Materials for teaching a course on research

Materials for teaching a course
on research in mathematics education
about students’ difficulties in mathematics
prepared by
Anna Sierpinska
© Anna Sierpinska, 2019

Table of contents – short version
Introduction...................................................................................................................................................................1
A brief history of the course..........................................................................................................................................1
Course outlines..............................................................................................................................................................2
The purpose of the course.........................................................................................................................................3
Texts ..........................................................................................................................................................................3
Assessment................................................................................................................................................................8
Examples of classroom activities and lectures ............................................................................................................10
Difficulties with the notion of directed number......................................................................................................11
Difficulties related to fractions in elementary schoolchildren and future teachers................................................24
Difficulties related to irrational numbers ................................................................................................................30
Difficulties in geometry ...........................................................................................................................................35
Epistemological obstacles, with examples from the history of the notion of function...........................................51
Further difficulties with the notion of function.......................................................................................................60
Difficulties with proofs by induction........................................................................................................................70
Affective sources of difficulty in mathematics ........................................................................................................74
Examples of participants’ research topics: titles and references ................................................................................77
Difficulties with proofs.............................................................................................................................................77
Difficulties in conceptualizing infinity......................................................................................................................78
Difficulties in probability..........................................................................................................................................79
Difficulties in algebra ...............................................................................................................................................80
Difficulties with multiplication.................................................................................................................................81
Difficulties in Problem Solving .................................................................................................................................82
Affective issues in learning mathematics ................................................................................................................83
References...................................................................................................................................................................84

A. Sierpinska Materials for a course on difficulties in mathematics
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Table of contents – long version, with titles of lectures and activities
Introduction...................................................................................................................................................................1
A brief history of the course..........................................................................................................................................1
Course outlines..............................................................................................................................................................2
The purpose of the course.........................................................................................................................................3
Texts ..........................................................................................................................................................................3
I. Theories about difficulties in studying mathematics..........................................................................................4
II. Difficulties in studying different areas of mathematics.....................................................................................4
III. Affective, socio-economic and cultural sources of difficulties in learning mathematics..................................8
IV. Methodological issues in researching students’ difficulties in mathematics ...................................................8
Assessment................................................................................................................................................................8
Examples of classroom activities and lectures ............................................................................................................10
Difficulties with the notion of directed number......................................................................................................11
Lecture 1: Getting a sense of students’ difficulties in mathematics on the example of difficulties with directed
numbers...............................................................................................................................................................11
Activity 1. Assessing the relative difficulty of problems about directed numbers ..............................................12
Lecture 2: Some objective (epistemological) reasons of one problem about directed numbers being more
difficult than another...........................................................................................................................................15
Lecture 3: Understanding directed numbers requires a reconceptualization of the concept of number...........17
Lecture 4: Levels of understanding of DN according to Coquin-Viennot ............................................................18
Lecture 5: Understanding directed numbers requires also a reconceptualization of operations on numbers. ..19
Activity 2. Multiplication in DN............................................................................................................................19
Lecture 6. Can the meaning of directed numbers be made easier by making it more “concrete”? ...................21
Lecture 7: Conceptualization of number in structural algebraic terms may not be convincing to the learners .21
Activity 3. Prove that −1 × −1 = +1. ................................................................................................................22
Lecture 8. How convincing axiomatic approach to DN can be? ..........................................................................22
Activity 4. Personal experience in understanding 𝐷𝑁.........................................................................................24

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Difficulties related to fractions in elementary schoolchildren and future teachers................................................24
Lecture 9. Epistemological and didactic sources of difficulties in learning fractions ..........................................24
Activity 5. Analyze an exercise about fractions. ..................................................................................................28
Difficulties related to irrational numbers ................................................................................................................30
Lecture 10 combined with Activity 6. (interactive lecture) Incommensurable quantities and irrational numbers
.............................................................................................................................................................................30
Lecture 11 combined with Activity 7. Why some students find proofs by contradiction unconvincing? Why
some proofs of irrationality of the square root of 2 are less convincing than others? .......................................32
Lecture 12. What makes geometry a difficult subject to learn (and to teach)?..................................................35
Lecture 13. The notion of “theoretical thinking”.................................................................................................36
Activity 8. Reflecting on one’s own process of solving a geometric problem .....................................................37
Activity 9. Analyzing a given solution of a geometric problem............................................................................38
Lecture 14. Different geometries require different ways of thinking .................................................................38
Activity 10. Experiencing the differences between realistic and axiomatic geometries .....................................42
Activity 11. The notion of angle – a concept difficult to define...........................................................................43
Activity 12. Experiencing thinking at the formal level in geometry. How difficult does it feel?..........................44
Lecture 15. The van Hiele theory of levels of knowing geometry .......................................................................44
Activity 13. Self-observation when solving a problem; deciding at which van Hiele level your reasoning was..47
Lecture 16 and Activity 14 The obstacle of “grid geometry” among future elementary teachers......................47
Epistemological obstacles, with examples from the history of the notion of function...........................................51
Lecture 17. Epistemological obstacles – a special class of sources of difficulty in mathematics ........................51
Activity 15. What does the phrase “analytic expression” mean?........................................................................54
Activity 16. The positive (knowledge building) aspects of the “analytic-expression” conception of function....56
Activity 17. Is a constant function “really” a function? .......................................................................................56
Activity 18. A dilemma caused by the “analytic-expression” conception of function.........................................57
Activity 19. Covariational conception: an epistemological obstacle? .................................................................57

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Lecture 18. The formal definition of function as an epistemological and a didactic obstacle. A more
fundamental obstacle to overcome at the beginning of learning functions. ......................................................57
Further difficulties with the notion of function.......................................................................................................60
Lecture 19. Difficulties in algebra have an impact on students’ learning of Calculus .........................................60
Activity 20. An exercise in analyzing students’ solutions ....................................................................................62
Activity 21. Analyzing diagnostic tools for detecting difficulties related to the notion of function....................66
Lecture 20. Standard “related rates problems” miss the opportunity to develop students’ covariational
thinking about functions......................................................................................................................................68
Activity 22. Discussion of the didactic value of offering a step-by-step procedure for solving a type of problems
.............................................................................................................................................................................69
Difficulties with proofs by induction........................................................................................................................70
Lecture 21 The illusory simplicity of proofs by induction ...................................................................................70
Activity 23. Finding a flaw in a reasoning seemingly by induction ......................................................................70
Activity 24. Analyzing students’ “proofs” by induction .......................................................................................71
Affective sources of difficulty in mathematics ........................................................................................................74
Lecture 22. Research on adult students’ frustration in pre-university level, prerequisite mathematics courses
.............................................................................................................................................................................74
Activity 25. Reflection on your own emotional experience when studying mathematics ..................................76
Examples of participants’ research topics: titles and references ................................................................................77
Difficulties with proofs.............................................................................................................................................77
Proofs in college level Calculus course ................................................................................................................77
Students' cognitive difficulties with mathematical proofs in high school ...........................................................77
Difficulties in distinguishing between the arbitrary and the necessary in mathematics.....................................78
Difficulties in conceptualizing infinity......................................................................................................................78
Difficulties in probability..........................................................................................................................................79
Modern probability and the illusion of linearity: Have graduate students met Kolmogorov?............................79
Difficulties in algebra ...............................................................................................................................................80
Students’ difficulties with algebraic notation......................................................................................................80

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Geometric transformations .................................................................................................................................81
Difficulties with multiplication.................................................................................................................................81
Misconceptions and difficulties that arise through the use of instructional situations that serves as a model for
the mathematical operation of multiplication. ...................................................................................................81
Difficulties in Problem Solving .................................................................................................................................82
Affective issues in learning mathematics ................................................................................................................83
Regulation of Emotions during Mathematical Activities in Adults ......................................................................83
References...................................................................................................................................................................84

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Introduction
This document is addressed to instructors of a university course on students’ difficulties in mathematics.
The participants in the course are assumed to have a bachelor’s degree in mathematics or at least 21
credits in undergraduate mathematics (Linear Algebra, Advanced Calculus, Probability, Statistics and
Abstract Algebra). They are assumed to be interested in doing research in mathematics education at
least at the master’s level.
Based on my own experience teaching such course in the Department of Mathematics of Concordia
University, the document contains information about the purpose of the course as it was presented to
the participants in the “Course outlines”, the forms of assessment used, lists of references to literature
on difficulties in different domains of mathematics, examples of lectures and classroom activities and
examples of topics on which participants chose to do their research.
I start by telling a brief history of the course I developed and taught in the years 2011-2018.
A brief history of the course
I started developing the course on students’ difficulties in mathematics as part of the Master in the
Teaching of Mathematics program (M.T.M.). It was first offered in the fall term of the academic year
2011/12 as
MATH 630-O Topics in the Psychology of Mathematics Education
Topic: Students’ Difficulties in Mathematics
The calendar description of the M.T.M. course MATH 630 was as follows:
MATH 630 Topics in the Psychology of Mathematics Education (3 credits)
This course studies epistemological, cognitive, affective, social and cultural issues
involved in mathematics.
Note: The content varies from term to term and from year to year. Students may re-
register for this course, provided the course content has changed. Changes in
content are indicated by the title of the course.
I have records of teaching the course on difficulties in 2011, 2013 and 2017. The present document is
based on materials developed for these versions of the course.
In the fall of 2017, the course became cross-listed with
MAST 652 Topics in Research in Mathematics Education
Topic: Research on Students’ Difficulties in Mathematics
in the MSc/MA in Mathematics program. This course was set up for MSc/MA students who chose
Mathematics Education as their special area of study. In the process of merging the M.T.M. and the

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MSc/MA programs, admissions to the M.T.M. program were suspended in January 2016 and candidates
interested in specializing in the area of Mathematics Education were being re-directed to apply to
MSc/MA in Mathematics, choosing, in their application, Mathematics Education as their preferred area
of study.
The calendar description of MAST 652 is as follows:
MAST 652 Topics in Research in Mathematics Education (3 credits)
The general aim of this course is to acquaint students with research problems
in mathematics education and ways of approaching them (theoretical frameworks
and research methodologies).
Note: The content varies from term to term and from year to year. Students may re-
register for this course provided the course content has changed. Changes in content
are indicated by the title of the course.
The M.T.M. course on students’ difficulties in mathematics was associated with a course on instructional
design – the art of designing mathematics lessons aimed at helping students overcome their difficulties
in the subject. The course on instructional design built on knowledge about students’ difficulties and
therefore it was recommended to take it after the course on difficulties. This was not always possible,
since only at most one of these courses could be offered in any academic year and students would
usually complete their studies in about two-and-half years. In the M.T.M. program, this course had a
different number: MATH 624 Topics in Mathematics Education. In the MSc/MA in Mathematics
program, the course on instructional design is taught as a special topic course under the same number,
MAST 652. The MAST 652 course on the topic of instructional design was first offered in the fall of 2018,
and was taught by a visiting professor, Dr. Analía Bergé. At this time, I had already retired from the
university (in June 2018). Dr. Bergé used some of the materials I developed for the MATH 624 course on
instructional design in the M.T.M. program. I am planning to make these materials freely available
online as well.
In this document, I will refer to students registered in the course on difficulties as “participants”,
reserving the name “students” to the learners of mathematics experiencing those difficulties.
Course outlines
A course outline normally provides information about
- the instructor’s name, forms of contact with him or her (telephone, email), office number and office
hours;
- texts for the course;
- the purpose of the course, and
- assessment,
in this order. Here, I start with information on the purpose of the course, as it has been presented in the
course outlines in the years 2013 and 2017. Then, I will present a (non-comprehensive) list of
bibliographical sources on the subject of students’ difficulties in mathematics (“Texts”), slightly larger

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than that given in the above mentioned course outlines. Finally, there will be information on the
assessment as presented to participants in the course outlines.
The purpose of the course
The description of the purpose in the 2013 and 2017 course outlines was as follows:
This course is premised on the assumption that there is no learning without
overcoming some difficulty. If you have experienced no difficulty in performing a task
then you have probably learned nothing by doing it. Therefore difficulties are treated
as something normal, expected and even welcome. The teacher must have a
sufficiently deep understanding of the possible epistemological, cognitive and
didactic sources of difficulties in mathematics to recognize those difficulties in
particular students. Based on this understanding, the teacher can create learning
situations that motivate students to overcome the difficulties associated with the
material.
In this course, you will have an opportunity to learn to anticipate the possible
difficulties students may have in learning different content areas in mathematics
such as arithmetic, high school algebra, geometry, college algebra, college level
calculus, and, depending on the interests of the participants, the more advanced
areas of mathematics. Difficulties related to the specificity of mathematical
reasoning and proving will also be tackled. Participants will choose topics to study
more in depth according to their own interests. They will inform the instructor about
their interests and the instructor will help them select relevant literature.
Texts
There has never been a “textbook” for the course on students’ difficulties in mathematics. A variety of
sources – journal articles, sections of books, reports in conference proceedings, etc. – were used. Two or
three weeks before the beginning of the course, participants were given a list of relevant readings,
organized along topics of research into difficulties in mathematics. Participants were not expected to
read them all, neither before nor during the course. The list was meant to help them choose a topic and
the literature they would want to study in more depth during the course and on which they would like
to make a reading presentation, write a literature review and do empirical research. They could choose
readings from those listed, but, since the reference list was not exhaustive, participants could choose
other readings related to the topic, depending on their specific interests. They were, however, required
to obtain the instructor’s approval of the sources they proposed to use in their reading presentations
and research, to avoid situations where, for example, Wikipedia articles would be the sole source of
information for the participant.
Below, I give a sample of topics and references, mostly based on the list used in the 2017 course. There
are four large categories of topics:
I. Theories about difficulties in studying mathematics

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II. Difficulties in studying specific areas of mathematics
III. Affective, socio-economic and cultural issues in learning mathematics
IV. Methodological issues in research on students’ difficulties in mathematics
Below, those large categories are split into more specific topics, and a sample of references is given for
each.
I. Theories about difficulties in studying mathematics
Statistical measures of difficulty: higher rate of failure on a survey item means that the item is more
difficult.
1) OECD Programme for International Student Assessment (PISA): (Kaur, 1997); (Saenz, 2009); (Olande, 2014)
Theories of levels of knowing mathematics
2) Bloom’s taxonomy: (Bloom, 1984); (Tallman, Carlson, Bressoud, & Pearson, 2016)
3) van Hiele levels:
- first applications to diagnostic tests (Burger & Shaughnessy, 1986; Crowley, 1987; Fuys, Geddes, & Tischler,
1988; Gutiérrez, Jaime, & Fortuny, 1991; Usiskin, 1982)
- updated van Hiele taxonomy (Rynhart, 2012)
4) Intra-, inter-, and trans-level of development of a mathematical domain (Piaget & Garcia, 1989)
Different sources of difficulties in mathematics
5) Cognitive, epistemological and didactic obstacles (Brousseau, 1997; Sierpinska, 1994; Sierpinska, 1992; Brown,
2008)
6) Pseudo-conceptual and pseudo=analytical ways of thinking in learning mathematics (Vinner, 1997)
Different ways of thinking in mathematics
7) Numerical vs Quantitative thinking (Thompson P. , 1993; 1994); (Brown, 2012); (Bran Lopez, 2015)
8) Practical vs Theoretical thinking (Sierpinska, 2005; Sierpinska, Nnadozie, & Oktaç, 2002)
9) Synthetic-Geometric, Analytic-Arithmetic and Analytic-Structural thinking in Linear Algebra (Sierpinska, 2000)
10) Structural reasoning (Harel & Soto, 2016)
11) Procept – concept theory of understanding mathematics (Gray & Tall, 1994)
II. Difficulties in studying different areas of mathematics
Fractions
12) High school students’ difficulties with fractions (Hart K. , 1981)
13) Fractions are difficult no matter how we teach them. Why? (Davydov & Tsvetkovich, 1991); (Bobos &
Sierpinska, 2017)

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Negative and positive (directed) numbers
14) Epistemological obstacles related to the notion of positive and negative numbers (Glaeser, 1981); (Hefendehl-
Hebeker, 1991)
15) A hierarchy of difficulties in learning integers (Coquin-Viennot, 1985)
16) High school students’ understanding of positive and negative numbers (Küchemann, Positive and negative
numbers, 1981)
17) Manipulatives – a potential source of misconceptions about directed numbers (Chilvers, 1985)
Problem solving
18) Literature review on difficulties in problem solving (Kaur, 1997); (Bran Lopez, 2015)
19) Problem solving in school (Graeber, Tirosh, & Glover, 1989); (Brown, 2012); (De Bock, Verschaffel, & Janssens,
2002); (Goldin, 2000); (Bednarz & Janvier, 1996); (Schmidt & Bednarz, 2002)
20) Undergraduate mathematics: Metacognitive skills lacking in novices (Schoenfeld, 1992)
21) Problem-solving difficulties in mature students re-taking an algebra course (Bran Lopez, 2015)
22) Intuition may be misleading (Avital & Barbeau, 1991)
23) Different research perspectives on problem solving (Silver, 1985)
Proving
24) Overview of research and reflection on mathematical proof (Hanna & Jahnke, 1996); (Douek, 2007); (Reid &
Knipping, 2010)
25) Proving is difficult and makes little sense to students (Schoenfeld, 1992)
26) Difficulty deciding whether an argument is a proof or not (Selden & Selden, 2003); (Alcock & Weber, 2005)
27) Sources of difficulties in undergraduates learning to write more formal proofs (Moore, 1994)
28) Students’ proof schemes (Harel & Sowder, 1998)
29) Knowing vs proving (Balacheff, 2010)
30) Difficulties with proofs by induction (Reid, 1992); (Brown, 2008)
31) Beginning university students rarely use mathematical proving to convince themselves of the validity of their
results in problem solving (Vasilyeva, 2018)
32) Role and function of proof in mathematics (De Villiers, 1990)
33) Difficulties in distinguishing between the arbitrary and the necessary in mathematics (Hewitt, 1999; Hewitt,
2001)
High school and college algebra
34) Students’ understanding letters in algebra (Küchemann, Algebra, 1981); (McGregor & Stacey, 1997);
35) Cognitive obstacles in learning algebra (Herscovics, 1989)

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36) Difficulties in understanding absolute value (Duroux, 1983); (Chiarugi, Fracassina, & Furinghetti, 1990);
(Sierpinska, Bobos, & Pruncut, 2011); (Almog & Bat-Sheva, 2012)
Linear Algebra
37) An overview of sources of difficulty in learning and teaching of linear algebra (Dorier & Sierpinska, 2001)
38) Understanding basic concepts of Linear Algebra after the first university level course (Stewart & Thomas,
2003)
39) Understanding representation of a vector in a basis (Hillel & Sierpinska, 1994)
40) Relationship between knowledge learned in linear algebra and teacher-student interactions (Sierpinska, 1997)
41) A large-scale study of 11-16 years old children’s understanding of vectors and matrices (Ruddock, 1981)
42) Difficulty with the language of vector space theory (Sierpinska, Understanding in mathematics, 1994, pp. 45-
48; 95-96)
43) Explanation of students’ difficulties in linear algebra in terms of Piaget & Garcia’s theory of intra-, inter- and
trans-levels of thinking about mathematical objects (Sierpinska, Understanding in mathematics, 1994, pp. 114-
117)
44) Difficulty to engage with theoretical thinking in learning linear algebra; incompatibility of thinking modes
between the teacher and the student as source of difficulty in linear algebra (Sierpinska, 2000); (Sierpinska,
Nnadozie, & Oktaç, 2002)
Functions
45) Diagnostic test of functions understanding (O'Shea, Breen, & Jaworski, 2016)
46) Students’ (mis)conceptions about functions (Vinner & Dreyfus, 1989)
47) Cognitive obstacles in learning functions (as part of algebra) (Herscovics, 1989)
48) Epistemological obstacles related to functions (Sierpinska, 1992)
49) College algebra and functions courses fail to teach students the essential meaning of functions, namely co-
variation of quantities; students’ conception of function is too narrow (mostly, function = algebraic expression)
and does not allow them to use functions to model physical phenomena (Carlson M. P., 1998; Carlson, Jacobs,
Coe, Larsen, & Hsu, 2002; Carlson, Oehrtman, & Engelke, 2010; Thompson & Carlson, 2017)
50) Difficulties in understanding representations of function (Janvier, 1986); (Kaldrimidou & Ikonomou, 1998)
51) Difficulties in understanding the formal concept of function (Sajka, 2003)
52) Practices related to teaching functions represented in textbooks as possible sources of didactic obstacles
(Mesa, 2004)
Calculus
53) Overview of research findings on students’ difficulties in calculus (Robert & Speer, 2001)
54) Misconceptions about a function and its derivative (Aspinwall, Shaw, & Presmeg, 1997)

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55) Assessing students’ readiness for Calculus (Carlson, Madison, & West, 2010)
56) Assessing students’ covariational understanding of functions in the context of coordination of rates of change
(Carlson M. P., 1998); (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002)
57) Comparison of USA and Korean teachers’ understandings of the average rate of change (Yoon, Byerley, &
Thompson, 2015)
58) Comparative study of US and Korean teachers’ understandings of mathematics (Thompson P. , 2015)
59) Didactic-institutional sources of students’ difficulties in learning limits of functions (Hardy, 2009; Hardy, 2009)1
60) Students’ difficulties in understanding related rates problems (Tziritas, 2011)
61) University students’ understanding of inflection points (Tsamir & Ovodenko, 2013)
62) Difficulties in algebra as an obstacle to learning Calculus (Giovaniello, 2017)
Analysis
63) Epistemological obstacles related to limits (Sierpinska, 1990)
64) Testing students’ understanding of limits (Przenioslo, 2004)
65) Didactical obstacles created by using visualization in teaching iterations of functions (Sierpinska, 2003)
66) Problems of understanding the continuity property of the Real Numbers system: (Dedekind, 1901); (Bergé,
2008; 2010); (Durand-Guerrier, 2016)
67) Problems of understanding the Fundamental Theorem of Calculus (Thompson P. , 1994)
Probability
68) Review of research in probability and statistics education (Shaughnessy, 1992)
69) Students’ beliefs and strategies in probability (Watson & Moritz, 2003)
70) Epistemological, cognitive and didactic obstacles to learning probability (Miszaniec, 2016)
71) The obstacle called “illusion of linearity” in probabilistic thinking (van Dooren, De Bock, Depaepe, Janssens, &
Verschaffel, 2002); (Miszaniec, 2016)
72) Students’ reasoning on coin tasks (Rubel, 2007)
73) Confusing probability with typicality (Lefebvre, 2010)
Statistics
74) Approaches and instruments for diagnosing students’ understanding and knowledge of statistics (Garfield &
Chance, 2000); see also (Shaughnessy, 1992)
75) Difficulties related to the concept of sample (Watson & Moritz, 2000); (Saldanha & Thompson, 2006)
Geometry
76) Overview of research on learning geometry (Owens & Outhred, 2006)
1
The PhD thesis, and a journal article summarizing its results.

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77) Difficulties in solving geometry problems, especially failure to apply already learned knowledge to a new
problem (Chinnappan, 1998)
78) The van Hiele levels of development in geometry (Usiskin, 1982); (Burger & Shaughnessy, 1986); (Fuys,
Geddes, & Tischler, 1988) ; (Gutiérrez, Jaime, & Fortuny, 1991); (Monaghan, 2000); (Munro, 2010)
79) Different paradigms of geometry (Houdement & Kuzniak, 2003)
80) A large-scale study of 11-16 years old children’s understanding of reflections and rotations (Küchemann,
Relections and rotations, 1981)
81) Prospective elementary teachers’ understanding of geometric transformations (Pelczer, 2017)
82) Students’ understanding of area (Jamski, 1978); (Woodward & Byrd, 1983); (Simon & Blume, 1994)
83) Difficulties with the notion of angle: (Smart, 2008) and references included in the bibliography of this thesis
III. Affective, socio-economic and cultural sources of difficulties in learning mathematics
84) Socio-economic reasons for the USA falling behind other countries in international assessments of
mathematical aptitude (Berliner, 2013)
85) Frustration with having to (re)take elementary math courses at a mature age (Sierpinska, Bobos, & Knipping,
2008); (Sierpinska, Bobos, & Knipping, 2007)
86) Weak motivation to study mathematics (Beddard, 2012)
87) Interactions between teachers and students as factors in students’ achievement in mathematics (Heyd-
Metzuyanim, 2013)
IV. Methodological issues in researching students’ difficulties in mathematics
88) Researching teachers’ knowledge of mathematics (Thompson P. , 2016);
89) Problems inherent in studying students’ mathematical thinking (Vasilyeva, 2018)
Assessment
Assessment in all versions of the course was based on:
 the quality of participants’ participation in classroom activities and discussions (10% of the final grade),
 an oral presentation of readings on students’ difficulties in a chosen area of mathematics (10% or 15%),
and two major written assignments:
 a literature review (30% or 25%), and
 a research report (50%).
There were also some minor, ad hoc, written assignments, rewarded by some bonus marks or counted
as part of the 10% for participation. For example – completing an activity started in class, or reflecting
on a question that arose in classroom discussions. In the 2017 version of the course, in the first class, I

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asked participants to write a reflection on their own difficulties in mathematics. The assignment was
formulated as follows:
(Assignment 1 – 2017) Before you can understand and describe how other people feel and
think about mathematics (which you will have to do in your research for this course), it is
useful to do some introspection and try to describe how you yourself feel and think about
mathematics. Reflect on your own learning of mathematics.
a. Do you remember something you found particularly difficult to understand?
What was it?
b. Do you now know why you found it so difficult?
c. Have you managed to overcome this difficulty? In what way?
Submitting this assignment was rewarded by 2 bonus marks.
At the beginning of the course, participants were asked to inform the instructor about the area of
mathematics in which they wanted to study the learners’ difficulties. They were asked to do so no later
than by the third week of classes. They then received the instructor’s advice on the selection of
readings. Presentations of readings were normally scheduled in the 5th
to 8th
weeks of classes.
The description of the literature review assignment was approximately as follows:
The participant will read at least four articles or book sections on students’
difficulties in learning their chosen area of mathematics. In an essay of about 3000
words, the participant will identify and describe the difficulties, their nature, their
sources, their symptoms in the form of, for example, certain recurrent errors or
behaviors. The participant will also describe, a) the diagnostic instruments that
allowed researchers to reveal these difficulties and, b) the theoretical frameworks
used to analyze and explain them.
This assignment was due in the 9th
week of classes, i.e., one week after the last oral reading
presentation. After each oral presentation the participant received feedback from their peers and the
instructor that they could use to improve their literature review.
The Research Report was the most important assignment in the course. It was, for some of the
participants, a first experience in conducting research in the area of mathematics education and in
writing a paper about it. With ample feedback and advice from the instructor, they were thus being
prepared to write their Projects or Theses later on in their program.
In this assignment, the participant was expected to design a questionnaire (to be passed in a class of
students) or an interview (to be conducted with a smaller number of students, but at least 2 students),
aimed at revealing their difficulties with a particular mathematical area (or a concept or a process). The
student was then expected to describe and analyze the responses obtained in a report of about 6000
words.
The report had to have a certain form and contain certain parts that, in the 2017 course outline, were
described approximately as follows:

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1. Introduction
Brief presentation of the aim of the research, the students’ backgrounds, and a
preview of the results obtained.
2. The research instrument
How was the interview or the questionnaire structured? What were the questions?
How can they be justified from the point of view of the goals of the research: what
made the questions suitable for the task of revealing a particular set of difficulties?
3. Results
The students’ responses are described and analyzed.
4. Discussion
The results are discussed in the light of what has already been found about students’
difficulties in the literature. Do the results corroborate those found by other
researchers? Do they refine them? Contradict them? How and why?
5. Conclusions and recommendations
5.1 Summary of the revealed difficulties
5.2 Could the research instrument and/or technique be improved for
diagnostic purposes? How?
6. References
At the end of the report, references must be listed in alphabetical order, using the
APA (preferably the Sixth Edition) style. References within the main body of the text
should have the format (Author’s last name, YEAR), e.g., (Ainley, 1994).
Participants were given an opportunity to present their initial ideas for their research in the class
sessions of weeks 9 to 12 of the course, and obtain feedback and advice from their peers and the
instructor.
The deadline for submission of completed reports was usually 10 to 12 days after the end of classes.
Examples of classroom activities and lectures
The first four weeks of classes were teacher-led. The teacher gave short lectures, organized activities
and discussions. Some activities were designed to prepare the participants for analyzing actual
mathematics students’ solutions, and detecting and describing the nature of their difficulties, thus
preparing them for their research.
Following are some examples of the lectures and activities drawn on those given in the 2011, 2013 and
2017 versions of the course. The examples are organized mainly according to difficulties with a
particular mathematical concept (directed numbers, fractions, irrational numbers, and functions), area
(geometry) or process (proving, proving by induction). There is also a series of lectures on the notion of
epistemological obstacle but the examples and activities connected with this lecture are mostly about

11
epistemological obstacles related to functions. One lecture has been devoted to affective issues related
to studying mathematics. I tried to avoid lecturing on topics chosen by the participants to do their
reading presentations and research on, or at least base my lectures on different readings than they
listed in their topic proposals. I received the topic proposals by the third week of classes only. So in
preparing lectures for the first three weeks, I would choose topics that I considered important but that I
believed unlikely to be covered in the participants’ reading presentations.
There are altogether 22 examples of lectures and 25 – of activities. I did not give all these lectures and
activities in one year, of course.
Difficulties with the notion of directed number
In the 2011 version of the course, the first two weeks were devoted to the notion of positive and
negative or “directed” numbers. The notion is considered to be “elementary”, assumed to be acquired
already in elementary school, and therefore easy, so one may wonder why talk about it in a course on
difficulties in mathematics for graduate students in mathematics who are probably going to be teaching
at college or university level. The reason is that the notion is not as easy as it may seem and it is good for
a future teacher to realize that the most elementary notions in mathematics often hold unsuspected
depth and complexity. In the history of Western mathematics, as late as in the nineteenth century,
mathematicians held heated debates on whether to even consider the negative numbers as numbers at
all [(Pycior, 1984); see also (Glaeser, 1981) and (Hefendehl-Hebeker, 1991)]. Today’s students may be
used to seeing negative numbers but this does not always mean that they understand them correctly.
Even college students happen to believe that “−𝑥” denotes a negative number, and have a lot of
difficulties with problems involving absolute value (Gagatsis & Thomaidis, 1994).
The lectures and activities on the topic served also some more general purposes of the preparation of
graduate students to teach and do research in mathematics education. Novice teachers tend to judge
their students in terms of binary distinctions: understands/does not understand, knows/does not know,
etc., and the content of teaching or a problem in terms of difficult/easy. We prepare them to get used to
asking not IF a student understands a given notion but HOW do they understand it; not IF a problem is
difficult (or easy) but, for example, WHAT makes many students fail (or succeed) on it, or WHAT makes a
problem more difficult than another.
Lecture 1: Getting a sense of students’ difficulties in mathematics on the example of difficulties
with directed numbers
Our thinking about students’ difficulties in mathematics will be organized around the hypothesis that
these difficulties may have three kinds of sources:
- in the student’s thinking, behavior and knowledge (cognitive sources);
- in mathematics itself (epistemological sources), and
- in the way mathematics is taught (didactical sources).
Before I go on to discuss these sources in any systematic way, I’d like us to collectively reflect on
students’ difficulties with one very fundamental notion in mathematics, called “directed numbers” in

12
mathematics education literature, that is, numbers endowed with a plus or a minus sign. This activity
will hopefully convince you that difficulties in learning any piece of mathematics may stem from all
these sources.
The reflection will be provoked by an activity based on an article by Danièle Coquin-Viennot (1985), in
which she proposes a hierarchy of levels of understanding directed numbers, and claims that the
difficulty of passing to a higher level can be partly explained by certain “didactical obstacles”, i.e., the
way the notion is taught in school.
The activity will be composed of 4 parts.
In the first part, we will look at the set of problems that the Author used in her research to assess
students’ level of understanding of directed numbers (this was her “research instrument”). There were
two groups of students: 12 years old (beginning of high school) and 16 years old (finishing high school).
We will solve these problems and rank them from the problem that we think is the easiest to one that
we think would be the most difficult for each of these groups of students. In this first part, each of you
will work alone. In the second part, you will compare your ranking with another participant and you will
discuss any discrepancies until you come to a consensus. In the third part of the activity, we will
compare our predictions with the results obtained by Coquin-Viennot: the difficulty of a problem for a
group is measured by the percentage of students in that group having solved it correctly. The smaller
the percentage, the higher the difficulty. In the last, 4th
part of the activity, we will debate on possible
sources of the difficulty of learning directed numbers.
Activity 1. Assessing the relative difficulty of problems about directed numbers
Part I.
Rank the following problems according to difficulty for students
(a) finishing their high school (16 years old);
(b) after the first year of their high school (12 years old).
Note: For ease of reference, the problems were given short mnemonic labels in square brackets after
the number of the problem.
1. [SOM 1] Calculate 𝑆 = 2 − 2 + 0 − 1 − 3 + 5
2. [RANG] Put the following numbers in increasing order: −7, 5, −2, 6
3. [ORD 1] 𝑥 and 𝑦 are two integers such that 𝑥 ≥ 𝑦. What is the order relation between 𝑥 − 2 and 𝑦 − 3?
4. [ORD 2] 𝑥 and 𝑦 are two integers such that 𝑥 ≥ 𝑦. What is the order relation between (−3)𝑥 and (−3)𝑦?
5. [SOM 2] Calculate: 𝑆 = −12 + 14 − 15 + 13 − 11 + 11 − 13 + 12 − 15 + 12 − 11 + 16
6. [EQUA] Solve the equation: 2𝑥 − 15 = 𝑥 + 30
7. [RELA] If I add an integer to the number 𝑎, is the result always greater than 𝑎? Yes or No? Explain.
8. [VALA] Knowing that 𝑥 ≥ +2 , write |𝑥 − (+2)| without using the absolute value bars.
9. [ING] For admission into an engineering school, the candidate must pass 6 tests. Each test is worth 120
marks. To be admitted, the candidate must obtain an average of at least 60 on 120. Student A got the

13
following grades: Algebra: 63; Geometry: 57; Physics: 60; Chemistry: 55; Science: 59; French: 64. Will he
be admitted? Justify your answer.
10. [SPOR] An athlete participates in a sports event. To win the general medal he has to obtain a certain
average on the set of competitions A, B, C, D, E, and F.
On A, he wins two points.
On B, loses two points.
On C, he wins no points.
On D, he loses one point.
On E, he loses three points.
On F, he wins 5 points.
Will the athlete win the general medal? Justify.
11. [TRAI] (The problem is read to the students who are asked to solve the problem mentally, and hand in the
answer to the teacher immediately afterwards) A train starts from the city A in the direction of the city B.
It stops at some stations between A and B. I will tell you how many people get off the train and how many
get on the train at these stations. I will tell you at the end how many people were on the train at A. You
will then write how many people arrived on the train at B.
Station Number of people getting off Number of people getting on
1 14 18
2 15 11
3 12 12
4 13 15
5 15 12
6 11 16
There were 120 people on the train at A.
12. [DUBO] There are 6 questions on an exam. Each question is worth 20 marks. To pass, one has to obtain an
average grade of at least 10 out of 20. Here are the results of the student Dubois: French: 12; English: 8;
Math: 10; Physics: 11; History: 6; Science: 12. Will this student pass? Justify your answer.
13. [SUCE] Peter has some money in his pocket. Should he want to buy one candy, he’d be left with 30 cents.
Should he buy two candies, he’d be short of 15 cents. What is the cost of one candy? Justify your answer.
14. [CERC] (a) There are two circles with centers at O and O’ on an oriented axis in the configuration below.
Given that: 𝑂𝐴̅̅̅̅ = 5, 𝐴𝐴′̅̅̅̅̅ = 4, 𝐴′𝑂′̅̅̅̅̅̅ = 3, 𝑂𝑂′̅̅̅̅̅ = 12, verify that 𝑂𝐴̅̅̅̅ + 𝐴𝐴′̅̅̅̅̅ + 𝐴′𝑂′̅̅̅̅̅̅ = 𝑂𝑂′̅̅̅̅̅

14
(b) Here is another figure:
Given that 𝑂𝐴̅̅̅̅ = 5, 𝐴′𝑂′̅̅̅̅̅̅ = 3, 𝑂𝑂′̅̅̅̅̅ = 6, find 𝐴𝐴′̅̅̅̅̅ .
Verify that 𝑂𝐴̅̅̅̅ + 𝐴𝐴′̅̅̅̅̅ + 𝐴′𝑂′̅̅̅̅̅̅ = 𝑂𝑂′̅̅̅̅̅
Part II
Get in groups of two or three and compare your ranking. Ask those who got a different ranking than you
why they thought a problem more difficult than another.
Part III
Compare your predictions about the difficulty of the problems with actual results obtained by a group of
researchers in France (Coquin-Viennot, 1985) when they asked high school students to solve these
problems.
Problem Relative frequency of correct
answers in last grade of high school
(N=101)
Relative frequency of correct
answers in the first grade of
high school (N=89)
1. RANG 99 42 (rank 5)
2. SOM 1 97 21 (9)
3. DUBO 94 83 (1)
4. CERC 93 25 (7)
5. ING 93 74 (2)
6. EQUA 89 0 (12)
7. SOM 2 88 17 (10)
8. TRAI 80 49 (3)
9. ORD 1 79 1 (11)
10. RELA 71 24 (8)
11. SPOR 63 35 (6)
12. SUCE 59 46 (4)
13. ORD 2 31 0 (12)
14. VALA 28 1 (11)
Total number of students 101 89

15
PART IV
Whole class discussion: What could be the possible sources of students’ difficulties with integers?
The teacher lets the participants express their views and conjectures and then offers some comments in
the form of a lecture.
Lecture 2: Some objective (epistemological) reasons of one problem about directed numbers
being more difficult than another
Let us compare two techniques of solving the ING problem. One of them is based on a conception of a
directed number as an absolute measure number preceded by a sign. The other requires a
reconceptualization of numbers as representing additive change or a difference relative to a certain
reference point (Sierpinska, Bobos, & Pruncut, 2011). The ING problem can be solved by either one of
these techniques. But the next problem, SPOR, must be solved using the more sophisticated notion of
directed number. This can be the reason why SPOR turned out to be solved correctly by less students
that ING.
[ING] For admission into an engineering school, the candidate must pass 6 tests. Each
test is worth 120 marks. To be admitted, the candidate must obtain an average of at
least 60 on 120. Student A got the following grades: Algebra: 63; Geometry: 57;
Physics: 60; Chemistry: 55; Science: 59; French: 64. Will he be admitted? Justify your
answer.
A solution based on the conception of directed number as absolute measure number preceded by a sign
can be:
Calculate the average and see if it is greater or smaller than 60.
Average is 59.6666… so the student will not be admitted.
A solution based on the conception of directed number as difference relative to a reference point can
be:
Calculate the sum of differences (Actual grade – 60). If the sum is less than 0, the
student will not be admitted.
In this case, the sum of differences is +3 − 3 + 0 − 5 − 1 + 4 = −2 < 0 so the
student will not be admitted.
By the way, we can prove that the technique in the second solution is valid in general. If 𝐺𝑖 denote the
grades and 𝑀 their mean, then assuming that there are 𝑛 grades, we have:

16
∑(𝐺𝑖 − 𝑀)
𝑛
𝑖=1
= ∑ 𝐺𝑖
𝑛
𝑖=1
− 𝑛𝑀 < 0 ⟺
∑ 𝐺𝑖
𝑛
𝑖=1
𝑛
< 𝑀
Now, back to the discussion. Let’s look at the SPOR problem:
[SPOR] An athlete participates in a sports event. To win the general medal he has to
obtain a certain average on the set of competitions A, B, C, D, E, and F.
On A, he wins two points.
On B, loses two points.
On C, he wins no points.
On D, he loses one point.
On E, he loses three points.
On F, he wins 5 points.
Will the athlete win the general medal? Justify.
The only way to solve this problem is to use the technique in the second solution of ING, because all we
are given are the differences. The sum of the differences is
+2 − 2 + 0 − 1 − 3 + 5 = +1
It is positive, so the athlete will win the general medal.
Now let us look at the SUCE problem.
[SUCE] Peter has some money in his pocket. Should he want to buy one candy, he’d
be left with 30 cents. Should he buy two candies, he’d be short of 15 cents. What is
the cost of one candy? Justify your answer.
It turned out to be quite difficult: only 59% of the older students solved it correctly Why? It can be
solved without using directed numbers or even algebra, using a diagram and arithmetic only:
Or it can be solved using algebra, but without directed numbers.
Let M stand for the amount of money in cents that Peter has in his pocket, and C the
price of one candy in cents. Then we can write a system of equations where all
variables denote positive numbers or simply the absolute measure numbers:
𝑀 − 𝐶 = 30
2𝐶 = 𝑀 + 15

17
The difficulty of this problem resides in the fact that all the givens are relations between quantities; we
are not given the value of any quantity. The quantities are unknown. This is enough to make a problem
more difficult, as stated by Bednarz and her collaborators (Bednarz & Janvier, 1996; Schmidt & Bednarz,
2002). Moreover, the given relations are directed numbers in the more sense of differences relative to a
reference point. This makes this problem objectively more difficult than problems with givens
understandable with the “sign + absolute measure number” conception.
This lecture ended the first week’s classroom session in 2011. In the second week the topic of difficulties
with directed numbers continued to be discussed in a series of lectures and activities.
Lecture 3: Understanding directed numbers requires a reconceptualization of the concept of
number
As suggested in the previous lecture, understanding directed numbers requires a reconceptualization of
the notion of number from number as absolute measure of something (𝐴𝑀𝑁, absolute measure
number) to number as representation of a directed change (decrease, increase) of the measure of
something (𝐷𝑁, directed number). Increase is indicated by the plus sign; decrease – by the minus sign.
𝐷𝑁 with a plus sign are called positive; 𝐷𝑁 with a minus sign are called negative.
The 𝐷𝑁 conception is necessary in the study of Mathematical Analysis – the mathematics of change. The
19th
century French mathematician Augustin-Louis Cauchy understood well the importance of this
reconceptualization and the challenge it presents to students of Analysis when he was writing his Cours
d’Analyse, the first systematic exposition of this domain of mathematics (Cauchy, 1821/1968)2
. The
beginning of the book is devoted to an introduction of a new kind of numbers, that he called
“quantités”, which correspond to what we have denoted here by 𝐷𝑁, reserving the name “nombres” to
what we have denoted by 𝐴𝑀𝑁.
There is an isomorphism between 𝐴𝑀𝑁 and a part of 𝐷𝑁 (the “positive” 𝐷𝑁, which we can denote by
𝐷𝑁+
). But isomorphism is not the same as identity of objects.
We can embed 𝐴𝑀𝑁 into 𝐷𝑁: 𝑓: 𝐴𝑀𝑁 → 𝐷𝑁, 𝑓(𝑥) = 𝑥. The image of the embedding is 𝑁+
.
But to go from the image back to 𝐴𝑀𝑁, we need the absolute value function:
𝑔: 𝐷𝑁+
→ 𝐴𝑀𝑁, 𝑔(𝑥) = |𝑥|.
There is a structural similarity between 𝐴𝑀𝑁 and subset of 𝐷𝑁, but the elements of these structures
are different in nature. From the perspective of 𝐷𝑁, a symbol like, for example, “+3” represents one
single whole, a number in itself. From the perspective of 𝐴𝑀𝑁, this symbol represents two objects: a
number (3) and a sign (+) (Sierpinska, Bobos, & Pruncut, 2011). This subtle distinction often remains
unnoticed by students and teachers alike.
2
See also extensive quotations from Cauchy’s Cours d’Analyse in (Duroux, 1983) and Duroux’s commentaries on
them.

18
When students are first introduced to directed numbers, tasks they are given allow them to continue
thinking of number as absolute measure, and conceive of the new “directed number” not as an entity in
itself, but as a compound object, made of a sign and a number in the old sense (the “Sign+AMN”
conception of number). For example, students would be given a rule such as, “when adding two directed
numbers with different signs, subtract the one with smaller absolute value from the one with larger
absolute value and supply the result with the sign of the number with larger absolute value”.
Thinking of number as a compound object underlies common students’ mistakes such as interpreting a
letter in an algebraic expression as representing a non-negative number [ (Duroux, 1983); (Chiarugi,
Fracassina, & Furinghetti, 1990); (Gagatsis & Thomaidis, 1994)]. Number as absolute measure has no
sign since it ignores direction. Thus, the letter variable 𝑥, which appears to represent a single entity,
must refer to absolute measure, a number without a sign. The symbol “ x ” then necessarily refers to a
“negative number”, and the statement “|𝑥| = –𝑥, if 𝑥 < 0” could be understood as allowing the
absolute value to be negative sometimes. (Sierpinska, Bobos, & Pruncut, 2011).
Lecture 4: Levels of understanding of DN according to Coquin-Viennot
Coquin-Viennot (1985) identified four levels of understanding of integers in students based on a
mathematical analysis of the notion combined with analysis of the teaching of the concept in school and
the results of her empirical study.
The conception of number that students develop in elementary school is that of
Conception I: Number as Absolute Measure
From this point of view, 0 represents “nothing” and negative numbers do not exist.
So, 5 − 7 = 0.
In high school, this conception is extended to:
Conception II: 𝐴𝑀𝑁 endowed with a sign (±𝐴𝑀𝑁)
A number preceded by the minus sign represents a missing quantity, a lack of this quantity. For example,
−300 may represent a debt of three hundred dollars, where $300 is the amount of money (𝐴𝑀𝑁) one
will have to pay back. In this conception, positive and negative numbers constitute two separate
entities, and each has to be processed separately.
By practicing sums of numbers that can be positive and negative, at least some students develop the
third conception:
Conception III: Directed numbers form a unified system of numbers that can be added (𝐷𝑁, +)
This conception allows students to simplify longer arithmetic expressions by cancellations and
compensations. Interpretation of negative numbers as missing amounts or debts still makes sense
within the frame of this conception. But multiplying two debts and getting a positive balance doesn’t
make sense.
Conception IV: Directed numbers form a unified system of numbers where all four arithmetic operations
can be defined and make sense (𝐷𝑁, +,×)

19
This conception requires a structural understanding of the system 𝐷𝑁 which gives meaning to the
“minus times minus is plus” rule.
Conception IV was very rare among students who participated in Coquin-Viennot’s research.
Lecture 5: Understanding directed numbers requires also a reconceptualization of operations on
numbers.
Let us consider, for example, the operation of addition in 𝐴𝑀𝑁. If 𝑚, 𝑛 ∈ 𝐴𝑀𝑁, what can be the
meaning of 𝑚 + 𝑛?
If 𝑚, 𝑛 are measures in the same units of two objects of the same kind, 𝑚 + 𝑛 can be interpreted as
the measure of an object of the same kind made of the given two. For example, if the length of a
rectangle is 2.5 meters and the width is 5.8 meters, the perimeter is twice the sum of 2.5 and 5.8
meters.
On the other hand, addition in 𝐷𝑁 represents a composition of two additive changes (that is, changes
that consist in adding or subtracting something, not scaling). If:
𝑛: 𝐷𝑁 → 𝐷𝑁 𝑚: 𝐷𝑁 → 𝐷𝑁
𝑥 ↦ 𝑥 + 𝑛 𝑥 ⟼ 𝑥 + 𝑚
then the sum is the composition of these functions:
𝑛 + 𝑚: 𝐷𝑁 → 𝐷𝑁
(𝑛 + 𝑚)(𝑥) = (𝑥 + 𝑛) + 𝑚
Addition of directed numbers can be interpreted in applications of mathematics in terms of directed
quantities such as temperatures. For example, calculating an average temperature, as in the following
problem:
The daily low temperatures for Halifax were recorded for five consecutive days. The
temperatures (in degrees Celsius) were +6, −1, −2, −3, +5. Find the average low
temperature, over the five-day period.
It is much harder to give “situational” meaning to multiplication of two negative numbers.
(At this point the lecture is interrupted to give the participants an opportunity to reflect on this problem
on their own.)
Activity 2. Multiplication in DN
a) Can you think of a situational problem where multiplication of two negative numbers would be
necessary?
b) What do you think of the introduction to multiplication of integers in the textbook for Grade 9 by
Flewelling et al. (Flewelling, Carli, & Telfer, 1993, pp. 44-45) (See reproduction below)

20

21
Lecture 6. Can the meaning of directed numbers be made easier by making it more “concrete”?
There have been attempts to give directed numbers a more concrete meaning, especially for elementary
school students, e.g., (Chilvers, 1985). But these concretizations were either simplistic (and not
permitting an interpretation of multiplication of two negative numbers) or very complicated and not
facilitating the task of understanding at all. Besides, they could reinforce certain limited conceptions of
mathematical concepts; for example, Chilvers’ approach reinforces the idea that multiplication is simply
a shortcut for repeated addition.
Lecture 7: Conceptualization of number in structural algebraic terms may not be convincing to
the learners
After many unsuccessful attempts at giving negative numbers a concrete meaning, mathematicians
resigned themselves to accept an abstract-algebraic meaning by means of an axiomatic theory.
The basic condition was: we don’t want 𝐷𝑁 to be a completely new system of numbers with nothing in
common with 𝐴𝑀𝑁. Restricted to 𝐷𝑁+
, we want the operations in 𝐷𝑁 to have the same properties as

22
in 𝐴𝑀𝑁. 𝐷𝑁 was then defined axiomatically, by properties, and the properties were consistent with
those of operations in 𝐴𝑀𝑁.
With this approach, the rule, “minus times minus makes plus” did not have to make sense in terms of
“debts” or other interpretations of the kind we saw earlier; it was enough that it could be logically
derived from the axioms.
Let’s recall the axioms that have been postulated about properties of operations on numbers.
Axiom 1: Zero is a number such that, for any number 𝑎, 𝑎 + 𝑧𝑒𝑟𝑜 = 𝑎
Axiom 2: For any number 𝑎, there exists a number 𝑏 such that 𝑎 + 𝑏 = 𝑧𝑒𝑟𝑜; 𝑏 is called the “opposite”
of 𝑎.
Axiom 3: One is a number such that, for any number 𝑎, 𝑎 ∙ 𝑜𝑛𝑒 = 𝑎
Other axioms postulate commutativity and associativity of addition and multiplication, the
existence of the multiplicative inverse of any number other than zero, and the distributivity property.
Let us try to prove the rule “minus times minus makes plus” using these axioms.
Activity 3. Prove that (−1) × (−1) = +1.
(MSc/MA students should be left alone to write this proof. In some cases, a hint can be given.)
Hint: It will be useful to prove some more elementary propositions first.
Proposition 1. The zero number is unique. We denote it by “0”.
Proposition 2. Any number multiplied by 0 gives 0.
Proposition 3. Every number 𝑎 has a unique opposite. We denote it by −𝑎.
Proposition 4. −(−𝑎) = 𝑎
Lecture 8. How convincing axiomatic approach to DN can be?
Axiomatic reasoning about the minus times minus is plus rule may not work as a convincing explanation
to high school or even many undergraduate mathematics students.
In fact, this explanation was extremely hard to swallow for many mathematicians in the 19th
century,
when this kind of explanations first appeared in the context of the trend towards establishing
mathematics on rigorous and strictly formal foundations, without any reference to spatial intuitions,
physical metaphors and other worldly matters.
How hard it was and why is described in (Pycior, 1984). At that time, mathematicians considered
negative and imaginary numbers useful in solving equations, but only as computational tools in
intermediate calculations; the final solution had to be a positive real number; negative or complex
solutions were considered meaningless.

23
Some British mathematicians called for expulsion of these numbers from algebra on the ground that
there were no adequate definitions for them. Axiomatic definitions were far from satisfactory to people
who believed that the meaning of general terms can only be given by referring to some physical or ideal
reality, not by postulating that a term “X” be used to name any object that has such and such property
(or properties). When we are told what a term refers to, we can form a “general idea” of it. If we are
told what properties an object must have for us to name it in a certain way, we have no general idea of
what it would look like.
A British mathematician by the name of Frend (Frend, 1796-99) (quite forgotten today) criticized
negative numbers as
“jargon, at which common sense recoils…; like many other figments…. [negative
numbers find] the most strenuous support among those who love to take things
upon trust, and hate the labour of serious thought”.
Those who defended negative numbers and their explanation in structural-algebraic terms counter-
argued that “logic rather than meaning was crucial to sound reasoning” (Pycior, 1984, p. 430).
Other historical evidence of the difficulty to conceive of and teach and learn negative numbers is not
lacking. The citations below are taken from an article in the journal For the Learning of Mathematics on
difficulties (obstacles) in conceptualizing directed numbers (Hefendehl-Hebeker, 1991).
Citation 1: Blaise Pascal
I know people who cannot understand that when you subtract four from zero what is
left is zero. (Blaise Pascal)
Citation 2: Stendahl
[In 1797 when I was 14] I thought mathematics ruled out all hypocrisy…. Imagine
how I felt when I realized that no one could explain to me why minus times minus
yields plus (− × −= +)!... That this difficulty was not explained to me was bad
enough (it leads to truth and so must, undoubtedly, be explainable). What was worse
was that it was explained to me by means of reasons that were obviously unclear to
those who employed them.
M. Chabert, whom I pressed hard, was embarrassed. He repeated the very lesson
that I objected to and I read in his face what he thought: It is a ritual, everyone
swallows this explanation. Euler and Lagrange, who certainly knew as much as you
do, let it stand…. It is clear that you are just trying to be difficult…. [But how could I
be convinced by] the not-so-apt comparison made even more inept by M. Chaubert’s
hopelessly drawn out Grenoble accent, to assume that the negative quantities
represent someone’s debt. How can a person gain 5 000 000, that is, five million, by
multiplying a debt of 10 000 francs by [a debt of] 500 francs?...
It took me a long time to conclude that my objection to the theorem: minus times
minus is plus simply didn’t enter M. Chabert’s head, that M. Dupuy would invariably
answer with a superior smile, and that the mathematics luminaries that I approached

24
with my questions would always poke fun at me. I finally told myself what I tell to
this day: It must be that minus times minus is plus. After all, this rule is used in
computing all the time and apparently leads to true and unassailable outcomes.
(Stendhal, 1835)
Activity 4. Personal experience in understanding 𝐷𝑁
What is your position on the axiomatic definition of negative numbers? Do you find it convincing? Have
you always found it convincing?
Difficulties related to fractions in elementary schoolchildren and future
teachers
Lecture 9. Epistemological and didactic sources of difficulties in learning fractions
An opinion, popular in the US, is that fractions are difficult to learn because they are complex
mathematical objects. At first, influential researchers claimed that, in order to learn fractions, children
have to recognize them in six different contexts or interpretations, called “subconstructs”, and
understand the relations between these interpretations:
Analyses of the components of the concept of rational number (Kieren, 1976;
Novillis, 1976; Rappaport, 1962; Riess, 1964; Usiskin, 1979) suggest one obvious
reason why complete comprehension of rational numbers is a formidable learning
task. Rational numbers can be interpreted in at least these six ways (referred to as
subconstructs): a part-to-whole comparison, a decimal, a ratio, and indicated division
(quotient), an operator, and a measure of continuous or discrete quantities. Kieren
(1976) contends that a complete understanding of rational numbers requires not
only an understanding of each of these separate subconstructs but also of how they
relate. Theoretical analyses and recent empirical evidence suggests that different
cognitive structures may be necessary for dealing with the various rational number
subconstructs. (Behr, Lesh, Post, & Silver, 1983, pp. 92-93)
Later, the number of subconstructs has been reduced to 5:
“A factor contributing to students’ difficulties in learning fraction refers to its
multifaceted construct (Behr, Harel, Post, & Lesh, 1993; Kieren, 1992; Lamon, 1999).
In this respect, five subconstructs have been identified: part–whole, ratio, quotient,
measure, and operator. For instance, the fraction 3/4 can be conceived as a part of a
whole (three out of four equal parts), as a quotient (three divided by four), an
operator (three quarters of a quantity), a ratio (three parts to four parts), and finally
as measure (as a point on a number line).” (Pantziara & Philippou, 2012, p. 63)
This point of view on the meaning of fractions leads to designing separate teaching “units” for each of
the subconstructs. This makes it difficult for students to understand how they are all about the same

25
concept. The “part-whole” subconstruct is often introduced in the context of sharing pizzas, where
“fraction” refers to a fraction of a quantity – thus a quantity and not a number. But the quantity is not
explicit, and it is in unspecified units (when we divide a pizza into “equal parts”, do we mean that the
pieces have to have equal weight? or equal surface area? or volume? or number of calories?). So
fraction of a quantity becomes, simply, an object (a piece of a pizza). The “part-whole” subconstruct is
transformed into a “part-of-a-whole” misconception of fraction. The common pictorial representations
reinforce this misconception. See, for example, the popular “fractions-fractions” song3
. How can the
student make a connection between, say, 3 slices of a pizza and the notion of ¾ as a number 𝑥 which,
multiplied by 4 gives 3 or the result of dividing 3 by 4?
Future elementary teachers often have the same misconceptions about fractions as children have when
they are subjected to the kind of “pizza-mathics” hinted at above. When asked “what is a fraction?” the
most common response is “a part of a whole”. This was my experience when I taught a course on
fractions for pre-service elementary teachers at Concordia University in three consecutive winters of
2013-2015. During these years, together with my collaborator, Georgeana Bobos (who was then my
doctoral student), we designed, and experimented with, an alternative approach to teaching fractions to
future elementary teachers, aiming at building a connection between their notion of fraction as a part of
a whole – a material object – and the notion of fraction as an abstract number. We described this
approach in (Bobos & Sierpinska, 2017). In the introduction of this paper, we summarized our approach,
called “Measurement approach”, as follows:
[T]he measurement approach [was] inspired by an approach under the same name,
but addressed to children, described in (Davydov & Tsvetkovich, 1991). (…) The
conception rests on the assumption, shared by many mathematics educators, that
future elementary teachers’ understanding of fractions is often made of two
disconnected parts: the material conception or the visually-based idea of fraction of
something, and the formal conception or the formal calculus on expressions of the
form
𝑎
𝑏
with 𝑎 and 𝑏 being whole numbers. We share with other mathematics
educators the conviction that it is important to help prospective teachers to connect
the material and the formal parts of their conceptions (Parker & Baldridge, 2003; Van
de Walle & Lovin, 2006; Reys, Lindquist, Lambdin, Smith & Colgan, 2010; Sowder,
Sowder & Nickerson, 2011; Lamon, 2012). But we propose to do it differently: by
means of a theory of fractions of quantities.
Fraction of a quantity is a mathematical theorization of the visual and intuitive idea
of fraction of something. In the material conception, one thing (pizza, cake, shaded
part of a diagram) is the fraction
𝑎
𝑏
of something (called a “whole” or a “unit”), when
the thing is made of 𝑎 equal parts and the whole is made of 𝑏 such parts4
. When
we make explicit the quality of the objects we are comparing in this situation (length,
area, volume, weight, or number of elements), and use an explicit unit to measure
this quality, then we are not comparing objects but quantities (Thompson, 1994).
When we replace the equal parts into which the objects are divided by the units in
3
Available on YouTube at https://www.youtube.com/watch?v=DnFrOetuUKg
4
We assume that 𝑎 and 𝑏 represent whole numbers, and 𝑏 is non-zero.

26
which their chosen quality is measured, we obtain the notion of fraction of a
quantity. This notion, defined as follows, constitutes the foundation of our proposed
theory:
Quantity 𝐴 is the fraction
𝑎
𝑏
of quantity 𝐵 if there is a common unit 𝑢 such that
𝐴 measures 𝑎 units 𝑢 and 𝐵 measures 𝑏 units 𝑢.
The plan of the course was as follows. We would start by questioning the common
textbook fraction exercises such as “How much pizza has been eaten?” accompanied
by drawings of circular pizzas cut into slices. We would ask, how is the amount of the
pizza measured? Is it the weight? The surface area? The number of slices? The
number of calories? We would bring a real pizza and kitchen scales to the classroom.
We would weigh the whole pizza and each slice in grams or ounces, and we would
find that each slice is a different fraction of the weight of the whole pizza, because a
real pizza is usually not a perfect circle and the slicing is also quite rough. For
example, if the whole pizza weighs 548 g, and has been cut into 8 slices, one slice
may weigh 72 g, and another – 68 g. Then the weight of the former slice is an “ugly”
fraction – 72/548 – of the weight of the whole pizza (or 18/137, if we use 4 g as the
common unit of measure). For 72 g to be the ideal one-eighth of the weight of the
whole pizza, the latter would have to be 576 g, not 548 g. Based on this and other
similar experiences (e.g., comparing the volumes of an apple and of its parts),
students would already have some idea of the notion of fraction of a quantity as we
intended it.
Our notion of fraction of quantity is even more primitive than the common material
conception described above (part-whole relationship) because there is no
intermediary abstraction implicitly made, as it is the case when one looks at a
diagram or a picture of a chocolate chip cookie cut into parts, and assumes the parts
to be “equal” in some implicit sense. That children do not spontaneously make these
idealizations is evident in how they would argue that a cookie cut “in half” is not
fairly shared if one of the sharers ends up with more chocolate chips (thus the object
at stake has not only area, but also number of chocolate chips, as measurable
qualities of interest).
Fraction of quantity is thus closer to the concrete, but it is also amenable to doing
authentic mathematics, we argue, via two mechanisms: quantification, which is,
essentially, looking at objects for their quantifiable aspects, and theorization, or
stabilizing meanings via definitions and reasoning within a system, based on agreed
upon rules. Both capture ways of inquiry into reality that mathematics, as a human
practice, has always been advancing.
The definition of fraction of a quantity is thus used to derive – by means of
theoretical thinking (Sierpinska, Bobos & Pruncut, 2011) and quantitative reasoning
(Thompson, 1993; 1994) – answers to problems about relations between quantities.
Generalizations of these solutions serve, later on, to motivate the rules of operations
on abstract fractions – the object of the aforementioned formal conception of

27
fraction – by showing reasons why these operations have been defined the way they
were.
The process of transition from fractions of quantities to abstract fractions is
supported by problems of comparison of fractions of quantities, similar to some of
the problems that are associated with the “ratio sub-construct” of fractions in the
“five sub-constructs” tradition5
. For example, given that there is so much water in
one bottle (e.g., 300 ml of water in a 750 mL bottle) and so much water in another
(e.g., 105 mL in a quarter-liter bottle), which bottle is fuller? Or, which mix of water
and syrup is sweeter? Or, which sea water has higher salinity? Or – a problem about
environmental consciousness such as this one:
‘If suburb A counts 2574 households and
2
3
of them compost their organic waste,
while suburb B counts 3878 households and
1
2
of them compost their organic
waste, which one is more environmentally conscious?’
In this context, fractions acquire the meaning of measures of qualities such as
fullness, sweetness, salinity, etc., or, in general, measures of how-much-ness of one
quality relative to another quality of the same kind. They can be thought of as single
numbers on a par with whole numbers, which, on their part, can be conceived of as
measures of qualities such as two-ness, three-ness, etc., of sets of objects. Then, one
can think of developing a theory of those numbers independently of quantities they
could be fractions of: defining their equivalence, order, operations, properties of
those operations, etc. A theory of abstract fractions is thus obtained6
.
Thus, what we call measurement approach to fractions for prospective teachers, is
this process of using the theory of fractions of quantities to go from the idea of
fraction of something to a theory of abstract fractions. (Bobos & Sierpinska, 2017)
The approach does not make understanding fractions “easy” for the future teachers. This was not the
objective of the experiment; there is no easy way to understand and learn fractions. The paper describes
the nature of the difficulties the future teachers experienced in re-learning fractions through the
measurement approach.
5
For example, Noelting’s (1978) problem about the “orangeiness” of a juice was classified as requiring the “ratio”
sub-construct in (Charalambous & Pitta-Pantazi, 2007).
6
Mathematics education literature to teaching fractions in elementary school sometimes claims that their
objective is to teach rational numbers or “rational number concepts”. But this is misleading, because the aim is
never to teach rational numbers in the sense that this term is understood in theoretical mathematics, that is, as a
theoretical system constructed algebraically (as the quotient field on the ring of integers) or analytically as a by-
product of more general issues such as conceptualizing rates as numbers rather than as pairs of numbers
(Thompson & Saldanha, 2003, pp. 8-10). This is why we prefer saying “abstract fractions” or “fractions as abstract
numbers” to saying “rational numbers”.

28
Activity 5. Analyze an exercise about fractions.
Here is an exercise from a book for elementary teachers:
Source: (Lamon, 2012, p. 122)
a) In what ways the exercise (i) fosters; (ii) hinders future teachers’ understanding of fractions?
b) Reformulate the exercise in the language of fractions of quantities, making the reference quantity
explicit.
c) Solve the exercise using the following definition: “Quantity 𝐴 is said to be a fraction
𝑎
𝑏
of a quantity 𝐵
if there exists a unit 𝑢 such that 𝐵 measures 𝑏 such units 𝑢 and 𝐴 measures 𝑎 units 𝑢.”
Suggestions for the instructor:
Participants should be given ample time to think about the questions and discuss them in small groups.
After ca. 15-20 mn, the instructor can engage a whole class discussion, inviting groups to share their
thoughts with others. Regarding question (ii), hopefully someone will mention the confusion between
fractions of quantities and abstract fractions (or, more generally, between things and numbers) as a
possible obstacle reinforced by the way the exercise is formulated. If not, the instructor can suggest the
idea in form of a question, or s/he can invite participants think some more about the question and share
their thoughts on the classroom online “forum”.
As regards question (ii), participants’ reformulations will be read out loud and the instructor will choose
some problematic ones and some reasonable ones to be written on the board. Those formulations will
then be discussed by the whole group of participants. The instructor can kindle the discussion using
questions such as, Is the formulation clear? Is it still essentially the same question?
In responding to this last question, it may be useful to make explicit the structure of the original
exercise. This structure can be represented as follows:
A given number 𝑁 of things is a given fraction
𝑎
𝑏
of an unknown number 𝑀 of things.
What number 𝑋 of things is a given fraction
𝑐
𝑑
of the same number 𝑀 of things?
[The numerical values are such that 𝑑 = 𝑏 and 𝑎|𝑁; moreover,
𝑎
𝑏
can be given as a
mixed number.]
In the given exercise, it is not quite clear that the reference quantity is of the number of things kind, and
that 𝑁 = 12; we are given the two fractions:
𝑎
𝑏
= 1
1
3
,
𝑐
𝑑
=
2
3
.

29
Having the structure explicit makes it easy to construct other similar problems and formulate them in
the language of fractions of quantities. For example, one possible reformulation can be:
If 12 baubles (Christmas decorations) constitute 1
1
3
of 1 boxful of such baubles,
how many baubles would constitute
2
3
of the boxful?
Another can be:
Amy has written 12 Christmas postcards. She decorated each postcard with one star
sticker, using up 1
1
3
of a sheet of such stickers. Betty used
2
3
of such a sheet of
stickers. How many star stickers did Betty use?
Regarding question c), participants will certainly come up with all sorts of solutions which will give the
correct result but may be conceptually incomplete or shaky and whose written record may be unclear or
ambiguous. Usually, the names of units in which quantities are measured will be omitted, and the
operation of unit conversion, so fundamental for the concept of fraction7
, will not be noticed and
highlighted in the solution. Some solutions may also be based on the conception of “fraction as a part of
a whole” and not on the definition of fraction of a quantity as requested in this Activity. It would be
good for the participants’ understanding of other people’s difficulties with fractions if they experienced
some difficulties themselves and became aware of them. Forcing themselves to write their reasoning in
a more intelligible way, and to reason based on a definition they are unfamiliar with may provide an
opportunity for such experience.
It is important, however, to let the participants invited to present their solutions on the board to write
their solution to the end without interrupting them mid-way with teacher’s interventions and
corrections. They have to experience and realize the ambiguity and incompleteness of their writing
themselves, from the reactions of their peers and from looking at their solutions again after having
finished writing.
They could be asked to solve the more general exercise as expressed by the structure of the problem
above:
A given number 𝑁 of things is a given fraction
𝑎
𝑏
of an unknown number 𝑀 of things.
What number 𝑋 of things is a given fraction
𝑐
𝑏
of the same number 𝑀 of things?
Using the definition of the fraction of a quantity, the reasoning can go approximately as follows:
7
“The object source of the operation of multiplication is considered to be the practice of change of unit in
measurement (Davydov V. V., 1991) and, since a fraction represents a multiplicative relationship between two
quantities of the same kind, change of unit is also an essential element of the object source of fraction (Davydov &
Tsvetkovich, 1991). If I want to measure a quantity 𝐴 with some unit 𝐵, more often than not neither 𝐵 is contained
in 𝐴 a whole number of times nor 𝐴 is contained in 𝐵 a whole number of times, but there is a remainder. If there
is a smaller unit 𝑢 that fits a whole number 𝑛 of times into 𝐴 and a whole number 𝑚 of times into 𝐵, then the ratio
of 𝐴 to 𝐵 is 𝑛 to 𝑚. In this case, there is a convention to treat this ratio as a new kind of number, called “fraction”,
denote it by the symbol
𝑛
𝑚
and say that 𝐴 measures
𝑛
𝑚
units B.” (Bobos & Sierpinska, 2017)

30
By definition, the expressions:
“𝑁 things is
𝑎
𝑏
of 𝑀 things”, and
“𝑋 things is
𝑐
𝑏
of 𝑀 things”
mean that for some unit 𝑢, 𝑀 things measure 𝑏 units 𝑢, 𝑁 things measure 𝑎 units 𝑢 and 𝑋 things
measure 𝑐 such units.
We note that two units are used: “thing” and 𝑢.
We are given the number 𝑁, and 𝑎. So we can calculate how many things is one unit 𝑢 worth.
We know that 𝑎 units 𝑢 are equivalent to 𝑁 things. Proportionally, 1 unit 𝑢 is equivalent to 𝑎 times less
things:
1 𝑢 = (𝑁 ÷ 𝑎) things.
(We have assumed that 𝑎|𝑁, so 𝑁 ÷ 𝑎 is a natural number and the result makes sense.)
Having this “conversion equation”, we can calculate the number 𝑋.
𝑋 things = 𝑐 units 𝑢 = c × (N ÷ a) things
For = 12 , 𝑐 = 2 , and 𝑎 = 4 (since 1
1
3
=
4
3
), 𝑋 = 2 × (12 ÷ 4) = 6.
Difficulties related to irrational numbers
Lecture 10 combined with Activity 6. (interactive lecture) Incommensurable quantities and
irrational numbers
Suppose I ask, Is the diagonal of a square a fraction of its side? and I get these answers:
A. No, because the diagonal is longer than the side.
B. No, because the diagonal is not a part of the side. Neither is the side of the square a fraction of is
diagonal, for the same reason. They are two separate objects.
What do you think about these answers?
What conceptions of fraction underlie these answers? [Teacher allows participants to express their
views.)
How about this answer:
C. No, because the ratio of the length of the diagonal of a square to the length of its side is an irrational
number.
Suppose I am not quite satisfied with this answer and demand further explanations: Can you prove that
there is no common unit, however small, that would fit a whole number of times into the lengths of

31
both the diagonal and the side, i.e., that they are incommensurable? (Participants propose proofs.
Teacher will not be satisfied with standard textbook proofs of the irrationality of √2 and will demand
making an explicit link between irrationality and incommensurability.)
(All proofs proposed by participants will probably be proofs by contradiction.)
There is a widespread opinion that proofs by contradiction are usually not very convincing to “common
mortals”. We will reflect on possible reasons for that in the next lecture. For now, let’s try to prove the
incommensurability of the diagonal of a square with its side by some direct argument.
Here is an idea.
How does one find a common measure of two segments? Think of Euclid’s Algorithm for finding the GCD
of two natural numbers and adapt it to finding the greatest segment fitting a whole number of times
into two segments. Apply it to the diagonal and the side of a square. Construct the first two remainders
and then notice (and prove) that the consecutive remainders can always be constructed by cutting off
the length of the side of some square from its diagonal. Since the diagonal is longer than the side, the
remainder is never 0. So there is no segment that would fit a whole number of times into both the
diagonal and the side.
The next figure shows the construction of the first two remainders:
The second remainder r2 can be constructed as a cut-off of the length of the side on the diagonal in a
smaller circle, as shown in the figure below:

32
Show that the two thick segments labeled r2 in the figure above are congruent.
Observe yourself trying to do it. Have you experienced any difficulties? Can you describe their nature?
Lecture 11 combined with Activity 7. Why some students find proofs by contradiction
unconvincing? Why some proofs of irrationality of the square root of 2 are less convincing than
others?
Recall the schema of the proof by contradiction of a statement 𝑆.
Assume ¬𝑆 is true.
Use correct reasoning to prove ¬𝑆 ⇒ 𝐶 , where 𝐶 is some contradiction (i.e., an
obviously false statement).
Conclude ¬𝑆 must be false. Therefore 𝑆 must be true.
One reason for such reasoning being unconvincing can be that proofs by contradiction are based on the
Law of Excluded Middle: if a statement is false, its negation must be true. This law is valid only in
mathematics. In real life, there are many shades of truth. “John is not unhappy” does not necessarily
mean the same as “John is happy”. Another is that a proof by contradiction uses the rather non-
intuitive logical property that if something false is derived by correct reasoning from some assumption
𝐴, then this assumption must be false: falsity cannot be derived from truth, or: if the truth value of the
implication 𝑝 ⇒ 𝑞 , where 𝑞 is false, is true, then 𝑝 must be false.
But there may be some specific reasons why a given proof by contradiction is unconvincing. For
example, it may be doing a decent job of proving THAT a certain statement must be true, yet fail to
explain WHY it is true.
If students perceive proofs as an exercise in just another mathematical technique that they have to
learn and will be tested on, and do not have an opportunity to appreciate the explanatory function of at
least some proofs, they may lose motivation to learn to prove or to learn mathematics in general.
People, young and old, usually don’t want to learn what they perceive as meaningless or useless. De
Villiers addresses this source of difficulties in learning to prove in mathematics in his often cited article
on different functions of proof (De Villiers, 1990). The article starts like this:
The problems that pupils have with perceiving a need for proof is well-known to all
high school teachers and is identified without exception in all educational research as
a major problem in the teaching of proof. Who has not yet experienced frustration
when confronted by pupils asking ‘why do we have to prove this?’ (…) [P]upils’
problems with proof should not simply be attributed to slow cognitive development
(e.g. an inability to reason logically), but also that they may not see the function
(meaning, purpose and usefulness) of proof. (De Villiers, 1990, p. 17)
Here is the Activity. Let us look at three proofs of the irrationality of √2, and compare them from the
point of view of how convincing and explanatory they are.

33
For ease of reference in the discussion, the lines of the proofs have been numbered.
Statement: There is no rational number 𝑟 such that 𝑟2
= 2.
Proof 1.
[1] By contradiction, assume there exists a rational number 𝑟 such that 𝑟2
= 2.
[2] Since 𝑟 ∈ ℚ, there exist integers 𝑝, 𝑞, with 𝑞 ≠ 0 , such that 𝑟 =
𝑝
𝑞
.
[3] Without loss of generality, we may assume that 𝑝 > 0, 𝑞 > 0. So 𝑝 > 𝑞 ≥ 1.
[4] Since 𝑟 =
𝑝
𝑞
then 𝑝2
= 2𝑞2
.
[5] So 2|𝑝2
and, since 2 is prime, 2|𝑝.
[6] According to the Fundamental Theorem of Arithmetic, 𝑝 has a unique prime factorization:
𝑝 = 2 𝛼
∙ 3 𝛽
∙ …
[7] We know that 𝛼 > 0 because there is a 2 in the prime factorization of 𝑝.
[8] Then 𝑝2
= 22𝛼
∙ 32𝛽
∙ …
[9] So there is also a 2 among the prime factors of 𝑝2
. In fact, there must be a double number of twos in 𝑝2
:
an even number of twos.
[10] But on the right side of the equality 𝑝2
= 2𝑞2
there is an odd number of twos: if there are any number of
twos in 𝑞, there is a double of them in 𝑞2
, and there is the extra two in front.
[11] This is a contradiction with the Fundamental Theorem of Arithmetic: one single number has an even
number of twos in one prime factorization and an odd number of twos in another. This violates the
theorem’s claim that the prime factorization of a natural number is unique.
Proof 2.
[1] By contradiction, assume there exists a rational number 𝑟 such that 𝑟2
= 2.
[2] Since 𝑟 ∈ ℚ, there exist integers 𝑝, 𝑞, 𝑞 ≠ 0 such that 𝑟 =
𝑝
𝑞
. Without loss of generality we can assume
that 𝑝 and 𝑞 are relatively prime.
[3] Since 𝑟 =
𝑝
𝑞
then 𝑝2
= 2𝑞2
.
[4] So 2|𝑝2
and, since 2 is prime, 2|𝑝.
[5] Therefore, there exists 𝑘 ∈ ℤ such that 𝑝 = 2𝑘.
[6] Then 4𝑘2
= 2𝑞2
, which implies 2𝑘2
= 𝑞2
; so 2|𝑞2
.
[7] Again, since 2 is prime, 2|𝑞.
[8] So 2 is a factor of both 𝑝 and 𝑞, a contradiction with the assumption that they are relatively prime.
Proof 3.
This is Dedekind’s proof of a more general statement:

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