{"id":8179,"date":"2018-10-19T15:26:30","date_gmt":"2018-10-19T15:26:30","guid":{"rendered":"https:\/\/citejournal.org\/\/\/"},"modified":"2019-03-12T16:25:30","modified_gmt":"2019-03-12T16:25:30","slug":"novice-secondary-mathematics-teachers-evaluation-of-mathematical-cognitive-technological-tools","status":"publish","type":"post","link":"https:\/\/citejournal.org\/volume-18\/issue-4-18\/mathematics\/novice-secondary-mathematics-teachers-evaluation-of-mathematical-cognitive-technological-tools","title":{"rendered":"Novice Secondary Mathematics Teachers\u2019 Evaluation of Mathematical Cognitive Technological Tools"},"content":{"rendered":"
The National Council of Teachers of Mathematics (NCTM, 2014) and the Association of Mathematics Teacher Educators (AMTE, 2015) stated that technology tools are essential and indispensable resources for teaching and learning mathematics in the 21st<\/sup> century. Mathematics teachers who use technology effectively \u201cmaximize the potential of technology to develop students\u2019 understanding, stimulate their interest, and increase their proficiency in mathematics\u201d (NCTM, 2008, p. 1).<\/p>\n Even though many technology tools are available to teachers, however, numerous teachers have difficulty taking advantage of these tools and effectively incorporating them into their curricula (Koehler, Mishra, Kereluik, Shin, & Graham, 2014; Niess, 2011). In their recommendations on the uses of educational tools for professional development of mathematics teachers, the authors of the Mathematical Education of Teachers II<\/em> stated, \u201cTeachers need to develop the ability to critically evaluate the affordances and limitations of a given tool, both for their own learning and to support the learning of their students\u201d (Conference Board of the Mathematical Sciences [CBMS], 2012, p. 34).<\/p>\n Teacher\u2019s decisions on which tools to use and how to use the tools in a particular learning situation can provide students opportunities or constraints to develop mathematical understanding (CBMS, 2012). Therefore, investigating the ways teachers make these decisions is important. Yet, relatively few studies have been conducted on the ways mathematics teachers evaluate technology tools specifically designed to teach a particular mathematics concept (e.g., Battey, Kafai, & Franke, 2005; Johnston & Suh, 2009; Smith, Shin, & Kim, 2017a).<\/p>\n In fact, most of the research on mathematics teachers\u2019 evaluation of technology has focused on the teachers\u2019 criteria to evaluate technology (e.g., Smith et al., 2017a) or the type of technology they used in their lesson plans (e.g., Johnston & Suh, 2009). However, these researchers did not examine the processes used by teachers to evaluate the technology. The purpose of this study was to examine the ways in which novice secondary mathematics teachers evaluated four online technological tools, each of which included an online dynamic geometry applet designed to have students explore the same geometric concept.<\/p>\n As technology has become more prevalent in today\u2019s classroom, teachers are using it in a variety of ways, including having students do research on the internet; assessing students\u2019 skills, knowledge, and understanding using digital devices; having students collaborate and interact with each other and the content using interactive digital displays; communicating with each other and the teacher both in and out of the classroom; and developing understanding of content to name a few.<\/p>\n This paper focuses on technology that teachers and students can use to develop students\u2019 understanding of mathematics, specifically mathematical cognitive technologies (MCTs). MCTs are digital technological tools \u201cthat [help] transcend the limitations of the mind (e.g., attention to goals, short-term memory span) in thinking, learning, and problem-solving activities\u201d (Pea, 1987, p. 91).<\/p>\n Popular MCTs include dynamic geometry environments (e.g., GeoGebra), graphing utilities (e.g., Desmos), dynamic data analysis and statistical environments (e.g., Fathom), and many online applications designed for teaching and learning of particular mathematics concepts. MCTs have the potential to help students \u201cbecome more fluent in performing routine mathematical tasks that could be laborious and counterproductive to mathematical thinking\u201d (Pea, 1987, p. 106). Rather than being bogged down by routine tasks and computations, students focus on problem solving and developing mathematical thinking skills.<\/p>\n Moreover, MCTs are environments that afford students the opportunity to recognize patterns and discover mathematical properties by developing and testing conjectures, exploring various mathematical characteristics, and discovering theorems. All MCTs are not the same, each having their own affordances, limitations, and features that influence students\u2019 learning and understanding of mathematics. Thus, teachers\u2019 evaluation and selection of which MCT(s) to use is not trivial.<\/p>\n Previous research on the ways teachers evaluate MCTs has primarily focused on the criteria created and analysis performed by prospective elementary teachers (e.g., Battey et al., 2005; Johnston & Suh, 2009). In both studies, the researchers found, in general, that prospective elementary teachers seem to select and use technology based on student engagement, surface features of the software, and motivation, rather than on student thinking and accurately representing the mathematics. Evaluation, in general, is often defined as the systematic assessment of the merits and value of objects (e.g., Scriven, 1991). The process of evaluating any object, behavior, or idea requires a person to consider the object, or some aspect of it, and make a judgment based on a set of criteria, which could be either implicit or explicit. In general, some goal guides the evaluation, and the criteria usually stems from meeting this goal. To evaluate MCTs, teachers consider the technological tool and its features and interpret them according to a set of criteria in order to determine whether to use this particular tool to teach a particular concept. Their decision on which technological tool to use is based on their evaluation.<\/p>\n Many mathematics education researchers (e.g., Jacobs, Lamb & Philipp, 2010; Star & Strickland, 2008; van Es & Sherin, 2002) have examined teachers\u2019 decisions and their decision-making processes to gain a better understanding of what teachers attend to in a classroom situation and how they make sense of what they attend to using their knowledge of teaching, learning, and the context of the situation. As students engage in a particular classroom activity, what teachers attend to is important in order to understand the complicated mechanisms of teachers\u2019 decision making, which influences student learning (Star & Strickland, 2008).<\/p>\n Other researchers (e.g., Jacobs et al., 2010) have suggested that the way teachers respond to what they observe seems to be just as important as what<\/em> they attend to and how<\/em> they interpret it. Jacobs et al. argued teachers\u2019 decisions about what to do next are an integrated teacher move almost simultaneously taking place with their attentions to what has previously occurred and their interpretations of these occurrences. Thus, identifying and examining teachers\u2019 three actions \u2013\u2013 attending, interpreting, and deciding how to respond \u2013\u2013 in a classroom activity could provide insight into teachers\u2019 decision making processes.<\/p>\n These three actions comprise Jacobs et al.\u2019s professional noticing of children\u2019s mathematical thinking framework. Researchers in mathematics education have extended or adapted Jacobs et al.\u2019s (2010) noticing framework to connect to other areas of mathematics education, including how teachers examine children\u2019s participation in classrooms (Wager, 2014), and teachers\u2019 noticing of curricular materials (Males, Earnest, Dietiker, & Amador, 2015).<\/p>\n Even though much of the mathematics education research on teacher noticing is situated in the context of the classroom or artifacts of the classroom, and teachers\u2019 evaluation of technology is done outside the classroom, the processes teachers use to evaluate technology are similar to noticing. When evaluating and noticing, teachers perceive and attend to some specific object or behavior in the specific context, interpret the object or behavior, and possibly respond to their interpretation of the object or behavior if warranted.<\/p>\n Two distinct differences appear between evaluation and teacher noticing, however. First, evaluation uses a set of criteria to assess an object or situation, whereas noticing seems to examine the teachers\u2019 attentions free from any sort of guidelines or criteria. When teachers notice, they focus on what seems interesting about the situation.<\/p>\n Second, teacher noticing does not seem to include any sort of judgment, whereas the point of evaluation is to determine the merits and value of the object. Yet, Sherin, Russ, and Colestock (2011) contended that the way a teacher perceives a classroom situation is influenced by the teacher\u2019s expectations, knowledge, and understanding of the situation. When teachers attend to and interpret a student\u2019s mathematical thinking, they are focusing on what the student knows and understands, which will be based on the teacher\u2019s understanding of the mathematical content and how students think about the content. Thus, teachers make comparisons between what they expect the students to know and what the students actually know.<\/p>\n One could consider these expectations a form of criteria. When noticing, the teacher may not be making a merit-based judgment, but the teacher is judging the students\u2019 knowledge and understanding. In some ways, evaluation could be viewed as a specific type of noticing. Thus, using a noticing lens to examine teachers\u2019 evaluation of MCTs seems reasonable.<\/p>\n Based on the works of Jacobs et al. (2010) and Males et al. (2015), we developed the mathematical cognitive technology noticing (MCTN) framework (Smith, Shin & Kim, 2017b) in order to use a noticing lens to examine how mathematics teachers evaluate MCTs. The MCTN framework consists of the three noticing actions (attending<\/em>, interpreting<\/em>, and responding<\/em>) in which teachers could engage when evaluating MCTs and the types of features or activities within each of these actions (see Figure 1). In the following sections, we provide descriptions of each of the activities within each of the three actions of attending, interpreting, and responding.<\/p>\nLiterature Review<\/h2>\n
Teachers\u2019 Evaluation of Mathematical Cognitive Technology<\/h3>\n
\nIn Smith et al.\u2019s (2017a) study of secondary mathematics teachers\u2019 criteria to evaluate MCTs, both prospective and in-service teachers created criteria focused on how well an MCT represented the mathematical concepts, whether the MCT included supportive features to aid in developing the appropriate mathematics concept, how students interacted and engaged with the mathematics concepts when using the MCT, and whether the MCT afforded all students the opportunity to learn. The differences between the criteria used by the secondary and elementary teachers were likely related to the differences between the teachers\u2019 knowledge base, which consisted of their knowledge of mathematics, teaching, and technology.<\/p>\nEvaluation and Noticing<\/h3>\n
Mathematical Cognitive Technology Noticing Framework<\/h3>\n