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PAGLIARO IS AN ASSOCIATE PROFESSOR IN THE
PROGRAM IN EDUCATION OF DEAF/HARD OF
HEARING STUDENTS, DEPARTMENT OF
INSTRUCTION AND LEARNING, UNIVERSITY OF
PinsBURGH, PinsBURGH, PA. KRITZER IS A
DOCTORAL STUDENT IN SPECIAL EDUCATION,
DEPARTMENT OF INSTRUCTION AND LEARNING,
UNIVERSITY OF PITTSBURGH.
HE STUDY DOCUMENTS what deaf education teachers know about discrete mathematics topics and determines if these topics are present in the mathematics curriculum. Survey
data were collected from 290 mathematics teachers at center and public school programs serving a minimum of 120 students with hearing loss, grades K-8 or K-12. in the
United States. Findings indicate that deaf education teachers are familiar with many discrete mathematics topics but do not include them in instruction because they
consider the concepts too complicated for their students. Also, regardless of familiarity level, deaf education teachers are not familiar with discrete mathematics
terminology; nor is their mathematics teaching structured to provide opportunities to apply the real-world-oriented activities used in discrete mathematics
instruction. Findings emphasize the need for higher expectations of students with hearing loss, and for reform in mathematics curriculum and instruction within deaf
education.
Hcnv should buses he routed and scheduled so that ail students arrive at school on rime and in the most cost-effective way? How can Rnal exams be scheduled so that no
student has a conflict? Who actually won the 2000 presidential election?
What do these questions have in common? They all incorpt)rate discrete mathematics. Discrete mathematics is a contemporary branch of mathematics, widely used in
business and industry and having practical applications in everyday life. It includes such tasks as finding shortest routes be
tween locations, scheduling tournaments, and conducting elections. It incorporates real-world problems and [irovides opportunities to apply traditional mathematical
concepts from various branches of mathematics in meaningful ways. In addition, this type t)f mathematics is engaging for students of all ability levels. There are no
data, however, on the inclusion of discrete mathematics in tbe deaf education classroom. In the present study, we sought to document what deaf education teachers know
about discrete mathematics topics, and to document the presence of such topics in the curriculum.
iMF. ISO, No. 3. 2005 AMERICAN ANN.\I,S OP nil-: D!i-\i
DISCRETE MATHEMATICS IN DEAF EDUCATION
Background General Mathematics Education As stated in a publication of the National Research Council, Helping Children Lear/j Mathematics, “Success in tomorrow’s job
market will require more than computational competence. It will ret|uire the ability to apply mathematical knowledge to solve problems” (Mathematics Learning Study
Committee, 2002, p. 3). This involves going beyond the traditional, fact-based mathematics instruction to a reformed, problem-basetl, authentic curriculum. The
National Council of Teachers of Mathematics (NCTM) strongly recommends that educators actively engage students in solving “real-world” problems that are purposeful
anti worthwhile (NCTM, 2000), in order to adequately prepare them for productive roles in s(x:iety. Instruction, therefore, niList jirovide opportunities for students
to experience mathematics as a valuable tt)ol with which to solve problems, communicate, and reason (NCTM, 1989). In such a curriculum, the use of discontinuous, or
discrete, mathematics is essential.
Discrete mathematics jircjvides teachers with a practical way to think about traditional mathematics topics and with new strategies for enhancing student engagement. 1
brough authentic problem-solving ex[ieriences, students use a synthesis of mathematical concepts toward a purposeful goal. They investigate the mathematics behind
political elections, for example, exj^lore repetiti\’e patterns and processes in nature, and navigate paths and networks in order to make snow removal or garbage
collection more efficient. Such problems may not have one specific “right” answer, but rather several answers that students evaluate based on given limits and other
circumstances such as budget restraints or time factors.
The NCTM (2000) supports the use of discrete mathematics as “an integral part t)f the school mathematics curriculum K-12″ (p. 31) and a means by which all other
branches of mathematics can be included. Although traditionally used with students at the high school level, discrete mathematics can easily be woven into a refi)rm-
based mathematics curriculum at the early grade levels. For example, students in elementary’ school classes can learn to determine the shortest routes to various
locations around the countiy while simultaneously learning about geography, practicing computation skills, and discovering algorithms (Kiitzer ik Pagliaro, 2003)- In
addition, low-achieving or math-anxious students can be introduced tf) motivating projects that involve more than dull, repetitive practice exercises (Kenney, 2001).
Despite the pronounced merits of discrete mathematics, most teachers have not had sufficient preparation in the content of discrete mathematics or in the teaching of
its topics (Rosenstein & Kovvalczyk, 2002). Since 1989, the Leadership Program in Discrete Mathematics at Rutgers University has sought to ameliorate this deficiency
in mathematics preparation among teachers by conducting professional development institutes each summer. The success of the program is evidciued not only by the
repeated funding it has received from the National Science Foundation but by the more than 1,200 participants who have incorporated discrete mathematics concepts into
their instruction and who have shared their knowledge with colleagues, locally and nationally (Rosenstein ik Kowalczyk, 2002).
Mathematics Education for Deaf and Hard of Hearing Students Although the National Actiim Plan for Mathematics Education Reform for the
Deaf (Dietz. 1995) has called for reform in the mathematics education of deaf and hard of hearing students that matches the NCTM recommendations (including the u.se of
discrete mathematics), studies have shown that K-12 deaf education teachers are more traditional in their instruction, making infrequent use of reform-based activities
(Pagliaro, 1998b; Pagliaro & Ansell, 2002). Other studies more specifically questitin the problem solving that teachers of deaf and hard of hearing students profess to
include in instruction across grade levels and across settings such as schools for the deaf, resource r<OTms, and mainstream programs (Kelly Ung. ik Pagliaro, 2003;
Pagliaro & Ansell, 2002). For ex:\mple, while teachers in the study by Kelly and colleagues indicated that they spent almost 4 hours per week on problemsolving
activities, they also indicated that this time was spent on superficial “practice exerci.ses.” Such exercises iTierely require students to apply a learned procedure or
algorithm to several similar problems, in contrast to true problem-solving tasks for which students must analyze, reason, and think logically while determining and
applying appropriate strategies to find a solution.
The disappointing results of this traditional approach to mathematics instruction are evident in recent data on student performance. Data from tbe Stanford Achievement
Test (9th ed.) show that half of deaf and hard of hearing students, on average, achieve no better than just under the sixthgrade level in mathematics computation and
only at the fifth-grade level in problem solving by the end t)f high school (Traxler, 2000). In addition, these students exhibit great difficulty in tasks involving
reasoning (Allen, 1995).
A factor in the paucity of reformbased instruction among deaf edu
VoLLiME 150, No. 3. 200S A.MERICAN A,\NA[,,S OF IHR
cation teachers is said to lie these teachers’ own limitet! experience and insufficient preparation in mathematics in general (Kluwin & Moores, 1985; Pagliaro, 199Ha).
Studies have shown that teacher preparation in mathematics and mathematics education positively influences the quality of instruction and, therefore, student
achievement (Falkner, Levi, & Carpenter, 1999; Pagliaro, 1998b; Pagliaro & Ansell, 2002). However, few deaf education teachers htild degrees or certification in
mathematics or mathematics education (Kelly et al., 2003; Pagliaro, 1998a). In addition, most are not active in mathematics-related {professional development
(Pagliaro, 199Ha; Pagliaro & Ansell, 2002).
No study has looked specifically at deaf education teachers’ knowledge and instructi(.)n with regard to discrete mathematics. In light of this fintiing, we conducted a
study to determine the extent of teachers’ kntjwiedge of discrete mathematics and how they used it in the education of tieaf and hard of hearing students. F<jur
research questions guided the study:
1. How familiar are deaf education teachers with discrete mathematics topics and terms? 2. How often do deaf education teachers include discrete mathematics topics in
their mathematics instruction? 3. How often do deaf education teachers use instructional techniques that support the use of discrete mathematics, such as open-ended
investigations and real-world problem solving, in their mathematics instruction? 4. What are the relationships among teachers’ knowledge of discrete mathcMiiatics
topics, their inclusion of these topics in the classroom, and fttur background
variables: grade level, mathematics {preparation, teaching experience, and classroom setting?
Methodology The research was conducted to determine how familiar teachers of deaf and hard of hearing students are with discrete mathematics concepts, and the
fretjuency with which they present these concepts in their mathematics instruction of students with hearing loss in graties K-12.
Data Collection Data were collected by means of a survey sent to all schools for the deaf in the United States, and to I’.S. public school programs serving a minimum
of 120 deaf or hard of hearing students in gratles K-S or K-12, as listed in the Programs and Services Chart of the 2002 reference issue of the American Armals of the
Deaf. Based on these criteria, a total of 149 schools and programs were included in the initial mailing. Administrators of these programs were asked to distribute one
survey wiih ivturn envelope to the individual they considered the most effective mathematics teacher in their programs in each of four grade ranges: K-2, 3-5, 6-8, and
(if applicable) 9-12. Each school or program that returned at least one survey received a copy of Helping Children Learn Mathematics (Mathematics Learning Study
Committee. 2002).
The sLii-vey consisted of questions related to teacher familiarity with and use of the most common discrete mathematics topics and terminology (listed in Tables 1 and
2, respectively). In addition, because discrete mathematics involves the use of authentic problems, teachers were asked to complete a rating scale indicating how often
they inclutietl instructional techniques that support the use of discrete mathematics in their overall mathematics instruction. The survey was an adaptation of one
designed and used by the Leadership Program in Discrete Mathematics at Rutgers L’niversity (Rtisenstein & Kowaiczyk, 2002).
Table 1 Teachers’ Familiarity With Discrete Mathematics Topics
Topic
Scheduling events/resolving conflicts (W= 283)
Routes {N= 283)
Combinations {W= 285) ^ – , ,, .
Elections/voling’ (A/- 284) ‘Matrices-‘” (A/=279) Repeating patterns [N – 285) Generating number patterns (W = 285) Generating geometric patterns [N = 283) Chance of
an event happening (N = 284J Logic problems {N = 285) Strategic planning (A/= 282) Venn diagrams (W = 283)
Games related to logic or strategic planning {N = 284) •’Significant difference by teaching experience. “Significant difference by grade-ievel category. ‘Significant
difference by mathematics preparation. ‘Significant difference by classroom setting.
Not familiar (%) 40 12
7 38 30′ f 7 K 5 , 8 ‘ ” ‘•14 . . ,
11 52 12 31
Familiar (%)
60 ^
88 . 93 i 62 m m 93 95”i|H 92
89 78 ^ 88 69 .^H
ISO, No. 3, 2005 MtEKICAN ANNAI.S OK THE
DISCRETE MATHEMATICS IN DEAF EDUCATION
Table 2 Teachers’ Familiarity With Discrete Mathematics Terms
Term Not at all familiar familiar
Euler paths/circuits’ : Fibonacci numbers'” “^^^^^^
Fractals * Graphs (vertex and edge graphs)” ~ ‘ Graph coloring (vertex and edge graphs)” ^B^BJMP Tower of Hanoi” Chaos theory ~^^^^^^^^ Spanning trees Hamittonian
paths/circuits Pascal’s triangle^”-” Patterns. Hll^mH^ft_i Probability Recursion*^ ‘ ^^^^^^^^^^!^^ Traveling salesperson problem * Fair divisioh”‘ •^gj^m^gggg^^
Tesselations’
91
59 21 fl
^^ 85
85 96 ^IB 46
6 5 84 84
50
Note. Some rows do not total 100 because of rounding. ‘Significant difference by mathematics preparation. ‘Significant difference by grade-level category ‘Significant
difference by teaching experience
Somewhat familiar
(%)
6
27
10
11 ••••• 34
30
t4
26
Very
3
13
4
4
^0 20 Bfr5 65
2
24
Sample A total of 290 surveys were returned from 96 schools and programs serving deaf anti liard of hearing students. These included 71 from teat hers working in the
K-2 grade range. 77 from teachers in the 3-5 grade range. 73 frotn teachers in the 6-8 grade range, and 69 from teachers in the 9-12 grade range. Sixty-five percent of
returned surveys came from teachers working in center schools for deaf and hard of hearing students; 22% were from teachers working in self-contained classrooms within
public schools; 8% were from resource room teachers (in public schools); 3% were from mainstream settings. The remaining 2% of surveys came from teachers who did not
indicate a primar>’ setting. Ninety percent of the responding teachers
held a deaf education degree either at the bachelor’s or master’s level, and 92% held deaf education teachitig certification. Fourteen percent of the teachers held a
mathematics-related (either mathematics or mathematics education) degree or certification (or both), with 63% of these at the 9-12 grade-level range atid another 33%
at the 6-8 gi”ade-level range. Twelve percent of the responding teachers held both a deaf education degree and a mathematics-related degree; 8%j held both deaf
education and mathematicsrelated certifications. The majority of teachers holding a deaf education or mathematics-reiatetl degree or certification (or both) taught in
the 9-12 grade range. Overall. 6% of the teachers stated that they had taken a specific course in discrete mathematics;
all of these teachers worked in the 6-12 grade range (the majority in grades 9-12). Fifty-eight percent of the res|)onding teachers had more than 11 years’ experience
teaching students with hearing loss. Fourteen percent had 6-10 years’ experience; 25% had 1-5 years’ experience; 4% had less than 1 year of experience. (The total
percentage exceeds 100 because of rounditig.) There wete no statistically significant tiifferences in experience, setting, or deaf education degree by grade-level
category; Significant differetices, however, were found between grade-level categories with regard to posse.ssion of a tiiathematics degree (x2 = 48.523, /; < .01),
possession of a mathematics certification (x = 26.471,p < .01), and cotnpletion of a discrete mathematics course (x2 = 33.367.p < .01); more teachers in the 9-12 grade
range were represented in each of these categories.
Data Analysis Data were coded and analyzed using descriptive statistics and correlational summaries. Specifically, frequencies of response were determined for each
survey item, with chi-square and correlation analyses calculated to deterrnine interactions between background variables (grade level, mathematics preparation,
teaching experience, and classroom setting) and responses. On tests in which a significant difference was found, apprtjpriate post hoc tests (Goodman’s gamma and
MannWTiitney nonparametric tests) were performed to detertnine where the difference occurred.
Results Familiarity ^Xlth Discrete Mathematics Topics and Terms Familiarity with discrete mathetiiatics was measured in two ways: by topic
VOLUME 150. N(j. 3, 200^ AMERICAN A.NNALS OF THE
and by term. First, teachers were asked to indicate whether they were fatiiiliar with 13 of the most common discrete mathematics topics. These topics, and teachers’
responses to the question, are .shown in Table 1. More than three fourths of the teachers were familiar with all but four of the topics. Topics familiar to less than
75% of the teachers were (in de.scending order of farniliarity) mathces, games related to logic or strategic planning, elections/voting, and scheduling
events/resolving con/licts. Matrices was the oril>’ topic iov which a significant difference by gradelevel category was found (x2 = 13-644, p < .01), with 5H% of K-2
teachers, 72% of 3-5 teachers. 66/b of 6-8 teachers, and 86% of 9-12 teachere reporting that they were fatniliar with the topic.
Second, teachers were questioned on their level of |-amiliarity (“not at all familiar,” “somewhat familiar,” or “ver>’ familiar”) with 17 common discrete mathematics
terms. These terms and teachers’ levels of familiarity with them are j:)rovitied in Table 2. Of the 17 terms, 10 were unfamiliar to more than two thirds of the
reporting teachers. Less than 10% of the teachers were very familiar with each of these same 10 terms. Only five of the terms were either s<imewhat familiar or very
familiar to a majority of the teachers t)verall, and only two of those (patterns and prohahilit}-) were ver\’ familiar to a majority As indicated in Tal’)le 2, there
were statistically significant differences in familiarity by grade-level category: ‘lower of Hanoi and Tesselations at the .05 level; Fibonacci numbers, fractals,
recursion, and traveling salesperson problem at the .01 level; and Pascal’s triangle at the .001 level. Post hoc analyses of each significant relationship showed that,
overall, teachers at the lower grades tended to be less familiar with discrete mathematics terms thati teachers at the higher grades.
Table 3 Extent to Which Teachers Familiar With Discrete Mathematics Topics Used Them in Instruction
Topic Scheduling (N = 67) :..gMi^^HH^^^& Routes (W=143) Combinatorics (N=163) ^BHHHHI Elections/voting (N = 60)
Matrices’ [N^ 76) ^^^^^^^^^ Repeating patterns’ (A/= 216) Generating number patterns”” [N = 229) Generating geometric patterns (W=ia51 Ctiance of an event happening’
(/V=140) Logic problems {N-160)
«5^tfategic planning (W = 96) Venn diagrams’ (/V-159) ‘Games (N= 66) ^^^^j^^^^^^^^^^jj^ • Significant difference by mathematics preparation. ” Significant difference
by grade-level category. ‘• Significant difference by classroom setting. ” Significant difference by teaching experience.
Teachers who included topic ‘ in instruction foj
^ 39
57 IB 31 34
39
82 84
71
57 63 42
64
44
Use of Discrete Mathematics Topics in Instruction Teachers were asked how often they included discrete tnathematics topics in their instruction. Table 3 shows, by
topic, the percentage of those teachers familiar with discrete mathematics topics who included the topic in their instruction. Eight of the 13 topics were included by
more than 50% of the teachers, and just 3 were iticluded by more than 66%. Thus. 10 of the 13 tnost common topics in discrete mathematics were not presented to deaf
antl hard of hearing students in more than two thirds of the classrooms, even though the teachers themselves may have been familiar with the topics. Of course, this
result is cotiipoundc’d when those teachers who originally re|”)orted chat they were not familiar with tlistfete mathematics topics are included. When these teachers
are accounted for as well, just two topics—repeating patterns imi\ generating geometric patterns—were in
cludetl in more than two thirds of the c!assrot)ms. Further analyses by gradelevel category showed statistically significant differences for just two topics:
generating in/mber patterns :\m\ generating geomet ric patterns, with more teachers in the lower grades including these topics in instruction. The most c<.)mmon reason
given by K-i2 teachers for not including discrete mathematics topics in instruction was that the mathematics level of discrete mathematics concepts was too high for
their students. Table 4 is a list oi’ the reasons given by teachers, from most to least prevalent.
Use of instructional Techniques That Support Inclusion of Discrete Mathematics Topics Because certain instructional techniques, such as open-etided investigations and
reai-wodd problem-solving activities, lend them.selves to the use of discrete mathematics topics rnore than
Mi- 150. No. 3. 200S A.MnRiCA.N AI-S OF I Hi-. Dr.-\i
DISCRETE MATHEMATICS IN DEAF EDUCATION
Table 4 Teachers’ Reasons for Not Using Discrete Mathematics Activities in Instruction
Reason
Math^natics level too high Not included in curriculum Don’t have tjme *i Don t know how to apply at my grade level
Other : Don’t have in ili ii il il^B^M Don’t see relevance
Number of citations
65 36
24
16
5 Notes. N = 201. Percentages do not totai 100 because of rounding.
-aa—^ 18
12
8
2
Table 5 Teachers’ Rate of Use of instructional Strategies That Support or Inhibit Inclusion of Discrete Mathematics
Rate of use
Supportive instructional technique •B of writing as a reguiar part of .c| (e.g., math journals Use of calculators and/or computers”” Use of alternative assessments
{e.g., portfolios or self-assessment) Use of mathematics topics integrated with other curriculum areas’ Use of mathematics manipulatives” Small groups working
collaboratively on problems, challenges, or projects Students inventing their own methods tor solving problems” Students working on open-ended investigations
Student-led discussions and/or presentations” Students working on real-world problems or applications in mathematics’
Students working on a problem fof ti aore class peri
Never(%)
Twice a Three times week or a week or less (%) more (%) Daily (%)
16
Inhibitive instructional technique Never(%)
Teapher lectures/presentations’ Teacher-led whole-class discussions
Indivtduai students working on worksheet, exercises, and problem s Students following detailed instructions to complete structured projects 41
Note. Some rows do not total 100 because of rounding. ‘ Significant difference by grade-level category. ” Significant difference by mathematics preparation. •=
Significant difference by teaching experience.
Twice a Three times week or a week or less (%) more (%) Daily (%)
30 25 ” 37’^ 31 27 34
41 10
others, the survey included questions on teachers’ instructional pedagogy. Table 5 lists techniques that either support or inhibit the teaching of discrete
mathematics. A strategy that is used three titiies per week or more is defineti in the present study as being frequently used. We found that 10 of 11 instructional
techniques that support the use of discrete mathematics were used frequently by less than 50% (if the surveyed teachers. Use ofmatfoematics manipulatives was the only
technique employed frequently by more than half of the teachers. By contrast, more than 60% of teachers reported that they tnade freqiient use of three of four
strategies that inhibit the inclusion of discrete mathematics instruction: teacher lectures/presentations; teacher-led whole-class discussions; and individual students
working on u <ork sheets, e.xercises. and problem sets. Only one of the inhibitive strategies, students following detailed instruction to complete structured projects,
was applied frequently by less than half of the respondents. It should al.so be noted that more than 40% of teachers overall reported that they never applied four
techniques supportive of di.screte mathematics: use of writing as a regular part of class; use of alternative assessments; students working on open-ended
investigations; and student-led discussions and/or presentations. N(ir did they allow their students to use the technique students working on a problem for two or more
class periods. Further analyses indicated significant differences by grade level for i4se of mathematics topics integrated with other curriculum areas {p < .01);
students working on real-world problems or applications in mathematics (p < .01); teacher lectures/presentations (p < .05); use of calculators and/ or computers (p <
.000); and use of tnathematics manipulatives (p <
VOLUME 150, No. 3, 2005 AMERICAN AN’N.M.S OF THE DEAF
.000). The analyses showed that K-2 teachers were more likely to integrate mathematics topics with other curriculum areas; that teachers in the higher grades used
real-worki problems, lectures and presentations, and calculators or computers more frequently than those teaching in the lower grades; and that teachers in the higher
grades usetl mathematics mani|iulatives less frequently that those in the lower grades.
Familiarity With and Use of Discrete Mathematics Topics in Relation to Background Variables CotreiationaJ analyses were performed to determine the relationship between
familiarity with and use of discrete iTiathematics topics, terms, and instructional techniques, and the background variables mathematics preparation, teaching
experience, and classroom setting. Very few of the analyses indicated significant differences with respect to these three background variables (see Tables 1, 2, and
3). Post hoc analyses were performed it! instances when significance was shown, to further investigate the tiature of the difference.
Mathematics Preparation (The results reported in the present .section should be regarded with caution, as few teachers in the sample had mathematics preparation.) Data
indicatetl that preparation in mathematics made little difference in teachers’ familiarity with discrete mathematics topics, their use of these topics in instruction,
or their pedagogy. Those teachers without mathematics preparation were just as likely to be familiar with discrete mathematics topics as those with preparation. The
one exception was with the topic matrices. Approximately 20% more teachers with preparation were familiar with
this topic than teachers without preparation (p < .05). No statistically significant differences were evident in teachers’ use of 11 of 13 topics of discrete
tnathematics. The exceptions were matrices and repeating patterns. Both of these topics were included by more teachers without mathematics preparation {j) < .05) than
with. In addition, there were no statistically significant differences on the basis of mathematics preparation for 10 of the 15 given instructional techniqties. The
five exceptions were that teachers with preparation were tnore likely t(i use lectures and presentations (p < .05); more likely to use calculators and computers (/; <
.05); less likely to use mathematics manipulatives (p < .001); less likely to use student-invented methods for problem solving (p < .05); and less likely to use –
Students-led discussions or presentations (p < .05). It should be noted, however that teachers with mathematics preparation taught at the higher grades, which may have
been a factor in the results.
Regarding familiarity with common terms of discrete mathematics, differences were founti, however. More teachers with preparation than those without were familiar with
approximately half of the terms given. These terms included combinatorics, Euler paths/circuits. Fibonacci numbers, fractals, Pascal’s triangle, recursion, trai’eling
salesperson problem, and fairdivision (p < .001). Statistical analyses were performed on the responses of those teachers who had completed a course specifically in
discrete mathematics. Results showed these teachers to be more familiar with the topic matnces (p < .05), and more familiar with the terms combinatorics (p < .000),
Euler paths/circuits (p < .000), fractals (p < .01), Fibonacci numbers {/} < .000), Tower of Hanoi (p < .05),
Pascal’s triangle ip < .000), recursion (p < .000), traveling salesperson problem (p < .O’S), fair division ip < .000), Tessellations (p < .05), and spanning trees (p
< .000). Also, they were more likely to use calculators or computers in instructif)n (p < .01) and to integrate mathematics with other topic areas (p < .01).
Teaching Experience In most instances, no statistically significant difference was found in the data regarding years of ex|)erience teaching deaf and hard of hearing
students. All exceptions involved teachers with less than 1 year of experience. It should be noted, however, that teachers with less than 1 year of experience
constituted just 4% of the sample.
No statistically significant differences were found in familiarity with 12 of the 13 topics of discrete mathematics with regard to teachers’ experience. The one
exception was elections/voting (p < .01). More teachers with less than 1 year and greater than 10 years of experience were familiar with this topic. There were no
statistically significant differences for teachers’ use of 12 of the 13 topics of discrete mathematics. The one exception was that teachers with less than 1 year of
experience were less likely to include Venn diagrams in instruction (p < .05). While no statistically significant differences occurred for teachers’ familiarity with
13 of the 17 terms of discrete mathematics, teachers with less than 1 year of experience were significantly more familiar with the 4 remaining terms: graphs (vertex
and edge graphs), p < .01; fair division, p < .0\; graph coloring (vertex and edge graphs), p < .05; and Pascal’s triangle, p < .05.
There were no statistically significant differences for 14 of the 15 instructional techniques, the exception being that teachers with less than 1 year of experience
made more fre
F 150, No. 3,2005 AMERICAN ANNALS OF TfiE DEAE
DISCRETE MATHEMATICS IN DEAF EDUCATION
t|uent use ofcalculatt)rs or computers (Jj < .05).
Classroom Setting As with teaching experience, no statistically significant differences were found in the majority of analyses for classroom setting. There were no
statistically significant differences for teachers’ fatiiiliarity with 12 of the 13 topics of discrete mathematics. The one exception w^as matrices: More mainstream
teachers were not familiar with this topic (p < .05). There were no statistically significant differences for teachers” use of 11 of the 13 topics of discrete
mathematics in the classroom. The exceptions were generating number patterns and chance of an event happening (p < .05). Mainstream teachers used both these topics
less frequently in their instruction. Teachers in self-contained classrooms were less likely to use chance of an event happening than teachers in other settitigs.
There were also no statistically significant differences in familiarity for 16 of the 17 terms. The exception was Pascal’s triangle. Teachers at center schools were
more familiar with this term (p < .05).
No statistically significant differences were found by classroom setting for any of the 15 instructional techniques. Regardless, within each setting work sheets were
usetl fre(.|uent!y by more than 60% of teachers, and lectures and presentations were used frequently by at least 50% of the teachers. By contrast, open-ended
investigations were used frequently within each setting by just 10%-17% of teachers, and real-world problems were used frequently by less than 50% of the teachers.
Discussion Results from the present study show that although deaf education teachers are familiar with manv discrete mathe
matics topics, they do not include them in instruction primarily because they feel that the concepts are too “high level” for their students. In addition, regardless
of their levels of familiarity and use. tleaf education teachers are not fatniliar with the termintilogv’ of the discipline; nor is their mathematics teaching
structured in a manner that wt)uid provide opportunities for the use of discrete mathematics topics and activities. With ver>’ few exceptit;)ns, possessit)n of a
mathematics-relateti degree or certificate, experience in teaching deaf and hard of hearitig students, and classroom setting yielded no statistically significant
differences in the results. Teacher familiarity and use of discrete mathematics tt^pics did tend to vary somewhat based on grade level, however, with teachers at the
upper grade levels (9-12) being more familiar with these concepts and including them more frequentK’ in itistruction. Overall, teachers at the lower grades tended tt>
have no fatniliarity with discrete mathematics terms, t)r less familiarity with discrete mathematics terms than teachers at the higher grades. These fintlings have
implications for those involved in the education of deaf and hard of hearing students in three areas: curriculum, teacher preparatitin, and teacher expectations.
The study results ct)nfirm the ct)ntinued use of a traditional mathematics curriculum that is both outdated and inapprtipriate ft)r preparing deaf and hard of hearing
students for today’s world. If deaf education teachers hope to improve the performance of their students in tnathematics, they must change the curriculum and
instructional habits that have failed students for so lotig, and take heed of the NCTM recommendations and research on best practices. Teachers must embrace an
instructional stvle
that is reform based, less teacher dependent, and more responsive to students’ developmental levels and interests, with the curriculutn including tliscrete mathematics
topics at all levels. Within this domain, other, more traditit)nal mathematical concepts and algorithms can be used in a |iurj)t)seful nianner, thus showing students
the application of mathematics to authentic situations and preparing them for life in the real wodd. In order to transititin from the use of a traditional mathematics
curriculum anti practices, teachers iti bt”)th inservice and preservice settings must be properly prepared. Although the study results did not show broad, significant
increases in use t)f and familiarity with tliscrete mathematics among teachers with preparation, other studies do show that teacher preparation and professional
develt)pment practices do make a difference in mathetnatit:s instruction and pedagog\’ (Kelly et al., 2003; Pagliaro & Ansell, 2002). The contrary results found in the
present study may indicate, then, that teachers have not had the adequate preparation, particularly in discrete tnathematics, required to develt)p an understantling of
the relevant topics, nt)r a knt)wledge of how to incorptjrate them into instruction. Support ftir this conclusion cotnes frt^m results showing that those teachers who
ctjmpleted a course in discrete mathematics were significantly more familiar with several topics and terms, and made frequent use of twt”) instructional techniques
that support discrete mathematics, including integrating mathematics with other ttjpic areas.
Teacher |ireparatit)n programs in {.leaf education, thereft^re, must require courses in tnathetnatics and tnathematics education for preservice teachers that are
taught by quality instructors who model the instructional
VOLUME 150, No. 3. 2UU5 AMERICAN ANNALS OF THH DEAE
pedagtigy that teachers at-e expected to use in their classrooms. These courses shoukl include topics of discrete mathematics, t!iscussion pursuant to the methodology
tif constructivist teaching, and specific information related to the unit|ue neetls of deaf anti hard of hearitig students. In additit)n, those teachers alreatly in
the classroom should partake of prt)fessional develtipment programs in tnathematics, particularly discrete mathematics: not simply 2-ht)ur workshtips or “one-shot”
seminars, but well-designeti institutes of learning that includes intense stutty, classrootn support, anti j^rofcssional guitlance by experts in the field. During such
professional development, teachers shtnild have opptjrtunities to learn mathetnatics topics presented at a challenging level in content-based workshops. Teachers
shoulti have oppt)rtunities in stutly-group settings to engage in thought-provoking problem-solving activities that draw upon such t(.)pics. This shoulti be f<.>llowed
by oppt)rtunities for teachers tt) discuss ways to integrate the topics into instruction at various grade levels. Through this process, teachers will ctime to
utitlerstand how concepts can be presented at lower grade anti developmental levels in a pedagogically appropriate manner that will build tin experience and establish
a solid foundation for mtire cotnplex concepts.
One finding frt)m the present study is that teachers, even those who were familiar with discrete mathematics topics, tlid not include these topics in itistruction
because they believetl the mathematics ievel to be HM high ft)r their students with hearing loss. This belief mav he indicative t)f the low ex
[ deaf etkicatioti teachers have of their students. Therefore, along with im[-)lementing changes in curriculum anti preparation, teachers need to recognize that their
students, at any le\’el, can do mathematics inckitled in discrete mathematics. Classroom evitience indicates that students are indeed capable of using and
understantling discrete mathematics topics. Ft)r exatnpie, Simmt atid Davis (199H) showetl how an acti\ity that invoKt^s students in a fractal-generatitig exercise can
be used at the seccjntlaiy level to explore number systems, sequences, recursion, iteratit>n, fractitinal ditnensions, and measutvtnetit concepts. We have offeretl an
additional exatnpie of the use of discrete mathetnatics at the elementary level (Kritzer ik Pagliaro, 2003), describing a series tif discrete mathematics activities
that were dtine with a group of elementary-school deaf atitl hard of hearing students.
With raised expectations, well-prepared teachers, and a curriculum that inclutles discrete mathematics and in* structional techniques that afft)rd intellectual gi’owth
and mathematical development, deaf and hard of hearing students will be better equipped to solve the challenges of our wtirltl, today and tomt)rrow.
Note We wish to thank Dr. Elaine Ruhenstein kw her statistical assistance with the present stutly.—The Authors
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