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53

For the Learning of Mathematics 29, 3 (November, 2009)

FLM Publishing Association, Edmonton, Alberta, Canada

“The use of mathematical knowledge in teaching is often

taken for granted” (Ball & Bass, 2000, p. 86). Only a small

number of experiences of such usage remain memorable.

They begin with what Mason (2002) refers to as ‘distur-

bance’:

Most frequently there is some form of disturbance

which starts things off. It may be a surprise remark in

a lesson, […] or a moment of insight (p. 10).

The story ‘Sean vs. Cantor’ presents two such experiences

of disturbance that we analyse from mathematical and ped-

agogical perspectives.

Background and setting

Our story is situated in the course ‘Foundations of Mathe-

matics’, a Master’s course for practicing secondary

mathematics teachers. The main character is Lora, an expe-

rienced instructor who has taught several offerings of this

course. ‘Foundations of Mathematics’ introduces students

to several fundamental ideas and ‘big theorems’ in mathe-

matics, which either were long forgotten or were not

encountered in students’ undergraduate studies. Infinity and

Cantor’s method of corresponding infinite sets were among

the topics explored in the course.

The idea of infinity was introduced in a friendly and

‘playful’ manner via the exploration of famous paradoxes,

such as Hilbert’s Hotel Infinity and the Ping-Pong Ball

Conundrum (Mamolo & Zazkis, 2008), before introducing

students to the conventional mathematical understanding of

the presented ideas. The discord between intuitions and for-

mal mathematics – such as reasonable intuitive beliefs that

the set of even numbers is smaller than the set of natural

numbers, or that the set of natural numbers is smaller than

the set of rational numbers – was explicitly acknowledged.

However, this practical intuition, based on the reasoning of

inclusion appropriate for finite sets, is inconsistent with the

normative standard for comparing infinite sets. Formally,

these sets are considered equinumerous, all having the same

cardinality, denoted by ℵ0. The proofs are based on estab-

lishing a one-to-one correspondence between the pairs of

sets. For the latter case, setting up a one-to-one correspon-

dence relies on a systematic listing of rational numbers so

that they are all guaranteed to appear on the list, as illus-

trated in figure 1.

Unwinding the spiral produces a list in which each ratio-

nal number is paired with exactly one natural number (0g1,

1/1g2, -1/1g3, 2/1g4, etc.), and as such demonstrates that

the cardinality of the two sets is the same.

Having explored a variety of sets with cardinality equal

to ℵ0, students were introduced to Cantor’s theorem, that

the cardinality of the set of real numbers is greater than ℵ0,

or, in informal terms, that there are more real numbers than

natural numbers. The proof of this theorem is often referred

to as Cantor’s diagonal method (see, e.g., Burger & Star-

bird, 2000, for a proof).

Sean’s correspondence

Following a discussion of different infinities, Sean sug-

gested a correspondence between real and natural numbers

that he argued was one-to-one. (In fact, Sean’s suggestion

attends only to real numbers in the interval (0,1), but so does

the conventional presentation of Cantor’s proof of the theo-

rem in question.) Sean suggested considering numbers in

their decimal representation, as follows:

Start with numbers that have only one (non-zero) digit

after the decimal point, and correspond them to the first

nine natural numbers: 0.1g1, 0.2g2, … 0.9g9. Then

look at numbers with 2 digits after the decimal point,

avoiding those with 0 at the end, and correspond them

to the natural numbers from 10 to 99:

SEAN vs. CANTOR: USING

MATHEMATICAL KNOWLEDGE

IN ‘EXPERIENCE OF DISTURBANCE’

RINA ZAZKIS, AMI MAMOLO

Figure 1. Corresponding rational and natural numbers

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54

0.01g10, 0.02g11, … , 0.09g19, 0.11g20, … ,

0.99g99

Then take all the numbers with 3 digits after the deci-

mal point, avoiding those with 0 or 00 at the end, and

correspond them to the next ‘bunch’ of natural num-

bers, and so on. This method presents an ‘ordering’ of

real numbers, and so their cardinality is ℵ0.

This suggestion of Sean’s created an initial ‘experience of

disturbance’ for Lora. She presented various arguments to

refute Sean’s claim, however he resisted accepting them.

Sean’s persistence directed Lora to seek other, more con-

vincing, refutations.

The fact that it is impossible to find a natural number that

corresponds, for example, to 1/3(as it has infinite non-zero

decimal representation) did not convince Sean. He insisted

that when corresponding the natural numbers and rational

numbers in class, an explicit ‘match’ for each fraction was

not specified, but instead the emphasis was on the possibil-

ity of listing the rational numbers ‘in order’. Sean claimed

that his method presented a possible ordering, that “eventu-

ally will get to 1/3or any other number”. There was an

apparent confusion in his reasoning – alerted to in previous

research (e.g., Mamolo & Zazkis, 2008) – between a very

large number of digits after the decimal point, and an infi-

nite decimal representation.

What finally convinced Sean was Lora’s observation that

when the rational numbers were ‘ordered’ by unwinding the

spiral it was possible to determine what came before and

after each number. Lora asked Sean to determine what real

number was placed before or after 1/3in his ordering. Failing

to determine this convinced Sean of the inappropriateness of

his correspondence. However, Lora was unhappy with her

approach of immediately refuting a student’s suggestion,

rather than giving him, and others, an opportunity to explore

it in greater detail.

Creating a conflict

It was a year later teaching the same course that Lora pre-

sented ‘Sean’s correspondence’ to a new group of students.

Following the class’ exposure to Cantor’s theorem and

proof, Lora introduced for the students’ scrutiny Sean’s sug-

gestion for a correspondence between sets of real and natural

numbers. Her goal was to provoke a cognitive conflict in

her classroom, a goal motivated by the abundant research

suggesting that this is a powerful pedagogical approach

(e.g., Tirosh & Graeber, 1990). However, it is only after stu-

dents realise the existence of a conflict, a potential conflict

created or recognised by the instructor becomes a cognitive

conflict for students (Zazkis & Chernoff, 2007). To Lora’s

surprise, the conflict – presented by the fact that Sean’s cor-

respondence contradicted the established proof of Cantor’s

theorem – was not immediately recognised.

“Cool!” – was the reaction voiced by one of the students,

and several nods around the table indicated other students’

agreement. Lora was extremely surprised with this reaction.

She expected immediate recognition that something was

wrong with Sean’s suggestion. She further expected that

such recognition would be followed by a search for a flaw

in his argument, and this might require some time and ‘scaf-

folding’. What initially appeared as mutual acceptance of

the argument was totally unexpected; it contradicted not

only the specific proof of Cantor’s, but also the essence of a

‘mathematical theorem’. This was yet another ‘experience

of disturbance’ for Lora. So both confused and ‘disturbed’

Lora did what she found, as most teachers, very difficult to

do: she said nothing and waited.

Conflict recognition

As implied earlier, the first step toward conflict resolution

is conflict recognition. This step was made by a very quiet

remark of one student: “So are you saying that Cantor was

wrong and Sean should get a Fields medal?”

Yet again, Lora said nothing, restraining a smile. The next

set of claims came simultaneously from different corners of

the class:

• “Wait, there must be something wrong.”

• “It is more likely that Sean is wrong rather than

Cantor.”

• “We must be missing something. It is unlikely that

no one thought of disproving this for the last two

hundred years.”

Conflict resolution

Once the conflict was acknowledged, the flaw in Sean’s argu-

ment was not difficult to detect. The first realisation towards

refuting the argument was that the real number π– a generic

example for an irrational number – does not have a match

within natural numbers, as its decimal expansion is infinite.

However, since πis not included in the interval (0,1), this

observation was modified, recognising that none of the irra-

tional numbers between 0 and 1 would have a ‘natural match’

in Sean’s correspondence because of their infinite decimal

expansion. While this was sufficient to conclude, in a student’s

semi-cynical words: “Sean was wrong, Cantor was right, sur-

prise, surprise”, Lora sought a more detailed ‘resolution’.

If not, what yes?

Following consensus on the refutation of Sean’s method,

Lora posed the following question: “So we agree that Sean’s

correspondence does not prove that real numbers and nat-

ural numbers have the same cardinality. But what does it

prove, if anything?”

This question was inspired by Koichu’s (2008) ‘If not,

what yes?’ extension of the famous pedagogical strategy

‘What if not’, described by Brown and Walter (1993).

Koichu’s approach is based on presenting students with a

mathematical claim that has to be refuted, and then asking,

‘Since this statement is wrong, which one would be cor-

rect?’ inviting gradual modifications of the presented claim.

In our case, if Sean’s correspondence does not demonstrate

what was initially intended, what (if anything) does it prove?

The initial suggestion to ‘what yes?’ was that Sean’s cor-

respondence presented a matching of rational numbers with

natural numbers. Some students considered this as a ‘sim-

pler’ correspondence than the ‘conventional’ one introduced

in class. However, once again, the ‘wisdom of Sean’ was

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55

called into question. The sceptical voice suggested that if it

were that simple, then why would textbooks introduce the

spiral method?

Eventually, the students realised that only a subset of the

set of rational numbers was included in Sean’s correspon-

dence – namely, the set whose elements have a finite

decimal representation. Thus, the correspondence that Sean

created proves that the set of rational numbers with a finite

decimal representation (in the interval (0,1) ) has the same

cardinality as the set of natural numbers. While finding yet

another set with cardinality ℵ0was not a very exciting math-

ematical conclusion, it was a worthwhile mathematical

engagement for students.

Mathematics in teaching: examining complexities

Growing attention in mathematics education research to

teachers’ knowledge has led to revisiting and refining the clas-

sical categories of subject matter knowledge and pedagogical

content knowledge (Shulman, 1986). On one hand there are

attempts to ‘zoom out’ and consider more general constructs,

such as knowledge of mathematics for teaching (Ball & Bass,

2000) or simply mathematics-for-teaching (Davis & Simmt,

2006). On the other hand there are attempts to ‘zoom in’ and

refine the deeply intertwined notions of mathematics and

mathematical pedagogy. This results, for example, in a more

detailed examination of the relationship among mathematical

and pedagogical goals (Liljedahl, Chernoff, & Zazkis, 2007)

and in introducing additional sub-categories of teachers’

knowledge (Hill, Ball, & Schilling, 2008). Such refinements

sharpen the lens through which the complexities of mathe-

matics in teaching can be viewed and analysed.

Mathematics in teaching: complexity of using

Liljedahl et al. (2007) illustrated a way of examining the use

of tasks in teacher education with a 2 ×2 array, presented in

figure 2. They suggested reading the content of the four cells

as “The use of xto promote understanding of Y”. The array

disaggregates the “knowledge of mathematics and use of ped-

agogy from the mathematical and pedagogical understandings

we wish to instil within our students” (p. 240).

Though this framework was developed for the analysis

of task design in teacher education, it serves well in

analysing the classroom experiences reported above from

the perspective of each cell.

mM: The use of mathematics to promote understanding

of Mathematics

Lora’s original goal was to demonstrate the difference in

cardinality between the sets of real and natural numbers.

Lora’s deep understanding of mathematics equipped her

with tools for refuting Sean’s argument. However, it was

the creation of a ‘convincing argument for Sean’ that exem-

plified the usage of mathematical knowledge in teaching

that enhanced students’ understanding of mathematics. It

was the search for additional arguments, building upon the

claims presented by Sean regarding the issue of ‘ordering’

numbers as the basis for the argument of correspondence,

that turned Lora’s mathematics to Sean’s eventual mathe-

matical understanding.

When Lora revisited Sean’s argument with a new group

of students, it was again her sophisticated mathematical

understanding which enabled her to engage students effec-

tively in the ‘if not, what yes’ investigation, and to develop

their own mathematical understanding of the relevance and

scope of Sean’s argument.

pM: The use of pedagogy to promote understanding of

Mathematics

Liljedahl et al. (2007) recognise that teachers “need not only

to be aware of the mathematics embedded within the task,

but [also] … need an understanding of how to mobilize this

knowledge for their students’ learning” (p. 240). This aware-

ness and understanding of Lora’s resulted in her developing

two different pedagogical approaches: creation of cognitive

conflict and ‘if not, what yes’ investigation. Through both of

these approaches, students developed specific mathematical

knowledge regarding properties of infinite sets, in particular

the sets of real and rational numbers. Further, Lora’s usage of

pedagogy promoted students’ understanding of mathematics

more broadly. After Lora’s surprising experience regarding

students’ initial acceptance of Sean’s argument despite the

conflict with established mathematical knowledge, a new

mathematical goal developed – one which related to the

meaning of a mathematical theorem. This goal was met

through students’ experiences and resolutions of cognitive

conflict, and through their own mathematical endeavours to

refine the claim associated with Sean’s argument.

mP: The use of mathematics to promote understanding

of Pedagogy

Though the tasks themselves did not have explicit pedagog-

ical goals, these goals are always present when working with

teachers. Creating a cognitive conflict can be seen as exem-

plification of general pedagogy. It was appreciated by

students, who are practicing teachers, as a powerful strat-

egy for some of their future endeavours. Further, in

resonance with Koichu’s (2008) encounters, students also

recognized the ‘if not, what yes’ pedagogical approach as

“an experience in ‘genuine’ doing of mathematics” (p. 450),

and as such saw it as a compelling pedagogical tool.

pP: The use of pedagogy to promote understanding of

Pedagogy

While explicit pedagogical moves have been acknowledged

earlier, there is a simple one that may not be noticed with-

out drawing attention to it. This is Lora’s move of remaining

silent, saying nothing. If used only occasionally, it serves as

a powerful tool in promoting exchange of mathematical

Figure 2. Goals and usage grid

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56

ideas among students. In working with teachers, it exempli-

fies a very simple technique, and experiencing its effects

invites teachers to try this in their own classroom.

Mathematics in teaching: complexity of knowing

We introduced and analysed two examples of ‘experience

of disturbance’ in the described events. The first was an

unusual idea presented by Sean, suggesting an inappropri-

ate correspondence between the sets of real and natural

numbers. The second was an unexpected first reaction from

a class of mathematics teachers, initially accepting Sean’s

correspondence. In both cases the disturbance resulted in a

positive twist: it triggered further ‘unpacking’ and expan-

sion of Lora’s knowledge into its new forms.

The first disturbance enriched Lora’s understanding of

students’ potential difficulties and triggered the search for

and development of additional refuting arguments, where

the explanatory power of different arguments – though

equivalent mathematically – was perceived differently by a

student. Using the terms developed by Hill, Ball, and

Schilling (2008), this disturbance helped Lora acquire a

more profound knowledge of content and students (KCS) as

it enriched her repertoire of possible incorrect ideas and

unhelpful intuitions that may be held by students. In partic-

ular, it enriched Lora’s “knowledge of how students think

about, know or learn this particular content” (ibid., p. 375).

As a result, Lora’s common content knowledge (CCK) and

her specialized content knowledge (SCK) developed into

knowledge of content and teaching (KCT), as evidenced in

the enhanced variety of explanations that she developed.

This disturbance also resulted in developing tasks for

another group of students, implementing the pedagogy of

‘cognitive conflict’ and ‘if not what yes’, which can be seen

as further indicators of extending Lora’s KCT.

The second disturbance resulted in a new understanding

of the mathematical dispositions of a group of practicing

teachers. The fact that students’ reactions were unexpected,

but dealt with skilfully, is a clear sign of Lora’s enhance-

ment of her KCS. It also shows the strong interrelationship

between KCS and KCT, as deeper understanding of the for-

mer contributes to growth in the latter.

Conclusion

We agree with Ball and Bass (2000) that “[n]o repertoire of

pedagogical content knowledge, no matter how extensive,

can adequately anticipate what it is that students may think,

how some topic may evolve in a class, the need for a new

representation or explanation for a familiar topic” (p. 88)

and that “[b]eing able to use mathematical knowledge

involves using mathematical understanding and sensibility

to reason about subtle pedagogical questions” (p. 99).

The story of ‘Sean vs. Cantor’ illustrates exactly this:

unanticipated students’ thinking and the teacher’s skill and

sensibility in developing new explanations and new instruc-

tional engagements. Our main contribution is in exemplifying

how ideas developed and adopted by studying the work of

elementary school teachers are transferable to teaching

undergraduate mathematics. Lora’s experience further

demonstrates the importance of ‘noticing’ and ‘disturbance’

(Mason, 2002) in transferring personal knowing of mathe-

matics into using it in teaching.

References

Ball, D. and Bass, H. (2000) ‘Interweaving content and pedagogy in teach-

ing and learning to teach: knowing and using mathematics’, in Boaler,

J. (ed.), Multiple perspectives on mathematics teaching and learning,

Westport, CT, Ablex, pp. 83–104.

Brown, S. and Walter, M. (1993) The art of problem posing, Mahwah, NJ,

Erlbaum.

Burger, E. and Starbird, M. (2000) The heart of mathematics: an invitation

to effective thinking, Emeryville, CA, Key College.

Davis, B. and Simmt, E. (2006) ‘Mathematics-for-teaching: an ongoing

investigation of the mathematics that teachers (need to) know’, Educa-

tional Studies in Mathematics 61, 293–319.

Hill, H., Ball, D. and Schilling, S. (2008) ‘Unpacking pedagogical content

knowledge: conceptualizing and measuring teachers’ topic-specific

knowledge of students’, Journal for Research in Mathematics Educa-

tion 39(4), pp. 372–400.

Koichu, B. (2008) ‘If not, what yes?’, International Journal of Mathemati-

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Liljedahl, P., Chernoff, E. and Zazkis, R. (2007) ‘Interweaving mathemat-

ics and pedagogy in task design: a tale of one task’, Journal of

Mathematics Teacher Education 10(4–6), pp. 239–249.

Mamolo, A. and Zazkis, R. (2008) ‘Paradoxes as a window to infinity’,

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Mason. J. (2002) Researching your own practice: the discipline of notic-

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