____ THE
______ MATHEMATICS ___
_________ EDUCATOR _____
Volume 18 Number 1
Summer 2008
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editors
Kelly W. Edenfield
Ryan Fox
Dear TME Reader,
Associate Editors
Tonya Brooks
Eric Gold
Allyson Hallman
Diana May
Kyle Schultz
Susan Sexton
Catherine Ulrich
Advisor
Dorothy Y. White
MESA Officers
2007-2008
President
Rachael Brown
Vice-President
Nick Cluster
Secretary
Kelly Edenfield
Treasurer
Susan Sexton
Colloquium Chair
Dana TeCroney
NCTM
Representative
Kyle Schultz
Undergraduate
Representative
Dan Davis
Kelli Parker
Along with my co-editor Kelly Edenfield and the editorial staff, it is with great excitement
that I welcome you to the first issue of the 18th volume of The Mathematics Educator. I hope that
you will find the following articles appealing to your professional and intellectual interests.
The articles you will see in this issue cover a variety of topics within mathematics education
from various perspectives. In the Guest Editorial, Carla Moldavan reflects on the recent National
Mathematics Advisory Panel report, using her experiences as professor and classroom teacher as
a basis for her reflection. In his article, Erdogan Halat investigates potential differences that might
exist among groups of secondary teachers in their own geometric reasoning. S. Asli OzgunKoca’s article details a connection between properties of a computer software program and
student’s understanding of linear relationships. Bobby Ojose provides an analysis of Piaget’s
stages of cognitive development, using mathematics education at the elementary level to develop
his thoughts. Lastly, Zachary Rutledge and Anderson Norton detail how they, as researchers, can
gain understanding of pre-service teachers’ and high school students’ knowledge of school
mathematics through a letter-writing activity.
There are many people who are responsible for compiling a work such as this and I do want
to give credit to those who have made this issue possible. I want to thank our reviewers and
associate editors for the tireless efforts. Although this is the first issue of the new volume, it is the
last issue of the 2007-2008 school year. This year TME had two co-editors. It has been a true
privilege to work with Kelly this past school year as a fellow editor. With all sincerity I thank
Kelly for the great work she has done for the journal; I literally could not have done this without
her. For the upcoming school year, I will remain as a co-editor of the journal, and Diana May will
join me as a co-editor. I am grateful for the opportunities to work with TME this year and looking
forward to new opportunities I will have in the upcoming year.
Ryan Fox
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@uga.edu
www.coe.uga.edu/tme
About the Cover
The great stellated dodecahedron is a three-dimensional solid formed by extending the edges of the regular
dodecahedron, a Platonic solid. This particular great stellated dodecahedron was created by Nicholas Cluster, Colleen
Garrett, Ronnachai Panapoi, and Dana TeCroney.
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
Summer 2008
Volume 18 Number 1
Table of Contents
2 Guest Editorial… Ruminations on the Final Report of the National Mathematics
Panel
CARLA MOLDAVAN
8 In-Service Middle and High School Mathematics Teachers: Geometric Reasoning
and Gender
ERDOGAN HALAT
15 Ninth Grade Students Studying the Movement of Fish to Learn about Linear
Relationships: The Use of Video-Based Analysis Software in Mathematics
Classrooms
S. ASLI ÖZGÜN-KOCA
26 Applying Piaget’s Theory of Cognitive Development to Mathematics Instruction
BOBBY OJOSE
31 Preservice Teachers’ Mathematical Task Posing: An Opportunity for
Coordination of Perspectives
ZACHARY RUTLEDGE & ANDERSON NORTON
41 Upcoming Conferences
42 Submissions information
43 Subscription form
© 2008 Mathematics Education Student Association
All Rights Reserved
The Mathematics Educator
2008, Vol. 18, No. 1, 2–7
Guest Editorial…
Ruminations on the Final Report of the National Mathematics
Panel
Carla Moldavan
On March 13, 2008, The Final Report of the
National Mathematics Advisory Panel (NMAP, 2008)
was released. President George W. Bush had
established the Panel and charged its members to use
the best available scientific research to give advice on
how to improve mathematics education. The Panel
“found no research or insufficient research relating to a
great many matters of concern in educational policy
and practice” (p. xv). The Panel acknowledged that, in
light of the perceived lack of high-quality research,
instructional practice should also be informed by “the
best professional judgment and experience of
accomplished classroom teachers” (p. xiv).
My goal in this article is to illustrate some of the
points made in The Final Report using my own
experiences. I have taught in a two-year public college,
a four-year public college, a four-year private college,
a public high school, and a public middle school. Much
of my career was devoted to helping students in
developmental studies or learning support (remedial)
mathematics classes. Also much of my effort has been
and still is in working with pre-service and in-service
teachers. During the 2005-2006 academic year, I took a
leave of absence from teaching pre-service and inservice teachers in order to teach seventh-grade
mathematics. Many, but not all, of the illustrations
used in this article will refer to the experiences in the
middle school (hereinafter referred to as SMS). The
illustrations represent principles I especially would like
to convey to pre-service teachers.
The main findings and recommendations of the
Panel are organized into seven areas: Curricular
Since finishing her Master’s degree in mathematics education from
the University of Georgia in 1973, Carla Christie Moldavan has
taught mathematics at three different colleges: Dalton College,
Kennesaw State College, and Berry College. In addition she has
three years experience teaching in high school and middle school.
She has accepted a new position as chair of the division of
mathematics at Georgia Highlands College. She completed her
Ed.D. in Curriculum & Instruction at the University of Georgia in
1986.
2
Content, Learning Processes, Teachers and Teacher
Education, Instructional Practices, Instructional
Materials, Assessment, and Research Policies and
Mechanisms. This article does not attempt to address
all of these, nor does it claim to elevate the status of the
principles illustrated here above other principles. The
reader will want to read the entire Final Report to be
well-informed about what it has to say to mathematics
educators.
Disparities in Mathematics Achievement Related to
Race and Income
The first chapter of The Final Report provides
background for the President’s charge to the National
Mathematics Advisory Panel. It cites the United
States’ performance on international tests and the vast
demand for remediation in mathematics in college. It
goes on to address disparities in achievement:
Moreover, there are large, persistent disparities in
mathematics achievement related to race and
income—disparities that are not only devastating
for individuals and families but also project poorly
for the nation’s future, given the youthfulness and
growth rates of the largest minority populations.
(NMAP, 2008, pp. 4–5)
The National Assessment of Educational Progress
clearly demonstrates the disparity in performance on
mathematics test items when students are categorized
by race or income (U. S. Department of Education,
1990/2007). In addition, the state of Georgia provides
another source of information about disparity in
performance through the results of the state’s highstakes Criterion-Referenced Competency Test (CRCT)
(The Governor’s Office of Student Achievement,
2007). Student performance on the CRCT for a school,
a system, or the entire state is summarized by showing.
what percent of students are placed in each of three
categories: Does Not Meet, Meets, and Exceeds.
During the 2005-2006 school year, for example, 19%
of the seventh-graders in Georgia did not meet
expectations; however 28% of the black seventhRuminations on National Mathematics Panel
Table 1
Mathematics Criterion-Referenced Competency Test Data
Group
2005 – 2006 7th Grade Georgia
2005 – 2006 Black 7th Grade Georgia
2005 – 2006 White 7th Grade Georgia
2005 – 2006 7th Grade Georgia Economically Disadvantaged
2005 – 2006 7th Grade Georgia Not Economically Disadvantaged
2006 – 2007 SMS 7th Grade
2005 – 2006 SMS 7th Grade
2004 – 2005 SMS 7th Grade
2006 – 2007 SMS Black 7th Grade
2005 – 2006 SMS Black 7th Grade
2004 – 2005 SMS Black 7th Grade
2006 – 2007 SMS 7th Grade Economically Disadvantaged
2005 – 2006 SMS 7th Grade Economically Disadvantaged
2004 – 2005 SMS 7th Grade Economically Disadvantaged
2006 – 2007 SMS 7th Grade Not Economically Disadvantaged
2005 – 2006 SMS 7th Grade Not Economically Disadvantaged
2004 – 2005 SMS 7th Grade Not Economically Disadvantaged
graders did not meet expectations (See Table 1).
Twenty-three percent of the seventh-graders exceeded
expectations, whereas 32% of white seventh-graders
exceeded expectations, compared to 10% of black
seventh-graders exceeding expectations.
The
state’s
statistics
for
economically
disadvantaged students were practically identical to
statistics for black seventh-graders for the 2005–2006
school year. SMS is a school with approximately 20%
black students and 75% students who qualify for free
or reduced lunch. Students with an identified disability
comprise 16% of the school. With this demographic
information, it is interesting to note some of the test
results for the school. For example, 18% of the
seventh-graders at SMS exceeded expectations on the
2006 CRCT, compared to 7% the previous year. Of the
black seventh-graders in 2006 at SMS, 16% exceeded
expectations, up from 6% of the seventh-graders in
2005. Only 20% of the black seventh-graders did not
meet expectations on the 2006 CRCT, down from 32%
the previous year. As for the economically
disadvantaged, the percent not meeting expectations
declined from 38% in 2005 to 27% in 2006. During the
same time period the percent of economically
disadvantaged students exceeding expectations rose
from 7% to 15%. Of those students in seventh-grade at
SMS who were not economically disadvantaged, 27%
exceeded expectations in 2006, compared to 8% in
2005 and 2% in 2007.
In presenting this data I emphasize that teachers
should not be content with examining data that only
Carla Moldavan
Does Not Meet
Standards
19%
28%
11%
28%
11%
34%
26%
32%
32%
20%
32%
34%
27%
38%
36%
23%
21%
Meets Standards
58%
61%
57%
62%
55%
58%
56%
61%
64%
64%
61%
56%
58%
56%
62%
50%
71%
Exceeds
Standards
23%
10%
32%
11%
34%
7%
18%
7%
4%
16%
6%
10%
15%
7%
2%
27%
8%
looks at an overall pass rate on tests such as CRCT;
they must question why disparities exist and then work
for equitable instruction. There should be high
expectations and the opportunity for all students to
learn mathematics. A primary reason for my teaching a
year at SMS was to reach the disadvantaged students. I
am encouraged by the results as indicated by CRCT
data, both looking at patterns across years at SMS, as
well as looking at SMS compared to state data.
Conceptual Understanding, Procedural Fluency,
and Automatic Recall of Facts
The report of the Panel provides more emphasis on
the hierarchical nature of mathematics than has been
evident in the past two decades. It calls for a “focused,
coherent progression” (NMAP, 2008, p. xvi) and for
avoiding any approach that “continually revisits topics
year after year without closure” (p. xvi). The Georgia
Performance Standards (Georgia State Department of
Education, 2008) also operate on the assumption that
students have mastered content from previous grade
levels.
Students’ lack of prerequisite skills is a major
challenge to teachers. For example, the Algebra I
teachers surveyed by the Panel sent a strong message
that a source of concern for them was their students’
inability to work with fractions (NMAP, 2008, p. 9).
The National Mathematics Advisory Panel saw this
particular difficulty as “a major obstacle to further
progress in mathematics, including algebra” (p. xix).
However, the difficulty with fractions is persistent long
3
after students have completed four years of
mathematics in high school, including calculus for
some students. Recently I gave a pre-assessment on
fractions to a class of pre-service teachers. This
assessment consisted of six questions: simplifying a
fraction, adding a mixed number and fraction,
subtracting two mixed numbers, multiplying two
fractions, multiplying two mixed numbers, and
dividing two mixed numbers. For this recent class of
pre-service teachers, the mean and median number of
questions correct was three. In addition to numerous
errors in the whole number arithmetic (e.g. 20 – 6 = 4),
several students treated multiplication of fractions as a
proportion and tried to work with cross-products.
The knowledge about fractions is an example that
illustrates complex issues related to teaching. Will a
focused curriculum result in no longer having a need to
revisit fractions? Do the Algebra I teachers revisit
fractions? Do calculus teachers revisit fractions? If so,
how is the re-teaching different from the initial
exposure? The fact that pre-service teachers have
made it through four years of high school mathematics
and still cannot perform operations on rational numbers
shows that the problem is much wider than the
concerns of Algebra I teachers. Ma (1999) provided
the classic example of lack of conceptual knowledge
about fractions in her data on the inability of U. S.
elementary teachers to give an example of an
application that called for division of fractions. Clearly
the procedural knowledge about fractions of preservice teachers is also lacking.
Actually many of these pre-service teachers have
always been tracked with the students who excel, even
though there are significant gaps in their knowledge.
As a result of tracking, they have not encountered
students who experience real difficulties with
mathematics. To illustrate the reality of the gaps in the
younger students’ mathematical understanding, I will
relate some of my seventh-graders’ lack of
prerequisites.
One of our first new topics in my seventh-grade
class was signed numbers. As students began to work
on some exercises I had given them, I went around
observing their work. As I stood over one girl, I saw
many tally marks, and at first I had the impression she
was just doodling. However, as I lingered over her, I
saw that she was doing 92 + -17, by making 92 tally
marks and crossing out 17 of them. At a later date, I
asked the same girl to identify which digit in a numeral
I had written was in a certain place value; her response
was an incorrect one, representing a digit on the
opposite side of the decimal point from the correct
4
answer. I should probably add that the response was
not just a matter of whether the word had a –ths
ending. These are only a couple of examples of the
missing pre-requisites that surfaced for a student
entering the seventh grade.
Another girl was having difficulty with signed
numbers. One strategy I use in making sense out of
addition of integers is to relate it to a “common-sense
example.” In her case I tried to get her to find -5 + 1 by
asking her if I owed her $5 and paid back $1, how
much would I still owe her? It was her third attempt to
answer the question before she gave a correct response.
With a different student who was learning-disabled, I
attempted to do a task analysis to enable him to
multiply decimals. However, I met with extreme
difficulty when he did not seem to be able to count the
number of decimal places to the right of the decimal.
As I pointed to the digits to the right of the decimal to
get him to count them, he shook his head as if he did
not know what I was asking. With both this student and
the girl attempting to answer -5 + 1, there was certainly
a lack of self-efficacy in addition to the lack of
prerequisites. Later in the year when the class was on a
totally different topic, I saw a smile come over the
girl’s face as she exclaimed, “That’s easy!” and
realized she could expect herself to be able to do
mathematics.
On the other hand, some of the more capable
students with prerequisite skills did not exhibit
conceptual understanding. For example, even though
he knew most of his multiplication facts and could
perform the standard algorithm for multiplication, one
boy resorted to finding the product of 92 x 8 by
repeated addition (92 + 92 + 92 + 92 + 92 + 92 + 92 +
92). One of the questions teachers must struggle with is
how much time and energy to spend on procedural
learning. Several years ago I had read a question that
haunted me: Is the goal of school mathematics to make
children as good as a $5 calculator? I firmly believe
that conceptual understanding is important and that
calculators can free people up to concentrate on the
steps of how to solve a problem. However, my
seventh-grade experience made me feel terrible when I
thought about the fact that my students were not as
capable of doing arithmetic as a $5 calculator. In
addition to not knowing basic facts, they did not
possess conceptual understandings, such as when to
use multiplication and where to place a decimal point
in a product.
Some students with disabilities are allowed to use
calculators as an accommodation. In dealing with some
mathematical content, the calculator does not provide
Ruminations on National Mathematics Panel
the desired efficiency. When students do not know
multiplication facts, divisibility rules are not really
shortcuts. When divisibility is not recognized, trying to
find the prime factorization of a number can become
even more challenging. Similarly, simplifying fractions
is an arduous task.
The National Council of Teachers of Mathematics
(NCTM, 1989), in its first standards document, called
for calculators to be available to all students at all
times (p. 8). The revision a decade later (NCTM, 2000)
clarified that there are times calculators are to be put
away (pp. 32–33). Use of calculators in prior grades
was a concern expressed by Algebra I teachers
surveyed by the National Math Panel (2008). The
Panel’s response was to caution “that to the degree that
calculators impede the development of automaticity,
fluency in computation will be adversely affected” (p.
50). The Panel summarized the balance among
conceptual understanding, computational fluency, and
problem-solving skills: “Debates regarding the relative
importance of these aspects of mathematical
knowledge are misguided. These capabilities are
mutually supportive, each facilitating learning of the
others” (p. xix).
Affective and Motivational Factors
The Panel (2008) cited empirical evidence that
children’s focus can be shifted from their innate ability
to their engagement in mathematics learning, and that
this will improve their meeting the learning outcomes
(p. xx). The Panel called for educators to help students
and parents understand this relationship between effort
and performance.
These points are certainly important, but the
assumption is being made that all parents and students
value education. At a parent-teacher conference close
to the beginning of the year, one mother of a seventhgrader told me she did not believe in teaching “that
higher math.” The “higher math” consisted of negative
numbers and percents. Checkbooks and shopping were
not sufficient examples to convince her otherwise.
About two weeks before the end of the year, the stepfather of another student told me that a high-school
diploma was just a piece of paper. He had dropped out
at age 15 and worked in the textile mill ever since and,
in his mind, had never needed a formal education.
Note that a distinction must be made between
recognizing different views towards education and
blaming the home environment/parents’ lack of value
toward education for low academic performance. As
pointed out by DeCastro-Ambrosetti and Cho (2005),
as long as a rift between home and school exists,
Carla Moldavan
communication between parents and teachers will
continue to be strained and hindered.
Other affective aspects that must be considered in
working with students include situations that make it
hard to focus on schoolwork. Emotional difficulties
students might experience include depression, abuse,
and separation from family members. Because we as
educators believe that education is the ticket out of bad
situations, the stress of daily dealing with these kinds
of problems can sap one’s strength and make the value
of adding mixed numbers, for example, seem
questionable. Of the algebra teachers participating in
the survey commissioned by the National Mathematics
Advisory Panel (2008, p. 9), 62% rated working with
unmotivated students as the single most challenging
aspect of teaching Algebra I successfully.
Individual Students’ Achievement Gains and
Instructional Practices
Although the difficulties of emotional baggage and
lack of prerequisites pose challenges for high
achievement in mathematics, the gains of individual
students can be staggering. In addition to the analysis
of how SMS seventh-graders performed on CRCT as a
whole, by race, and by economic status, I scrutinized
the performance of my students individually by
checking records to determine their past scores.
CRCT scores were reported for individuals for the
years cited in this article on a scale that sets 300 as the
minimum score to earn “meets expectations.” Scores of
at least 350 were categorized as “exceeding
expectations.” The minimum mathematics score of one
of the students I had in the seventh grade had been 255
the previous year. The minimum for 2006 was 15
points higher at 270. The highest score for 2006 was
375, whereas the highest score for those students in
2005 had been 359. One of the students who exceeded
expectations posted a 49-point increase from his score
the previous year. Other students had increases of 26
points and 34 points over their 2005 scores. The
average gain for students in my inclusion class was 14
points per student. The National Mathematics Advisory
Panel (2008) acknowledges “little is known from
existing high-quality research about what effective
teachers do to generate greater gains in student
learning” (p. xxi). I present the data from my seventhgraders’ CRCT scores in order to encourage teachers to
study measures of achievement of individuals and to
identify promising practices.
Teacher-education programs may espouse what
constitutes best practice to the point that a pre-service
teacher might accept that training without question.
5
Similarly, teachers may find themselves in systems
where they are required to use a certain curriculum and
follow a certain format, be that a scripted lesson or a
work period with closing presentations by students.
Teachers need to have the freedom to use their
professional judgment and to develop and assess
effective techniques.
Assessment
The Panel (2008) does state that teachers’ regular
use of formative assessment improves students’
learning (p. xxiii). Pre-service teachers may
particularly need help in learning to “kid-watch,”
informal assessment, to assess their students’ learning.
Often pre-service teachers are tied up in the content or
the activity of a lesson and cannot identify students
who have misconceptions. I offer some examples of
the informal assessment that took place in my seventhgrade mathematics classes.
At the beginning of the year, I asked the students
to make a poster about their favorite number. I told
them that my favorite number was eight. I showed
them several ways of performing arithmetic operations
to get a result of eight. They were to show on their
posters many ways to write their chosen number.
Interestingly, only one student wrote an equation
involving a fraction. One student was able to show that
she understood she could make an infinite number of
equations to get her number by continuing to increase
the minuend and subtrahend by one each time.
Another beginning of the year activity I chose was
to give the students old calendars, have them choose a
3 x 3 square on it, and add the nine numbers in the
square. I hoped to use the activity to show them how I
could tell in advance what the sum would be, using n
to represent the middle number, finding algebraic
expressions for each of the other numbers in terms of
n, and finding that the sum would always be 9n.
However, I realized that even though understanding
variables was a sixth-grade Georgia Performance
Standard (Georgia Department of Education, 2007),
the students were not comfortable with using a variable
in the calendar context. Moreover, I realized that
perhaps only one or two in a class of sixteen students
could accurately add these nine numbers!
I found myself using few formal assessments
throughout the year of teaching seventh grade.
Assessment was something that took place every day,
and it was used to inform instruction. So to new
teachers I repeat the rhyme: “A teacher on her feet is
worth two in her seat.” The teacher should not just be
maintaining discipline and monitoring seatwork, but
6
should be questioning and probing for students’
understanding.
Conclusion
The Final Report of the National Mathematics
Advisory Panel (2008) is the most recent document
published that offers advice on how to improve
mathematics education. The Panel’s task was arduous,
involving reviewing 16,000 research publications. It is
somewhat disheartening to know that, in the Panel’s
assessment, there is insufficient high-quality research
to draw many conclusions. Nevertheless, mathematics
educators will continue to search for meaning from
their own experiences and to conduct research.
Qualitative research, which seemed to take root in
light of the fact that in education it is too difficult to
control all the variables, will be overshadowed once
again by quantitative research with experimental and
control groups. Perhaps the Panel could take on the
role of setting up experimental designs and hypotheses
and then recruiting researchers to carry out the studies.
Just as mathematics specialists may be needed in
elementary schools rather than trying to increase the
mathematical proficiency of all teachers, the research
specialists are needed to insure that investigations will
meet the tests to be considered rigorous.
Meanwhile, the pendulum will swing on many
facets of mathematics education, but teachers will
continue to go to classrooms every day and offer
stability and safety to all. They will provide their
students with multiple opportunities to succeed one
class period at a time, but have high-stakes assessments
like the CRCT looming over them every day. They will
do their best to show the relevance of mathematics.
They will want their students to understand why, even
when the students do not care why. They will keep
searching for the answers for not only how to best
teach mathematics, but also how to best teach students.
References
DeCastro-Ambrosetti, D., & Cho, G. (2005). Do parents value
education? Teachers’ perceptions of minority parents.
[Electronic version]. Multicultural Education, 13, 44–46.
Georgia State Department of Education. (2008). Georgia
mathematics performance standards. Retrieved March 25,
2008 from http://www.georgiastandards.org/math.aspx
The Governor’s Office of Student Achievement. (2007). State of
Georgia K-12 report card. Retrieved March 25, 2008 from
http://www.gaosa.org/report.aspx
Ma, L. (1999). Knowing and teaching elementary mathematics:
Teachers’ understanding of fundamental mathematics in
China and the U. S. Philadelphia: Lawrence Erlbaum.
Ruminations on National Mathematics Panel
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.
National Mathematics Advisory Panel. (2008). Foundations for
success: The final report of the National Mathematics
Advisory Panel. Washington, DC: U.S. Department of
Education. Retrieved March 21, 2008 from
www.ed.gov/MathPanel
Carla Moldavan
U. S. Department of Education National Center for Educational
Statistics. (1990/2007). National Assessment of Educational
Progress: The nation’s report card. Retrieved January 31,
2008 from http://nces.ed.gov/nationsreportcard/naepdata
.
7
The Mathematics Educator
2008, Vol. 18, No. 1, 8–14
In-Service Middle and High School Mathematics Teachers:
Geometric Reasoning Stages and Gender
Erdogan Halat
The purpose of this current study was to investigate the reasoning stages of in-service middle and
high school mathematics teachers in geometry. There was a total of 148 in-service middle and high
school mathematics teachers involved in the study. Participants’ geometric reasoning stages were
determined through a multiple-choice geometry test. The independent samples t-test with α = 0.05
was used in the analysis of the quantitative data. The study demonstrated that the in-service middle
and high school mathematics teachers showed all the van Hiele levels, visualization, analysis,
ordering, deduction, and rigor, and that there was no difference in terms of mean reasoning stage
between in-service middle and high school mathematics teachers. Moreover, there was no gender
difference found regarding the geometric thinking levels.
Introduction
Various studies have documented that many
students encounter difficulties and performed poorly in
both middle and high schools geometry classrooms
(e.g., Fuys, Geddes, & Tischler, 1988; Gutierrez,
Jaime, & Fortuny, 1991). Usiskin (1982) has found that
many students fail to grasp key concepts in geometry
and leave their geometry classes without learning basic
terminology. Moreover, research shows a decline in
students’ motivation toward mathematics (Gottfried,
Fleming, & Gottfried, 2001). According to Billstein
and Williamson (2003), “declines in positive attitudes
toward mathematics are common among students in
the middle school years” (p. 281). Among the
variables that affect student learning, researchers have
suggested that the teacher has the greatest impact on
students’ motivation and mathematics learning (e.g.,
Wright, Horn, & Sanders, 1997; Stipek, 1998). Burger
and Shaughnessy (1986), along with Geddes and
Fortunato (1993), claim that the quality of instruction
is one of the greatest influences on the students’
acquisition of geometry knowledge in mathematics
classes. The students’ progress from one reasoning
(van Hiele) level to the next also depends on the
quality of instruction more than other factors, such as
students’ age, environment, and parental and peer
support (Crowley, 1987; Fuys et al., 1988).
Erdogan Halat is currently an Assistant Professor in the
Department of Secondary Science and Mathematics Education at
Afyon Kocatepe University, Turkey. He received his Ph.D. in
Mathematics Education from Florida State University. His
interests include van Hiele theory, motivation, webquests, and
gender.
8
According to Stipek (1998), teachers’ content
knowledge has a significant impact on students’
performance. Mayberry (1983) and Fuys et al. (1988)
state that geometry content knowledge among preservice and in-service middle school teachers is not
adequate. Chappell (2003) says, “Individuals without
sufficient backgrounds in mathematics or mathematics
pedagogy are being placed in middle school
mathematics classrooms to teach” (p. 294). In this
study the researcher will investigate the argument that
insufficient geometry knowledge of in-service
mathematics teachers might be another reason behind
students’ poor performance in geometry.
The van Hiele Theory and its Philosophy
Level-I: Visualization or Recognition
At this level students recognize and identify certain
geometric figures according to their familiar
appearance. However, students do not perceive the
geometric properties of figures. When students call a
figure a square, they react to the whole figure and not
to its right angles, equal side lengths, and equal
diagonal lengths. For example, at this level students
can recognize certain squares very easily because they
look like the outline of a window or frame (Figure 1
left). However, they do not call the second shape in
Figure 1 a square because it does not look like the
outline of a window or frame.
Figure 1. Two perspectives of a square.
Mathematics Teachers’ van Hiele Levels
Level-II: Analysis
At this level students analyze figures in terms of
their components and relationships among these
components. For instance, a student’s analysis may
assert that opposite sides of a rectangle are congruent
or all of its angles are right angles. Students can also
identify and name geometric figures by knowing their
properties. They would correctly identify only the
second and fourth shapes in Figure 2 as parallelograms.
Although at this level the students are able to
acknowledge various relationships among the parts of
the figures, they do not perceive any relationship
between squares and rectangles or rectangles and
parallelograms; students perceive properties of one
class of shapes empirically, but can not relate the
properties of two different classes of shapes. For
example, students would not see rectangles or squares
as parallelograms because they do not see one set of
figures as a subset of another.
Figure 2. Examples of parallelograms.
Level-III: Ordering
At this level students logically order and interrelate
previously discovered properties by giving informal
arguments. Logical implications and class inclusions
are understood and recognized. At this level the
students are able to see the relationships among the
quadrilaterals in Figure 3: they can easily say that a
square is also a rectangle and a rectangle is also a
parallelogram. Students are aware of relationships
among different types of figures. These relationships
may have been unclear to the students at level-II
(Analysis). According to Hoffer (1988),
they even may be able to observe various such
relationships themselves and they only have an
implicit understanding of how these relationships
link to justify their observations. In other words,
the students have not yet developed the ability to
prove theorems. (p. 239)
Figure 3. An example of ordering parallelograms.
Level-IV: Deduction
At this level students can analyze and explain
relationships between figures. They can prove
theorems deductively, supply reasons for statements in
formal proofs, and understand the role of axioms and
definitions. In other words,
the students can follow the line of argument in
proofs of statements presented to them, and they
can develop sequences of statements to deduce one
statement from another. What may have been an
implicit understanding at the previous level,
Ordering, of why certain statements were true now
develops into reasoning patterns that enable the
students to create sequences of statements to
formally explain, or prove, why a statement is true
[see Figure 4] (Hoffer, 1988, p. 239)
Students operating at level-IV can state that if a
figure is a rhombus and a rectangle then it must be a
square and prove this statement deductively. Students
cannot analyze or compare various deductive systems.
For example, students cannot establish theorems in
different axiomatic systems.
Figure 4. Showing that a rhombus is also a square.
Level-V: Rigor
At this level students are able to analyze and
compare various deductive systems. A student should
be able to know, understand, and give information
Erdogan Halat
9
about any kind of geometric figures (e.g., Fuys et al.,
1988). Moreover, Hoffer (1988) says, “this is the most
rigorous level of thought- the depth of which is similar
to that of a mathematician” (p. 239).
Empirical Research on the van Hiele Theory
Since the proposal of the van Hiele theory, studies
have focused on various components of this learning
model at different grade levels. Wirzup (1976)
conducted several studies and introduced the van Hiele
theory in the United States. His work caught the
attention of educators and researchers; four major
studies were initiated by Hoffer (1988), Burger and
Shaughnessy (1986), Usiskin (1982), and Fuys et al.
(1986). Where Hoffer described and identified each
van Hiele level, Burger and Shaughnessy focused on
the characteristics of the van Hiele levels of reasoning.
Usiskin affirmed the validity of the existence of the
first four levels in high school geometry courses. Fuys
et al. examined the effects of instruction on a student’s
predominant Van Hiele level.
These research findings provide mathematics
teachers insight on how students think and what
difficulties they face while learning geometry. Several
textbook writers have based their geometry sections or
books on the van Hiele theory, such as Michael Serra’s
(1997) geometry book and Connected Mathematics
Project’s “Shapes and Designs” (Lappan, Fey,
Fitzgerald, Friel & Phillips, 1996). The writers of both
textbooks claimed that they implemented the van Hiele
theory in their writings and designed their instructional
approaches based on this theory.
Moreover, studies determined van Hiele reasoning
stages in geometry of middle, high and college level
students. For instance, Burger and Shaughnessy (1986)
and Halat (2006, 2007) found mostly level-I
(Visualization) reasoning in grades K–8. Fuys et al.’s
(1988) interviews with sixth and ninth grade students
classified as average and above average found none
performed above level-II (Analysis). This finding
supports the idea that many high school students in the
United States reason at level-I (Visualization) or levelII (Analysis) of Van Hiele theory (Usiskin, 1982;
Hoffer, 1988). These findings imply that neither
middle nor high school students meet the expectations
of NCTM (2000). At the end of 8th grade, students
should be able to perform at level-II (Analysis) and at
the end of 12th grade, students should be able to
perform at level-III (Ordering) or level-IV (Deduction)
(Usiskin, 1982; Mayberry, 1983; Crowley, 1987;
Knight, 2006)). Usiskin, Mayberry, Burger and
10
Shaughnessy, and Fuys et al. agreed that the last level
(Rigor) is more appropriate for college students.
Some
researchers
have linked
students’
mathematics performance to teachers’ content
knowledge. For example, Chappell (2003) claims that
high school students’ less than desirable background in
geometry is due to middle school mathematics
teachers’ superficial geometry knowledge. According
to Gutierrez, Jaime, and Fortuny (1991), Duatepe
(2000) and Knight (2006), pre-service elementary
school mathematics teachers’ reasoning stages were
below level-III (Ordering). Likewise, Mayberry (1983)
stated that the 19 pre-service elementary school
teachers involved in her study were not at a suitable
van Hiele level to understand formal geometry and that
their previous instruction had not help them to attain
knowledge of geometry consistent with level-IV
(Deduction).
Knight’s (2006) study with pre-service elementary
and secondary mathematics teachers found that their
reasoning stages were below level-III (Ordering) and
level-IV (Deduction), respectively. These results are
consistent with the findings of Gutierrez, Jaime, and
Fortuny (1991), Mayberry (1983), Duatepe (2000), and
Durmuş, Toluk, and Olkun (2002). None of these preservice elementary and secondary mathematics
teachers demonstrated a level-V (Rigor) reasoning
stage in geometry. This is surprising because the van
Hiele levels of pre-service elementary and secondary
mathematics teachers are lower than the expected
levels for students completing middle school and high
school, respectively (Crowley, 1987; Hoffer, 1988;
NCTM, 2000). Although most of these studies
mentioned above were done with students, this study
will investigate in-service middle and high school
mathematics teachers.
Gender Differences in Mathematics
Research indicates gender should be included as a
variable in analysis, even if it is not the main focus of a
study (Forgasız, 2005; Armstrong, 1981; Ethington,
1992; Grossman & Grossman, 1994; Lloyd, Walsh &
Yailagh, 2005). Over the past few decades, research
suggests a difference between the achievement of male
and female students in many content areas of
mathematics, including spatial visualization, problem
solving, computation, and measurement (e.g.,
Grossman and Grossman, 1994; Lloyd, Walsh and
Yailagh, 2005). According to Armstrong, female
students performed better at computation and spatial
visualization than males. Fox and Cohn (1980) found
males performed significantly better than females on
Mathematics Teachers’ van Hiele Levels
the mathematics section of the Scholastic Aptitude
Test. Similarly, Smith and Walker (1988) concurred
with this finding in their study of tenth grade geometry
students. According to Hyde, Fennema and Lamon
(1990) and Malpass, O’Neil and Hocevar (1999), there
is a significant increase in the gender gap among gifted
or high scoring students on mathematics tests. Factors
explaining gender differences in mathematics include
prior achievement, attitudes towards mathematics, and
support from others (Becker, 1981; Ethington, 1992;
Grossman & Grossman, 1994; Fan & Chen, 1997).
However, in recent years there is a considerable
decrease in the difference of the mean scores between
male and female students’ achievement (Halat, 2006).
Although in the past female students had negative
attitudes towards mathematics, today they are less
likely to perceive mathematics as a male domain
(Friedman, 1994; Fennema & Hart, 1994; Halat, 2006).
For example, Fennema and Hart (1994) claimed that
interventions designed to address inequalities in middle
or high school mathematics classrooms played
important role in the establishment of gender equity in
learning mathematics. Likewise, Halat (2006) found no
difference in the acquisition of the van Hiele levels
between male and female students using van Hiele
theory based-curricula. Instruction influenced by the
van Hiele theory-based curricula may cause changes in
females’ attitudes towards mathematics courses (Halat,
2006).
The Purpose of the Study
The current study focuses on the reasoning stages
of in-service middle and high school mathematics
teachers in geometry. The following questions guided
this study:
1. What are the reasoning levels of in-service middle
and high school mathematics teachers in
geometry?
2. What differences exist in terms of geometric
reasoning levels between in-service middle and
high school mathematics teachers?
3. Is there a difference in terms of geometric
reasoning levels between male and female inservice mathematics teachers?
Method
Participants
In this study the researcher followed the
convenience sampling procedure, defined as “using as
the sample whoever happens to be available” (Gay,
1996, p.126). According to McMillan (2000), this is
Erdogan Halat
the most common procedure in today’s educational
research environment because of the difficulty of
finding volunteers to participate and obtaining
permission from schools and parents. The data was
collected during the Spring and Summer of 2006. Of a
total of 384 in-service secondary school mathematics
teachers in a city located in the western part of
Anatolia in Turkey, 148 teachers (39%) agreed to take
the Van Hiele Geometry Test (VHGT). (See Table 1
for the grade level and gender of the teachers.)
Of the participating teachers, 110 were in-service
middle school mathematics teachers—49 male and 61
female—and 38 were in-service high school
mathematics teachers-31 male and 7 female. The
participants teaching at the middle school level
represented 54% of in-service middle school
mathematics teachers in the city and the participants
teaching at the high school level represented 21% of
those in the city. The sample includes public and
private school teachers of both geometry and algebra.
The years of mathematics teaching experience varied
from 1 to 26 years. The high school mathematics
teachers took the test at their work places during the
school day. However, the middle school teachers took
the test at the end of an educational seminar. This
seminar conducted by the researcher did not relate to
van Hiele levels.
Data Collection
The researcher gave participants a geometry
test called the Van Hiele Geometry Test (VHGT),
consisting of 25 multiple-choice geometry questions.
The VHGT was taken from a study by Usiskin (1982)
with his written permission and is designed to measure
the subject’s van Hiele level when operating in a
geometric context. This test was translated to Turkish
by the investigator. Five mathematicians reviewed the
Turkish version of VHGT in terms of its language and
content. All participants’ answer sheets from VHGT
were read and scored by the investigator. Each
participant received a score referring to a van Hiele
level, guided by Usiskin’s grading system.
Analysis of Data
The data were responses from the in-service
middle and high school mathematics teachers’ answer
sheets. The criterion for success for attaining any given
van Hiele level in this study was four out of five
correct responses. The investigator constructed a
frequency table to display the distribution of the
mathematics teachers’ van Hiele level. Independent
samples t-test with α = 0.05 was used to compare the
11
Table 1
Frequency Table for In-service Middle and High School Math Teachers’ van Hiele Levels
Groups
N
A
B
Total
110
38
148
Level-I
(Visualization)
n
%
19
17.3
0
0
Level-II
(Analysis)
n
19
14
Level-III
(Ordering)
%
17.3
36.8
geometric reasoning levels between teachers’ genders
along with level of students taught. The Levene’s test
with α = 0.05 showed no violation of the equality of
variance assumption in all the ANCOVA and the
independent-samples t-test tables used in the study.
Results
In determining the reasoning levels of middle and
high school mathematics teachers in geometry, Table 1
provides a summary of the distribution, indicating that
van Hiele levels I through V were present. The most
common stage for middle school mathematics
teachers’ reasoning stages was level-III (Ordering)
(49.1%), but some showed a level-IV (Deduction)
(10.9%) or level-V (Rigor) (5.5%) performance.
According to table 1, none of the high school
mathematics teachers showed level-I (Visualization)
reasoning stage on the test; most were at level-II
(Analysis) (36.8%) and level-III (Ordering) (47.4%).
However, there were some performing at a level-IV
(Deduction) (7.9%) or level-V (Rigor) (7.9%) of
geometric reasoning.
Table 2 displays the mean score of in-service
middle and high school teachers in order to help
determine the differences that might exist between
these two groups. High school mathematics teachers’
van Hiele levels (2.87) was greater than that of the
middle school mathematics teachers (2.70). The mean
score difference in terms of reasoning stages was not
statistically significant [t = –0.88, p = 0.37 > 0.05].
Table 3 presents the descriptive statistics for the
mathematics teachers’ van Hiele levels by gender. The
table shows that the male mathematics teachers’ mean
Table 2
n
54
18
%
49.1
47.4
Level-IV
(Deduction)
n
%
12
10.9
3
7.9
Level-V
(Rigor)
n
6
3
%
5.5
7.9
score (2.88) is greater than that of the female
mathematics teachers (2.59). However, according to
the independent samples t-test, the mean score
differences between male and female mathematics
teachers on the Van Hiele Geometry Test (VHGT) is
not statistically significant, [t = 1.73, p = 0.086 > 0.05].
Discussions and Conclusion
This study revealed that the in-service middle and
high school mathematics teachers showed all reasoning
stages described by the van Hieles. Although the
proportion of mathematics teachers showing level-V
(Rigor) was low in comparison to other levels, it is
important to see some teachers operating at this level.
This is important in a theoretical perspective because
Usiskin (1982), Mayberry (1983), Burger and
Shaughnessy (1986) and Fuys, Geddes and Tischler
(1988) agreed that the last level, rigor (level-V), was
not appropriate for high school students. It was more
appropriate for college students or mathematics
teachers. However, some studies noted that none of
their
pre-service
elementary
and
secondary
mathematics teachers indicated level-V (Rigor)
reasoning stages in geometry (e.g., Mayberry, 1983;
Gutierrez, Jaime, & Fortuny, 1991; Durmuş, Toluk, &
Olkun, 2002; Knight, 2006). This study found that
there were some mathematics teachers who operated at
level-V (Rigor) on the test. Therefore, the finding
supports the idea that level-V reasoning may be a
realistic expectation of secondary teachers.
Table 3
Descriptive Statistics and the Independent Samples tTest for the In-service Teachers’ van Hiele Levels
Descriptive Statistics and Independent Samples t-Test
for the In-service Mathematics Teachers’ van Hiele
Levels
Group
Group
N
Male
Female
Total
80
68
148
N
van Hiele Geometry Test
Mean
SD
SE
df
2.70
1.05
0.10
146
2.87
0.87
0.14
t
-0.88
A
110
B
38
Total
148
Note. A – In-service middle school mathematics teachers,
B – In-service high school mathematics teachers.
12
p
0.37
van Hiele Geometry Test
Mean
SD
SE
2.88
0.97
0.10
2.59
1.04
0.12
df
146
t
1.73
p
0.086
Mathematics Teachers’ van Hiele Levels
The study found that almost 83% of the middle
school mathematics teachers’ van Hiele levels were at
or above level-II (Analysis). The reasoning stages of
the middle school mathematics teachers involved in
this study were higher than the level of their students;
research has shown that most middle school students
reason at level-I (Visualization) or at most level-II
(Analysis) (Burger & Shaughnessy, 1986; Fuys et al.,
1988; Halat, 2006). Sixty-three percent of the high
school mathematics teachers were at or above level-III
(Ordering), and only 15.8 percent of the secondary
mathematics teachers were at or above level-IV
(Deduction), the level at which high school students
should be (NCTM, 2000). Mathematics teachers must
have strong geometry knowledge and reasoning skills
themselves in order to help high school students meet
this expectation. The findings of this study imply that
high school mathematics teachers’ van Hiele levels
may not be adequate for teaching geometry at the
secondary level. This should be of particular interest
to those charged with the task of preparing teachers of
mathematics. The results of this study suggest
mathematics teacher educators should assess the
geometry knowledge of their pre-service teachers and
modify programs to encourage growth in their
geometric reasoning.
Furthermore, the study showed that there was no
statistically significant difference with reference to
geometric thinking levels between male and female
mathematics teachers on the geometry test. As
discussed earlier, research has documented that
although there is a difference between the achievement
of males and females in many content areas of
mathematics (Grossman and Grossman, 1994; Lloyd,
Walsh and Yailagh, 2005), there is no difference with
respect to gender in reference to motivation and
achievement in mathematics (Friedman, 1994;
Fennema & Hart, 1994; Halat, 2006). The findings of
the current study might support the latter group of
research.
Limitations and Future Research
According to Mayberry (1983), students can attain
different levels for different concepts. Likewise,
Burger and Shaughnessy (1986) found that students
may exhibit different levels of reasoning on different
tasks. Because the researcher tested teachers using only
questions on quadrilaterals, the results of this study
should not be generalized to all geometry topics.
Moreover, the results of the study should not be
generalized to all in-service middle and high school
mathematics teachers because of the differences in
Erdogan Halat
teacher preparation. Furthermore, the convenience
sampling procedure followed in the study may limit the
generalization of the findings. Additional research
studies done with pre-service elementary and
secondary mathematics teachers would be necessary in
order to make a more general statement about teachers’
geometric reasoning.
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14
Mathematics Teachers’ van Hiele Levels
The Mathematics Educator
2008, Vol. 18, No. 1, 15–25
Ninth Grade Students Studying the Movement of Fish to Learn
about Linear Relationships: The Use of Video-Based Analysis
Software in Mathematics Classrooms
S. Asli Özgün-Koca
The use of technology to create multiple representations of a concept has become one of the significant
instructional environments that the National Council of Teachers of Mathematics (2000) suggests
strongly for mathematics teachers to consider. One example of this type of environment is educational
software with linked multiple representations. An activity for both linked and semi-linked versions of
multi-representational software which was used in a dissertation study is presented along with two
ninth grade algebra students’ responses in order to provide an example of possible uses and effects of
semi-linked and linked computer software in mathematics classrooms. It was also aimed to make
connections between practice and research. The conclusion of this study was that semi-linked
representations could be as effective as linked representations and that there was a role for each in
different situations, at different levels, and with different mathematical concepts.
All aspects of a complex idea cannot be adequately
represented within a single notation system, and
hence require multiple systems for their full
expression,
means
that
multiple,
linked
representations will grow in importance as an
application of the new, dynamic, interactive media.
(Kaput, 1992, p. 530)
The utilization of technology for exploring
multiple representations has received increased
attention in mathematics education in the last decade.
The National Council of Teachers of Mathematics
(NCTM, 2000) states, “new forms of representation
associated with electronic technology create a need for
even greater instructional attention to representation”
(p. 67). By implementing advanced technologies, like
movies, new forms of representations are possible in
mathematics classrooms. Interactive and dynamic
linkages among multiple representations provide new
capabilities that traditional environments, such as
blackboard and paper-and-pencil, cannot (Ainsworth &
Van Labeke, 2004). Linked multiple representations
are a group of representations in which altering a given
representation automatically updates every other
representation to reflect the same change; semi-linked
S. Asli Ozgun-Koca obtained her doctoral degree from the Ohio
State University in 2001. Currently, she teaches mathematics and
secondary mathematics education courses at Wayne State
University, Detroit, MI. Her research interests focus on the use
of technology in mathematics instruction and understanding
secondary mathematics teachers' views about teaching and
learning of mathematics.
S. Asli Özgün-Koca
representations are defined as those for which the
corresponding updates of changes within the
representations are available only upon request and are
not automatically updated (Rich, 1996). Educational
software is one environment that allows for these
linkages (Hegedus & Kaput, 2004).
VideoPoint (Luetzelschwab & Laws, 2000) is a
video-based motion analysis software tools that allows
users to collect data from digital movies and perform
calculations with that data, such as finding the distance
between points (see Figure 1). To accommodate the
request of this author, the software developer made
changes to create the fully linked and semi-linked
versions of VideoPoint. VideoPoint links traditional
representations–graphs, tables, equations–but also
offers a novel representation–the movie. Although
VideoPoint was designed as linked representational
software, the linkage for the table representation was
not two-way. When the user makes changes in one
representation, the table as an example, another
representation, like the graph, is highlighted to reflect
the change. At this stage the linkage between the table
and the graph is one-way. In order to make this linkage
two-way, the user should also be able to make a change
in the graph and see its effects on the table. The graph,
table, and movie representations are linked two-way in
the fully linked version of the software. Thus, when the
user clicks on a point in those representations, the
corresponding data points in the other two
representations are highlighted. This can be observed
in Figure 1 among the table, the graph, and the movie.
15
Figure 1. A screenshot from VideoPoint (Fish movie is obtained from Graph Action Plus)
In this study, a movie of two fish swimming
towards each other from opposite sides of the screen
was used. The distances between the fish’s head and
the left hand side of the screen were measured over
time and reported in the four representations. The
movie, graph and table show the fish’s positions at one
second. When the user makes a change in any of these
representations, all other representations are updated to
reflect the change. For instance, when one clicks on a
different cell in the table or advances to the next frame
in the movie the corresponding points at the new
position are highlighted in all other representations.
When the user of the linked version clicks to see
the algebraic form (the equation of best fit) of the
phenomena, the line of best fit is graphed in the graph
window and its equation appears above the graph
automatically (see Figure 1). On the other hand, the
user of the semi-linked version is not able to see any
updates when he or she clicks on one representation or
makes changes in any representation. The only linkage
that is available in the semi-linked version is between
the graph and equation forms. When the user estimates
16
the coefficients in the algebraic form, he or she has an
option to see the graph of the predicted equation (see
Figure 2).
In this example, the student is creating a best-fit
line for one set of data points (with a positive slope) on
the graph by modeling. She changed the slope from 70
to 100 in order to obtain a steeper slope. Being able to
see the graph of the previous model with the current
model helped the student relate the algebraic form and
the graph more effectively by comparing before and
after pictures.
Review of Literature
Research studies and practitioner articles indicate
that the use of multiple representations with or without
technology may help students to construct
mathematical concepts in more empowering ways.
Articles without the use of technology emphasize the
use of various representations during instruction
(Clement, 2004; Harel, 1989; Knuth, 2000; Suh &
Moyer, 2007). Technology oriented studies, on the
other hand, utilized computer (Harrop, 2003; Hegedus
& Kaput, 2004; Jiang & McClintock, 2000; Noble,
Video-Based Analysis Software
Figure 2. A screenshot from VideoPoint—Estimating the Coefficients
Nemirovsky, Wright, & Tierney, 2001; Suh & Moyer,
2007; van der Meij & de Jong, 2006) or calculator
(Herman, 2007; Piez & Voxman, 1997; Ruthven,1990)
technology in order to investigate the effects of
multiple representations on learning.
Goldenberg et al. (1988) argue that “multiple
linked representations increase redundancy and thus
can reduce ambiguities that might be present in any
single representation” (p. 1). Therefore, multiple
representations can facilitate understanding. Kaput
(1986) also notes,
By making visually explicit the relationships
between different representations and the ways that
actions in one have consequences in the others, the
most difficult pedagogical and curricular problem
of building cognitive links between them becomes
much more tractable than when representations
could be tied together only by clumsy, serial
illustration in static media. (p. 199)
Goldenberg (1995) describes other advantages of
computer-based multiple linked representations:
• The interactive nature of the computer allows
students to become engaged in dialogue with
themselves.
• Raising conflict and surprise [which leads to
more thinking].
• Affirming (if not paralleling) students’ own
internal multiple representations.
S. Asli Özgün-Koca
• Helping us [educators and researchers]
distinguish between students’ expressed models
and the ones they act on.
• [Providing] immediacy and accuracy with
which the computer ties two or more
representations together.
• [Helping] students themselves
represent their concepts. (pp. 159–161)
multiply
In these kinds of environments, the computer
performs the required computations, thus leaving the
student free to alter the representations and to monitor
the consequences
of those actions across
representations. Moreover, the ability to represent the
same mathematical concept using many representations
and to make explicit the relationships among these
representations by dynamically linking them to each
other have been discussed as reasons for the
effectiveness of these learning environments (Kaput,
1986, 1994).
Research utilizing linked multiple representational
software creates two groups of studies: comparative
studies and case studies. The former ones (Rich, 1996;
Rosenheck, 1992) compared groups of students by
using different technologies in treatments. Due to the
crucial differences in the environments (e.g., computer
versus non-computer or calculators versus computers),
it is difficult to draw clear conclusions. In fact, results
of these studies showed no significant differences
between groups. On the other hand, the case studies
indicate more encouraging results because of the use of
linked multiple representation software (Borba, 1993;
17
Borba & Confrey, 1993; Lin, 1993; Rizzuti, 1992;
Yerushalmy & Gafni, 1992). The linkage among
representations in the computer-based environment
was a powerful tool that provided a visual environment
for students to develop and test their mathematical
conjectures. However, Ainsworth (1999) and van der
Meij and de Jong (2006) discussed possible
disadvantages of multiple linkage representations:
dynamic linking may put students in a passive mode by
doing too much for them and a cognitive overload may
result from providing too much information.
Theoretical Framework
Dienes’ multiple embodiment principle is a
prominent theory emphasizing multiple representations
in mathematics education. The multiple embodiment
principle suggests that conceptual learning of students
is enhanced when students are exposed to a concept
through a variety of representations (Dienes, 1960).
Additionally, Kaput (1995) notes the relationship
between external and internal representation: when one
moves
from
mental
operations
(internal
representations) to physical operations (external
representations), “one has cognitive content that one
seeks to externalize for purposes of communication or
testing for viability” (p. 140). On the other hand, in
moving from physical operations to mental operations,
“processes are based on an intent to use some existing
physical material to assist one’s thinking” (p. 140).
Students’ pre-existing knowledge structures influence
the external representational tools they use to perform
mathematics
tasks
and
to
communicate
mathematically. Conversely, the representational tools
available to students influence their mathematical
knowledge.
Now the question is how understanding across
multiple representations can be improved with
educational technology. Kaput (1994) notes that
physical links, such as those provided by a graphing
calculator or a computer, could be beneficial in
supporting students’ construction of cognitive links:
The purpose of the physical connection is to make
the relationship explicit and observable at the level
of actions in order to help build the integration of
knowledge structures and coordination of changes.
(p. 389)
Furthermore, Goldin and Kaput (1996) note,
By acting in one of the externally linked
representations and either observing the
consequences of that action in the other
representational system or making an explicit
prediction about the second representational system
18
to compare with the effect produced by actions in
the linked representation, one experiences the
linkages in new ways and is provided with new
opportunities for internal constructions. (p. 416)
According to Piaget’s theory (Piaget, 1952; Piaget
& Inhelder, 1969), cognitive development is described
as a process of adaptation and organization driven by a
series of equilibrium-disequilibrium states.
If
everything is in equilibrium, we do not need to change
anything in our cognitive structures. Adaptation occurs
when the child interacts with his or her environment.
The child is coping with his or her world, and this
involves adjusting. Assimilation is the process whereby
the child integrates new information into his or her
mental structures and interprets events in terms of the
existing cognitive structure, whereas accommodation
refers to changing the cognitive structure. Adaptation
is achieved when equilibrium is reached through a
series of assimilations and accommodations.
Organization is a structural concept used to describe
the integration of cognitive structures.
Linked representational software gives students
immediate feedback on the consequences of their
mathematical actions with machine accuracy, but it
may or may not engender the disequilibrium necessary
for learning. Semi-linked software, by not showing the
corresponding changes in other representations, forces
students to resolve the dissonance in their cognitive
structures by giving time to reflect or to ask questions
about what kind of changes could result from a change
in any representation. If their organization of
knowledge is well established, they can deal with the
question. However, if not, then they will need
accommodations in their cognitive structures. Thus, a
semi-linked representational environment puts students
in a more active role as learners.
Purpose and Rationale
The studies reviewed above investigated various
effects of multiple linked representation software.
However, the present study focused directly on the
effects of the linking property of the software on
students’ learning. Two groups of students in a
classroom environment used different versions of the
same computer software: fully linked and semi-linked.
The goal was to see how this feature of the software
affected their learning and understanding of the
relationships between the representations and the
mathematics content itself.
The major aim of this paper, however, is to present
an activity for both linked and semi-linked versions of
the software in order to demonstrate the use of the
Video-Based Analysis Software
software with the aim of connecting practice with
research. The purpose here is to offer an activity using
video-based motion analysis software in a mathematics
classroom and to suggest how to handle multiple
representations within the activity. While doing that,
the results of the dissertation research study are also
provided in order to emphasize the connection between
research and practice.
Methodology
In an eight-week period, 20 Algebra I students,
separated into two groups, used VideoPoint: one group
used linked representation software and the second
group used semi-linked representation software. Four
computer lab sessions were spaced out during the data
collection period. Because this particular school
schedules its classes for 78 minutes, one group was
taken out of the classroom for a 35-minute computer
lab during the first part of the class; then during the
second part, the other group went to the computer lab.
This study used a mixed method design. Its aim is
to “provide better (stronger) inferences and the
opportunity presenting a greater diversity of divergent
views [explanations]” (Tashakkari & Teddlie, 2003,
pp. 14–15). Tashakkari and Teddlie (1998) note that
“the term mixed methods typically refers to both data
collection techniques and analyses given that the type
of data collected is so intertwined with the type of
analysis that is used” (p. 43). Data collection methods
included mathematics pre-and post-tests, follow-up
interviews after the mathematics post-test, clinical
interviews in the computer lab at the end of the
treatment, classroom and lab observations and
document analysis (classroom materials, computer
dribble files, exams, and assignments). A grounded
survey was conducted at the end of the study in order
to see students’ opinions about mathematics,
representations in general, and the computer
environment.
A panel of experts assured the researcher of both
the content and face validity of the instruments.
Instruments were continuously updated according to
feedback from students both during the pilot and
throughout the actual study. As Tashakkari and Teddlie
(1998) argued, the use of a mixed method design led
this study to have data and methodological
triangulation. Other techniques used to provide
trustworthiness of this study were member check and
peer debriefing.
The data obtained through clinical interviews will
be used in this paper to provide an example of possible
uses and effects of semi-linked and linked computer
S. Asli Özgün-Koca
software in mathematics classrooms. The analysis of
the data obtained from these clinical interviews was
based on categorizing in order to investigate the
emerging themes throughout.
The Use of Semi-linked and Linked Software in
Mathematics Classroom
This section presents an activity using multiplerepresentational software. The parallel tasks for the
linked and semi-linked versions are displayed in two
columns. The activity included five main sections:
namely an introduction section; three sections that
focus on the graphical, tabular, algebraic
representations separately; and a general questions
section at the end. Two students’ responses are
provided, one using the linked version, the other
student using the semi-linked version of the software.
Even though just two students’ responses are displayed
below in the tables, general conclusions from the larger
study and general comments about the use of different
versions are also included in the narrative.
The lab activity started by watching a movie: two
fish swimming at a constant rate across the screen
towards each other. The fish movie was obtained from
Graph Action Plus. A grey fish (the fish at the bottom
of the screen) swims from right to left and a striped
fish (the fish at the top of the screen) swims from left
to right (see Figure 1). Students were asked general
questions about the movie, such as, “How does the
distance between the two fish change as they swim?”
At this point, only the movie window was open on the
screen. Typical responses included, “At first they got
closer and closer together and then they got farther and
farther away.”
The second part of the activity focused on the
graphical representation (see Table 1). First students
were asked to create the graph of the phenomenon by
paper and pencil after watching the movie. As Goldin
and Kaput (1996) mentioned, asking students to “make
an explicit prediction” (p. 416) before seeing the
computer-produced result could be very effective in
creating environments for students to construct
linkages among representations (i.e., between the
movie and the graph in this case). This approach was
used throughout the activity with all representations.
When creating a graph after watching this movie
or, more generally, when making predictions about a
representation by using another representation,
students need to recognize the outside information,
select an appropriate schema, and create an answer to
the question. This assimilation is described as
recognitive assimilation in Piaget’s theory, defined as
19
Table 1
Graphical Part of the Activity for Linked and Semi-Linked Versions of Video Point
Linked Version
Semi-Linked Version
Students were asked to predict the graphs of each fish’s distance from the left-hand side of the screen versus time.
“As the time increase, both of the fish’s distance increases actually.
Her graphs are switched. “As time went, [the grey fish] started farther
They both increase.”
away and it got closer and closer and the striped fish started out really
close and it got farther and farther away.”
After sketching their graph, students opened a window to see what the computer-produced graph looked like and compared their graph to the
computer’s graph.
After seeing that the graphs were not as expected, we started
discussing what was happening. The student focused on the distance
between the fish instead of their distances from the left-hand side of
screen separately. After this new information he said, “The striped
fish’s distance increases. This would be the decreasing one [showing
the gray fish’s graph].”
When she opened the computer graph, she used the linkage to find out
which line belongs to which fish. She saw that her prediction was not
correct by clicking on a data point on the graph. Then VideoPoint
showed her a movie screen where the fish’s labels were shown. She
needed to accommodate this new information.
The next task was to identify and describe the point on the graph that represents where the two fish meet.
To find out where two fish meet on the graph, she used the linkage.
There she could see that at the intersection of the two graphs, the
movie frame showed that the two fish met.
In this version, when the student clicked on the frame where the two
fish meet, he could not see the corresponding data points on the graph.
considering reality and choosing an appropriate
scheme (Montangero & Maurice-Naville, 1997). After
making their predictions, students opened and observed
the computer-produced graph. This gave them a chance
to check their work; differences existed between both
students’ hand-produced graphs and the computergenerated graphs (see Table 1). The linked group
student’s graphs were switched. The grey fish was the
one whose distance was decreasing whereas the striped
fish was the one with increasing distance. The semilinked group student, on the other hand, provided
increasing graphs for both fish. Because of these
discrepancies, the students may have experienced
disequilibrium and needed to accommodate this new
information. With the help of the software, the linked
group student used the linkage and accommodated the
new information. Even if the version of the software
did not provide linkages among representations for the
semi-linked group student, the information provided by
the representations helped him to re-think his
prediction and compare them with the computerproduced representation (see Table 1).
In the next task, students were asked to identify
and describe the point on the graph that represents
20
Video-Based Analysis Software
Table 2
Tabular Part of the Activity for Linked and Semi-Linked Versions of VideoPoint
Linked Version
Semi-Linked Version
First, students were asked their predictions about the table of values.
“The gray fish’s distance is going to decrease. The striped fish’s
“The gray fish’s distance will decrease. The striped fish’s will
distance will increase because it is going away from the starting point.”
increase.”
Students were asked to fill out a table by using the graph.
Time (Seconds)
Grey Fish Distance
Striped Fish
Distance
50 pixels
130 pixels
1 second
He had the same trouble as the linked group student when reading the
values from the graph. At this point, the linkage could be helpful for
him to correct that mistake.
She had trouble reading values from the graph. After finding when the
striped fish was 50 pixels away [point #1], to find what the distance of
the other fish was at that time [point #2], she moved horizontally to the
point #3 and read the distance of the other fish at another time. The use
of linkage could be helpful. If she clicked on the point #1, she would
see other fish’s data point highlighted [point #2].
We checked their answers with the computer-produced table in order for them to have feedback.
Students were asked to identify and describe the point in the table which represents where the two fish meet.
“Because those numbers are where they were the closest, like at .7333
seconds, the striped fish was still like closer to the…[left hand side of
the screen] than the grey fish was and by .8 seconds it was farther
away than the grey fish was.”
where the two fish meet. The linked group students
could use the linkage and identify the point on the
graph without needing more explanation (see Table 1).
They sometimes only referred to the movie, saying
something like, “Just look at the movie. This is the
point where the two fish meet,” after double clicking
on the graph. The semi-linked group students, on the
other hand, did not have this kind of opportunity. This
lack of linkage between the movie and the graphical
representation seemed to force some of the semi-linked
S. Asli Özgün-Koca
“So like right there [showing .8 second on the table] 130 [pixels].
Right on the graph they crossed about right here [he reads the distance
130 from the graph] and then I just looked that closer to that [on the
table].” At this point he constructed the linkage between the table and
the graph by himself.
group students to provide richer explanations such as,
“They are at the same place at the same time.”
A similar approach was followed for the tabular
representation (see Table 2). First students were asked
their predictions about the table of values. Then they
were asked to complete a table by reading values for
the graph (see Table 2). Both students in Table 2 had
the same trouble—reading values from the graph. For
instance, after the linked group student located the time
that the striped fish was 50 pixels away on the graph
successfully (labeled as point # 1 on the graph in Table
21
Table 3
Symbolic Part of the Activity for Linked and Semi-Linked Versions of VideoPoint
Linked Version
Semi-Linked Version
In the third section, students’ were asked to make predictions about the slope and y-intercept of the algebraic form.
“The grey fish has negative slope and the striped fish has positive
He thought that the striped fish would have positive slope and the grey
slope”
fish would have negative slope. Predictions for the y-intercept were not
What about the y-intercepts?
clear.
“I do not know”
She accessed the equation of the line of best fit immediately with the F
button next to the graph. The equation is shown at the top of the graph
and also the graph of the line of best fit is sketched on the data points.
He predicted the coefficients of the equation with the modeling button.
It did not take long for him to predict the equations; he used the
computer feedback and proceeded accordingly. He could interpret how
the changes in the algebraic form would affect the graph.
The next task was interpreting the differences in the equations of the two fish.
She thought that the slopes of the two fish had different signs,
“The striped fish went away…so it [the distance] increased that has
“because one keeps getting closer to the point and the other one keeps
positive [slope] but the grey fish’s distance decreased; that has
getting farther away.”
negative slope.”
Students were also asked to use the equations to determine the time that represents where the two fish meet.
“I am not sure how to do it”
“I do not get [understand] the equation.”
2), she was asked to locate the distance of the grey fish
at that time (approx. 0.42 seconds). Instead of moving
vertically to the point labeled # 2, she moved her
cursor horizontally to the point labeled as #3 to read
the distance of the grey fish at 1 second. The linkage
could be helpful to both students. If the linked group
student used the linkage and clicked on the point # 1,
point #2 would be circled. However, the linked group
student did not think of using the linkage at this point,
and the semi-linked group student did not have this
option.
When students were asked to identify and discuss
the point in the table that represents where the two fish
meet, the semi-linked group student was able to
construct a linkage by using the information provided
by the multiple representations. He used the graphical
representation (that he used previously to answer a
similar question) in order to interpret the tabular
representation more effectively. The linked group
student, on the other hand, attended to the distances of
22
both fish from the left-hand side of the movie screen.
At one data point, one fish was closer to the left-hand
side of the screen and then at the next data point the
other fish was closer to the left-hand side of the screen.
So she concluded that the fish should meet between
those data points.
The third section of the activity focused on the
algebraic representation (see Table 3). Students were
asked to make predictions about the symbolic
representation of this phenomenon. This part was the
most difficult section for the students. The two students
in Table 3 were representative of many students who
could not predict or even start to think about the
symbolic form. Whereas linked group students had
easy access to the algebraic form, the semi-linked
group students needed to predict the coefficients of the
equation. When the semi-linked group students entered
their predictions, the line for their last two predictions
appeared on the graph window. This feature of
VideoPoint showed students how well their predictions
Video-Based Analysis Software
fit the data points and how the changes in the algebraic
form affected the graph (see Table 3). Because the
computer software creates linkages between the
symbolic and graphical representations, students can
focus on how manipulating the algebraic form in a
specific way causes changes in the graph (Kaput,
1995).
Many students had difficulties interpreting the
algebraic form and using the equations to predict the
time where the two fish meet. When finding the time
that the two fish meet by using the graph or the table,
students were able to make connections to the context
(movie) more easily than when asked to use the
algebraic form. The interpretation of the algebraic
model and the symbolic manipulation required are
possible reasons that students struggled more in this
part of the activity.
The final section included a general question, such
as identifying the distance between the two fish at the
beginning of the movie. The students were allowed to
use any representation they wished to answer this
question. Students reported the representation they
used and were encouraged to use other representations
to answer the same question. The linked-group student
used a table effectively to answer this question. When
asked to use the graph, she used the linkage, clicked on
the data point on the graph and saw both points circled
at the same time (see Figure 3). The semi-linked group
student also approached the question using a tabular
representation: “You could subtract these two distances
[pointing to the first distances at the table].” When he
was asked to use the graph, he said, “You would take
the beginning from right here [pointing to the first data
point of the striped fish on the graph] and that
beginning right up there [pointing to the first data point
of the gray fish on the graph] and subtract them.”
Figure 3. Using linkages to find two fish’s distance at
the beginning
S. Asli Özgün-Koca
Conclusions and Discussion
This study focused on the effects of the use of a
multiple representational computer environment on
students’ learning. In the linked computer
environment, students either used the linkage directly
to answer the question or they assimilated the
information provided through the linkage, using their
previous knowledge to choose an existing, appropriate
schema to answer the question. If they used the linkage
directly, the software was the basis of their
explanation. Students who chose not to use the linkage
provided explanations for their answers based more on
the mathematical aspects of the question. In either
case, when the computer feedback contradicted their
predictions, disequilibrium occurred, and the students
needed to re-interpret this new information through
their existing knowledge; that is, they assimilated the
new information. If they could not interpret this new
information with their present schema, they needed to
accommodate their preexisting knowledge in order to
reach equilibrium; that is, in Piaget’s words, they
modified “internal schemes to fit reality” (Piaget &
Inhelder, 1969, p. 6).
There were students who trusted their own
knowledge and answers. They did not use the linkage
at all. An interpretation of Kaput’s theory (1995)
discussing the relationship between external and
internal representations could be helpful in interpreting
this issue. Students who trusted their internal
representations (mental operations) might not need to
test their knowledge for viability with the software;
they preferred not to use the linkage. However, there
were students who ignored the linkage or did not use
the linkage when they could have benefited from using
the linkage. Now, the other direction in Kaput’s theory,
moving
from
physical
operations
(external
representations) to mental operations, could be used.
Here, the linkage, if used, could have served as the
“existing physical material” (p. 140) to help students
further construct their incomplete schema.
Results suggested that in a semi-linked
environment, students seemed to rely mainly on their
own existing knowledge with the help of the software
to respond to a question. Although this environment
did not provide such rich feedback as in the linked
environment, ready-made graphs or tables presented
powerful visual information/feedback for students to
use while answering the questions. The software could
have served as helper, record keeper, or representation
provider for the students. Without the linkage, students
seemed to provide more mathematically based
explanations rather than movie-based explanations and
23
constructed the linkages between representations for
themselves. They were seen to be in a more active role
mentally as learners. However, some students were not
able to discern the relationships among representations.
They could have used the linkage, if it had been
available, in order to construct more empowering
concepts.
Having access to multiple mathematical
representations provided by VideoPoint enabled
students to choose the types of representation with
which they were most comfortable. Another advantage
was the increased attention to the relationships among
representations and the mathematical content instead of
computation, manipulation, or drawing. Moreover, the
software offered an environment with resources and
constraints for students to construct new schema or
change their existing ones by passing through a series
of equilibrium-disequilibrium states.
Semi-linked representations can be as effective as
linked representations for mathematical concept
development. Being able to switch between the linked
and semi-linked versions would be invaluable because
the linked and semi-linked versions have their own
benefits and limitations. Mathematics teachers might
prefer linked or semi-linked versions of software for
different age groups or grade levels. The most
beneficial usage could come from using a linked
version to introduce a mathematical idea and help
students construct their schema. Once accomplished,
the linkage could be removed and the semi-linked
version could be turned on in order to make students
use their newly constructed schema. This emphasizes
the importance of the teacher’s role in the classroom.
Technology, if used appropriately, is a very effective
tool in the process of teaching and learning of
mathematics. However, there are many important
decisions to be made by the teacher, such as when and
how to use technology and with whom.
This article provides an example of utilizing linked
and semi-linked representational software in
mathematics teaching. The existing theories and the
results of this research study were used to discuss the
advantages, disadvantages, roles and effects of both
types of technological environments in students’
learning of linear relationships. Research in
mathematics education allows us to improve the
teaching and learning processes in mathematics
classrooms. When strong bridges are constructed
between the practice of teaching mathematics and
research in mathematics education, they might serve
educators, researchers and teachers in more
empowering ways.
24
References
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S. Asli Özgün-Koca
25
The Mathematics Educator
2008, Vol. 18, No. 1, 26–30
Applying Piaget’s Theory of Cognitive Development to
Mathematics Instruction
Bobby Ojose
This paper is based on a presentation given at National Council of Teachers of Mathematics (NCTM)
in 2005 in Anaheim, California. It explicates the developmental stages of the child as posited by Piaget.
The author then ties each of the stages to developmentally appropriate mathematics instruction. The
implications in terms of not imposing unfamiliar ideas on the child and importance of peer interaction
are highlighted.
Introduction
Underlying Assumptions
Jean Piaget’s work on children’s cognitive
development, specifically with quantitative concepts,
has garnered much attention within the field of
education. Piaget explored children’s cognitive
development to study his primary interest in genetic
epistemology. Upon completion of his doctorate, he
became intrigued with the processes by which children
achieved their answers; he used conversation as a
means to probe children’s thinking based on
experimental procedures used in psychiatric
questioning.
One contribution of Piagetian theory concerns the
developmental stages of children’s cognition. His work
on children’s quantitative development has provided
mathematics educators with crucial insights into how
children learn mathematical concepts and ideas. This
article describes stages of cognitive development with
an emphasis on their importance to mathematical
development and provides suggestions for planning
mathematics instruction.
The approach of this article will be to provide a
brief discussion of Piaget’s underlying assumptions
regarding the stages of development. Each stage will
be described and characterized, highlighting the stageappropriate mathematics techniques that help lay a
solid foundation for future mathematics learning. The
conclusion will incorporate general implications of the
knowledge of stages of development for mathematics
instruction.
Piaget believed that the development of a child
occurs through a continuous transformation of thought
processes. A developmental stage consists of a period
of months or years when certain development takes
place. Although students are usually grouped by
chronological age, their development levels may differ
significantly (Weinert & Helmke, 1998), as well as the
rate at which individual children pass through each
stage. This difference may depend on maturity,
experience, culture, and the ability of the child (Papila
& Olds, 1996). According to Berk (1997), Piaget
believed that children develop steadily and gradually
throughout the varying stages and that the experiences
in one stage form the foundations for movement to the
next. All people pass through each stage before starting
the next one; no one skips any stage. This implies older
children, and even adults, who have not passed through
later stages process information in ways that are
characteristic of young children at the same
developmental stage (Eggen & Kauchak, 2000).
Dr. Bobby Ojose is an Assistant Professor at the University of
Redlands, California. He teaches mathematics education and
quantitative research methods classes. His research interests
encompass constructivism in teaching and learning mathematics.
26
Stages of Cognitive Development
Piaget has identified four primary stages of
development: sensorimotor, preoperational, concrete
operational, and formal operational.
Sensorimotor Stage
In the sensorimotor stage, an infant’s mental and
cognitive attributes develop from birth until the
appearance of language. This stage is characterized by
the progressive acquisition of object permanence in
which the child becomes able to find objects after they
have been displaced, even if the objects have been
taken out of his field of vision. For example, Piaget’s
experiments at this stage include hiding an object
under a pillow to see if the baby finds the object.
.
Applying Piaget’s Theory
An additional characteristic of children at this
stage is their ability to link numbers to objects (Piaget,
1977) (e.g., one dog, two cats, three pigs, four hippos).
To develop the mathematical capability of a child in
this stage, the child’s ability might be enhanced if he is
allowed ample opportunity to act on the environment
in unrestricted (but safe) ways in order to start building
concepts (Martin, 2000). Evidence suggests that
children at the sensorimotor stage have some
understanding of the concepts of numbers and counting
(Fuson, 1988). Educators of children in this stage of
development should lay a solid mathematical
foundation by providing activities that incorporate
counting and thus enhance children’s conceptual
development of number. For example, teachers and
parents can help children count their fingers, toys, and
candies. Questions such as “Who has more?” or “Are
there enough?” could be a part of the daily lives of
children as young as two or three years of age.
Another activity that could enhance the
mathematical development of children at this stage
connects mathematics and literature. There is a
plethora of children’s books that embed mathematical
content. (See Appendix A for a non-exhaustive list of
children’s books incorporating mathematical concepts
and ideas.) A recommendation would be that these
books include pictorial illustrations. Because children
at this stage can link numbers to objects, learners can
benefit from seeing pictures of objects and their
respective numbers simultaneously. Along with the
mathematical benefits, children’s books can contribute
to the development of their reading skills and
comprehension.
Preoperational Stage
The characteristics of this stage include an increase
in language ability (with over-generalizations),
symbolic thought, egocentric perspective, and limited
logic. In this second stage, children should engage with
problem-solving tasks that incorporate available
materials such as blocks, sand, and water. While the
child is working with a problem, the teacher should
elicit conversation from the child. The verbalization of
the child, as well as his actions on the materials, gives
a basis that permits the teacher to infer the mechanisms
of the child’s thought processes.
There is lack of logic associated with this stage of
development; rational thought makes little appearance.
The child links together unrelated events, sees objects
as possessing life, does not understand point-of-view,
and cannot reverse operations. For example, a child at
this stage who understands that adding four to five
Bobby Ojose
yields nine cannot yet perform the reverse operation of
taking four from nine.
Children’s perceptions in this stage of development
are generally restricted to one aspect or dimension of
an object at the expense of the other aspects. For
example, Piaget tested the concept of conservation by
pouring the same amount of liquid into two similar
containers. When the liquid from one container is
poured into a third, wider container, the level is lower
and the child thinks there is less liquid in the third
container. Thus the child is using one dimension,
height, as the basis for his judgment of another
dimension, volume.
Teaching students in this stage of development
should
employ
effective
questioning
about
characterizing objects. For example, when students
investigate geometric shapes, a teacher could ask
students to group the shapes according to similar
characteristics. Questions following the investigation
could include, “How did you decide where each object
belonged? Are there other ways to group these
together?” Engaging in discussion or interactions with
the children may engender the children’s discovery of
the variety of ways to group objects, thus helping the
children think about the quantities in novel ways
(Thompson, 1990).
Concrete Operations Stage
The third stage is characterized by remarkable
cognitive growth, when children’s development of
language and acquisition of basic skills accelerate
dramatically. Children at this stage utilize their senses
in order to know; they can now consider two or three
dimensions simultaneously instead of successively. For
example, in the liquids experiment, if the child notices
the lowered level of the liquid, he also notices the dish
is wider, seeing both dimensions at the same time.
Additionally, seriation and classification are the two
logical operations that develop during this stage
(Piaget, 1977) and both are essential for understanding
number concepts. Seriation is the ability to order
objects according to increasing or decreasing length,
weight, or volume. On the other hand, classification
involves grouping objects on the basis of a common
characteristic.
According to Burns & Silbey (2000), “hands-on
experiences and multiple ways of representing a
mathematical solution can be ways of fostering the
development of this cognitive stage” (p. 55). The
importance of hands-on activities cannot be
overemphasized at this stage. These activities provide
students an avenue to make abstract ideas concrete,
27
allowing them to get their hands on mathematical ideas
and concepts as useful tools for solving problems.
Because concrete experiences are needed, teachers
might use manipulatives with their students to explore
concepts such as place value and arithmetical
operations. Existing manipulative materials include:
pattern blocks, Cuisenaire rods, algebra tiles, algebra
cubes, geoboards, tangrams, counters, dice, and
spinners. However, teachers are not limited to
commercial materials, they can also use convenient
materials in activities such as paper folding and
cutting. As students use the materials, they acquire
experiences that help lay the foundation for more
advanced mathematical thinking. Furthermore,
students’ use of materials helps to build their
mathematical confidence by giving them a way to test
and confirm their reasoning.
One of the important challenges in mathematics
teaching is to help students make connections between
the mathematics concepts and the activity. Children
may not automatically make connections between the
work they do with manipulative materials and the
corresponding abstract mathematics: “children tend to
think that the manipulations they do with models are
one method for finding a solution and pencil-and-paper
math is entirely separate” (Burns & Silbey, 2000, p.
60). For example, it may be difficult for children to
conceptualize how a four by six inch rectangle built
with wooden tiles relates to four multiplied by six, or
four groups of six. Teachers could help students make
connections by showing how the rectangles can be
separated into four rows of six tiles each and by
demonstrating how the rectangle is another
representation of four groups of six.
Providing various mathematical representations
acknowledges the uniqueness of students and provides
multiple paths for making ideas meaningful.
Engendering opportunities for students to present
mathematical solutions in multiple ways (e.g.,
symbols, graphs, tables, and words) is one tool for
cognitive development in this stage. Eggen & Kauchak
(2000) noted that while a specific way of representing
an idea is meaningful to some students, a different
representation might be more meaningful to others.
Formal Operations Stage
The child at this stage is capable of forming
hypotheses and deducing possible consequences,
allowing the child to construct his own mathematics.
Furthermore, the child typically begins to develop
abstract thought patterns where reasoning is executed
using pure symbols without the necessity of perceptive
28
data. For example, the formal operational learner can
solve x + 2x = 9 without having to refer to a concrete
situation presented by the teacher, such as, “Tony ate a
certain number of candies. His sister ate twice as many.
Together they ate nine. How many did Tony eat?”
Reasoning skills within this stage refer to the mental
process involved in the generalizing and evaluating of
logical arguments (Anderson, 1990) and include
clarification, inference, evaluation, and application.
Clarification. Clarification requires students to
identify and analyze elements of a problem, allowing
them to decipher the information needed in solving a
problem. By encouraging students to extract relevant
information from a problem statement, teachers can
help
students
enhance
their
mathematical
understanding.
Inference.
Students
at
this
stage
are
developmentally ready to make inductive and
deductive inferences in mathematics. Deductive
inferences involve reasoning from general concepts to
specific instances. On the other hand, inductive
inferences are based on extracting similarities and
differences among specific objects and events and
arriving at generalizations.
Evaluation. Evaluation involves using criteria to
judge the adequacy of a problem solution. For
example, the student can follow a predetermined rubric
to judge the correctness of his solution to a problem.
Evaluation leads to formulating hypotheses about
future events, assuming one’s problem solving is
correct thus far.
Application. Application involves students
connecting mathematical concepts to real-life
situations. For example, the student could apply his
knowledge of rational equations to the following
situation: “You can clean your house in 4 hours. Your
sister can clean it in 6 hours. How long will it take you
to clean the house, working together?”
Implications of Piaget’s Theory
Critics of Piaget’s work argue that his proposed
theory does not offer a complete description of
cognitive development (Eggen & Kauchak, 2000). For
example, Piaget is criticized for underestimating the
abilities of young children. Abstract directions and
requirements may cause young children to fail at tasks
they can do under simpler conditions (Gelman, Meck,
& Merkin, 1986). Piaget has also been criticized for
overestimating the abilities of older learners, having
implications for both learners and teachers. For
example, middle school teachers interpreting Piaget’s
work may assume that their students can always think
Applying Piaget’s Theory
logically in the abstract, yet this is often not the case
(Eggen & Kauchak, 2000).
Although not possible to teach cognitive
development explicitly, research has demonstrated that
it can be accelerated (Zimmerman & Whitehurst,
1979). Piaget believed that the amount of time each
child spends in each stage varies by environment
(Kamii, 1982). All students in a class are not
necessarily operating at the same level. Teachers could
benefit from understanding the levels at which their
students are functioning and should try to ascertain
their students’ cognitive levels to adjust their teaching
accordingly. By emphasizing methods of reasoning,
the teacher provides critical direction so that the child
can discover concepts through investigation. The child
should be encouraged to self-check, approximate,
reflect and reason while the teacher studies the child’s
work to better understand his thinking (Piaget, 1970).
The numbers and quantities used to teach the
children number should be meaningful to them.
Various situations can be set up that encourage
mathematical reasoning. For example, a child may be
asked to bring enough cups for everybody in the class,
without being explicitly told to count. This will require
them to compare the number of people to the number
of cups needed. Other examples include dividing
objects among a group fairly, keeping classroom
records like attendance, and voting to make class
decisions.
Games are also a good way to acquire
understanding of mathematical principles (Kamii,
1982). For example, the game of musical chairs
requires coordination between the set of children and
the set of chairs. Scorekeeping in marbles and bowling
requires comparison of quantities and simple
arithmetical operations. Comparisons of quantities are
required in a guessing game where one child chooses a
number between one and ten and another attempts to
determine it, being told if his guesses are too high or
too low.
Summary
As children develop, they progress through stages
characterized by unique ways of understanding the
world. During the sensorimotor stage, young children
develop eye-hand coordination schemes and object
permanence. The preoperational stage includes growth
of symbolic thought, as evidenced by the increased use
of language. During the concrete operational stage,
children can perform basic operations such as
classification and serial ordering of concrete objects. In
the final stage, formal operations, students develop the
Bobby Ojose
ability to think abstractly and metacognitively, as well
as reason hypothetically. This article articulated these
stages in light of mathematics instruction. In general,
the knowledge of Piaget’s stages helps the teacher
understand the cognitive development of the child as
the teacher plans stage-appropriate activities to keep
students active.
References
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.
29
Appendix A: Children’s Literature Incorporating Mathematical Concepts and Ideas
Anno, M. (1982). Anno’s counting house. New York: Philomel Books.
Anno, M. (1994). Anno’s magic seeds. New York: Philomel Books.
Anno, M., & Anno, M. (1983). Anno’s mysterious multiplying jar. New York: Philomel Books.
Ash, R. (1996). Incredible comparisons. New York: Dorling Kindersley.
Briggs, R. (1970). Jim and the beanstalk. New York: Coward–McCann.
Carle, E. (1969). The very hungry caterpillar. New York: Putnam.
Chalmers, M. (1986). Six dogs, twenty-three cats, forty-five mice, and one hundred sixty spiders. New York:
Harper Collins.
Chwast, S. (1993). The twelve circus rings. San Diego, CA: Gulliver Books, Harcourt Brace Jovanovich.
Clement, R. (1991). Counting on Frank. Milwaukee: Gareth Stevens Children’s Book.
Cushman, R. (1991). Do you wanna bet? Your chance to find out about probability. New York: Clarion Books.
Dee, R. (1988). Two ways to count to ten. New York: Holt.
Falwell, C. (1993). Feast for 10. New York: Clarion Books.
Friedman, A. (1994). The king’s commissioners. New York: Scholastic.
Gag, W. (1928). Millions of cats. New York: Coward-McCann.
Giganti, P. (1988). How many snails? A counting book. New York: Greenwillow.
Giganti, P. (1992). Each orange had 8 slices. New York: Greenwillow.
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Hoban, T. (1981). More than one. New York: Greenwillow.
Hutchins, P. (1986). The doorbell rang. New York: Greenwillow.
Jaspersohn, W. (1993). Cookies. Old Tappan, NJ: Macmillan.
Juster, N. (1961). The phantom tollbooth. New York: Random House.
Linden, A. M. (1994). One sailing grandma: A Caribbean counting book. New York: Heinemann.
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Mathews, L. (1979). Gator pie. New York: Dodd, Mead.
McKissack, P. C. (1992). A million fish…more or less. New York: Knopf.
Munsch, R. (1987). Moira’s birthday. Toronto: Annick Press.
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Pluckrose, H. (1988). Pattern. New York: Franklin Watts.
San Souci, R. (1989). The boy and the ghost. New York: Simon-Schuster Books.
St. John, G. (1975). How to count like a Martian. New York: Walck.
Schwartz, D. (1985). How much is a million? New York: Lothrop, Lee, & Shepard.
Sharmat, M. W. (1979). The 329th friend. New York: Four Winds Press.
Tahan, M. (1993). The man who counted. A collection of mathematical adventures. New York: Norton.
Wells, R. E. (1993). Is the blue whale the biggest thing there is? Morton Grove, IL: Whitman.
Wolkstein, D. (1972). 8,000 stones. New York: Doubleday.
30
Applying Piaget’s Theory
The Mathematics Educator
2008, Vol. 18, No. 1, 31–40
Preservice Teachers’ Mathematical Task Posing: An
Opportunity for Coordination of Perspectives
Zachary Rutledge
Anderson Norton
This article provides detailed analysis, from a radical constructivist perspective, of a sequence of
letter-writing exchanges between a preservice secondary mathematics teacher and a high school
student. This analysis shows the ways in which the preservice teacher gained understanding of the
high school student’s mathematics and attempted to pose tasks accordingly, leading to a fruitful
mathematical exchange. In addition, this article also considers the same exchange from what could be
considered broadly as a situated perspective towards learning. We conclude by suggesting that these
perspectives could be considered compatible within this study if a distinction is made between the
student’s point of view and the researcher’s.
The purpose of this article is two-fold. The first is
to provide a detailed analysis of one sequence of letterwriting exchanges between a preservice teacher (PST)
and a high school algebra student. These exchanges,
which were part of the methods course the PST took,
involved posing mathematical tasks to high school
students. The rationale for this project was to provide
PSTs with an opportunity to learn the practice of
posing tasks and assessing students’ mathematics; this
work builds upon research conducted by Crespo (2003)
who analyzed the mathematical communication
between elementary students and PSTs. In expanding
on Crespo’s work, we developed several measures for
gauging cognitive activity and showed that in many of
these measures the PSTs posed better tasks., We
demonstrate with sample exchanges that the PSTs
learned to amend a single task in order to make it more
accessible to the student.
The second purpose of this article is to examine
our previous work from a perspective that might
broadly be considered socio-cultural or situated.
Following the recommendations of other researchers
(Cobb, 2007; Lester, 2005), the sample analysis
included in this paper suggests a way to coordinate
psychological and sociological perspectives on
learning. In particular, we examine the various social
contexts in which the letter-writing interactions were
situated while considering cognitive activities that each
Zachary Rutledge is a doctoral student at Indiana University,
Bloomington. In addition to working with preservice teachers in
developing their task-posing abilities, he is also involved in
analyzing data from the National Assessment of Education
Progress.
Anderson Norton is Assistant Professor in the Mathematics
Department at Virginia Tech. He teaches math courses for future
secondary school teachers and conducts research on students'
mathematical development.
Zachary Rutledge & Anderson Norton
participant brought to bear on those situations.
We conducted our analysis of the entire body of
data using a constructivist lens. Afterwards, we
examined one example in detail and wanted to extend
the discussion by considering what another theoretical
perspective suggests. This post-hoc discussion does not
have a clear method as it is meant to be suggestive of
several methods available to the researcher. Any one of
these methods could ultimately be used to examine
these data in more detail from a situated perspective.
However, despite the post-hoc nature of the situated
analysis, we conclude that this extended discussion and
coordination of cognitive and situated perspectives has
enriched our understanding of the letter-writing
interactions. To support this, we provide detailed
analyses of a sequence of interactions in which a
particular letter-writing pair maintained socio-cultural
boundaries, a process in which the student’s individual
understanding played a central role.
Method and Theoretical Orientation
From a psychological perspective, we were
concerned with the kinds of elicited cognitive activity
that we could infer from the task exchanges. We relied
on descriptions of cognitive processes described in
three main sources: Bloom’s taxonomy as described by
Kastberg (2003), Principles and Standards of School
Mathematics (National Council of Teachers of
Mathematics [NCTM], 2000), and a chapter on
“cognitively complex tasks” by Stein, Smith,
Henningsen, and Silver (2000). From Bloom’s
taxonomy we borrowed the four highest levels of
cognitive activity: Application, Analysis, Synthesis, and
Evaluation. We borrowed the process standards—
Communication, Connections, Problem Solving,
Reasoning and Proof, and Representations—from the
31
NCTM document. Finally, we borrowed Stein et al.’s
levels
of
cognitive
demand—Memorization,
Procedures without Connections, Procedures with
Connections, and Doing Mathematics—and used all of
these as descriptors of elicited cognitive activity (see
Appendix).
Our use of Stein et al.’s four levels of cognitive
demand do not differ significantly from the
descriptions provided by Stein et al. with the exception
of the category created to describe the highest level of
cognitive demand: Doing Mathematics. First, we
briefly review the other three categories. Memorization
is precisely as it sounds. If someone asked a student to
state the definition of an acute triangle and the student
responded, then this mathematical task would be
inferred by the researcher as Memorization. We
provide a full discussion of these measures in Norton
and Rutledge (2008).
On the other hand, Procedures with and without
Connections both involve the use of a mathematical
procedure to accomplish the task objective. For
example, should a student be confronted with the task
of solving a system of two linear equations, the student
may readily solve for one variable in one equation and
then substitute into the other, or perhaps the student
would be more inclined to use a matrix and row
reduction methods. In this example, we would likely
infer from the student’s behavior that a procedure was
definitely used, but we would not be able to infer that
conceptual understanding accompanied this activity.
This does not mean that the student does not
understand linear equations but rather we could not
infer this understanding from the student’s interaction
with this particular activity.
Continuing this vein of thought, suppose the
student were given the same task, but in addition to
solving the task using a procedure, the student sketched
the graphs of the two functions in order to verify the
reasonability of the answer. In doing so, the student
would be indicating that the correct answer is the point
at which the two lines intersect. This would show, not
only skill with the procedure, but a more robust
understanding of what it really means to solve for two
equations with two unknowns. We would likely infer
from this behavior that the student had engaged in
Procedures with Connections.
Doing Mathematics as defined by Stein et al.
(2000) was too vague for our purposes; therefore we
turned to Schifter (1996) who considered conjecturing
as a part of doing mathematics. We adapted her
definition by adding the additional requirement that the
student had to give some evidence that they had made
32
a conjecture and then tested the conjecture. If we saw
evidence of both, then we classified that particular
exchange as a case of Doing Mathematics.
Problem Solving was a difficult measure to define
and operationalize. However, we ultimately found the
definition as provided by Lester and Kehle (2003) to be
of use.
Successful problem solving involves coordinating
previous
experiences,
knowledge,
familiar
representations and patterns of inference, and
intuition in an effort to generate new
representations and patterns of inference that
resolve the tension or ambiguity (i.e., lack of
meaningful representations and supporting
inferential moves) that prompted the original
problem solving activity. (p. 510)
Therefore, we only considered an exchange to have
elicited Problem Solving if we found some degree of
struggle from the student. It is important to note that
this is specific to the student and the problem with
which they are engaging, regardless of how difficult
the problem seemed to a third-person observer.
With these definitions in mind, it is now critical to
consider the ways we could identify these various
processes in the interactions. To this end, we adopted a
radical constructivist perspective that highlighted the
researchers’ inferences about students’ mathematical
activity (von Glasersfeld, 1995). This perspective
places certain demands on the way we identified these
measures in practice. For example, consider the
measure Analysis. We had to keep in mind that
students construct their own meanings for
mathematical situations and analyze mathematical
situations in a way that is different from our own
analyses. Thus, to infer that a student had engaged in
Analysis, we had to be able to imagine a reasonable
and consistent way of operating in which the student
broke down the situation into constituent mathematical
parts to better understand it. We were conservative in
making such inferences; we needed to be able to find
clear indications of Analysis that fit with the totality of
the student’s written response.
In the next section, we describe the interactions of
Ellen and Jacques (both pseudonyms) and our
inferences about the cognitive activities that those
interactions elicited from Jacques. As with all of the
letter-writing pairs at the time, Ellen was a PST
enrolled in the first of two mathematics methods
courses, and Jacques was completing the final weeks of
his second trimester of Algebra I. The elicited activities
found in the following analyzed sequence between
Preservice Teachers’ Task Posing
Figure 1. Ellen’s initial task to Jacques.
Jacques and Ellen represent the kinds of activities
found across all the letter-writing pairs. We noticed an
overall increase in cognitive level from purely
procedural to Procedures with Connections, and we
determined that many of the PST’s tasks elicited
Communication, Application, and Analysis. Across all
exchanges between PSTs and high school students, we
rarely identified instances of Problem Solving in our
data. Therefore, the presence of this particular measure
in Jacques and Ellen’s exchanges indicates the
fruitfulness of their interaction. To clarify these
statements, we explain how we inferred cognitive
activity from Jacques’ written responses in each of four
exchanges with Ellen.
Analysis
Figure 1 illustrates Ellen’s initial task1 to Jacques,
one that we might formally recognize as an analytic
geometry problem. Note that Ellen made all of the
algebraic manipulations after she received Jacques’
response.
Jacques’ response (Figure 2) indicates that he was
unable to engage meaningfully in the task of finding
equations for lines meeting the specified geometric
conditions. However, he was able to assimilate the
situation as one involving solutions to systems of
equations. He had a lot of experience working with
Zachary Rutledge & Anderson Norton
systems of equations in his algebra class, and the
situation described in Figure 1 might appear to have
many familiar features from such experiences
(intersecting lines on a graph, coordinates, questions
about linear equations, etc.). From his activity of
manipulating two linear equations and their graph, we
inferred that Jacques’ knowledge of solving systems of
linear equations was procedural only. Therefore, we
would expect that Jacques would be able to manipulate
symbols (solve for ‘x’, substitute values, etc.), but he
might not have a more connected understanding that
would link the equations to graphical representation.
Jacques might have had a more connected
understanding of the concepts underlying the
procedure, but there was no clear indication from
which to infer this. Therefore, we coded the elicited
activity as Procedures without Connections.
Figure 2. Jacques’ response to Task 1.
33
Other codes applied to Jacques’ response included
Application and Communication. The former was
based on our inference that Jacques used existing ideas
in a novel situation. He assimilated his knowledge of
systems of equations to a situation involving finding
equations of intersecting lines. He effectively applied
an algebraic procedure to a new domain which
observers might call analytic geometry. We based the
latter code on our inference that Jacques’ written
language was intended to convey a mathematical idea
that systems of equations can be used to find points of
intersection.
In her next letter, Ellen affirmed Jacques’ response
but turned the conversation back to her original intent
for the task. After restating the task, Ellen attempted to
focus Jacques’ attention on the angles, the lengths of
the sides, and the type of triangle that she had drawn.
Jacques responded (Figure 3) by pointing out that the
marked angle was 90 degrees. In addition, he supplied
three equations without work, thus rendering it more
difficult to infer how he was operating. In writing the
slopes as fractions, including the whole numbers, he
was likely focusing on slope as rise over run. He
accurately identified the slope of one line and the signs
on all lines. We inferred that his procedure for
computing slope as rise over run was connected with a
graphical understanding of slope as the gradient of the
line. Because there was evidence of cognitive activity
that went beyond the application of procedures we
identified this as Procedures with Connections.
In addition to coding Application and
Communication, we also found indication of Analysis
in Jacques’ response because Jacques seemed to break
down the graph to obtain specific mathematical
information, such as the signs of the slopes of the lines.
Finally, we inferred that Jacques had engaged in
Problem Solving because he seemed to struggle (as
indicated by his question, “is this correct?”) and yet
made progress in resolving the novel situation.
Ellen returned to the same problem situation again
in posing Task 3. She attempted to focus Jacques’
attention on the lower triangle in Figure 4. She wanted
Jacques to connect the coordinate point (6,1) with
distances on the triangle. She asked him if he could use
those lengths to compute the length of l1. (Note that
Ellen defined l1, l2, and l3 in her original letter to be the
line and referred to the sides of the triangle in Figure 1
as the segments of those lines respectively. In Task 3,
Ellen mixes the notation and uses l1 to refer to the
segment associated with the line l1. We could not infer
that this caused Jacques confusion.
Jacques took advantage of Ellen’s questions and
applied the Pythagorean Theorem to the new situation
(see Figure 5). Because he used the procedure to
calculate a distance without clear prompting from
Ellen, we coded this interaction as having elicited
Application and Procedures with Connections. In other
words, we inferred from Jacques’ novel use of the
Pythagorean Theorem that he understood it beyond
rote computation and could use it flexibly in new
situations. He had developed a kind of efficacy in
using it, purposefully transforming information from
the coordinate pair (6,1) into information about side
lengths, a and b, of a right triangle. We also inferred
from those actions that Jacques had attempted to
communicate a mathematical idea (as indicated by his
writing at the top of Figure 5) and that he had broken
down (analyzed) the situation into constituent
mathematical ideas (x=6 and y=1).
Figure 3. Jacques’ response to Task 2.
34
Preservice Teachers’ Task Posing
Figure 4. Ellen’s third task for Jacques.
Discussion: Summary of Exchanges
Figure 5. Jacques’ response to Task 3.
Zachary Rutledge & Anderson Norton
Although not part of this particular report, Ellen
used the same problem stem in the next task with
similar results, in terms of elicited cognitive activity
(we inferred from his response Procedures with
Connections, Application, and Communication).
Jacques seemed to respond well affectively; he
commented in his response that, “this is really fun
doing this, u [sic] are making it very understandable.”
We inferred from the pair’s interactions that Jacques
had constructed ways of using procedures, like the
Pythagorean Theorem, that were connected to
meaningful concepts. He had also constructed a
tenuous grasp of formalized linear equations; that is, he
35
seemed able to generate only linear equations that went
through the origin. From such inferences, we argue that
Ellen had successfully engaged Jacques in a variety of
high-level processes such as Problem Solving and
Analysis. Across all four tasks in the sequence, Ellen
consistently elicited Application and mathematical
Communication from him. In addition, she and Jacques
engaged in activities that started as procedural, but
quickly progressed to and remained at Procedures with
Connections.
Extended Discussion from a Situated Perspective
Broadly speaking, a situated view of learning
would include what Wenger (1998) refers to as
apprenticeship forms of learning or ideas about
learning in communities of practice. These forms of
learning, as Lave (1997) states, “are likely to be based
on assumptions that knowing, thinking, and
understanding are generated in practice, in situations
whose specific characteristics are part of practice as it
unfolds” (p. 19). In other words, learning mathematics
is about learning the social practices of school
mathematics, often including the establishment of
norms about what constitutes appropriate mathematical
activity and mathematical learning.
Reconsidering the teacher-student interactions
from this point of view, we argue that two main issues
emerged during the task iterations. The first issue is
that the PST and student possessed compatible
understandings about their roles in relation to one
another concerning mathematical activity. In addition,
and non-trivially, they both agreed to take up these
roles. We will support this idea by showing that,
although the tasks seemed formal and lacking in reallife relevance, the student readily engaged with them.
The second issue is that that the way in which these
two took up their respective roles could be associated
with what it traditionally means to engage in
mathematical activity in the classroom. The PST reconstituted this in more and less obvious ways
throughout the exchanges. One example of how we
support this second idea is by showing that the PST
never incorporated any of the student’s personal
interests into the mathematics.
Considering the first issue, Lave (1997) contends
that practices in school can remove ownership of
mathematics from students. In other words, practices in
school mathematics classrooms encourage students to
learn the practices of schooling, which may not be the
same as the practices of mathematics. For example,
students may be inducted into the practice of
completing pre-designed steps in a problem from the
36
teacher. The practice then becomes the generation of
the steps on the part of the teacher and the working of
each step independently on the part of the students.
The overall mathematical meaning or goal may be lost
or, as Lave would contend, ownership of the problem
is taken from the students.
It is interesting to consider the task-posing
sequence between Ellen and Jacques at this level.
Using a situated perspective, we can see that Jacques
and Ellen were “on the same page”. Expectations about
what constitutes mathematical activity seemed agreed
upon by both participants. Specifically, the interactions
could be viewed, as Lave (1997) described above, as a
series of often procedural steps designed by the PST to
guide the student through this pre-conceived task, and
this constituted the agreed upon mathematical activity
in which the two engaged.
Furthermore, this kind of agreement could be
interpreted in several ways. For example, some
researchers refer to scripts and would argue that both
participants were using a dominant script for
communication (Gutierrez, Rymes, & Larson, 1995)
where the participants can be viewed as using the
standard way of talking and acting in certain situations.
In this case, it could be argued that Ellen adopted the
standard way of interacting with the student and
likewise that Jacques took up a typical way of
interacting with someone in an instructional role.
Similarly, others might argue that they were both
participating in the same or similar (big-D) Discourse
(Gee, 1999). In either case, this tacit agreement
between the members of the pair could be a strong
factor in determining the kinds of psychological
activities that we inferred from the exchanges. In
particular, had the student not agreed, it possibly would
have impacted the mathematical interactions, perhaps
leading to disengagement from either party.
As indicated earlier, the nature of the mathematics
was one that was likely disconnected from the
student’s personal experience in many ways. It was an
“abstract” situation that provided little intrinsic
motivation. In other words, the student could question
why there would ever be lines moving around to create
an isosceles triangle, like in Figure 1. This kind of
question from the student is important as it highlights
the nature of the mathematics in which we expect our
students to engage. It is not to say that such an activity
is necessarily “bad” or “good.” With this in mind, the
power of the teacher script is palpable as the student, in
good humor, engages with this task despite the lack of
introduction to the purpose behind the task and despite
the PST not providing any sort of motivation for it.
Preservice Teachers’ Task Posing
Figure 6. Jacques’ introductory letter to Ellen.
Considering the second issue, however, we see
something a little different. Although we hypothesize a
certain level of agreement between the two at the
Discourse level (or similarly, we could suggest that
they both are adhering to a dominant script about
school mathematics), we also hypothesize that the PST
demanded that the student speak the Discourse of
school mathematics. To support this claim, we consider
the initial introductory letter where Jacques stated “I
want to be both president of the United States and the
owner of my own fast food franchise chain.” He then
closes the letter by providing a great deal more
information about himself (see Figure 6). Here Jacques
shares that, among other things, he is interested in
science and works at a fast-food restaurant. However,
Ellen, for purposes of task selection, ignored these
facts. She picked a task that was what some may
consider “de-contextualized.” Yet, researchers such as
Lave (1996) would consider these kinds of tasks highly
contextualized in certain “socially, especially
politically, situated practices” (p. 155).
Further, by ignoring where the student was
“coming from” in this way, the PST potentially lost the
opportunity to establish a Third Space (Gutierrez et al.,
1995). It is in this space where student backgrounds
and interests can meet with teachers’ learning
objectives and provide for fruitful collaboration. In this
series of tasks, we could ask, “Could Ellen have
incorporated the student’s interest in business?” or,
“Could she have embedded the task in a political
context?”
Before concluding this section, it is important to
note that a situated perspective does not demand that
teachers include students’ personal information into
tasks. As cited above, some authors associated with the
situated perspective have questioned the value of
Zachary Rutledge & Anderson Norton
teaching what would traditionally be considered purely
“abstract” mathematics. By considering the example of
Ellen and Jacques, we show the degree to which Ellen
maintains her dedication to the “abstract” task. By
considering Jacques’ personal information (his home
life, cultural background, and interests), we show that
Ellen did have at least one other option for a type of
task besides an “abstract” one.
In her final letter, Ellen explicated a certain set of
values and ways of looking at mathematics, stating to
Jacques that he should be pleased with his
“perseverance” and that she hoped he had learned “to
approach a complicated problem as a series of smaller,
easier problems.” This view of mathematics
exemplifies what could be called the dominant
Discourse of traditional mathematics teaching. It is
what Lave (1997) might consider to be the kind of
practice that can remove ownership of the subject from
the student. Situated theorists such as Lave may argue
that it perpetuates a way of teaching mathematics that
can limit student agency and the role of the student in
generating new ideas—a practice that is not consistent
with the actual practice of research mathematicians
(Boaler & Greeno, 2000).
Coordination and Conclusion
This article has described two perspectives on the
same set of interactions between a PST and a high
school student. One details the individual at work, and
the other gives a “bigger picture” of the world in which
the individual operates. However, we encourage a
more careful consideration of how these two tracts of
analysis are related. For example, when Jacques
applied the Pythagorean Theorem to Task 3, we
explained his actions as an assimilation of the situation
into his conceptual understanding of the procedure
(von Glasersfeld, 1995). Alternatively, we might have
37
explained his actions as resulting from an identification
of common attributes across the new situation and
previous situations in which Jacques had used the
Pythagorean Theorem (Greeno, 1997). Both
explanations seem valid, and even compatible, as long
as we clarify issues involving the observer and points
of view.
We suggest that the two explanations are indeed
compatible if we attempt to adopt the student’s point of
view in both cases. Although there may be many
commonalities between the situation described in Task
1 and our observations of a student’s previous
experiences with the Pythagorean Theorem, our
observations are a poor substitute for the student’s
lived experience. Otherwise, we should have expected
Jacques to apply the Pythagorean Theorem in his
response to Task 1, as Ellen clearly expected him to do
(as indicated by her markings in Figure 1). This
necessitates the kind of inference we made about
Jacques’ actions; we have no access to students’ lived
experiences, and so we must make inferences based on
our observations, knowing full well that students’
points of view and, thus, students’ mathematics may be
quite different from our own. So, if we take “common
attributes” to describe commonalities from the
student’s point of view of the new situation and
previous experience, the situated perspective
complements a cognitive perspective.
Consider our operationalization of Problem
Solving as another example in which we might
reconcile perspectives. Problem Solving required that
the student lack a readily defined way of resolving the
task; we had to be able to infer that he experienced
some threshold of cognitive struggle. From a situated
perspective, we were not simply measuring whether
the student was able to deal with novel situations, but
we were also measuring the degree to which the
student had experienced these kinds of situations
before. In particular, if a student had experienced the
same situation many times before, then we would be
unlikely to assess this as Problem Solving (as the
solution/procedure would likely be generated
effortlessly by the student); however, we would still be
unlikely to assess Problem Solving if the student had
little experience with similar problems, as the student
would likely be unable to engage. The likely cases
where we would assess Problem Solving would be
between these two. It would have to be a case where,
from his point of view, the student had relatively
similar experiences, but yet different enough that we
would detect cognitive struggle. For example, with
Ellen’s second task, Jacques showed some familiarity
38
with the set-up (the part that called for linear
equations), yet he struggled with the novel parts of it
(the parts that required him to deduce the missing
points so that he could use a point-slope formula, for
instance).
This discussion of Problem Solving does offer
some instance of how these two perspectives could be
compatible; however, while analyzing this measure
along side the others, it became apparent to us that this
coordination also presented productive criticisms of the
two perspectives. For example, as mentioned above,
Lave (1997) states that certain practices in mathematics
can remove student agency—practices such as
breaking down problems into multiple steps. This
practice is a common practice amongst mathematicians
as described by Polya (1973). It is likely that Lave
indicated something more subtle than just the mere act
of breaking down a problem into steps; yet, it is not
clear how to interpret this statement in the situation
with Jacques. Was Ellen supporting the kind of
mathematics to which Lave was referring or was she
moving the student towards something that resembled
the practice of mathematicians like Polya?
On the other hand, the constructivist perspective
has a heritage of recursive model building with
subsequent refinements of these models (Steffe &
Thompson, 2000). Unfortunately, given the constraints
of our study as well as its purpose, we did not revise
our models through recursion. In other words, we
formed models of the students’ cognitive processes as
indicated by their written responses to tasks, but we did
not have opportunities to test and revise purposefully
those models through continuing interaction with the
students. We could have emulated recursive model
building by looking back through previous responses
from the student in an attempt to identify a consistent
way of operating across the tasks, but our methods
dictated that we assess each week’s responses
separately. Therefore we did not fully utilize the tools
available to the radical constructivist. This is a
weakness to our approach and one that became clear to
us as we compared other perspectives to our own. In
particular, our analysis of Problem Solving could have
looked substantially different if we had built a stable
model of Jacques’ mathematics. This would have been
a model we could likely have used to interpret his
response to the task more insightfully. Moreover, it is
not clear to us what kinds of model building heuristics
are available to researchers using such a perspective,
leading to a further divergence from the situated
perspective.
Preservice Teachers’ Task Posing
In conclusion, we have shown how one PST
elicited cognitive activity from a student over the
course of letter-writing exchanges. We have also
indicated how our measures could be viewed as
indications of prior experiences and dependent upon
the student’s comfort with certain norms associated
with traditional mathematics teaching. In addition,
considering mathematics as situated in larger sociocultural structures, we have been able to critique our
own analysis and ultimately suggest paths for further
exploration.
More generally, we have given a suggestive way in
which two competing lenses can be used to consider
data and create a conversation between two competing
paradigms. This conversation provided alternative
ways to view the same data, but also generated fruitful
criticisms of the approaches. These alternative ways of
viewing the data hinged largely on the perspective
adopted by the researchers in considering their data.
For example, if a situated perspective focuses upon the
opportunities presented in a certain task, then it
becomes an important issue as to who is identifying
these opportunities. If the opportunities are considered
to be from the learner’s perspective, then such a
perspective may have many pragmatic commonalities
with radical constructivism.
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1
In the case of the exchanges described here, the PST
built all subsequent tasks as modification of an original
problem (what we call “Task 1”), so in many ways, there
was only one primary task in these exchanges. This did not
always have to be the case however, and the PSTs were free
to change tasks completely between exchanges. For the sake
of coding and analysis, we called any mathematical request
from the PST a “task” regardless as to whether it was a
brand new task or a modification of a previous task. So, in
this paper, “Task X” means “the mathematical task request
made by the PST during week X.”
Kastberg, S. (2003). Using Bloom’s taxonomy as a framework for
classroom assessment. The Mathematics Teacher, 96, 402–
405.
Lave, J. (1996). Teaching, as learning, in practice. Mind, Culture,
and Activity, 3, 149–164.
Lave, J. (1997). The culture of acquisition and the practice of
understanding. In D. Kirshner & J. A. Whitson (Eds.), Situated
cognition: Social, semiotic, and psychological perspectives
(pp. 17–35). Mahwah, NJ: Lawrence Erlbaum Associates.
Zachary Rutledge & Anderson Norton
39
Appendix
Table A1
Descriptions of Cognitive Processes Described in Bloom’s Taxonomy
Cognitive Process
Short Definition
Application
Using previously learned information in new and concrete situations to solve problems
Analysis
Breaking down informational materials into their component parts so that the hierarchy of ideas is clear
Synthesis
Putting together elements and parts to form a whole
Evaluation
Judging the value of material and methods for given purposes
Note. From “Using Bloom’s Taxonomy as a Framework for Classroom Assessment,” by S. K. Kastberg, 2003, Mathematics Teacher, 96 (6), p.
403. Adapted with permission of the author.
Table A2
Description of Cognitive Processes Described by the NCTM Process Standards
Cognitive Process
Communication
Short Definition
Expressing mathematical ideas in words to clarify and share them, so that “ideas become objects of reflection” (p.
60)
Connections
Relating mathematical ideas to each other, and to previous experiences in other domains, such as science
Problem Solving
“Engaging in a task for which the solution method is not known in advance” (p. 52), which involves the use of
strategies in struggling toward a solution.
Reasoning and Proof
Making analytical arguments, including informal explanations and conjectures
Representation
The “process and product” (p. 67) of modeling mathematical ideas and information in some form, in order to
organize, record, and communicate.
Note. Summarized from Principles and Standards for School Mathematics (NCTM, 2000).
Table A3
Cognitive Processes Described by Smith, Stein, Henningsen, and Silver
Cognitive Process
Memorization
Procedures without
Connections
Procedures with
Connections
Doing Mathematics
Short Definition
Memorizing or reproducing “facts, rules, formulae, or definitions” (2000, p. 16) without any apparent connection
to underlying concepts
Using a procedure or algorithm that is implicitly or explicitly called for by the task, without any apparent
connection to underlying concepts
Using procedures to deepen understanding of underlying concepts
Investigating complex relationships within the task, its solution, and related concepts, often involving
metacognition, analysis, and problem solving
Note. These definitions are summarized from Implementing Standards-Based Mathematics Instruction (Stein et al., 2000).
40
Preservice Teachers’ Task Posing
CONFERENCES 2008-2009…
ICME11
International Congress on Mathematical Education
Monterrey, Mexico
July 6–13, 2008
Morelia, Michoacán,
Mexico
July 17–21,
2008
Denver, CO
August 3–7,
2008
Washington, DC
January 5–8,
2009
Orlando, FL
February 5–7,
2009
Rome, GA
March 5–7,
2009
San Diego, CA
April 13–17,
2009
Washington, DC
April 20–22,
2009
Washington, DC
April 22–25,
2009
Toronto, Canada
May 2009
http://www.icme11.org
PME-32
International Group for the Psychology of Mathematics Education
http://www.igpme.org
JSM of the ASA
Joint Statistical Meetings of the American Statistical Association
http://www.amstat.org/meetings/jsm/2008/
MAA-AMS
Joint Meeting of the Mathematical Association of America and the American
Mathematical Society
http://www.ams.org
AMTE
Association of Mathematics Teacher Educators
http://www.amte.net
RCML
Research Council on Mathematics Learning
http://www.unlv.edu/RCML/
AERA
American Education Research Association
http://www.aera.net
NCSM
National Council of Supervisors of Mathematics
http://www.ncsmonline.org/
NCTM
National Council of Teachers of Mathematics
http://www.nctm.org
CMESG
Canadian Mathematics Education Study Group
http://cmesg.math.ca
41
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In this Issue,
Guest Editorial… Ruminations on the Final Report of the National Mathematics Panel
CARLA MOLDAVAN
In-Service Middle and High School Mathematics Teachers: Geometric Reasoning and
Gender
ERDOGAN HALAT
Ninth Grade Students Studying the Movement of Fish to Learn about Linear
Relationships: The Use of Video-Based Analysis Software in Mathematics
Classrooms
S. ASLI ÖZGÜN-KOCA
Applying Piaget’s Theory of Cognitive Development to Mathematics Instruction
BOBBY OJOSE
Preservice Teachers’ Mathematical Task Posing: An Opportunity for Coordination of
Perspectives
ZACHARY RUTLEDGE & ANDERSON NORTON