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Sean vs. Cantor: Using mathematical knowledge in 'experience of disturbance'

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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 0. 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 0, students were introduced to Cantor’s theorem, that
the cardinality of the set of real numbers is greater than 0,
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 0.
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 0was 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,
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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-
cal Education in Science and Technology 39(4), pp. 443–454.
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’,
Research in Mathematics Education 10, pp. 167–182.
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ing, London, UK, RoutledgeFalmer.
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ing’, Educational Researcher 15(2), pp. 4–14.
Tirosh, D. and Graeber, A. O. (1990) ‘Evoking cognitive conflict to explore
preservice teachers’ thinking about division’, Journal for Research in
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plary?’, Educational Studies in Mathematics 68(3), pp. 195–208.
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... Whereas ritual has served for us to raise new and important questions for the activity in mathematics classrooms, improvisation points toward ways of teaching, ways of researching, and ways of being. Zazkis and Mamolo (2009) argued that mathematics educators need to facilitate "unanticipated students' thinking and the teacher's skill and sensibility in developing new explanations and new instructional engagements" (p. 56). ...
... Mamolo and Zazkis (2008). 8 See Mamolo and Zazkis (2008, p.176). 9 See Mamolo and Zazkis (2008, p.179). 10 See Zazkis and Mamolo (2009) B: Of course, there is abundant research on students' understanding of infinity and infinite sets. For example Dina Tirosh and Pessia Tsamir extended the work of Fischbein, pointing to how our intuitive ideas get challenged when the notion of infinity is introduced. ...
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We illustrate how the classical dialogues – Galileo’s Dialogue on Infinity from Dialogues Concerning Two New Sciences, Plato’s Meno, and Lakatos’ Proofs and Refutations – can be used in teacher education. By re-capturing our conversation, we demonstrate the use of the classical dialogues to revisit mathematical notions, such as infinity, or to highlight meta-mathematical issues, such as definitions and proofs. We share several scripting assignments used with teachers and several student-written scripts produced in response to such assignments. We elaborate on the benefits of bringing classical dialogues for discussion in classes of mathematics teachers. These include, but are not limited to, enculturation by exposure to historical context, reinforcement of mathematical ideas and concepts, introduction to subsequent readings and assignments, and extended variety of tasks for the use in mathematics teacher education.
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