STUDY 1: CHILDREN'S UNDERSTANDING OF MULTIDIGIT NUMBERS This
study, by Jones et al. (1996), validated a framework that described and
predicted various levels of young children's thinking related to multidigit
number sense. The framework itself was used to generate and evaluate two
different versions of an instructional program emphasizing number sense
in general education classrooms that included students with identified
learning disabilities. The instructional approach based on the framework,
grounded in social constructivism (e.g., Cobb & Bauersfeld, 1995), took
the position that students' opportunities to construct mathematical knowledge
arise from attempts to resolve conflicting points of view in a group,
from attempts to reconstruct and verbalize a mathematical idea or solution
and, more generally, from attempts to reach consensus with others.
The
instructional program was consistent with the recommendations of Englert
et al. (1992) and Heshusius (1991), who suggested that students with disabilities
should be challenged with meaningful problem tasks that promote multiple
solutions and strategies. Throughout the study, all students were encouraged
and given time to work collaboratively to solve problems at whatever levels
they could attain. A further expectation was that all students would share
and justify their thinking in different ways. Differences in the understandings
demonstrated by children in the two instructional groups were attributed
largely to the quality of the problem-solving experiences and the level
of student interactions.
STUDY
2: THE USE OF REFLECTIVE ANALYSES ON THE INSTRUCTIONAL PRACTICES
OF PROSPECTIVE ELEMENTARY TEACHERS. This research, by Langrall, Thornton,
Jones, and Malone (1996), used a case study approach to investigate the
effects of reflection on prospective elementary teachers' instructional
practices in mathematics. Recommendations from the NCTM (1989, 1991) Standards
documents provided the basis for the teachers' reflections on their instructional
experiences in elementary classrooms. Six prospective teachers participated
in a series of 11 instructional classroom experiences. In the first and
last of these sessions, the teachers instructed small groups of students,
including children with learning disabilities. Video analysis and transcripts
of these lessons were supplemented with other data, including semistructured
interviews, stimulated-recall sessions, written documents submitted by
the teachers, and researcher field notes. These data sources became the
context for six case studies that documented the changes in instructional
strategies adopted by the teachers. This intervention resulted in powerful
changes in the instructional practices of these prospective teachers,
including changes in the way they related to students with learning disabilities.
Major
changes included (a) greater use of problem-driven tasks and open-ended
questions, (b) increased expectations for student reasoning and multiple-solution
strategies, (c) greater emphasis on student dialogue and collaboration;
and (d) less teacher-directed instruction. Although the focus of this
study was on prospective teachers, the research also captured rich vignettes
of student interactions with each other and with their teacher.
STUDY
3: SUPPORTING MIDDLE SCHOOL STUDENTS WITH LD IN THE MAINSTREAM
MATHEMATICS CLASSROOM The major goal of this research project, by Borasi,
Packman, and Woodward (1991), was to institute a comprehensive professional
development program that would encourage and support middle school mathematics
teachers in rethinking their teaching goals and practices so as to better
respond to the learning needs of all their students, with special attention
to those with learning disabilities.
The
project was developed by an interdisciplinary team comprising mathematics
educators, an expert in learning disabilities, and both mathematics and
special education teachers. Its theoretical framework was characterized
by a constructivist perspective on knowledge and learning, an information-processing
model of learning disabilities, and an "inquiry approach" to teaching
mathematics. At the core of this project was the development of three
units intended to illustrate an "inaction" inquiry approach in middle
school classrooms that included students with learning disabilities. Extensive
case study data on the students with LD were collected during the implementation
of these units in different instructional settings (a private school for
students with learning disabilities, and both urban and suburban public
school general education classrooms). The results suggest that an inquiry
approach, complemented by appropriate instructional modifications and
adaptations, can help mathematics teachers meet the challenges that accompany
a diverse student population. Furthermore, in mathematics classrooms informed
by such an approach, students' learning differences could be capitalized
upon and turned into assets in the learning setting.
STUDY
4: MATHEMATICAL PROBLEM-SOLVING PROCESSES OF PRIMARY-GRADE
STUDENTS IDENTIFIED WITH LD Behrend's (1994) study examined the problem-solving
processes of five second- and third-grade students identified as learning
disabled. In accordance with the recommendations of their Individualized
Education Programs (IEPs), these students received daily mathematics instruction
in an LD resource room. Cognitively Guided Instruction (Fennema & Carpenter,
1985) provided the framework for assessing the children's independent
and assisted problem-solving abilities.
Data
were gathered during individual interviews and small-group sessions. During
the group sessions, students were presented with word problems, given
time to solve them, and encouraged to share their strategies in whole-group
discussions. Prompts, from general to more explicit, were given only when
needed. Behrend (1994) found that, given the opportunity, the students
in her study were capable of sharing their strategies, listening to other
children's strategies, discussing similarities and differences among strategies,
justifying their thinking, and helping each other understand word problems.
Although students modeled their solutions for each other, she observed
that teacher modeling of solution strategies was rarely needed and generally
did not promote better problem solving among the children. All five students
could solve a variety of problems, including difficult addition, subtraction,
multiplication, and division word problems; problems with extraneous numerical
information; and multiple-step problems. Behrend (1994) found that the
students in her study were capable of generating and utilizing their own
problem-solving strategies and did not need to be taught specific strategies.
Based on this finding, she questioned the need for explicit strategy instruction
in mathematics for students with learning disabilities and recommended
instructional approaches that utilize students' available problem-solving
processes.
CASE
ILLUSTRATIONS OF THE FOUR THEMES Four case illustrations from
the studies described above exemplify the themes presented in this article.
Although some of these cases could illustrate more than one theme, the
discussion highlights salient features of each theme in turn.
PROVIDING
A BROAD AND BALANCED MATHEMATICS CURRICULUM Overview of Theme.
Baroody and Hume (1991) noted that many children who experience learning
difficulties in mathematics, including those who have learning disabilities,
are "curriculum disabled." For such students, Trafton and Claus (1994)
recommended a broader and more balanced curriculum, in contrast to a more
traditional curriculum, with its repetitive and unnecessary emphasis on
computation. A broader curriculum can be established by utilizing problem-driven
instruction that incorporates a greater emphasis on number sense and estimation,
data analysis, spatial sense and geometric thinking, patterns and relationships
leading to algebraic understandings, and the supportive use of technology
(National Council of Teachers of Mathematics, 1989). Broadening the curriculum
to include a variety of mathematical domains does not preclude, but actually
encourages, the development of appropriate mathematical skills. In essence,
it presents opportunities for different kinds of thinking and success
beyond numerical reasoning. Such a curriculum should be reflected in children's
IEPs. Although it does not entail disregarding computation in the IEP,
a broadened thrust enables students with learning disabilities to use
mathematics more flexibly, insightfully, and productively (Bley & Thornton,
1994; Borasi, in press; Englert et al., 1992).
Case
Illustration. To exemplify the "broad and balanced curriculum" theme,
this section draws on case data from two students with identified learning
disabilities. These illustrations include an episode involving a student
we will call Jana, focusing on mental math (Jones et al., 1996), and an
episode documenting "Terrell's" thinking about a geometry task (Langrall
et al., 1996). On the Wechsler Intelligence Scale for Children-III, 9-year-old
Jana scored well below average on the Verbal Comprehension and Mathematics
subtests. Because she also had severe receptive language and auditory
memory difficulties, she was placed in a self-contained learning disabilities
program. Despite her difficulties with word problems, her good visual
memory and interest in mathematics led to her being jointly placed in
a general classroom for mathematics, where the following episode occurred.
The
activity had started the moment Mrs. Tate's class entered the room. Each
pair of children selected a card that showed an amount of money they could
spend. The task was to "purchase" items from the garage sale mural on
the wall, spending as much of their money as possible. The children worked
in pairs for a short time before Mrs. Tate brought them together to share
their thinking. Over the past 2 weeks the garage sale activity had constituted
a daily math problem for the children--one that had grown out of earlier
problem-solving experiences with addition and money. Jana spoke for herself
and her partner: "We picked the picture frame for 38¢ and the poster for
15¢--and we just have 7¢ change." When asked to explain how they knew
they would have 7¢ change, Jana said, "We just thought about the 100s
Chart. We started with 38 and went down to 48, then counted 5 more. So
we paid 53¢ -- that gives us 7¢ back because we had 60¢ to spend."
This
episode illustrates how the 100s Chart enabled Jana to move beyond paper-and-pencil
computation. During early instruction with this graphic aid, Jana was
encouraged to move a finger along the chart as she counted. She then grew
able to visualize the counting-on process just by thinking of the 100s
Chart. In this case the chart was an appropriate compensatory tool that
enabled Jana to compute two-digit sums mentally.
The
second case centers on Terrell, a student with learning disabilities enrolled
in a general fifth-grade class. He showed little ability to reason abstractly
and had visual-perception problems, but retained information once he had
internalized it. The following episode illustrates how Terrell, his learning
difficulties notwithstanding, manipulated pattern blocks and drew on his
understanding of a high-dive flip to reason about angle measures in a
meaningful way. Terrell pointed to the three trapezoids he had arranged
around a dot on the overhead [see Figure 1].
He
explained what Duane had told his group about how a "360 flip" off the
high dive "goes all the way around." "Here three of these [trapezoids]
go all the way around. So we divided 360 by 3 and got 120 for the big
angle." As he observed groups at work, the teacher, Mr. Adams, had not
been sure that Terrell understood Duane's explanation of a 360-degree
turn, so he was pleased when he heard Terrell rephrase the explanation
to Duane and later volunteer to present the group's solution. Mr. Adams's
expectation that all group members would be able to present the group's
solution set the stage for Terrell to verbalize his solution strategy
within his working group. Expectations like this that encompass opportunities
for children with learning disabilities to articulate their thinking have
been shown to help with learning and retention (see Montague, this issue).
Other groups in the class had found different ways of showing that the
obtuse angle of the pattern block was 120 degrees. The initial task had
challenged each group of four students to determine the measures of each
of the angles of the pattern blocks. As part of the ensuing discussion,
a summary chart was made to organize the class' findings. These case illustrations
document how two students with learning disabilities were successful in
mathematics programs that emphasized a broad, balanced curriculum. When
instruction is consistent with the tenets of such a curriculum and different
approaches are valued, it is possible for children to achieve success
within their specific limitations (Bulgren & Montague, 1989; Cawley, Fitzmaurice-Hayes,
& Shaw, 1988; Ginsburg, this series).
ENGAGING
STUDENTS IN RICH, MEANINGFUL PROBLEM TASKS Overview of Theme.
Recent recommendations (e.g., National Council of Teachers of Mathematics,
1989, 1991; National Research Council, 1990) highlight the need for relevant,
problem-driven instruction. A central thesis of these recommendations
is that all students should become confident "doers" of mathematics and,
consequently, be capable and resourceful problem solvers. This requires
that all students have the opportunity to explore many different types
of mathematical problems and that they be both expected and encouraged
to use a variety of strategies in solving them (National Council of Teachers
of Mathematics, 1989).
Although problem solving traditionally has been a difficult area for many
students with learning disabilities (Montague & Bos, 1986; Wansart, 1990),
Bulgren and Montague (1989) reported that these students can succeed beyond
current expectations if they are exposed to developmentally appropriate,
meaningful problem tasks that are complemented by appropriate instructional
modifications. Moreover, children experiencing difficulties with formal
computation or basic fact recall should not be prohibited from engaging
in more challenging problem-solving tasks (Cawley & Miller, 1989; Ginsburg,
this series).
In fact, a substantial body of research highlights the effectiveness of
using problem solving as the vehicle for learning mathematics, including
basic facts and computation (e.g., Carpenter & Moser, 1984). When problem
tasks are sufficiently complex, rich, and open-ended, they can be explored
at different levels of understanding. Stenmark (1991) characterized a
"rich" problem in three ways: (a) The problem leads to other problems,
(b) the problem raises other questions, and (c) the problem has many solution
approaches. We would add a fourth criterion: that a problem makes multiple
connections. Case Illustration. One example of a complex, rich problem
task, which was presented to a self-contained class of nine students classified
as severely learning disabled, is the following triangle--rectangle problem:
Is every triangle 1/2 of a rectangle? Yes or no? Prove it.
The following excerpt from the teacher's journal provides insight into
both the nature of the problem task and the thinking and physical representations
the students used to solve it. Three boys, working together, cut out the
colored triangles [from the Triangle Worksheet (TWS; see Figure 2)], taped
them to the triangles on the white TWS [See Figure 3], and formed parallelograms.
Their premise was, "No--two equal triangles do not form rectangles. The
shapes formed are not rectangles because they do not have 90 degree angles."
(Stone, 1993, p. 54) A [second] group of two boys gave themselves permission
to cut the ... triangles [see Figure 3] on the altitude and tape the two
[colored] triangles, one on either side, to the white triangle. They had
a little difficulty with #3, the obtuse triangle. They cut off a piece
along a line at the end. After they taped the two pieces to the existing
triangle they had a small piece sticking out on the left and a small hole
on the right. They asked if it was O.K. to cut off the piece and move
it. They ended up with a perfect rectangle with a base of 4 and a height
of 3 (very ingenious!). (Stone, 1993, p. 56)
One
girl was working by herself due to absenteeism. She also cut the cut-out
triangles. She worked totally independently. Her first conjecture was
that all triangles except #3 could form rectangles. She was very proud
of herself when she finally figured out how to do #3 like the other boys
did. (Stone, 1993, p. 54) Within problem contexts such as this, students
with learning disabilities are able to draw upon their diverse strengths
as they solve problems using different parameters and achieve success
"within their specific limitations" (Borasi, in press).
This
kind of exploration addresses the broader need to challenge students to
think beyond common expectations. When viewed according to the characteristics
proposed by Stenmark (1991), the triangle--rectangle problem meets the
criteria for a rich problem in that it (a) generated extension problems,
(b) raised questions about shapes, (c) generated different solutions by
redefining the parameters of the problem, and (d) set the stage for exploring
further connections. In relation to extension problems, the group of children
in the first scenario just mentioned correctly reasoned that every triangle
is not half a rectangle but is half a parallelogram. This raised a further
problem that was pursued in a later lesson, "Is every triangle half a
parallelogram?"
The second group of three boys redefined the problem in their own way
and, in essence, investigated an extension problem: "Can a rectangle be
formed by physically changing two congruent triangles?" The problem raised
questions about the defining properties of shapes. For example, when is
a parallelogram a rectangle? The problem also provided an opportunity
for the teacher to follow up on the distinction between congruent shapes
and shapes that have the same area. Because of differing interpretations
and reasoning, the problem gave rise to two different but valid solutions.
In one case, assuming that the shape of the triangle could not be changed,
the children concluded that it was not possible for every triangle to
be half a rectangle. In the other case, the children made a different
assumption--that the shape of the triangle could be changed as long as
areas remained the same. In this situation it was possible to construct
a rectangle that was twice the area of the given triangle. In terms of
connections, the triangle--rectangle problem task set the stage for the
teacher to make the link between areas of triangles and areas of rectangles.
The natural connection between the visualization of length--width measures
of a rectangle and the corresponding base--height measures of a triangle
could be highlighted in such instruction. Further, it would be possible
to make connections among the areas of a triangle, a rectangle, and a
parallelogram. When students are given ongoing opportunities to engage
in rich problem tasks, as in this case, the results can be quite dramatic.
This success is consistent with research documenting the fact that students
learn what they have an opportunity to practice. Students who have had
many opportunities to solve mathematical problems become better at problem
solving (e.g., Carpenter et al., 1989; National Council of Teachers of
Mathematics, 1989; Silver, 1985).
ACCOMMODATING
THE DIVERSE WAYS IN WHICH CHILDREN LEARN Overview of Theme.
Mathematics today is viewed as a "sense-making experience" involving numerical,
logical, and spatial concepts and relationships. Because sense-making
is idiosyncratic, students with learning disabilities generally need considerable
time to understand problem situations and construct strategies.
Further,
if these students are to develop higher levels of mathematical thinking
and more positive dispositions toward mathematics, they need ongoing opportunities
to explore mathematical tasks in ways that match their learning strengths
(Speer & Brahier, 1994). For example, learning groups might be formed
on the basis of complementary learning styles. With this approach, students
with different strengths can find their niches and achieve success within
their specific limitations (Borasi, in press).
Case
Illustration. An episode involving Dan (a pseudonym; Behrend, 1994) is
a case in point. Dan, a 9-year-old, received mathematics instruction in
a learning disabilities resource room. On the Wechsler Intelligence Scale
for Children--Revised, his Full Scale IQ score was average, although he
had difficulty processing multiple pieces of information. At the time
of the study he was on medication to control his attention-deficit disorder.
Dan was the most inconsistent student with respect to mathematical performance
in Behrend's (1994) study. His inconsistency was readily apparent in routine
computational problems, where he attempted to apply learned rules in a
nonmeaningful way. For example, when asked which of two ways (see Figure
4a) would be the better way to find the total, Dan selected the example
on the left because it corresponded to his interpretation of the teacher's
rule for adding: "The ones are first" (p. 74). Dan believed that 78 was
a reasonable answer because the 4 was "where it's supposed to be.... Because
that's where you mostly put the first number for numbers" (p. 75). However,
when Dan was faced with a nonroutine problem and was allowed the flexibility
to solve it in his own way, he demonstrated surprising insight, as illustrated
by his solution to the following problem. 19 children are taking a bus
to the zoo. They will have to sit either two or three in a seat. The bus
has 7 seats. How many children will have to sit three to a seat, and how
many can sit two to a seat? (Behrend, 1994, p. 77) Dan quickly drew seven
lines to represent seats, drew a circle for each seat, and repeated the
process until he had accounted for 19 circles (see Figure 4b). This kind
of modeling and counting strategy exemplifies Dan's thinking in problem
situations for which a known procedure was not readily available. Not
only was he capable of solving nonroutine problems like this, but Behrend
reported that he was also capable of correctly solving problems that included
extraneous information. In effect, when the teacher accommodated Dan's
distinctive learning style, Dan was successful; when he felt compelled
to use the teacher's algorithmic approach, he invariably failed. The inflexibility
of a meaningless procedure appeared to inhibit his ability to recognize
the reasonableness of an answer or his attempts at alternative strategies.
As this case illustrates, accommodating the diverse ways in which children
learn does not always require proactive strategies on the part of the
teacher. Rather, there are times when the teacher needs to step back and
observe and listen to children's thinking patterns so that he or she can
respond to and maximize the children's strengths. In her study, Behrend
(1994) found that students with learning disabilities constructed and
utilized their own strategies to solve a wide variety of problem types.
She concluded that instruction should build on children's current understandings
and promote the development of increasingly more efficient problem-solving
strategies, rather than emphasizing specific rules and procedures. In
finding implications for instruction, Behrend generates a powerful message
for teachers of diverse learners: A model of instruction which involves
posing problems, allowing students time to solve the problems in their
own way, listening to students' strategies, assisting only when necessary,
and discussing similarities and differences between strategies provides
several advantages over the other forms of instruction. Teachers are able
to make assessment an integral part of instruction, students are given
more control over their learning, and mathematics is seen as a process
of making sense of number relationships. Instruction becomes less a matter
of following directions, or imitating what has been modeled, and more
a way of making connections to what is already known. (p. 109)
ENCOURAGING
STUDENTS TO DISCUSS AND JUSTIFY THEIR PROBLEM-SOLVING STRATEGIES AND SOLUTIONS
Overview of Theme. Research has suggested that classrooms in which students
"discuss, critique, explain, and when necessary, justify their interpretations
and solutions" (Cobb et al., 1991, p. 6) are effective at nurturing mathematical
thinking. Such inquiry-oriented approaches are consonant with Scheid's
(1990) review of research in special education, which has also emphasized
the importance of children's thinking about their solutions and justifying
them. Students can communicate and justify their thinking and reasoning
through journal writing, partner sharing, or whole-class discussion, depending
on the situation and individual student needs. After completing a problem
task, teachers could invite students to share their thinking or journal
entry with a partner or small group. In this way, all students have an
opportunity to communicate their thinking in some way, whether or not
they subsequently share their ideas with the larger group.
This
Think-Pair-Share approach (McTighe & Lyman, 1988) increases the kinds
of personal communications that are necessary for students to internally
process, organize, and retain ideas (Pimm, 1987). Whole-class discussions
in which students explain and justify their solutions to problems provide
a rich forum in which students develop their understanding of mathematics.
In sharing their ideas, students take ownership of their learning and
negotiate meanings, rather than relying solely on the teacher's authority
(Cobb et al., 1991). Lo, Wheatley, and Smith (1991) also reported positive
changes in students' dispositions and self-esteem when they were expected
to listen to each other and respect others' ideas. Students with diverse
learning needs gain credence with their peers by reporting to the whole
class what they have learned from participating cooperatively in group
work or journal writing. Reporting sessions also provide opportunities
for less articulate students to learn from their peers who, in a sense,
serve as role models for higher level thinking. Repeated exposure to such
experiences enhances the likelihood that students with specific disabilities
will begin to independently think at higher levels (Scheid, 1990).
Case Illustration. Borasi, Kort, Leonard, and Stone (1993), reporting
on a class of nine children with severe learning disabilities, noted how
these students frequently wrote to explain to others what they did, and
then paired up for sharing. In fact, the two children who also had attention-deficit/hyperactivity
disorder were often invited to share their thinking while walking down
the hall so as to "get rid of some excess energy" (p. 143).
In another example from Borasi et al. (1993), students were asked to write
a journal entry describing their processes for finding the number of tiles
needed to tessellate the classroom floor. One student, whom we will call
Todd, had a severe motor disability in writing as well as a "numerical"
disability. He was helped by Borasi, a participant observer in the class,
to first reconstruct and then record his solution in a journal. Prompted
by the researcher's questions, Todd explained how he had solved the tiling
problem. He declined her offer to scribe for him, preferring to do it
himself. He described each step of his solution process aloud before writing
it down. The journalizing task was completed over a 2-day period, with
support and questioning from Borasi. In the end Todd produced a well-organized
and understandable entry, which Borasi later transcribed on a computer
for sharing with other students. Reflecting on Todd's experience, the
researcher commented that this one on one work seemed really important
and productive with a student with severe writing disabilities such as
[Todd]; it was our hope that this experience would show [Todd] what he
could really do, and provide a model for the future; we do not expect
him now to be able to do similar writing on his own yet, but perhaps he
might be able to do it a second time around with less help, and gradually
learn to do the same without the adult support. (Borasi et al., 1993,
p. 152)
CONCLUDING COMMENTS Consistent with recent recommendations
of the National Council of Teachers of Mathematics (1989, 1991), this
article has presented and illustrated four promising themes for mathematics
instruction that have emerged from recent mathematics studies involving
students with LD.
These
themes--(a) providing a broad and balanced mathematics curriculum; (b)
engaging students in rich, meaningful problem tasks; (c) accommodating
the diverse ways in which children learn; and (d) encouraging students
to discuss and justify their problem-solving strategies and solutions--suggest
ways for rethinking the teaching and learning of mathematics for students
with learning disabilities. Case data exemplifying these themes provide
a vision of what can happen when teachers nurture mathematical thinking
and provide time and opportunity for students to engage in and share their
solutions to rich, meaningful problems. Students with cognitive and processing
disabilities deserve--and have the potential--to be empowered mathematically.
In the field of learning disabilities, relatively few studies reflect
the instructional themes identified in this article. This article has
highlighted four studies that illustrate successful engagement of students
with LD in problem solving and higher level thinking. Consistent with
the findings of these studies, we are recommending a broadened approach
to curriculum and instruction that accommodates and capitalizes on diversity
in thinking and learning. Although further research is needed, the studies
outlined in this article suggest that the mathematical abilities of students
with LD can be accommodated and capitalized upon when these students have
pervasive opportunities to learn in challenging, broad, and well-balanced
programs. The fact that students may need appropriate compensatory techniques
notwithstanding, our thesis is that programs based on the themes presented
in this article can raise the mathematical thinking of these students
to levels previously considered to be beyond their reach. If you don't
let your grasp extend your reach, then you'll never extend your reach.
--Woody Allen, 1992
|
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AUTHOR:
Carol
A. Thornton, Cynthia W. Langrall, and Graham A. Jones
Carol
A. Thornton, PhD, is a distinguished university professor in the Department
of Mathematics at Illinois State University. She has conducted extensive
research on children's mathematics learning in the context of classroom
instruction, has special interests in curriculum and staff development
involving both general and special education teachers, and is the co-author
of Teaching Mathematics to Students with Learning Disabilities (published
by PRO-ED) and the coeditor of a special education book for the National
Council of Teachers of Mathematics.
Cynthia
W. Langrall, PhD, is an assistant professor in the Department of Mathematics
at Illinois State University, where her research focuses on the teaching
and learning of mathematics in the elementary and middle grades, with
a special emphasis on instructional practices that accommodate student
diversity.
Graham
A. Jones, PhD, is a visiting professor in the Department of Mathematics
at Illinois State University. Dr. Jones's interests include research on
children's mathematical thinking in number, probability, and problem solving.
He has also conducted and reviewed research in teacher education and staff
development that have an impact on both general and special education.
Address: Carol A. Thornton, Illinois State University, 4520 Mathematics
Department, Normal, IL 61790-4520.
SOURCE:
Journal of Learning Disabilities 30 142-50 Mr/Ap '97. Reproduced
with permission from Journal of Learning Disabilities, copyright 1997
by the National Council of Teachers of Mathematics. All rights reserved. |