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Introducing Young Children

to the Role of Assumptions inProvingAndreas J. Stylianides

a

aUniversity of Oxford , United Kingdom

Published online: 05 Dec 2007.

To cite this article:Andreas J. Stylianides (2007) Introducing Young Children to theRole of Assumptions in Proving, Mathematical Thinking and Learning, 9:4, 361-385,

DOI: 10.1080/10986060701533805

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MATHEMATICAL THINKING AND LEARNING, 9(4), 361385

Copyright 2007, Lawrence Erlbaum Associates, Inc.

Introducing Young Children to the Roleof Assumptions in Proving

Andreas J. Stylianides

University of Oxford, United Kingdom

The notion of assumptions permeates school mathematics, but instruction tends to

highlight this notion only in the advanced grades. In this article, I argue that it is

important for even young children to develop a sense of the role of assumptions

in proving, and I investigate what it might mean and look like for instruction

to promote this goal. Toward this end, I study an episode from third grade that

describes the first time that the students in the class were introduced in a deliberate

and explicit way to the role of assumptions in proving. The central role of themathematical task in the episode is identified, and features of mathematical tasks

that can generate rich mathematical activity in the intersection of assumptions

and proving are discussed. In addition, issues of the role of teachers in fostering

productive interactions between students and mathematical tasks that have those

features are considered.

Almost every conclusion in life depends on a set of statements to which people

have agreed or which they accept (implicitly or explicitly); these statements

may be called assumptions (Fawcett, 1938). For example, a group of children

who say that the librarian of their school can win the best teacher award are

operating under the assumption that the librarian is a teacher. The notion of

assumptions permeates almost every mathematical activity, both in school and

in the discipline. Consider, for example, a class working on the task of showing

how many different addition sentences there are for 5. Do the sentences 1 + 4=5

and 4 + 1=5 count as different? Ones interpretation of what counts as different

Correspondence should be sent to Andreas J. Stylianides, University of Oxford, Department

of Education, 15 Norham Gardens, Oxford OX2 6PY, United Kingdom, E-mail: andreas.

[email protected]



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362 STYLIANIDES

in this task reflects ones assumptions. In the context ofproving1 in particular,

assumptions play a crucial role as the building blocks of arguments, proofs,and mathematical theories. One cannot make sense, or examine the validity, of

arguments and proofs unless one understands the (stated or unstated) assumptions

that underlie them and support their conclusions. To continue with the number

sentences example, consider a child who concludes that there are exactly four

different addition sentences for 5: 1+4=5, 2+ 3=5, 3+ 2=5, and 4 + 1=5. Is

this conclusion true? It would be true under a specific set of assumptions: (1)

the task refers to two-addendnumber sentences over the set of positive integers

from 1 to 5, and (2) commutative number sentences count as different. Yet,

the conclusion would be false under a different set of assumptions that wouldpermit, for example, multi-addendnumber sentences such as 1+1+3=5.

Although assumptions permeate students mathematical activity in all grades,

instruction tends to highlight them only in advanced grades. This raises the

question of whether it would be meaningful for instruction to help even young

children (i.e., children in the early elementary grades) develop a sense of the

role of assumptions in their mathematical activities, especially in the context

of proving. There are at least three (interrelated) reasons that it is important to

introduce even young children to the role of assumptions.

First, given the growing appreciation of the idea that doing and knowing mathe-matics is a sense-making activity (e.g., Fennema & Romberg, 1999; Hiebert &

Carpenter, 1992; Mason, Burton, & Stacey, 1982; National Council of Teachers

of Mathematics, 2000), explicitness on the role of assumptions can allow children

to understand and examine critically the conclusions that they accept based on the

grounds that support them. According to Fawcett (1938), proving situations offer

a natural context for children to understand the relationship between conclusions

and their underlying assumptions, thereby fostering childrens ability for reflective

thinking (Dewey, 1910); that is, active, persistent and careful consideration of

any belief or supposed form of knowledge in the light of the grounds that support

it and the further conclusions to which it tends (quoted in Fawcett, 1938, p. 6).

1I use the term proving to describe the activity associated with the search for a proof. In turn,

I use the term proof to describein the context of a classroom community at a given timea

mathematical argument that fulfills three criteria: (1) it builds on true statements that are accepted

by the community and that can be used without further justification, (2) it uses valid modes of

argumentation that are known or conceptually accessible to the community, and (3) it employs

appropriate modes of representation that are known or conceptually accessible to the community (see

Stylianides, 2007, for elaboration). The terms true, valid, and appropriate that are used in the

descriptions of the three criteria for a proof should be understood in the context of what is typically

agreed on in the field of mathematics within the domain of particular mathematical theories. In

addition, it should be noted that the notion of assumptions corresponds to the first criterion regarding

the statements on which a given argument or a proof is based.



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THE ROLE OF ASSUMPTIONS IN PROVING 363

Second, the fact that assumptions tend to be highlighted only in the advanced

grades creates a discontinuity in students mathematical experiences, as it

involves a didactical break (Balacheff, 1988) represented by the requirement

for a new way of thinking about mathematics. Studies such as Fawcett (1938)

and Schoenfeld (1985) at the high school and university levels, respectively,

point to the systematic work that is generally required by instruction at these

levels to help students develop an appreciation of important mathematical ideas

and processesrelated to assumptions, proving, and problem solvingthat

instruction at the lower levels tends not to highlight. If instruction in the lower

grades highlighted these ideas and processes, then more advanced mathematics

would not seem so alien to students and the remedial part of advanced instructionwould not be so necessary.

Third, given the central role that assumptions play in mathematical practice

(see, e.g., Fawcett, 1938; Kitcher, 1984), one may claim that we cannot have a

viable school mathematics curriculumor opportunities that have integrity for

students to learn itunless we help even young children develop an appreci-

ation of the role of assumptions in mathematics. This claim finds support from

ideas that were advanced by educational scholars such as Bruner (1960), Schwab

(1978), and Lampert (1992) about how instruction can organize students experi-

ences with disciplinary concepts in school. Bruner (1960) asserted that thereshould be a continuity between what a scholar does on the forefront of his

discipline and what a child does in approaching it for the first time (pp. 2728).

Likewise, Schwab (1978) argued for a school curriculum in which there is,

from the start, a representation of the discipline (p. 269), and in which students

have more intensive encounters with the inquiry and ideas of the discipline as

they progress through school. The idea expressed in these quotations is not that

instruction should treat students as little mathematicians. Rather, the idea is

that instruction should help students learn how to do what Lampert (1992) called

authentic mathematics; that is, participate in activities that are genuinely mathe-matical and learn from those activities. According to Lampert, [c]lassroom

discourse in authentic mathematics has to bounce back and forth between being

authentic (that is, meaningful and important) to the immediate participants and

being authentic in its reflection of a wider mathematical culture (p. 310).

These reasons on the importance of helping even young children develop

a sense of the role of assumptions in mathematical activityespecially in the

context of provingraise the following question: What might it mean and look

like for teachers to help young children develop a sense of the role of assumptions

in proving? In this article, I investigate this question, with particular attentionto how young children can be assisted to develop a sense of the fundamental

mathematical idea that the truth of conclusions depends on the assumptions

supporting these conclusions. To promote my research goal, I will study an

episode from a third-grade class that describes the first time when the children



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364 STYLIANIDES

in this class were introduced in a deliberate and explicit way to the role of

assumptions in proving.

Before presenting the episode, I will use an example from Fawcetts (1938)

high school geometry class to exemplify the idea that the truth of conclusions

depends on the assumptions that support them. In addition, the example will offer

an image of how instruction can introduce advanced students to this idea through

experiences that center on the concept of proof and will set the stage for my

investigation of how instruction can promote the same goal in the early grades.

EXEMPLIFYING THE IDEA THAT THE TRUTH OF CONCLUSIONSDEPENDS ON THE ASSUMPTIONS THAT SUPPORT THEM

Fawcetts (1938) high school geometry class at Ohio State Universitys

laboratory school was organized as a community of mathematical discourse in

which students were invited to explore interesting situations while the teacher

assumed the role of a facilitator, guiding students explorations and directing

them toward discovery and proof of important mathematical results. A key idea

discussed in Fawcetts class was that conclusions are true only within the

limits of the assumptions on which they depend (p. 71). In a series of lessons,the teacher engaged the students in a number of explorations, which culmi-

nated in a task asking the students to analyze a proof of the following theorem:

The sum of the interior angles of a triangle is 180. The students analyses

revealed that the proof depended on several assumptions. Fawcett focused the

students attention on one of the assumptions, the parallel postulate (i.e. the

fifth postulate of Euclidean geometry), which denotes that through a given point

not on a given line one and only one line can be drawn parallel to the given

line. As a result of their engagement in this task, and under the guidance of their

teacher, the students were introduced to the non-Euclidean worlds of Elliptic(or Riemannian) and Hyperbolic (or Lobatchewskian) geometries. Mathemati-

cians developed these geometries essentially by changing one of the assumptions

of Euclidean geometry; namely, the parallel postulate. For the development of

Elliptic geometry, Riemann replaced the parallel postulate with a statement that

asserted the existence ofnoparallel lines; whereas for the development of Hyper-

bolic geometry, Lobatchewsky replaced the same postulate with a statement

that asserted the existence ofmany parallel lines. This difference in the sets of

assumptions of the three geometries supports some different conclusions. For

example, although the sum of the interior angles of a triangle in Euclideangeometry is equal to 180, in Elliptic geometry the sum is more than 180 and

in Hyperbolic geometry the sum is less than 180.

According to Fawcett, the analysis of the proof of the theorem concerning

the sum of the interior angles of a triangle, followed by a consideration of the



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mathematical developments connected with the change of the parallel postulate,helped his students realize that Euclids assumptions were not inherent in thenature of space and that the choice of assumptions by Euclid, Riemann, andLobatchewsky offered accurate accounts of particular kinds of geometricalspaces. For example, Euclidean geometry describes the planar space, whereas

Elliptic geometry describes the surface of a sphere. Thus, this activity helpedthe students develop a sense of the fact that different sets of assumptionscan give rise to self-consistent and useful theories that promote understandingof different aspects of mathematics. Even though the theories yield someconclusions that appear to be contradictory across theories, the theories are inpeaceful coexistence.

The example from Fawcetts high school class illustrates how a teacher can helpadvanced mathematics students understand the role of assumptions in proving and,in particular, the idea that the truth of conclusions depends on the assumptionsthat support them. Fawcett promoted his students understanding of this idea byengaging them with problems and issues faced by mathematicians in the historical

developmentofgeometry.Theadvancedmathematicallevelofhisstudentsallowedsuch an instructional approach. The question that is raised at this point is: Howmight teachers promote much younger students understanding of the same idea?My discussion of the episode from third grade offers insights into this question.

THE EPISODE

Background

The episode is derived from a large longitudinal database of the MathematicsTeaching and Learning to Teach Project at the University of Michigan. This

database documents an entire year of the mathematics teaching of Deborah Ball,

a well-known teacherresearcher (see, e.g., Ball, 1993; 2000), in a third-gradeclass in a U.S. public school. The records collected across that year includevideotapes and audiotapes of the classroom lessons; observation notes; classroomand interview transcripts; copies of students work in their notebooks, homeworkassignments, and quizzes; and copies of the teachers journal entries with her

lesson plans and teaching reflections.The class was socioeconomically, ethnically, and racially diverse, with 22

students of multiple ability levels. The mathematics period in the class wasapproximately one hour long, five days per week. During each period, the class

worked on one or two tasks that were carefully selected by the teacher to engagestudents in rich mathematical activity. The period often began with the students

exploring a task individually or in pairs, then in small groups, and ultimatelyin the whole group. The curriculum was organized around units on generalmathematical topics such as number theory, integer arithmetic, and probability.



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366 STYLIANIDES

Deborah Balls teaching of the third-grade class was organized as a year-long

teaching experiment. One of the goals of this experiment was to explore what it

would mean and look like to help young children become skilled mathematical

reasoners. To promote this goal, Ball modeled her classroom as a community

of mathematical discourse, in which the validity for ideas rest[ed] on reason and

mathematical argument, rather than on the authority of the teacher or the answer

key (Ball, 1993, p. 388). Inspired by Bruners (1960) notion of intellectual

honesty, Balls teaching was a continuous struggle to achieve a defensible balance

between two (often competing) considerations: mathematics as a discipline and

children as mathematical learners.

I must consider the mathematics in relation to the children and the children in

relation to the mathematics. My ears and eyes must search the world around us,

the discipline of mathematics, and the world of the child with both mathematical

and child filters. (Ball, 1993, p. 394)

Balls pedagogical commitment to the two considerations is important in under-

standing her decisions in the episode I describe later.

The above description of Balls teaching practice, as well as research reports

on this practice (see, e.g., Ball & Bass, 2003; Stylianides, 2007), indicate its non-typical character. This non-typical character is, from a methodological point of

view, a necessity for the purposes of this article: Given that in most elementary

classrooms today there is little attention to issues of proving (see, e.g., Ball,

Hoyles, Jahnke, & Movshovitz-Hadar, 2002, for an account of the current place

of proving at the elementary school level in different countries), to study what it

might mean and look like for teachers to help young children develop a sense of

the role of assumptions in proving, researchers need to examine teaching practices

that promoted this goal with considerable success. By so doing, researchers can

develop a better understanding of what is entailed in introducing young childrento the role of assumptions in proving, thereby setting the foundations for the

design of means that can support large numbers of elementary school teachers

to promote this goal effectively among students.

The episode occurred on October 3 and describes part of the work of the

class on integer addition and subtraction, which began on September 26. What

prompted the class to investigate aspects of integer arithmetic was students

conceptions that one cant take nine away from zero, which on September 25

made many students think that 300 190=290. Ball noted in her journal that

day: [E]xpanding [students] working domain for numbers seems a reasonablepriority before either estimation competence (number sense) or precision with

computation (Teachers Journal, September 25, p. 26).

To support her students thinking about integers, Ball made a representational

model available to them that she called the building model (see Figure 1). The



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0 (ground floor)

Roof

12

11

10

9

8

7

6

5

4

3

2

1

1

2

3

4

5

6

7

8

9

10

11

12

FIGURE 1 The building model.

model consisted of a building with 12 floors below ground, the ground floor (calledthe 0th floor), 12 floors above ground, and a roof. Ball used the circumflex

() above the numerals in the place of the minus sign of negative integers. Herdecisiontonotfollowthestandardmathematicalnotationwasbasedonpedagogical

reasons. She believed that substituting the circumflex for the minus sign wouldhelp the students focus on the idea of a negative number as a number, not as an

operation (i.e., subtraction) on a positive number (Ball, 1993, p. 380; emphasis inoriginal). For consistency, I use the same notation in this article.

The class used the building model to figure out answers to number sentenceswith integer addition and subtraction in which negative integers appeared only

at the beginning of the sentences. To interpret and figure out answers to suchnumber sentences, the students imagined that each number sentence represented

the trip of a person in the building. For example, the number sentence 5+21=?

would be interpreted and solved as follows.

The first term indicates the persons starting position. So, in this case, the person

begins five floors below the ground floor. The addition operation indicates that the

person has to go up the building. The second term indicates that the person has



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368 STYLIANIDES

to travel two floors in this direction. The subtraction operation indicates that the

person has to go downthe building. The third term indicates that the person has to

travel one floor in this direction. The person ends up being four floors below the

ground floor, so the answer to the number sentence is 4.

On October 2, the day before the episode, the students worked on findingdifferent ways for a person to get to the second floor.

Description

On October 3, the teacher gave the following task to the class: How many ways

are there for a person to get to the second floor? Prove your answer. In herjournal entry after class that day, the teacher noted about the task:

1. The emphasis here, compared to yesterday, was on figuring out and

justifying how many different ways there are for a person in the building to

get to the second floor. I knew that those kids who only wrote two-addendnumber sentences (e.g., 4 + 6=2) would have 25 answers but those, like

Lisa, who wrote multi-addend number sentences (e.g., 6 + 10 + 3 2 + 1

4=2) would have infinite solutions. (Teachers Journal, October 3, p. 37)

The class period began with the students working on the task individually or

in their small groups. At some point, Ball called them back to the whole group

to clarify the task:

2. Ball: [ ] Maybe you finished finding all the ways, maybe you didnt.

I want you to think right now about how many ways there are.

Did you find them all? If you found them all, figure out how manythere are and prove thats all there are. If you didnt find them all,

write down what you think about how many there should be andwhyyou think that.

The students continued their work on the task for 10 more minutes. Then, Ball

called them back to the whole group and asked them to share their work on the

task. Among several volunteers, Ball chose Riba to share her work. Riba went

up to the board, stood by the building model, and said:

3. Riba: See, look [pause] arent there 25 numbers [she points to the floors

at the building model]? Then there have to be 25 cause [pause] 25answers because you cant make more because there are only 25.

Nathan pointed out that one would have to also count the roof to find 25 floors,

but Riba said that there were 25 floors without counting the roof as a floor. Ball



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asked Riba to show Nathan why she thought there were 25 floors. Riba counted

the floors one by one, beginning from the floor minus twelve and ending at thetwelfth floor. Nathan admitted that Riba was right. Riba then offered to give a

different explanation, as she said, about why there were 25 ways for a person

to get to the second floor:

4. Riba: This is twelve below zero [points to the lowest floor]. If you writetwelve below zero in your notebook [writes 12 on the board], you

would [pause] Im saying, look, take twelve below zero. Then

you take [counts floors up from minus twelve to two] one, two,

three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen,

fourteen! Plus fourteen equals two [completes the following numbersentence on the board: 12+14=2].

Riba then noted that the same idea would apply for all 25 floors, not just thefloor minus twelve. Ball asked Riba to write down some more number sentencesand helped her record the sentence that corresponded to the trip beginning atfloor minus eleven: 11+13=2. Riba took over from there.

5. Riba: And there could be another one and another one. You see, ten belowzero plus [pause] this would have to be twelve equals two [writes10+12=2 on the board]. You have to keep on going like that, soyou will finish at one below zero, and then go to ground and yousay: zero plus two equals two [writes 0 + 2=2 on the board]. Likethat. And then keep on going, keep on going, keep on going until

this, and after you finish you go to twelve [points to the highestfloor] and then you go this way and you get to two [moves her handdownwards from the twelfth floor to the second]. Ill show you thisother way. Twelve [writes 12 on the board] there is [counts floorsdown from the eleventh floor to the second] one, two, three, four,five, six, seven, eight, nine, ten! Then it will be twelve take away tenequals two [completes the following number sentence on the board:12 10=2]. I stopped at two, and thats why it equals two! [ ]

6. Ball: So you are saying that there are 25 answers?7. Riba: Yeah!

Ball invited comments from the other students. Betsy asked for some clarification

and Riba explained her thinking again. Ball then summarized Ribas argument:

8. Ball: [

] What she is saying is that you can start in every floor andthen add up to two [shows up the building to the second floor] orsubtract to two [shows down the building to the second floor]. And

there are 25 floors, so she got 25 answers. Thats all she is saying

right now. Does that make sense or not?



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370 STYLIANIDES

The class seemed to have a good understanding of Ribas argument and Ball

decided to move on, asking whether other students found a different answer to

the task.

Several students said that they found more than 25 ways. Jeannie said that

she found 26 ways while Lucy said that she could not read the number of ways

she thought there were because the number was big. Ball decided to give the

floor to Lucy. In her journal entry after the class period that day, Ball explained

the rationale for her decision.

9. I deliberated for a moment about whether to have Jeannie write all her 26ways

up so we could see how she got more than Ribas 25. Since I knew shed only

written two-addend number sentences, I decided it would not be a fruitful use

of time just to discover that shed written one down more than once. Still, this

is a dilemma, for will Jeannie understand that it has to be 25 if you only write

two-addend sentences? Or will she not understand that, logically, there can only

be 25. (Teachers Journal, October 3, p. 38; emphasis in original)

When Lucy came up to the board, she said that she had written number sentences

such as 12 + 24 10=2, and this was how she decided that there was a big

number of ways. Lisa was the next person to come up to the board. She said

she had followed an approach similar to Lucys, and she recorded the numbersentence 11+ 11+ 2=2.

Ball then asked the class to think about what was the same between the

solutions of Lucy and Lisa, and how this was different from what Riba had

presented earlier. After eliciting some ideas from the students, Ball pointed out

that in Ribas solution the person in the task went to the second floor in a single

stop, whereas in the solutions of Lucy and Lisa the person made more than one

stop en route to the second floor. Balls journal entry after the class period that

day is illuminating of her thinking:

10. I decided to push the point that Lucy and Lisa had both made the assumption

that the person in the problem could make more than one stop en route to the

second floor (e.g., 5+ 9 2=2). This seemed, for the moment, the more important

issue relative to the problem at hand: of how many solutions there are to the

problem how many ways are there to get to the second floor. With each of the

assumptions kids have made, the answer would be different (25 vs. infinite) and

I wanted the kids to see that, and to begin to develop a sense for the role of

assumptions. (Teachers Journal, October 3, p. 39; emphasis in original)

In trying to ensure that the students understood what Lucy and Lisa did, Ball

invited the class to describe a trip with multiple stops en route to the second

floor. The class as a group made up a trip that corresponded to the following

number sentence: 6 1+7 8 2=2. As soon as the class concluded the



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description of the trip, Lucy observed that the person could keep going up and

down many times before finishing his trip at the second floor. This observationprovoked Betsys reaction. Betsy objected to the way Lucy and Lisa solved the

task by saying: The person wants to go to the second floorhe doesnt want

to go all over the building. Lisa defended hers and Lucys approach on the

grounds that, even if the person made multiple stops, he wouldeventually get to

the second floor.

Ball asked the other students in the class what they wrote in their notebooks

in solving the task. Four students said that they had solutions like those of Lucy

and Lisa. The other students in the class said that they had only two-addend

number sentences like Riba did. Ball said the following:

11. Ball: Neither one of these is right or wrong. Both are okay, but

they are important. There is an important thing here for you

to look at. Everyone look up here, please. Lucy and Lisa and

Mei and, I dont remember who else [students mention Seans

name] and Sean made a different, what we call, assumption. Im

going to write the word on the board because its an important

word for us [writes the word assumption on the board]. Thatmeans that they thought something a little bit different than what

Betsys assumption was. Betsys assumption was that the person

wanted to go as fast as possible to the second floor. The problem

doesnt say that, but its okay that she thought that. That was her

assumption. That means: thats what she thought. And then she

did all her work on the problem because of that way of thinking.

Right, Betsy? [Betsy affirms.] So did some of the rest of you. Like

Cassandra assumed the same thing. [ ] She assumed that the

person wanted to get there quickly. What was Lisas assumption?What did Lisa think about the person? Betsy?

12. Betsy: That he wanted to take the whole day going from place to place.

[Students laugh.]

13. Ball: Maybe he wanted to visit people, or wasnt in a rush. The problem

doesnt say that, does it? [Students agree that the problem does

not say that.] But it also doesnt say that he didnt. So they made

twodifferentassumptions. They had two different ideas. Lisa did

her work after she made her assumption. You [turning to Betsy]

did your work after you made yours [i.e., your assumption]. Andboth are right, but they are different assumptions. If you make

Betsys assumption, if you assume that the person wants to get to

the second floor as fast as possible, how many ways are there for

the person to get to the second floor? [pause] If the only thing you



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372 STYLIANIDES

are going to let the person do is start somewhere and go right tothe second floor, in how many ways can the person do that? Chris?

14. Chris: 26.15. Ball: Why 26?

16. Chris: Because he can also start from the roof.17. Ball: Okay. He can start from the roof or every other floor and go to

the second floor.

Ball remarked that the proof for 26 would be similar to the proof that waspresented earlier by Riba, but, in this case, one would also have to consider the

roof as a possible starting point for the trips. She also pointed out that Chris

made a different assumption. The class stated Chriss assumption: that you canalso count the roof. Then Ball raised the question:

18. Ball: If you assume what Lisa and Lucy and Sean assumed, that the

person wanted to travel around, how many ways are there for the

person to do it?

Ball called on Ofala to answer this question because, as she noted in her journalentry after the class period that day, she knew that Ofala had in her notebook an

idea about afinidy (meaning to say infinity). However, Ofala did not seem toremember this idea. Ball then called on Lucy, who had said earlier that she couldnot read the number of ways she thought there were because the number wasbig. Ball asked her to come up to the board to write this number. Lucy wrote

the number 8,000,000,000,000,000,000,000,000 on the board, commenting thatshe wanted to add more zeroes to the number but there was not enough space inher notebook.

19. Ball: What are you trying to say with this number?20. Lucy: Im trying to say that there is a lotof them!

Then Ball called on Jeannie, who said:

21. Jeannie: I think there are as much as you want because [pause] he canspend months going up and down if he wanted to! [Students

laugh.]

The discussion continued for a few more minutes before the class periodended. Several students tried to explain how many ways there would be if one

assumed that a person could travel around. Just before the end of the period Meisaid, The answer goes on for ever. However, there was not enough time leftfor her to explain what she meant by that.

Ball concluded her journal entry about the class period that day with areflection on the students emerging understanding of the notion of infinity:



15/28


22. I think people got the sense that there were a LOT of answers, given the second

assumption [i.e., that the person could travel around]. But a LOT can mean 28 or

1,000 or 9,000,000,000,000 to an eight-year-old. And endless doesnt necessarily

mean something different from a very big numberbig numbers are themselves

endless to these kids, I think. (This is a new twist on the idea of the confounding

of infinity as a very big number.) (Teachers Journal, October 3, p. 41)

Discussion

My discussion of the episode is organized into four sections. In the first, I specify

the two main assumptions one could make about the conditions of the task in

the episode and I connect them with the proving activity that was generatedin the class. In the second, I elaborate further on the role of assumptions in

the episode. In the third, I identify analogues between students experiencewith assumptions in the episode and mathematicians work with assumptions inthe discipline. In the last section, I discuss features of mathematical tasks that

can generate rich mathematical activity in the intersection of assumptions andproving. This discussion raises issues about the relationship between assumptionsand definitions, and about the role of teachers in fostering productive interactions

between students and mathematical tasks that have those features.

Two different assumptions for the tasks conditions and proving activity

generated in the class. The mathematical task in which the teacher engagedher students in the episode (figuring out and proving how many ways there

are for a person to get to the second floor of the building) was purposefullyambiguous, thus allowing the formulation of different assumptions about itsconditions. In particular, there were two main assumptions a student could make

about the nature of the persons trips in the task: direct route to the second floorversus multiple stops en route to the second floor. Depending on the assumption

a student would make (consciously or unconsciously), the student would engagein a proving activity that was expected to result in a different conclusion: 25ways (if the roof was not taken as a possible starting point of the persons trips)

versus infinitely many ways.In implementing the task in her class, Ball had two primary goals. First, she

aimed to help her students develop arguments (and, if possible, proofs) for the

number of different ways, based on each of the two main assumptions aboutthe tasks conditions (line 1). Second, she aimed to help her students understandthat the apparent conflict between the different conclusions students could reach

(25 ways versus infinitely many ways) was due to the different assumptions thatsupported these conclusions (line 10).Ribas approach to the task was based on the assumption that the person in

the task would go directly to the second floor, even though she did not seemto be conscious of the fact that she was making this assumption. Riba provided



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374 STYLIANIDES

two related arguments about why, given her assumption, there were exactly 25ways (lines 35).2 Ribas first argument, summarized also by Ball (line 8), wentas follows: The person can begin from any floor and travel directly from thereto the second floor; because there are 25 floors, there are 25 (different) ways forthe person to get to the second floor. This argument was based on the statement

that there were 25 floors in the building, which was not readily accepted bystudents such as Nathan who claimed that one would also need to count theroof to find 25 floors. To refute Nathans claim, Riba counted, one by one, allthe floors in the building. Ribas second argument took the form of systematicenumeration of all the possible trips to the second floor, but, instead of listing thenumber sentence corresponding to each trip, she described in some detail how

one would go about generating these number sentences and gave representativeexamples on the board (lines 45).

After the presentation of Ribas arguments, Ball directed the studentsattention to the other main assumption one could make about the tasks condi-tions, namely, that the person could make multiple stops en route to the secondfloor. She gave the floor to Lucy and Lisa, who wrote multiterm number

sentences on the board to illustrate what made them think that the number of waysfor the person to get to the second floor was big. The two different approachesto the task that were followed by different groups of students generated somecontroversy in the class. Some students (e.g., Betsy) thought that the directroute approach was the only legitimate approach. Other students (e.g., Lisa)thought that the multiple stops approach was also legitimate because the person

does eventually get to the second floor. This controversy naturally raised theneed for the teacher to highlight the role of assumptions in what the students hadbeen arguing about. Ball explained to the students that each approach was basedon a different assumption (lines 11 and 13). She called the students attention tothe termassumptionsand tried to help them understand the relationship betweendifferent assumptions and different conclusions by asking them to consider what

would be the answer to the task given each assumption (lines 13 and 18).The students easily addressed the case in which the person in the task would

follow a direct route to the second floor. They did so by referring to the argumentsthat Riba presented at the beginning of the episode. However, the students faceddifficulties in dealing with the case in which the person in the task could makemultiple stops en route to the second floor, primarily because the solution setin this case had infinite cardinality. Although several students expressed ideas

that approximated the notion of infinity, they could not articulate and explaintheir thoughts clearly (lines 1921). For example, Jeannie claimed there are as

much [ways] as you want (line 21). She tried to justify her claim by sayingthat the person in the task could spend months going up and down if he wanted

2These arguments can be considered as proofs in the given context.



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to (line 21). This argument could potentially be the basis of a proof that would

develop along the following lines: The person can go up and down, visitingfloors in the building before he ends up at the second floor, so that in each

new trip the person makes a different number of stops than in any previous

trip. Because the number of stops the person can make extends ad infinitum, the

number of possible trips is infinite.3 Yet, it is an empirical question whether

third graders can formulate or understand an argument such as this.

The role played by assumptions in the episode. Figure 2 summarizes

the work of the class on the mathematical task in the episode, indicating the

two-fold role that assumptions played in the episode.

How many ways are there for aperson to get to the second floor?

Prove your answer.

Assumptionsabout theconditions of themathematical task

Mathematical task

Proving activity

The person needs tofollow a direct route to the

second floor

Development of argumentsbased on the assumptionabove. The developmentof these arguments has toconsider a finite number of

possible cases

The person can makemultiple stops en route to

the second floor

ConclusionsFinite number of ways

(25 ways, if the roof is not taken

as a possible starting point)

Infinite number of ways

Development of argumentsbased on the assumptionabove. The developmentof these arguments has toconsider an infinite number

of possible cases

Conflict

ResolutionDifferent (legitimate) assumptions gave rise to

different arguments, which in turn resulted in differentconclusions. So, although the two conclusions seem

to be contradictory, both of them are legitimate.

FIGURE 2 The work of the class on the mathematical task in the episode.

3For example, the person in the problem can make his trips according to the following process.

For the first trip, the person goes up and down oncebetween the third floor and the fourth floor, and

then ends up at the second floor. For each new trip, the person goes up and down one time more

than in the previous trip between the third floor and the fourth floor, and then ends up at the second

floor. This procedure contains no terminating condition and enters a process of infinite recursion.



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376 STYLIANIDES

The first role that assumptions played was as a tool in the hands of the teacher

to engage students in two broad categories of proving activity: activity that

involves a finite number of cases, and activity that involves an infinite number

of cases. Accordingly, the students had an opportunity to develop a sense of the

fundamental mathematical idea that some proving strategies that can be used to

solve tasks that belong to the former category are not applicable in solving tasks

that belong to the latter category (Stylianides & Ball, in press). For example,

if a student assumes that the person has to follow a direct route to the second

floor, then the systematic enumeration of cases is a valid strategy for proving

that there are exactly 25 ways for the person to get there. This is an important

strategy that can also be used by students to solve tasks on combinatorial ideas(see, e.g., English, 1991; 2005; Inhelder & Piaget, 1958). However, this strategy

is not as useful if a student tries to prove that there are infinitely many ways for

the person to get to the second floor, operating under the assumption that the

person can make multiple stops en route to the second floor.

Besides the two main assumptions described in Figure 2 about the nature of

the persons trip, there were some other elements in the tasks conditions that

in some students eyes were also subject to different assumptions. For example,

even though almost all students assumed that the 25 floors of the building

were the only possible starting points for the persons trips, Chris assumedthat the roof of the building was another possible starting point (lines 1317).

Nevertheless, the teacher did not have students explore the implications of the

different assumptions about the possible starting points for the persons trips.

Specifically, she did not ask them to produce a new proof for 26; rather, she

simply remarked that the proof for 26 would be similar to Ribas proof for

25. Had the teacher decided to have students explore the implications of these

assumptions, the students would engage in the same type of proving activity;

namely, activity that involves a finitenumber of cases. Thus, the understandings

that the students could develop about proving strategies from engaging in thisactivity would likely not make a significant difference to the understandings that

they developed in the activity in the episode. Using Daviss (1992) notion of

residue as a way of describing the student understandings that remain after an

activity, we may say that the activity of exploring different assumptions about

the possible starting points for the persons trips would not offer the chance of

leaving behind any important residue in addition to the residue that remained

from the activity in the episode.

The second role that assumptions played in the episode was as a means for

resolving a problematic situation; namely, the apparent conflict between thedifferent conclusions reached by different groups of students for the number

of possible trips to the second floor. This was a situation that allowed and

encouraged students to problematize what they stud[ied], to define problems that

elicit[ed] their curiosities and sense-making skills (Hiebert et al., 1996, p. 12).



19/28


Specifically, the class examined the arguments that gave rise to the conflicting

conclusions, tried to examine the validity of these arguments, and searched

for resolutions. The teacher, consistent with the role of the representative of

the mathematical community in the classroom (see Ball, 1993; Lampert, 1992;

Stylianides, 2007; Yackel & Cobb, 1996), aimed to help students understand

that the situation could be resolved by reference to the different (legitimate)

assumptions that constituted the basis of the different arguments and conclusions.

As a result, the students were offered an opportunity to develop a sense that the

truth of conclusions depends on the assumptions that support them.

Analogues between students experiences with assumptions in theepisode and mathematicians work with assumptions in the discipline. It

may seem inappropriate to consider students experiences with assumptions in

the episode in relation to mathematicians work with assumptions in the disci-

pline. After all, [c]hildren are different than mathematicians in their experiences,

immediate ambitions, cognitive processing power, representational tools, and

so on (Hiebert et al., 1996, p. 19). Nevertheless, it is worth describing some

analogues that exist between students work in the episode and mathematicians

work as it relates to the development of Euclidean and non-Euclidean geome-

tries (presented earlier in the example from Fawcetts class). I will describe theanalogues and then explain what these analogues suggest about the students

experiences.

Figure 3 presents these analogues. In sum, different groups of students in the

episode made (unconsciously) different assumptions about the tasks conditions

(analogously to the adoption of different sets of axioms by Euclid, Riemann,

and Lobatchewsky); they attempted to build valid arguments based on these

assumptions (analogously to the attempts of Euclid, Riemann, and Lobatchewsky

to build self-consistent geometrical theories based on different sets of axioms);

they resolved, with their teachers help, the issue of contradictory conclusionsof their arguments by reference to the different assumptions that supported

them (analogously to the fact that the different sets of axioms of Euclidean,

Riemannian, and Lobatchewskian geometries account for their contradictory

conclusions for the sum of the interior angles of a triangle); and were offered

opportunities to promote their understanding of different proving strategies and to

develop a sense of the role of assumptions in proving (analogously to the fact that

mathematicians development of different geometrical theories based on different

sets of axioms promoted the fields understanding of the geometrical space).

These analogues suggest that, in supportive classroom environments, evenyoung children can engage in authentic mathematical activity (Lampert, 1992)

that is related to assumptions and proving. The analogues suggest further that

young children, similar to mathematicians, can problematize mathematical situa-

tions (Hiebert et al., 1996) that are related to assumptions and proving, with



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Groups of students used asstarting points of their provingactivity different assumptions

about the conditions of the task

The groups produced differentarguments that were legitimate,

but the conclusions of these

arguments appeared to becontradictory across groups

The groups engaged inimportant proving activity,

which differed among groups

The discussion of the differentarguments surfaced the

different assumptions onwhich they were based and

resolved the issue ofcontradictory conclusions

Students were offeredopportunities to: (1) advancetheir understanding of different

proving strategies, and (2)develop a sense of the roleof assumptions in proving

Groups of mathematiciansused as starting points of theirtheory-building in geometry

different sets of axioms

The groups produced differentgeometries that are self-

consistent and useful, but some

of their conclusions appear tobe contradictory across theories

The groups engaged inimportant theory-building,

which differed among groups

The different sets of axiomson which the geometries arebased helps explain why the

geometries yield someconclusions that appear to becontradictory across theories

The development of differentgeometries based on

different sets of axiomsadvanced mathematicians

understanding of thegeometrical space

The work of mathematicians indeveloping the Euclidean and

non-Euclidean geometries

The work of the class on the task

presented in the episode

FIGURE 3 Analogues between the work of the class in the episode and the work of

mathematicians.

378



21/28


the goal of resolving these situations in sensible ways. Thus, we may say that

childrens engagement with assumptions in the context of proving can serve,

similarly to mathematicians engagement, as a vehicle to deep understanding of

mathematics.

Finally, it may be useful to think of students engagement with assumptions in

the episode as a genuine yet rudimentary version (Bruner, 1960) of mathemati-

cians engagement with assumptions in the discipline. A possible instructional

sequence on students engagement with assumptions in school mathematics may

begin with students engagement with assumptions in the local setting of

mathematical tasks with ambiguous conditions (lower grades) and conclude with

students engagement with assumptions in the global setting of mathematicaltheories that are supported by different sets of axioms (advanced grades).

Mathematical tasks that can generate rich mathematical activity in

the intersection of assumptions and proving. Several researchers have

elaborated on the role that different kinds of mathematical tasks can play in

classroom activity (e.g., Christiansen & Walther, 1986; Doyle, 1988; Leinhardt,

Zaslavsky, & Stein, 1990; Zaslavsky, 2005). In this section, I aim to contribute

to this body of research by discussing features of mathematical tasks that

can generate rich mathematical activity in the intersection of assumptions andproving, using the task in the episode as a basis for my discussion.

The mathematical task in the episode had three primary features: (1) the

conditions of the task were ambiguously stated and, therefore, were subject to

different legitimate assumptions; (2) each of the main assumptions students could

make about the conditions of the task led to important mathematical activity that

called for the use of different proving strategies; and (3) the conclusions of the

arguments or proofs that students constructed (or could construct) based on each

assumption appeared to be contradictory, thus surfacing the role of assumptions.

I do not argue that these three features are necessary for a mathematical taskto generate rich mathematical activity in the intersection of assumptions and

proving. Rather, I argue that tasks with these features have strong potential to

generate such activity, given that the implementation of the tasks preserves their

nature and cognitive demands (see, e.g., Stein, Grover, & Henningsen, 1996). In

addition, as was illustrated earlier by the mathematical activity in the episode,

tasks with these features can help students engage in authentic mathematical

activity (Lampert, 1992) and problematize mathematical situations (Hiebert et al.,

1996), thereby offering the chance of leaving behind important residue (Davis,

1992).Next I present two examples of mathematical tasks that have the three features

that I previously described and that can be used (such as the task in the episode)

in the early grades. With these examples, I aim to clarify further the three features

and raise issues about the relationship between assumptions and definitions, and



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380 STYLIANIDES

about the role of teachers in fostering productive interactions between students

and mathematical tasks that have these features.The first example relates to the mathematical task from the introductory part

of the article on addition sentences for 5. In this task, students are asked to

explore different addition sentences for 5 and then to show how many such

sentences there are. The domain of discourse is not specified for the students,

thus allowing at least two different assumptions about what this set might be. If

a student makes (consciously or unconsciously) the assumption that the domain

of discourse is the set of positive integers from 1 to 5, then there are finitely

many different number sentences (the exact number depends on what counts as

different and on the number of addends that one allows; cf. introductory part ofthe article). This claim can be proved by systematic enumeration of all possible

cases. However, if a student makes the assumption that the domain of discourse

is the entire set of integers, then there are infinitely many number sentences. This

claim cannot be proven by systematic enumeration of all possible cases; rather, it

requires the development of a general argument that describes, for example, the

solution set of the algebraic sentence x+ x+5 = 5, where xcan be any integer.4

The apparent conflict between the different conclusions of the two arguments

(finitely many versus infinitely many number sentences) raises the need for an

explanation about why both conclusions are legitimate, thus surfacing the roleof assumptions.

The second example is a mathematical task that first engages students in

exploring properties of quadrilaterals and then asks students to consider the truth

or falsity of the statement: Every rectangle is a trapezoid. Depending on what

one takes the definition of a trapezoid to bea quadrilateral with at least one

pair of parallel sides versus a quadrilateral with exactly one pair of parallel

sidesthe statement is true or false, respectively. In each case, the statement can

be verified or refuted using different proving strategies such as the development

of a direct proof or the construction of a counterexample, respectively.This example raises the interesting issue of the relationship between assump-

tions and definitions. The concept of assumption intersects with the concept of

4Evidence from Deborah Balls third-grade class shows that young children can produce, after

appropriate teacher scaffolding, such general arguments. In particular, a student in Balls class

developed the following generic proof (Balacheff, 1988; Harel & Sowder, 1998; Rowland, 2002)

for the claim that there are infinitely many number sentences for 10: We would take any number,

it wouldnt matter what number, say 200. And then we would minus 200, then we would plus 10,

and it would always equal 10. So you could go on for a long, long time, just keep on doing that.

[ ] So, since numbers they never stop, you could go on and on and on and on and on and on and

on [ ] With Balls help, the students represented this argument algebraically, using the sentence

x x + 10=10, and noting that xcan stand for any number. For elaboration on this classroom

episode (including an analysis of the students proof and of the teachers actions that supported the

reformulation of the proof in algebraic form), see Stylianides (2007).



23/28


definition in complicated ways. It is beyond the scope of this article to sort out

the relationship between assumptions and definitions, but I discuss two aspects

of this relationship in the context of the particular example.

First, the example shows that one possible form assumptions can take in the

context of proving relates to the choice of definition. Zaslavsky (2005) noted that

the choice of definition need not be connected to correctness; rather, [i]t could

be related to personal preferences, beliefs, values or the theoretical framework

or context to which one refers (p. 301). For example, the choice of the inclusive

definition of a trapezoid (i.e., the definition according to which a rectangle is a

special case of a trapezoid) may reflect a value of the fundamental mathematical

idea of generalization(see, e.g., Kitcher, 1984).Second, a classroom community who shares a clear definition of a trapezoid

would most likely not engage in discussions about the role of assumptions in

verifying or refuting the statement that every rectangle is a trapezoid, because

the clear definition would rule out the ambiguity of the situation. This remark

suggests that clear definitions of mathematical terms can influence the interpre-

tation of mathematical tasks that could allow different hidden assumptions, and

raises the issue of how teachers can manage the tension between clear definitions

and ambiguous situations that can support different assumptions. It is reasonable

to say that teachers decisions in managing this tension should be guided by thegoals that they try to accomplish in different situations. In situations in which

teachers aim to offer their students opportunities to develop a sense of the role of

assumptions in proving, unclear definitions and ambiguous task conditions might

be appropriate. In situations in which the emphasis of instruction is more on

introducing students to new proving strategies (e.g., proof by exhaustion through

systematic consideration of all possible cases) and less on increasing students

understanding of the elements that constitute the foundation of an argument or a

proof (assumptions, axioms, etc.), clear definitions and unambiguous task condi-

tions might be appropriate. Finally, teachers might also engage students in situa-tions that promote bothan understanding of the role of assumptions in resolving

conflicting conclusions in mathematical tasks with ambiguous conditions and

an appreciation of the role of clear definitions in ensuring unambiguous interpre-

tation of a tasks conditions. An instructional sequence that could promote this

expanded set of goals could begin with posing a task with ambiguous conditions

that would allow the generation of conflicting conclusions based on different

assumptions, continue with helping students make their assumptions explicit to

resolve the issue of different conclusions, and end by asking students to think

what kind of definitions would rule out the ambiguity in the tasks conditions sothat everybody would be working on the same problem.

To recapitulate, if and when teachers consider it important to help their

students develop a sense of the role of assumptions in proving, it seems fruitful

for teachers to use mathematical tasks that have the three features described



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382 STYLIANIDES

earlier. Teachers have an important role in identifying situations in which it

is appropriate to implement such mathematical tasks in classrooms given their

instructional goals. There may be situations in which teachers do not realize that a

particular mathematical task they implement in their classrooms has the features

described earlier. For example, a task may have an ambiguity that the teacher

fails to notice but that causes students to make different assumptions about the

conditions of the task. Even though it may not be in a teachers original intent to

discuss the multiple assumptions that students can make on the conditions of a

mathematical task, it is important that the teacher be able to recognize that some

student approaches to the task that appear faulty may in fact be mathematically

sound and based on an unforeseen set of legitimate assumptions. By recognizing

the value of such approaches (with reference to the assumptions on which they

are based), teachers not only help students develop their understanding of the

role of assumptions in proving, but also support students in making sense of the

mathematics involved.

CONCLUSION

The notion of assumptions permeates school mathematics, especially the activityof proving, but it tends to be highlighted by instruction only in the advanced

grades. In this article, I argued for the importance of helping even young children

develop a sense of the role of assumptions in proving, primarily because explic-

itness on the role of assumptions can allow children to make sense of and examine

critically their conclusions based on the grounds that support them. In addition, if

instruction in the lower grades highlighted the notion of assumptions, then more

advanced mathematics would not seem so alien to students. Thus, the bulk of the

article was an investigation of what it might mean and look like for instruction

to highlight the role of assumptions in proving in the context of the early grades.In conducting this investigation, I discussed features of mathematical tasks that

can generate rich mathematical activity in the intersection of assumptions and

proving. I also considered issues of the role of teachers in fostering productive

interactions between students and mathematical tasks with those features, and of

the relationship between the concept of assumption and definition.

This article identifies several directions for future research. For example, it

is important to investigate students and teachers understandings of the role of

assumptions in proving. The article has prepared the ground for such investi-

gations by identifying features of mathematical tasks that can be used (e.g., inspecially designed tests or clinical interviews) to elicit students and teachers

understandings in this domain. These investigations can have implications for

teacher education and professional development programs that aim to equip

teachers with necessary knowledge to help students develop an appreciation



25/28


of the role of assumptions in proving. Furthermore, these investigations can

broaden and advance the existing research base on individuals understandings

of proof. To date, most research studies in this domain have focused primarily

on individuals understandings of logical principles, of proof methods, and

of the distinction between empirical and deductive arguments (e.g., Chazan,

1993; Harel & Sowder, 1998; Healy & Hoyles, 2000; Hoyles & Kchemann,

2002; Knuth, 2002; Martin & Harel, 1989; Simon & Blume, 1996; Stylianides,

Stylianides, & Philippou, 2004; 2007).

Finally, the article makes the point that elementary mathematics is real

mathematics, or has the potential to be, if we look at it as a sense-making

activity. For this activity to be authentic to students and authentic in its reflectionof the wider mathematical culture (Lampert, 1992), instruction needs to help

students understand the grounds that support the conclusions they draw and

the means by which these conclusions are derived and represented (Stylianides,

2007). Accordingly, notions such as assumptions, proof, and proving should be

seen as critical components of students interactions with mathematics and high

priorities of instruction.

ACKNOWLEDGMENTS

The preparation of this article was supported in part by funds from the National

Science Foundation to the Diversity in Mathematics Education Center for

Learning and Teaching. The opinions expressed in the article are those of the

author and do not necessarily reflect the position, policy, or endorsement of the

National Science Foundation or of the aforementioned Center. The description

of the episode in the article is derived from the authors dissertation thesis, which

was completed at the University of Michigan under the supervision of Deborah

Ball; nevertheless, the discussion of the episode, with a focus on the role ofassumptions in proving, is new. The author wishes to thank Lyn English, Alan

Schoenfeld, Gabriel Stylianides, and three reviewers (Ken Clements and two

anonymous) for useful comments on an earlier version of the article.

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