Nuts and Bolts: Student Team to Design MATH 115,

Mathematics through Its History, Summer 1998

 

Using University of Redlands Hewlett Foundation funds, I was able to pay small stipends to six Southern California based students to work part-time to prepare activities for a new elementary mathematics history course. All six students were about to begin their junior year. Three seniors also helped with the course. All of the students had studied mathematics history with me in MATH 245, Number Theory / History of Mathematics the previous semester; most of them were preparing to be high school teachers.

After I provided the students with dozens of sample activities, we held a series of joint meetings in order to hammer out an outline for the course and then to divide up the topics among two-student teams. After that, I worked with the students individually or in pairs. Students prepared from two to four units each, centered around an activity or set of activities; each student ended up presenting or co-presenting at least one unit in the class. Units and activities were to be designed to help MATH 115 students discover, explore, and understand the earliest uses of number systems and counting, arithmetic, fractions, geometry, algebra, probability, number theory, and infinite sums in civilizations around the world. For the most part, students organized the sample activities I had provided for them into a coherent whole, often following suggestions I made, but they also found resources on the Web and in books and journals they found on their own. They found the background information they needed in mathematics history texts borrowed from me and from their local libraries.

During the fall semester of 1998, we met weekly to organize and practice our class presentations. During the first offering of the course (Interim 1999), I met nightly with my co-instructors to fine-tune (OK, micro-manage) their presentations. (Students earned 1 unit of directed study credit for fall preparation and another unit for Interim preparation and teaching.) During the second offering of the course (Interim 2000), five student co-instructors participated. I would say that student co-instructors participated in at least some part of the class session for at least two-thirds of the class sessions the first year and for at least one-half of the class sessions the second year.

This was a great experience for the students involved, and was very rewarding for me as well. Eight of my eleven co-instructors now teach mathematics in high school or college (the latter as teaching assistants in their graduate programs). One of the student instructors prepared and taught a three-week, intensive mathematics history course for the very bright fifth and sixth grade students in the Johns Hopkins Center for Talented Youth program at Stanford University for two summers in a row.

 

See proposal for funding and invitation to potential student instructors, below.

 

 

PROPOSAL FOR HEWLETT FUNDS

FOR STUDENT/FACULTY TEAM TO DESIGN AN ACTIVITY-BASED

ELEMENTARY MATHEMATICS HISTORY COURSE

Summer, 1998

Project Director: Dr. Janet L. Beery, Mathematics

 

 

My own relatively recent interest in mathematics history reflects a healthy and growing trend in the mathematics community. At long last, mathematics faculty and students are becoming interested in the history of their own subject, and are seeking ways to incorporate historical knowledge and ideas into their mathematics courses and curriculum. What I hereby propose is funding for a cadre of students to work with me this summer to design a new Liberal Arts Foundation (LAF) course featuring the history of mathematics. This course would be an elementary course open to all students having the basic algebra skills required for admission to the University, and would be activity-based.

 

How project meets departmental objectives

The project would meet mathematics department objectives of (1) offering a wider variety of Liberal Arts Foundation (LAF) mathematics courses for University of Redlands students, especially non-science students, and (2) providing meaningful summer programs for mathematics majors, especially those students who intend to pursue careers as secondary school teachers.

 

(1) During the past year, especially, the mathematics department has had several requests from students for additional elementary mathematics courses satisfying the "M3" portion of the LAF Mathematics and Science (MS) requirement. These requests stem both from students' interest in mathematics and from the dearth of such courses offered by other Science Center departments due to staffing shortages. I probably would have to teach the course as overload, either in Spring 1999 or Fall 1999 (in which case it could be a First Year Seminar).

 

(2) Our very best students participate in externally funded national summer mathematics programs. However, we have many excellent students, including many students who are strongly interested in mathematics teaching careers, who either are not chosen for these programs or are not interested in them. These students have chosen a challenging major and career and need all the preparation, motivation, and encouragement we can give them.

 

Expectations for student participants

 

Students would design activities and projects to be used in the mathematics history course. They would begin by reviewing samples of activities that I already have collected. They also would spend some time searching for samples of such exercises at relevant websites and in relevant journals, most notably the journal of the National Council of Mathematics Teachers (NCTM), Mathematics Teacher, a publication with which students interested in teaching careers should become familiar! For the most part, though, the students would design activities based on descriptions of historical mathematical concepts found in original mathematical sources (some in languages other than English) and in mathematics history textbooks.

 

The piece of the project on which we would work first would make use of the six early arithmetic texts in our own Armacost Library Rare Book Room and the two early arithmetic texts in the Heritage Room of the Smiley Library. We would design exercises to guide students as they themselves examine these locally available early arithmetic (and algebra) texts. For instance, students would be led to compare algorithms for the basic arithmetic operations and some algebraic operations in these texts with present day algorithms.

 

Other projects would be designed to help future students discover and understand the earliest uses of counting, fractions, geometry, algebra, probability, infinite sums, etc. in civilizations around the world. Student participants would be encouraged to generate their own ideas for projects, but also could be assigned activities to write. They would be closely supervised by me; I would discuss with them frequently the general outline for the course and how the individual activities fit into or modify this outline.

 

I would encourage students to work in pairs and would expect each pair of students to complete three to five activities/projects, depending on length. Most, if not all, work on activities would be done on campus so that I could supervise it. Each student would complete about 50 hours of work, but this could vary. Since students would have full-time jobs near their permanent residences, each student would set up a work schedule with me at the start of the summer. Schedules could include work on several consecutive weekends, or during one or two weeks at the start of the summer, or etc.

 

Outcomes, including dissemination

 

The student-designed activities would be used in the elementary mathematics history course, and also could be used in our MATH 100/101, Finite Mathematics / Mathematics for Liberal Arts (equivalent courses in that students may receive credit for at most one of the two) and MATH 102, our mathematics course especially for (and open only to) prospective elementary school teachers (liberal studies majors). Projects not only would be used in mathematics courses, but could be disseminated by the students who designed them at local (and perhaps national) mathematics meetings and in publications such as Mathematics Teacher. The National Science Foundation sponsored Institute for the History of Mathematics and its Use in Teaching, of which I am a member, plans to publish volumes of classroom history units, so students might have the opportunity to disseminate their work there, too. Unpublished activities would be available at my website.

 

In addition, students would serve as teaching assistants for the course in which the activities would be used, so would have the chance to facilitate their own activities. Since most of the students currently are sophomores, they would be available to help with the course even if it were not taught until Fall 1999. Student participants also would have the opportunity to present and use their work in the MATH 100/101/102 courses described above.

 

Qualifications of project director

 

I have taught our sophomore/junior level mathematics history course for mathematics majors and minors for two years, and have participated in the National Science Foundation sponsored Institute for the History of Mathematics and its Use in Teaching for two years. Last summer, I worked with a student on various mathematics history projects, most curriculum-related. Two of these projects involved early mathematics texts in English and in Spanish; the current project would make use of this earlier work.

 

Qualifications of student participants

 

The students who would work on this project would be sophomore and junior students who

(a) have studied mathematics history with me, so have a framework on which to build activities,

 

(b) have a strong interest in teaching mathematics, usually at the high school level, after graduation,

(c) have a grade point average (well) over 3.0 and a demonstrated ability and willingness to work hard, and

 

(d) are able to commute to Redlands daily or for a few days at a time (permanent residence in Southern California).

I would select these students from a group of eight students who meet the above criteria and are interested in the proposed project. Four of these students applied to participate in national summer mathematics programs and still are waiting to hear if they are accepted. I could name these students and provide more information about each one, if you like.

 

Budget

 

Ideally, six students would be funded at $400 each, to include stipend, travel, and food, for a total of $2400. Travel to Southern California libraries, photocopying fees, and purchase of any necessary texts should run no more than $200. Total: $2600. Of course, I'll use whatever you are able to give me: funding at a lower level would mean fewer students and/or a smaller stipend for each student.

 

A single student funded under the Science Center summer research model would cost $2400 (stipend) + $600 (housing) = $3000. I am requesting funds for several students, rather than for a single student as I did last summer, because (A) more students would be able to participate and (B) I think students would be even more productive than my student last summer if they worked together and if they were assigned even more focused projects with a shorter timeframe for completion.

 

 

 

YOU ARE INVITED

TO JOIN A STUDENT TEAM FORMED TO DESIGN

AN ELEMENTARY MATHEMATICS HISTORY COURSE

Summer, 1998

Project Director: Dr. Janet L. Beery, Mathematics

 

 

 

OUR GOAL: Design a new Liberal Arts Foundation (LAF) course featuring the history of mathematics. This course would be an elementary course open to all students having the basic algebra skills required for admission to the University (meaning, they've passed the Math Placement Exam), and would have a strong activity component. The course would be at least as mathematical as it is historical; students would earn M3 LAF credit (rather than, say, HH). The typical student in the course probably would be a student who has taken MATH 100 or 101 for M2 and a lab science for M1, and would prefer to fulfill the M3 requirement with a math course rather than a science course.

 

 

What would YOU do?

 

        Decide on an outline for the course---that is, decide what to include in the course. We'd put together a draft outline at the start, with the understanding that we'd probably modify it a lot as we go. You'd review comprehensive mathematics history texts in order to put together the draft outline.

        Design units of study and especially students activities and projects to be used in the mathematics history course. Topics might include counting, fractions, geometry, algebra, probability, infinite sums, and calculus. You would be encouraged to generate your own ideas for topics and activities, but don't worry: lots of resources for ideas would be available, such as the following ones.

o       You could begin by reviewing samples of activities that I already have collected.

o       You could search for samples of math history exercises at relevant websites and in relevant journals, most notably the journal of the National Council of Mathematics Teachers (NCTM), Mathematics Teacher.

o       You could get ideas for activities from descriptions of historical mathematical concepts found in mathematics history textbooks and in original mathematical sources (some in languages other than English: Spanish, French, German, Hawaiian, etc.).

o       Some activities (or one big activity) would make use of the six early arithmetic texts in our own Armacost Library Rare Book Room and the two early arithmetic texts in the Heritage Room of the Smiley Library. We would design exercises to guide students as they themselves examine these locally available early arithmetic (and algebra) texts. For instance, students would be led to compare algorithms for the basic arithmetic operations and some algebraic operations in these texts with present day algorithms. We also could include arithmetic algorithms from even earlier European and New World texts.

When working on teaching units and on activities/projects, you probably would work in pairs, with each pair completing three to five units/ activities/projects, depending on length. Most work on activities would be done on campus, with each student completing about 50 hours of work, but this could vary. Since students would have full-time jobs near their summer residences, each student would set up a work schedule with me at the start of the summer. Schedules could include work on several consecutive weekends, or during one or two weeks at the start of the summer, etc., but it would be most effective for us to choose as many common times as possible.

        Finally, you would have the opportunity to serve as a teaching assistant for the math history course, so would have the chance to teach and/or facilitate your own units and/or activities. You could earn work-study money and/or academic credit and/or glory for this experience. You might also have the opportunity to present your work in the MATH 100/101/102 courses described above.

 

 

When would this course be offered, anyway?

 

As early as next fall, probably next spring (Spring 1999), possibly the following fall

 

The activities you design also might be used in our MATH 100/101, Math for Liberal Arts / Finite Mathematics courses (equivalent courses in that students may receive credit for at most one of the two) and MATH 102, our mathematics course especially for (and open only to) prospective elementary school teachers (liberal studies majors).

 

Furthermore, you might be able to present especially good activities at local mathematics meetings and/or publish them in teaching magazines.

 

 

Are you qualified?

 

The six (6) students who would work on this project ideally would be sophomore and junior students who

(a) have studied mathematics history with me, so have a framework on which to build activities,

 

(b) have an interest in teaching mathematics, perhaps at the high school level, after graduation,

(c) have a grade point average over 3.0 and a demonstrated ability and willingness to work hard, and

 

(d) are able to commute to Redlands periodically (summer residence in Southern California). You'd either commute from home, or, if that's too big of a pain, stay with me or with friends in Redlands for a few days at a time.

 

 

What would you get out of the project besides an expanded knowledge of mathematics history, personal satisfaction, and a great entry for your resume?

 

Cold, hard cash, of course! Now, we're not talking about a whole lot of cash--- you'd still have to get a summer job---but I should be able to pay you about $400 each. My grant would cover special travel to Southern California libraries, photocopying fees, and purchase of any necessary texts, but, unfortunately, I don't have money for your travel to and from Redlands.

 

 

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