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Teachers' Reasons for Instructional DecisionsAuthor(s): PETER L. GLIDDENSource: The Mathematics Teacher, Vol. 84, No. 8 (NOVEMBER 1991), pp. 610-614Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27967332 .
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Teachers' Reasons for
Instructional Decisions By PETER L GLIDDEN
One
of the lessons learned from the "new math" reform movement of the sixties
is that effecting lasting change requires more than developing curriculum materials at a national level for adoption at the local level (National Research Council 1989;
NCTM 1989; Mumme and Weissglass 1989). Lasting reform also requires directly involv
ing teachers in curriculum development so
that they have "ownership" of the product (National Research Council 1989). This
ownership is necessary because teachers act as curriculum filters (Holmes Group 1986; Porter et al. 1988; Romberg 1988).
If teachers should be involved in mathe matics curriculum development because
they filter the curriculum, then an impor tant question arises: "What criteria do teachers use to decide what and how math ematics should be taught?" As a practical matter, teachers are more likely to adopt and teach curricula (including those based on the Curriculum and Evaluation Stan
dards) that meet their criteria than curric ula that do not. Data from the Second Inter national Mathematics Study (International Association for the Evaluation of Educa tional Achievements, Second International
Mathematics Study 1985a, 1985b, 1989) of fer reasons behind teachers' instructional decisions.
Source off the Data
In the Second International Mathematics
Study eighth- and twelfth-grade mathemat ics teachers completed questionnaires about
Peter Glidden teaches undergraduate and graduate mathematics education courses at the University of Illi
nois at Urbana-Champaign, Champaign, IL 61820. His
research interests include cognitive modeling of mathe
matics learning, curriculum development, and teachers'
instructional decisions.
their teaching decisions. Both eighth- and
twelfth-grade teachers were asked what
teaching methods they used and why they used them. (See fig. 1 for examples of the
questions asked.) Additionally, twelfth
grade teachers were asked if they covered certain topics and why they covered them. Teachers could select more than one reason
in response to each question.
Eighth-grade teachers used different criteria than twelfth-grade teachers.
Teachers who used a particular teaching method or covered a particular topic were
asked if they did so because it was (a) treated in the textbook, (6) included in the
syllabus or on an external examination, (c) well known to the teacher, (d) easy to teach, (e) easy for students to understand, ( f) en
joyed by students, (g) related to prior math
ematics, or (h) useful later. These reasons
for using a teaching method or covering a
topic are classified as positive reasons.
Teachers who did not use a particular method or cover a particular topic were
asked to indicate why not, choosing from the reverse of the reasons listed in the previous paragraph. For example, a teacher could re
port that a particular method was not used because it is not in the textbook or it is difficult for students to understand. Teach ers' reasons for not using a teaching method or not covering a topic are classified as neg ative reasons.
Results
To make the results easier to understand, we report the mean number of times a rea
610 Mathematics Teacher
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Eighth-Grade Teachers
Measurement: Value of I had my students measure and find the ratio of the circumference to the diameter of a number of circular objects, and ap
proximate Cid for any circle.
Geometry: Sum of angles in a triangle is 180?
My students cut the angles off a triangle and arranged them in a straight line.
Algebra: Addition of integers Addition by number line. I used the num ber line to add integers.
Twelfth-Grade Teachers
Algebra: How logarithms were introduced Inverse Function Base. A logarithmic function is defined as the inverse of the
exponential function
fix) ? 10*.
Consider the graph of the log function. It is observed for several specific problems that the ordinate at* = ab is equal to the sum of the ordinates at = a and at* =
b. Thus
log ab = log a + log 6.
Source: International Association for Evaluation of Educational Achievement, Second international Mathematics Study. Instrument Book: Classroom Process Questionnaires. Technical report 5,
Fig. 1, Examples of the questions asked eighth- and twelfth-grade teachers about the teaching methods they used.
son was cited as a percent of the total num
ber of questionnaires. (The reader may no
tice that the mean number of reasons
reported by twelfth-grade teachers is consid
erably greater than the mean number of
reasons reported by eighth-grade teachers.
This difference is attributed to the instruc
tions on the questionnaires: twelfth-grade teachers were asked to "circle as many [rea
sons] as apply" for covering topics and to
"indicate the reason(s)" for using particular
teaching methods; eighth-grade teachers
were asked to mark the "primary rea
son(s).") In figure 2, for example, "Easy to
Understand" has a value of about 15 percent for positive reasons. In other words, eighth
grade teachers reported that when they used
a particular teaching method, about 15 per cent of them used it because it was easy for
students to understand. Also in figure 2,
"Easy to Understand" has a value of about 5
percent for negative reasons. In other words,
eighth-grade teachers reported that when
they did not use a particular teaching
method, about 5 percent of them did not use
it because it was difficult for students to understand.
Eighth-grade teachers: Teaching methods
Figure 2 illustrates eighth-grade teachers'
reasons for using particular teaching meth
ods listed in decreasing order of positive rea
sons from left to right. From this figure we see that although "Text" (inclusion in the
Source: International Association for Evaluation of Educational Achievement, Second International Mathematics Study. Uniteti States?Population A Fre quencies. Vol. 2 (USAA2)
Fig. 2. Eighth-grade teachers' reasons for using par ticular teaching methods (N teachers = 235, United States)
November 1991 611
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textbook) is an important reason for using a
teaching method, it is not the most impor tant reason. "Well Known," "Easy to Under
stand," and "Useful Later" are cited more
frequently than "Text." In other words, in
selecting particular teaching methods,
eighth-grade teachers selected methods that
were, in order of reported frequency, (a) well
known to the teacher, (b) easy for students to understand, and (c) useful later. The fourth most frequently reported reason for
using a teaching method was that the method was used in the textbook.
Eighth-grade teachers' most frequently reported reasons for not using particular teaching methods are that these methods were (a) not treated in the textbook or (6) not well known to the teacher. Teachers also
reported that students' enjoyment of a teach
ing method had little bearing on its use in the classroom.
Twelfth-grade teachers: Teaching methods
Twelfth-grade teachers used teaching meth
ods because they were, in order of reported frequency, (a) useful later, (6) treated in the
textbook, and (c) well known to the teacher
(see fig. 3). Twelfth-grade teachers' most fre
quently reported reasons for not using teaching methods are that the methods (a) were not treated in the textbook, (b) did not appear in the syllabus or on an external
-g Positive
g 40 -
Negative
?lni.
50 h
Note: Not all teachers completed all question naires. The mean number of teachers com
pleting each questionnaire was 156 for the United States. Source: International Association for Evaluation of Educational Achievement, Second International Mathematics Study. Classroom Process Question naires: item Level Data. United States?Population technical report 3B.
Fig. 3. Twelfth-grade teachers' reasons for using
particular teaching methods
Positive
Negative
<2
Note: Not all teachers completed all question naires. The mean number of teachers com
pleting each questionnaire was 156 for the United States. Source: International Association for Evaluation of Educational Achievement, Second International Mathematics Study. Classroom Process Question naires: Item Level Data. United States?Population technical report 38.
Fig. 4. Twelfth-grade teachers' reasons for covering particular topics.
examination, or (c) were not well known to the teacher. Teachers also reported that stu
dents' enjoyment of a teaching method had little bearing on its use in the classroom.
Twelfth-grade teachers: Topic coverage
Figure 4 illustrates that twelfth-grade teachers reported various reasons for cover
ing topics. Their most frequently reported reasons for selecting topics are that the top ics were (a) useful later, (b) covered in the
textbook, (c) included in the syllabus or on an external examination, (d) well known to
the teacher, or (e) related to the students'
prior mathematics. Conversely, twelfth
grade teachers' most frequently reported reasons for not teaching topics are that the
topics were (a) not included in the syllabus or (b) not covered in the textbook. Twelfth
grade teachers reported that whether a topic was easy to understand, easy to teach, or
enjoyed by students had little bearing on whether it was taught.
Discussion
Negative reasons
Considerable agreement is evident between
eighth-grade and twelfth-grade teachers in
their reasons for not using a particular method or covering a specific topic. The data
suggest that the textbook and syllabus or
612 Mathematics Teacher
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external examination exert a strong limit
ing influence on teachers. Teachers report that they do not use methods or teach topics that are not in the textbook or are not in
cluded in the syllabus or on an external
examination. The data also suggest that
teachers, particularly eighth-grade teach
ers, do not use teaching methods if they are
unfamiliar with them. Additionally, eighth
grade teachers report that they do not use
methods believed to be difficult for students.
Positive reasons
Much agreement can be found between
eighth-grade and twelfth-grade teachers in
their reasons for using a particular method.
Both eighth- and twelfth-grade teachers re
port that they use certain methods because
they are well known, useful later, or in
cluded in the textbook. Moreover, twelfth
grade teachers report that they cover certain
topics for these same reasons. Note, how
ever, that although the text is influential, it
is not the most frequently cited positive rea
son (further confirming the findings of Por
ter et al. [1988]). Teachers displayed some differences in
their reasons for selecting topics and meth
ods. "Useful Later" was the most frequently cited reason for teaching twelfth-grade top ics. As for negative reasons, "Syllabus/ External Exam" and "Text" were also more
influential in selecting twelfth-grade topics than eighth- and twelfth-grade methods.
"Easy to Understand" was more frequently cited by eighth-grade teachers than by
twelfth-grade teachers. The importance of "Useful Later,"
"Syllabus/External Exam," and "Text" for
twelfth-grade topics is probably due to the
nature of twelfth-grade mathematics courses in the United States. Twelfth-grade students will probably study postsecondary
mathematics and revisit twelfth-grade top ics in more advanced mathematics courses.
Advanced Placement courses follow a sylla bus in preparation for an external examina
tion. Other twelfth-grade mathematics
courses are likely closely to follow a syllabus or textbook because little consensus has
evolved on what content (e.g., analytical ge
ometry, finite mathematics, functions, lim
its, differentiation, integration, probability and statistics) is appropriate for a twelfth
grade mathematics course.
The preponderance of eighth-grade teachers citing "Well Known" and "Easy to
Understand" can be attributed to the popu lation they serve and the curricula they use.
Mathematics is required of all students in
the eighth grade, and it is a gateway to high school mathematics. Therefore these teach
ers are likely more concerned with methods
that are easy for the majority of students to
understand not only to promote learning but
also to reduce frustration, thereby helping to
maintain order in the classroom. The em
phasis in the eighth-grade mathematics cur
riculum on reviewing arithmetic rather
than presenting new material is well docu
mented (Flanders 1987; McKnight et al. 1987). It is plausible, therefore, that eighth
grade teachers teach what is well known, that is, what they have taught many times
before and what students have seen many times before.
Developing curricula is only part of what is needed*
Implications for curricular reform
Overall, these data confirm the view that
developing curricula based on the Curricu
lum and Evaluation Standards is only one
part of the total effort required to reform
school mathematics. Moreover, these data
shed new light on what criteria teachers use
in their instructional decisions. Teachers
need a sound mathematical understanding of the proposed new mathematics content
(e.g., discrete mathematics, probability, and
statistics), and they need to know how this
new content is useful later, both in applica tions and for higher mathematics. Teachers
also need to see that the proposed new teach
ing and evaluation methods (e.g., use of cal
culators, cooperative groups, and mathe
matical modeling and multiple means of
assessment) make understanding mathe
matics easier for students.
November 1991 613
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We suggest, therefore, that school dis
tricts, state boards of education, professional organizations, colleges, universities, and others take the lead in offering opportunities for teachers to increase their mathematical and pedagogical understanding so that they can make informed curricular decisions. These opportunities can consist of academic
courses, in-service workshops, summer in
stitutes, conferences, and making accessible to classroom teachers results of relevant re search on teaching and learning mathemat ics.
If the vision of mathematics education
presented in the Curriculum and Evaluation Standards is to become a reality, then teach ers need to make informed decisions about what mathematics they teach and how they teach it. Empowering teachers will not be
easy, and it cannot be done overnight. As
Romberg (1988) discussed so well, mathe matics teachers need not only to act as pro fessionals but to be treated as professionals as well. Teachers filter the mathematics cur riculum. Understanding and addressing their instructional decisions is necessary to
effecting lasting curricular reform.
REFERENCES Flanders, James R. "How Much of the Content in Math
ematics Textbooks Is New?" Arithmetic Teacher 35
(September 1987):18-23.
Holmes Group. Tomorrow's Teachers. East Lansing, Mich.: The Holmes Group, 1986.
International Association for the Evaluation of Educa tional Achievement, Second International Mathe
matics Study. Classroom Process Questionnaires: Item Level Data. United States?Population tech nical report 3. Champaign, 111.: The Association, 1985a.
-. Instrument Book: Classroom Process Question naires. Technical report 5. Champaign, 111.: The As
sociation, 1985b.
-. United States?Population A Frequencies. Vol. 2 (USAA2). Champaign, 111.: The Association, 1989.
McKnight, Curtis C, F. Joe Crosswhite, John A. Dos sey, Edward Kifer, Jane O. Swafford, Kenneth J.
Travers, and Thomas J. Cooney. The Underachieving Curriculum: Assessing U.S. School Mathematics from an International Perspective. Champaign, 111.:
Stipes Publishing Co., 1987. Mumme, Judith, and Julian Weissglass. "Implement
ing the Standards: The Role of the Teacher in Imple menting the Standards." Mathematics Teacher 82 (October 1989): 522-26.
National Research Council. Mathematical Sciences Ed ucation Board. Everybody Counts: A Report to the
Nation on the Future of Mathematics Education.
Washington, D.C.: National Academy Press, 1989.
National Council of Teachers of Mathematics, Commis sion on Standards for School Mathematics. Curricu lum and Evaluation Standards for School Mathemat ics. Reston, Va.: The Council, 1989.
Porter, Andrew, Robert Floden, Donald Freeman, William Schmidt, and John Schw?le. "Content De terminants in Elementary School Mathematics." In
Perspectives on Research on Effective Mathematics
Teaching, vol. 1, edited by Douglas A. Grouws and Thomas J. Cooney, 96-113. Reston, Va.: Lawrence Erlbaum Associates and National Council of Teach ers of Mathematics, 1988.
Romberg, Thomas A. "Can Teachers Be Professionals?" In Perspectives on Research on Effective Mathematics
Teaching, vol. 1, edited by Douglas A. Grouws and Thomas J. Cooney, 224-44. Reston, Va.: Lawrence Erlbaum Associates and National Council of Teach ers of Mathematics, 1988.
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