2013 Conference Proceedings - University of Nevada, Las Vegas

they believe the operation of averaging two numbers possesses the group properties of closure,associativity, identity, and inverses. Mevarech concludes that students’ misconceptions cannot becorrected simply by repeatedly demonstrating the correct procedure in a lecture and discussionsetting. Rather, students must have the opportunity to receive feedback and engage in correctiveactivities.Li & Li (2008) note a shift in focus for mathematics education misconception studies from afocus on deficiencies in reasoning (Brown & Burton, 1978) to a focus on the learning processitself. They suggest that advances in science education misconception research have paved theway for mathematics education researchers to better understand why student misconceptions areso resistant to change. Specifically, studies suggest that students’ initial mathematicalunderstandings often exhibit process-like thinking (e.g., conceptual understanding of numberbegins with counting). As students’ mathematical learning progresses, their understanding beginsto exhibit object-like thinking (e.g., a more abstract notion of number as a mathematical object).Li & Li challenge researchers to develop theories of mathematical misconceptions that take thisshift in understanding into account. Perhaps misconceptions formed during process-like thinkingpersist into students’ object-like conceptualizations of mathematics.Even as we progress in our understanding of what underlies student misconceptions ofmathematics, just knowing about misconceptions is not enough. In an effort to answer Shulman’s(1986) charge to develop a coherent framework of the knowledge necessary for effectiveteaching, Ball, Thames, & Phelps (2008) propose the domains of mathematical knowledge forteaching (MKT) (Figure 1). According to the primary MKT domains, successful teachers mustpossess different types of content knowledge. First, teachers must know the content they areteaching to students in a way that most people knowledgeable in the content area do. Thisknowledge is referred to as common content knowledge (CCK). In addition to CCK, teachersmust also have knowledge of the content that enables them to understand the multiple waysstudents interact with that content as they are learning. This type of knowledge is specializedcontent knowledge (SCK) because it consists of unique knowledge of the content that teachersmust possess but a typical person knowledgeable of the content area would likely not.The MKT framework divides content knowledge closely related to the teaching and learningprocess (pedagogical content knowledge) into two primary domains: knowledge of content andstudents (KCS) and knowledge of content and teaching (KCT). KCS pertains to the ways thatProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 77