Developing Understanding in Mathematics “If the creation of the conceptual networks that constitute each individual’s map of reality - including her mathemtical understanding - is the product of constructive and interpretive activity, then it follows that no matter how lucidly and patiently teachers explain to their students, they cannot understand for their students” (Schifter & Fosnost, 1993, p, 9). Thus, first and foremost goal among mathematics educators is that students should “make sense” of mathematics. Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics Constructivism is currently the most widely accepted theory of how children develop understanding. It suggests that children must be active participants in the development of their own understanding It is a theory, but if it is true, it is the way ALL learning takes place - even rote memorization Constructivism rejects the “blank slate” notion of learning Current understanding of the biology of the brain supports this. Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics 2581114172023 7 x 8 = ? Talk about how you “learned” it. Try to come up with as many “good” ways of thinking of the answer as you can. How do your ways relate to the red and blue dot metaphor? Okay, let’s recall the number sequence. How did you do it? Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics Instrumental Understanding Relational Understanding QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Continuum of Understanding Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics: Benefits of Relational Understanding It is Intrinsically Rewarding It Enhances Memory There is Less to Remember It Helps with Learning New Concepts and Procedures It Improves Problem-Solving Abilities It is Self-Generative It improves Attitudes and Beliefs Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics: Types of Mathematical Knowledge Conceptual Knowledge Relationships or logical ideas Procedural Knowledge Knowledge of rules and symbolic representations Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics: Role of Models in Understanding Mathematics Concepts are abstract Models are ways of representing concepts. One Bean is not the concept “1” but represents the concept “1” Although models (such a manipulatives) have become very popular, there are other ways of representing mathematics concepts Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics: Role of Models in Understanding Pictures Written symbols Manipulative models Real-world situations Oral language Lesh, Post, and Behr (1987) Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics: Using Models Models are “thinker” toys Help children develop new concepts Help children make connections between concepts and symbols “write an equation to tell what you just did” “how would you go about recording what you did?” Assess children’s understanding Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics: Incorrect Use of Models When teacher says, “Do as I do” It is possible for children to mindlessly “manipulate” models (just as they might mindlessly “invert and multiply” fractions) Children can be “on-task” with manipulatives, but “off-task” with mathematics Over directed use of models can result in them ceasing to be “thinker” tools, and become “answergetters.” When this is the focus, little reflective thought occurs which results in little real growth Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics: Teaching Developmentally Children construct their own knowledge and understanding; we cannot transmit ideas to passive learners. Knowledge and understanding are unique for each learner. Reflective thinking is the single most important ingredient for effective learning. Effective teaching is a child-centered activity. Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics: Effective Strategies Effective teaching strategies are meant to promote, "purposeful mental engagement or reflective thought about the ideas we want students to develop" which he indicates is the "single most important key to effective teaching"(Van de Walle, p. 32). Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005 Developing Understanding in Mathematics: Effective Strategies Creating an Effective Mathematical Environment Posing Worthwhile Mathematical Tasks Using Cooperative Learning Groups Using Models as Thinking Tools Encouraging Student Discourse Requiring Justification of Student Responses Listening Actively Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005

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# Value of Student-Invented Algorithms