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  • More Reflections on Teaching Mathematics for Elementary School TeachersAuthor(s): Rick Billstein and Johnny W. LottSource: The Arithmetic Teacher, Vol. 29, No. 5 (January 1982), pp. 37-38Published by: National Council of Teachers of MathematicsStable URL: .Accessed: 13/06/2014 10:04

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  • More Reflections on Teaching Mathematics for Elementary

    School Teachers By Rick Billstein and Johnny W. Lott

    In his article, "Some Reflections on Teaching Mathematics for Elemen- tary School Teachers," (November 1980 issue of the Arithmetic Teacher), Lowell Leake suggested that mathe- matics educators share teaching ideas, successes, and failures in courses for elementary school teach- ers. This article is a response to his suggestion.

    For many years now at the Univer- sity of Montana, we have had a se- quence of three one-quarter, mathe- matics content courses for elementary school teachers. The courses, which developed along the lines of the rec- ommendations of the Committee on the Undergraduate Program in Mathe- matics (CUPM), now include many of the facets described by Leake, but are handled in a somewhat different man- ner.

    In their methods classes, which are separate from the content courses, our students are required to start a card file of activities developed through the use of articles in the Arithmetic Teacher. In our opinion, the card file is more important, in the long run, than reaction papers.

    For the content courses, we have found that by supplying the students Rick Billstein in an associate professor of math- ematics and associate chairman of the mathe- matics department at the University of Mon- tana in Missoula. His principal responsibilities are in the area of elementary school mathemat- ics education. Johnny Lott is an associate professor of mathematics on the same campus. His specific responsibilities are in the area of mathematics education for elementary and sec- ondary school teachers. He has taught in the secondary schools and during the current aca- demic year he is on sabbatical leave from the university to teach in the Pelican, Alaska, Schools.

    January 1982

    with lists of questions that could have come from the classroom, similar to those posed by Crouse and Sloyer (1977), and a bibliography with refer- ences to the AT and other sources, we can require that the students write responses to the specific mathemati- cal questions by doing research in both the content and method areas. These questions and bibliographies are chosen in conjunction with the topic being discussed and are contin- ually updated.

    The following is an example of a question we might ask when integers are the subject of study:

    A student says that "5 > "2, since a debt of $5 is larger than a debt of $2. What error is being made by the students and how can it be correct- ed?

    We think that by using the AT and other library resources our students will realize that they can find answers and help for classroom questions. To further emphasize the usefulness of these resources, we strongly encour- age our students to take advantage of their student status to become mem- bers (at reduced rates) of both the NCTM and the Montana Council of Teachers of Mathematics (MCTM).

    Occasionally we invite a practicing elementary school teacher to teach our classes and to answer questions for our students. This practice, along with that of bringing various textbook series to class, serves to give us more credibility concerning much of the material being covered. Our students are also asked to review at least one textbook in the lower grades (K through 4) and one in the upper grades

    (5 through 8) each quarter. In this way, they can see just how the mathe- matics we are covering in our classes is presented at those levels.

    Calculators have been an integral part of our courses since the MCTM conducted its Calculators in Schools Project in 1977. Since most of our students come to us with little knowl- edge of calculating devices, we start with the very basics - how calculators can be used in the classroom, which types of calculators are available, the type of logic the calculator should have, the uses of various keys, and the things to be aware of when order- ing a classroom set of calculators.

    Forty calculators are available for use with our classes and we encour- age students to buy their own ma- chines only after we have discussed the pros and cons of various calcula- tors. Once the preliminaries are com- pleted, we give our students a set of selected problems for various grade levels both to work and to discuss in class. The current research on the use of calculators in the elementary school is also investigated. Micro- computer and computer demonstra- tions are also arranged as time per- mits. An eventual goal is to integrate their use more fully into our courses.

    Two years ago we made problem solving the focus of our courses. The first topic of the first quarter is a four- step, Polya-type approach to solving problems. The following four steps are used: understanding the problem, devising a plan, carrying out the plan, and looking back. Initially, the prob- lems investigated draw on few mathe- matical skills, but do require creative thinking. The various problem-solving


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  • strategies, such as looking at patterns and reducing the problem to a simpler problem, are discussed and empha- sized. The following is a typical exam- pie:

    It is the first day of a mathematics class and twenty people are present in the room. To become acquaint- ed, each person shakes hands just once with everyone else. How many handshakes take place ,

    In this example, the strategies dis- cussed are trying a simpler problem, making a table, looking for a pattern, and drawing a model.

    After the initial presentation of problem solving, we introduce each new unit with a somewhat sophisticat- ed preliminary problem, stated in sim- ple terms, which can be solved using problem-solving techniques along with the mathematical content of the unit to be taught. Students are encour- aged to try to solve the problem im- mediately, but very seldom is this done. As the students work through the block of material in the unit, they are to keep the preliminary problem in mind and whenever they think they have learned enough content to solve the problem, they are to try it again. If the problem is not solved by the time the unit is completed, then the prob- lem is discussed and solved in class.

    The following is an example of a preliminary problem for a geometry unit:

    Suppose a stained-glass-window maker needed assorted triangular pieces of glass for a project. One rectangular plate of glass that she planned to cut contained ten air bubbles, no three of which were in a straight line. To avoid having an air bubble in her finished product, she decided to cut triangular pieces by making the air bubbles and the cor- ners of the plate the vertices of the triangle. How many triangular pieces did she cut?

    In the teaching of probability, we have found that hands-on experience with devices that generate random events is important and that teaching probability using a formula-oriented approach is not as effective as devel- oping almost all the material via tree


    diagrams. The Monte Carlo approach is very helpful in developing many of the ideas in the course. Even when students find that they cannot work many problems theoretically, they find ways to simulate the problem using random-digit tables. Simulation techniques are very powerful and can be reinforced even more if computer simulation techniques are added to the course.

    For the geometry units, we have chosen a construction approach. By having students involved in paper- folding activities, and compass and straightedge and MIRA construc- tions, most plane geometry notions can be motivated. We find the MIRA is a very valuable tool in developing geometrical concepts (see Woodward 1977). All compass and straightedge constructions, except drawing a cir- cle, can be investigated using a MIRA. MIRAs are also especially helpful when studying line symme- tries and motion geometry.

    A discussion and a showing of drawings by M. C. Escher are valu- able motivators in the study of isomet- ric transformations. Many drawings are analyzed and transformations are used by students to develop Escher- type drawings of their own.

    In our classes, the study of metric measure is integrated into the course. It is introduced in a hands-on setting and is not taught in terms of conver- sions from the customar