to compare playgrounds and decide which were
"better" than others. Because the student liked slides and seesaws,
she brought the teacher to several playgrounds and showed her which slides she
liked best and why. Together, they decided that speed and length of sliding
time were important for judging slides, so they used a stopwatch to find ways
to measure these, using the student s body length to measure the distance
covered in a slide. After comparing data for several slides, they then
discussed possible ways to compare the quality of seesaws at playgrounds in the
future. To a large extent we see the positive scaffolding effect described by
Vygotsky (1986) wherein the more experienced learner is able to act as a
catalyst and support for the less experienced one. An interview of this type involves
some aspect of discovery of a new way to represent and solve a mathematical
problem using more intuitive notions than school-taught procedures or at least
combining intuitive notions with school-taught procedures.
This positive outcome, however, is not always the case when the interviewer
approaches the interview with such a close connection to mathematics as an
object. Too often, the child is so removed from the experience of such a
relationship to mathematics that the outcome of the interview is either a
series of questions to which the child is unable to make any reasonable
response or more likely, the interviewer is forced to detach him or herself
from the mathematics and pursue a line of questioning that taps more closely
into the routinized outlook of the child. In this situation, the engaged
teacher in the process of reintegration of self with and through mathematical
object relations may in effect be no different in a therapeutic sense for the
student than a mathematical dictatorial egoist or passive egoist.
NEITHER TOUGH-LOVE NOR DREAM OF LOVE
In general then, the clinical interview is a function of the interaction
between student and teacher- interviewer in which the individual levels of
attachment to mathematics are the main issue. At times the teacher and student
may seem well matched and the interview appears to be comfortable for both,
whether or not the participants learn anything about mathematics or each other.
When the types "clash," the interview is mostly a study in negotiation
of purpose for the interview, subsequently moving the discussion to be
ostensibly about mathematics but more clearly about interpersonal dynamics.
Underlying this interaction are both the student's and the teacher's
performance expectations for the mathematics. It is for this reason that
"listening" teachers commonly foster an atmosphere about rules and
procedures despite their own object-relations with mathematics as an object of
the self. Similarly, the student's conception of the role of authority and how
to behave in the company of teachers colors the quality of what he or she is
willing to put forth regardless of the child's own object-relations with
mathematics. What we note in this context is that the participants'
relationship and expectations for the interview as an object is as important as
their relationships with the mathematics. A focus on the teachers, then, raises
the experience of the interview as paramount and crucial in the on-going
development of object relations with both the experience of learning
mathematics in school and mathematics itself.
Clinical interviewing in the classroom context indeed requires substantial
reframing of the teaching/learning encounter. A teacher must be able to talk with
an individual student while the rest of the class is supposedly "on
task" in some "unsupervised" way. Teachers find a variety of
options, including: interviewing in the context of a class run on extensive
small group work that requires consistent but not persistent monitoring by the
teacher; exhaustive use of aides and adult volunteers that work with small
groups of students while the teacher interviews a student; interviewing while
the rest of the class performs "seatwork" such as "worksheets;"
inviting small groups of students to lunch-time or after-school interviews; and
interviewing within the context of whole-class discussion, during which the
teacher facilitates an interview with the entire class involved in the
questioning an elaboration of each others' thoughts. Regardless of the
interview context, most of the pre-service and experienced teachers we work
with find that experiences with individual students are helpful in reflecting
on interviewing in general; they typically describe their relationship with
mathematics as drastically changed.
In the context of assessment it is evident that the teacher can not soundly
aim to identify and satisfy every need of the student (despite a desire for
teachers to view their jobs in heavily medical terms such as diagnosis and
prescription). To say that a student "needs" any particular skill or
fact is as ludicrous as saying that a child "needs" to have a
particular concept readily available for interpreting "reality"
mathematically. Equally inapprpriate would be to give over the interview as an
environment devoted to the child and its need for "holding." The
purpose of the clinical interview may be seen as a component of a social space
known in practice as a classroom. In this space, it is important for the teacher
to continue to reflect on his or her relationship with the mathematics, perhaps
even more so than to "assess" the needs of a particular child. To
eroticize the individual child or the mathematics of a conversation as an
object of self would be to come too close to a dream-of-love fantasy commonly
represented in popular culture media images of teaching and learning. The
vision is consistent with fantasies of devotion and self-sacrifice, and, as
Judith Robertson (1997) writes, with the rhetoric of scientific efficiency
maximized through individualized child-centered learning. Robertson notes,
"Child-centered discourses position the ... teacher as benevolent overseer
of a landscape of elemental and natural goodness." She continues,
What is forgotten in the fiction is that centering children in this way does not in any natural way best develop their potential as learners. Children not only consistently circumvent teachers' intentions in child-centered activities, but the very notion that children can "discover" the truth through "individualized activities" is a denial on many levels. It denies that any discursive system (including those that posture as "individualized") produces its own particular truths, no matter how these truths are veiled or fictionalized. It denies that the "rational" learner who is ostensibly a product of child-centered practices is not in mastery of either knowledge or the self. Finally, it withholds from learners the legitimate right to expect that teachers will intervene in learning in ways that may feel discomfiting, that may not always be easily understood, that may be insistently directive, and that are not always experienced as ego affirming. (Robertson, 1997, p. 91)
"Dreams of love," writes Robertson, "do not resolve the difficulties
of teaching, nor ultimately, do they increase its pleasures." (p.91) The
educational encounter is not the satisfaction of personal drives deriving from
particular selves, but an ongoing establishment of relations with objects,
within a particular social practice; an educational encounter is an evocation
of selves. This is where critics of recent reform efforts in mathematics
education may be wrong: by taking a tough-love stance they misinterpret
reform-based mathematics as abandoning skill drill whereas reform-based
curriculum merely intends to place its meaning in a new context. On the other
hand, where the dream-of-love, child-centered assessment interpretation of
reform-based mathematics goes wrong is in its fear of challenging the mathematics
itself, and thus failing to enable more substantive changes. It avoids
confrontation of the self and its relationship with the mathematics in its
desire not to disrupt the dream. In our imagined curriculum, an experientially
and reflectively aware aspect of the person is called into existence as the
object of his/her own unconscious ego processes; that is, students and teachers
become "subjects." Paramount are the perpetually changing object
relations that make up a self, and how a teacher's relationship with this
theory would change as her or his own object relations alter and coalesce in
new ways through work with children.
Our imagined curriculum shifts the emphasis from skill levels and covered
topics toward relationships with mathematics objects and the subsequent uses of
objects. As Block writes, "Our lives may be said to derive from the uses
of objects, and this is ultimately a creative process. How objects are used
derives from the effects of a facilitating environment that enables the child
to actually find what the child creates, to create and to link up that creation
with the Real." (p. 27) Our interpretation of curriculum and the clinical
interview dwells on the seriousness of what is at stake: The teacher's
relationship with mathematics is crucial to the experience the child has, and
therefore effects the child's use of objects. The child's mathematical
development is the history of many internal relations, expressed through uses
of objects. Assessment, as in for example clinical interviewing, is not
necessarily about the child, but is the teacher's use of objects, an
expression of the teacher's self as object relations.
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1 The authors thank the reviewers for feedback which moved us to a much better understanding of the issues of this article.