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Concept Mapping Through the Conceptual Levels of Understanding

Concept Mapping Through the Conceptual Levels of Understanding. Presented by Michael Sanchez Weslaco Independent School District The University of Texas – Pan American. About Me. Mathematics Teacher Algebra I / Math Modeling Algebra II Pre – Calculus Weslaco, Texas.

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Concept Mapping Through the Conceptual Levels of Understanding

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  1. Concept Mapping Through the Conceptual Levels of Understanding Presented by Michael Sanchez Weslaco Independent School District The University of Texas – Pan American

  2. About Me • Mathematics Teacher • Algebra I / Math Modeling • Algebra II • Pre – Calculus • Weslaco, Texas

  3. Why Action Research? • Graduate Student • University of Texas – Pan American • Mathematics and Science Teacher Preparation Academy • Encourages Action Research in the Classrooms

  4. Research Problem • Algebra II Students enter the course with incomplete understanding of the algebraic topics that were previously taught in previous grades. • Students lack connection between multiple representations. • Students struggle with the vocabulary behind the mathematics. • Students have difficulty understanding the concept of function and function transformations.

  5. Reasons for Problem • Students are not taught algebra concepts as a whole. • Not enough emphasis on the use of vocabulary in instruction. • Instruction does not utilize the use of the conceptual levels of mathematics understanding.

  6. Hypothesis • Using concept maps to support lessons based on the APOS theory and the Psychological Models of Mathematics Understanding will help students learn mathematical concepts at a higher level.

  7. Theoretical Framework • APOS Theory • Dubinsky & McDonald • Constructivist Approach in constructing mathematical understanding • The Psychological Models of Mathematical Understanding • Kalchman, Moss, & Case (2001) • Describes levels of understanding through the use of digital and analogic schemata

  8. Overview of APOS Theory • Four Levels of Understanding • Action • Process • Object • Schema

  9. Action • Concrete processes • Counting • Grouping

  10. Process • Collection of Actions • Formation of Patterns

  11. Object • Using the Actions and Process to create a mathematical model to represent them.

  12. Schema • Combining the Action, Process, and Object levels to create a complete understanding. • A mental framework used to solve similar situations.

  13. Overview of the Psychological Models • Four Levels of Understanding built upon to primary schemata: digital (sequential) and analogic (spatial). • Level 1 • Level 2 • Level 3 • Level 4

  14. Level 1 • Basic Computations. • Digital and Analogic Schemata done in isolation.

  15. Level 2 • Digital and Analogic are used together and mapped to each other.

  16. Level 3 • Students gradually categorize different contexts of application and are able to create to some extent different representations.

  17. Level 4 • Students can explain the process and make differentiations of several “what if” situations. They see the concept as a whole rather than in isolation.

  18. How can we expect our students to achieve this? • Concept Mapping as a tool to support the construction of conceptual understanding.

  19. Why Concept Mapping? • Helps organize vocabulary and concepts by making connections between them. • Can create a hierarchy of concepts and make connections between them.

  20. Implementation • Initial Topic • Functions • Function Transformations • Beginning Phases • Designing Skeletons to Foster Class Discussion • Mapping Representations • Matching Process Tables, Graphs, Symbolic Representations, and Verbal Descriptions.

  21. The Shift • There was a need to adapt implementation. • District Timelines • Student Difficulties • Adapted use of Skeletons

  22. Implementation • Quadratics • Lessons using the theoretical framework • Concept Mapping used to support understanding of process tables to symbolic representations. • Placed Emphasis on linking actions and process to object levels.

  23. Results • Students enjoyed activities. • Liked the idea of starting at a concrete level and building upwards. • In initial phase students had difficulty creating their own maps. • Fostered whole group discussion through the use of the skeleton concept maps.

  24. Reflections • Adapting to Change • Planning according to District Timelines and Benchmarks. • Learning Experience • Data Collection • Potential • Changed the way I create my lessons. • Anticipating Questions. • The Future! • Continuing Action Research past Graduate work.

  25. Questions?

  26. Thank You!

  27. References • Baralos, G. (n. d.). Concept Mapping as Evaluation Tool in Mathematics. Centre for Educational Research, Greece. • Baroody, J. & Bartels, B. (2000) Using Concept Maps to Link Mathematical Ideas. Mathematics Teaching in Middle School. Volume 9, No. 5, May 2000. • Dogan-Dunlap. H., (2007). Reasoning With Metaphors and Construction an Understanding of the Mathematical Function Concept. In Woo, J. H., Lew, H. C., Park, L.S. & Seo, D. Y. (Eds). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 209-216 • Dubinsky, E. & McDonald, M (n. d.) APOS: A Constructivist Theory of Learning in Undergraduate Education Research. • Gagatsis, A., & Shiakalli, M. (2004). Ability to Translate from One Representation of the Concept of Function to Another and Mathematical Problem Solving. Educational Psychology, 24(5), 645-657. doi:10.1080/0144341042000262953. • Jones, M. (2006). Demystifying Functions: The Historical and Pedagogical Difficulties of the Concept of the Function. The Rose Hulman Undergraduate Mathematics Journal: Volume 7, Issue 2, 2006. Retrieved from http://www.rose-hulman.edu/mathjournal/archives/2006/vol7-n2/paper5/v7n2-5pd.pdf • Llinares, S., (2000) Secondary School Mathematics Teacher’s Professional Knowledge: a case from the teaching of the concept of function. Teachers and Teaching: theory and practice, Vol. 6, No. 1, p. 41-62. • Kalchman, M., Moss, J., Case, R. (2001) Psychological Models for the Development of Mathematical Understanding: Rational Numbers and Functions. In Cognition and Instruction: Twenty-Five Years of progress. Carver, M. & Klahr, D. • Mwakapenda, W. (2003). Concept Mapping and Context in Mathematics Education. The Mathematical Education into the 21st Century Project. • O’Conner, J & Robertson (2005) The Function Concept. The University of St. Andrews, Scotland Website. Retrieved from • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Functions.html • Schmittau, J. (2004). Uses of Concept Mapping in Teacher Education in Mathematics. Concept Maps: Theory, Methodology, Technology. • Schoenfeld, A. H.(1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York; MacMillan. • Zazkis, R., Liljedahl, P., & Gadowsky, K. (2003). Conceptions of Function Translation: Obstacles, intuitions, and rerouting, Journal of Mathematical Behavior, 22(4), 435, doi: 10.1016/j.jmathb.2003.09.003. Retrieved from www.sfu.ca/~zazkis/publications/FunctionTranslation.pdf

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