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Kaye Stacey
University of Melbourne, Australia
This paper and the accompanying presentation has a simple message, that
mathematical thinking is important in three ways.
Mathematical thinking is an important goal of schooling.
Mathematical thinking is important as a way of learning mathematics.
Mathematical thinking is important for teaching mathematics.
Mathematical thinking is a highly complex activity, and a great deal has been written
and studied about it. Within this paper, I will give several examples of mathematical
thinking, and to demonstrate two pairs of processes through which mathematical
thinking very often proceeds:
Specialising and Generalising
Conjecturing and Convincing.
Being able to use mathematical thinking in solving problems is one of the most the
fundamental goals of teaching mathematics, but it is also one of its most elusive goals.
It is an ultimate goal of teaching that students will be able to conduct mathematical
investigations by themselves, and that they will be able to identify where the
mathematics they have learned is applicable in real world situations. In the phrase of
the mathematician Paul Halmos (1980), problem solving is “the heart of
mathematics”. However, whilst teachers around the world have considerable
successes with achieving this goal, especially with more able students, there is always
a great need for improvement, so that more students get a deeper appreciation of what
it means to think mathematically and to use mathematics to help in their daily and
working lives.
The ability to think mathematically and to use mathematical thinking to solve
problems is an important goal of schooling. In this respect, mathematical thinking
will support science, technology, economic life and development in an economy.
Increasingly, governments are recognising that economic well-being in a country is
underpinned by strong levels of what has come to be called ‘mathematical literacy’
(PISA, 2006) in the population.
Mathematical literacy is a term popularised especially by the OECD’s PISA program
of international assessments of 15 year old students. Mathematical literacy is the
ability to use mathematics for everyday living, and for work, and for further study,
and so the PISA assessments present students with problems set in realistic contexts.
The framework used by PISA shows that mathematical literacy involves many
components of mathematical thinking, including reasoning, modelling and making
connections between ideas. It is clear then, that mathematical thinking is important
in large measure because it equips students with the ability to use mathematics, and
as such is an important outcome of schooling.
At the same time as emphasising mathematics because it is useful, schooling needs to
give students a taste of the intellectual adventure that mathematics can be. Whilst the
highest levels of mathematical endeavour will always be reserved for just a tiny
minority, it would be wonderful if many students could have just a small taste of the
spirit of discovery of mathematics as described in the quote below from Andrew
Wiles, the mathematician who proved Fermat’s Last Theorem in 1994. This problem
had been unsolved for 357 years.
One enters the first room of the mansion and it’s dark. One stumbles around bumping
into furniture, but gradually you learn where each piece of furniture is. Finally, after six
months of so, you find the light switch, you turn it on, and suddenly it’s all illuminated.
You can see exactly where you were. Then you move into the next room and spend
another six months in the dark. So each of these breakthroughs, while sometimes they’re
momentary, sometimes over a period of a day or two, they are the culmination of, and
couldn’t exist without, the many months of stumbling around in the dark that precede
them. (Andrew Wiles, quoted by Singh, 1997, p236, 237)
At the APEC meeting in Tokyo in January 2006, Jan de Lange spoke in detail about
the use of mathematics to equip young people for life, so I will instead focus this
paper on two other ways in which mathematical thinking is important.
Since mathematical thinking is a process, it is probably best discussed through
examples, but before looking at examples, I briefly examine some frameworks
provided to illuminate mathematical thinking, going beyond the ideas of
mathematical literacy. There are many different ‘windows’ through which the
mathematical thinking can be viewed. The organising committee for this conference
(APEC, 2006) has provided a substantial discussion on this point. Stacey (2005)
gives a review of how mathematical thinking is treated in curriculum documents in
Australia, Britain and USA.
One well researched framework was provided by Schoenfeld (1985), who organised
his work on mathematical problem solving under four headings: the resources of
mathematical knowledge and skills that the student brings to the task, the heuristic
strategies that that the student can use in solving problems, the monitoring and
control that the student exerts on the problem solving process to guide it in
productive directions, and the beliefs that the student holds about mathematics, which
enable or disable problem solving attempts. McLeod (1992) has supplemented this
view by expounding on the important of affect in mathematical problem solving.
In my own work, I have found it helpful for teachers to consider that solving
problems with mathematics requires a wide range of skills and abilities, including:
Deep mathematical knowledge
General reasoning abilities
Knowledge of heuristic strategies
Helpful beliefs and attitudes (e.g. an expectation that maths will be useful)
Personal attributes such as confidence, persistence and organisation
Skills for communicating a solution.
Of these, the first three are most closely part of mathematical thinking.
In my book with John Mason and Leone Burton (Mason, Burton and Stacey, 1982),
we provided a guide to the stages through which solving a mathematical problem is
likely to pass (Entry, Attack, Review) and advice on improving problem solving
performance by giving experience of heuristic strategies and on monitoring and
controlling the problem solving process in a meta-cognitive way.
We also identified four fundamental processes, in two pairs, and showed how
thinking mathematically very often proceeds by alternating between them:
specialising – trying special cases, looking at examples
generalising - looking for patterns and relationships
conjecturing – predicting relationships and results
convincing – finding and communicating reasons why something is true.
I will illustrate these ideas in the two examples below. The first example examines
the mathematical thinking of the problem solver, whilst the second examines the
mathematical thinking of the teacher. The two problems are rather different – the
second is within the mainstream curriculum, and the mathematical thinking is guided
by the teacher in the classroom episode shown. The first problem is an open problem,
selected because it is similar to open investigations that a teacher might choose to use,
but I hope that its unusual presentation will let the audience feel some of the mystery
and magic of investigation afresh.
In this section, I will illustrate these four processes of mathematical thinking in the
context of a problem that may be used to stimulate mathematical thinking about
numbers or as an introduction to algebra. If students’ ability to think mathematically
is an important outcome of schooling, then it is clear that mathematical thinking must
feature prominently in lessons.
Number puzzles and tricks are excellent for these purposes, and in the presentation I
will use a number puzzle in a format of the Flash Mind Reader, created by Andy
Naughton and published on the internet (HREF1). The Flash Mind Reader does not
look like a number puzzle. Indeed its creator writes:
We have been asked many times how the Mind Reader works, but will not publish that
information on this website. All magicians […] do not give away how their effects work.
The reason for this is that it spoils the fun for those who like to remain mystified and
when you do find out how something works it's always a bit of a let-down. If you are
really keen to find out how it works we suggest that you apply your brain and try to work
it out on paper or search further afield. (HREF1)
As with many other number tricks, an audience member secretly chooses a number
(and a symbol), a mathematical process is carried out, and the computer reveals the
audience member’s choice. In this case, a number is chosen, the sum of the digits is
subtracted from the number and a symbol corresponding to this number is found from
a table. The computer then magically shows the right symbol. The Flash Mind
Reader is too difficult to use in most elementary school classes, the target of this
conference, but I have selected it so that my audience of mathematics education
experts can experience afresh some of the magic and mystery of numbers. As the
group works towards a solution, we have many opportunities to observe
mathematical thinking in action.
Through this process of shared problem solving as we investigate the Flash Mind
Reader, I hope to make the following points about mathematical thinking. Firstly,
when people first see the Flash Mind Reader, mathematical explanations are far from
their minds. Some people propose that it really does read minds, and they may try to
test their theory by not concentrating hard on the number that they choose. Others
hypothesise that the program exerts some psychological power over the person’s
choice of number. Others suggest it is only an optical illusion, resulting from staring
at the screen. This illustrates that a key component of mathematical thinking is
having a disposition to looking at the world in a mathematical way, and an attitude of
seeking a logical explanation.
As we seek to explain how the Flash Mind reader works, the fundamental processes
of thinking mathematically will be evident. The most basic way of trying to
understand a problem situation is to try the Flash Mind Reader several times, with
different numbers and different types of numbers. This helps us understand the
problem (in this case, what is to be explained) and to gather some information. This
is a simple example of specialising, the first of the four processes of thinking
mathematically processes.
As we enter more deeply into the problem, specialising changes its character. First
we may look at one number, noting that if 87 is the number, then the sum of its digits
is 15 and 87 – 15 is 72.
Beginning to work systematically leads to evidence of a pattern:
87 8 + 7 = 15 87 – 15 = 72
86 8 + 6 = 14 86 – 14 = 72
85 8 + 5 = 13 85 – 13 = 72
84 8 + 4 = 12 84 – 12 = 72
and a cycle of experimentation (which numbers lead to 72?, what do other numbers
lead to?) and generalising follows.
Of course, at this stage it is important to note the value of working with the unclosed
expressions such as 8+7 instead of the closed 15, because this reveals the general
patterns and reasons so much better. Working with the unclosed expression to reveal
structure is an admirable feature of Japanese elementary education.
87 87 – 7 = 80 80 – 8 = 72
86 86 – 6 = 80 80 – 8 = 72
85 85 – 5 = 80 80 – 8 = 72
84 84 – 4 = 80 80 – 8 = 72
It is also worthwhile noting at this point, that although we are working with a specific
example, the aim here is to see the general in the specific.
This generalising may lead to a conjecture that the trick works because all starting
numbers produce a multiple of 9 and all multiples of 9 have the same symbol. But
this conjecture is not quite true and further examination of examples (more
specialising) finally identifies the exceptions and leads to a convincing argument. In
school, we aim for students to be able to use algebra to write a proof, but even before
they have this skill, they can be produce convincing arguments. An orientation to
justify and prove (at an appropriate level of formality) is important throughout school.
If students are to become good mathematical thinkers, then mathematical thinking
needs to be a prominent part of their education. In addition, however, students who
have an understanding of the components of mathematical thinking will be able to
use these abilities independently to make sense of mathematics that they are learning.
For example, if they do not understand what a question is asking, they should decide
themselves to try an example (specialise) to see what happens, and if they are
oriented to constructing convincing arguments, then they can learn from reasons
rather than rules. Experiences like the exploration above, at an appropriate level build
these dispositions.
Mathematical thinking is not only important for solving mathematical problems and
for learning mathematics. In this section, I will draw on an Australian classroom
episode to discuss how mathematical thinking is essential for teaching mathematics.
This episode is taken from data collected by Dr Helen Chick, of the University of
Melbourne, for a research project on teachers’ pedagogical content knowledge. For
other examples, see Chick, 2003; Chick & Baker, 2005, Chick, Baker, Pham &
Cheng, 2006a; Chick, Pham & Baker, 2006b). Providing opportunities for students
to learn about mathematical thinking requires considerable mathematical thinking on
the part of teachers.
The first announcement for this conference states that a teacher requires
mathematical thinking for analysing subject matter (p. 4), planning lessons for a
specified aim (p. 4) and anticipating students’ responses (p. 5).These are indeed key
places where mathematical thinking is required. However, in this section, I
concentrate on the mathematical thinking that is needed on a minute by minute basis
in the process of conducting a good mathematics lesson. Mathematical thinking is
not just in planning lessons and curricula; it makes a difference to every minute of the
The teacher in this classroom extract is in her fifth year of teaching. She stands out in
Chick’s data as one of the teachers in the sample exhibiting the deepest pedagogical
content knowledge (Shulman, 1986, 1987). Her pupils are aged about 11 years, and
are in Grade 6. This lesson began by reviewing ideas of both area and perimeter. We
will examine just the first 15 minutes.
The teacher selected an open and reversed task to encourage investigation and
mathematical thinking. Students had 1cm grid paper and were all asked to draw a
rectangle with an area of 20 square cm. This task is open in the sense that there are
multiple correct answers, and it is ‘reversed’ when it is contrasted to the more
common task of being given a rectangle and finding its area. The teacher reminded
students that area could be measured by the number of grid squares inside a shape.
In terms of the processes of mathematical thinking, the teacher at this stage is
ensuring that each student is specialising. They are each working on a special case,
and coming to know it well, and this will provide an anchor for future discussions
and generalisations. I make no claim that the teacher herself analyses this move in
this way.
As the teacher circulated around the room assisting and monitoring students, she
came to a student who asked if he could draw a square instead of a rectangle. In the
dialogue which follows, the teachers’ response highlighted the definition of a
rectangle, and she encouraged the student to work from the definition to see that a
square is indeed a rectangle.
S: Can I do a square?
T: Is a square a rectangle?
T: What’s a rectangle?
T: How do you get something to be a rectangle? What’s the definition of a rectangle?
S: Two parallel lines
T: Two sets of parallel lines … and …
S: Four right angles.
T: So is that [square] a rectangle?
S: Yes.
T: [Pause as teacher realises that student understands that the square is a rectangle, but
there is a measurement error] But has that got an area of 20?
S: [Thinks] Er, no.
T: [Nods and winks]
Other responses to this student would have closed down the opportunity to teach him
about how definitions are used in mathematics. To the question “Can I do a square?”,
she may have simply replied “No, I asked you to draw a rectangle” or she might
have immediately focussed on the error that led the student to ask the question.
Instead she saw the opportunity to develop his use of definitions. When the teacher
realised that the student had asked about the square because he had made a
measurement error, she judged that this was within the student’s own capability to
correct, and so she simply indicated that he should check his work.
In the next segment, a student showed his 4 x 5 rectangle on the overhead projector,
and the teacher traced around it, confirmed its area is 20 square cm and showed that
multiplying the length by the width can be used instead of counting the squares,
which many students did. In this segment, the teacher demonstrated that reasoning is
a key component of doing mathematics. She emphasised the mathematical
connections between finding the number of squares covered by the rectangle by
repeated addition (4 on the first row, 4 on the next, …) and by multiplication. In her
classroom, the formula was not just a rule to be remembered, but it was to be
understood. The development of the formula was a clear example of ‘seeing the
general in a special case’. The formula was developed from the 4 x 5 rectangle in
such a way that the generality of the argument was highlighted.
The teacher paid further attention to generalisation and over-generalisation at this
point, when a student commented: ‘That’s how you work out area – you do the length
times the width’. The teacher seized on this opportunity to address students’ tendency
to over-generalise, and teased out, through a short class discussion, that LxW only
works for rectangles.
S1: That’s how you work out area -- you do the length times the width.
T: When S said that’s how you find the area of a shape, is he completely correct?
S2: That’s what you do with a 2D shape.
T: Yes, for this kind of shape. What kind of shape would it not actually work for?
S3: Triangles.
S4: A circle.
T: [With further questioning, teases out that LxW only applies to rectangles]
In the next few minutes, the teacher highlighted the link between multiplication and
area by asking students to make other rectangles with area 20 square centimetres.
Previously all students had made 4 x 5 or 2 x 10, but after a few minutes, the class
had found 20 x 1, 1 x 20, 10 x 2, 2 x 10, 4 x 5 and 5 x 4 and had identified all these
side lengths as the factors of 20. Making links between different parts of the
mathematics curriculum characterises her teaching.
Then, in another act of generalisation, the teacher begins to move beyond whole
T: Are there any other numbers that are going to give an area of 20? [Pauses, as if
uncertain. There is no response from the students at first]
T: No? How do we know that there’s not?
S: You could put 40 by 0.5.
T: Ah! You’ve gone into decimals. If we go into decimals we’re going to have heaps,
aren’t we?
After these first 15 minutes of the lesson, the students found rectangles with an area
of 16 square centimetres and the teacher stressed the important problem solving
strategy of working systematically. Later, in order to contrast the two concepts of
area and perimeter, students found many shapes of area 12 square cm (not just
rectangles) and determined their perimeters.
Even the first 15 minutes of this lesson show that considerable mathematical thinking
on behalf of the teacher is necessary to provide a lesson that is rich in mathematical
thinking for students. We see how she draws on her mathematical concepts, deeply
understood, and on her knowledge of connections among concepts and the links
between concepts and procedures. She also draws on important general mathematical
principles such as
working systematically
specialising – generalising: learning from examples by looking for the
general in the particular
convincing: the need for justification, explanation and connections
the role of definitions in mathematics.
Chick’s work analyses teaching in terms of the knowledge possessed by the teachers.
She tracks how teachers reveal various categories of pedagogical content knowledge
(Shulman, 1986) in the course of teaching a lesson. In the analysis above, I viewed
the lesson from the point of view of the process of thinking mathematically within the
lesson rather than tracking the knowledge used. To draw an analogy, in researching a
students’ solution to a mathematical problem, a researcher can note the mathematical
content used, or the researcher can observe the process of solving the problem.
Similarly, teaching can be analysed from the “knowledge’ point of view, or analysed
from the process point of view.
For those us who enjoy mathematical thinking, I believe it is productive to see
teaching mathematics as another instance of solving problems with mathematics.
This places the emphasis not on the static knowledge used in the lesson asabove but
on a process account of teaching. In order to use mathematics to solve a problem in
any area of application, whether it is about money or physics or sport or engineering,
mathematics must be used in combination with understanding from the area of
application. In the case of teaching mathematics, the solver has to bring together
expertise in both mathematics and in general pedagogy, and combine these two
domains of knowledge together to solve the problem, whether it be to analyse subject
matter, to create a plan for a good lesson, or on a minute-by-minute basis to respond
to students in a mathematically productive way. If teachers are to encourage
mathematical thinking in students, then they need to engage in mathematical thinking
throughout the lesson themselves.
APEC –Tsukuba (Organising Committee) (2006) First announcement. International
Conference on Innovative Teaching of Mathematics through Lesson Study. CRICED,
University of Tsukuba.
Chick, H. L. (2003). ‘Pre-service teachers’ explanations of two mathematical concepts’
Proceedings of the 2003 conference, Australian Association for Research in Education.
Chick, H.L. and Baker, M. (2005) ‘Teaching elementary probability: Not leaving it to
chance’, in P.C. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce
& A. Roche (eds.) Building Connections: Theory, Research and Practice. (Proceedings
of the 28
annual conference of the Mathematics Education Research Group of
Australasia), MERGA, Sydney, pp. 233-240.
Chick, H.L., Baker, M., Pham, T., and Cheng, H. (2006a) ‘Aspects of teachers’ pedagogical
content knowledge for decimals’, in J. Novotná, H. Moraová, M. Krátká, & N.
Stehlíková (eds.), Proc. 30
conference e International Group for the Psychology of
Mathematics Education, PME, Prague, Vol. 2, pp. 297-304.
Chick, H.L., Pham, T., and Baker, M. (2006b) ‘Probing teachers’ pedagogical content
knowledge: Lessons from the case of the subtraction algorithm’, in P. Grootenboer, R.
Zevenbergen, & M. Chinnappan (eds.), Identities, Cultures and Learning Spaces (Proc.
annual conference of Mathematics Education Research Group of Australasia),
MERGA, Sydney, pp. 139-146.
Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly, 87(7), 519
– 524.
HREF1 CyberGlass Design - The Flash Mind Reader. Accessed
28 November 2006.
Mason, J. Burton, L. and Stacey, K. (1982) Thinking Mathematically. London: Pearson.
(Also available in translation in French, German, Spanish, Chinese, Thai (2007))
McLeod, D.B. (1992) Research on affect in mathematics education: a reconceptualisation.
In D.A. Grouws, Ed., Handbook of research on mathematics teaching and learning, (pp.
575–596).New York: MacMillan, New York.
PISA (Programme for International Student Assessment) (2006) Assessing Scientific,
Reading and Mathematical Literacy. A Framework for PISA 2006. Paris: OECD.
Schoenfeld, A. (1985) Mathematical Problem Solving. Orlando: Academic Press.
Shulman, L.S. (1986) Those who understand: Knowledge growth in teaching, Educational
Researcher 15 (2), 4-14.
Shulman, L.S. (1987) Knowledge and teaching: Foundations of the new reform, Harvard
Educational Review 57(1), 1-22.
Singh, S. (1997) Fermat’s Enigma, New York: Walter
Stacey, K. & Groves, S. (1985) Strategies for Problem Solving. Lesson Plans for
Developing Mathematical Thinking. Melbourne: Objective Learning Materials.
Stacey, K. & Groves, S. (2001) Resolver Problemas: Estrategias. Madrid: Lisbon.
Stacey, Kaye (2005) The place of problem solving in contemporary mathematics curriculum
documents. Journal of Mathematical Behavior 24, pp 341 – 350.
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In this paper we present a framework for investigating teachers' mathematical pedagogical content knowledge (PCK). The components of the framework permit identification of strengths and weaknesses in the PCK held by various teachers on different topics. It was applied to teachers' written and interview responses addressing a student's difficulties with the standard subtraction algorithm. This enabled comparison of the PCK held by individual teachers, together with analysis of the teachers' PCK as a group. Most understood the issues and knew suitable representations, but occasionally lacked key understanding of students' misconceptions and how to help students recognise them. Investigating teachers' pedagogical content knowledge (PCK) is challenging for many reasons. PCK reveals itself in many places—in teachers' planning, classroom interactions, explanations, mathematical competency, and so on—and a study of only one environment will give a limited perspective. In addition, PCK's multifaceted nature makes considering all aspects time-consuming and complex. This report uses a framework that provides a set of lenses for studying PCK to address this second problem. At the same time we show that one carefully chosen environment can still reveal detail about teachers' PCK.
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This paper considers the role of content knowledge and pedagogical content knowledge in the teaching of probability to Grade 5 students. Lessons of two teachers were studied to determine the activities and teaching strategies used to bring out ideas associated with chance, and examine how probability understanding is developed in class. The lessons are shown to be rich in deep probabilistic ideas. The complex interplay between these concepts was sometimes handled well by the teachers, whereas on other occasions gaps in content and pedagogical content knowledge had the potential to cause misconceptions for students. Since the 1990s much research attention has focused on the pedagogical content knowledge of teachers. At the same time, chance and data have become more significant components of the curriculum, particularly at the primary school level. This report takes a closer look at the way two teachers use both content and pedagogical content knowledge to help Year 5 students develop key concepts in probability.
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This paper reviews the presentation of problem solving and process aspects of mathematics in curriculum documents from Australia, UK, USA and Singapore. The place of problem solving in the documents is reviewed and contrasted, and illustrative problems from teachers’ support materials are used to demonstrate how problem solving is now more often treated as a teaching method, rather than a goal in itself. The paper also analyses how the curriculum documents describe the growth of students’ abilities in the process areas of mathematics, and assesses the guidance that this provides for teachers. At each stage, the paper suggests directions for research that would be useful in assisting curriculum documents to promote the fundamental but elusive goal of making students better problem solvers.
Lee S. Shulman builds his foundation for teaching reform on an idea of teaching that emphasizes comprehension and reasoning, transformation and reflection. "This emphasis is justified," he writes, "by the resoluteness with which research and policy have so blatantly ignored those aspects of teaching in the past." To articulate and justify this conception, Shulman responds to four questions: What are the sources of the knowledge base for teaching? In what terms can these sources be conceptualized? What are the processes of pedagogical reasoning and action? and What are the implications for teaching policy and educational reform? The answers — informed by philosophy, psychology, and a growing body of casework based on young and experienced practitioners — go far beyond current reform assumptions and initiatives. The outcome for educational practitioners, scholars, and policymakers is a major redirection in how teaching is to be understood and teachers are to be trained and evaluated. This article was selected for the November 1986 special issue on "Teachers, Teaching, and Teacher Education," but appears here because of the exigencies of publishing.
begins by considering alternative theoretical foundations for research on affect / Mandler's (1984) theory, an approach to research on affect that is based on cognitive psychology, is selected for further discussion, particularly because it illustrates how affect can be incorporated into cognitive studies of mathematics learning and teaching / presents a framework for research on affect that reorganizes the literature into three major areas: beliefs, attitudes, and emotions / research on a number of topics from the affective domain is summarized, and linked to the proposed framework / explores how qualitative as well as quantitative research methods can be used in research on affect (PsycINFO Database Record (c) 2012 APA, all rights reserved)