v18n1

____ THE ______MATHEMATICS___

_________EDUCATOR _____ Volume 18 Number 1

Summer 2008 MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff Editors Kelly W. Edenfield Ryan Fox 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

A Note from the Editor Dear TME Reader,

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 Ozgun-Koca’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 [email protected] The University of Georgia www.coe.uga.edu/tme Athens, GA 30602-7124

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

2 Ruminations on National Mathematics Panel

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 in-service 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.

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 high-stakes 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 seventh-

Carla Moldavan 3

Table 1

Mathematics Criterion-Referenced Competency Test Data Group Does Not Meet

Standards Meets Standards Exceeds Standards

2005 – 2006 7th Grade Georgia 19% 58% 23% 2005 – 2006 Black 7th Grade Georgia 2005 – 2006 White 7th Grade Georgia

28% 11%

61% 57%

10% 32%

2005 – 2006 7th Grade Georgia Economically Disadvantaged 2005 – 2006 7th Grade Georgia Not Economically Disadvantaged

28% 11%

62% 55%

11% 34%

2006 – 2007 SMS 7th Grade 2005 – 2006 SMS 7th Grade 2004 – 2005 SMS 7th Grade

34% 26% 32%

58% 56% 61%

7% 18% 7%

2006 – 2007 SMS Black 7th Grade 2005 – 2006 SMS Black 7th Grade 2004 – 2005 SMS Black 7th Grade

32% 20% 32%

64% 64% 61%

4% 16% 6%

2006 – 2007 SMS 7th Grade Economically Disadvantaged 2005 – 2006 SMS 7th Grade Economically Disadvantaged 2004 – 2005 SMS 7th Grade Economically Disadvantaged

34% 27% 38%

56% 58% 56%

10% 15% 7%

2006 – 2007 SMS 7th Grade Not Economically Disadvantaged 2005 – 2006 SMS 7th Grade Not Economically Disadvantaged 2004 – 2005 SMS 7th Grade Not Economically Disadvantaged

36% 23% 21%

62% 50% 71%

2% 27% 8%

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

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

Ruminations on National Mathematics Panel 4

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 pre-service 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

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

Carla Moldavan 5

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 seventh-grader 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 step-father 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,

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 seventh- graders’ 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.

Ruminations on National Mathematics Panel 6

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 seventh-grade 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

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.

Carla Moldavan 7

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

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

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8 Mathematics Teachers’ van Hiele Levels

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).

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 pre-service 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.

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.

Erdogan Halat 9

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

10 Mathematics Teachers’ van Hiele Levels

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 level-II (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

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 pre-service 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

Erdogan Halat 11

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 in-service 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

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

Mathematics Teachers’ van Hiele Levels 12

Table 1

Frequency Table for In-service Middle and High School Math Teachers’ van Hiele Levels Groups N Level-I

(Visualization) Level-II

(Analysis) Level-III

(Ordering) Level-IV

(Deduction) Level-V (Rigor)

n % n % n % n % n % A 110 19 17.3 19 17.3 54 49.1 12 10.9 6 5.5 B 38 0 0 14 36.8 18 47.4 3 7.9 3 7.9

Total 148

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

Descriptive Statistics and the Independent Samples t-Test for the In-service Teachers’ van Hiele Levels Group N van Hiele Geometry Test

Mean SD SE df t p A 110 2.70 1.05 0.10 146 -0.88 0.37 B 38 2.87 0.87 0.14

Total 148 Note. A – In-service middle school mathematics teachers, B – In-service high school mathematics teachers.

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 Independent Samples t-Test for the In-service Mathematics Teachers’ van Hiele Levels Group N van Hiele Geometry Test

Mean SD SE df t p Male 80 2.88 0.97 0.10 146 1.73 0.086

Female 68 2.59 1.04 0.12 Total 148

Erdogan Halat 13

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

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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Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17, 31–48.

Chappell, M. F. (2003). Keeping mathematics front and center: Reaction to middle-grades curriculum projects research. In S. L. Senk & D. R. Thompson (Eds.), Standards-based school mathematics curricula. What are they? What do students learn? (pp. 285–298). Mahwah, NJ: Lawrence Erlbaum Associates.

Crowley, M. (1987). The van Hiele model of development of geometric thought. In M. M. Lindquist, (Ed.), Learning and teaching geometry, K-12 (pp.1–16). Reston, VA: National Council of Teachers of Mathematics.

Duatepe, A. (2000). An investigation of the relationship between van Hiele geometric level of thinking and demographic variables for pre-service elementary school teachers. Unpublished Masters’ Thesis, Middle East Technical University.

Durmuş, S., Toluk, Z., & Olkun, S. (2002). Sınıf öğretmenliği ve matematik öğretmenliği öğrencilerinin geometrik düşünme düzeyleri. Orta Doğu Teknik Üniversitesi’nce düzenlenen 5. Ulusal Fen Bilimleri ve Matematik eğitimi Kongresi’nde sunulmuş bildiri, 16-18 Eylül: ODTÜ, Ankara.

Ethington, C. A. (1992). Gender differences in a psychological model of mathematics achievement. Journal for Research in Mathematics Education, 23, 166–181.

Fan, X., & Chen, M. (1997). Gender differences in mathematics achievement: Findings from the National Education Longitudinal Study of 1988. Journal of Experimental Education, 65(3), 229–242.

Fennema, E., & Hart, L. E. (1994). Gender and the JRME. Journal for Research in Mathematics Education, 25, 648–659.

Forgasız, H. (2005). Gender and mathematics: Re-igniting the debate. Mathematics Education Research Journal, 17(1), 1–2.

Fox, L., & Cohn, S. (1980). Sex differences in the development of precious mathematical talent. In L. Fox, L.A. Brody, & D. Tobin (Eds.), Women and the mathematical mystique. Baltimore, MD: Johns Hopkins University Press.

Mathematics Teachers’ van Hiele Levels 14

Friedman, L. (1994). Visualization in mathematics: Spatial reasoning skill and gender differences. In D. Kirshner (Ed.), Proceedings of the Sixteenth Annual Meeting North American Chapter of the International Group for the Psychology of Mathematics Education, (Vol.1, pp. 211–217). Baton Rouge: Louisiana State University.

Fuys, D., Geddes, D., & Tischler, R. (1988). The Van Hiele model of thinking in geometry among adolescents. (Journal for Research in Mathematics Education Monograph No. 3). Reston, VA: National Council of Teachers of Mathematics.

Gay, L. R. (1996). Educational research: Competencies for analysis and application (5th ed.). Upper Saddle River, NJ: Merrill Prentice Hall.

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Grossman, H., & Grossman, S. H. (1994). Gender issues in education. Needham Heights, MA: Allyn & Bacon.

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Halat, E. (2006). Sex-related differences in the acquisition of the van Hiele levels and motivation in learning geometry. Asia Pacific Education Review, 7(2), 173–183.

Halat, E. (2007). Reform-based curriculum & acquisition of the levels. Eurasia Journal of Mathematics, Science and Technology Education, 3(1), 41–49.

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Knight, K. C. (2006). An investigation into the change in the van Hiele level of understanding geometry of pre-service elementary and secondary mathematics teachers. Unpublished Masters Thesis,. University of Maine.

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S. Asli Özgün-Koca 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

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

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.

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.

Video-Based Analysis Software 16

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

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,


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.

• 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 multiply represent their concepts. (pp. 159–161)

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;


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

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


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

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 multiple-representational 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


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.

Her graphs are switched. “As time went, [the grey fish] started farther away and it got closer and closer and the striped fish started out really close and it got farther and farther away.”

“As the time increase, both of the fish’s distance increases actually. They both increase.”

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.

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.

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].”

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 computer-generated 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 semi-linked 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 computer-produced representation (see Table 1).

In the next task, students were asked to identify and describe the point on the graph that represents


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 will decrease. The striped fish’s will

increase.” “The gray fish’s distance is going to decrease. The striped fish’s

distance will increase because it is going away from the starting point.” 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

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].

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.

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.”

“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.

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

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


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

slope” What about the y-intercepts?

“I do not know”

He thought that the striped fish would have positive slope and the grey fish would have negative slope. Predictions for the y-intercept were not clear.

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, “because one keeps getting closer to the point and the other one keeps

getting farther away.”

“The striped fish went away…so it [the distance] increased that has positive [slope] but the grey fish’s distance decreased; that has

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

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


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

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


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.

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26 Applying Piaget’s Theory

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 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 stage-appropriate 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.

Underlying Assumptions 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).

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.

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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.

Bobby Ojose 27

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

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,


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

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

Bobby Ojose 29

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

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 Anderson, J. R. (1990). Cognitive psychology and its implications

(3rd ed.). New York: Freeman. Berk, L. E. (1997). Child development (4th ed.). Needham Heights,

MA: Allyn & Bacon. Burns, M., & Silbey, R. (2000). So you have to teach math? Sound

advice for K-6 teachers. Sausalito, CA: Math Solutions Publications.

Eggen, P. D., & Kauchak, D. P. (2000). Educational psychology: Windows on classrooms (5th ed.). Upper Saddle River, NJ: Prentice Hall.

Fuson, K. C. (1988). Children’s counting and concepts of numbers. New York: Springer.

Gelman, R., Meck, E., & Merkin, S. (1986). Young children’s numerical competence. Cognitive Development, 1, 1–29.

Johnson-Laird, P. N. (1999). Deductive reasoning. Annual Review of Psychology, 50, 109–135.

Kamii, C. (1982). Number in preschool and kindergarten: Educational implications of Piaget’s theory. Washington, DC: National Association for the Education of Young Children.

Martin, D. J. (2000). Elementary science methods: A constructivist approach (2nd ed.). Belmont, CA: Wadsworth.

Papila, D. E., & Olds, S. W. (1996). A child’s world: Infancy through adolescence (7th ed.). New York: McGraw-Hill.

Piaget, J. (1970). Science of education and the psychology of the child. New York: Viking.

Piaget, J. (1977). Epistemology and psychology of functions. Dordrecht, Netherlands: D. Reidel Publishing Company.

Thompson, C. S. (1990). Place value and larger numbers. In J. N. Payne (Ed.), Mathematics for young children (pp. 89–108). Reston, VA: National Council of Teachers of Mathematics.

Thurstone, L. L. (1970). Attitudes can be measured. In G. F. Summers (Ed.), Attitude measurement (pp. 127–141). Chicago: Rand McNally

Weinert, F. E., & Helmke, A. (1998). The neglected role of individual differences in theoretical models of cognitive development. Learning and Instruction, 8, 309–324.

Wise, S. L. (1985). The development and validity of a scale measuring attitudes toward statistics. Educational and Psychological Measurement, 45, 401–405

Zimmerman, B. J., & Whitehurst, G. J. (1979). Structure and function: A comparison of two views of the development of language and cognition. In G. J. Whitehurst and B. J. Zimmerman (Eds.), The functions of language and cognition (pp. 1–22). New York: Academic Press.

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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. Greenfield, E. (1989). Aaron and Gayla’s counting book. Boston: Houghton Mifflin. 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. Lobal, A. (1970). Frog and toad are friends. New York: Harper-Collins. 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. Myller, R. (1990). How big is a foot? New York: Dell. Norton, M. (1953). The borrowers. New York: Harcourt Brace. Parker, T. (1984). In one day. Boston: Houghton Mifflin. 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.


Zachary Rutledge & Anderson Norton 31

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 letter-writing 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

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

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.

32 Preservice Teachers’ Task Posing

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

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


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

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.


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 4. Ellen’s third task for Jacques.


Discussion: Summary of Exchanges 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


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 real-life 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 re-constituted 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

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.


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

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


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

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.


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 socio- cultural 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.

References Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing

in mathematical worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 45–82). Stamford, CT: Ablex.

Cobb, P. (2007). Putting philosophy to work: Coping with multiple theoretical perspectives. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning. Greenwich, CT: Information Age Publishing.

Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in preservice teachers' practices. Educational Studies in Mathematics, 52, 243–270.

Gee, J. P. (1999). An introduction to discourse analysis: Theory and method. London: Routledge.

Greeno, J. G. (1997). Response: On claims that answer the wrong questions. Educational Researcher, 26(1), 5–17.

Gutierrez, K., Rymes, B., & Larson, J. (1995). Script, counterscript, and underlife in the classroom: James Brown versus Brown v. Board of Education. Harvard Educational Review, 65, 444–471.

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.

Lester, F. K., Jr. (2005). On the theoretical, conceptual, and philosophical foundations for research in mathematics education. Zentralblatt fur Didaktik der Mathematik: International Reviews on Mathematical Education, 37, 457–467.

Lester, F., & Kehle, P. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. Lesh & H. M. Doerr (Eds.), Beyond Constructivism (pp. 501–518). Mahwah, NJ: Lawrence Erlbaum Associates.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

Norton, A., & Rutledge Z. (2008). Measuring responses to task-posing cycles: Mathematical letter writing between algebra students and preservice teachers. Manuscript submitted for publication.

Polya, G. (1973). How to solve it. Princeton, NJ: Princeton University Press.

Schifter, D. (1996). A constructivist perspective on teaching and learning mathematics. In C. T. Fosnot (Ed.), Constructivism: Theory, perspectives, and practice (pp. 73–80). New York: Teachers College Press.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing Standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.

Steffe, L., & Thompson, P. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelley & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Mahwah, NJ: Lawrence Erlbaum Associates.

von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London: The Falmer Press.

Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge, UK: Cambridge University Press.

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.”


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 Short Definition Communication 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 Short Definition Memorization Memorizing or reproducing “facts, rules, formulae, or definitions” (2000, p. 16) without any apparent connection

to underlying concepts Procedures without Connections

Using a procedure or algorithm that is implicitly or explicitly called for by the task, without any apparent connection to underlying concepts

Procedures with Connections

Using procedures to deepen understanding of underlying concepts

Doing Mathematics 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).

41

CONFERENCES 2008-2009…

ICME11 International Congress on Mathematical Education http://www.icme11.org

Monterrey, Mexico July 6–13, 2008

PME-32 International Group for the Psychology of Mathematics Education http://www.igpme.org

Morelia, Michoacán, Mexico

July 17–21, 2008

JSM of the ASA Joint Statistical Meetings of the American Statistical Association http://www.amstat.org/meetings/jsm/2008/

Denver, CO August 3–7, 2008

MAA-AMS Joint Meeting of the Mathematical Association of America and the American Mathematical Society http://www.ams.org

Washington, DC January 5–8, 2009

AMTE Association of Mathematics Teacher Educators http://www.amte.net

Orlando, FL February 5–7, 2009

RCML Research Council on Mathematics Learning http://www.unlv.edu/RCML/

Rome, GA March 5–7, 2009

AERA American Education Research Association http://www.aera.net

San Diego, CA April 13–17, 2009

NCSM National Council of Supervisors of Mathematics http://www.ncsmonline.org/

Washington, DC April 20–22, 2009

NCTM National Council of Teachers of Mathematics http://www.nctm.org

Washington, DC April 22–25, 2009

CMESG Canadian Mathematics Education Study Group http://cmesg.math.ca

Toronto, Canada May 2009

42

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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

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