Learning to Notice Student Thinking About the Equal Sign: K-8 Preservice Teachers’ Experiences in a Teacher Preparation Program

Abstract

What is teacher noticing? Is teacher noticing different than simply good teaching? Is teacher noticing only an in-the-moment occurrence within a whole class setting or is it also akin to a one-on-one clinical interview? Is the framing of this practice as teacher noticing affording us an opportunity to reexamine, or more deeply examine, a crucial aspect of teaching? These questions continue to be discussed and passionately debated among our research team and colleagues. When one colleague was discussing professional noticing within mathematics, another was interpreting it as professional vision, while still another was attempting to envision it beyond a mathematics content perspective. These discussions caused us pause and we had to ask ourselves, "How IS all of this different? Or, is it?" So, "is it noticing, or is it….?" In Sherin, Jacobs, and Philipp's (2011) edited volume, several authors described the foundations of noticing. The roots are varied and the interpretations of the construct are many. Foundational research in professional vision (Goodwin, 1994) and the discipline of noticing (Mason, 2002) set the stage for additional research in this field. Further frameworks of noticing have surfaced, such as noticing (van Es, 2011; Sherin & Star, 2011), teacher noticing (Sherin, Jacobs, & Philipp, 2011), and professional noticing of children's mathematical thinking (Jacobs, Lamb, & Philipp, 2010). As a result, one goal of this monograph is to seek clarification of the construct and its related branches. Our monograph explores recent developments in noticing and responds, in part, to the challenges Alan Schoenfeld put forth in the final commentary of the aforementioned volume by Sherin et al. In his commentary, Schoenfeld (2011) left us with multiple questions to pursue with the goal of applying our researched knowledge to the development of effective teachers. Sherin et al.'s and Schoenfeld's questions and our own successes and vii

Pre-service Teachers’ Noticing of 7th Grade Students’ Errors and Misconceptions about the Subject of Equations

“…Cebir öğrenme alanı, matematik eğitiminde önemli bir yere sahiptir. Literatürde bu öğrenme alanına ilişkin çok sayıda çalışma yer almaktadır (Dede, 2004;Masal ve Sert-Çelik, 2018;van den Kieboom, Magiera ve Moyer, J., 2017;Yaman, Toluk ve Olkun, 2003). Bu çalışmalar, öğrencilerin denklem kurma ve çözme, cebirsel ifadelerin kullanımı ve problem çözme gibi pek çok konuda güçlükler yaşadığını göstermektedir.…”

“…Çalışmaların önemli bir bölümü de, öğrencilerin değişken ve eşitlik kavramlarında yaşadıkları güçlükler ve kavram yanılgıları üzerinedir. (Dede, 2004;Mac Gregor ve Stacey, 1997;Masal ve Sert-Çelik, 2018;van den Kieboom, Magiera ve Moyer, 2017;Knuth, Alibali, McNeil, Weinberg ve Stephens, 2005).…”

“…Literatürde, eşitlik kavramı hakkında öğrencilerin yaklaşımları incelendiğinde, yaklaşımların ilişkisel ve işlemsel düşünce olmak üzere iki ana kategoride yoğunlaştığı görülmektedir. (Knuth vd., 2005;van den Kieboom, Magiera, & Moyer, J., 2017). İşlemsel düşünen öğrenciler, eşitliğin sadece sonuç üretmesi gerektiğine inanırlarken, ilişkisel düşünen öğrenciler ise eşittir işaretinin, iki taraftaki aynı nicelikleri ayıran bir sembol olduğunu düşünürler (van den Kieboom vd., 2017).…”

“…Literatürde, öğrenci çalışmalarına ve öğrencinin düşünce sürecine ilişkin öğretmen ve öğretmen adayı farkındalığını inceleyen çok sayıda çalışmaya rastlanmaktadır. (Schack, Fisher, Thimas, Eisenhardt, Tassell ve Yoder, 2013;Bartell, Webel, Bowen ve Dyson, 2013;van den Kieboom, Magiera ve Moyer, 2017;Callejo ve Zapatera, 2017;Fernández, Llinares ve Valls, 2012, Fernández, Llinares ve Valls, 2012. Örneğin, Bartell, Webel, Bowen ve Dyson (2013) öğretmen adaylarının, çocukların kavramsal anlama kanıtlarını tanıma yeteneklerini ve öğretmen adaylarının alan bilgilerinin öğrencinin matematiksel anlamalarını teşhis etmedeki rolünü incelemiştir.…”

The evolution of a child study assignment: a potential approach to developing elementary mathematics teachers’ professional noticing

“…It proposes that "effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning" (p. 10). Indeed, a growing body of empirical studies have demonstrated that teacher noticing and using students' mathematical thinking helps improve the quality of teaching and promote students' conceptual understanding and achievement (Carpenter, Fennema, Franke, Levi, & Empson, 2000;Crespo, 2000;Jacobs, Lamb, & Philipp, 2010;Sleep & Boerst, 2010;Swan, 2001;van den Kieboom, Magiera, & Moyer, 2017;Wells, 2017). Yet, noticing is not a simple task (Choy, 2016;van Es & Sherin, 2008).…”