A proof ought to explain: A classroom teacher-researcher, a mathematics educator, and three cohorts of fifth graders seek to make meaning of a non-obvious algebraic expression.

Vicki Zack, St. George's Elementary School, Montreal, Quebec
David A. Reid, Acadia University, Wolfville, Nova Scotia
This paper details one aspect of the collaboration between a classroom teacher and researcher (Vicki) and a university mathematics educator and researcher (David) who have been working over a number of years (in particular 1995-1996, 1998-1999, 1999-2000) with Vicki's fifth grade students on a variant of the chessboard task, and on the inquiry which has arisen therefrom. Vicki's original challenge to her students was that they construct a general procedure, which some succeeded in doing. In the midst of seeking to encode the general procedure into an algebraic expression, the fifth graders were blocked; teacher and children together bumped up against the realisation that they could not do so be given the state of their current knowledge. Reciprocal adult-child nudging and challenges were in evidence. Vicki sought out and offered the students a 'non-obvious expression' which worked -- n(n+1)(2n+1) ÷ 6 (Anderson, 1996). The children, in 1996, in turn raised the bar: they saw that it worked but asked why it worked as it did (Zack, 1997), a question Vicki could not answer. The students' challenge evoked a longstanding search, namely our (Vicki and David's) search for an explanation which made sense to fifth graders, in answer their need to understand why the non-obvious expression worked as it did.The paper will deal for the most part with the 'tri-tower pyramid' proof presented by David to two groups of fifth grade students in May 1999 and in May 2000, and with whether it answered the question posed by the fifth grade cohort in 1996.
 

Establishing the context: The school and classroom setting, the task, and the trajectory of the inquiry by adults and children

The school is private and non-denominational, with a population that is ethnically, religiously and linguistically mixed. Most students come from English speaking, middle class backgrounds. Non-routine problem solving is central to the mathematics curriculum at all grade levels. The school and classroom learning site is one in which the children are expected to publicly express their thinking, and engage in mathematical practice characterized by conjecture, argument, and justification (Cobb, Wood, & Yackel, 1993, p. 98).Mathematics is studied for 45 minutes each day (and twice a week extended to 90 minutes), and extended investigations of non-routine problems take up the entire lesson three times a week. There are usually about 25 students enrolled in the class, but mathematics in done as a half class of 12 or 13. The students are grouped heterogeneously in groups of four or five. When working on a problem they first work in twos and threes, then come together in their groups to compare solutions, then report to the half-class. (Other episodes and interpretations involving some of the same children can be found in Zack, 1997, 1999.)

The students were videotaped throughout their group and half-class discussions. In addition, their written work in their “math logs” was photocopied, written responses to questions focussed on particular aspects of their activity were collected, and they were interviewed and videotaped reflecting on their past activity.
 

The trajectory of this inquiry: Vicki nudges the children, and the children in turn nudge her

One part of this investigation, that of the search for a visual proof, originated in the first CMESG meeting Vicki attended, in Halifax, in May 1996. In her working group Bill Higginson asked the participants to think of an idea they wanted to share, and she chose to speak about one of the 30-some non-routine problem-solving assignments she has assigned to her students -- namely a variant of the chessboard problem. However, she had never appreciated the richness inherent in that task until she heard the responses of CMESG members, among them, Bill Higginson, Harry White, and others. Due to the plenary address given by Celia Hoyles that year on proof, Vicki began thinking about young children and their notions of proof. Serendipity led her to select this task and present it at that CMESG working group session that year, but her study of this mathematical task and of questions about proving has since exploded into investigations into children's notions of proof (Zack, 1997), their use of everyday and mathematical language in their arguments about proving (Zack, 1999), and the linguistic coherence in the students' arguments (Zack, 1998).
 

The task

The task and its extensions, Vicki’s own questions and personal confusion, and the children's challenges and questions, led us (Vicki and David) and the children to push forward in our investigation in unanticipated but fruitful ways.Here (Fig. 1) was the task as Vicki presented it to the class in the first year (May, 1994):
 
Find all the squares in the figure on the left. Can you prove that you have found them all?

Figure 1: The Count the Squares Problem

Vicki expanded the process each year, adding extensions in order to see whether and how the children could see and generalize various patterns, primarily the sum of squares. The problem evolved in terms of extended questions posed and class time spent from the first year, 1994 (one 90-minute session) to the most recent, 1999 (three 90 minute sessions, then a full week of interviews: children responding to segments of videotape featuring themselves discussing the tasks and talking about some aspect of proving, discussions about proving with small group of 2, 3, 4, to 6 people, then David presenting two visual proofs to small and larger groups).

Originally, Vicki’s expectation was that some children might see the pattern of the sum of the squares —, and express their hypothesis that the pattern would continue, i. e.,12+22+32+42+52. . . . Thus, if the children were able to generalize and say that the pattern would continue thus, that was good enough; indeed she felt that was sufficient for fifth graders. When she posed the “What if it were a 60 by 60 square?” question in 1996, what she expected the children to say was that 'the pattern just continues'. Vicki did not expect, nor did she want, the children to work out the actual numerical answer. She even tried to stop them from working it out. They ignored her. This unexpected development actually led to a number of surprises. In one instance the interaction between the members of two teams holding opposing positions about the answer to the 60 by 60 led to a discussion about proving, and prompted one of the teams to construct three counter-examples in order to try to convince the other team that their position was untenable (Zack, 1997). In another instance, rather than working with the numbers, thinking that there must of course be an 'easier way' than working through to the 60 by 60, a few children endeavored to construct an algebraic expression but came up empty (Alan and Keiichi in 1996, and Walt in 1999). Vicki was not aware in 1995/6 that it was not possible for the children to derive an algebraic expression for this generalization, and so this occurrence was a surprise for her as well. She was pushed to seek out other sources, people as well as book sources.

Vicki was pleased to discover, and brought back to the children, the Johnston Anderson (1996) formula — n(n+1)(2n+1) ÷ 6 — which Anderson himself called a 'non-obvious expression'. Her expectation was that she and the class would use it and see that it fit all of the examples which the children had calculated concretely. That was good enough for Vicki as she did not know why the expression worked; she only knew that it did. However, the students raised the bar again; they wanted to know why it worked as it did (Zack, 1997). They also wanted to know how anyone could come up with that expression. It did not fit the kind of algebraic expression which some of them had been able to derive in other instances, where there was a meaningful connection and transition between the concrete examples, and the general algebraic expression (Zack, 1995; Graves & Zack, 1996). Seeking an explanation to bring back to the students, Vicki was told that fifth graders could not construct or understand the mathematical approach to the construction of Anderson's algebraic expression given the state of their current mathematical knowledge. It seemed at that time a dead end. Thus, in 1997, the paper written for the PME conference of that year ended with a list of the children's emergent definitions of what they felt proof ought to be, among them that the proof must make sense and that the person presenting it must say why it works. When asked “What do you think of Johnston Anderson's rule?”, the children responded that explanations and proofs should make sense. Ross, for example, said that Anderson's rule was “brilliant, but he should explain why it works.” Lew said that “if the Johnston rule had evidence, if Johnston himself explained why it worked it would be more convincing.” And Rina felt that Anderson's expression was “a great way to figure out the problem but it doesn't make sense. . . . I think a mathematical proof is when you say why it works and if it works for everything show why” (Zack, 1997, p. 297).  Perhaps due to the in-classroom emphasis on explaining oneself, the children pushed to know the whys, and hows. Hanna has suggested that proofs which explain ought to be favoured above those which merely prove (1995, p. 48). All but one of the fifth graders polled (of a total of 10 that year) stated unequivocally that a proof ought to explain.

Their questions in turn pushed David and Vicki to embark on an investigation to find an explanation, in response to the children's need to know why. Over the next while (1996-2000), David, building on his long-standing interest in proof (e.g., Reid, 1992, 1995, 1997, 1998) explored and deliberated, at times with others (Cf, e-mail communication with Tommy Dreyfus), but most often alone. He decided on three possibilities and at various times showed and discussed with the children one of three visual proofs, each meeting with some measure of success and a number of unanswered questions: See Figures 2, 3, 4.

Figure 2: The wrapping proof presented to small groups in November 1996

Figure 3: Man-Keung Siu’s visual proof (from Nelsen, 1993, p. 77), the basis for the tri-tower visual proof presented in 1999 and 2000

Figure 4: Martin Gardner and Dan Kalman’s visual proof (from Nelsen, 1993, p. 78), the basis for the odd number visual proof presented to small groups in 1999 and 2000

Having summarized what brought us to this point, the next part of this paper will deal with the tri-tower visual proof presented in 1999 and 2000 (akin to Figure 3 and as shown in the photos below), and a number of challenging ideas embedded therein (Figure 5). We will also reflect upon whether David's proof did or did not answer the question posed by the students in 1996, namely that a proof ought to explain.
 

David's presentation of the tri-tower proof, presented in May 1999 and May 2000

 
Begin by representing the sum of the square numbers as a set of squares made from multi-link cubes:
These squares are put together to make a “pyramid”:
Two more pyramids are assembled:
The pyramids are put together with the aim of making something as close to a rectangular block as possible:
The block is then examined to see how many cubes are in each layer, and how many layers there are:
There are 4 and a half layers, each of which is 4 by 5.Note that it is easy to see that if pyramids of size 5 had been used the resulting block would be 1 larger in all three dimensions, because a 5 by 5 square would be added onto the bottom of each size 4 pyramid.
Finally note that three pyramids were used, so the number of cubes in one pyramid is the number of cubes in the block, divided by 3,
The number of cubes in one pyramid is the same as the sum of the square numbers, which establishes this formula (and because it is easy to see how the same construction would be done with pyramids of any size, the generalization of the formula is established as well).
For some students the connection between the two formulae is clear. For others it isn’t but that doesn’t undermine the formula derived in the explanation.

 

Crucial ideas

There are a number of crucial ideas (see Figure 5) that must be understood in order to understand the explanation as a whole. Not all the children understood all these ideas, and those that did understand them did not all understand them at the same time. Nonetheless, they were able to attain partial understandings of the explanation that were good enough to support their continuing engagement in the mathematical activity of the class.
Crucial ideas that they might not understand but hold and wait for understanding.

 
1
To count squares you add 1+4+9+16+…These numbers are SQUARE numbers in the sense of being NxN.
2
The NUMBER of blocks in a pyramid of height N is the SAME AS the NUMBER of squares in a NxN grid (1+4+9+…+NxN)
3
Assembling three pyramids always produces the same three dimensional object. (note: I used induction, Jackie does better in 9.1)
4
One face of the object is made up of an entire NxN square, plus the edge of another one, forming an Nx(N+1) rectangle.
5
The top layer contains HALF as many blocks as the other layer, which is the same as a FULL layer of HALF blocks. 
6
Arrays: That a three dimensional box is composed of AxBxC little cubes
7
If you use three pyramids you have to divide by 3 later.
8
You can use a letter to stand for a variable in a formula.

Figure 5: Crucial ideas

Is this proving?

Did David's visual proof answer the question posed by the students in 1996, namely that a proof ought to explain? When working with the fifth graders in the May 2000 cohort, in the discussion following the presentation the question was raised: Is this proving? The question can be rephrased and then answered in a number of ways.
 

Did the children feel that it was a proof?

In the class of May 2000, nine students responded “No” to the question “Think of what David did yesterday with the block towers. Was it proving?”, fifteen responded “Yes,” and three responded “Yes, partly”.  On the face of it this indicates that most of the children did feel that the manipulation of the blocks and the accompanying commentary constituted proving or a proof.  It should be noted however that their reasons for feeling so varied widely, as did their personal definitions of “proof” and “proving”.Among those who responded “No” reasons included:
“He was just showing us a way that it worked but he didn't prove that it worked.”“He wasn't proving he was finding out the answer.” (4 similar responses)

“I don't think it was proving because we were not trying to prove the answer.”

“It didn't prove the answer was right.”

“It's a formula not a way to prove.”

Among those who responded “Yes” reasons included:
“When he was using the blocks he showed evidence that his answer was correct.” (2 similar responses)“Yes, it was proving because we got the answer for the 4 by 4 and for the 5 by 5.” (3 similar responses)

“Yes, it was proving because it was make a math sentence and show what I did” (2 similar responses)

“Yes, it was proving because he was showing us how you get all the squares.” (3 similar responses)

“He showed us an easier way to do the problems and he proved that it works when we tested it.  Also he showed us why it worked.”

“He showed how he did it and why it worked.”

“He showed exactly what he was doing and because he explained why he was doing it.”

The range of reasons for accepting or rejecting the argument using blocks as a proof leave us in the situation of claiming that in some cases a child who accepted the argument as a proof, in fact did not understand it to be a proof for the kinds of reasons we would like, and so we might assert that it wasn’t a proof for that child.  For example, the children who said it was a proof because a couple of examples were shown (“Yes, it was proving because we got the answer for the 4 by 4 and for the 5 by 5.”) seem only to have seen the argument as the production of some empirical evidence, not as a generic example that could apply to any number of squares.  Similarly, those who emphasized the production of a formula (“Yes, it was proving because it was make a math sentence”) seem also to have missed the point we would have liked them to have understood.This raises the question of what we feel is a proof.  Our research, while focussed on the children, tells us about ourselves. By considering that responses that do not fit our expectations we can identify some of those expectations, specifically that we expect a proof to be:

Would a mathematician accept the argument as a proof?

As with the children the answer depends on the mathematician you ask.  Presumably, Man-Keung Siu, creator of the visual proof on which the blocks demonstration was based, Roger Nelsen, editor of the book in which it was found (Nelsen, 1993) and Phillip J. Davis, who asserts that such arguments should play a more significant role in mathematics education (1993), are three mathematicians who would call it a proof.  But others might disagree.
 
 

Do we accept the argument as a proof?

Yes and no. In the abstract, it is a proof, but in practice it matters how the audience reacts.  Is a play a comedy if no one laughs, even if the author expected them to?

 

References:

Anderson, J. (1996). The place of proof in school mathematics. Mathematics Teaching 155, pp. 33-39.

Cobb, Paul, Wood, Terry, & Yackel, Erna. (1993). Discourse, mathematical thinking, and classroom practice. In E. Forman, N, Minick, & C. A. Stone (Eds.), Contexts for learning: Sociocultural dynamics in children's development (pp. 91-119). N. Y.: Oxford University Press.

Davis Ph. J. (1993). Visual Theorems. Educational Studies in Mathematics 24(4) 333-344.

Graves, B., & Zack, V. (1996). Discourse in an inquiry math elementary classroom and the collaborative construction of an elegant algebraic expression. In Puig, L., & Gutiérrez, A. (Eds.), Proceedings of the Twentieth Conference of the International Group for the Psychology of Mathematics Education (PME 20) (pp. 27-34). Valencia, Spain.

Hanna, G. (1995). Challenges to the importance of proof. For the learning of mathematics, 15(3), pp. 42-49.

Nelsen, R. (1993). Proofs without words: Exercises in visual thinking. Washington, DC: Mathematical Association of America.

Reid, D. A. (1992). Mathematical induction: An epistemological study with consequences for teaching. Unpublished master’s thesis, Concordia University, Department of Mathematics and Statistics.

---------. (1995). The need to prove. Unpublished doctoral dissertation, University of Alberta, Department of Secondary Education.

---------.(1997). Constraints and opportunities in teaching proving.In Erkki Pehkonnen (Ed.)Proceedings of the Twentieth-first Annual Conference of the International Group for the Psychology of Mathematics Education, (Vol. 4, pp.49-55). Lahti, Finland

---------. (1998). Why is proof by contradiction difficult? In Alwyn Olivier and Karen Newstead (Eds.) Proceedings of the Twentieth-second Annual Conference of the International Group for the Psychology of Mathematics Education, (Vol. 4 pp. 41-48 ) Stellenbosch, South Africa.

Zack, V. (1995). Algebraic thinking in the upper elementary school: The role of collaboration in making meaning of 'generalisation'. Proceedings of the Nineteenth International Conference of the International Group for the Psychology of Mathematics Education (PME 19) (Vol. 2, pp. 106-113). Recife, Brazil.

---------. (1997). “You have to prove us wrong”: Proof at the elementary school level. In E. Pehkonen (Ed.), Proceedings of the Twenty-First Conference of the International Group for the Psychology of Mathematics Education (PME 21) (Vol. 4, pp. 291-298). Lahti, Finland.

--------- . (1998). Coherence in five fifth grade boys' argumentation about proof: A sociolinguistic study of the role of repetition and logical structure in the boys' talk. Paper and presentation, NCTM Research Presession, National Council of Teachers of Mathematics (NCTM) 76th Annual Meeting, Washington, D. C., April 2-4, 1998.

----------. (1999). Everyday and mathematical language in children's argumentation about proof. Educational Review, 51(2). Special issue: Culture and the mathematics classroom. Guest editor: Leone Burton.pp. 129-146.


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