Challenging Preservice Teachers' Mathematical Understanding: The Case of Division by Zero

Abstract

84Today we showed our elementary preservice teachers a video where an interviewer was asking young students what the answer to 5 ÷ 0 was. The students in the video consistently responded that the answer was 0, and when asked how they knew this, many responded: "my teacher taught me that." We asked our preservice teachers to imagine that these were students in their 5th grade class, and to discuss in their groups how they would respond to and challenge such students' thinking. The response was a complete silence for what seemed to be an eternity, until one preservice teacher sheepishly asked: "Do you mean to say that the answer is not 0?" (Instructor journal, Fall '93). Our interest in preservice teachers' understanding of division by 0 began over a decade ago while coteaching a mathematics methods course for elementary teacher candidates. The episode in the quote above relates what happened the day we came to class with the question of division by 0 as a context for exploring the challenges of listening and responding to students' mathematical ideas. We had planned to engage the class in thinking about how they might respond to children's mathematical ideas in ways that respect but still challenge and extend their reasoning.Preservice elementary school teachers' fragmented understanding of mathematics is widely documented in the research literature. Their understanding of division by 0 is no exception. This article reports on two teacher education tasks and experiences designed to challenge and extend preservice teachers' understanding of division by 0. These tasks asked preservice teachers to investigate division by 0 in the context of responding to students' erroneous mathematical ideas and were respectively structured so that the question was investigated through discussion with peers and through independent investigation. Results revealed that preservice teachers gained new mathematical (what the answer is and why it is so) and pedagogical (how they might explain it to students) insights through both experiences. However, the quality of these insights were related to the participants' disposition to justify their thinking and (or) to investigate mathematics they did not understand. The study's results highlight the value of using teacher learning tasks that situate mathematical inquiry in teaching practice but also highlight the challenge for teacher educators to design experiences that help preservice teachers see the importance of, and develop the tools and inclination for, mathematical inquiry that is needed for teaching mathematics with understanding.As the journal excerpt suggests, this particular class did not turn out at all as we had expected.We had imagined that the preservice teachers would be challenged to think of ways of explaining to young students why the answer to dividing by 0 is not 0 using mathematical arguments rather than appeal to authority. We thought they could find it difficult to explain why division by 0 might be "undefined," "impossible," or "not allowed" without using language or ...

Leveraging variation of historical number systems to build understanding of the base-ten place-value system

“…PTs struggle to understand the content of elementary school (Ball, 1988;Ma, 1999;Simon, 1993;Simon & Blume, 1992;Zazkis & Campbell, 1996) including whole number and operation (Chapman, 2007;Crespo & Nicol, 2006;Kaasila, Pehkonen, & Hellinen, 2010;Thanheiser, 2009Thanheiser, , 2010. Many PTs can perform the algorithms for addition, subtraction, multiplication, and division of whole numbers (Harkness & Thomas, 2008;Khoury & Zazkis, 1994;Menon, 2009;Zazkis & Khoury, 1993) but struggle when asked to explain the mathematics underlying the algorithms (Ball, 1988;Ma, 1999;Simon, 1993;Thanheiser, 2009Thanheiser, , 2010.…”

Choices of Pre-service Elementary Mathematics Teachers’ Instructional Explanations: Division by Zero

“…Sıfıra bölme konusuna yönelik yapılan çalışmalarda (Ball, 1990;Cankoy, 2010;Crespo & Nicol, 2006;Quinn ve ark., 2008) sıklıkla öğretmen ve öğretmen adaylarının bilgileri, anlamaları ve kavram yanılgıları ortaya konulmuştur. Ayrıca öğrencilerin sıfıra bölme konusundaki öğretimsel açıklamalara ilişkin tercihlerini belirleyen bazı çalışmalara da rastlanmaktadır.…”

“…Yapılan birçok çalışmada öğretmen adaylarının üniversite öncesinden ve üniversite derslerinden getirdikleri matematiksel anlamaların ilköğretim düzeyinde öğretim yapabilmeleri için yetersiz olduğunu göstermektedir (Ball, 1990;Ma, 1999;Toluk Uçar, 2011). Ayrıca sıfıra bölme konusuna yönelik yapılan çalışmalarda (Ball, 1990;Crespo & Nicol, 2006;Çelik ve Akşan, 2013;Even & Tirosh, 1995;Tsamir, Sheffer & Tirosh, 2000;Wheeler & Feghali, 1983) ise öğretmen adaylarının açıklamalarının yetersiz ve daha çok kural temelli olduğu ifade edilmektedir. Bu durum öğretimini yapacağı konuda yeterli alan bilgisine sahip olmayan öğretmen adaylarını daha çok kendilerinin anlayacağı ve geçmiş deneyimlerinde karşılaştıkları açıklamaları tercih etmelerine yönlendirmiş olabilir.…”

“…The prior studies (e.g. Ball, 1990;Crespo & Nicol, 2006;Wheeler & Feghali, 1983) have indicated that students, preservice teachers and teachers all have misconceptions about division by zero. Division by zero is one of the most notable topics in studies intended to define teachers' and pre-service teachers' understanding and misconceptions about zero.…”

Matematik Öğretmenlerinin Sonsuzluk, Belirsizlik ve Tanımsızlık Kavramlarına ilişkin Öğretimsel Açıklamaları

“…İlgili alanyazın incelendiğinde, bu kavramlar üzerine yapılan çalışmaların genellikle öğrenciler (Aztekin, 2008;Ervynck, 1994;Reys & Grouws, 1975;Singer & Voica, 2007) ve öğretmen adayları (Aztekin, Arikan, & Sriraman, 2010;Crespo & Cynthia, 2006;Çelik & Akşan, 2013) ile yürütüldüğü dikkat çekmektedir. Öğretmenlerin bu kavramlara ilişkin anlamalarını inceleyen sınırlı sayıda çalışmaya (Cankoy, 2010) rastlandığı görülmektedir.…”