Session title
• Science and Mathematics Education
Lecture title
• Mathematics Teacher Candidates’ Approaches to Mathematical Proof
Name of presenter
• Mehtap TAŞTEPE
• Sinop University, Turkey
Date of presentation
• Tuesday, July 2
Time of presentation
• 10.00-11.30
Room
• D205
1. Introduction
• Doing proof has an important role in mathematics education. Although
mathematical proof is so important for mathematics education, it is pointed
out that undergraduate students have difficulties in proving, and they find it
meaningless
(exp. Moore, 1994; Harel&Sowder, 1998; Dreyfus, 1999; Almeida, 2000; Jones, 2000; Weber, 2001;
Recio&Godino, 2001; Weber, 2004b; Stylianides, Stylianides ve Philippou, 2007; akt. Sarı, Altun, Aşkar; 2007; Bastürk, 2010).
• In international literature, a large number of research has been conducted in
order to uncover students’, teacher candidates’ and teachers' views on proof
and processes of proving
(exp. Jones, 1997; Harel&Sowder, 1998; Almeida, 2000; Jones, 2001; Recio&Godino, 2001; Raman,
2001; Weber, 2001; Knuth, 2002; Raman, 2002; Almeida, 2003; Raman, 2003; Solomon,2006; Stylianides, Stylianides and Philippou, 2007).
• We think that mathematic teacher candidates’ view about proof is important
for their students.
• In this study, Balacheff taxonomy was based on.
Balacheff 's taxonomy
• Balacheff divided mathematical proof into three
levels as pragmatic proof, intellectual proof and
demonstration proof. Most low-level 'pragmatic
proofs' are made by giving examples.
m, n ∈ N be. P: (2 │ n ∧ 5 │ m), q: (2 + 5) │ (n + m) Recommend (p ⇒ q) ≡ 1
whether the search.
Proof: For this reason, when p ≡ q ≡ 0 so that one at least one of each m, n ∈ N,
the number will be enough to find. Accordingly,'''' n = 4 and m = 5, p
proposition is true, but (2 + 5) │ (4 + 5), the incorrect premise. Then (p ⇒ q) ≡
0 is the (Irmak, p. 44, 2008).
Balacheff 's taxonomy
• Mid-level 'intellectual proofs' are based on the
formulation proofs;
Any A, B, and C clusters AU (B ∩ C) = (AUB) U (AUC) show that
Proof: Let x ∈ A U (B ∩ C) ⇔ v x B x ∈ A ∩ C
⇔ x ∈ A v (x ∈ B ∧ x C)
⇔ (x ∈ A v x B) ∧ (x A v × C)
∧⇔x∈AxAUB UC
⇔ x ∈ (A U B) U (A U C)
AU is (A ∩ B) = (AUB) U (AUC) is obtained. (Çallıalp, p. 18th,
1999)
Balacheff 's taxonomy
• and the most advanced level 'demonstration proofs' have to be
organized by a theory or proofs using information that adopted by a
community (Özer and Arıkan, 2002).
N is a positive integer different from infinity to I = {1, 2, 3, ..., n} get. in this instance
Show that U ... UA_n UA_i = A_1UA_2.
Proof: Set family reunion
UA_i = {x │ (i) (i ∈ I} A_i ∧ x ∈
Format is defined, the combination elements (any i) (i ∈ I ∧ x ∈ A_i) suggest to all my objects. A_i sets
so that there is at least one consists of all objects. Provides all the elements of A_i this requirement
because each x ∈ I and x ∈ A_i A_i'' 1'' for the proposition to be true is evident. Similarly, A_2, A_3,
A_4, ..., a_n seen in a similar manner in the propositions to be true. (To put it briefly, k, from 1 to n 'up,
no matter what the integer x ∈ A_kiken
K ∈ I'''' ∧ x ∈ A_k proposition is true). For this reason, A_1, A_2, A_3, A_4, ..., a_n combination of sets
of the entire family, there are cluster members. I index set 1, 2, 3, ..., n from the combination of other
elements not set all the elements family consists only of those elements. As a result, the UA_i =
A_1UA_2 U ... UA_n is obtained (Şahin, p. 149, 2010).
1.1 The Purpose of The Study
• The Purpose of this study, in general, is to determine
the perspectives of teacher candidates on proof.
• To determine their views about requirement for proof
at primary school, high school or universty.
• In particular, it is to determine students’ perspectives
within the scope of Balacheff's taxonomy.
2.Method
• The research design used for the study was a descriptive
survey method.
• a questionnaire consisting of 11 questions made up from a
list of question types and categories question.
• Questions was applied to a group of 32 mathematics
teacher candidates living in İstanbul, who make up of 16
girls and 16 boys of senior students.
• The data were analyzed by statistical methods and content
analysis.
3. Findings
• General Findings
• Firstly students’ interests in mathematics and their achievement levels
were found significant.
• The importance of proof
• 2 (6, 25%): «Must be present in the literature as literary, but an unused item»
• 6 (18,75%): «In some cases they are used to convince involved individuals»
• 6 (18,75%): «for individuals interested in mathematics and mathematics
education, it is an important element»
• 18 (56,25%): «An item which ensures that mathematics is math and a science»
The Importance of proof
The degree of importance
ensures that mathematics is math
an important element
Series1
İn some cases are used
unused item
0
10
20
30
%
40
50
60
3. Findings
• Using of proof
▫ high school level
 2 (6,25%): «the high school level is not necessary»
 21 (65,62%): «sometimes necessary for the high school level»
 8 (25,00%): «necessary»
 1 (3,12%): « certainly necessary»
 «To teach the subject by proving is boring and pointless for students who
do not know the methods of proof»
 «I think that should be used to avoid the discussion of the subject. It is
important to grasp the logic of mathematics.»
 «We are dealing with proof heavily in university. So we must use proof in
high school. Moreover, it is required to understand better .»
Using of Proof/ High School
certainly necessary
Series1, 3.12
necessary
sometimes necessary
not necessary
0
10
20
30
40
%
50
60
70
3. Findings
• The relationship between the doing proof at university and the doing
proof at high school
▫ «actually they are basically same proofs. The proof done at university is a
bit more extensive»
▫ «Definitely they are very different. the proof done at high school is very
simple.»
• The conceptualizing of proof at University
• 3 (% 9,37): «I am memorizing»
• 22 (% 68,75): «I am memorizing a fraction of proofs, I understand a
portion of the proof»
• 7 (% 21,87): «first time I see it, I can understand the logic of the proof
and I can do proof myself.»
The Conceptualizing of Proof at Universty
prove myself
memorize a fraction of proof
memorize
0
10
20
30
40
%
50
60
70
80
3. Findings
• Balacheff Taxonomy
• 3 (% 9,37): pragmatic proof
• 17 (% 53,12): intellectual proof
• 8 (% 25): demonstration proof
«Simple (pragmatic), have limited the meaning (intellectual),
more meaningful (demonstration)»
«It is more understood and mathematical (intellectual), it is
as at primary level but understandable (pragmatic), it consists
of characters more than numbers (demonstration).»
Balacheff Taxonomy
Demonstration Proof
İntellectual Proof
Pragmatic Proof
0
10
20
30
%
40
50
60
4. Conclusion and Discussion
• there are many important results of my study to be discussed. But
time is limited.
• Students thought that proof was important but they thought it was
difficult so it wasn’t necessary at high school. But for this reason
they said that they couldn’ t understand the proof at university. So
it is thought that they couldn’t learn it easily. And for this reason
they memorized the proof.
4. Conclusion and Discussion
• To Bastürk (2010; 2012) , in his studies carried out to primary and secondary
mathematics teacher candidates, he examined the concept about students’ proof and
about how to do proof. Although students were aware of the importance of proof in
mathematics, he revealed that students prepared courses based on mimetic thinking
so they were memorizing proofs at textbooks or personal notes
• In another study with candidate teachers, it is observed that teachers' attitudes don’t
occur exactly about proving and don’t be known fully that proving is the importance
of mathematics and mathematics teaching (Moralı, Uğurel, Türnüklü ve Yeşildere,
2006).
4. Conclusion and Discussion
• There are different outlooks about relationship between the
proof done at universty and the proof done at high school.
They couldn’t remember the proof done at high school
because they memorized proofs. Their excuses were
university entrance exams. According to us, proof can use at
university entrance exams.
• Balacheff’s taxonomy; They have chosen intellectual proof.
We think that type of proofs used by teachers or books is
intellectual proof (you can see, Tastepe, (2012), my master
of science thesis).
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