encouraging preservice mathematics teachers as mathematicians
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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 .
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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
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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 <n*antity
l3 + 23 + 33 + ..- + ife3 + 0fe + l)^ = (1 + 2 + 3 + ?? + *)2 + (Jfe + 1)3 by tto Induction Hypothesis
by Gauss's formula
by expanding terms
by factoring terms
= (1 + 2 + 3 + ??? + ? + (& +1))2 by Gauss's formula
Because l3 + 23 + 33+-+ fe3 +(* +1)3 = (1 + 2 + 3+ "?"+*.+ (k +l))2, (fe +1) is an element of S. Then, by induction,
l3 + 2s + 33 + ? + n3 = (1 + 2 + 3 + ??? + nf for all natural numbers ?.
Fig. 1 The step of expanding terms and then factoring (in red) does not provide any insight into why the (n + 1)st step follows from the nth. Compare this to the improved induction proof in figure 3.
Vol. 100, No. 7 ? March 2007 | MATHEMATICS TEACHER 465
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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 is a visual model of the initial step in the induction proof: l3
= l2.
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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
ranged to show 12 + 22 + 22 (c) The same rods rearranged to show (1 + 2)2
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
along the length of the base and the final one along the width of the prism (fig. 4c).
In proving that l3 + 23+ 33 = (1 + 2 + 3)2, we first
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,
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
and a cube with side length 3, represented by nine
green rods (fig. 5a). To rearrange these into a rect
angular prism of height 1 and square base (1 + 2 +
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).
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
adding a cube with odd side length and a cube with even side length, we will continue for another step.
In proving that l3 + 23 + 33 + 43 = (1 + 2 + 3 + 4)2,
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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
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
be divided in two and used to create the larger square, (c) The same rods rearranged to show (1 + 2 + 3 + 4)2
we consider the sum l3 + 23 + 33 + 43. We have a rect
angular prism of height 1 with a square base (1 + 2 +
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
angular prism of height 1 and square base (1 + 2 + 3 +
4) (1 + 2 + 3 + 4), place one layer of purple rods
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
along the length and two along the width (fig. 6c). Now, let's generalize. (The reader might refer to
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
height 1 and square base (1 + 2 + ??? + n+ (n+1)) x
(1 + 2+ ??? + ?+( +1)). For even values of n, we
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.
For odd values of n, we have an even number of
layers of integer rods in the cube. We again add to the
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
We place
layers
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
larger square and half are placed along the length. The key to the proof by induction is to use
Gauss's formula for the sum of the first integers,
1 + 2 + 3 + ??? + ? = n[n +1)
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
into +1 layers of {n +1) x (n +1) squares; this is the expansion of ( +1)3 = (n + l)\n + 1) found in the
improved proof by induction. We always place one
468 MATHEMATICS TEACHER | Vol. 100, No. 7 ? March 2007
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layer along the block diagonal and then distribute the
remaining rods. The layer along the block diagonal is
the latter quantity in the expansion [n + l)\n +1) =
n(n +1)2 + (n +1)2. How do we view the quantity n(n +1)2? Physically, we take half the layers of
[ +1)2 for both the length and width:
n(n + lf = - + -
2 2 (rc + 1)2
Rewrite this as
t n(n + l) n(n + l) rc(rc + l)2 ='
' in+ 11
and we find the quantity
n(n + l)
that makes the connection to Gauss's formula in
the proof by induction.
CONCLUSIONS AND CHALLENGES The goal of this assignment was to encourage
preservice teachers to think as mathematicians by asking them to reflect on connections between the mathematics they know and the mathematics they will teach. One preservice teacher arrived at a new
(at least, new to her) physical representation of a
number theory result. It was particularly exciting for Sanchez Biesecker
to experience the joy of discovery of the physical technique. Reflecting on the assignment, she says, "It really did make me feel like a mathematician."
By completing only the proof by induction, Sanchez Biesecker did not feel that she had gained enough understanding, and in fact felt that she had produced the proof by following a rote algorithm. By using the manipulatives to illustrate the relationship, she invented her own representation of a solution to the
problem and realized a greater understanding of how the sum of cubes is related to the square of the sum.
Both Sanchez Biesecker and Quillen say that this exercise changed the way they think about induc tion. A secondary benefit of the assignment was
that both preservice teachers gained firsthand expe rience of the power of investigating mathematics
through manipulatives in addition to paper-and-pen cil calculations. Quillen reflected that while she has not used this specific activity in her teaching, she does rely on using integer rods in her classroom.
Sanchez Biesecker submitted her induction proof and physical representation as fulfillment of the
assignment, and she suggested that a prealgebra class would benefit from expanding the original activity (where students investigated the relationship l3 + 23 +
33 + ? ? ? + n3 = (1 + 2 + 3 + ? ? ? + r?f by comparing tables
of values) to include creating a physical representation
of the relationship using integer rods. She realized that
her understanding of the relationship was strength ened by "seeing" the result using the integer rods, and
this convinced her that students should "see" it, too.
But because of the timing of her student-teaching expe
rience, she did not revisit this result in her classroom.
She and I invite readers to conduct such an activity with their own students and share the results with us.
Sanchez Biesecker's results pose questions for
the mathematician in all of us. Other number the
ory results have induction proofs that can be visual ized with integer rods. Can you construct a physical representation of Gauss's formula with integer rods, for example? We challenge you to generate
physical representations of other results and invite
you to share those results with us, too. This assignment asks preservice teachers to con
nect higher mathematics to school mathematics. All of us who are responsible for the education of future teachers should look for ways to encourage student thinking about such connections.
REFERENCES Conference Board of the Mathematical Sciences
(CBMS). The Mathematical Education of Teachers Part I. Washington, DC: Mathematical Association of America, 2001.
Cuoco, Al. "Mathematics for Teaching." Notices of the AMS48 (February 2001): 168-74.
-. "Teaching Mathematics in the United States."
Notices oftheAMS 50 (August 2003): 777-87.
Fry, Alan. "Sums of Cubes III." In Proofs without Words: Exercises in Visual Thinking, edited by Roger B. Nelsen. Washington, DC: Mathematical Association of America, 1993.
National Council of Teachers of Mathematics
(NCTM). Professional Standards for Teaching Math ematics. Reston, VA: NCTM, 1991.
-. Principles and Standards for School Mathemat ics. Reston, VA: NCTM, 2000. oo
For more on the mathematical relationship at the cen ter of this article, see Angelo S. Di Domenico, "A Gen eralization of Two Ancient Formulas,
" Mathematics
Teacher 100 (September 2006): 114-19.?Ed.
ELIZABETH BURROUGHS, burroughs? humboldt.edu, is an assistant profes sor in the Department of Mathemat ics at Humboldt State University,
Arcata, CA 95521, and a former high school mathematics teacher. She is grateful to stu dents like Alyssa Sanchez Biesecker and Mela nie Ouil?en who inspire her investigations into connections between undergraduate mathemat ics and mathematics for schoolchildren.
Vol. 100, No. 7 ? March 2007 | MATHEMATICS TEACHER 469
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