Encouraging Preservice Mathematics Teachers as Mathematicians

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<ul><li><p>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</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms &amp; Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp</p><p> .</p><p>JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.</p><p> .</p><p>National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.</p><p>http://www.jstor.org </p><p>This content downloaded from 195.178.73.237 on Thu, 24 Apr 2014 14:38:45 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>Encouraging Preservice </p><p>Mathematics Teachers </p><p>as Mathematicians </p><p>Elizabeth A. Burroughs </p><p>important component of a math ematics teacher education program is </p><p>convincing preservice teachers that </p><p>they are mathematicians. After all, school students' views of mathemat </p><p>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 </p><p>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 </p><p>mathematical thinkers by looking for "attributes like </p><p>linearity, periodicity, continuity, randomness and </p><p>symmetry,... take actions like representing, experi </p><p>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 </p><p>investigations in mathematics are intimately involved with their success as mathematics teachers. </p><p>464 MATHEMATICS TEACHER | Vol. 100, No. 7 ? March 2007 </p><p>This content downloaded from 195.178.73.237 on Thu, 24 Apr 2014 14:38:45 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>THE ASSIGNMENT First, the prospective secondary teachers were asked to read two articles, "Mathematics for Teach </p><p>ing" (Cuoco 2001) and "Teaching Mathematics in the United States" (Cuoco 2003), from Notices, the monthly journal for members of the American </p><p>Mathematical Society. I felt it was important for the </p><p>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 </p><p>professional development. Cuoco says that the "best </p><p>high school teachers are those who have a research like experience in mathematics" (2001, p. 171). </p><p>Cuoco talks of a "vertical disconnect"?the dif ference between the mathematics that students </p><p>study to become qualified to teach and the math ematics that they will be expected to teach. The </p><p>assignment (which follows) is designed to address that disconnect: </p><p>Choose a topic from higher mathematics?as a </p><p>general guide, a class "higher" than second-year calculus. Describe how knowing more about that </p><p>topic can enrich your teaching of school math ematics. You should? </p><p>introduce the mathematical topic in a general way. Keep in mind your audience is others like you, those who have completed the nec </p><p>essary mathematics courses or studied the </p><p>necessary material to obtain a credential. address how the topic relates to school </p><p>mathematics. </p><p>think of an activity or lesson involving the advanced topic that would be appropriate to use in a school mathematics course. </p><p>The preservice teachers were challenged by the </p><p>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 </p><p>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. </p><p>One Preservice Teacher's Problem </p><p>l3 + 23 + 33 + ??? + n3 = (1 + 2 + 3 + ??? + nf </p><p>For one preservice teacher, this assignment resulted </p><p>in a physical representation of the statement that the sum of the cubes of the first natural numbers is </p><p>equal to the square of their sum. This physical result </p><p>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 </p><p>preservice teacher, Alyssa Sanchez Biesecker, was </p><p>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+ </p><p>?'? + )2. Around the same time as she was observing this, Sanchez Bie secker was thinking about the assignment from the </p><p>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. </p><p>In her discussion of the mathematical topic, Sanchez Biesecker wrote an induction proof of this </p><p>particular number theory result (fig. 1), but she </p><p>* Sanchez Blesecker's froof by Induction CkHteider the set S of all natural numbers such that </p><p>l3 + 23 + 33+;.. + ??^? +2 + 3+??? + ?)2. </p><p>Because 1 is a natural mmb6t such that l9 = l2,1 is an element of S. </p><p>Suppose that k is an elem?nt of &amp; Show that k +1 is also an element of &amp; Consider the </p></li><li><p>Flg. 2 (a) Integer rods showing 13 + 23 + 33 + ??? + n3 for = 4 (b) The same rods rearranged to show (1 + 2 + 3 + </p><p>??? + nf </p><p>Improved Proof by Induction Collider the set S of ah natural numbers ? stidith^t </p><p>il+?;+?+^+rfs(i + 2+^ </p><p>Because 1 is a natural number sudi that I3 = l2,1 is an element of S. </p><p>Suppose thatk is an element of&amp; Show that k +1 is also an element of S. Consider the quantity </p><p>1'*2,.*8,.+-.??+ + + ) = (l + 2 + 3+- + fc)?+{* + I)s fay the Induction Hypothesis </p><p>fay Causes formula </p><p>by factoring </p><p>by factoring </p><p>."fay comhitjing terms </p><p>fay factoring </p><p>= (1 + 2 + 3 + ??? + * + (fc +1))2 by Gause's formula </p><p>Because Is+ 2*+ 33 +??'? + *s+ (* + l)s - (1 + 2 + 3 + ??? +.* +.&lt; </p><p>(fe +1))2, (fe +1) is an element of S whenever h is an element ?f S. Then, induction, </p><p>Is + 2s + 33 + - + n3 = (1 + 2 + 3 + ? + nf for all natural numbers n. </p><p>Fig. 3 An improved proof by induction that connects to the physical representation </p><p>466 MATHEMATICS TEACHER | Vol. 100, No. 7 ? March 2007 </p><p>struggled with making a clear connection between </p><p>constructing a proof by induction and leading a </p><p>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 </p><p>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. </p><p>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 </p><p>provide a description of the physical strategy. This </p><p>strategy provides the reason why the statement is true and gives an indication of how to go about pro viding a rigorous proof. </p><p>The proof by induction that Sanchez Biesecker </p><p>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 </p><p>Thinking (1993, p. 86), though perhaps Fry's is not as explicit. Formalizing Sanchez Biesecker's </p><p>strategy for the physical representation leads to an </p><p>improved proof by induction (fig. 3). </p><p>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 </p><p>having a volume of 1 and as having a base that cov ers a square lxl area. This is a visual model of the initial step in the induction proof: l3 = l2. </p><p>This content downloaded from 195.178.73.237 on Thu, 24 Apr 2014 14:38:45 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>Fiq. 4 (a) Integer rods showing 12 + 23. We have a square of area 12 plus a cube of volume 23. (b) The same rods rear </p><p>ranged to show 12 + 22 + 22 (c) The same rods rearranged to show (1 + 2)2 </p><p>Next we consider l3 + 23 = (1 + 2)2. The sum l3 + 23 is represented by one white integer rod and a cube with side length 2, constructed using four red rods (fig. 4a). To rearrange these into a rectangular prism of height 1 and square base (1 + 2) x (1 + 2), adjoin a pair of red rods to the white rod as shown in the left of figure 4b. Place one of the other rods </p><p>along the length of the base and the final one along the width of the prism (fig. 4c). </p><p>In proving that l3 + 23+ 33 = (1 + 2 + 3)2, we first </p><p>represent the sum l3 + 23 + 33. We have a rectangu lar prism of height 1 with a square base (1 + 2) (1 + 2), which by the previous case is equal to l3 + 23, </p><p>Fig. 5 (a) Integer rods showing (1 + 2)2 + 33. We have a square of area (1 + 2)2 plus a cube of volume 33. (b) The same rods rearranged to show (1 + 2)2 + 32 + 2(3)2 (c) The same rods rearranged to show (1 + 2 + 3)2 </p><p>and a cube with side length 3, represented by nine </p><p>green rods (fig. 5a). To rearrange these into a rect </p><p>angular prism of height 1 and square base (1 + 2 + </p><p>3) (1 + 2 + 3), place one layer of green rods along the block diagonal (fig. 5b). Divide the remaining two layers so that one layer is along the length of the base and one layer is along the width (fig. 5c). </p><p>Notice that the strategy here differs from the previ ous case, because we have added a cube with odd side length. To emphasize the difference between </p><p>adding a cube with odd side length and a cube with even side length, we will continue for another step. </p><p>In proving that l3 + 23 + 33 + 43 = (1 + 2 + 3 + 4)2, </p><p>Vol. 100, No. 7 ? March 2007 | MATHEMATICS TEACHER 467 </p><p>This content downloaded from 195.178.73.237 on Thu, 24 Apr 2014 14:38:45 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>Fig. 6 (a) Integer rods showing (1 + 2 + 3)2 + 42 + 3(4)2. The goal is to rearrange the three layers of 4 4 squares to include in the larger square, (b) The same rods rearranged to </p><p>show the placement of two of the remaining layers of 4 4 squares using one 4x4 square along both the length and the width of the larger square. One 4x4 square remains. It will </p><p>be divided in two and used to create the larger square, (c) The same rods rearranged to show (1 + 2 + 3 + 4)2 </p><p>we consider the sum l3 + 23 + 33 + 43. We have a rect </p><p>angular prism of height 1 with a square base (1 + 2 + </p><p>3) (1 + 2 + 3), which by the previous case is equal to l3 + 23 + 33, and a cube with side length 4, represented by sixteen purple rods. To rearrange these into a rect </p><p>angular prism of height 1 and square base (1 + 2 + 3 + </p><p>4) (1 + 2 + 3 + 4), place one layer of purple rods </p><p>along the block diagonal (fig. 6a). Notice how the case shown in figure 6 differs from the case shown in fig ure 5: Now there is an odd number of layers left. Take two of the remaining three layers and place one along the length and one along the width (fig. 6b). Divide the final remaining layer in half and place two rods </p><p>along the length and two along the width (fig. 6c). Now, let's generalize. (The reader might refer to </p><p>figs. 5 and 6 to help visualize these generalizations.) Starting with a prism of height 1 and square base (1 + 2 + ? ? ? + ) (1 + 2 + ? - ? + n), add a cube of volume ( +1)3 and rearrange the rods to result in a prism of </p><p>height 1 and square base (1 + 2 + ??? + n+ (n+1)) x </p><p>(1 + 2+ ??? + ?+( +1)). For even values of n, we </p><p>have an odd number of layers of integer rods in the cube. Place one of the layers on the '"block diagonal" of the new square. There are layers of (n + 1) ( + 1) squares remaining. Place nil of the layers along both the length and width of the larger square. </p><p>For odd values of n, we have an even number of </p><p>layers of integer rods in the cube. We again add to the </p><p>existing square one layer of rods from the cube along the block diagonal. But now we have an odd number of remaining (n +1) (n +1) square layers. Divide by two and round down to the nearest integer, denoted by </p><p>We place </p><p>layers </p><p>along both the length and the width of the square. The remaining one layer is split so that half the rods from that layer are placed along the width of the </p><p>larger square and half are placed along the length. The key to the proof by induction is to use </p><p>Gauss's formula for the sum of the first integers, </p><p>1 + 2 + 3 + ??? + ? = n[n +1) </p><p>How does Gauss's formula relate to the physical rep resentation? As we consider adding the rods that form the cube to the existing rods that form the square, we are representing the algebraic quantity (1 + 2 + 3 + ? ? + nf + ( +1)3. Then we physically split the cube </p><p>into +1 layers of {n +1) x (n +1) squares; this is the expansion of ( +1)3 = (n + l)\n + 1) found in the </p><p>improved proof by induction. We always place one </p><p>468 MATHEMATICS TEACHER | Vol. 100, No. 7 ? March 2007 </p><p>This content downloaded from 195.178.73.237 on Thu, 24 Apr 2014 14:38:45 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>layer along the block diagonal and then distribute the </p><p>remaining rods. The layer along the block diagonal is the latter quantity in the expansion [n + l)\n +1) </p><p>= </p><p>n(n +1)2 + (n +1)2. How do we view the quantity n(n +1)2? Physically, we take half the layers of </p><p>[ +1)2 for both the length and width: </p><p>n(n + lf = - + - 2 2 (rc + 1)2 </p><p>Rewr...</p></li></ul>