Providing Meaningful Fieldwork for Preservice Mathematics Teachers: A College-School Collaboration

Providing Meaningful Fieldwork for Preservice Mathematics Teachers: A College-SchoolCollaborationAuthor(s): Frances R. Curcio, Alice F. Artzt and Merna PorterSource: The Mathematics Teacher, Vol. 98, No. 9 (MAY 2005), pp. 604-609Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27971824 .

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teacher education

Providing Meaningful

Fieldwork for Preservice

Mathematics Teachers:

A College-School Collaboration Frances R. Curcio and Alice F. Artzt, with Merna Porter

One of the greatest challenges for second

ary mathematics teacher-educators is

preparing future teachers to support re

form efforts that lead to high-quality teaching. In particular, careful lesson

planning, anticipation of student misconceptions, and constructive reflection on a lesson after instruction are

critical concerns not only for novice teachers but also for experienced teachers. One way to help preservice teachers begin to appreciate the importance of plan ning and reflecting entails college faculty collaborating with exemplary school teachers in integrating and con

necting learning theories with teaching practice. Teacher preparation programs have traditionally in

corporated fieldwork as a critical component in sup

porting the validity of recommended approaches. How

ever, finding field placements that support the

philosophy of reform-based teacher preparation pro grams is not always possible. In fact, recent evidence

suggests that incongruent field placements maybe counterproductive and damaging in developing open

minded attitudes toward reform among preservice teachers (Philipp et al. 2002). In agreement with Eisen

hart and her colleagues (1993), we believe that for

fieldwork to be most effective, it needs to take place in an environment in which the philosophy is aligned with that of the teacher preparation program. Also, fieldwork must be carefully designed to focus on par

604 MATHEMATICS TEACHER | Vol. 98, No. 9 ? May 2005



ticular aspects of mathematics instruction. Moreover,

since teaching is more than meets the eye, preservice teachers need guidance in interpreting what they ob serve and in understanding the decision-making processes of the classroom teacher. Although the main

purpose of this article is to describe a school-college col laboration in the design and implementation of preser vice fieldwork, we hope that this article will help read ers examine the importance of carefully preparing for

instruction, anticipating student misunderstandings, and reflecting on instruction after the lesson. A brief

description of the required preservice college course, details related to observing instruction with the advan

tage of getting into the mind of the teacher before and after the lesson, an explanation of unexpected out

comes, and college student testimony follow.

BRIEF DESCRIPTION OF THE COURSE Unlike traditional secondary mathematics teacher

preparation programs in which students do not take education courses until the first semester of their

junior year, we have instituted a course, the Psy chology of Learning Mathematics, which is offered in the first semester of the freshman year.

The course, designed by Alice Artzt and Eleanor

Armour-Thomas, and the supporting fieldwork de scribed in this article could be adapted for preservice teachers at any level, not just for freshmen. (More in formation about the secondary mathematics teacher

preparation program at Queens College is available at

www.qc.edu/time2000 and www.qc.edu/SEYS/ matheducation.) This experience early in a student's

college education serves many purposes. First, the course is designed to build on the college freshman's own past and present experiences in learning mathe matics. Through the students' own personal reflec tions on how they learn, the course develops the psy chological principles of learning mathematics. In turn,

by understanding these principles, the freshmen are able to improve their own learning in their often-de

manding and challenging college mathematics courses.

Furthermore, the student-centered approach to in

struction employed in this course serves as a model for these students to use while they reflect on the teach

ing and learning that they experience throughout their

undergraduate study. One of the most powerful com

ponents of this course?and the focus of this article? is the field-experience component, which was de

signed in collaboration with an in-service teacher. This early participation in schools helps the

freshmen begin to feel like professionals, develop a better understanding of the complexities of learning and teaching, witness how the theories that they learn in their college class are applied in the school

classroom, and reaffirm their commitment to the

teaching profession. The details of this fieldwork

experience follow.

OBSERVING MATHEMATICS INSTRUCTION One day each week for ten weeks, the freshmen visit a local urban public school that has a collabo rative relationship with the college. A mathematics education faculty member provides onsite support. For each of the ten visits, the freshmen attend?

a 45-minute session to discuss the elements and content of the carefully planned lesson, with the teacher describing her thinking about the content, her knowledge about her students, her expectations for the students, and her thoughts about anticipated student responses to the planned activities; a 45-minute class, during which the college stu dents observe the lesson while considering spe cific guidelines related to the weekly topic of discussion in the Psychology of Learning Mathe matics course; and a 45-minute follow-up that allows the freshmen to interact with the teacher while she debriefs and reflects on the lesson.

Before the lesson Before the first-year college students observe the les

son, the eighth-grade teacher, Merna Porter, meets with them to discuss the intended lesson and to re view her aim, objectives, and tasks. So that the dis cussion connects with their college assignment, Porter asks the freshmen to explain the week's topic for study in their Psychology of Learning Mathemat ics class. (The appendix lists all observation topics.)

Even though Porter asks the students to describe the topics for observation, she is fully aware of all the assignments because of her collaboration with the college professors. She has, in fact, kept up with the readings from Ormrod (2003) that have been

assigned to the college freshmen. Before discussing the lesson with Porter, the

freshmen work on the motivational mathematical

problem planned for the lesson that they will ob serve so that they can better understand the con

cepts to be developed. Porter encourages the stu dents to ask questions or contribute their own ideas. A discussion of eighth graders' possible solution

Vol. 98, No. 9 ? May 2005 | MATHEMATICS TEACHER 605



Porter said, "I have the algebra in the back of my head, and some of the students might be ready to move into it," but she reminded the college fresh men that the eighth graders had not yet studied

algebra. Sara thought that the eighth graders might think of the proportion 3/5 = n/40 and get 24. Porter asked the freshmen to think about what

might be confusing about this approach for some

eighth graders. Luis said that he might expect

eighth graders to wonder where the 5 comes from. Danielle thought that students might list

equivalent ratios, as follows:

2:3 8:12 14:21 4:6 10:15 16:24 6:9 12:18

and find an answ?r of 24 dogs. David noticed that Danielle's list could also include the sums 5,10, 15, 20,25, 30,35,40, to help the students see the total number of animals.

When Porter asked the freshmen to predict mis

conceptions that the eighth graders might have,

they were unable to do so. Porter said that she often thinks about typical mistakes and misconcep tions so she can anticipate them, be on the lookout for them, and offer ways to help her students avoid such pitfalls. Many times, for one reason or another, students do not read the problem or understand the relationship among the elements in a problem. In this problem, she suggested that some students

might try to create an incorrect proportion, 2/3 =

n/40. Porter said that she would try to help stu dents recognize the part-to-part relationship of 2:3 and the part-to-whole relationship of n:40 and ask whether equating these two ratios makes sense.

She cautioned the freshmen to avoid using terms that might not be familiar to the eighth graders and

encouraged the freshmen to help the eighth graders build their mathematical vocabularies.

Porter explained that she plans to encourage her students to use any approach that makes sense to them, as long as they can defend the mathematics that they are applying.

Vignette 1 Excerpt of prelesson discussion

strategies and misconceptions, as well as how they relate to the psychological principles under study, is an important element of the prelesson meeting.

When Porter greeted the first-year college stu

dents and asked them to describe the topic of the week for the Psychology of Learning Mathematics

class, one of the students, Rebecca, indicated that

they were expected to focus on problem solving during their observation, as well as observe how the

teacher taps into the eighth-grade students' higher order thinking. The freshmen were to write a re

port in which they described the types of mathe matics problems presented, the strategies students used to solve the problems, opportunities students

had to "learn how to learn," and how they assisted in developing the eighth graders' critical-thinking skills in their work with individual students.

Always mindful of tapping students' higher-order thinking, Porter stated that she usually begins a les son with a problem to provide a context for the

mathematics that she expects her students to learn or to review. On this day, she planned to give several

problems. The following introductory problem was

planned to review ratio and proportion, leading into a lesson on fraction-decimal-percent equivalences:

The ratio of cats to dogs in a shelter is 2 to 3. If you wish to adopt a dog and if 40 dogs and cats are at

the shelter, from how many dogs can you choose?

To help the college freshmen understand her

thinking about how she planned the lesson, Porter asked them first to solve the problem in as many ways as they could. Most of the freshmen thought of an algebraic solution, letting 2x represent the number of cats and 3x represent the number of

dogs; then solving the equation Sx = 40, so = 8 and the number of dogs is 24.

Vignette 1 describes the conversation that Porter had with the freshmen before the middle school students entered the classroom. This vignette

clearly shows that Porter is trying to encourage the

college freshmen to expand their own problem solving approaches by predicting the strategies that the middle school students will use and the prob lems that they might encounter.

Observing the lesson Before the eighth graders arrived, the approxi mately 25 college freshmen arranged their seats in the back and along the sides of the classroom. With notebooks in hand, they recorded the specific inter actions that support the guidelines set forth in the

their class. At certain points during the lesson, the

college freshmen walked around the room to ob serve the eighth-grade students working. On some

occasions, the teacher organized the class in small

groups, and the college freshmen were invited to in

teract with the students. Vignette 2 describes the

ideas that the eighth-grade students came up with when they tried solving the problem. The strategies that some of the college freshmen predicted the stu

dents would use were actually used. Also, the diffi

culties that Porter and some of the freshmen antici

pated that the eighth-grade students would have did

surface during the class problem-solving session.




After the lesson Once the lesson was over and the eighth graders had left the classroom, the teacher and the college fresh

men took about ten minutes to review their notes and collect their thoughts. Porter began the session

by sharing her reactions to the lesson and her stu dents' behaviors. Her reflecting aloud gave the fresh men insight into her thinking about her students'

learning and the follow-up approaches that she

planned to employ. Porter invited comments, ques tions, and suggestions from the freshmen, expecting them to use examples of student work and students' comments as their evidence. The freshmen usually questioned Porter about her thoughts regarding the

psychological concepts that they were studying in their college class.

On this day, Porter began the postlesson reflection

by describing her reaction to the work of the eighth grade students and noting that her prediction about what would be confusing to the students was correct:

As I walked around to see how the students were

handling the problem, I noticed that just as we had discussed before the lesson, some students set

up the proportion incorrectly as x/40 = 2/3.1 asked them questions to find out what they thought they could get by setting it up that way. They were saying, "Well, if there were two cats there were three dogs, so for cats there would be

forty dogs," so I led them through that reasoning and they realized that way was not the way to

proceed. Then I suggested that they work with a

ratio; I had to give that clue. But you can see from Caroline's work that they still found the part-to part versus the part-to-whole concept confusing.

As Porter spoke, the college freshmen realized that recognizing these relationships constitutes an

example of the type of higher-order thinking that

they had been discussing in their college class and were required to write about in their observation

report. Vignette 3 describes the postlesson com ments that the college freshmen made when they reflected on the lesson, giving further evidence that

The freshmen observed the excited and noisy eighth graders enter the classroom. Porter welcomed them and calmly helped them settle down, reminding them of the class routines and pointing to the problems on the chalkboard. The eighth graders responded respectfully and cooper atively and copied the problems into their notebooks and worked on them. Porter and the freshmen walked around the room, observing the

eighth graders while they worked. When Porter interacted with the

eighth graders, the freshmen took notes on what the students were

doing and how Porter reacted to the students and their work. Porter called on Rory and Caroline to put their work on the chalk

board to share with the class, so that she could highlight two solutions.

Rory wrote the following on the chalkboard and explained her thinking.

2:3 4:6

8:12 16:24 There are 24 dogs.

Caroline wrote the following on the chalkboard and explained her thinking:

2/5 = n/40 (2/5X8/8)= n/40

= 16 There are 16 dogs.

Most of the class agreed with Rory's answer but could not figure out what Caroline had done wrong. Porter used the opportunity to bring attention to the part-to-whole relationship and helped the class realize that 2:5 was the ratio of cats to the total number of animals. Caroline immediately recognized that she had found the number of cats rather than the number of dogs.

Porter encouraged the class to look more carefully at Rory's list of ra tios and asked whether any other number pairs could have been included.

Vignette 2 Excerpt of the lesson

they were able to examine the higher-order think

ing skills that this class was developing.

UNEXPECTED OUTCOMES After the college freshmen spent one period each week for ten weeks in the middle school classroom, the eighth graders and Porter wanted to visit the

Psychology of Leaning Mathematics class on the

college campus. Arrangements were made to ac

commodate their wishes. A special session was

planned so that the college freshmen and the

eighth-grade students could discuss the attributes of a great mathematics teacher. Before the session, the

college students were asked to predict what the

eighth-grade students would say. Their list included the following attributes: nice, fair, caring, explains things well, has patience, does not give much home

work, and gives good grades. The college freshmen were pleased to hear the

eighth graders express all the same characteristics, ex

cept for those related to homework and grades. The

eighth graders clearly understood the value of home work and the importance of earning grades that they deserved. Everyone agreed that Porter exemplifies all that they envision of a great mathematics teacher.

Vol. 98, No. 9 ? May 2005 | MATHEMATICS TEACHER 607



Danielle stated, "There were some students who were playing around with equivalent ratios?2:3,4:6, da-da-da, just like I had predicted. Some of them had found quite a few equivalent ratios, but they didn't quite know when to stop. So Rory came up to me, and she had the solution. I

asked her how she knew when to stop. She explained that in the first

ratio, 2:3, there are five animals represented, and in the 16:24 ratio there are forty animals represented, so she knew that she had to stop there. But I noticed that she only considered some of the ratios." Porter pointed out, "Students typically feel comfortable with doubles and have greater difficulty when dealing with other multiples." David noticed that a few stu dents created charts, and others had written out the ratios in order.

Kristen asked whether any students had tried to solve the problem

algebraically. None of the freshmen saw any evidence of algebraic methods. Porter said that an algebraic solution is expected later on.

Jenny asked, "When you begin algebra work, will you bring this back and have the students use algebra to solve it?"

Porter responded, "Yes, right, because we are at a prealgebra stage and we are phasing them into algebra right now. They should be able to reason

out a problem like this one by using proportions. We will return to a problem like it later on. We will come back to it and say, 'Here is how we solved it be fore. This approach is another one. How do the two methods compare?'"

Vignette 3 Excerpt of postlesson reflection

COLLEGE STUDENT TESTIMONY The following typical reaction written in a college freshman's journal convinces us of the value of this

experience.

Being able to observe a math class was really great and unlike anything I've experienced before. To see

all those theories and principles come to life was

amazing. Seeing them in a real-life situation was

helpful in showing us that these theories really do have practical applications, and they're not just something you read about in a textbook. And to have such an accommodating teacher discuss the lesson with us beforehand and afterward was a real treat. She would tell us her expectations for the les son?how she thought the students would grasp the concept and what difficulties she was anticipat ing. I found that pretty helpful, especially when the

things she predicted came true! But it was the post lesson discussion that was the most helpful. We

were able to discuss with the teacher what we ob

served, and she gave us her own reflection on the lesson. She addressed many of the questions that

we were required to answer in our observation re

port, and so we were able to take her view into ac count when we were writing it. She also pointed out to us some important things that happened that we might have failed to pick up on.

CLOSING COMMENTS Having experienced collaboration, in-depth plan ning, observation, focus on student thinking, and

reflection, the preservice teachers learned about the

power of a different culture of teaching. The preser vice teachers, many of whom are products of tradi tional lecture methods of instruction in high school, experienced reform-based teaching and student centered instruction supportive of the approach es

poused by the college faculty. As a result, they be came aware of what is expected of them as they embark on the journey toward becoming secondary mathematics teachers. Furthermore, the qualities of the observations and discussions that allow partici pants to focus on the teachers' thoughts behind their instruction provide a natural setting for mean

ingful fieldwork that bridges theory and practice. We are not familiar with any other secondary

mathematics teacher-preparation program that offers a course for college freshmen that is similar to Psy chology of Learning Mathematics and that integrates a field component collaboratively designed and imple mented with an exemplary school teacher. The au

thors are very interested in hearing about such efforts and welcome an interchange of ideas with readers.

REFERENCES Eisenhart, Margaret, Hilda Borko, Robert Underhill,

Catherine Brown, Doug Jones, and Patricia C. Agard. "Conceptual Knowledge Falls through the Cracks:

Complexities of Learning to Teach Mathematics for

Understanding." Journal for Research in Mathematics

Education 24 (January 1993): 8-40.

Ormrod, Jeanne Ellis. Educational Psychology: Developing Learners. 4th ed. Upper Saddle River, N.J.: Merrill Prentice Hall, 2003.

Philipp, Randy, Lisa Clement, Jennifer Chauvot,

Cheryl Vincent, and Eva Thanheiser. Mathematical Early Field Experiences for Preservice

Elementary School Teachers: Promoting Change or

Confirming Tradition? Paper presented at the research presession of the 80th annual meeting of the National Council of Teachers of Mathematics, Las Vegas, April 21, 2002. oo

FRAN CURCIO, Frances_Curcio@ qc.edu, and ALICE ARTZT, QCArtzt

d>aol.com, teach at Queens College of the City University of New York, Flushing, NY 11367. They are currently conducting a longitudinal study ana

lyzing the development of secondary mathematics teachers from their freshman year in college into their first three years as teachers. MERNA

PORTER, [email protected], teach es eighth-grade mathematics at the

Louis Armstrong Middle School, East Elmhurst, NY 11369. Her instructional strategies embody a

reform-based, student-centered philosophy.




APPENDIX: FIELDWORK ASSIGNMENTS 1. Role of behaviorism in learning

a. Describe how the teacher uses behav

iorist principles to help the students

learn the content.

b. Describe how the teacher uses behav

iorist principles to help the students

learn classroom rules and procedures. c. If you are working with an individual

or a small group of students, describe

ideas from behaviorist perspectives of

learning that you are using to help the

students learn.

d. Give your reactions to the effectiveness

of the behaviorist approaches that you ob

served or used. Did they work better with

some students than with others? Explain. 2. Role of cognitive development in learn

ing: information processing and memory a. What role did memory play in the stu

dents' learning of the new concepts dis

cussed during the class?

b. Describe the strategies that the teacher

used to help the students remember the

concepts discussed during the class.

c. If you are working with an individual

or a small group of students, describe the

strategies that you used to help the stu

dents remember the new concepts that

they were learning. d. Give your reactions to the effectiveness

of the memory strategies that you ob


some students than with others? Explain. 3. Learning concepts

a. Describe one of the concepts that stu

dents were required to learn during this

lesson.

b. Describe the defining attributes of the

concept. c. Give positive and negative examples of

this concept. d. Give an example of an irrelevant di

mension of the concept. e. Give a schematic drawing of the con

cept with examples at different hierarchi

cal levels.

/ To what extent did the teacher incorpo rate these elements of concept learning

during the lesson? Explain.

g. With regard to the students that you are tutoring or with whom you aje work

ing in small groups, how do you use the

elements of concept learning to facilitate

students' understanding? h. Give your reactions to the effectiveness

of these elements on students' learning of

concepts. Do they appear to have different

effects on different students? Explain. 4. Role of cognitive development in learn

ing: stages

a. Describe the tasks that the students

were engaged in during class.

b. Identify the level of cognitive develop ment required for these tasks (for example,

pTeoperational, concrete operational^ formal

operational). Justify your categorization. c. For those of you who are observing a

geometry class, identify the van Hiele

level required of the tasks and justify your categorization. d. With regard to the students that you are observing or tutoring, what ideas

from cognitive development are being considered to help the students learn?

e. Give your reactions to the appropriate ness of the level of cognitive development

required for the task for the specific stu

dents you observed or with whom you worked. Was it less or more appropriate for different students? Explain. Role of cognitive development in learn

ing: social interaction

a. Describe the nature of the dialogue be

tween the teacher and the students and

between the students themselves that

took place during the lesson.

b. To what extent were intersubjectivity or

scaffolding evident in the dialogue? Give

specific examples to justify your claim.

c. To what extent were efforts made to

work within the students' zone of pf?xi mal development? Describe.

d: With regard to the students that you are tutoring or with whom you work in

small groups, howT are you using the con

cepts of intersubjectivity, scaffolding, and zone of proximal development to fa

cilitate student learning? e. Give your reactions to the effectiveness

of these concepts on student learning. Do

they appear to have different effects on

different students? Explain.

Learning through exposition a. Describe how the teacher uses strate

gies of exposition to help the students

learn the content.

b. If you are working with an individual

or a small group of students, describe any

strategies of exposition that you are

using to help the students learn. c. Give your reactions to the effectiveness

of these expository strategies that you ob


some students than with others? Explain. Role of constructivism in learning a. Describe any incidences in the class

when students were given the opportu

nity to create or develop a concept. b. What kinds of tasks actively engaged the

students in their learning of the concepts?

c. With regard to the student whom you are tutoring, what ideas from the con

structivist perspective are you using to

help the student learn?

d. Give your reactions to the effective

ness of the consrructivist approaches that

you observed or used. Did they work bet

ter with some students than with others?

Explain. 8. Role of higher-order thinking skills in

learning a. In the class that you observed, how

would you describe the nature of the

problems students were engaged in (well

defined or ill-defined)? Explain. b. What strategies were students ex

pected to use in solving the problems (al

gorithms or heuristics)? Explain. c. What metacognitive strategies were

students expected to use in solving the

problems during class?

d. What opportunities were students

given to learn "learning to learn" skills?

If opportunities were not provided, how

could they have been?

e. In your own work with an individual

or a small group of students, describe

what you are doing to develop the critical

thinking skills of your students.

9. Role of personal/social development in

learning a. Describe any incidences that occurred

during the class that may affect how the

students feel about themselves as learn

ers (for example, any disparaging re

marks or cotnpHments from either the

teacher or peers). b. With regard to the student that you are ttitoring, what ideas from

personal/social development are you

using to help the student learn? c. Give your reactions to the effectiveness of

the strategies that you have seen used or

that you have used yourself from

personal/social development Did they work better with some students than with

others? Explain. 10. Role of motivation in learning

a. What intrinsic motivations did you no

tice during classroom instruction?

b. What extrinsic motivations did you no

tice during classroom instruction?

c. With regard to the individual or small

group of students with whom you are

working, what motivational strategies did you use to help the students learn?

d. Give your reactions to the effectiveness of

the motivational strategies that you observed

or used. Did they work better with some

students than with others? Explain.

Ol. 9S, No. 9 ? May 2005 | MATHEMATICS TEACHER 609



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Providing Meaningful Fieldwork for Preservice Mathematics Teachers: A College-School Collaboration