A mathematics laboratory for prospective teachers

A mathematics laboratory for prospective teachersAuthor(s): DAVID M. CLARKSONSource: The Arithmetic Teacher, Vol. 17, No. 1 (JANUARY 1970), pp. 75-78Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186129 .

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Forum on teacher preparation Francis /. Mueller

A mathematics laboratory for prospective teachers DAVID M. CLARKSON

Syracuse University, Syracuse, New York

David Clarkson recounts one of his projects as a visiting lecturer at the State University College at New Paltz, New York. At the present, he is associate director of the Madison Project and is teaching at Syracuse University.

A he Report of the Cambridge Conference on the Correlation of Science and Mathe- matics in the Schools recommends that schools of education plan programs of "apprentice teaching in the schools, includ- ing work with materials of the sort being developed in new curriculum projects."1 A group of mathematics educators in Eng- land has urged the use of courses empha- sizing problem solving: "It is the explora- tion of these more open problems which we feel to be the essential characteristic of real mathematical activity."2 A loud chorus of opinion suggests that courses in methodology should be jointly planned and executed by both mathematicians and educators and that they should involve practical work with children. When the opportunity to design an experimental elementary mathematics methods course was offered the writer, he decided to em- phasize the mathematics laboratory ap- proach which gives an important role to

1 Cambridge Conference, Goals for the Correlation of Elementary Science and Mathematics (Boston: Houghton Mifflin Co. for Education Development Center, 1969).

2 Mathematics Section of the Association of Teach- ers in Colleges and Departments of Education, Teaching Mathematics, Main Courses in Colleges of Education (London: A.T.C.D.E., 1967).

problem solving. Conferences with mem- bers of the mathematics and education departments, as well as with school offi- cials, paved the way for the experiment; the sympathetic support of the chairman of the division of education at the college made it possible financially.

A few teachers in neighboring school districts had been using activity methods in their classes. Because there were not enough of them to provide an in-school laboratory for the course sections, it was decided to bring children onto the campus. A block of time - most of a morning - - was reserved in the college schedule so that every college student would have a substantial experience working with a child. The children came from four schools, in- cluding a residential school for delinquents. Their ages ranged from five to twelve years. Each child, and each college student, had an opportunity to participate in from five to ten laboratory sessions during the semes- ter. No attempt was made to create a typi- cal elementary classroom scene; rather, about sixty children and an equal number of students worked together in three or four college classrooms, halls, and out- doors. The sessions constituted about one- third of the work of the college course.

Excellence in Mathematics Education - For All 75

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Theoretical discussions in the college course were always closely followed by practical work with materials in class, and subsequently with children. Because of time limitations many topics could not be included. Students were encouraged to develop their own interests and to pursue topics in depth rather than to survey the field superficially. Behind this decision was the belief that if a student has really worked out the problem of discussing measurement with a child, say, he may develop skills and approaches that will enable him to do a similar job with many other topics of elementary mathematics. If a good method of attack is gained by several such experiences it should be appli- cable to many more. In any event, there simply isn't enough time in a semester course to cover even a majority of the current topics.

The mathematics laboratory approaches that formed the focus of the course have only recently come into vogue in the United States. While they have some roots in the developmental psychology of Piaget and others, they have also developed in response to the heuristic, as opposed to the formalist, school of mathematics educators. One of the best statements of this view- point may be found in the book Freedom to Learn? A part of the early meetings of the course was devoted to a discussion of Piagetian tests, particularly those related to the idea of conservation. The college students then had opportunities to admin- ister some of these tests to children in the laboratory setting and to look critically at them. A few became so fascinated with the results that they devoted part of their vacation time to testing children in their neighborhoods, and several worked up major reports on this aspect of the course. A video-tape recorder was used during laboratory periods to record some of this activity so that it could be shared with the rest of the students later.

Some mathematics educators are wor- ried that an excessive emphasis on mate- rials - messing about with "things" - will detract from the development of mathema- i tical content in the laboratory situation. Some of the activities, such as playing with "Tangrams," may seem to bear only a trivial relation to the study of serious mathematics. A major effort of the course J was to relate the activities to significant mathematics, but this was done in the informal context of the laboratory. A vari- ety of texts was used; the Nuffield Guides and other recent publications were particu- larly helpful.4' 5 Students in the course kept logs of each session at which they worked with children, and these were read and commented upon by the teacher. For pur- poses of evaluation, the logs proved to be even more valuable than the usual "proj- ects." Motivation of both students and chil- dren was extremely high and was main- tained throughout the course. The following selection of excerpts from student logs conveys some of its flavor:

One girl and I took a yardstick and we mea- sured anything that she wanted in the building. She liked to measure long things because then she could add the numbers. It was interesting to note that she liked to add the numbers but she was too busy to write them out on a chart. She enjoyed the chance to move about the building freely but disliked the idea of sitting down and working with the information she had acquired. But then this little girl noticed that another child was putting a chart up for display and she wanted me to make one for her. We com- promised, and I wrote the words while she com- pleted the mathematical details. When she was done she really seemed to get enjoyment because she could see the comparison between the [mea- sure of the] door size and the water fountain.

In the lab I noticed a child's response to the free atmosphere. Three boys were working with tangrams. Two of the boys were very persevering in their attempt to form a square and worked at it for more than half an hour even though they could easily have chosen something else. The third boy just could not catch on to the tangrams and without any embarrassment or feeling of failure he was able to go to another activity.

76 The Arithmetic Teacher /January 1970

s Edith E. Biggs and James R. MacLean, Freedom to Learn: An Active Learning Approach to Mathe- matics (Reading, Mass.: Addison-Wesley, 1969).

4 Nuffield Mathematics Project, Guides (New York: John Wiley & Sons, 1968).

s Association of Teachers of Mathematics, Notes on Mathematics in Primary Schools (New York: Cambridge University Press, 1967).

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Some bright eighth grade boys wanted to make graphs and, since I was free, I agreed to help them although I don't know much about graph- ing. We went to a building on the campus which has an elevator and used a stop watch to collect data on its motion. The boys thought of col- lecting data that I would have ignored. For ex- ample, they compared the time it took to get from floors 2 to 3 with the time it took to go from 3 to 2. We were all surprised to find it takes longer to go down than it does to go up. Afterwards they made graphs of the data and were perplexed to find that only insignificant differences showed up. The next session they experimented with changing the scales on the graphs and found that this could make differ- ences appear significant. By letting them take the lead I learned a lot!

The content of the course and the lab- oratory sessions accompanying it was eclec- tic and open-ended. Structural materials such as Cuisenaire rods, multibase blocks, attribute blocks, and student-made ma- terials were available, as were a number of suggestions for activity cards. Much work was done with nailboards and shapes. Graphing activity was everywhere in evi- dence. Balances, tape measures, stop- watches, and other measuring instruments, including a spate of homemade trundle wheels, got extensive play. Some of this equipment was expensive, but many cheaper substitutes could have been made had there been more planning time.6 The total cost of a well-equipped mathematics laboratory at the elementary level should be less than $500, and can be considerably less. Fur- thermore, most of the expensive items are permanent acquisitions. Students were en- couraged to make their own materials and, where appropriate, to involve the children in this also.

Five basic content areas were developed: graphing, measurement, geometrical rela- tions, number patterns, and reasoning. Of course, these areas overlapped. For ex- ample, in the work with graphs, students and children progressed from simple charts based on counting (histograms, etc.) to

6 Patricia S. Davidson, "An Annotated Bibliog- raphy of Suggested Manipulative Devices," The Arithmetic Teacher 15 (October 1968): 509-24.

empirically derived graphs (spring stretch, ball bounce, etc.) to graphs of functions (guess my rule, etc.). Questions of inter- polation and extrapolation were raised, as well as simple concepts of analytic geom- etry. Functions relating the number of nails on a nailboard to the areas of shapes stretched on it (Pick's theorem) and the functions that emerge from games were also discussed. Opportunities to strengthen computational skills and explore the struc- tures behind those techniques were not ignored, but no attempt was made to develop conventional lessons in the skills.

There are obvious defects in a program of teacher education which waits until the students are in their last two years of col- lege to give them direct exposure to work with children. Moreover, many educators now question the value of student teaching when it is not preceded by extensive ex- perience with individual pupils and small groups. For most prospective teachers, the job of classroom organization and manage- ment is all-consuming. Little effort may be reserved for the kind of observation, anal- ysis, conversation, and evaluation which comes naturally when one student is work-, ing with one child. We speak of the values of sensitivity training, and particularly of the close observations of individuals this implies, but we often fail to make enough provision for this kind of experience. Even further, we speak of instructional objectives often without giving our college students sufficient opportunity to try them out in the microcosm of a one-to-one confrontation. It was to meet these obvious needs that the laboratory sessions were organized on a one-to-one basis.

Because of the "free" atmosphere of the sessions, many students had a chance to work with children of widely varying ages and abilities. Some of them discovered they preferred to work with younger or older children before they were locked into a student teaching assignment that might have proved uncongenial for them. Some stu- dents had a chance to match their abstract idealistic desire to teach in the big city

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ghetto with their experience of trying to communicate with just one child temporar- ily removed from that environment. (Hap- pily, in most cases, the experience increased their desire to serve education in this capac- ity.) It was mainly because of these ad- vantages that the alternatives of using chil- dren of one age level, or background, or ability, or by classes were rejected for the program. Had transportation not been available for the children, the plan would have been to send students into schools to work with individual children in the "back of the room" if possible, or in some other informal situation. In our local case, it would have meant placing students in class- rooms where activity methods were already being used, although there is still a short- age of such places.

An informal evaluation of the experi- mental program proceeded on several levels. Most obviously it was evaluated, and positively, by the children and their parents. There was never any difficulty in obtaining the children; they were always "waiting for the bus" on lab mornings. Participating administrators fed back favor- able reports from parents and teachers, and some of the latter became interested in the laboratory activities themselves. Since the children were involved for only a few ses- sions, it was not feasible to attempt any substantial evaluation of what they learned. Anecdotal records and observations of the instructor indicated some increase in at- tention span, particularly among the aca- demically deprived children. The general consensus of the administrators, teachers, students, college colleagues, and occasional visitors to the lab sessions was that it was certainly not a harmful experience for the children who missed perhaps a half-dozen of their regular morning programs in order to attend. The college students were dem- onstrably grateful for the opportunity to do some real teaching before they faced the moment of truth with their first class in student teaching. Their observations and techniques improved during the course, but, perhaps most important, they began

to work harder at mathematics as they discovered the need while attempting to keep up with their eager pupils.

This brief account is by no means in- tended to convey the impression that we didn't make mistakes; far from it! For ex- ample, in the first semester trial, the stu- dents and children were put together very early in the course before the students had had enough time to become familiar with the materials. This was not a disadvantage to students who were strong in their self- concept as teacher, but it was traumatic to some students who found themselves behind the children they were working with. Occasionally the classes were rather chaotic, and some children wasted quite a bit of time before they caught on to the lack of externally imposed discipline. Rec- ord keeping left something to be desired at the beginning, and the instructor was so busy with the over-all scene that some stu- dents were denied direct evaluation during the laboratory periods. The administrative detail, particularly getting the pupils on and off campus, was extensive at first. Ad- vance planning was necessary to obtain a workable schedule, and it took persever- ance to keep it. Yet, for all these faults, the program was sufficiently successful to warrant continuation and the recommen- dation for expansion and adaptation to other circumstances and subject areas.

Much more remains to be learned about both the use of laboratory methods in ele- mentary school mathematics instruction and in teacher preparation programs. One thing is clear, however, from even this small venture: It can work. A half year after the first laboratory sessions one of the students in the course called the in- structor long distance to tell him that she was engaged in student teaching and was introducing laboratory methods in her class. The reception of her effort was so positive that she had been asked to help the school's curriculum committee prepare an order for laboratory equipment so that the other teachers could introduce the lab- oratory method the following year.

78 The Arithmetic Teacher /January 1970

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