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.

21. 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.

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

25. 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.

28. 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.

29. 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.

30. Questions?

31. Thank You!

32. 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.pdfhttp://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 www.sfu.ca/~zazkis/publications/FunctionTranslation.pdf