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Encouraging Preservice Mathematics Teachers as MathematiciansAuthor(s): Elizabeth A. BurroughsSource: The Mathematics Teacher, Vol. 100, No. 7 (MARCH 2007), pp. 464-469Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27972301 .Accessed: 24/04/2014 14:38

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Encouraging Preservice

Mathematics Teachers

as Mathematicians

Elizabeth A. Burroughs

important component of a math ematics teacher education program is

convincing preservice teachers that

they are mathematicians. After all, school students' views of mathemat

ics and mathematicians are shaped primarily by the mathematics teachers they have. NCTM's P?nciples and Standards for School Mathematics articulates in the Teaching Principle that to "be effective, teachers must know and understand deeply the mathematics they are teaching and be able to draw on that knowledge with flexibility in their teaching tasks" (NCTM 2000, p. 17). In addition, NCTM's Professional Standards for Teaching Mathematics calls for the education of teach ers to include "school mathematics within the disci

pline of mathematics" (NCTM 1991, p. 132). In order to include school mathematics within the discipline of mathematics, university mathematicians are urged to help prospective teachers of mathematics become

mathematical thinkers by looking for "attributes like

linearity, periodicity, continuity, randomness and

symmetry,... take actions like representing, experi

menting, modeling, classifying, visualizing, comput ing, and proving" (CBMS 2001, p. 8). The following assignment for students in a secondary mathematics methods course was designed to encourage preservice teachers to think as mathematicians because their

investigations in mathematics are intimately involved with their success as mathematics teachers.

464 MATHEMATICS TEACHER | Vol. 100, No. 7 ? March 2007

This content downloaded from 195.178.73.237 on Thu, 24 Apr 2014 14:38:45 PMAll use subject to JSTOR Terms and Conditions

THE ASSIGNMENT First, the prospective secondary teachers were asked to read two articles, "Mathematics for Teach

ing" (Cuoco 2001) and "Teaching Mathematics in the United States" (Cuoco 2003), from Notices, the monthly journal for members of the American

Mathematical Society. I felt it was important for the

preservice teachers to read material from a primary source whose intended audience is mathematicians, because for this assignment I wanted them to think as mathematicians. I also wanted their reflections on these readings to give them a perspective from which to view their beginning teaching years?that they are not expected to know it all and that they should regularly look for ways to continue their

professional development. Cuoco says that the "best

high school teachers are those who have a research like experience in mathematics" (2001, p. 171).

Cuoco talks of a "vertical disconnect"?the dif ference between the mathematics that students

study to become qualified to teach and the math ematics that they will be expected to teach. The

assignment (which follows) is designed to address that disconnect:

Choose a topic from higher mathematics?as a

general guide, a class "higher" than second-year calculus. Describe how knowing more about that

topic can enrich your teaching of school math ematics. You should?

introduce the mathematical topic in a general way. Keep in mind your audience is others like you, those who have completed the nec

essary mathematics courses or studied the

necessary material to obtain a credential. address how the topic relates to school

mathematics.

think of an activity or lesson involving the advanced topic that would be appropriate to use in a school mathematics course.

The preservice teachers were challenged by the

requirement to focus on the connection between the mathematical topic and school mathematics; their inclination was to write a lot about the mathemat ics (as they would write a report for a mathematics

course) and pay only cursory attention to its relation to school mathematics. I limited their explanation of the mathematics to one page to encourage them to focus on the connections to school mathematics.

One Preservice Teacher's Problem

l3 + 23 + 33 + ??? + n3 = (1 + 2 + 3 + ??? + nf

For one preservice teacher, this assignment resulted

in a physical representation of the statement that the sum of the cubes of the first natural numbers is

equal to the square of their sum. This physical result

provided new mathematical insight for her. Like a research mathematician, she invented a mathemati cal representation of the solution to the problem (it happened to be a physical representation). This

preservice teacher, Alyssa Sanchez Biesecker, was

assisting regularly as a student teacher in a prealgebra classroom. She observed a classroom activity where students constructed tables of the first cubes and tables of the square of the sum of the first natural numbers. From this exercise with tables, students were to inductively reason the relationship that l3 + 23 + 33 + + rc3 = (l + 2 + 3+

?'? + )2. Around the same time as she was observing this, Sanchez Bie secker was thinking about the assignment from the

methods class. She asked if this relationship from number theory would be an appropriate topic, and I indicated it would. We did not imagine it would lead to such a rich mathematical discovery.

In her discussion of the mathematical topic, Sanchez Biesecker wrote an induction proof of this

particular number theory result (fig. 1), but she

* Sanchez Blesecker's froof by Induction CkHteider the set S of all natural numbers such that

l3 + 23 + 33+;.. + ??^? +2 + 3+??? + ?)2.

Because 1 is a natural mmb6t such that l9 = l2,1 is an element of S.

Suppose that k is an elem?nt of & Show that k +1 is also an element of & Consider the

Flg. 2 (a) Integer rods showing 13 + 23 + 33 + ??? + n3 for = 4 (b) The same rods rearranged to show (1 + 2 + 3 +

??? + nf

Improved Proof by Induction Collider the set S of ah natural numbers ? stidith^t

il+?;+?+^+rfs(i + 2+^

Because 1 is a natural number sudi that I3 = l2,1 is an element of S.

Suppose thatk is an element of& Show that k +1 is also an element of S. Consider the quantity

1'*2,.*8,.+-.??+ + + ) = (l + 2 + 3+- + fc)?+{* + I)s fay the Induction Hypothesis

fay Causes formula

by factoring

by factoring

."fay comhitjing terms

fay factoring

= (1 + 2 + 3 + ??? + * + (fc +1))2 by Gause's formula

Because Is+ 2*+ 33 +??'? + *s+ (* + l)s - (1 + 2 + 3 + ??? +.* +.<

(fe +1))2, (fe +1) is an element of S whenever h is an element ?f S. Then, induction,

Is + 2s + 33 + - + n3 = (1 + 2 + 3 + ? + nf for all natural numbers n.

Fig. 3 An improved proof by induction that connects to the physical representation

466 MATHEMATICS TEACHER | Vol. 100, No. 7 ? March 2007

struggled with making a clear connection between

constructing a proof by induction and leading a

prealgebra class through the inductive activity using tables. A set of integer rods was available, and Sanchez Biesecker and another student, Mela nie Quillen, started to use the rods to represent the

quantities involved. Within a few minutes, they had constructed a physical model using the rods that made clear the connection between the sum of cubes and the square of a sum.

When Sancehez Biesecker turned in her assign ment, she presented a strategy for constructing the wth case of her physical representation and a proof by induction. To provide a mathematical result, Sanchez Biesecker needed to do more than just provide a physical representation?she needed to

provide a description of the physical strategy. This

strategy provides the reason why the statement is true and gives an indication of how to go about pro viding a rigorous proof.

The proof by induction that Sanchez Biesecker

presented did not fully connect to the physical strategy, but her work on this problem inspired me to take a closer look at the connection between the two. Sanchez Biesecker's physical representation (fig. 2) resembles one created by Alan L. Fry, pub lished in Proofs without Words: Exercises in Visual

Thinking (1993, p. 86), though perhaps Fry's is not as explicit. Formalizing Sanchez Biesecker's

strategy for the physical representation leads to an

improved proof by induction (fig. 3).

ANALYSIS OF THE PHYSICAL REPRESENTATION I encourage the reader to have a set of integer rods in hand while analyzing the physical representation. Starting with one white integer block, we view it as

having a volume of 1 and as having a base that cov ers a square lxl area. This

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