____ THE
______ MATHEMATICS ___
_________ EDUCATOR _____
Volume 16 Number 2
Fall 2006
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editor
Kyle T. Schultz
Dear TME Reader,
Associate Editors
Rachael Brown
Na Young Kwon
Catherine Ulrich
Production
Ginger A. Rhodes
Advisor
Dorothy Y. White
MESA Officers
2005-2006
President
Rachael Brown
Vice-President
Filyet Asli Ersoz
Secretary
Eileen Murray
Treasurer
Jun-ichi Yamaguchi
NCTM
Representative
Ginger A. Rhodes
Undergraduate
Representative
Laine Bradshaw
Rachel Stokely
Along with the editorial team, I am very happy to present a new issue of The Mathematics Educator,
the second and final issue of Volume 16. In it, I hope you will find research and commentary that spark new
ideas and build upon the existing dialogue and research of the mathematics education community.
In accordance with the mission of TME, this issue presents a variety of viewpoints on a broad spectrum
of issues in mathematics education. From his observation of middle school students working on algebraic
tasks, David Slavit explores the mechanisms through which these students create meaning from their
endeavors and notice algebraic properties. Mehmet Ozturk and Kusum Singh share research on the effects of
various socioeconomic factors, such as parental involvement, educational aspirations, and mathematics selfconcept, upon the choices students make with regard to taking advanced high school mathematics courses.
This issue also features articles and editorials focusing upon teacher education. Lovemore Nyaumwe
and David Mtetwa present their investigation of collaborative assessment of student teachers in Zimbabwe.
In this model, both the college lecturer and a fellow student teacher are responsible for assessing a student
teacher’s classroom practice. Alison Castro discusses her efforts to improve, through interventions in content
and methods courses, elementary preservice teachers’ conceptions of mathematics curriculum materials.
In an invited editorial, Lee Peng Yee reflects upon how he and other educators in Singapore are
addressing the issue of mathematics subject knowlege in teacher training programs. In this issue’s ‘In Focus’
piece, Ginger Rhodes, Jeanette Phillips, Janet Tomlinson, and Martha Reems, who have worked together in
mentoring student teachers, share two important ideas for improving the student teaching experience. In
addition, they pose three questions that give teacher educators an opportunity to reflect upon their practice.
The production of this issue is not a solo effort. I would like to thank the reviewers, authors, and faculty
members whose work and advice have shaped the presentation of ideas found in the pages that follow. In
particular, I extend my gratitude to our editorial and production staff, who has made this experience
rewarding and memorable.
On a personal note, I would like to take this opportunity to remember John Van de Walle, a colleague
and leader within the mathematics education community, who recently passed away. John may be best
known for his mathematics methods textbooks that are used in pre-service education programs across the
United States and beyond its borders. He also authored numerous articles and chapters in NCTM
publications and frequently shared his vision of mathematics education at conferences.
John served in numerous capacities within NCTM, including as a director and as a committee member.
It is through this latter capacity that I had the opportunity to meet and work with him, as we served together
on NCTM’s Learning, Teaching, Curriculum, and Assessment Committee. During this time, I found John to
be insightful and passionate in his work. I admired his leadership, vast wealth of knowledge, and sense of
humor. As a mentor, his guidance and encouragement was invaluable as I transitioned from the high school
classroom to graduate school. I, along with many others, will miss John and cherish fond memories of him.
I am dedicating this issue in John’s memory and wish to extend condolences from us here at TME to his
family, friends, and colleagues.
Kyle T. Schultz
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@uga.edu
www.coe.uga.edu/tme
About the Cover
This issue’s cover features a photograph of a stellated octahedron, an octahedron with a tetrahedron constructed on
each of its faces. Joseph Harris, an Algebra 1 student at Clarke Central High School in Athens, Georgia, constructed
this polyhedron from folded construction paper. A talented artist, Harris has constructed a number of polyhedra under
the guidance of his teacher, Stephen Bismarck.
This publication is supported by the College of Education at The University of Georgia
____________ THE ________________
___________ MATHEMATICS ________
______________ EDUCATOR ____________
An Official Publication of
The Mathematics Education Student Association
The University of Georgia
Fall 2006
Volume 16 Number 2
Table of Contents
2 Guest Editorial… Mathematics for Teaching or Mathematics for Teachers?
LEE PENG YEE
4 Uncovering Algebra: Sense Making and Property Noticing
DAVID SLAVIT
14 Preparing Elementary Preservice Teachers to Use Mathematics Curriculum
Materials
ALISON M. CASTRO
25 Direct and Indirect Effects of Socioeconomic Status and Previous Mathematics
Achievement on High School Advanced Mathematics Course Taking
MEHMET A. OZTURK & KUSUM SINGH
35 Efficacy of College Lecturer and Student Peer Collaborative Assessment of InService Mathematics Student Teachers’ Teaching Practice Instruction
LOVEMORE J. NYAUMWE & DAVID K. MTETWA
43 In Focus… Mentor Teachers’ Perspectives on Student Teaching
GINGER A. RHODES, JEANETTE PHILLIPS, JANET TOMLINSON, MARTHA
REEMS
47 Upcoming conferences
48 Submissions information
49 Subscription form
© 2006 Mathematics Education Student Association
All Rights Reserved
The Mathematics Educator
2006, Vol. 16, No. 2, 2–3
Guest Editorial…
Mathematics for Teaching or Mathematics for Teachers?
Lee Peng Yee
In Singapore, one topic of great interest in
mathematics teacher education is the subject
knowledge (SK) in mathematics of teachers. This topic
has arisen in other places, often under the name
“mathematical knowledge for teaching”, or “MKT.”
Many mathematicians and mathematics educators
believe that teachers should be given more content
knowledge in mathematics. Such content knowledge
should not be just higher mathematics but mathematics
that is connected to mathematics teaching. In other
words, it should be mathematics for teaching.
Many books have been published on this subject in
the United States during the past few years. McCrory
(2006) provides a review of twenty such books. In
addition, the Fall 2005 issue of American Educator
features the topic of mathematics for teaching (for
example, see Ball, Hill, & Bass, 2005).
Issues
At Singapore’s National Institute of Education
(NIE)1, at least 8 years ago, we introduced a course on
SK in mathematics for elementary school preservice
teachers in one program. It has presently been extended
to all three training programs for elementary preservice
teachers. For the past few years, we have engaged in a
discussion on what to teach in SK, how to teach it, and
who should be teaching it. It is a controversial subject,
at least among some mathematicians and mathematics
educators in Singapore. I shall list the five issues
currently under discussion.
Issue 1: Is SK mathematics for teaching or
mathematics for teachers?
It is generally agreed that we should give teachers
more mathematics than what they need. The question is
how far we should go. For example, should we teach
the topic of base-five numbers? Some say yes, some
say no, and others a qualified yes, meaning just the
concept and not technical aspects such as the four basic
Lee Peng Yee has taught at universities in different countries for
more than 40 years. He supervised more than 20 PhD students in
mathematics. He was heavily involved in planning the school
mathematics syllabus in Singapore and the degree program for
teachers at the National Institute of Education, Singapore.
2
operations. The argument is whether SK should be a
course on mathematics for teaching or mathematics for
teachers. If it is the former, we cover only the topics
taught in elementary school. If it is the latter, we
should go beyond what is being taught in the
classroom. The idea behind mathematics for teachers is
not only to learn mathematics but also to learn enough
mathematics so that teachers become more confident
when teaching.
Currently, the SK course being taught at NIE is
meant to be a course on mathematics for teachers. One
rationale is if we want our school pupils to attempt socalled challenging problems, then teachers themselves
should have tried such problems. Furthermore, the
problems should be challenging to teachers; that is,
challenging at the level of teachers, not pupils. If
teachers have never tackled problems challenging to
them, how will they understand the difficulties that
their pupils may encounter? Hence we must give them
more mathematics beyond the elementary school.
Issue 2: Is SK a course in mathematics or a course in
mathematics education?
By a course in mathematics, I mean the emphasis
is on mathematics and rigor. By a course in
mathematics education, I mean topics in the course
may include problem solving heuristics. This does not
mean that the two do not intersect. Of course, they do.
The difference is in the approach. The mathematical
approach may make use of heuristics implicitly,
whereas the pedagogical approach may begin with
heuristics and illustrate the heuristics using examples
in mathematics. At NIE, it depends on who is teaching
the course. Since the SK course at NIE is often taught
by someone whose background is more mathematical
than pedagogical, the approach naturally tends to be
more mathematical.
Issue 3: Who should be teaching the SK course?
This issue is connected to Issues 1 and 2 above.
Ideally, the SK course should be shared between a
mathematician and a mathematics educator. In reality,
it is often taught by one or the other and not jointly.
The background training of the instructor seems to
determine how the course will be taught. Though the
Mathematics for Teaching or Mathematics for Teachers?
course outline may be the same, once inside the
classroom, the lessons look different. Sometimes we
wonder whether it is necessary for it to be taught in a
uniform manner. Perhaps not.
Furthermore, the ancient Chinese introduced the
concept of diameter before radius. Indeed, there is a
word for diameter in Chinese, but not for radius.
Radius in Chinese is simply called half-diameter.
Issue 4: Is rigor a major learning objective for the SK
course?
Example 2
There are different possibilities for the degree of
mathematical rigor used with preservice teachers. For
example, should we use different notations for an angle
and the measure of an angle? Should we distinguish
between a line segment and the length of a line
segment? Should we define an edge on a solid?
Apparently, rigor means different things to different
people. Some feel that we should be sufficiently
rigorous when introducing concepts. At the same time,
we want to keep the language dynamic and less rigid.
For a mathematician, rigor comes before the Concrete,
Pictorial, and Abstract teaching strategy (CPA).2 I
guess the issue is not rigor, but the degree of rigor.
Issue 5. Does every preservice elementary teacher
need to take the SK course?
Another way of stating the issue is whether some
preservice teachers can be exempted from taking the
course. Perhaps we can have a placement test to
determine who does not need to take the SK course.
Perhaps we should practice differentiated teaching,
meaning we may teach different versions of SK to
preservice teachers under different training programs.
A course on subject knowledge
It was a long intellectual discussion at NIE on what
to teach in SK, and how to teach it. The course has
evolved over the years to what it is today. It is
generally agreed that we should have a course on SK in
mathematics for elementary school teachers under
training and a certain amount of rigor should be
maintained. It is also agreed that we should make it
refreshing so that students find it useful and have a
greater motivation to learn. Let me give two examples
to explain what I mean by refreshing.
Example 1
The area of a circle can be given by the formula
! 2
d where d is the diameter of the circle. If we
4
take ! to be 3 then the inscribed circle of a square has
A=
an area that is about 3/4 of the area of the square. This
is more instructional than the formula A = ! r 2 . I saw
a display of such a model in a department of
mathematics at an old university in Rome.
Lee Peng Yee
The inequality a 2 + b 2 ! 2ab can be verified
algebraically and geometrically. An alternative form is
(a + b) 2 ! 4ab . Consider a rectangle with area A and
perimeter l . If the length and the width of the
rectangle are respectively a and b, then A = ab and l =
2(a + b). Hence the above inequality becomes
2
'l$
% " ! 4 A . Fix l , then A is maximized when the
&2#
equality holds, that is, the rectangle is a square.
Similarly, fix A and we can find the minimum l in
terms of A.
As we can see, the topics covered in the examples
are school related. The presentation is accessible to
teachers, and language used is familiar. The examples
provide a glimpse of what I mean by mathematics for
teachers.
Conclusion
I have written here what has been discussed at NIE
on the subject of SK for mathematics teachers. While I
may have personal views, I do not have answers for all
the questions asked. In my view, the key factors in
mathematics for teachers are “rigor” and “refreshing”.
Our direction should be to document what we have
done, to build up a closer link between mathematics
and pedagogy, and to nourish a better understanding
between mathematicians and mathematics educators.
Hopefully, in time, SK will become an integrated part
of teacher training.
References
Ball, D. L., Hill, H.C., & Bass, H. (2005, Fall). Knowing
mathematics for teaching: Who knows mathematics well
enough to teach third grade, and how can we decide?
American Educator, 14–22, 43–46.
McCrory, R. (2006). Mathematicians and mathematics textbooks
for prospective elementary teachers. Notices of the American
Mathematical Society, 53(1), 20–29.
1
The National Institute of Education is the sole teachertraining institute in Singapore. Aligned with initiatives set
forth by Singapore’s Ministry of Education, NIE administers
preservice, in-service, and graduate programs for teachers.
2
For more information on this strategy, see: Lee, P. Y.
(Ed.). (2007). Teaching primary school mathematics, A
resource book. Singapore: McGraw Hill.
3
The Mathematics Educator
2006, Vol. 16, No. 2, 4–13
Uncovering Algebra: Sense Making and Property Noticing
David Slavit
This paper articulates a perspective on learning to discuss ways in which students develop personal sense and
negotiate meaning in a middle school algebra context. Building on a sociocultural perspective that incorporates
mental objects, learning is described as a mutually dependent process involving personal sense making and the
public negotiation of meaning. Analysis of student problem solving is focused on the development of taken-asshared meaning through an individual and collaborative analysis of the properties of various conceptual entities.
The results suggest that functional properties inherent in linear relationships were more supportive in eliciting
meaning making exchanges than were algebraic properties associated with generalized arithmetic, although the
contextual nature of the linear tasks may have also supported the meaning making activity.
What is the difference between sense and
meaning? Drawing on the philosophies of Vygotsky
and Leont’ev, Wertsch (1991) distinguishes between
sense and meaning by focusing on the personal and
public aspects of activity. Lave, Murtaugh, and de la
Rocha (1984) concur: “sense designates personal
intent, as opposed to meaning, which is public,
explicit, and literal (p. 73).” The personal act of
reflection and the public act of communication relate in
a manner that allows one’s personal reflections to
mediate and be mediated by one’s interactions with the
environment (Bauersfeld, 1992; Hiebert, 1992).
From this perspective, sense is based on one’s
individual reflections, whereas meaning has both a
personal and public dimension. Throughout the paper,
we will take the perspective that sense refers to the
current status of cognitive acts constructed within an
individual’s mental plane, and that an individual’s
meaning is the inferred result of the intention or
process of making one’s sense knowable within a
social environment. In other words, one’s meaning is
an intent to articulate one’s sense, but meaning is also a
public, “taken-as-shared” (Cobb, Yackel, & Wood,
1992) construct developed in a social context.
Because this paper focuses on learning and
knowledge construction, we turn our attention
Dr. David Slavit is Associate Professor of Mathematics
Education at Washington State University Vancouver in the
Department of Teaching & Learning and the Department of
Mathematics. His research interests focus on preservice and
inservice teacher development as well as student understandings
of mathematical content, particularly algebra. Past Director of
the Masters in Teaching program at WSUV, he has been
principal investigator of numerous National Science Foundation
grants, including the current Sustaining Teacher Research:
Inquiry, Dialogue, and Engagement (STRIDE) research project.
Dr. Slavit has also done extensive research on teaching and
learning with instructional technology. Email: dslavit@wsu.edu
4
explicitly to the differences between sense making and
meaning making, with the latter explored in a social
dimension. The philosophical complexities in the
analysis of collective meaning making, as opposed to
an individual’s isolated meaning making, are greatly
expanded due to the interactions between the personal
and added social dynamics at play.
For example, a radical constructivist perspective
emphasizes the interpretations of one’s experiences in
framing realties, so the distinctions between sense
making and meaning making become quite complex.
As Lerman (1996) states:
Rejecting a picture theory of mind, that mental
representations of reality are exact replicas of the
real world, leads, for the radical constructivist, to
the conclusion that one can only argue that all
representations are constructed by the individual,
and hence meanings are ultimately those in the
individual’s mental plane. (p. 137)
The cognizing individual constructs her or his own
world (i.e., makes sense) out of the articulated
meanings put forth by others in social interactions. But
the meaning making of others is filtered by the
organization of an individual’s own experiences and
sense-making processes, and any attempt to convey
one’s meaning is, in turn, filtered by the sense-making
processes of others (von Glaserfeld, 1995). Thus, sense
and meaning are quite blurred, as the result of attempts
at collective meaning making are always unknown
across the individuals’ own mental planes. One’s own
meaning, as well as the perceived meaning of others,
must remain in the internal realm of sense making.
Others, however, draw more separation between
the notions of sense and meaning. For example, many
researchers have articulated a view of learning in
which “mental objects” are constructed through
reflections on actions and activity. This mental
Uncovering Algebra
abstraction has been called reification (Sfard and
Linchevski, 1994), encapsulation (Dubinsky, 2000),
and verb-noun status (Davis, 1984). For these
researchers, sense making leads to distinct acts of
meaning making where individuals construct and
publicly share clear descriptions of these mental
objects.
Positioned perhaps between the above two
approaches, Cobb et al. (1992) describe a sociocultural
perspective of learning in which students make use of
experiences
and
interactions,
mediated
by
developmental interventions of the instructor, in order
to construct understandings that are either principally
generated by one’s sense-making activities or that
approximate and expand on the taken-as-shared
meanings of a community or society. This approach
goes beyond the acquisition and participation
metaphors of learning and includes aspects of
knowledge creation (Paavola, Lipponen, and
Hakkarainen, 2004). Here, it is possible that the
approximation of others’ meaning in an individual’s
sense-making activity can be thought to be negligible.
Slavit (1997) has built on the above perspectives to
articulate a theory involving the development of
meaning in the context of algebraic ideas through an
awareness of the properties of publicly negotiated and
taken-as-shared mental objects. The ability to make
sense of the properties that are embedded within and
help define a situation, activity, symbol system, or idea
can lead to the development of richer forms of senseand meaning-making activity. These developments
commonly occur in the cognitive act of formalization
(Kieren, 1994), which allows an individual to make
sense of mathematical constructs. For example,
students over time might recognize that linear
functions grow at a constant rate (from a numeric or
graphic perspective), are continuous, and have exactly
one x- and y-intercept (excepting vertical and
horizontal lines). These and other properties lead to an
understanding of linear functions borne from the sensemaking process. Further, these understandings can then
be weighed against one’s perceptions of the taken-asshared meanings of society as a whole, including
negotiation with one’s peers; these understandings
could also be initially generated out of interactions
with one’s peers. In this perspective, the learner
develops mental constructs associated with established
mathematical objects and ideas by focusing on his or
her interpretation and awareness of the properties that
define these objects and ideas.
David Slavit
Therefore, constructing sense and meaning in
algebra can be approached by focusing on the
properties associated with the objects and ideas that
help define this specific area of mathematics. Although
algebra is a multi-faceted mathematical area, this study
is primarily concerned with the two areas defined as
“generalizing and formalizing patterns and constraints”
and “study[ing] of functions, relations, and joint
variation” (Kaput, 1995). Patterns in arithmetic
computations can be identified by noting properties
that these computations share, which can then be
generalized to formal algebraic rules.1 These
generalizations form the basis of understanding algebra
as generalized arithmetic and illustrate the
development of sense and meaning in this area of
algebra through a property-noticing process. Similarly,
one can experience growth relationships between two
varying quantities in a variety of situations to support
the development of more abstract notions of functions,
including the concept of covariation (a patterned
change in one variable due to a patterned change in
another; Kieran & Sfard, 1999; Slavit, 1997).
Method
The purpose of the study is to examine the sensemaking and meaning-making activities of pairs of
students engaged in problem-solving episodes related
to the algebraic topics of generalized arithmetic and
function. While not attempting to prove that
collaborative learning environments are more effective
than individual settings, the study investigates the
manner in which pairs of students use specific
cognitive and social processes in algebraic problemsolving environments. Data consist of 15 videotaped
interviews involving problem-solving episodes with
fourteen 7th grade and sixteen 8th grade students from
two middle school classrooms in a rural, mediumsocioeconomic status school in the northwestern
United States. The students worked in pairs on two
tasks (detailed in Figure 1) for approximately 20–30
minutes. Approximately half of the students in each of
the two classrooms were randomly selected to
participate, and all interviews were transcribed. The
eighth grade students had a limited amount of formal
exposure to algebra prior to the study, consisting
mainly of equation solving, whereas the seventh grade
students had no previous formal algebra instruction.
Although whole-class discussion of problem solving
strategies and solutions were common in both
classrooms, the students had limited prior exposure to
working in collaborative pairs.
5
Task 1) Two carnivals are coming to town. You and your friend decide to go to different carnivals. The carnival that you
attend charges $10 to get in and an additional $2 for each ride. The carnival your friend attends charges $6 to get in, but
each additional ride costs $3. If the two of you spent the same amount of money, how many rides could each of you have
ridden?
.
.
Task 2) What digit is in the one’s place of the number: 29 34 56
Figure 1. Problem-solving tasks used to measure students’ sense of functional algebra and generalized arithmetic, respectively.
The interviews were conducted in a small area of a
quiet room, and the students were under no time
constraints to complete the tasks. The students were
given one copy of the first task and were told to “work
the problem anyway you wish, but you may wish to
work together.” Only pencil and paper were provided.
When it seemed that the students were at the end of
their solution attempt, they were questioned on the
manner in which they approached the task, and asked
to think about other ways to solve the problem. This
looking back stage was intended to promote critical
reflection by the students on their solution strategy.
These procedures were repeated for the second task.
Analysis was conducted on the videotaped
segments as well as the students’ written work. The
analysis centered on the kinds of algebraic
understandings (Kaput, 1995) the students seemed to
bring into the problem-solving situation, and the kinds
of algebraic understandings that the pairs utilized and
constructed in their solution. Hence, the social,
external, and mathematical constraints inherent in
problem solving were present, but the analysis centered
on the construction of understandings. Particular
attention was given to the kinds of properties that the
students attached to the algebraic ideas and mental
objects that helped support their investigation.
Numerous
researchers
have
extended
a
sociocultural view of learning to research on teaching,
grounded in participant’s actions and perspectives
(Cobb et al., 2003; Simon and Tzur, 1999). Researcher
participation, and the methodology itself, can be jointly
negotiated with the participants. Kieren et al. (1995)
took a similar approach to research on learning,
describing an enactive learning environment that
attempts to balance the overall aspects of the learning
situation with the cognitive and social backgrounds of
the participants as they engage in mathematical
activity. Kieren et al. used a group interview format
where the mathematical activity and research focus are
mediated by researchers, participants, and setting.
Kieren and his colleagues prefer to balance the role of
one’s sense-making activity and one’s ability to
6
negotiate and construct meaning. As they state, instead
of situated cognition or situated cognition, research
should focus on situated cognition. Hence, analysis of
student mathematical activity should be concerned
with understanding the students’ individual cognitive
processes in the context of the entire setting, including
the genesis and nature of interactions that lead to
knowledge construction. Likewise, a discussion of the
setting should be framed by the activity that occurs
within. This perspective was the lens through which
the interviews in this study were both conducted and
analyzed. Discussion of results will focus on overall
trends in the data, followed by a microanalysis of
problem-solving interactions.
Results
Task 1 was designed to allow students to approach
the problem either arithmetically or from a more
algebraic perspective. Students can solve the problem
by simply adding the cost of a ride or rides to each
admission and finding the amount of money where
Person
10
12
14
16
18
20
22
24
26
28
30
A
Person
6
9
12
15
18
21
24
27
30
33
36
B
Number
of Rides
0
1
2
3
4
5
6
7
8
9
10
Figure 2: Amounts of money spent by carnival attendees for a
given number of rides.
Uncovering Algebra
these totals are the same (see Figure 2). However,
students could also extend this strategy and find many
other combinations where the amount of money spent
is the same. Specifically, the students could express
their answer in a manner that describes a variable
relationship between the amount of rides and money
spent. Students could discuss the general relationships
in the numeric solution pairs that were obtained and
then set the expressions 10 + 2a and 6 + 3b equal to
each other, or they could draw the graph of this linear
relationship. These strategies would suggest a more
formal algebraic approach to the task than the
computational method described above because of the
greater degrees of abstraction present in the
representation and solution process.
Task 2 could be worked in a purely computational
way, where the value of 29 ⋅ 34 ⋅ 56 is computed and
the value of the one’s place identified. However,
students could also identify a factor of 10 present in the
product, either through initial computation or by an
examination of the factors, and recognize that the value
of the product must end in zero.
Task 1
Overall analysis revealed that sense-making and
property-noticing activities of some students occurred
in this setting, which may not have been constructed if
working alone. However, slightly less than half of the
pairs of students worked almost independently of one
another, and very little knowledge was shared.
Students who worked collaboratively constructed
solution strategies that involved meanings introduced
by both students, and the congruencies and
incongruencies in the individual strategies seemed to
advance the collective approach in most, but not all,
working pairs (see Table 1). The solution of Tim and
Molly2, discussed later, provides a clear example of a
pair of students whose collective solution strategy
yielded results that may not have evolved if working
alone.
Every group began the problem by listing dollar
amounts for a given number of rides, and every group
but one found either the solution (one ride for Person
A, two rides for Person B, $12) or the solution (4, 4,
$18). Overall, more than half of the student pairs
attended to covariance properties in regard to the
number of rides or money spent (see Table 1). This
analysis either involved the relationship between the
increase in the number of rides of each carnival
attendee for every new solution (three additional rides
for Person A and two additional rides for Person B), or
David Slavit
an increase in the amount of money spent for every
new solution (one new solution for every $6 increase).
A greater number of eighth grade pairs than
seventh grade pairs performed an analysis of the rate at
which new solutions were found in regard to money
spent (see Table 1). This result was due to the fact that
the seventh grade student pairs were more likely to
focus only on situations in which both riders went on
the same number of rides. Finding the amount of
money each person would spend for the same number
of rides does not include an analysis of the times when
each spends the same amount of money but enjoys a
different number of rides. Students who only focused
on a uniform increase in the number of rides across
riders found the solution (4, 4, $18), but failed to find
solutions such as (1, 2, $12).
Students investigating Task 1 in ways that did not
involve a uniform increase in the number of rides had
to perform a complex analysis of the covariance
properties present in the situation. The students first
had to recognize that the two quantities relating to the
amounts of money spent by each attendee increase at
different rates as more rides are taken. Then the
students had to realize that this difference impacts
another form of covariance—each person’s number of
rides is different for each solution. The latter difference
is due to a 3:2 ratio between the number of rides of the
two attendees from one equal spending value to the
next (see Figure 2). Therefore, to make these kinds of
generalizations, the students had to simultaneously
negotiate two different covariance situations and relate
them to their analysis.
The eighth grade students were more likely to
discuss a general solution to the task by finding a
pattern in the solutions displayed in a numeric table of
values they created. An example of this kind of
analysis is provided below. Many of these groups took
a more formal approach by making explicit notice of
the manner of covariation in the number of rides and
money spent, as just described. These kinds of
observations are pivotal when attempting to generalize
a solution for an arbitrary number of rides.
Two other groups made explicit mention of
covariation that was inappropriate to the problem
context, and four groups made no explicit mention of
any notion of covariance. One pair of students explored
these notions using algebraic symbols, but the rest of
the students did not go beyond the numeric table and
verbal descriptions.
7
Table 1
Algebraic understandings of covariance and solution possibilities exhibited by students on Task 1
No. groups
7th grade
(7 groups)
3
8th grade
(8 groups)
3
Made explicit notice of covariation in regard to amount spent (a new solution
for every 6 dollar increase)
1
4
Made no explicit mention of any notion of covariance
3
1
Stated or suggested that they realized that their were a theoretically infinite
number of solutions to the task
3
4
Stated or suggested that they realized that there were “a
lot” of solutions to the task
2
4
Found only one solution and did not seem to think others could be found
2
0
Group’s algebraic understanding
Made explicit notice of the differences in covariation in the number of rides
and money spent between the two carnival attendees (3:2 increase in number
of rides per additional solution)
Analyzing the property of covariance (i.e., slope)
from multiple situational perspectives eluded many of
the student pairs, most of whom were in the seventh
grade (see Table 1). Overall, some general differences
existed in the level of algebraic reasoning and
awareness of algebraic properties between the students
in the two grade levels. This may have been due to the
eighth grade students’ prior exposure to algebraic
ideas, including notions of variable and equation
solving.
Task 2
Collaborative problem-solving behaviors by the
student pairs on Task 2 occurred less frequently than
on Task 1. All but two of the student pairs either
worked in isolation, with each student computing the
value of the product individually, or with no real
collaborative problem-solving activity. The latter
involved negotiation of individual computational
duties, such as one person calculating 29 and the other
person finding the values of the other two factors. All
students who worked in isolation compared their
answers to the calculations of their partner. Overall,
there was very little interaction between the students
on this task.
The solution strategies on Task 2 were also more
uniform. Only one seventh-grade pair and one eighthgrade pair who successfully answered the question did
not calculate the value of the expression. These two
student pairs noted the presence of the factors two and
8
five in the product, recalled the property of a factor of
10 producing a zero in the one’s place, and then
utilized this property to construct a solution. However,
unlike Task 1, simple probing questions, such as “Is
that the only way you could have done it?”, elicited
generalizations in many of the student pairings who
had already solved the problem through direct
calculation. In particular, the realization of the
presence of a factor of 10 led to the development of
strategies in three of the seventh grade pairs and three
of the eighth grade pairs similar to that provided by the
two pairs of students mentioned above. Therefore, this
task did not initially elicit algebraic solution strategies,
but these behaviors did arise when prompted by the
researcher.
Enactive Learning: A Closer View
This section will focus on the problem-solving
strategies of Tim and Molly, two eighth graders, as
they worked together on Task 1. It will explore the
contextual, social, and cognitive factors that may have
played a role in their problem-solving processes. The
transcription of the solution strategy developed by Tim
and Molly illustrates how they initially constructed an
understanding of the problem based on the properties
of the two functions that represent the amounts of
money spent by the two carnival attendees. These
properties involved the initial amount (admission) and
rate of increase (cost per ride). Analysis centered on
their individual sense-making and collective meaningUncovering Algebra
making activities as they engaged in a solution attempt.
This interview was chosen because of the degree of
collaboration and collective sense making that
occurred. Although shared meaning developed
between Tim and Molly, the previously discussed data
indicates that such activity was not found in all of the
interviews.
After Tim found an initial solution of one ride and
two rides (1, 2, $12), Molly conjectured that more
solutions would be possible. They discussed the
covarying properties of the two dependent variables,
and this led Tim to the solution (4, 4, $18). At this
point, the two participants realized that infinitely many
solutions were possible, but they had no means of
describing what these solutions would be. Therefore,
the students approached the problem from a
computational perspective by making use of the
functional notion of covariance to find alternate
solutions, but they were unable to generalize to the
arbitrary case. Hence, their solution showed aspects of
algebraic thought, but they failed to develop functional
properties associated with arbitrary quantities. Instead,
the idea of covariance was contemplated from an
arithmetical perspective. Further, the students’ ability
to articulate their sense-making processes led to
additional meaning making between the pair, leading to
the acquisition of higher degrees of understanding of
the mathematical situation.
A closer look at specific portions of the interview
reveals particular instances of sense making and
instances where the students noticed properties that led
to their solution. The following segment occurred at
the onset of their solution attempt on Task 1:
multiple, almost, or something like that,
but anyway.
Molly: You
actually
(gibingly).
Tim:
learned
something
That wasn’t funny (good-humoredly).
Molly: One of my family’s jokes.
The beginning portion of the interview illustrates
several important aspects present throughout the
interview. First, the two students felt comfortable with
one another and did not appear to be nervous or
affected by aspects of the interview setting (e.g., the
video camera, presence of researcher, pressure of
solving the problem). Second, the students were
individually engaged in the task and actively sought an
understanding of the context and solution strategy.
The next few lines of the interview illustrate that
the two were beginning to make use of each other’s
sense-making activities.
Molly: This actually depends on how many rides
you went on, to go on, if you wanted to go
on like two rides, you could spend
Tim:
It depends, no, OK
Molly: for each
Tim:
OK, if I wanted, if I was here and you
there, right, if I wanted to go on two rides
that would be a total of 14 dollars, for me,
if you wanted to go on two rides then it
would only be 12 dollars for you, so it
would end up costing
T & M: (read problem, mumbling)
Molly: Oh, I was mixed up, OK (laughs) well,
one of them, one person didn’t have to
ride at all to get 10 dollars, no
Molly: I don’t know.
Tim:
Tim:
Wow (exasperated). (pause) Do you have
a calculator?
DS:
No.
Tim:
OK, so right now we know that each
person paid at least 10 and 6 dollars.
Molly: And then they could also go on two times,
but it depends how many rides he’d want
to go to.
Tim:
How many rides can each—
Molly: Actually, it depends on how many rides
he’d want to go on.
Tim:
David Slavit
No actually, um, actually it’s pretty much
asking what, like, sort of asking, it’s
almost like asking lowest common
OK, so what we are trying to figure out is,
they spent the same amount
Molly: Yeah I know, OK, and six dollars, what,
what adds up to being, lets see, 10, 20,
(long pause), OK
Molly began to explore the problem by advancing
on her initial sense of the situation, which involved an
understanding of the need to consider “how many
rides.” She made use of the cost of admission and ride
price, situational properties that correspond to the
linear functional properties of y-intercept and slope.
They were beginning to construct and make use of
shared meanings of the situation.
Tim utilized Molly’s remarks regarding the need to
consider the case of going on two rides to begin his
numerical analysis. After these computations, Molly
recognized that this would not be a desired solution
9
and said, “I was mixed up.” In the last two comments,
they returned to their individual sense-making
activities. Tim then made the following key insight:
Tim:
I could go on one and you could go on
two and then we’d each spend 12 dollars
(very confidently). That’s the answer
(chuckles). And 12 dollars is one answer,
or, ‘cause
Molly: Yeah that’s true.
Tim:
You’d go twice (writes) 12
Molly: So you’d
Tim:
No, one person would go. Yeah (put
pencil down) I get it, there’s that one.
Using the covariance properties previously
explored, Tim decided to vary the number of rides for
each person, changing the (2, 2) case to (1, 2). This
produced a solution that Molly was able to
immediately verbalize.
Tim’s personal solution became a shared and
meaningful one, although Tim was clearly the initiator
of nearly all of the public meaning. But this
immediately changed when I became part of the
interaction after assuming from Tim’s last comment
that the problem-solving activity had ended:
DS:
You spent 12? OK, so the one was two
and the other was one ride?
Tim:
(writes) 6 times 30
DS:
Or what makes you say that?
Tim:
and if the other person goes on four, or
wait a minute, this person goes on two.
Molly: It would be the same amount just as the
first time.
DS:
By the first time you mean when the one
person rides one and the other person
rides two?
Molly: OK, one person
Tim:
No it wouldn’t (confidently). OK, wait, so
you’re saying, so you’re saying
Molly: OK, you go
Tim:
one person goes four times, goes on four
rides
T & M: and the other person goes on two
Tim:
that’s 12, 14, this one’s 9, 12, 15, 18, so,
no, that’s not exactly true.
Molly: What (contentiously)?
Tim:
You said one person goes on two and
another person goes on four.
Molly: So, but wait.
Tim:
‘cause this person has to pay six just to get
in and three for each ride
Molly: Well you could spend more time and
Molly: yeah
Tim:
Tim:
You could spend a lot more money, and
then
Molly: Yeah
Tim:
You could
Molly: One person could go on two rides and the
other person could go on four rides and
you’d still get the same thing.
Molly’s current sense of the solution allowed her
to expand the situation by linearly increasing their
solutions, with some minor prompting from me.
However, Molly’s generalization was inaccurate, as it
did not include a proper analysis of the rates of
increase in the number of rides between solutions.
While Molly was verbalizing the meanings she
constructed that led to this conjecture, Tim makes
10
OK, what would they spend in that case?
Molly: Well, if you had
So
Molly: one person would go on two rides and the
other person would go on one.
Tim:
DS:
I’d go once,
Molly: You’d go once
Tim:
sense of Molly’s remarks and challenges the meaning
put forth by Molly after conducting a few
computations:
that’s four rides, that’s a total of 12 right
there, plus another six is 18, so that’s not
necessarily true.
Molly: Well, um, if you take, it’s 10 and six, and
then the first time, one person goes on one
once
Tim:
that’s two dollars right there
Molly: yes, that’s two dollars, and then another
person goes on another time, that’s two
times, and then you go on it again, six,
and another two, I guess that doesn’t
work. It’s worked in the past for me.
Using the meanings put forth by Tim, Molly
recognized the faults in her sense of the situation and
altered her belief in her own solution of (2, 4). The pair
Uncovering Algebra
constructed a collective sense of the precise nature of
the covariation in the situation and used this to further
their understanding and solution of the problem,
including a greater awareness of the property of
covariance. Eventually, their meaning making
exchanges collectively led to the (4, 4) solution, but
either participant made no further generalizations.
However, Tim does suggest that he believes more
solutions are possible:
Tim:
You could find other answers if you’d
keep going but, if you had all the time in
the world I’m sure you could find a lot
more answers.
Molly: If you were an old fogey I bet you could,
because you’d have a lot of time.
Despite failing to advance the solution, the above
dialogue illustrates how the articulation of sense and
joint construction of meaning are interdependent
processes that can lead to knowledge acquisition.
Molly and Tim successfully completed Task 2, but
their analysis was not as thorough and did not employ
any algebraic methods. Their solution strategy
involved performing the entire multiplicative
computation stated in the problem, and then examining
the digit in the one’s place in their final answer. The
pair shared the computation and writing duties
throughout, but did not construct a solution in a truly
collaborative manner.
Molly: It’s two to the ninth, so one more
(multiplication of two).
Tim:
OK, that’s 512. Here, you do this for a
while. (hands Molly the paper)
Molly: OK.
After the pair completed the computation, I
initiated a discussion that did not generate a more
algebraic approach, with Tim concluding by stating, “I
think there’s a shorter way but we don’t know.” The
use of the term “we” suggests that Tim was viewing
the pair as a fully functioning team throughout this task
as well, even though no real collaboration occurred.
However, unlike Task 1, the pair was unable to utilize
algebraic methods in this solution because the ability to
identify and make use of the effect of multiplication by
10 was not apparent. As discussed before, this result
was typical across all but two of the groups without
prompting from the researcher.
Analysis of the work of another eighth-grade pair
provided a different perspective on the nature of
collaboration and the use of algebraic properties. Like
Tim and Molly, Bill and Gloria collaboratively
David Slavit
explored Task 1 and arrived at multiple solutions and a
detailed analysis of the 3:2 ratio that existed between
the increase in rides. On Task 2, both students initially
asked for calculators, and then divided up the
computation.
Bill:
I’ll do two to the ninth and you do three to
the fourth.
Gloria: OK.
The pair continued computing silently for a very
long time, with both eventually working on 56 together.
However, before this computation was completed,
Gloria stated, “Wouldn’t we get zero?” Bill either
ignored or dismissed this comment and continued his
computation. After several more minutes, the pair
arrived at an answer of zero.
Gloria: Zero will be there, and if you multiply
anything by zero you get zero.
DS:
Can you look at the problem and see why?
Bill:
(long pause) Not really.
Gloria:
We got zero here (at one point in the
computation) and it stayed zero after that.
Although Bill and Gloria began working
collaboratively on the computational aspect of the
problem, they did not share in any meaning-making
activity regarding the conceptual aspects embedded in
this task. Gloria began to explore this, but did not
progress either individually or collaboratively with
Bill. As stated above, this lack of group meaning
making was true of the majority of paired groupings on
this task.
Conclusion
In many situations, learning is a collective process
of privately constructed personal sense and publicly
negotiated meaning. The ability to utilize one’s sense
to articulate meaning, as well as make sense of other’s
stated meanings, enriches the taken-as-shared network
being constructed. Learning is a dynamic interplay
between one’s sense, one’s stated meanings, and the
sense one makes out of other’s stated meanings. This
study provides evidence where these three aspects of
learning can combine to form a collective, rich
understanding of a problem-solving situation, or a
situation where sense- and meaning-making do not
fully develop.
This study does not intend to make the case that
students are more successful working together. Rather,
it tries to articulate how specific conceptual processes
are utilized when working on algebraic tasks in both an
individual and collaborative environment, and what
11
kinds of algebraic tasks might elicit individual and
collaborative problem-solving behavior. The main
mathematical object of analysis in Task 1 was the
linear growth relationship that contained the property
of constant rate of change and a covariance property
between the two variables. In the case of Tim and
Molly, the taken-as-shared meanings that developed
were certainly a product of the two individual’s sensemaking activities, but the development of these
meanings also effected future sense-making and
meaning-making activities in the task. These students
were both willing and able to participate in this
personal and shared process, and their collective senseand meaning-making activities were at the heart of
their learning experiences.
Overall, the tasks and learning environment were
able to elicit sense-making activities based upon the
individual students’ understandings that were then
transferred into meaning-making activities shared by
the student pairs. These activities led to solution
strategies that involved the generalization of arithmetic
constructs into more algebraic realms. This occurred in
the context of interactions between students in Task 1,
but interactions with the researcher also led to these
occurrences in Task 2 for several student pairs. The
most significant advancements made by the student
pairs appeared to have occurred in Task 1, where the
students explored aspects of the covariance to advance
their understandings of the task, the number of
solutions found, and their ability to articulate a
generalized solution. The fact that there was more
interaction on Task 1 may have been due to the fact
that the linear growth properties inherent in this task
were more apparent to the students than the
multiplicative or number theory properties inherent in
Task 2. The use of a context involving two people may
have also made Task 1 easier to model than Task 2. As
a result, the students were able to participate in richer
meaning making exchanges about Task 1, and develop
more advanced solutions.
It appears that, for these middle school students,
algebraic understandings of the property of covariance
on Task 1 were readily available. These understandings
allowed many of the student pairs to approach the task
in algebraic realms, making use of the properties of
covariance, correspondence, and slope to identify
multiple solutions to the task. Understandings of
appropriate arithmetic properties which may have led
to a more generalized approach to Task 2 were not as
apparent. Hence, in this study, the students were better
able to utilize their sense-making activities to uncover
properties related to aspects of functional algebra than
12
with properties associated with algebra as generalized
arithmetic. But the limited scope of this study does not
begin to allow for generalizations of this result. Further
research is needed to discuss differences in students’
facility with various algebraic properties and the ability
to work with these properties at various levels of
generality (Kaput, 1995).
Moreover, students with procedural problemsolving tendencies (which many of the students
appeared to possess) would not be expected to utilize
algebraic understandings on Task 2 because a solution
strategy requiring direct computation is immediately
recognized. But, in Task 1, a student who begins to
compute the number of rides associated with various
amounts of money spent by the carnival attendees is
generating information that can lead to a discovery of
notions of slope and covariance, which happened
frequently in these student pairs. Therefore, the
problem-solving behaviors of the students may have
led to differences in their ability to recognize
properties of the mathematical situations and to solve
problems with various degrees of generality across the
two tasks. These data illustrate the complexities
inherent in the personal and public interplay of
students’ knowledge construction processes that often
go unnoticed in an interactive, problem-solving
environment.
References
Bauersfeld, H. (1992). Classroom cultures from a social
constructivist’s perspective. Educational Studies in
Mathematics, 23, 467–481.
Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist
alternative to the representational view of mind in
mathematics education. Journal For Research In Mathematics
Education, 23(1), 3–31.
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L.
(2003). Design experiments in educational research.
Educational Researcher, 32(1), 9–13.
Davis, R. B. (1984). Learning mathematics: The cognitive science
approach to mathematics education. London: Routledge.
Dubinsky, E. (2000). Mathematical literacy and abstraction in the
21st century. School Science and Mathematics, 100(6), 289–
297.
Hiebert, J. (1992). Reflection and communication: Cognitive
considerations in school mathematics reform. International
Journal of Educational Research, 17(5), 439–456.
Kaput, J. J. (1995). A research base supporting long term algebra
reform? In D. T. Owens, M. K. Reed, & G. M. Millsaps
(Eds.), Proceedings of PME-17 (Vol. 1), (pp. 71–94).
Columbus, OH: ERIC Clearinghouse for Science,
Mathematics, and Environmental Education.
Kieran, C. & Sfard, A. (1999). Seeing through symbols: The case
of equivalent expressions. Focus on Learning Problems in
Mathematics, 21(1), 1–17.
Uncovering Algebra
Kieren, T. E. (1994). Bonuses of understanding mathematical
understanding. In D. F. Robitaille, D. H. Wheeler, & C.
Kieran (Eds.), Selected lectures from the 7th International
Congress on Mathematical Education (pp. 211–228). SainteFoy, Quebec: Les Presses de L’Universite’ Laval.
Kieren, T. E., Calvert, L. G., Reid, D. A., & Simmt, E. (1995,
April). Coemergence: Four enactive portraits of mathematical
activity. Paper presented at the Annual Meeting of the
American Educational Research Association, San Francisco.
Lave, J., Murtaugh, M., & de la Rocha, O. (1984). The dialectical
construction of arithmetic in grocery shopping. In B. Rogoff &
J. Lave (Eds.), Everyday cognition: Its development in social
context (pp. 67–94). Cambridge, MA: Harvard University
Press.
Lerman, S. (1996). Intersubjectivity in mathematics learning: A
challenge to the radical constructivist paradigm? Journal for
Research in Mathematics Education, 27(2), 133–150.
Paavola, S., Lipponen, L., & Hakkarainen, K. (2004). Models of
innovative knowledge communities and three metaphors of
learning. Review of Educational Research, 74(4), 557–576 .
Sfard, A. & Linchevski, L. (1994). The gains and pitfalls of
reification: The case of algebra. Educational Studies in
Mathematics, 26, 191–228.
David Slavit
Simon, M. & Tzur, R. (1999). Explicating the teacher's perspective
from the researchers' perspectives: Generating accounts of
mathematics teachers' practice. Journal for Research in
Mathematics Education, 30(3), 252–264.
Slavit, D. (1997). An alternate route to the reification of function.
Educational Studies in Mathematics, 33(3), 259–282.
von Glaserfeld, E. (1995). Radical constructivism: A way of
knowing and learning. London: RoutledgeFalmer.
Wertsch, J. V. (1991). Voices of the mind: A sociocultural
approach to mediated action. Cambridge, MA: Harvard
University Press.
1
For example, a2 - b2 = (a + b)(a - b) expresses the
relationship between the difference of the squares of any two
numbers and the product of their sum and difference. An
additional example involves the discovery that the result of
any product that contains a factor of 10 must have a zero in
the one’s place..
2
All participant names are pseudonyms.
13
The Mathematics Educator
2006, Vol. 16, No. 2, 14–24
Preparing Elementary Preservice Teachers to Use Mathematics
Curriculum Materials
Alison M. Castro
Learning how to use mathematics curriculum materials to create learning opportunities is an important part of
the work of teaching. This paper presents findings from a study involving 15 elementary preservice teachers
enrolled in, first, a content and, then, a methods course, and discusses the extent to which three curriculum
interventions influenced their conceptions of how math curriculum materials are used. Additionally, this paper
discusses the implications of this research for mathematics teacher education programs and proposes a
framework for integrating work around curriculum materials into mathematics content and methods courses in
order to prepare preservice teachers for using these materials effectively.
Recent efforts by the National Council of Teachers
of Mathematics (NCTM) to improve the way that K-12
mathematics is taught and learned have implications
for mathematics teacher education. As teacher
education programs aim to develop teachers’
knowledge of mathematics and their knowledge of
students as learners, these programs “should [also]
develop teachers’ knowledge of and ability to use and
evaluate instructional materials and resources”
(NCTM, 1989, p. 151). In particular, “teachers need a
well-developed framework for identifying and
assessing instructional materials…and for learning to
use these resources effectively in their instruction” (p.
151). Although using mathematics curriculum
materials effectively is an important part of teachers’
work, it is an aspect of practice that is often overlooked
in teacher education programs.
Prevalent in classrooms across the country,
curriculum materials are important and can be
influential resources for teachers. Mathematics
curriculum materials, in particular, are potentially
influential given the challenging nature of mathematics
instruction espoused under recent reform efforts. In
response to NCTM’s (2000) recommendations
regarding the improvement of mathematics instruction,
some mathematics curriculum materials have become
highly designed and very detailed sources of both
Dr. Alison Castro is Assistant Professor of Mathematics
Education and Learning Sciences at the University of Illinois at
Chicago where she also serves as the Research Director for the
Teaching Integrated Mathematics and Science (TIMS) Project,
which investigates the impact of the Math Trailblazers
elementary curriculum on teachers’ practice and student
understanding in the classroom. Her research interests focus on
preservice and in-service teachers’ use of mathematics
curriculum materials to create opportunities to learn and the
factors that influence this use.
14
content and pedagogical information (Trafton, Reys, &
Wasman, 2001). From homework and grouping
suggestions to examples of student errors and
alternative solution strategies, such innovative
mathematics curricula provide a potential wealth of
information and instructional support for teachers.
Given the quantity of information and pedagogical
suggestions in these innovative materials, however,
teachers can potentially use mathematics curriculum
materials in a number of different ways. Whereassome
teachers tend to follow the suggestions in mathematics
curriculum materials almost as a script for instruction
(Graybeal & Stodolsky, 1987; McCutcheon, 1981),
other teachers do not rely on the suggestions in teacher
materials to the same extent, but rather adapt the
suggestions and activities as they see fit (Stake &
Easley, 1978). Furthermore, it is possible that teachers
use curriculum materials to only a limited extent, if at
all.
The increasing use of innovative mathematics
curricula combined with the aforementioned research
raises concerns about teachers’ use of curriculum
materials—that teachers can use these materials with
an inattention to the actual content and nature of the
tasks, activities, and pedagogical suggestions contained
in these resources. Research illustrates that the moves
and decisions of teachers during instruction influence
the nature of students’ work, often reducing the
complexity and challenge of the tasks and activities
(Stein & Lane, 1996; Stein, Smith, Henningsen, &
Silver, 2000). Given that mathematical tasks lie at the
center of innovative curriculum materials, teachers’
perceptions and use of these materials can potentially
have a strong influence on how these tasks are enacted.
If teachers are inattentive to the nature of the given
tasks in the curriculum materials and their enactment
of these tasks, they can undermine the task complexity.
Preparing Elementary Preservice Teachers
On the other hand, if teachers are deliberate and
purposeful in their use of curriculum materials to
inform their moves and decisions around tasks, they
may be better able to maintain students’ engagement in
complex, intellectual mathematical work.
Given the need for teachers to be informed users of
mathematics curriculum materials, it is important to
understand how teachers think about and learn to use
these materials. Preservice teachers’ conceptions of
and experiences with such materials provide insight
into this process. Little has been written about how
preservice elementary teachers learn to use
mathematics curriculum materials. Ball and FeimanNemser (1988) found that, during student teaching,
preservice teachers varied in their use of these
materials. The researchers attributed this differentiated
use to what students’ teacher education programs were
advocating about mathematics curriculum materials.
However, little more is known about how preservice
teachers learn to use curriculum materials and even
less is known about how teacher education programs
develop preservice teachers’ skills at using these
resources effectively. Thus, this study aims to (a)
explore preservice elementary teachers’ conceptions of
mathematics curriculum materials, (b) analyze what
and how preservice teachers learned from activities
that were designed to help them learn to use
mathematics curriculum materials effectively, and (c)
discuss the implications of this research for preservice
teacher education programs.1
Conceptual Framework
Teachers can use curriculum materials in a number
of different ways. Some teachers rely heavily on
curriculum materials, following the suggestions in an
almost prescriptive manner (Remillard, 1992), while
other teachers modify and adapt curriculum materials
in the course of planning for and teaching a lesson
(Stake & Easley 1978). Other researchers have
explored different factors that impact teachers’ use of
curriculum materials. From policy guidelines (Floden
et al., 1980; Kuhs & Freeman, 1979) and teachers’
interpretations of policies (Cohen et al., 1990) to
teachers’ ideas about the purpose of education and
nature of learning (Donovan, 1983; Stephens, 1982),
teachers’ use of curriculum materials is influenced by a
number of different factors.
Although the research on teachers’ use of
curriculum materials and the factors influencing this
use primarily focus on inservice teachers, the extant
research can be used as a framework for understanding
how preservice teachers interpret and use these
Alison M. Castro
materials. Just as inservice teachers draw upon
resources when making decisions about curriculum
materials, preservice teachers rely on similar types of
resources to make decisions about their practice.
Addressing the different resources that teachers draw
upon when teaching, Cohen, Raudenbush, & Ball
(2002) highlight teachers’ knowledge, skill, and will.
In their framework, resources such as teachers’
knowledge and skill can influence how curriculum
materials are utilized. “For when teachers…use
resources, they make judgments about which to use,
how to use them, with whom, and to what end. They
base these judgments on what they know and believe
about themselves, one another, and the content” (p.
104).
However, resources such as knowledge and skill
may be limited for preservice teachers. Due to their
inexperience, preservice teachers may have a narrow
view of teaching and classroom practice, and limited or
incomplete conceptions of the ways in which
curriculum materials can be utilized. Moreover,
preservice teachers bring preconceptions about
teaching into their teacher education programs
(Lampert & Ball, 1998). They have spent years as
students in the classroom watching their teachers and
developing ideas about good teaching, but they know
little about the decisions and challenges teachers
actually face in the classroom. In particular, preservice
teachers are most likely unaware of the ways in which
teachers use mathematics curriculum materials to make
decisions.
As preservice teachers have very limited personal
resources to draw upon when making teaching
decisions, teacher education programs can aim at
influencing such resources. That is, elementary
education programs can develop preservice teachers’
knowledge and skill at using mathematics curriculum
materials in ways that develop and further students’
understanding. As Cohen et al. (2002) describe,
Though some teachers judge with great care and
seek evidence with which they might revise, others are
less careful. In either event, teachers calibrate
instruction to their views of their capacities and their
students’ abilities and their will to learn. (p. 104)
Indeed, Simon & Schifter (1993) reported that
teacher education programs in mathematics education
can help teachers develop a conception of teaching and
learning that is consistent with recent reform ideas.
And Ball (1990) argues that mathematics methods
courses, in particular, can influence preservice
teachers’ knowledge, assumptions, and beliefs about
mathematics. Methods and content courses, taken
15
together, are a potentially useful venue for providing
preservice elementary teachers with the necessary
knowledge, skills, and conceptions of mathematics
curriculum materials to enable them to use these
materials effectively.2
Methodology
This study took place in the context of two
required courses in a year-long Master’s and
certification program in Elementary Education at a
large, midwestern university. The first course, a
mathematics content course for elementary teachers,
was an 8-week summer course designed to prepare
preservice teachers for teaching the core elementary
mathematical domains of number and operations. The
second course in this sequence was an elementary
mathematics methods course, which took place the
following semester.3 Notably, much of the underlying
philosophy of these two courses drew on NCTM’s
(1989, 2000) recommendations regarding the teaching
and learning of mathematics, including activities with
manipulatives that drew heavily on NCTM’s
recommendations regarding the use of these materials.
Fifteen students volunteered to participate in this
study. Because this study solicited participants on a
volunteer basis, this is not a random sample, and thus,
may not be representative of the population of
preservice elementary teachers at this institution. Many
of the students were considered non-traditional college
students in that they left full-time employment in order
to enroll in the Master’s program. In addition, several
of the students had a variety of informal teaching
experiences prior to entering the program, though it
was not required for admission.
Students’ notebooks comprise the first data source
for this study. As part of their grades for each course,
students were required to keep a notebook. These
notebooks provided a space for them to write any inclass notes and annotations of course readings, as well
as to complete weekly assignments. In addition to
recording students’ ideas and responses to the
curriculum activities, students’ notebooks also
provided a space for them to answer questions related
to their conceptions of mathematics curriculum
materials at the start of the content course (referred to
hereafter as pre-sequence).4
Individual interviews comprise the second data
source for this study. Using semi-structured interviews,
students were interviewed once following the
completion of the content course (referred to hereafter
as mid-sequence), and again following the completion
of the methods course (referred to hereafter as post16
sequence). During both of these interviews, students
were asked about their conceptions of mathematics
curriculum materials, the perceived role of these
materials in the classroom, and their thinking about the
curriculum activities. The interview protocols for both
sets of interviews included, among other questions, the
same questions that were asked at the pre-sequence
time point. See Appendix A for the mid-sequence
interview protocol.5 Both the student notebooks and
interview transcripts were analyzed and coded. From
these codes, analytic documents were compiled, which
were then used to generate inferences, and eventually
hypotheses, to understand how students thought about
math curriculum materials and the curriculum activities
(Erickson, 1985; Strauss & Corbin, 1985).
Curriculum Activities
Over the two courses, three interventions were
designed and administered to students to help them
develop their capacities to use mathematics curriculum
materials in deliberate and skillful ways. These
curriculum activities were designed to draw specific
attention to the nature and extent of the mathematical
and pedagogical information in curriculum materials.
Activity 1 took place during the third week of the
content course, and comprised about 60 minutes of
class time. The purpose of this activity was to first
determine the mathematical goals of a textbook lesson
in order to better understand how the lesson fit into the
larger unit, and then to determine the mathematics that
children were expected to learn from the lesson. In
particular, students were asked to read a specific
textbook lesson from a teacher’s guide and think about
the following questions: What is this lesson about?
What is the mathematics that children are supposed to
be learning? What do the problems or exercises seem
to intend children to do?6
Activity 2 took place during the fourth week of the
content course, in which students were discussing
mathematical proofs and methods of proving. This
activity comprised about 60 minutes of class time.
Specifically, students were discussing proofs of
statements involving even and odd numbers. They
were presented with four different definitions of even
numbers taken from four different elementary
mathematics curriculum programs. The purpose of this
activity was to first understand the different
definitions, and then to determine which types of
numbers would be considered even given the
definitions. For each definition, students were asked to
(a) determine whether it is mathematically valid, (b)
discuss whether it would be usable by third graders,
Preparing Elementary Preservice Teachers
and, in the case that the definition is not usable, (c)
revise the definition to make it appropriate for third
graders.
Finally, Activity 3 took place during the eighth
week of the methods course, in which students were
beginning to discuss the elements of lesson planning.
This activity comprised about 75 minutes of class time.
The purpose of this activity was to analyze a textbook
lesson with careful attention to not only the larger
lesson goals, but also to the other elements of the
lesson, such as the tasks, examples, language, problem
contexts, and mathematical representations.7 The
underlying rationale was that, by analyzing the
sequence of tasks, the language used, and the different
mathematical representations included, students would
be better able to make informed decisions about
instruction and then modify the lesson where
necessary. In short, these interventions were designed
to help students learn different aspects of mathematics
curriculum material use. These interventions also
provided opportunities for students to focus on the
mathematical and pedagogical aspects of a lesson, and
make determinations and assessments of the nature and
extent of the information accordingly.8
materials. Their responses were grouped into the
following categories: textual materials, which includes
teacher guides, transparencies, assessment resources,
textbooks, student notebooks and journals, etc.; nontextual materials, which includes pencils, paper,
calculators, and other materials or items that can be
used by students during a mathematics lesson; and
manipulatives, which includes Base 10 blocks, pattern
blocks, Unifix cubes, and any other commercially
made materials, as well as any teacher-made
manipulatives.9 Table 1 displays students’ responses.
As Table 1 illustrates, throughout the two courses,
students had varied conceptions as to what constitutes
mathematics curriculum materials. Some students
considered mathematics curriculum materials to be
exclusively textual materials, non-textual materials, or
manipulatives. Other students considered these
materials to be some combination of the three different
categories. Although students’ conceptions of
curriculum materials consistently fell into these three
categories at pre-, mid-, and post-sequence, the
distribution of their conceptions changed. The number
of students who included non-textual materials as
curriculum materials decreased from pre- to midsequence, and then again from mid- to post-sequence.
As these numbers decreased, the number of students
citing some combination of textual materials and
manipulatives increased throughout the two courses.
Analysis of Students’ Understandings of Math
Curriculum Materials
This section is divided into three parts. To
understand how students’ conceptions changed over
time, the first part describes what items and materials
students considered to be mathematics curriculum
materials. The second part discusses how students
envisioned these different items and materials being
utilized in the classroom. Finally, the third part
examines how students understood the three
curriculum activities.
How Can Mathematics Curriculum Materials Be
Used?
In addition to being asked what constitutes
mathematics curriculum materials, students were also
asked to describe how these materials can be used in
the classroom. Students’ responses varied along two
dimensions. While some students thought that
curriculum materials could be used to help students
learn, others saw these materials as tools that can
support teachers’ instructional decisions.
What Constitutes Mathematics Curriculum Materials?
The students in this study considered a variety of
materials and resources as mathematics curriculum
Table 1
Students’ Conceptions of Mathematics Curriculum Materials
Textual
Materials
Presequence
Midsequence
Postsequence
Alison M. Castro
Manipulatives
Textuals &
Manipulatives
2
Nontextual
Materials
0
Manipulatives
& Non-textuals
3
Textuals
& Nontextuals
2
2
Textuals,
Non-textuals,
Manipulatives
5
1
4
0
1
5
1
0
4
3
0
2
6
1
1
2
17
Table 2
Students’ Conceptions of How Curriculum Materials Can Be Used
Pre-sequence
Mid-sequence
Post-sequence
Help children learn
9
2
0
As Table 2 illustrates, students’ conceptions of
curriculum materials varied along two primary
dimensions. While at mid-sequence some students saw
these materials as being used by children, others
viewed curriculum materials as tools that support
teachers’ decisions. By post-sequence, students’ views
shifted, and all fifteen students viewed mathematics
curriculum materials as supporting teachers’ decisions.
In addition, students said that teachers can use these
materials in one of three different ways: (a) scripted
use, where a teacher relies heavily on the materials; (b)
modified use, where a teacher modifies or adapts
curriculum materials as they see fit; and (c) limited
use, where a teacher uses the curriculum materials to
only a limited extent. Although students did not appear
to possess these views of curriculum use at presequence, these three distinct views emerged at midand post-sequence. At post-sequence, more students
said that curriculum materials can be adapted and that
teachers do not necessarily have to, as one student
stated, “follow exactly what is stated in the book.”
Influence of Curriculum Activities
Throughout the study, students were asked
specifically about the three curriculum activities and
the extent to which they found the different activities
useful in learning how to use mathematics curriculum
materials. Table 3 displays students’ responses.
At mid-sequence, students were asked specifically
about Activity 1. In response to this question, only six
students specifically mentioned this activity as useful
in learning how to use mathematics curriculum
materials. Whereas two of these students said Activity
1 was useful insofar as it introduced and exposed them
Table 3
Students’ Perceptions of Utility of Curriculum
Activities
Student Learning
Curriculum
Activity
1
2
3
18
Supported
6
4
12
Not Supported
2
1
0
No
mention
7
10
3
Support teachers
4
13
15
Both
2
0
0
to different mathematics curricula, the other four
students cited this activity as particularly useful to their
learning. One of these four students stated the
following:
I think it was a helpful activity because maybe as a
new teacher you would just kind of, oh well this is
the teacher’s guide, this is how I need to teach this.
But by thoroughly examining it and you know,
looking at the math that’s going on and … maybe
you would see the faults in the book.
All four of these students also had similar
conceptions of how curriculum materials can be used.
Specifically, they said that teachers should modify and
adapt curriculum materials in order to meet the needs
of their particular classrooms.
In contrast to these six students, two of the
remaining nine students specifically stated that this
activity was not useful for learning how to use
curriculum materials because they felt too
inexperienced to make such decisions. In particular,
one student said, “I found it very difficult to get access
to. And I think I also thought of myself at that point in
the course as someone who didn’t really have the
background to decide whether the math was good
enough.” Similarly, the other student said that he did
not have enough experience to summarize the
mathematics in a lesson. Thus, these students felt they
lacked the background and experience to properly
summarize the mathematical ideas in curriculum
materials.
At mid-sequence, students were also asked
specifically about Activity 2. As with Activity 1, this
activity was mentioned by only a few students.
Specifically, four students mentioned that this activity
was helpful for learning how to use mathematics
curriculum materials. Three of these students stated
that this activity helped them to think about definitions
in textbooks differently than they had before. “I can
see some of the deficits of stuff that’s out there…. For
instance, I don’t think, from what I’ve seen so far, the
[name of a curriculum program] is not particularly
strong in definitions.” Another student stated that she
Preparing Elementary Preservice Teachers
found this activity particularly useful because it helped
her to think about mathematics in a different way.
And I think that you can’t just look at—I think
some teachers take this teacher guide and that’s
what they teach and this is what they talk about
versus looking in it and saying well what else do
we have about that … Because sometimes people
get caught up with reading what’s in front of them
and that’s how it is … It was eye-opening what we
did.
In contrast to these four students, one student said
that they were not experienced enough to evaluate
mathematical definitions in curriculum materials. It is
not clear whether the inexperience felt by this student
was due to a lack of strong content knowledge or to the
overall design of the activity.
Finally, at post-sequence, students were asked
about the extent to which Activity 3 helped them learn
how to use mathematics curriculum materials. In
response to this question, an overwhelming number of
the participating students viewed this activity as not
only a useful activity to include in such a course, but as
an important part of teachers’ work in the classroom as
well. In contrast to the first two curriculum activities,
twelve of the students in this study found this activity
very useful for learning how to use curriculum
materials. For example, when asked about the utility of
this activity, one student replied as follows:
I had to think about … different angles … that
helped me look at the curriculum in a very detailed,
deliberate way … I needed to go back and see what
they [referring to children] had done before,
needed to see what they were going to do next.
Similarly, another student stated the following:
I think it was useful because I don’t know that I
looked at the lesson plans so critically before …
it’s kind of one of those things that you just took
their word for it, if it was in the book you should
teach it … I got the message [from Activity 3] that
you take the lesson plan and kind of alter it
according to your students.
Notably, eight of these twelve students adhered to
the same view of curriculum use—that teachers should
modify and adapt curriculum materials according to
children’s different abilities.
In short, students generally saw Activity 3 as
useful for learning how to use mathematics curriculum
materials. Furthermore, they felt that it was applicable
to teachers’ daily practice. However, only a small
proportion of the students in this study viewed
Activities 1 and 2 as helpful for learning how to use
Alison M. Castro
curriculum materials, and important to their own
learning as future teachers. Students’ perceptions of the
utility of a curriculum activity seemed markedly
related to how they viewed curriculum use—several
students who cited Activity 3 as useful to their learning
thought that teachers should appropriately modify and
adapt curriculum materials for their classroom.
Discussion
As their conceptions of curriculum materials
shifted throughout the two courses, students formulated
and reformulated ideas about the particular ways in
which teachers can use mathematics curriculum
materials. Ranging from strict use to modified use to
no use at all, students seemed to have very clear
notions of the different ways in which teachers can and
should use curriculum materials. Generally, these
findings indicate that across the two courses, the focus
of students’ conceptions of mathematics curriculum
materials shifted. Although many students focused at
pre-sequence on how children interact with curriculum
materials, that focus changed by post-sequence to
highlight and include teachers’ interactions with these
materials.
The shift in students’ conceptions of what
constitutes mathematics curriculum materials clearly
illustrates their shift to a knowledge-based conception
of curriculum materials. At pre-sequence, several
students considered non-textual materials such as
paper, pencils, and rulers to be curriculum materials.
Although these materials can be used by children in a
variety of activities, both during and not during
instruction, they do not directly support children’s
learning of mathematics. At post-sequence, students’
conceptions of curriculum materials focused more
heavily on textual materials and manipulatives. Textual
materials and manipulatives are both predominately
used during or in preparation for instruction, and in
comparison to non-textual materials, textual materials
and manipulatives are more directly involved with
children’s acquisition of knowledge. Thus, over the
course of the sequence, students’ conceptions of
mathematics curriculum materials seemed to have
shifted to a more knowledge-based conception.
This shift in students’ conceptions of curriculum
materials also indicates a move to a more teacherbased conception of these materials. At pre-sequence,
students’ views of curriculum materials focused
primarily on children’s use of these materials. At postsequence, however, students’ views of curriculum
materials primarily focused on how teachers can use
such materials. This shift makes sense in light of the
19
shift described in the above paragraph. Knowledgebased curriculum materials support teachers’ planning
and instruction more directly (and perhaps to a greater
extent) than non-textual materials. Knowledge- and
teacher-based materials can be used to embody
mathematical content, which is precisely what is taught
during a lesson.
As students moved towards a more knowledgeand teacher-based conception of curriculum materials,
they began formulating particular views of how
teachers can use these materials. At pre-sequence,
students’ responses did not clearly indicate that they
had considered how teachers can use such materials.
By mid-sequence, students had formulated somewhat
concrete views of curriculum use—scripted use,
modified use, and limited use.10 Most students thought
teachers should modify and adapt curriculum materials.
By post-sequence, these newly formulated ideas had
even begun to shift as more students thought that
teachers should modify these materials. So, although
students held different views at mid- and post-sequence
regarding teachers’ use of curriculum materials,
students almost universally began to develop their
conception of such use over the span of the two
courses. This trend comports well with the shift in
students’ conceptions of what constitutes curriculum
materials described above. As students began to
consider curriculum materials as tools that support
instruction, they also began to consider how teachers
use such tools to inform their teaching.
From the research conducted in this study, it is not
clear why students formulated their particular views of
how teachers should use curriculum materials. That is,
it is unclear why many students believed that teachers
should modify and adapt mathematics curriculum
materials, whereas other students believed teachers
should either strictly use these materials or not use
them at all. Students certainly received messages about
curriculum materials from their cooperating teachers
and other sources external to the two courses, such as
periodicals. Moreover, students may have received
implicit (and explicit) messages about how to use
curriculum materials from these two courses.11
Nevertheless, the findings indicate that students were
in fact formulating concrete views of how teachers
should use curriculum materials during the two
courses.
Moreover, the precise effect of the curriculum
activities on students’ conceptions of curriculum
materials and how these materials can be used is not
immediately evident. Students indicated that the first
two activities were not very useful. However, students
20
generally viewed Activity 3 as a practical activity that
teachers would do on a regular basis. Activity 3 is
designed to help students focus on the mathematical
content embodied in curriculum materials, such as
textual materials and manipulatives. The finding that
students found this activity useful comports with
students’ knowledge- and teacher-based conceptions of
curriculum materials. Students considered the activity
useful precisely because it could support instruction.
Although on one hand it is possible that the lesson
analysis activity contributed to students’ conceptions
of curriculum materials and curriculum use, it is also
possible that students found the lesson analysis useful
because it supported their already existing conceptions.
In all likelihood, both of these possibilities are
simultaneously true. During the two courses, students
were constantly formulating and reformulating their
conceptions. The effect of any given curriculum
activity, in part, depends on students’ conceptions of
curriculum use before the activity begins.
Implications
In short, the trends that emerged in this study
indicate that students moved to more teacher-based
conceptions of what constitutes mathematics
curriculum materials and how these materials can be
used. Also, students’ conceptions of curriculum
materials shifted over the two courses to include more
knowledge-based materials, such as teacher’s guides,
assessment resources, and manipulatives. Despite these
changes in students’ conceptions of curriculum
materials, it is unclear to what extent these changes can
be attributed to the curriculum activities. Moreover, as
the curriculum activities did not directly influence
students’ conceptions, it is unclear whether the two
courses together impacted how students thought about
using mathematics curriculum materials.
However, mathematics content and methods
courses are able to provide students with at least some
conceptions of curriculum materials to enable them to
use these materials in skillful ways. To be sure, it
seems unreasonable to think that three curriculum
activities will equip students with all of the necessary
skills to enable them to use curriculum materials
effectively. The curriculum activities in this study did
not seem to influence students’ conceptions of
curriculum materials nor did these activities broaden
students’ potentially limited resources, as described by
Cohen et al. (2002). However, what is evident is that
students’ coursework can, in part, influence such
resources. For this reason, further work needs to be
done to create a more cohesive framework for
Preparing Elementary Preservice Teachers
mathematics content and methods courses that
integrates curriculum materials into the coursework to
a greater extent. Such a framework for content and
methods courses should include several components
that are crucial to helping students learn to use
curriculum materials effectively. Table 4 displays the
different components of this framework.
First, content and methods courses should expose
students to different mathematics curricula and provide
opportunities for students to learn about and familiarize
themselves with the potential resources that are
available to them. When describing their thoughts
about the different curriculum activities, three students
in this study stated that these activities were only
useful to them insofar as the activities exposed them to
different mathematics curricula. Second, students
should develop a discriminating eye towards math
curricula. That is, students should have opportunities to
look across an entire program; assess what information
is provided for teachers, how the lessons are structured
over the school year, and how the various curricular
components are related; and also to evaluate the extent
to which the program is aligned with different
standards and frameworks (when applicable).
Third, students should have opportunities to select,
develop, and possibly adapt mathematical tasks and
appropriate instructional strategies that are typically
provided
in
curriculum materials.
Although
mathematical tasks are important to children’s
learning, the work teachers do with tasks is even more
important. Teachers’ decisions and actions influence
the nature and extent of children’s engagement with
challenging tasks, and ultimately affect children’s
opportunities to learn (Stein et al., 2000). Students
need to learn to assess the difficulty of mathematical
tasks provided in curriculum materials in order to
implement tasks appropriate for children’s current
mathematical ability, and then, when necessary,
modify or adapt tasks in ways that maintain the
integrity of the task. Also, students should learn to
determine whether the given instructional suggestions
are appropriate, and, if not, to identify and employ
instructional strategies that will better facilitate
children’s learning.
Another important element of students’
coursework is to consider the use of manipulatives.
Throughout the two courses, more than half of the
students included manipulatives in their conceptions of
mathematics curriculum materials. Moreover, as the
students in this study described how manipulatives can
be used in the classroom, a majority of these students
stated that manipulatives can not only be used to
Alison M. Castro
Table 4
Framework for Mathematics Content and Methods
Courses
Component
Exposure to curriculum
materials
Purpose
Expose students to potential curricular
resources they may use in the future.
Developing
discriminating eye
Help students develop an overall
understanding of a math curriculum
program (what is important, valuable,
and needs to be modified) and recognize
alignment with state standards and
curriculum frameworks.
a
Math task analysis
Help students select, develop, and
possibly adapt tasks in ways that
maintain
task
integrity;
identify
appropriate instructional strategies.
Effective manipulative
use
Help students use manipulatives in ways
that support and maintain children’s
understandings of concepts.
accommodate children’s different abilities, but also to
make mathematics fun for and applicable to children.
As noted by several researchers, manipulatives can
often be used in unsystematic and unproductive ways
(Ball, 1992; Moyer, 2001; Stein & Bovalino, 2001).
Although teachers may have well-designed lessons
incorporating manipulative-based tasks, children’s
work may not automatically develop in ways that
support their understanding of the mathematics (Stein
& Bovalino, 2001). In addition, children often learn to
use manipulatives in a rote fashion, with little emphasis
and understanding of the mathematical concepts
behind the procedures (Hiebert & Wearne, 1992).
Thus, students need to learn to use manipulatives that
support and scaffold children’s learning, as opposed to
simply making mathematics fun and applicable to
children’s everyday lives, as mentioned by several
students in this study.
By redesigning mathematics content and methods
courses to prepare prospective teachers to use these
resources effectively in their instruction, we can enable
future teachers to more effectively provide students
with a high quality education. If prospective teachers
were better prepared to use mathematics curriculum
materials to create learning opportunities for students,
they would potentially be better prepared to manage
the complexities of teaching.
While [new] teachers may not be able to act upon
such [curriculum] knowledge immediately, it gives
them a mindset to inform their deliberations about
teaching, to view the issues of classroom … in a
21
larger context, and to be dissatisfied with the
compromises and survival tactics of the first year
as they continually reassess their own teaching in
an attempt to provide an appropriate learning
environment for their students. (Zumwalt, 1989,
p.182)
By designing mathematics content and methods
courses that prepare preservice teachers to use
curriculum materials, we are preparing them to become
knowledgeable professionals that are part of a larger
community of educators.
In addition to outlining a framework for preservice
programs, this study raises important issues that should
be taken into consideration when integrating
curriculum material-related coursework into content
and methods courses. First, the findings draw attention
to the influential role of students’ cooperating teachers,
as several students mentioned their cooperating teacher
in their field placement when describing how
mathematics curriculum materials can be used. Some
students seemed to be influenced by what they saw and
heard from their cooperating teachers in their field
placement, a phenomenon identified by other
researchers (Ball & Feiman-Nemser, 1988). Thus, it is
certainly possible that some students receive messages
regarding curriculum materials from cooperating
teachers that are inattentive to the nature of the
contents and suggestions in curriculum materials. At
the same time, they simultaneously receive conflicting
messages from their teacher education programs that
promote careful and deliberate use of these materials.
Although it is not clear how to respond to such a
situation, it is important to be aware of any external
and opposing influences on students’ coursework.
In closing, this study raises several important
issues related to preservice teachers’ conceptions and
use of mathematics curriculum materials. By
understanding the conceptions and assumptions
preservice teachers bring to teacher education
programs about mathematics curriculum materials,
teacher educators can become better able to design
coursework and implement activities that will help
students learn to use these materials in skillful ways.
References
Ball, D. (1990). Breaking with experience in learning to teach
mathematics: The role of a preservice methods course. For the
Learning of Mathematics, 10(2), 10-16.
Ball, D. (1992). Magical hopes: Manipulatives and the reform of
math education. American Educator, 16(2), 14-18, 46-47.
22
Ball, D., & Feiman-Nemser, S. (1988). Using textbooks and
teachers' guides: A dilemma for beginning teachers and
teacher educators. Curriculum Inquiry, 18, 401-423.
Cohen, D., Raudenbush, S., & Ball, D. (2002). Resources,
instruction, and research. In F. Mosteller & R. Boruch (Eds.),
Evidence matters: Randomized trials in education research
(pp. 80-119). Washington DC: Brookings Institution Press.
Cohen, D., Peterson, P., Wilson, S., Ball, D., Putnam, R., Prawat,
R., et al. (1990). The effects of state-level reform of elementary
mathematics curriculum on classroom practice (Final Report
to OERI – Elementary Subjects Center Series No. 25). East
Lansing, MI: Michigan State University, Center for Learning
and Teaching of Elementary Subjects. (ERIC Document
Reproduction Service No. ED323098)
Donovan, B. (1983). Power and curriculum in implementation: A
case study of an innovative mathematics program.
Unpublished doctoral dissertation, University of Wisconsin,
Madison.
Floden, R., Porter, A., Schmidt, W., Freeman, D., & Schwille, J.
(1980). Responses to curriculum pressures: A policy-capturing
study of teacher decisions about content. Journal of
Educational Psychology, 73, 129-141.
Erickson, F. (1985). Qualitative methods in research on teaching
(Occasional Paper No. 81). East Lansing, MI: Michigan State
University, Institute for Research on Teaching. (ERIC
Document Reproduction Service No. ED263203)
Graybeal, S., & Stodolsky, S. (1986, April). Instructional practice
in fifth-grade math and social studies: An analysis of teacher’s
guides. Paper presented at the annual meeting of the American
Educational Research Association, Washington D.C. (ERIC
Document Reproduction Service No. ED276614)
Hiebert, J., Wearne, D. (1992). Links between teaching and
learning place value with understanding in first grade. Journal
for Research in Mathematics Education, 23, 98-122.
Kuhs, T., & Freeman, D. (1979, April). The potential influence of
textbooks on teachers' selection of content for elementary
school mathematics (Research Series No. 48). Paper presented
at the annual meeting of the American Educational Research
Association, San Francisco, CA. (ERIC Document
Reproduction Service No. ED175856)
Lampert, M., & Ball, D. (1998). Teaching, multimedia and
mathematics. New York, NY: Teachers College Press.
McCutcheon, G. (1981). Elementary school teachers’ planning for
social studies and other subjects. Theory and Research in
Social Education, 9, 45-66.
Moyer, P. (2001). Are we having fun yet? How teachers use
manipulatives to teach mathematics. Educational Studies in
Mathematics, 47, 175-197.
National Council of Teachers of Mathematics. (1989). Curriculum
and evaluation standards for school mathematics. Reston,
VA: Author.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics. Reston, VA: Author.
Remillard, J. (1992). Teaching mathematics for understanding: A
fifth-grade teacher's interpretation of mathematics policy. The
Elementary School Journal, 93, 179–183.
Remillard, J. (1996). Changing texts, teachers, and teaching: The
role of curriculum materials in mathematics education reform.
Unpublished doctoral dissertation, Michigan State University,
East Lansing, MI.
Preparing Elementary Preservice Teachers
Remillard, J. (2004). Teachers' orientations toward mathematics
curriculum materials: Implications for teacher learning.
Journal for Research in Mathematics Education, 35, 352-388.
Remillard, J. (1999). Curriculum materials in mathematics
education reform: A framework for examining teachers'
curriculum development. Curriculum Inquiry, 23, 315-342.
Simon, M., & Schifter, D. (1993). Towards a constructivist
perspective: The impact of a mathematics teacher inservice
program on students. Educational Studies in Mathematics, 25,
331-340.
Stake, R., & Easley, J. (1978). Case studies in science education.
Urbana: University of Illinois.
Stein, M., & Bovalino, J. (2001). Manipulatives: One piece of the
puzzle. Mathematics Teaching in the Middle School, 6, 356359.
Stein, M., & Lane, S. (1996). Instructional tasks and the
development of student capacity to think and reason: An
analysis of the relationship between teaching and learning in a
reform mathematics project. Educational Research and
Evaluation, 2(1), 50-80.
Stein, M., Smith, M., Henningsen, M., & Silver, E. (2000).
Implementing standards-based mathematics instruction. New
York: Teachers College Press.
Stephens, W. (1982). Mathematical knowledge and schoolwork: A
case study of the teaching of developing mathematical
processes. Unpublished doctoral dissertation, University of
Wisconsin, Madison.
Strauss, A., & Corbin, J. (1990). Basics of qualitative research:
Grounded theory procedures and techniques. Newbury Park,
CA: Sage.
Trafton, P., Reys, B., & Wasman, D. (2001). Standards-based
mathematics curriculum materials: A phrase in search of a
definition. Phi Delta Kappan, 83, 259-264.
Zumwalt, K. (1989). Beginning professional teachers: The need for
a curricular vision of teaching. In M. C. Reynolds (Ed.),
Knowledge base for the beginning teacher (pp. 173-184).
Oxford, England: Pergamon Press.
1
Given the nature and focus of this study, it is important
to define what is meant by curriculum, as this term often has
multiple meanings. In some cases, curriculum refers to the
overarching national, state, and district-level frameworks
that specify what is to be taught in classrooms. Curriculum
can also refer to the resources teachers use to plan for and
guide their instruction. For the purposes of this study,
curriculum refers only to the resources used by teachers.
Consequently, curriculum materials refer to items such as
teacher guides, assessment resources, manipulatives, and any
other materials that accompany a particular mathematics
curriculum program.
ways intended by curriculum developers per se. Instead, the
author argues that students can learn how to discriminately
use these resources to select, develop, and/or adapt the
features within these materials, such as mathematical tasks
and suggested instructional strategies, in order to create
effective learning opportunities for students.
3
The author was the instructor for the content course and
was not the instructor for the methods course.
4
The primary reason for soliciting students’ initial
conceptions of math curriculum materials in this fashion was
that it may have been uncomfortable for them to be
interviewed upon immediately starting the program.
5
The mid- and post-sequence interview protocols include
the same subset of questions. The questions regarding the
curriculum materials are similar across both protocols, but
the wording is specific to the curriculum activities in
question when appropriate.
6
Students were given the same textbook lesson.
7
Students were given the same three textbook lessons.
8
It is important to note that the different textbook lessons
used in the curriculum activities came from both nonStandards-based and Standards-based mathematics curricula.
The latter, in this case, refer to curricula that were supported
by the National Science Foundation (NSF) funds in the early
1990s that were commissioned to create mathematics
programs that were aligned with the ideas put forth by the
NCTM Standards (1989).
9
Although, arguably, some of the materials included as
non-textual materials (e.g., rulers, protractors, calculators,
etc.) could be considered manipulatives, the materials listed
as manipulatives were considered to be (and were used in
both courses as) manipulatives. Furthermore, the non-textual
materials listed were not considered manipulatives in these
courses. These categories were identified based on students’
responses and the different materials that were used and
discussed in both courses.
10
These three categories of curriculum use comport with
inservice teachers’ use of curriculum materials as found by
Remillard (2004).
11
As the instructor for the content course, the author
supported a modified view of curriculum use. That is, using
curriculum materials in a modified or adaptive fashion.
However, it is not clear how the instructor for the methods
course discussed curriculum use.
2
It is important to note that the author is not implying that
students should learn how to use curriculum materials in the
Alison M. Castro
23
Appendix A
Mid-sequence Interview Protocol
Introduction:
As you know, I am conducting a study of preservice elementary teachers’ views of mathematics curriculum materials.
In this interview, I will be asking you questions about how you think about mathematics curriculum materials in
general, the role they play in the classroom, and how teachers can use these materials to help students learn
mathematics. Finally, I will be asking you questions about the curriculum materials activities from your class this
semester.
Mathematics Curriculum Materials:
1) What do you think of when you hear the phrase “mathematics curriculum materials?” (Issue is how respondent
defines what constitutes mathematics curriculum materials)
2) Why do you think of [list items, ideas mentioned by respondent in previous question] when you think of
mathematics curriculum materials?
3) What role do you think mathematics curriculum materials play in the classroom?
4) Why do you think so?
5) In what ways do you think teachers can use mathematics curriculum materials to help students learn mathematics?
(Issue is how respondent thinks mathematics curriculum materials can and should be used in the classroom,
regardless of their experience with these resources)
EDUC 518 Class:
1) Throughout your class this semester, you talked about and engaged in activities that were directly related to
mathematics curriculum materials. In particular, you worked on analyzing a textbook lesson, which included
examining the tasks, examples, language, representations, as well as the overall mathematical ideas embedded in the
lesson. What are your thoughts about this activity? If respondent asks for clarification: Did you find this activity useful
or not useful? If so, in what ways?
2) Over the course of this semester, what do you think you learned about mathematics curriculum materials?
3) How do you think you learned about mathematics curriculum materials?
4) Were there other activities or discussions, either in this course or in Math 485, that you think helped you learn about
mathematics curriculum materials? (Issue is whether respondent thinks of other activities from either course that
impacted how they think about mathematics curriculum materials)
Conclusion:
1) I really appreciate you taking the time to talk with me. Is there anything else you would like to add to what we have
already talked about here?
2) Do you have any questions for me before we finish this interview?
24
Preparing Elementary Preservice Teachers
The Mathematics Educator
2006, Vol. 16, No. 2, 25–34
Direct and Indirect Effects of Socioeconomic Status and
Previous Mathematics Achievement on High School Advanced
Mathematics Course Taking
Mehmet A. Ozturk
Kusum Singh
Direct and indirect effects of socioeconomic status (SES) and previous mathematics achievement on high
school advanced mathematics course taking were explored. Structural equation modeling was carried out on
data from the National Educational Longitudinal Study: 1988 database. The two variables were placed in a
model together with the mediating variables of parental involvement, educational aspirations of peers, student’s
educational aspirations, and mathematics self-concept. A nonsignificant direct effect of SES on course taking
suggests the lack of an ‘automatic’ privilege of high-SES students in terms of course placements. The
significant indirect effect of previous mathematics achievement tells that it needs to be translated into high
educational aspirations and a strong mathematics self-concept to eventually lead to advanced course taking.
In view of the freedom students in public high
schools have in choosing the courses they take, course
taking is a critical aspect of the students’ education.
Research consistently suggests that this is especially
true for mathematics. Course taking was found to have
the largest effect on academic achievement in
mathematics among the academic subjects examined
(e.g., Jones, Davenport, Bryson, Bekhuis, & Zwick,
1986; Schmidt, 1983). Researchers found that, even
when students’ social background and previous
academic achievement were controlled, course taking
was the single best predictor—twice as strong as any
other factor—of achievement in mathematics (Lee,
Burkam, Chow-Hoy, Smerdon, & Geverdt, 1998). In a
study on National Educational Longitudinal Study:
1988 (NELS: 88) data, results indicated that the
achievement growth differences in mathematics and
science among high- and low-socioeconomic status
students completing the same numbers of courses were
small. This was particularly true in mathematics, where
none of the socioeconomic status (SES) comparisons
were significant among students taking the same
numbers of high school mathematics courses (Hoffer,
Mehmet A. Ozturk is an assistant professor of educational
research at Cleveland State University. His research interests
include secondary school mathematics and science achievement,
psychosocial and educational consequences of adolescent parttime work, and academic expectations in elementary and
secondary schools.
Kusum Singh is a Professor of Educational Research and
Evaluation at Virginia Tech. Her research focuses on family,
social and school factors that affect academic achievement,
particularly in mathematics and science. She teaches graduate
courses in statistics and research methods.
Mehmet K. Ozturk & Kusum Singh
Rasinski, & Moore, 1995). Researchers hypothesize
that mathematics is almost exclusively learned in
school and that background factors do not exert much
of an influence through out-of-school learning (Jones
et al.; Schmidt). A parallel interpretation would be that
much of the SES differences in mathematics
achievement gains over the high school grades are due
to the different numbers of mathematics courses that
high- and low-SES students complete during high
school (Hoffer et al.).
Advanced level mathematics bears importance for
almost all students, regardless of their plans for the
future. For college admissions and success, these
courses play a critical role. Data collected from
students admitted to four-year colleges and universities
show the high numbers of advanced mathematics
courses completed by these students (Owings,
Madigan, & Daniel, 1998; U.S. Department of
Education, 1997). For students who want to enter the
job market after high school, these courses are also
beneficial. Many jobs that once required little
knowledge of mathematics now call for various skills
in algebra and measurement. According to an industrywide standard, an entry-level automotive worker
should have the knowledge to apply formulas from
algebra and physics in order to properly wire the
electrical circuits of any car (U.S. Department of
Education). These advanced courses have also been
shown to improve students’ fundamental mathematical
skills such as problem solving (Jones, 1985; Rock &
Pollack, 1995).
Two fundamental background variables that are
related to advanced mathematics course taking are SES
25
and previous mathematics achievement. SES has been
a significant variable in studies that looked at equity in
advanced course taking. There has been a concern
regarding any possible discrimination in course
placements based on students’ SES or minority status
(e.g., Calabrese, 1989; Lareau, 1987; Useem, 1991). It
may be that parents with higher income and
educational levels are at an advantage in terms of being
able to support their children toward advanced courses.
This can happen in a number of legitimate ways such
as role modeling and providing help with homework.
However, the intriguing question is whether or not
higher levels of SES bring about an ‘automatic’
privilege in course placements. Any such practice by
school administrators or teachers, such as allowing
parents of higher SES to have a greater say in their
children’s course placements, would be a form of
discrimination.
Previous mathematics achievement is also closely
related to advanced level mathematics course taking,
since it has almost always been a consideration in
advanced mathematics course placements. The critical
question about previous mathematics achievement is
whether students who have succeeded in prerequisite
courses automatically enroll in more advanced courses
or whether such achievement, by itself, is not enough
for these students to further their studies in advanced
mathematics. For example, there are studies suggesting
that many students will not proceed with advanced
courses—despite their proven record—unless they are
encouraged to do so or feel confident about their
mathematical abilities (e.g., Lantz & Smith, 1981).
Almost none of the mathematics courses that would be
classified as advanced are required for graduation, and
around 60 percent of high school students graduate
without having taken any of these advanced courses
(Council of Chief State School Officers, 2002; Finn,
Gerber, & Wang, 2002). Given these circumstances, it
is important to translate previous achievement into
advanced course taking.
Literature Review
SES and Advanced Mathematics Course Taking
The term socioeconomic status is used by
sociologists to denote an individual or family’s overall
rank in the social and economic hierarchy (Mayer &
Jencks, 1989). In most research, including national
studies, SES has been measured as a combination of
parents’ education, parents’ occupational prestige, and
family income (Mayer & Jencks; White, 1982).
Researchers have explored a number of
mechanisms through which SES exerts its influence on
26
course taking. The first one is, undoubtedly, role
modeling. Students from middle or high SES families
constantly see, in their parents and neighbors, social
and economic payoffs that good education could
provide, while many minority children in high-poverty
areas have few—if any—role models who have
succeeded in school or who have translated school
success into economic gain (Lippman et al., 1996;
Oakes, 1990). These claims suggest an indirect effect
of SES on advanced course taking through students’
educational aspirations.
Another mechanism in the relationship between
SES and advanced mathematics course taking is the
high educational expectations middle- and high-SES
families have for their children (Khattri, Riley, &
Kane, 1997; Muller & Kerbow, 1993), and the
conveyance of these expectations to them (Anderson,
1980; Hossler & Stage, 1992; Seginer, 1983; Wilson &
Wilson, 1992). Lantz and Smith (1981) found
significant positive relationships of educational
aspirations and parental encouragement to taking
nonrequired mathematics courses in high school.
Indicators of SES (parents' education and parents'
occupation) were only weakly related to nonrequired
mathematics course taking, when other important
variables, such as educational aspirations and parental
encouragement, were entered in the same regression
analysis. Findings from another study indicated that
well-educated parents were far more inclined to
pressure their children to take demanding mathematics
courses (Useem, 1991). Altogether, these findings
illustrate an indirect path from SES to advanced course
taking through parental expectations and involvement,
which influence the student’s own educational
aspirations.
Peers’ educational aspirations may also play a role
in the indirect effect of SES on advanced course
taking. Previous research suggests that parents want
and encourage their children to have friends with
similar educational aspirations, since peers’ aspirations
will be influential on the student’s own aspirations
(Cooper & Cooper, 1992). Davies and Kandel (1981),
in a study of adolescents, found a significant
relationship between parents' educational aspirations
for their children and aspirations of their children's best
friends. Another finding by Lantz and Smith (1981)
was that perceived peer attitudes were significantly
related to election of nonrequired mathematics courses,
even when it was entered in the regression analysis
together with parents' education, parents' occupation,
parental encouragement, and some other variables.
Direct and Indirect Effects
This group of findings assigns importance to peers’
educational aspirations.
An important issue is the difference between
parental influence and peer influence on the student’s
educational aspirations, which will have an impact on
course taking decisions. As for the parent or peer
influence in general, peers may have a greater
influence in some areas of the student's life, such as
types of behavior determining the current adolescent
lifestyle, whereas parents may have a higher influence
in other issues, such as those relevant to future goals
(Biddle, Bank, & Marlin, 1980; Davies & Kandel,
1981). However, with regards to the parent or peer
influence specifically on the student's educational
aspirations, Davies and Kandel found that parental
influence was much stronger than peer influence and
did not decline over the adolescent years.
Guidance and counseling policies and practices in
schools can either magnify or reduce the effect of SES
on course taking. There has been an ongoing concern
about the availability of counseling services to students
and parents, mostly due to high student-to-counselor
ratios and counselors’ record keeping duties (Martin,
2002; Powell, Farrar, & Cohen, 1985). A study on a
nationally representative sample of public school
students in grades five through eleven revealed that a
considerable number of students were not told about
the academic implications of their course taking
decisions in mathematics. Parents did not have any
access to counseling services either (Leitman, Binns, &
Unni, 1995). In the absence of proper counseling in
school, well-educated parents are still able to guide
their children through important course taking
decisions, since they are knowledgeable about courses.
On the other hand, those with low levels of education
cannot be of any help to their children in this matter
(Useem, 1991). What makes this problem worse is
even less availability of these services to low-SES or
minority students, who cannot get sufficient—if any—
guidance from their families or communities and are in
need of guidance the most. Many of these students are
concentrated in high-minority, high-poverty schools
(Lippman et al., 1996) and are less likely to have
access to guidance counseling for course taking
decisions (Lee & Ekstrom, 1987; Leitman et al.).
The mechanisms mentioned above are ways in
which parents’ socioeconomic characteristics can
legitimately play an indirect role in students’ course
taking. However, there are also concerns sounded by
researchers that imply a direct effect of SES, which can
be seen as a form of discrimination based on SES. For
example, some researchers assert that there is a
Mehmet K. Ozturk & Kusum Singh
differential treatment of minority students by school
counselors and teachers through counseling them into
undemanding nonacademic courses and discouraging
them from academic ones (Calabrese, 1989; Leitman et
al., 1995). Still others believe that the extent to which
schools allow parental involvement in course
placements is another factor that works against lowSES or minority students. In schools that try to
constrain parental intervention in course taking
decisions, parents with high SES or members of the
dominant culture have more of the social, intellectual,
and cultural resources to acquire the crucial
information about course placements and the courage
to take initiative. On the other hand, parents with low
levels of education remain uninformed and
discouraged by the school personnel to take initiative
in this matter (Lareau, 1987; Useem, 1991).
Previous Mathematics Achievement and Advanced
Mathematics Course Taking
Findings about the direct effect of previous
mathematics achievement on subsequent mathematics
course taking are inconclusive. Some studies found
that mathematics achievement still had a significant
effect on subsequent mathematics course taking, even
after taking into account the effect of mathematics selfconcept (e.g., Marsh, 1989); whereas others found that
previous mathematics achievement did not consistently
predict subsequent mathematics course taking. For
example, Lantz and Smith (1981) found that when the
last grade earned in mathematics was entered into
regression analysis with several other variables, such
as parents' education, parents' occupation and the
student’s mathematics self-concept, it predicted the
participation in nonrequired mathematics courses in
only one sample out of three. Students' subjective
comparisons of their mathematics performances with
those of other students as well as with their own
performances in other subjects were better predictors
of mathematics participation than the last grade earned.
The direct effect of previous mathematics
achievement on subsequent mathematics course taking,
especially at an advanced level, may be expected, since
previous achievement, measured either by grades or
standardized test scores, is a widespread criterion in
high school mathematics course placements (Oakes,
Gamoran, & Page, 1992; Useem, 1991). However, note
that its use as a criterion does not necessitate successful
students’ enrollment in subsequent nonrequired
advanced level mathematics courses.
27
Research Questions
The present study tries to answer two important
questions: First, is there any discrimination in
advanced mathematics course placements based on
students’ socioeconomic status? Second, do students
who succeed in previous mathematics courses
automatically enroll in the more advanced ones? Put
differently, is previous mathematics achievement a
necessary and sufficient condition for taking advanced
mathematics courses; and to what degree do some
variables, such as mathematics self-concept and
educational aspirations, mediate the relationship
between previous mathematics achievement and
advanced mathematics course taking?
Method
Study Design
The present study focused on the effects of two
background variables—namely, SES and previous
mathematics achievement—on advanced mathematics
course taking. The purpose of the study is to see how
much of the effect of each of these two variables is
direct and how much is an indirect effect mediated
through parental involvement, educational aspirations
of peers, educational aspirations of the student, or
mathematics self-concept. This has been done through
testing a causal model that includes these secondary
variables. The hypothesized relationships are
illustrated in Figure 1.
SES
School-level variables reflecting school policies
that are influential on students’ course taking were
controlled by keeping them constant. These variables
are type of school, variety and level of mathematics
courses offered, graduation requirements, and type of
tracking. The following is a description of how and
why these variables were kept constant.
In this study, only students in public high schools
were selected. Two major differences between public
and private high schools influenced our decision to ask
the research questions stated above for public high
school students only. The first difference is in the
curriculum. Most private high schools are
characterized by a narrow academic curriculum, where
all students complete a narrow set of mostly academic
courses, almost all of which are required for
graduation. Catholic schools, a large sector of private
schooling, are a good example of this type of
curriculum (Lee et al., 1998; Lee, Croninger, & Smith,
1997). In contrast, public high schools predominantly
feature a differentiated curriculum approach (Lee et
al.). In these schools, students take a subset of courses
among a variety of available offerings and are free to
take elective courses beyond graduation requirements.
Both research questions stated above are related to
students’ course selections and placements. In view of
the above-mentioned policies and practices in most
private schools, these research questions are primarily
relevant to public high school students.
EDUC.
ASPR.
PARENT
INVOLVE
MENT
MATH
COURSE
TAKING
EDUC. ASPR.
OF PEERS
PREV.
MATH
ACH.
MATH
SELF
CONCEPT
Figure 1. The Causal Model in the Present Study
28
Direct and Indirect Effects
In the present study, course taking in Algebra II,
Geometry, Trigonometry, and Calculus is investigated.
Accordingly, only students in schools that offered all
of these courses were selected.1
School graduation requirements have at least some
impact on students’ course taking, since these
requirements set the minimum for course taking. Since
requiring a minimum of exactly two years of
mathematics for graduation was the most common
practice in the schools that participated in the NELS:
88 study, only students in such schools were selected
in the present study.2
Broadly, tracking can be defined as a way of
grouping students, in which students enroll in different
programs of study and take different sequences of
courses based on their ability and success levels. The
multitude of tracking practices in schools makes it
difficult for researchers to employ a precise measure of
this variable. In much survey research, the three-track
categorization (i.e., academic, general, and vocational)
has been employed as a crude measure of tracking
(Oakes et al., 1992). Literature on the effect of tracking
on students’ course taking is confusing and
contradictory, since the term ‘tracking’ does not have a
uniform meaning. In the present study, only students
who were in any one of the three traditional programs
(general, academic, or vocational) were selected, which
excluded the ones who were in special or innovative
programs.
Data Source and Sample
In this study, the data were drawn from the base
year and first and second follow-up of NELS: 88. In
the base year, a two-stage stratified sample design was
used, with schools as the first-stage unit and students
within schools as the second-stage unit (Ingels et al.,
1994). The schools were stratified based on type
(public versus private), geographic region, urbanicity,
and percent of minority enrollment (Spencer et al.,
1990). Within each stratum, schools were selected with
probabilities in proportion to their estimated eighth
grade enrollment, which led to a pool of approximately
1000 schools. In the second stage of sampling, an
average of 23 students was selected randomly from
each school, producing a total sample of approximately
23,000 eighth-graders for the base year. The first and
second follow-up data were collected from the same
cohort in 1990 and in 1992, when most of the students
were tenth and twelfth graders, respectively. The
cohort was freshened with proper statistical techniques
in 1990 and in 1992 to achieve a representative sample
of the nation’s sophomores in the first follow-up and
Mehmet K. Ozturk & Kusum Singh
seniors in the second follow-up, respectively (Ingels et
al.).
At the first step of sample selection for the present
study, students who were members of the NELS: 88
sample in all three waves of data collection (base year,
first follow-up, and second follow-up) and for whom
transcript data were available, were selected. This
group was readily defined by NELS: 88 as a subgroup
within the overall sample, and a sampling weight was
provided. This group is a nationally representative
sample of 1988 8th graders, regardless of whether they
graduated from high school four years later or not.
Among these students, only those who graduated from
high school in Spring 1992 were selected in the first
step of this study.
At the second step, relevant school variables were
kept constant by selecting students who met all of the
criteria below:
• Did not change their schools between base year
(8th grade) and second follow-up (12th grade);
• Enrolled in a public school;
• Enrolled in schools that offered the complete set
of mathematics courses of interest in the present
study (Algebra II, Geometry, Trigonometry, and
Calculus);
• Were in schools that required exactly two years
of mathematics for graduation; and
• Were either in general, academic, or vocational
programs.
All the analyses in this study were performed on
this sample after a listwise deletion of missing data,
which resulted in a sample size of 1,699.
Measures
Advanced mathematics course taking. This
variable was measured as the sum of total Carnegie
units earned in Algebra II, Geometry, Trigonometry,
and Calculus. A Carnegie unit is defined as “a standard
of measurement used for secondary education that
represents the completion of a course that meets one
period per day for one year” (Ingels et al., 1994, p. O1).
Socioeconomic status of the student. SES was
measured by F2SES1, a continuous composite variable
already available in the NELS:88 database. It was
constructed from the base year parent questionnaire
data using five items: father’s education level,
mother’s education level, father’s occupation, mother’s
occupation, and family income. Occupational data
were recoded using Duncan’s Socioeconomic Index
(SEI) (as cited in Ingels et al., 1994), which assigns
values to various occupational groups.
29
Student’s previous mathematics achievement. This
variable was measured by BY2XMSTD, a score from a
standardized mathematics test administered in the
spring of 1988 (spring of 8th grade for the sample).
Parental expectations. This variable was measured
by two first follow-up (10th grade) and two second
follow-up (12th grade) questions, asking the students
how far in school they think their father or mother (one
question for father and one for mother) wants them to
go. There are ten choices indicating different levels of
education.
Parental involvement. This variable was measured
by two questions asked both in the first follow-up and
the second. The first question asked how often the
student discussed selecting courses at school with
either or both parents or guardians in the first half of
the school year. The second question asked how often
the student discussed going to college with either or
both parents or guardians in the first half of the school
year. Choices were never, sometimes, and often.
Educational aspirations of peers. This variable
was measured by two first follow-up and two second
follow-up questions. One question asked the student
how important, within the student’s peer group, it is to
get good grades. The second question asked, in the
same context, how important it is to continue education
past high school. Choices were not important,
somewhat important, and very important.
Mathematics self-concept. This variable was
measured by four first follow-up questions, asking the
student to choose the best answer for the following
items:
1. Mathematics is one of my best subjects.
2. I have always done well in mathematics.
3. I get good marks in mathematics.
4. I do badly in tests of mathematics.
The available choices were false, mostly false,
more false than true, more true than false, mostly true,
and true. The fourth item was reverse coded for
consistency. These four questions in the NELS: 88
database come from the SDQ-II by Marsh (as cited in
Ingels, Scott, Lindmark, Frankel, & Myers, 1992).
Educational aspirations of the student. This
variable was measured by one base-year, one first
follow-up, and one second follow-up question asking
the students how far in school they think they will get.
There were six educational levels as choices for the
base-year question, nine for the first follow-up, ten for
the second follow-up.
30
Analytic Method
Structural equation modeling was used as the
analytic technique. Such modeling allowed studying
the relative importance of variables as well as their
direct and indirect effects on the outcome variable.
Recall the hypothesized relationships among the
variables in the model (Figure 1). All of the
relationships in the model are hypothesized to be
positive, meaning that an increase in one variable in a
hypothesized relationship leads to an increase in the
other.
In all analyses, data were weighted by
F2TRP1WT, the sampling weight in the NELS: 88
database, created specifically for the sample used in
this study. This subsample is described as the students
who were sample members in the base year, first
follow-up, and the second follow-up of data collection
and for whom high school transcripts were collected.
An alpha level of .05 was used for all statistical tests.
For creation of correlation matrices, standard
deviations, and means to be used in structural equation
model estimations, the computer program SPSS 10.1
(SPSS Inc., 2000) was used. For structural equation
model estimations, LISREL 8.54 computer software
(Joreskog & Sorbom, 2003) was employed.
Results
During confirmatory factor analysis of the
measurement model, parental expectations were
included as a separate latent variable. However,
estimation of this model yielded a high collinearity
between parental expectations and educational
aspirations of the student. As verification, factor scores
for the two constructs were created through exploratory
factor analysis, and the bivariate correlation between
the factor scores was calculated. The resulting
correlation was .798, again indicating a high
collinearity. As a result, parental expectations were
eliminated from the model, and educational aspirations
of the student were kept. Estimation of this revised
model yielded a good fit (CFI = .94; GFI = .92;
Standardized RMR = .05). With acceptable values of
fit indices and all loadings significant at p < .05 level,
this model was chosen to be the final measurement
model. Table 1 presents the factor loadings.
After deciding on the measurement model, the
causal relationships among the variables were specified
in an initial structural model. Overall, results indicated
a good fit with the hypothesized relationships (CFI =
.94; GFI = .92; Standardized RMR = .06). The
directions and magnitudes of path coefficients,
representing the hypothesized relationships between
Direct and Indirect Effects
Table 1
Factor Loadings for the Final Measurement Model
No. of
Math
Courses
Taken
TOTALMAT
Ed. Aspirations
of
the Student
Math SelfConcept
Ed.
Aspirations
of Peers
Parental
Involvemen
t
Previous Math
Ach.
SES
1.00
BYS45
.66
F1S49
.79
F2S43
.76
F1S63D
.90
F1S63J
.88
F1S63Q
.90
F1S63S
.69
F1S70D
.65
F1S70I
.72
F2S68D
.43
F2S68H
.47
F1S105A
.48
F1S105G
.58
F2S99A
.50
F2S99F
.64
BY2XMSTD
1.00
F2SES1
1.00
Completely Standardized Factor Loadings.
All loadings were significant at p < .05 level.
the variables, were in accord with theory and previous
research findings. Therefore, this model was decided to
be the final structural model. Total, direct, and indirect
effects for the final structural model are given in Table
2. The correlation between SES and previous
mathematics achievement was .40.
Discussion
Findings
During the development of the final measurement
model, a very high positive correlation was found
between parents' educational expectations for their
children and students' educational aspirations. This
finding supports the claim that, starting from early
childhood, children imitate, identify, and, finally,
internalize the values and attitudes of their parents
(Comer, 1990). It is also congruent with previous
findings (e.g., Davies & Kandel, 1981) that parental
influence does not decline over the adolescent years.
However, the finding in the present study is about the
parental influence in matters of future educational
Mehmet K. Ozturk & Kusum Singh
plans and should not be overgeneralized to all aspects
of adolescent life.
One of the two key questions in this study was
whether SES would still have a significant direct effect
on mathematics course taking after its indirect effects
were taken into account. This analysis found no direct
effect of SES on mathematics course taking; however,
its indirect effect was not trivial (.14). This finding
fails to support the claim that parents' SES plays a
direct role in students' course placements. It implies
that there is no automatic privilege of being a student
from a middle- or high-SES family; rather, parental
involvement is critical in students taking advanced
mathematics courses. When the total indirect effect of
SES is partitioned into its components, .09 belongs to
the indirect path from SES to educational aspirations of
the student to mathematics course taking and .05
belongs to the indirect path from SES to parental
involvement to educational aspirations of the student to
mathematics course taking. The significant relationship
between parental involvement and educational
31
Table 2
Total, Direct, and Indirect Effects for the Final Structural Model
Parental
Involvement
Previous
Math Ach.
SES
Parental
Inv.
Educational
Asp. of Peers
Math SelfConcept
Educational Asp. of
the Student
Total
.13*
.31*
--
--
--
--
Direct
.13*
.31*
--
--
--
--
--
--
--
--
--
--
Total
.06*
.14*
.47*
--
--
--
Direct
--
--
.47*
--
--
--
Indirect
.06*
.14*
--
--
--
--
Total
.40*
--
--
--
--
--
Direct
.40*
--
--
--
--
--
Indirect
Educational
Aspirations
Peers
Math
Concept
of
Self-
Indirect
Educational Asp.
of the Student
No. of Math
Courses Taken
--
--
--
--
--
--
Total
.40*
.35*
.46*
.18*
--
--
Direct
.34*
.21*
.38*
.18*
--
--
Indirect
.06*
.14*
.08*
--
--
--
Total
.53*
.14*
.19*
.07*
.17*
.41*
Direct
.30*
.00
--
--
.17*
.41*
.19*
.07*
--
--
Indirect
.23*
.14*
Standardized Total, Direct, and Indirect Effects.
* indicates significance at p < .05 level.
aspirations of the student coupled with the finding that
there was a very high correlation between parental
educational expectations for the student and the
student’s own educational aspirations stresses parents’
critical role in their children’s education. Furthermore,
our results support the findings of Davies and Kandel
(1981) that parental influence on the student’s
educational aspirations was much stronger than that of
peers and did not decline over the adolescent years.
The other key question in this study was to what
degree the relationship between previous mathematics
achievement and advanced mathematics course taking
was mediated by other variables. The direct effect of
previous mathematics achievement on the number of
mathematics courses taken was found to be .30, which
was significant. A direct effect of previous
achievement was expected in this study, above and
beyond its indirect effect. The finding that previous
achievement also had a significant indirect effect of .23
implies that, even though previous achievement may
be a necessary condition most of the time, it is not a
sufficient condition for students to take advanced and
more challenging mathematics courses. When the
indirect effect was partitioned, a major portion
belonged to the path from previous mathematics
32
achievement to educational aspirations to course taking
(.14). The second largest belonged to the path from
previous mathematics achievement to mathematics
self-concept to course taking (.07). These two
components made up almost all of the indirect effect.
Since knowledge of mathematics is cumulative and
mathematics courses are sequential, these findings
suggest that early and continued success in
mathematics is critical for maintaining high
educational aspirations as well as self-confidence in
mathematics.
Implications
Perhaps the most important finding is that parental
involvement plays a critical role in students’ advanced
mathematics course taking. Schools can play a major
role in improving this determining factor. In the case of
advanced mathematics course taking, the first step to
enhancing parental involvement is to inform parents
about the importance of these courses for the student’s
future. This should be done in as many ways as
possible, including advising, parent conferences, and
sending information to parents. A next step may be to
inform parents about the courses, their sequences and
prerequisites, and related policies. This, first of all,
Direct and Indirect Effects
requires schools being transparent in their course
placement policies. Such transparency may also help
eliminate claims of implicit discriminatory tracking in
many schools. Schools can also clearly state their
course placement policies on their websites, in parent
manuals, or other related publications. This will
encourage parents to get involved in their children’s
course taking.
Findings from this study also indicate that
successful students in mathematics need to translate
their achievement into high educational aspirations to
continue taking non-required advanced mathematics
courses. This translation naturally occurs at home for
students from families with a high level of education,
where examples of opportunities a strong background
in mathematics can provide are immediate. This issue,
however, is critical for students coming from
disadvantaged families and communities with little
appreciation for education and little knowledge of the
education system. In the absence of a push by school
policies towards advanced coursework, the only source
of guidance, encouragement, and support for these
students will be their teachers and school counselors.
Therefore, frequent individual advising should be
provided to such students in order to encourage and
motivate them to take advanced courses and alert them
to prerequisites and other course placement criteria.
Recommendations for Future Research
First, a useful follow-up study would check any
possible differential relationships among the variables
in the model for students in schools with different
demographic characteristics. For example, parental
involvement may be more important for minority
students living in high-minority, high-poverty inner
cities than it is for Whites living in low-minority, lowpoverty suburbs. The three school demographic
variables that need to be considered are poverty
concentration, minority concentration, and urbanicity.
Second, future research should also consider
perceived utility of mathematics as a possible
influential variable. Several studies revealed that the
perceived utility of mathematics by students in their
future career is a significant factor in shaping
educational aspirations based on previous mathematics
achievement (Lantz & Smith, 1981; Linn & Hyde,
1989; Reyes, 1984). Due to the limitations of the
NELS: 88 database, this variable could not be included
in the present study.
Finally, qualitative studies investigating the
nature of relationships between parents and school
Mehmet K. Ozturk & Kusum Singh
administrators may prove useful in improving students’
advanced course taking through parental involvement.
References
Anderson, K. L. (1980). Educational goals of male and female
adolescents. The effects of parental characteristics and
attitudes. Youth & Society, 12, 173-188.
Biddle, B. J., Bank, B. J., & Marlin, M. M. (1980). Parental and
peer influence on adolescents. Social Forces, 58, 1057-1079.
Calabrese, R. L. (1989). The discriminatory impact of course
scheduling on minorities. Peabody Journal of Education,
66(4), 30-41.
Comer, J. P. (1990). Home, school, and academic learning. In J. I.
Goodlad & P. Keating (Eds.), Access to knowledge: An
agenda for our nation's schools (pp. 23-42). New York:
College Entrance Examination Board.
Cooper, C. R., & Cooper, R. G., Jr. (1992). Links between
adolescents' relationships with their parents and peers:
Models, evidence, and mechanisms. In R. D. Parke & G. W.
Ladd (Eds.), Family-peer relationships: Modes of linkage (pp.
135-158). Hillsdale, NJ: Lawrence Erlbaum.
Council of Chief State School Officers. (2002). Key state education
policies on PK-12 education: 2002. Retrieved March 3, 2006,
from http://www.ccsso.org/content/pdfs/KeyState2002.pdf
Davies, M., & Kandel, D. B. (1981). Parental and peer influences
on adolescents' educational plans: Some further evidence.
American Journal of Sociology, 87, 363-387.
Finn, J. D., Gerber, S. B., & Wang, M. C. (2002). Course offerings,
course requirements, and course taking in mathematics.
Journal of Curriculum and Supervision, 17, 336-366.
Hoffer, T. B., Rasinski, K. A., & Moore, W. (1995). Social
background differences in high school mathematics and
science coursetaking and achievement (NCES No. 95-206).
Washington, DC: U.S. Department of Education, Office of
Educational Research and Improvement.
Hossler, D., & Stage, F. K. (1992). Family and high school
experience influences on the postsecondary educational plans
of ninth-grade students. American Educational Research
Journal, 29, 425-451.
Ingels, S.J., Dowd, K.L., Baldridge, J.D., Stipe, J.L., Bartot, V.H.,
& Frankel, M.R. (1994). National education longitudinal
study of 1988 second follow-up: Student component data file
user’s manual (NCES 94-374). Washington, DC: U.S.
Department of Education, Office of Educational Research and
Improvement.
Ingels, S. J., Scott, L.A., Lindmark, J. T., Frankel, M. R., & Myers,
S. L. (1992). National education longitudinal study of 1988.
first follow-up: Student component data file user’s manual
(NCES 92-030). Washington, DC: U.S. Department of
Education, Office of Educational Research and Improvement.
Jones, L. V. (1985, March-April). Black-White differences in
mathematics: Some new research findings. Paper presented at
the annual meeting of the American Educational Research
Association, Chicago, IL.
Jones, L. V., Davenport, E. C., Jr., Bryson, A., Bekhuis, T., &
Zwick, R. (1986). Mathematics and science test scores as
related to courses taken in high school and other factors.
Journal of Educational Measurement, 23, 197-208.
33
Joreskog, K., & Sorbom, D. (2003). LISREL 8.54 [Computer
software]. Chicago, IL: Scientific Software International.
Reyes, L. H. (1984). Affective variables and mathematics
education. Elementary School Journal, 84, 558-581.
Khattri, N., Riley, K. W., & Kane, M. B. (1997). Students at risk in
poor, rural areas: A review of the research. Journal of
Research in Rural Education, 13(2), 79-100.
Rock, D. A., & Pollack, J. M. (1995). Mathematics course-taking
and gains in mathematics achievement. Statistics in brief
(NCES 95-714). Washington, DC: U.S. Department of
Education, Office of Educational Research and Improvement.
Lantz, A. E., & Smith, G. P. (1981). Factors influencing the choice
of nonrequired mathematics courses. Journal of Educational
Psychology, 73, 825-837.
Schmidt, W. H. (1983). High school course-taking: Its relationship
to achievement. Curriculum Studies, 15, 311-332.
Lareau, A. (1987). Social class differences in family-school
relationships: The importance of cultural capital. Sociology of
Education, 60, 73-85.
Seginer, R. (1983). Parents' educational expectations and children's
academic achievements: A literature review. Merrill-Palmer
Quarterly, 29, 1-23.
Lee, V. E., Burkam, D. T., Chow-Hoy, T., Smerdon, B. A., &
Geverdt, D. (1998). High school curriculum structure: Effects
on coursetaking and achievement in mathematics for high
school graduates (NCES Working Paper No. 98-09).
Washington, DC: U.S. Department of Education, Office of
Educational Research and Improvement.
Spencer, B. D., Frankel, M. R., Ingels, S. J., Rasinski, K. A.,
Tourangeau, R., & Owings, J. A. (1990). National education
longitudinal study of 1988: Base year sample design report
(NCES 90-463). Washington, DC: U.S. Department of
Education, Office of Educational Research and Improvement.
Lee, V. E., Croninger, R. G., & Smith, J. B. (1997). Course-taking,
equity, and mathematics learning: Testing the constrained
curriculum hypothesis in U.S. secondary schools. Educational
Evaluation and Policy Analysis, 19, 99-121.
Lee, V. E., & Ekstrom, R. B. (1987). Student access to guidance
counseling in high school. American Educational Research
Journal, 24, 287-310.
Leitman, R., Binns, K., & Unni, A. (1995). Uninformed decisions:
A survey of children and parents about math and science.
NACME Research Letter, 5(1), 1-10.
Linn, M. C., & Hyde, J. S. (1989). Gender, mathematics, and
science. Educational Researcher, 18(8), 17-19, 22-27.
Lippman, L., Burns, S., McArthur, E., Burton, R., Smith, T. M., &
Kaufman, P. (1996). Urban schools: The challenge of location
and poverty (NCES 96-184). Washington, DC: U.S.
Department of Education, Office of Educational Research and
Improvement.
Marsh, H. W. (1989). Sex differences in the development of verbal
and mathematics constructs: The High School and Beyond
Study. American Educational Research Journal, 26, 191-225.
Martin, P. J. (2002). Transforming school counseling: A national
perspective. Theory into Practice, 41, 148-153.
Mayer, S. E., & Jencks, C. (1989). Growing up in poor
neighborhoods: How much does it matter? Science, 243, 14411445.
Muller, C., & Kerbow, D. (1993). Parent involvement in the home,
school, and community. In B. Schneider & J. S. Coleman
(Eds.), Parents, their children, and schools (pp. 13-42).
Boulder, CO: Westview Press.
Oakes, J. (1990). Opportunities, achievement, and choice: Women
and minority students in science and mathematics. Review of
Research in Education, 16, 153-222.
Oakes, J., Gamoran, A., & Page, R. N. (1992). Curriculum
differentiation: Opportunities, outcomes, and meanings. In P.
W. Jackson (Ed.), Handbook of research on curriculum (pp.
570-608). New York: Mc Millan.
Owings, J., Madigan, T., & Daniel, B. (1998). Who goes to
America’s highly ranked “national” universities? (NCES 98095). Washington, DC: U.S. Department of Education, Office
of Educational Research and Improvement.
Powell, A. G., Farrar, E., & Cohen, D. K. (1985). The shopping
mall high school. Boston: Houghton Mifflin.
34
SPSS Inc. (2000). SPSS 10.1 [Computer software]. Chicago, IL:
Author.
U. S. Department of Education. (1997). Mathematics equals
opportunity. White paper prepared for U.S. Secretary of
Education. Retrieved March 3, 2006, from
http://ed.gov/pubs/math/index.html
Useem, E. L. (1991). Student selection into course sequences in
mathematics: The impact of parental involvement and school
policies. Journal of Research on Adolescence, 1, 231-250.
White, K. R. (1982). The relation between socioeconomic status
and academic achievement. Psychological Bulletin, 91, 461481.
Wilson, P. M., & Wilson, J. R. (1992). Environmental influences
on adolescent educational aspirations. Youth & Society, 24,
52-70.
1
Research suggests that in schools where the variety in
low-end (basic, general) mathematics courses is limited,
students tend to take more advanced courses and the average
achievement in the school is higher (Lee et al., 1998).
Although selecting students in schools offering the courses
counted as advanced in this study provided control over
advanced level course offerings, inability to control for the
variety in lower level course offerings should be
acknowledged as a limitation of the study. Such control
would have significantly reduced the sample size due to the
variation in these courses among schools.
2
After the collection of data used in this study, there have
been changes in the graduation requirements imposed by
states, school districts, or individual schools. These ongoing
changes have especially gained momentum after the No
Child Left Behind Act of 2001. They include introducing
graduation (exit) examinations and requiring specific
courses to be taken for graduation. However, this study does
not investigate the effect of graduation requirements on
course taking, and only deals with it for control purposes.
We believe that there have not been any significant changes
in the variables investigated through the causal model in this
study.
Direct and Indirect Effects
The Mathematics Educator
2006, Vol. 16, No. 2, 35–42
Efficacy of College Lecturer and Student Peer Collaborative
Assessment of In-Service Mathematics Student Teachers’
Teaching Practice Instruction
Lovemore J. Nyaumwe
David K. Mtetwa
This study investigated the effectiveness of collaboration between college lecturers and student peers in
assessing the instructional practice of in-service student teachers (ISTs). The study was inspired by criticisms
that college lecturers’ assessments were not producing valid critiques of ISTs’ mathematical and pedagogical
competencies to implement strategies they learned in their coursework. Case studies of two pairs of ISTs, one
pair at a state high school and the other at a private high school, provided data for this study. During their
coursework, ISTs learned new pedagogical skills and upgraded their content knowledge. Findings indicate that
lecturer and peer assessment of the same lesson taught by an IST resulted in different but complementary
critiques. The lecturer’s critique highlighted both strengths and weaknesses of a lesson while the peer’s critique
refrained from pointing out weaknesses of a lesson. An important implication for the findings, in Zimbabwe, is
that the deployment of ISTs in pairs for teaching practice may be beneficial to their professional development.
The purpose of this study is to determine the
effectiveness of collaborative assessment of in-service
mathematics student teachers’ classroom practice by
both a lecturer and a peer. This assessment occurred
during the full-time teaching practice segment of their
program in which student teachers were encouraged to
implement
constructivist-inspired
instructional
strategies. An in-service student teacher (IST) is a
certified and experienced teacher who enters a collegebased program of professional study for the purpose of
improving their professional knowledge and skills
using specific mathematical reforms in pedagogy and
content.
Peers are fellow in-service student teachers
working at the same teaching practice school. Student
teachers acting as peers attended the Bachelor of
Science in Education (BScEd) in-service program
during the same period. A lecturer is a university
mathematics educator who teaches the ISTs during the
Lovemore J. Nyaumwe is a Mathematics Education lecturer at
Bindura University of Science Education, Zimbabwe. He taught
mathematics at high school before becoming a lecturer. His
research interest is teacher professional development in
preservice and in-service contexts.
David K. Mtetwa is a senior Mathematics Education lecturer at
the University of Zimbabwe. He has a passionate interest for
teacher professional development. He has made an impact on
improving mathematics teacher education in the Southern Africa
region through national and regional involvement as an external
examiner for several universities, mathematics education
consultant, author and plenary speaker at national and regional
conferences.
Lovemore J. Nyaumwe & David K. Mtetwa
residential portion of their program. Lecturer-peer
collaborative assessment can be viewed as the joint
operation of a lecturer and peer in assessing the level to
which an IST has developed attitudes, knowledge, and
skills to implement constructivist pedagogical
strategies in their teaching. In addition, they make
suggestions to improve the implementation strategies
specific to an assessed lesson.
Constructivist theories encourage the use of
learner-centered instructional pedagogies because,
from a fallibilist perspective, mathematical knowledge
is viewed as context-based. Knowledge is believed to
originate from observations, experimentation and
abstraction using specific senses and, therefore, is
fallible, tentative, intuitive, subjective and dynamic
(Nyaumwe, 2004). From a constructivist view, to teach
mathematics well is to equip learners with a conceptual
understanding of the process skills that enables them to
individually or collectively develop a repertoire for
developing powerful constructions that concur with
viable mathematical knowledge (Davis, 1990).
Lecturer-peer assessment involves a lecturer and a
peer simultaneously assessing the instructional practice
of an IST, or a peer alone making the assessment for
the purposes of formative evaluation. In the absence of
a lecturer, peer assessment is still collaborative because
a peer acts as a proxy for the lecturer and reports to a
lecturer when they meet. Reciprocal peer assessment of
lessons is a two-way collaborative process that helps
both the peer assessor and the IST generate ideas about
how to improve their teaching practices. Collaborative
evaluation of an IST’s implementation of a
35
pedagogical reform is essential in order to identify
strengths and weaknesses. If the IST is performing
unsatisfactorily, the evaluation can assist in upgrading
practice to appropriate standards (Ziv, Verstein, &
Tamir, 1993).
Assessment of ISTs’ professional competencies
during practice teaching is a polemical issue in
Zimbabwe. Teacher educators in the country take the
assessment as their privileged domain. They lament
subjectivity and the propensity toward bias when
school authorities get involved in the assessment
process. Usually school-based assessments and
lecturers’ assessments vary significantly (Nyaumwe &
Mavhunga, 2005). Despite the assessment differences,
Zindi (1996) suggested that schools be involved in the
assessment of student teachers. He argued that reliance
on a lecturer as the only assessor of student teachers’
practice was not fair or valid because several assessors
produce more objective assessments of student
teachers’ professional competencies than a single
source.
Mathematics curriculum reform in Zimbabwe
encourages teachers to adopt constructivist approaches
in their teaching because of the potential of these
approaches to enable learners to transfer school
mathematics to contextualized situations through
modeling and problem solving. Constructivist
strategies emphasize (a) linking content to learners’
prior knowledge, (b) analyzing and interpreting
learners’ thinking and understanding, (c) encouraging
learner construction of mathematical concepts and
negotiations, as well as (d) facilitating multiple
presentations of solutions to problems. Constructivist
strategies also involve ISTs’ abilities to experiment
with new approaches that require learners’ engagement
in well-developed, open-ended, and authentic
investigations. In these investigations, learners develop
and evaluate conjectures, explain their work, and
communicate their results.
Due to differences in learner characteristics and
difficulties in effective sampling of instructional skills
across the domain of constructivist tenets, arguably, the
traditional lecturer assessments of ISTs’ professional
competencies do not provide a valid measure of the
IST’s ability to implement constructivist strategies
(Watt, 2005). This observation suggests it may be
useful to incorporate peers in the assessment process in
order to capture a wider range of instructional abilities
viewed from different perspectives. The present study
was inspired by an interest to explore the relationship
between assessment measures from a variety of
assessors for ISTs’ classroom practice during the
36
student teaching segment of their BScEd program. It
attempts to contribute to the debate on this issue by
investigating the research question: Does lecturer-peer
assessment of classroom practice of ISTs enhance
implementation of constructivist-related strategies
when teaching?
Answers to this research question could inform
local and international discussion on promoting holistic
assessment of ISTs’ instructional practice. Findings
from this study could also have important implications
for the deployment of ISTs to teaching practice schools
in Zimbabwe and elsewhere.
Conceptual Framework from Theoretical
Considerations
When working in cooperative groups, the
involvement of peers in the acquisition of procedural
and conceptual understanding of mathematical content
is well documented (Lowery, 2003; Schmuck and
Schmuck, 1997). Watt (2005) suggested that, in
Australia, the use of peers to assess each other’s work
has potential for improving learner mastery of
mathematical knowledge and skills. Viriato, Chevane,
and Mutimucuio (2005) explored the degree to which
peer assessment could contribute to the acquisition of
generic competencies of post-graduate science learners
in Mozambique. Both studies concluded that peer
assessment
fosters deep understanding, increases learner
involvement in the academic life, contributes in the
development of reflective skills, increases
awareness of a broad range of possible solutions to
problems, contributes to the development of self–
reliant and self-directed learners, and increases
cooperation and social interaction by lessening
competition among learners. (Sivan, as cited in
Viriato et al., p. 23)
These studies depict peer involvement in learning as
having a positive effect because it enhances
understanding of subject matter content.
Morrison, Mcduffie, and Akerson (2005) proposed
that, when teachers work with peers, the application of
new knowledge in appropriate contexts is facilitated by
their negotiations of and active involvement in the
implementation of the knowledge, and in watching and
discussing the efficacy of the implementation strategies
used. This is particularly true for the development of
instructional skills that develop during active
implementation in real classrooms. Putnam and Borko
(2000) argue that, in order to construct new knowledge
about pedagogical reform, ISTs need to be situated in
authentic classroom contexts. Immersion in these
Efficacy of College Lecturer and Student Peer Collaborative Assessment
contexts when implementing a pedagogical reform
promotes the transfer of theoretical knowledge of the
reform to practice. Lecturer-peer assessment of ISTs’
implementation of instructional strategies encourages
ISTs to study how peers interpret and implement
pedagogical reform. Well-organized peer assessments
might not only focus on peer understanding of
instructional theory and practice but also enhance the
development of a repertoire of professional skills
through explanations, justifications of claims, and
communication with peers during post-lesson reflective
dialogues.
The use of peers in a learning environment has
been documented as beneficial to the development of a
deep understanding of what peers and ISTs learn
collaboratively (Watt, 2005; Viriato et al., 2005). Peers
have the potential to expose each other to reform
strategies and techniques, share personalized strategies
and techniques, and collaborate in the evaluation of
implementing a pedagogical reform. Spector (1999)
recommends having ISTs sit in peers’ lessons in order
to help each other to better understand and apply the
theories of a reform and implement them in their
teaching.
Context of the study
In-service education is an individual teacher’s
personal initiative in Zimbabwe. There are numerous
motivations for embarking on in-service training.
Teachers who obtained certificates or diplomas in
education from a teachers’ college are qualified to
teach middle secondary school mathematics. For those
teachers to teach high school mathematics, increase
chances for promotion, or get a higher remuneration,
they must enroll in a full-time undergraduate in-service
program at a state or private university (for more
information on the Zimbabwean educational system,
see Appendix A). The Ministry of Education, Sport
and Culture of Zimbabwe supports that initiative by
granting two-year leaves of study to tenured teachers.
This study focuses on mathematics ISTs enrolled
in a science education program at a state university
located in the northern part of Zimbabwe. To graduate
with the BScEd degree from this university, an IST has
to pass 24 content courses in mathematics and
complete a dissertation, a practicum, and four
professional courses in education and mathematics
pedagogy. The ISTs enroll in six mathematics content
courses per semester that are also offered to preservice
undergraduate majors. They study all of the
mathematics content courses in the undergraduate
Lovemore J. Nyaumwe & David K. Mtetwa
program in order to meet the certification requirements
of the BScEd degree offered by the university.
The ISTs enroll in one professional course per
semester separate from the preservice undergraduate
students. The ISTs are exempted from some
undergraduate education courses under the assumption
that they possess sufficient knowledge and skills
acquired during their initial training at a teachers’
college. The program has more content than pedagogy
courses because it is assumed that the ISTs already
have a pedagogical base and that they need to
transform it into a learner-centred orientation so as to
facilitate the implementation of constructivist
instructional strategies in their teaching. After
completing the Advanced Pedagogics course, the ISTs
participate in four weeks of teaching practice in
between semesters of the program’s second year. The
goals of the Advanced Pedagogics course are that inservice students develop (a) a theoretical framework
for teaching mathematics at the high school level, (b) a
repertoire of constructivist theories for teaching
mathematics,
(c)
favorable
attitudes
toward
mathematics and mathematics teaching, (d) an
understanding of the importance of modeling and
problem solving in a context accessible to the learner,
and (e) the ability to apply the knowledge and skills
acquired in the course.
ISTs that pass all prerequisite courses for the
teaching practicum independently look for and select
schools for this experience. Schools accept ISTs after
agreeing to the conditions that ISTs teach under a
qualified cooperating teacher for four weeks and that
they observe classes taught by their cooperating
teacher as well as other teachers in the mathematics
department. The participants in this study were
deployed within a 200 km radius of the university.
Method
A convenience sample (Watt, 2005) from a cohort
of 22 ISTs was used for data collection. The only
criterion for sampling was attendance of at least two
in-service mathematics student teachers at the same
high school. Both high schools that met the sampling
criterion were located in an urban setting.
In the first visit by a lecturer, a common
understanding of lecturer-peer assessment was made
by reviewing the previously described characteristics
of constructivist-inspired pedagogies in vogue at the
university. The lecturer-peer assessment was meant to
be formative rather than summative. To achieve this
goal, the assessments were made on the basis of each
assessor’s personal impression, understanding, and
37
perceptions of constructivist tenets. In addition, the
assessors were free to consult the constructivist
pointers on the official classroom observation
instrument used by the university (see Nyaumwe &
Mavhunga, 2005, for the instrument). Assessors were
encouraged to use their personal understanding of
constructivist instructional strategies since a standard
assessment instrument may force them to focus on a
uniformly restricted sample of instructional skills.
Two lecturer-peer assessments of lessons taught by
each of the two peers, making a total of four lessons,
were made at each of the participating schools. To
increase the reliability of the assessment process,
participants conducted multiple assessments of the
lessons before data collection. After reaching a
common understanding on how lessons could be
assessed, the lecturer and peers made independent
assessments.
A lecturer and a peer sat in the same lesson
delivered by an IST and produced individual lesson
assessments. Each assessor took detailed notes of
classroom episodes. These notes were used in the postlesson reflective dialogues to pose and support
assertions made during lesson observation. One copy
of the written field notes was given to the IST who
taught the lesson in order to facilitate personal
reflection. Another copy was given to the lecturer, who
was also the researcher, for the purposes of this study.
The post-lesson reflective dialogues were audio taped
and later transcribed. The assessed lessons were
typically 70 minutes long.
The researcher interviewed each student teacher
pair separately at the end of the four-week school
attachment to determine the professional benefits
gained from the lecturer-peer assessment.
Data were analyzed by interpretative and analytic
induction (Bogdan & Biklen in Morrison et al., 2005)
by judging the extent to which the instructional
practice was commensurate with constructivist
instructional strategies. Similarities and differences
between the instructional practice and constructivist
perspectives were evaluated and recorded on
assessment critique forms.
Results from Written Critiques
The critiques written by the lecturer and a peer on
the performance of an IST’s instructional practice
highlighted some similarities and differences. The
following written critiques of a lesson taught by James
involving arithmetic and geometric series serve as
examples of the lecturer-peer assessment data
produced during the study. Throughout the critiques
below, the IST’s instructional actions are described by
the assessor verbatim, whereas the assessor’s
interpretations of instructional actions using
constructivist tenets are presented in brackets. The
lecturer wrote the first critique:
Results
The IST went round the classroom to listen and
assess learners working in groups. [It can be
assumed that he evaluated learners’ comprehension
and how they were solving the problems]. Group
work was concluded by highlighting the steps that
can be taken when finding the solution for number
4 that read “the third term of a GP is 8 and the fifth
term is 32, find S5”. [Instead of the IST using a
question and answer session to formulate the
simultaneous equations and subsequently finding
1
A written critique from James and Elizabeth ,
student teachers attached at the government school,
and an excerpt of an interview of Beaven and
Munashe, student teachers based at the private high
school, are used to provide evidence of the efficacy of
lecturer-peer assessments.
38
The introduction was on the conceptual meaning of
arithmetic and geometric series. The learners were
asked to provide the definitions and formulae for
the nth terms and sums of an AP and a GP.
[Learners’ previous knowledge was determined in
the introduction of the lesson]. Learners were later
asked to solve problems on work cards in groups of
between 3 and 5 each. The learners were
encouraged to solve the problems on the work
cards using six pre-determined steps of
comprehension, identification of variables, the
question/task, rule to use, making substitutions and
solving the equation. [The idea of using groups was
good to enable learners to socially invent their
solutions but group sizes were too large for
meaningful learner trial and error and negotiations
of solution methods. The provision of several tasks
on a work card catered for learners’ individual
differences by outcome. The pre-determined steps
given for learners to use as guide to solve the
problems reveal the IST’s formalist conception of
mathematics learning where ‘correct’ answers are
obtainable from using formal rules, procedures or
formulae].
The tasks on the work cards were thought
provoking for example, suppose θ+ Φ+ В+ Ω + …
is an arithmetic progression with a common
difference , find S4. [This and similar problems
required more reasoning and understanding than
direct application of a formula. One can conclude
that they were more of problem-solving than
consolidation exercises].
Efficacy of College Lecturer and Student Peer Collaborative Assessment
the values of the first term, a, and common ratio, r,
different groups should have presented their
solutions on the board. This was going to allow the
emergence of possibly different solution strategies
that might provide similar viable values for S5].
The assessment made by a peer of James for the
same lesson is as follows:
There was a good use of learners’ prior knowledge
by asking the nth terms and sums of the arithmetic
and geometric progressions. [Learners’ current
knowledge was linked to new concepts to be
developed in the lesson. This enhances building of
new concepts on knowledge existing in learners’
memories]. Learners’ interests were aroused by
asking them to come to the board to write the
formulae for sums and terms to nth terms of
arithmetic and geometric progressions. The
learners were asked to explain the formulae that
they wrote on the board. [Probing of learners’
responses to ascertain their current understanding
of the formulae was appropriate. It assisted James
to ascertain learners’ current understanding of
arithmetic and geometric formulae].
The progression of the lesson was logically
arranged from the known - formulae of AP and GP
- to the unknown - application of the formulae in
novel questions in groups. [James utilized learners’
prior knowledge. This was revealed by the
formulae they recalled for solving problems on
work cards].
James used various methods to solicit learners’
understanding such as verbal, written work on the
board and in groups and listening to group
discussions. [James continuously evaluated
learners’ understanding from time to time during
the lesson]. James encouraged learners to debate
their solutions, for instance, when formulating the
simultaneous equations on the problem on GP.
[Debating and negotiating viable solution methods
characterized the norms of James’s instructional
practice].
The tasks on work cards were challenging and
generally they suited the competence levels of
learners. James was confident and knowledgeable
of the content under review. [The work covered in
the lesson catered for learners’ individual
differences because it seemed to suit their cognitive
levels. Teacher confidence and knowledge are
prerequisite for successful instruction].
Though there were some elements of noise and
movement of learners in the lesson, they were
justifiable as they facilitated discussions and
verifications of learners’ current understanding
with peers in different groups. [Learner-centered
Lovemore J. Nyaumwe & David K. Mtetwa
lessons are necessarily
characterized
by
disagreements in the initial stages that lead to
learner negotiations and finally to a consensus.
Some element of noise and movements are
permissible as learners consult, verify their
conjectures and explain to each other what they
think is viable and justifying their decisions].
James interacted with the learners individually, in
groups or at class level during presentations on the
board. [High teacher-learner interaction is
recommended for teachers to be aware of learners’
current thinking, understanding of concepts and
using it to develop new concepts].
Although the lecturer and peer’s critiques are
based on the same lesson, they highlight different
constructivist pedagogical skills evident in an IST’s
instruction. For instance, the lecturer did not comment
on the IST’s pedagogical and mathematical
competencies, but the peer highlighted them as
important to determine successful implementation of
constructivist instructional strategies. IST mastery of
content is critical because a knowledgeable teacher
anticipates alternative conceptions and solution
strategies by learners. Whereas the lecturer expected
learners to present their group work solutions on the
board as a way of soliciting multiple methods for
finding a solution of a problem, the peer highlighted
learners’ movement in the classroom during group
work. Both the lecturer and the peer concurred on
highlighting differentiation as a constructivist tenet
exhibited by James. The peer’s critique specifically
refrained from pointing out weaknesses in James’s
implementation of constructivist instructional practice.
These critiques demonstrate that the peer did not
identify weaknesses in the IST’s practice. Peers, in
general, were hesitant to identify weaknesses of a
lesson because they perceived such criticisms as more
summative than formative. In addition, the peer
identified constructivist tenets that were not identified
by the lecturer and vice versa, providing
complementary assessments.
Results from Interviews of Student Pairs
Both pairs of ISTs hailed lecturer-peer assessments
as a major benefit of teaching practice with peers. The
personal experiences described by Beaven, one of the
ISTs involved, were similar to those echoed by other
peers:
The collaborative lecturer and peer assessment
gave me a great learning experience. The lecturer
and the peer looked at different professional
competencies in the same lesson. The use of
episodes from the lesson during post-lesson
39
reflective dialogues and the constructivist/
absolutist theory that explains it from the
understanding of a peer and a lecturer provided me
with a wide perspective of how a teaching episode
can be a point of focus for one assessor and a
trivial event to another. Though my peer was a
novice in the area of classroom assessment he
provided constructive critiques to the lessons I
taught.
The joint assessments enabled the ISTs to gain
insight into the interplay of curriculum goals, school
contexts, content, learners, the learning milieu and the
constructivist/absolutist instructional strategies they
learned during teacher education coursework. The
lecturer-peer assessments and subsequent post-lesson
reflective dialogues exposed the ISTs to new strategies
and techniques of implementing constructivist
instructional strategies. Discussions in the post-lesson
reflective dialogues provided peers with a wide range
of interpretations of their instructional practices. The
suggestions made during these dialogues enhanced IST
understanding of the implementation strategies that
promote learner understanding.
Based on the interview excerpt and lesson
assessment critiques, the lecturer-peer assessment
enhanced ISTs implementation of constructivist-related
strategies when teaching mathematics during their
teaching practice.
Discussion
Findings from this study indicate that the use of
lecturers and peers to assess ISTs’ implementation of
instructional strategies is beneficial to the development
of their professional skills. While simultaneously
assessing a lesson, a lecturer and a peer focus upon and
interpret instructional actions differently because their
individualized beliefs and values about teaching and
learning are filtered through personal frames. For
instance, Munashe applauded Beaven for stating the
objectives at the beginning of a lesson as a
motivational strategy, while the lecturer perceived it as
a way of enabling learners to understand the sequence
of the lesson.
Joint assessment of a lesson involves a lecturer and
peer using individualized perceptions of an
instructional episode. The complexity of these
individualized perceptions make it impossible for any
two evaluators assessing the same lesson to see and
interpret the professional competencies in the same
way, even in the presence of an agreed common
instrument. Fortunately, any differences in the
assessments highlight complementary teaching skills
which, when combined, provide a synergy of an IST’s
40
pedagogical competence to implement desired
strategies.
Teaching mathematics is a complex interpretive
process that depends on the context of the learning
environment, nature of content, learner interest, school
ethos, and curriculum goals, among others. Based on
this complexity, the pedagogical competencies of an
IST do not rest on a universally “accepted set of facts,
rules and assumptions” (Steele, 2005, p. 295). The
pedagogical competencies are not static to allow the
use of predetermined indicators as they vary in
response to learner needs. Learner behaviors and a
teacher’s interpretation of the learning environment
makes teaching vary from one moment to another,
making replication of a teaching episode impossible
(Steele, 2005; Wilson, 2003). Lecturer-peer assessment
liberates IST assessments from the personalized beliefs
and expectations of lecturers. As it bases assessment on
a variety of opinions, helping ISTs to explore a variety
of pedagogical ideas, lecturer-peer assessment might,
in turn, enhance their implementation of desired
pedagogical practices.
The use of one source of assessment data on ISTs’
classroom practice is not adequate because teaching
mathematics is an interpretive act (Steele, 2005) that
depends on an individualized area of focus. Because
there are no clearly defined rules for assessing
pedagogical competencies, assessment of ISTs’
instructional practice is conducted with the cooperation
of different sources of players. Multiple assessors are
necessary because teaching is an interpretative act and
assessments are conducted using a frame that is
contextualized and individualized. Multiple sources of
assessment data on the instructional practice of ISTs
might facilitate an understanding and development of
instructional knowledge and skills (Peressini, Borko,
Romagnano, Knuth, & Willis, 2004).
Involvement of peers in assessment has
motivational and cognitive merits. From a motivational
perspective, peer collaborative assessments contribute
to feelings of control regarding how the ISTs learn,
gain confidence, and understand how to implement
constructivist instructional strategies in their teaching.
In the post-lesson reflective dialogue, a peer and a
lecturer identify an episode from the assessed lesson
and use personalized understanding of constructivist
tenets to interpret it. The interpretations and
explanations of an instructional episode lead to a
discourse.
Discourse
on
pedagogy
provides
opportunities for peers and the lecturer to reflect, make
and defend claims, exchange alternative perceptions,
Efficacy of College Lecturer and Student Peer Collaborative Assessment
and negotiate a consensus that can be generalized to
instructional practices of other concepts.
For instance, in a problem-solving task, Elizabeth
asked learners to share 17 cattle in three groups,
consisting of ½, ⅓ and 1/9 of the cattle, respectively.
The learners shared the cattle and obtained 8.5, 5.7 and
1.9 cattle, which they rounded off to 9, 6 and 2. In the
post-lesson reflective dialogue the lecturer and James
questioned the reasonability of a non-integer quantity
of cattle. They argued that the approach caused the
mathematical results to become abstract and
meaningless. A logical way of approaching the ratio
problem that maintains realistic mathematical results
was to make a total of 18 cattle by borrowing one cow
and adding it to the 17 that were available. Eighteen
cattle can be shared evenly using the fractions ½, ⅓
and 1/9 and the borrowed cow can be returned after
sharing.
Further discussions on the problem revealed that,
by rounding off results, learners could obtain viable
answers from flawed reasoning. Logical reasoning is
an important skill in the mathematics curriculum that
learners should be given opportunities to develop.
Arguments, like the one in the post-lesson reflective
dialogue, have potential to deepen ISTs’ practical and
theoretical understanding of constructivist instructional
strategies in ways that might enhance learner
achievement. For the purposes of formative evaluation,
the use of peers to complement lecturers’ assessments
of ISTs is a viable initiative for producing a holistic
picture of their classroom practice.
Lecturers expose the pedagogical strategies that
they wish ISTs to implement in the classroom, making
them pursue similar strategies. In contrast, school
authorities may need training in order to understand
these strategies that ISTs are required to adopt. A lack
of coherence between the university and schools
enables school authorities to emphasize teaching skills
that are in conflict with those encouraged by lecturers
(Nyaumwe, 2001; Nyaumwe, Mtetwa & Brown, 2005).
Reliance on lecturer assessments of teaching
practice has been justified on the grounds that lecturers
were perceived as impartial and that their assessments
would maximize reliability and ensure comparability
of ISTs’ attained instructional competencies. This
belief devotes most of the “investment in assessment
on certification and accountability to the neglect of
formative evaluation” (Black, 1998, p. 812). An
influential reason for Zimbabwean teacher educators’
resistance or indifference to allowing other forms of
school-based assessment is that they regard them as
highly subjective (Nyaumwe & Mavhunga, 2005).
Lovemore J. Nyaumwe & David K. Mtetwa
The peer’s assessment in this study did not critique
the ISTs’ lesson delivery because it was perceived as
summative rather than formative evaluation. Peer
assessments may not be valid for summative evaluation
where assessments are used to rank students according
to ability, certification or accountability purposes. This
study did not attempt to assess peers’ ability to
evaluate each other’s instructional practices because it
was concerned with formative evaluation only. The
extent to which peers’ evaluations are valid might form
the focus of another study. The view of lecturers as the
sole assessors and evaluators of ISTs’ classroom
practice in Zimbabwe is held at the expense of validity
because lecturers’ assessments may mask some
weaknesses in implementing desired instructional
practices.
Findings from this study have shown that the
lecturer-peer assessments are effective for the purpose
of formative evaluation of ISTs’ instructional practice.
Debates among Zimbabwean educators on whether or
not to adopt this model of assessment might be
informed by their preference to prioritize formative or
summative evaluation. One case study cannot amplify
all the merits and demerits of the lecturer-peer
assessment phenomenon and the variables that might
influence its success. A similar study on preservice
student teachers might be useful in order to begin to
see the phenomenon in a wider frame.
References
Black, P. (1998). Assessment by teachers and the improvement of
students’ learning. In B. J. Fraser & K. G. Tobin (Eds.),
International handbook of science education (pp. 811–822).
Dordrecht, Netherlands: Kluwer Academic.
Davis, R. B. (1990). Constructivist views on the teaching and
learning of mathematics. In R. B. Davis., C. A. Mahler, and N.
Noddings (Eds.), Constructivist views on the teaching and
learning of mathematics. Reston, VA: National Council of
Teachers of mathematics.
Lowery, N. V. (2003). Assessment insights from the classroom.
The Mathematics Educator, 13(1), 15–21.
Morrison, J. A., Mcduffie, A. R., & Akerson, V. L. (2005).
Preservice teachers’ development and implementation of
science performance assessment tasks. International Journal
of Science and Mathematics Education, 3, 379–406.
Nyaumwe, L. (2001). A survey of Bindura University student
teachers' perceptions of the mentoring model of teaching
practice. Zimbabwe Journal of Educational Research, 13(3),
230–257.
Nyaumwe, L. (2004). The impact of full time student teaching on
preservice teachers' conceptions of mathematics teaching and
learning. Mathematics Teacher Education and Development,
6, 23–36.
41
Nyaumwe, L. J., & Mavhunga, F. Z. (2005). Why do mentors and
lecturers assess mathematics and science student teachers on
school attachment differently? African Journal of Research in
Mathematics, Science, and Technology Education, 9(2), 135–
146.
Nyaumwe, L. J., Mtetwa, D. K., & Brown, J. C. (2005). Bridging
the theory-practice gap of mathematics and science preservice
teachers using collegial, peer and mentor coaching.
International Journal for Mathematics Teaching and
Learning. Retrieved July 14, 2005, from
http://www.ex.ac.uk/cimt/Ijmtl/nyaumwe.pdf
Peressini, D., Borko, H., Romagnano, L., Knuth, E., & Willis, C.
(2004). A conceptual framework for learning to teach
secondary mathematics: A situative perspective. Educational
Studies in Mathematics, 56, 67–96.
Putnam, R., & Borko, H. (2000). What do new views of knowledge
and thinking have to say about research on teacher learning?
Educational Researcher, 29, 4–15.
Schmuck, R. A., & Schmuck, P. A. (1997). Group processes in the
classroom. Boston: McGraw Hill.
Shulman, L. S. (1986). Those who understand: Knowledge growth
in teaching. Educational Researcher, 15(2), 4–14.
Steele, M. D. (2005). Comparing knowledge bases and reasoning
structures in discussions of mathematics and pedagogy.
Journal of Mathematics Teacher Education, 8, 291–328.
Viriato, N., Chevane, V., & Mutimucuio, I. V. (2005). Student peer
assessment in a science education competence-based course.
In C. K. Kasanda, L. Muhammed, S. Akpo, & E. Nyongolo
(Eds.), Proceedings of the 13th Annual SAARMSTE
Conference (pp. 119–125). Windhoek, Namibia: Namibia
University.
Watt, H. (2005). Attitudes to the use of alternative assessment
methods in
mathematics: A study with secondary
mathematics teachers in Sydney, Australia. Educational
Studies in Mathematics, 58, 21–44.
Wilson, S. M. (2003). California dreaming: Reforming
mathematics education. New Haven, CT: Yale University
Press.
Zindi, F. (1996). Towards the improvement of practical teaching
assessment. The Zimbabwe Bulletin of Teacher Education,
4(4), 26–37.
Ziv, S., Verstein, M. S., & Tamir, P. (1993). Discrepant evaluations
of student teacher performances. Education Research and
Perspectives, 20(2), 15–23.
Spector, B. S. (1999, March). Bridging the gap between preservice
and inservice science and mathematics teacher education.
Paper presented at the annual meeting of the National
Association for Research in Science Teaching, Boston, MA.
1
Pseudonyms are used for moral and ethical reasons to
protect the identities of the participants.
Appendix A
Table A1
Levels of Formal Education in Zimbabwe
Student Age
Education Level
Required Teaching Credential
3–5
Kindergarten
6–12
Primary
13–14
Secondary: ZJC
15–16
Secondary: ‘O’ Level
Diploma: Teachers’ College
17–18
Secondary: ‘A’ Level
Diploma: Teachers’ College
19–22+
Tertiary (Undergraduate)
Degree: University
Note. ZJC = Zimbabwe Junior Certificate, ‘O’ Level = Ordinary Level, ‘A’ Level = Advanced Level.
Candidates who pass ‘O’ Level but fail to enroll in ‘A’ Level or who pass ‘A’ Level but fail to enroll at a
university may study for a diploma in teaching at a teachers’ college. Those who pass ‘A’ Level and enroll
at a university may study for an undergraduate degree program..
42
Efficacy of College Lecturer and Student Peer Collaborative Assessment
The Mathematics Educator
2006, Vol. 16, No. 2, 43–46
In Focus…
Mentor Teachers’ Perspectives on Student Teaching
Ginger A. Rhodes, Jeanette Phillips, Janet Tomlinson & Martha Reems
As a beginning university supervisor, Ginger had
limited experience and thoughts about student
teaching. After three years of observing student
teachers, she’s come to realize that university
supervisors and mentor teachers have a range of ideas
about the purpose of student teaching, as evidenced by
their interactions with and requirements of student
teachers. In the past two years, she has observed
student teachers at one school where the mentor
teachers, Janet, Jeanette, and Martha, have made a
conscious effort to reflect upon their mentoring
strategies. The mentor teachers participated in the
project Partnerships in Reform in Mathematics
Education (PRIME), a component of the NSF-funded
Center for Proficiency in Teaching Mathematics
(CPTM).1 PRIME is a multi-level professional
development program for preservice teachers, mentor
teachers, and university supervisors at the University
of Georgia. Each of the three groups of educators—
preservice teachers, mentor teachers, and university
supervisors—investigated their own practice while
participating in this two-semester professional
development program. As a part of PRIME, the three
groups of educators met on a weekly basis for 30-60
minutes.
This paper presents a collection of the ideas and
beliefs of these mentor teachers who go beyond
providing the typical student teaching experience. In a
discussion Ginger had with the three mentor teachers,
two major ideas regarding the student teaching
experience emerged along with three questions for the
education community. Through this paper we hope to
stimulate discussion about the purpose of student
teaching and the role of the mentor teacher.
Ginger Rhodes is a doctoral student in Department of
Mathematics and Science Education at the University of Georgia.
Her research interests include teachers’ understanding and use
of students’ mathematical thinking in teaching practices, teacher
education, and secondary students’ geometric reasoning.
Jeanette Phillips, Janet Tomlinson, and Martha Reems are
Nationally Board Certified high school mathematics teachers in
Georgia.
Ginger A. Rhodes, Jeanette Phillips, Janet Tomlinson & Martha Reems
Two Important Ideas
First and foremost, student teachers should work in
a nurturing and supportive environment that
encourages them to experiment. Some have used the
phrase “sink or swim” to describe the situation student
teachers and beginning teachers sometimes face when
learning how to teach. This idea is grounded in the
belief that one learns best how to teach through
experience. We believe this perspective is detrimental
to the student teaching experience. We agree
experience provides enormous opportunities for growth
and development. Yet, giving full responsibility too
early to beginning teachers will influence what they
learn. For example, suppose a student teacher wants to
incorporate into a lesson a paper folding activity that
entails managing small groups and whole group
discussions. If the student teacher has not developed
the skills to observe students’ mathematical thinking
during this activity, then it is asking too much of that
student teacher to also manage supplies and monitor
student behavior.
Many times, mentor teachers and university
supervisors believe student teachers need to manage an
entire lesson on their own, a seemingly obvious
requirement for those aspiring to be a full-time teacher.
Indeed, many preparation programs require student
teachers to carry a full-load of classes for a minimum
of two weeks, but there is typically some flexibility. It
is our belief that student teachers should not be given
this full responsibility too quickly or left on their own
to figure out what teaching entails. In high schools, the
timeline for acquiring new classes should be
determined based on the readiness of the student
teacher.
When student teachers are learning to create and
implement lessons, it is acceptable to divide
responsibilities for the lesson between the mentor
teacher and student teacher. This provides an
opportunity for the student teacher to focus on a piece
of the lesson instead of being overwhelmed with all
aspects. For example, the student teacher could
facilitate the tasks in a lesson while the mentor teacher
walks around to manage behavior and classroom
organization (e.g., passing out supplies and equipment,
43
collecting papers, etc). This allows and encourages the
student teacher to incorporate various teaching ideas
(e.g., a hands-on activity, use of technology, and small
group work) without the pressures of managing the
entire lesson. It is important for student teachers to
develop strategies for managing behavior, but it is also
valuable for student teachers to notice their students’
mathematical thinking during activities. By removing
most behavior management issues during beginning
lessons, the student teacher is allowed to focus
attention on instructional decisions. As a student
teacher gains more experience with lessons and the
ability to develop and implement portions of lessons,
he or she should take on more responsibility, including
managing student behavior.
A criticism against mentor teachers taking an
active role with student teachers in lessons is that K-12
students may lose respect for the student teacher. We
believe this can be prevented in several ways. On their
first day in the classroom, student teachers should be
introduced to students as a co-teacher, or as another
teacher in the classroom. From that point on, they can
interact with students. For example, student teachers
can help students during activities, go over homework,
or work with small groups. We want to emphasize that
while they are not taking on large roles, they are, more
importantly, interacting with students. In addition to
student teachers immediately taking an active role, the
mentor teachers should be conscious of the ways they
interject comments while their student teacher is
interacting with students. In other words, if the mentor
teacher treats the student teacher as a colleague in front
of the students, the students will view the student
teacher as a teacher. This leads to our second major
idea.
Secondly, student teachers should be treated as
colleagues. This may seem obvious to some, as student
teachers have reached the end of their coursework and
the beginning of their teaching careers. To some, this
may seem surprising, as student teachers are
inexperienced in the classroom and lack the knowledge
gained from experience. We believe that treating
student teachers as colleagues provides them with
insight into the teaching profession that will support
their development as reflective practitioners. Mentor
teachers will also benefit, as these conversations will
most likely support their own professional growth.
As teachers, we make instructional decisions in the
classroom from moment to moment. To an observer,
these decisions (and their underlying rationale) may
not be obvious. It is important for mentor teachers to
make these decisions explicit in order for student
44
teachers to gain awareness of issues dealing with
mathematics, students, and pedagogy. Thus, mentor
teachers should engage student teachers as colleagues
in professional conversations. These conversations can
happen through co-planning and co-teaching activities,
analyzing student work, and attending and discussing
professional meetings.
As educators, we believe that telling is a less
effective way of teaching content to students. If we
assume students learn mathematics best when they are
engaged in mathematical thought, then it is only
natural to assume that student teachers should learn
about teaching mathematics in a similar manner.
Student teachers have a thirst for knowledge about
teaching mathematics. They are excited to learn and
want to be successful teachers. Often, they will search
for the “correct” way to teach and look to other
educators for answers. It is easy for mentor teachers to
fall into the trap of telling the student teacher to teach
this content in this particular way. Student teachers
benefit more from being involved in professional
conversations about planning, implementation, and
reflection. For example, when Janet discusses planning
lessons with her student teacher, she tends to ask for
his input. Instead of the student teacher implementing
lessons that Janet or he pre-planned individually, the
student teacher is implementing lessons that they
jointly planned.
In order to highlight what we mean be treating
student teachers as colleagues, we wish to share three
additional examples from our experiences with student
teachers. Last year, Janet and her student teacher
regularly graded tests and quizzes together. During this
time, Janet would verbalize her reflections on how her
students were thinking mathematically, question
particular solutions, question particular test items, and
connect students’ performance to classroom practices.
Through these conversations, her student teacher
developed similar habits of thought.
Another example we wish to share is Jeanette
inviting and encouraging her student teacher to attend
meetings and conferences with her. In particular,
Jeanette and her student teacher regularly attended
county meetings that focused on teaching AP Calculus.
These meetings consisted of a group of teachers who
were teaching AP Calculus and who came together for
the purpose to discuss and improve their teaching
practices. After these meetings, Jeanette and her
student teacher had conversations in order to share and
elicit thoughts. Not only did the student teacher
observe and participate in the meetings, she had
Mentor Teachers
professional conversations about the meetings with
Jeanette.
The final example we wish to share is Martha’s
student teacher videotaping a lesson on synthetic
division. Then, during our weekly PRIME meeting we
watched a portion of the video. The discussion of the
video began with the student teacher sharing her
thoughts and questions about the lesson. In the
conversation that followed, the mentor teachers did not
simply answer the student teacher’s questions or tell
her how they typically teach synthetic division. Rather,
the conversation focused on reflection. The group
discussed the purpose of the lesson and the ways high
school students engaged the mathematics found in the
lesson. The group then discussed the mathematics
behind synthetic division, which led to other ways to
introduce it to students. We believe this kind of
mathematical and pedagogical conversation can and
should exist between mentor teachers, student teachers,
and university supervisors.
The two ideas we presented in this section can be
viewed as contrasting ideas. One may question how a
mentor teacher can treat a student teacher as a
colleague and protect him from becoming
overwhelmed by the complexities of teaching. We
don’t believe a mentor teacher has to do one or the
other; she can support and nurture a colleague.
Teaching mathematics is complex and difficult at
times. Student teachers need to have a realistic view of
what teaching entails. We are suggesting that mentor
teachers, as well as university supervisors, take steps to
appropriately introduce student teachers to these
difficulties and complexities so that valuable learning
takes place.
Another critique of our two ideas is that mentor
teachers are ultimately responsible for students
learning mathematics in their classroom. When mentor
teachers take on the added responsibility of hosting a
student teacher, they must make decisions that are best
for their students. In some cases these decisions may
limit the extent a student teacher can be treated as a
colleague. For example, there may be instances where
a K-12 student’s personal or medical history may
prevent the mentor teacher from sharing the reasons for
making particular decisions. We recognize these
situations and suggest that mentor teachers use their
professional judgment to manage them.
Three Questions
Universities and K-12 schools share the goal of
providing meaningful learning experiences for
prospective teachers during student teaching. Yet, a
Ginger A. Rhodes, Jeanette Phillips, Janet Tomlinson & Martha Reems
closer look reveals possible differences in what one
means by a “meaningful learning experience.” These
differences tie into one’s beliefs about the purpose(s)
of student teaching. This leads to our first question:
What is the purpose of student teaching? The answer to
this question might be different for student teachers,
mentor teachers, and university supervisors. Possible
purposes include being enculturated into schools,
learning to manage students, practice ideas learned
from university courses, learn how students think
mathematically, and to experience all aspects of
teaching. In some instances, differences in purpose for
student teaching create a divide between mentor
teachers and the university. For example, some mentor
teachers may be concerned with prospective teachers
being successful in the present moment, whereas
universities maybe concerned with prospective
teachers being successful over their careers. We
believe a balance between preparing prospective
teachers for the moment and for the future is necessary.
Finding the balance is a negotiation that can only
happen when the vested parties consider and
communicate their beliefs about the purpose of student
teaching. Through these negotiations, a common
understanding for the purpose of student teaching can
be developed. We are not suggesting that the vested
parties agree on one common purpose, but they should
be aware and understand each other’s purposes.
A university teacher preparation program is the
initial training experience aimed at preparing teachers.
Many of these preparation programs expect student
teachers to teach in a manner that differs from their
prior conceptions of teaching. Likewise, mentor
teachers are asked to participate in a student teaching
experience that differs from their own experiences. For
example, mentor teachers are asked to have
professional conversations with student teachers in
order to make instructional decisions explicit. In many
instances, these conversations are new for mentor
teachers. Mentor teachers are also asked to co-plan and
co-teach lessons. The idea of co-planning and coteaching can be interpreted and used in several ways.
For example, in some co-teaching situations one
teacher may be responsible for leading some portions
of the lesson while the second teacher leads others. In
another version of co-teaching, one teacher maintains
the lead throughout the lesson while the second teacher
is a helper. The helper may pass out materials and
work with smaller groups of students. In a third way of
implementing co-teaching, both teachers can lead by
taking equal roles throughout the lesson. A way that
might help one think about both teachers leading is to
45
consider the teachers jointly having a conversation
with students. Sharing in professional conversations,
co-planning, and co-teaching are new ideas for many
teachers; therefore many teachers have limited views
of how to implement these ideas with student teachers.
Situations like this lead to our second question: How
can the university support mentor teachers in their
efforts to develop their own mentoring skills?
Schools are faced with the challenge of providing a
high-quality mathematics teacher in every classroom.
The shortage of such teachers is evident, as there are
many classrooms where teachers are either not certified
or have substitute status. Similarly, universities face
the challenge of placing student teachers in positive
learning environments supported by high-quality
teachers. Though student teachers can learn from both
good and bad experiences, the learning is different.
Learning what not to do in a classroom is different
from learning what to do. The assumption that good
university preparation trumps a poor student teaching
experience is unfounded, as Frykholm (1996) found
mentor teachers have a greater impact on instructional
practices of preservice teachers. This observation lays
the foundation for our final question for teacher
educators: What are some options if a student teacher
is placed in an environment that is not ideal for their
learning? Put another way, what options are available
to teacher educators if a mentor teacher immediately
gives full classroom responsibility to a student teacher
without providing any guidance? We can all agree that
this is not the kind of environment that we want for our
student teachers. Is the best option to remove the
student teacher? If so, where does the student teacher
go? Are there other ways to manage the situation?
Conclusion
At first glance, the ideas we highlight in this paper
do not seem difficult to implement, but there are many
instances where student teachers do not feel
comfortable trying new ideas or do not regularly
participate in professional conversations. We feel these
student teachers are disadvantaged.
46
When we consider the student teaching experience,
we recognize the larger purpose is to prepare
prospective teachers for their future work as teachers.
Many times, however, mentor teachers voice their
excitement for hosting a student teacher for the reason
that they are interested in learning new teaching
methods and technologies. We want to draw attention
to the professional growth opportunities for mentor
teachers and university supervisors during the student
teaching experience. Mentor teachers and university
supervisors can gain much more than new teaching
methods and technologies. For example, when mentor
teachers make explicit their instructional decisions, we
believe this encourages them to thoughtfully think
through and possibly reconsider those decisions.
Ultimately, student teaching should be viewed as a
learning opportunity for everyone involved.
In our final comments, we wish to emphasize the
importance of developing strong relationships between
the universities and K-12 schools. The answers to the
questions posed above should come from both
institutions, as both institutions make valuable
contributions to the student teaching experience. It is
our responsibility as educators to develop these
relationships so that we may provide each prospective
teacher with a worthwhile initial experience as a
mathematics teacher.
References
Frykholm, J. A. (1996). Pre-service teachers in mathematics:
Struggling with the Standards. Teaching and Teacher
Education, 12, 665–681.
1
This work was made possible in part by Grant ESI0227586 from the National Science Foundation to the Center
for Proficiency in Teaching Mathematics at the University of
Georgia and the University of Michigan. Any opinions,
findings and conclusions, or recommendations expressed in
this material are those of the authors and do not necessarily
reflect the views of the National Science Foundation (NSF).
Mentor Teachers
CONFERENCES 2007…
MAA-AMS
New Orleans, LA
January 5-8,
2007
Irvine, CA
January 25-27,
2007
San Diego, CA
February 22-25,
2007
Cleveland, OH
March 1-3, 2007
Atlanta, GA
March 19-21,
2007
Atlanta, GA
March 21-24,
2007
Chicago, IL
April 7-11, 2007
Keele University, UK
April 11-14,
2007
Fredericton, Canada
June 8-12, 2007
Penang, Malaysia
June 18-22,
2007
ICTMT8
8th International Conference on Technology in Mathematics Teaching
Hradec Králové, Czech
Republic
July 1-4, 2007
PME-31
International Group for the Psychology of Mathematics Education
Seoul, South Korea
July 8-13, 2007
Salt Lake City, UT
July 29-August
2, 2007
Warsaw, Poland
July 31-August
3, 2007
Joint Meeting of the Mathematical Association of America and the American
Mathematical Society
http://www.ams.org
AMTE
Association of Mathematics Teacher Educators
http://amte.net
SIGMAA on RUME
Special Interest Group of the Mathematical Association of American on Research in
Undergraduate Mathematics Education
http://cresmet.asu.edu/crume2007/
RCML
Research Council on Mathematics Learning
http://www.unlv.edu/RCML/conference2007.html
NCSM
National Council of Supervisors of Mathematics
http://www.ncsonline.org/
NCTM
National Council of Teachers of Mathematics
http://www.nctm.org
AERA
American Education Research Association
http://www.aera.net
Mα
The Mathematical Association
http://www.m-a.org.uk/resources/conferences/index.html
CMESG
Canadian Mathematics Education Study Group
http://cmesg.math.ca
ICMI – EARCOME4
The Fourth East Asia Regional Conference on Mathematics Education
http://www.usm.my/education/earcome4
http://www.igpme.org
JSM of the ASA
Joint Statistical Meetings of the American Statistical Association
http://www.amstat.org/meetings/jsm/2007/
First Joint International Meeting with the Polish Mathematical Society
http://www.ams.org
47
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In this Issue,
Guest Editorial… Mathematics for Teaching or Mathematics for Teachers?
LEE PENG YEE
Uncovering Algebra: Sense Making and Property Noticing
DAVID SLAVIT
Preparing Elementary Preservice Teachers to Use Mathematics Curriculum Materials
ALISON M. CASTRO
Direct and Indirect Effects of Socioeconomic Status and Previous Mathematics
Achievement on High School Advanced Mathematics Course Taking
MEHMET A. OZTURK & KUSUM SINGH
Efficacy of College Lecturer and Student Peer Collaborative Assessment of In-Service
Mathematics Student Teachers’ Teaching Practice Instruction
LOVEMORE J. NYAUMWE & DAVID K. MTETWA
In Focus… Mentor Teachers’ Perspectives on Student Teaching
GINGER A. RHODES, JEANETTE PHILLIPS, JANET TOMLINSON & MARTHA
REEMS