Teachers' Reasons for Instructional Decisions

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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