32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE ictcm.com | #ICTCM
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Engaging TPACK to Change How Future Teachers View Mathematics
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Introduction • Pedagogical and Content Knowledge; PCK (Shulman, 1986) – Type of teacher knowledge – Intersection of Pedagogical and Content Knowledge • TPACK (Mishra & Koehler, 2006; Niess, 2005) – Development of technology provides tools for teachers that requires its own type of knowledge – Technology not as much in the foreground for Shulman as it is today – Adds Technological Knowledge to PCK
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Introduction
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Methods • Qualitative Research Methodology – Social constructivism (Vygotsky, 1978) • TPACK is socially constructed knowledge • Naturalistic qualitative methods are appropriate for constructivist approach (Lincoln & Guba, 1985) – Study explores formation of knowledge in natural environment, i.e. the classroom – Inductive analysis following Miles & Huberman (1994), start with no codes
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Methods • Subjects – 8 Pre-service secondary mathematics teachers in a mathematics methods course – Participants had varying backgrounds in both mathematics and education courses – Course includes focus on use of technology in the teaching and learning of mathematics
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Methods • Data Collection – Survey • Modified from Schmidt et al, 2009 • Pre- and post-test at beginning and end of semester • Identify existing beliefs and self-assessment – Observation • “Naturalistic” classroom observation • Collected both notes and audio
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Methods – Interview • Interview Protocol – Questions were related to class events – Interpretive questions intended to clarify and member check • Elaborated on data gathered during observation and provided member check and triangulation • Subjects participated in two individual interviews and one group interview over the course of the semester.
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Methods • Analysis – Organization of Data • Synced observational notes with audio • Interview transcripts uploaded into qualitative analysis software – Inductive Approach • Observational notes and interview transcripts scanned for emerging themes in an iterative process • Developed codes from themes • Codes validated and applied to notes and transcripts
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Findings • Inverse Function Section – Beginning of first unit of classroom text • Approximately two weeks into the semester • In class observation recorded over several days – Outline of handout • Preliminary Discussion – Students define inverse function – Reflect on understanding and how it might be taught • Concept Map • Vignette – High School Algebra lesson on inverse functions – Reaction to Vignette
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Findings – Think-Pair-Share; What is an inverse function to you, and how did you come to understand it? • Responses indicated procedural understanding – Switch x and y then solve, reverse input/output on table, reflection over y = x – How they were taught - describe working on problems and checking that it meets criteria – Suggest teaching using t-chart, identifying function, flipping columns, identifying inverse function (tabular to algebraic representation) – suggested using y = x 2 (Lack of PCK) • Content Knowledge (CK) is lacking – Missing deeper connection to undergraduate level mathematics – Lack of conceptual understanding
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Findings – Response to Vignette • Pedagogical discussion is shallow at first – He was a jerk, did not explain, referred to definition, just how it is – Pedagogically recognized an issue, but cannot fix. – Lack of CK affecting how students can respond to situation (interaction) • Guided discussion (affecting PCK) – Ask Mia to elaborate on flip (3 interpretations; reciprocal, tabular, graphical) – Relating Inverses and Identities Structurally ; functions, multiplication, addition, and matrices (Development of CK) – Mia’s flip idea is a form of prior knowledge, inverse fractions – Kat’s idea is prior knowledge relating to the identity: “Why don’t we get 1?”
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Findings • CAS Calculator Exploration (Uses TK) – Define functions e(x) = x and f(x) = x 3 – 5 – e ( f ( x )) , f ( e ( x )) , f ( y ) , solve ( f ( y ) = x,y ) (affecting TCK) – Discussion on use of CAS in classroom (affecting TPK)
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Findings – Concept Maps Discussion • Measuring their own Mathematical Content Knowledge – Explain and “defend” their concept maps; Goal to alleviate misconceptions – Understanding definition of inverse versus consequence of definition – Developing Content Knowledge (CK) • What does it mean to “do” math? – Guided discussion on the nature of mathematics – Technology versus doing it by hand – Procedure versus concept – Growth seems to correspond to change in perception of the nature of mathematics. (PK, CK, and TK)
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Findings – The Nature of Mathematics • What does it mean to “do math”? – Important discussion motivate by subjects’ responses to inverse functions – Not just getting an answer – Key turning point that sets tone for the rest of the semester – Subjects view on nature of mathematics shifts from procedural meaning of “doing math” – Conceptual view that looks at why/how, deeper understanding and connections Tod: “There was a point where it came up, ‘What is mathematics to you then?’ Is it really just the doing stuff or is it understanding the logic behind it? And so, that was another big thing.”
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Findings • Previous research links growth of TPACK in pre-service mathematics teachers to change in perspective from student to teacher – Similar change observed in this study but seems to relate to something deeper – Change in how subjects perceive the nature of mathematics explains this and other observations • Subjects initially demonstrated mathematical content knowledge is procedural – Result of past procedurally based instructions – Intersection of pedagogical content knowledge is lacking – cannot resolve pedagogical situations » Resort to procedural teaching methods » Prompts classroom discussion and exploration » Subjects realize understanding not as deep as they thought
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Findings • Development of knowledge components themselves and at intersections – Recognizing own limited procedural understanding puts subjects into cognitive conflict as they develop mathematical content knowledge – Interaction between pedagogical and mathematical content knowledge as subjects redefine what it means to do mathematics – Change in how subjects see technology used for their own learning and in the classroom as developing mathematical content knowledge interacts with technological knowledge
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Some Conclusions • Important to explicitly address all three knowledge components in the development of TPACK in close temporal proximity • Many pre-service mathematics teachers may only have a procedural understanding of mathematics concepts – Put them in mathematical cognitive conflict – Force reorganization of mathematical understanding • Limited view on the nature of mathematics will affect the development of TPACK and corresponding knowledge components
32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Contact Information Dr. David Grollimund Assistant Professor Colorado State University - Pueblo david.grollimund@csupueblo.edu
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