EXPLORING PRESERVICE MIDDLE AND HIGH SCHOOL MATHEMATICS TEACHERS’ 
           UNDERSTANDING OF DIRECTLY AND INVERSELY PROPORTIONAL 
                       RELATIONSHIPS 
                          by 
                     MUHAMMET ARICAN 
                  (Under the Direction of Andrew G. Izsák) 
                        ABSTRACT 
          This study used hands-on and missing-value word problems to examine preservice 
       middle and high school teachers’ knowledge resources when inferring directly and inversely 
       proportional relationships between quantities. Additionally, the study examined preservice 
       teachers’ solution strategies and their difficulties when solving single and multiple proportion 
       problems. An explanatory case study with multiple cases was used to make comparisons within 
       and across cases. This study used the knowledge-in-pieces perspective in reporting preservice 
       teachers’ reasoning about ratios and proportional relationships. It appeared that the extent to 
       which the preservice teachers were successful in coordinating the directly and inversely 
       proportional relationships hinged on their attention to the specific features of the context. 
       Although the preservice teachers accurately inferred the relationships between two covarying 
       quantities as directly proportional or inversely proportional, their inferences were mainly based 
       on attending to qualitative relationships—two quantities are increasing together—and the 
       constancy of the rate of change. Thus, preservice teachers who relied heavily on the qualitative 
       relationships and the constancy of the rate of change often judged nonproportional relationships 
           that consisted of a constant difference or constant sum to be proportional, even after identifying 
           correct nonproportional relationships. The results showed that the contexts of the hands-on 
           problems facilitated the preservice teachers’ coordination of the directly and inversely 
           proportional relationships more than the contexts of the missing-value word problems. 
            
            
           INDEX WORDS:    Preservice teachers, Teacher knowledge, Teacher education, Proportional 
                           reasoning, Ratio, Proportional relationships 
            
                                 
                            
                            
       EXPLORING PRESERVICE MIDDLE AND HIGH SCHOOL MATHEMATICS TEACHERS’ 
           UNDERSTANDING OF DIRECTLY AND INVERSELY PROPORTIONAL 
                       RELATIONSHIPS 
                            
                          by 
                            
                     MUHAMMET ARICAN 
                  BS + MEd., Gazi University, Turkey, 2004 
                  MEd., The University of Georgia, 2010 
                            
                            
                            
                            
        A Dissertation Submitted to the Graduate Faculty of The University of Georgia in Partial 
                 Fulfillment of the Requirements for the Degree 
                            
                    DOCTOR OF PHILOSOPHY 
                            
                      ATHENS, GEORGIA 
                         2015 
                    
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
                         © 2015 
                       Muhammet Arican 
                      All Rights Reserved