Concept Mapping Through the Conceptual Levels of Understanding

Presented by

Michael SanchezWeslaco Independent School District

The University of Texas – Pan American

About Me

Mathematics Teacher Algebra I / Math Modeling Algebra II Pre – Calculus

Weslaco, Texas

Why Action Research?

Graduate Student University of Texas – Pan American

Mathematics and Science Teacher Preparation Academy Encourages Action Research in the Classrooms

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.

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.

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.

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

Overview of APOS Theory

Four Levels of Understanding Action Process Object Schema

Action

Concrete processes Counting Grouping

Process

Collection of Actions Formation of Patterns

141)3(4

91)2(4

51)1(4

Object

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

14)( nns

Schema

Combining the Action, Process, and Object levels to create a complete understanding.

A mental framework used to solve similar situations.

bmxxf )(

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

Level 1

Basic Computations. Digital and Analogic Schemata done in isolation.

5,4,3,2,1

Level 2

Digital and Analogic are used together and mapped to each other.

Level 3

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

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.

2)sin(2

1)(

43)(

)()(2

xxf

xxf

khxaxf

How can we expect our students to achieve this?

Concept Mapping as a tool to support the construction of conceptual understanding.

Why Concept Mapping?

Helps organize vocabulary and concepts by making connections between them.

Can create a hierarchy of concepts and make connections between them.

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.

The Shift

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

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.

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.

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.

Questions?

Thank You!

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Volume 9, No. 5, May 2000. Dogan-Dunlap. H., (2007). Reasoning With Metaphors and Construction an Understanding of the Mathematical Function

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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

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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,

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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