Portrait of the Construction of Mathematical Knowledge:
The Case of a Deaf AND Blind University Student
christine L. Ebert
University of Delaware
cebert@math.udel.edu
Abstract: This study describes the construction of
mathematical knowledge of a university student who is blind and deaf. Of particular significance is the issue of
mathematical representation both in terms of representing what the teacher was
talking about and providing a medium for the student to represent her
knowledge. An equally important issue
concerned understanding symbol systems and the degree to which Braille text
facilitated that understanding. For the
teacher, the issue of defining and assessing mathematical understanding in each
of the courses was particularly salient.
By examining these issues in the case of this unique individual, we may
gain new insights into the construction and organization of mathematical
knowledge for all learners.
This study, which is set
within the context of university-level pre-calculus and calculus courses, is
part of a larger study that spans approximately two years in which one teacher
worked with the student in the following courses: 1) pre-calculus; 2) calculus
(scientific); 3) discrete mathematics; and 4) statistics. Thus, the mathematics discussed represents a
transition to advanced mathematical thinking in which “the mind has
simultaneous concept images based on earlier experiences that interact with new
ideas based on definitions and deductions” (Tall, 1992, p.496). In particular, the mathematical discussions
described in the study reflect an approach that builds on concepts that are
familiar to the student and provide the basis for later mathematical
development. Thinking about and
communicating mathematical ideas requires some means to represent them. Communication between student and teacher
certainly requires that the representations be external, taking the form of
spoken language, written symbols, pictures, or physical objects. Given the perspective that “the nature of
the internal representation is influenced and constrained by the external
situation being represented” (Kosslyn & Hatfield, 1984), it seems
reasonable to assume that “the form of an external representation (physical
materials, pictures, symbols, etc) with which a student interacts makes a
difference in the way the student represents the quantity or relationship
internally” (Hiebert & Carpenter, 1992, p.66). “Conversely, the way in which a student deals with or generates
an external representation reveals something of how the student has represented
the information internally” (Hiebert & Carpenter, 1992, p.66). However, in the case of a deaf/blind
student, the choices of external representations are limited. This study describes the various manipulatives
and physical objects that were created and utilized to facilitate mathematical
communication in which spoken language, written symbols, and pictures were not
available. In particular, with
respect to the issue of representation, this study provides a careful
examination and analysis of how the teaching and learning process interact
The data source for this
study consists of a 24-year old university student, Beth, who is deaf and
blind. She was born deaf and became
blind when she was quite young, at approximately three years of age. She has some memories of colors and objects
and well-developed iconic knowledge of many common objects.
At the university, the ADA
(Americans with Disabilities Act) office coordinated all of the support staff
and efforts on behalf of Beth.
Communication was possible through a team of tactile interpreters, who
communicated with Beth through a tactile form of American Sign Language. Whatever was said or written, including
diagrams and graphs, was communicated as accurately as possible to Beth. During class, when she volunteered an answer
to a question, she signed her response to the interpreter and the interpreter
vocalized her response
Beth was assigned to my
pre-calculus class in the fall of 1998.
At that point in time, I had no specific training in teaching students
with any type of special needs. For all
of the subsequent courses, I served as her tutor, meeting 3 hours per week in a
one-to-one setting with the interpreters.
The rationale behind Beth’s class attendance was that she should receive
the information in “real time” just like the other students. However, since she had no way of taking
notes and creating her own record of what she deemed important, her attendance
was similar to a hearing and sighted person attending a general interest
lecture. Certainly, she remembered
generally what was discussed in the class but usually could not provide the
details.
For the first course,
pre-calculus, my notes were transcribed and sent to Beth via email with
descriptions of graphs provided. For
the two subsequent courses, both the calculus and discrete texts were
translated into Braille. Beth’s
equipment, both at home and at the university (in the library), allows her to
read an 80-character line of text in Braille.
The equipment does not support equations, tables, pictures, diagrams, or
graphs.
For this reason, even the class notes had to be re-written in a more descriptive and less symbolic mode. Thus, faced with these constraints, it was essential to provide concrete representations for the equations and graphs so that Beth did not have to retain everything in memory.
Methods of Inquiry
From the beginning of the
experience, I kept detailed notes of all the mathematical discussions that took
place during the individual meetings.
These notes were edited and transcribed. Several of the sessions were video taped and these were also transcribed. In addition, several of the other people involved were
interviewed. These include six
different interpreters, the ADA coordinator, Beth, and her mother. These interviews were conducted to provide
background information and to augment the existing data.
Throughout this work, many
manipulatives and concrete objects were created to facilitate the mathematical
communication. These have been
documented, photographed, and used in some of the videotapes, to illustrate how
they were used. In addition, after the
fact, I wrote commentaries describing how the manipulatives were chosen and
created. For the most part, these
manipulatives were created, as they were needed, to support the discussion of a
particular concept or to facilitate a useful representation. I used magnetic letters and numbers and a
magnetic board to represent equations.
Given that the intended use for these sets is certainly not typical
pre-calculus or calculus expressions and equations, I supplemented these with
pipe-cleaner symbols. For graphs, we
began with a small geo-board and rubber bands but that soon proved to be too
restrictive. I created the x-y plane
with push-pins on a large cork board (approximately 24 by 36 inches) and used
rubber bands and additional push-pins to create the graphs. This device for
representing and analyzing the graphs of functions served for pre-calculus, calculus,
and statistics (regression).
As a participant-observer,
the inquiry was shaped by a constructivist
interpretation methodology (Noddings, 1990) with each stage of the inquiry
iteratively informing the next.
Although the notes and commentaries initially provided a simple record
of the instruction, it was clear that they also documented information about
issues of representation and the mathematical connections between
representations. From these notes and commentaries, the videotapes, and
the interviews, it is possible to describe the construction and organization of
mathematical knowledge in the context of these courses for this unique
individual.
Writing a linear equation given two points or one point and the slope is a standard task in pre-calculus and calculus. From my teaching experience, I have always believed that the point-slope form is generally easier for students to understand and manage with the fewest opportunities for careless calculation errors. However, during discussions about writing linear equations given two points, I found that Beth could not seem to keep track of all the pieces of the point-slope form of the equation but experienced success with the slope-intercept form. Regardless of the numbers, she always used the slope-intercept form and easily performed whatever calculations were necessary to correctly determine the y-intercept. Beth’s preference for the slope-intercept form reflects a far more interesting issue of knowledge organization than simply a preference based on previous experiences. For a sighted student, writing out the point-slope form, replacing the slope and point with specific values and simplifying the result to the slope-intercept form is based on following the pattern in the external representation of the equation. The representation serves both as a record of the ongoing work and a cue for the subsequent steps. However, for a non-sighted student, the point-slope representation requires internally keeping track of several numerical and variable expressions as well as correctly combining the appropriate expressions. It is much easier to remember to multiply the value of the slope by the value of the x-coordinate of the point and then subtract this result from the value of the y-coordinate. Once this result is obtained, it is a fairly simple task to assemble the requisite pieces into the equation of the particular line.
Being able to determine
equations of linear functions does not, of course, imply that one understands
the nature of linear functions and in particular their graphs. Pre-calculus courses today include a
significant emphasis on understanding functions and their graphs and frequently
utilize graphing utilities to explore the properties of graphs. Given this perspective, it was important
that Beth be able to recognize, create, analyze, and understand the properties
of the graphs of the various classes of functions discussed in
pre-calculus. I would create the graph
on the corkboard and Beth would explore it by touch. However, since graphical understanding is one of the major
objectives of pre-calculus, frequently Beth would simply explore the
rubber-band graph and we would discuss it.
In discussions about the
intervals over which a polynomial graph was increasing or decreasing, Beth
exhibited a typical error frequently made by sighted students. Suppose that a function is decreasing over
the interval from 5 to infinity.
Because the graph is decreasing, students report the interval as 5 to
minus infinity. Beth made the same
mistake. For a sighted student, one
might conjecture that the visual cue of decreasing is so salient that the brain
has difficulty focusing on the interval over which these phenomena occur. However, with a non-sighted student, there
is no visual cue, only a tactile one.
This similarity suggests that a decreasing function over an interval
from some value “b” to infinity may create sufficient cognitive dissonance to
cause confusion, which leads to the error.
These examples provide only
a brief glimpse of how Beth’s mathematical knowledge was constructed and
organized. They do illustrate, however,
some of the representational issues that influenced the knowledge organization. In the case of determining the linear
equation, Beth’s ability to execute a well-known procedure did not rely upon
the representation that requires the least substitution. Rather, the salient representation for her
was the one that required the fewest calculations. This choice also indicates an efficiency that is characteristic
of knowledge that is part of a network of representations. In the case of the decreasing polynomial
function, it seems that both a visual and a tactile representation lead to the
same misconception. In this case, the
means of accessing the representation are independent of the conclusions that
are generated from the representation.
It would also seem reasonable that the cognitive dissonance is created
by the graphical representation. Thus,
for Beth as well as for sighted and hearing students, there is evidence that
the internal representation is very definitely influenced by the external
representation. These findings provide
insights into the interaction between the teaching and learning process and how
representations contribute to our mathematical understanding.
Hiebert, J. & Carpenter,
T.P. (1992). Learning and teaching with understanding. In D. J. Grouws (Ed.), Handbook of Research on Mathematics Teaching
and Learning (pp. 65-91). New York: MacMillan.
Kosslyn, S.M., & Hatfield,
G. (1984). Representation without symbol systems. Social Research, 51,
1019-1045.
Noddings, N. (1990). Constructivism in mathematics
education. In R.B. Davis, C.A. Mayer,
& N. Noddings (Eds.), Constructivist
views on the teaching and learning of mathematics (pp. 7-29). Reston, VA:
National Council of Teachers of Mathematics.
Tall, D. (1992). The transition to advanced thinking: functions, limits, infinity, and proof. In D.J. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 495 – 509). New York: MacMillan.