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Proof Understandings of Preservice Mathematics Teachers

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Abstract

Pre-service teachers' understanding of proof processes were analysed based on questions of instruments and common expressions were determined in a qualitative manner.
PROOF UNDERSTANDINGS OF PRESERVICE MATHEMATICS TEACHERS


Introducon: 
         !"
"    #$$%& '       
()   
 !"#$$%&
*+ +
) *   * +  *   +
 !" "    #$$,& * 
 +  *       
-      *  
   "   !#$$% #$$,& *     
  + +  '     
    "*  .( !/001& 
* *            
 *
23 
Method:            2 
( " 4 2 !#$/0-"$/,&  
*+5
 6    * 78 !1  91 *&  
:   *    ;+ 
<<+=<+ 
5*+!
;+&
'  6 3  *    
     '    
+  *   
 >  ? +  4 
@*<'A!4'&*?+
'<'A!''&* 
(  +
   *        
?@*;<'A!@;'&*
B        *
  5   ?C    DA ?' 
 *DA *    ''  4'   >+
*!-'&
<6
>(  *
+5) *

References
EFGH4!#$/#&FIJJEJ + JKEL@I
#$!/&#/0M#98
.!#$$/&  6<;4>+'" +4NG!&O
+64! /1%M#/#&<+)+ 
.H"*O!/001&" 64) -="'=,#97M#19
"<P"PH ;!#$$%& Q6C'"R "'-
+D' =7#7/M#71
"P"<PH ;!#$$,&Q* +P=/$
/7%M/88
4' >*/& *
#&)  *
5 9&) *
7&** 
 **%&
     +  & !*S
&
'' *     (   +
  !* & : 6 >
 *     *  "
6*
 !  +      !" #$$%
#$$,&
@;'  +*
  
/# >5!*S&
-' <B    9   
*   5   ?C   
DA?'*DA* '4'4'
! &
+/3 
 
Findings: > (     *
 +  5     ) 
*5
:4'6
     4'  *    *
* 5'5
  +
+'+
       3   + 
     +     +
!  &
:'4'6
True/
Paral/
False
Explanaon/Proof
Scheme
Deducon
Direct proof
Proof by trial
Proof by inducon
Reduco ad
absurdum
Counterexample
Contradicon
Reasoning
All possible
situaons
No response
1/0/43 ;  T 
 , /9 8 % # 9 / / $ %
18/6/20 ; * +
T  # /9 /$ 8 # $ / $ / ,
0/3/39 ; T 
 # 8 0 8 # 7 % $ / 0
0/0/39 ; T 
 9 , 1 / 9 % / / $ 1
0/0/42 ; T 
 / / /9 , # 0 9 $ $ 8
+#:4'
''U** 
*+ >*+
  V *   !   &U
+             *
    ?  !&  !W/&A !  &
* *        R   
*!& +
  '        
+ 
  !5&5
  '
         
++)+ * +
!"#$$%U#$$,&( 
!E#$/#U.#$$/U"#$$,&
:@;'6= !%%X&* 
 ?*  AU     *  *6 ? 
 !& !W/&A?A?GA?
A  ?  A <      *
 =     *    
?YA
Conclusion6 =2       + 
   +   ( *    
  Y         
' *
U    S    * *
     *   "   + 
 *) 
     *  

:4:"! & Z/ Z#
 ! & 7!X0& 8!X/9&
!  & /9!9$X& //!#%X&
"  ! & /1!7/X& /,!90X&
<+ * 0!#$X& /$!#9X&
+9:'4'
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Article
Full-text available
This paper investigates prospective elementary and secondary school teachers' understanding of proof in a case where the truth set of an open sentence is broader than the set covered by a valid proof by mathematical induction. This case breaks the boundaries of students' usual experience with proving tasks. The most important finding is that a significant number of students from both groups who recognized correctly the validity of the purported proof thought that it was not possible for the truth set of the open sentence to include any number outside its domain of discourse covered by the proof. The discussion of student difficulties provides insights into the development of instructional practices in teacher preparation programs aiming to uncover these aspects of students' knowledge fragility and address them accordingly. Proof is a defining feature of mathematics and, in current school reform recommendations in various countries, is considered a fundamental aspect of instructional programs in all grade levels. However, to have success in the goal to make proof central to all students' mathematical experiences, prospective teachers need to have solid understanding of this mathematical concept. If teacher preparation programs are to develop effective instructional practices that will help prospective teachers cultivate proof in their classrooms, it is essential that these practices be informed by research that illuminates prospective teachers' understanding of proof. Despite the importance of this kind of research, only few studies have investigated in- service or preservice teachers' knowledge of proof (Knuth, 2002; Martin & Harel, 1989; Movshovitz-Hadar, 1993; Simon & Blume, 1996; Stylianides, Stylianides, & Philippou, 2004). Also, these studies have focused more on the logical components of different proof methods than on other important features of the proving process, such as the relationship among the domain of discourse D and truth set U of an open sentence, and a proof that purports to show that the sentence is true in D. This paper contributes to this research area, focusing on the proof method of mathematical induction. Specifically, we examine what might be some common difficulties that prospective teachers have in dealing with a proof by mathematical induction that is not as encompassing as it could be (D is a proper subset of U). Based on anecdotal evidence that students' normal experience is of being given opportunities to engage in 'universal' proofs (D = U), this study aims to advance the field's understanding of possible issues of knowledge fragility by exposing prospective teachers to a case that falls outside the boundaries of what appears to constitute 'standard practice' for them.
Article
Full-text available
There is a growing effort to make proof central to all students’ mathematical experiences across all grades. Success in this goal depends highly on teachers’ knowledge of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating preservice elementary and secondary school mathematics teachers’ knowledge of proof by mathematical induction. This research can inform the knowledge about preservice teachers that mathematics teacher educators need in order to effectively teach proof to preservice teachers. Our analysis is based on written responses of 95 participants to specially developed tasks and on semi-structured interviews with 11 of them. The findings show that preservice teachers from both groups have difficulties that center around: (1) the essence of the base step of the induction method; (2) the meaning associated with the inductive step in proving the implication P(k) ⇒ P(k+1) for an arbitrary k in the domain of discourse of P(n); and (3) the possibility of the truth set of a sentence in a statement proved by mathematical induction to include values outside its domain of discourse. The difficulties about the base and inductive steps are more salient among preservice elementary than secondary school teachers, but the difficulties about whether proofs by induction should be as encompassing as they could be are equally important for both groups. Implications for mathematics teacher education and future research are discussed in light of these findings.
Öğretmen adaylarının matematiksel tümevarım yoluyla ispat becerileri matematiksel ispat hakkındaki görüşleri
  • G Güler
  • E Özdemir
  • R Dikici
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