Information from Van de Walle (2004) Jamar Pickreign, Ph.D. 2005

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 2 5 8 1 1 1 4 1 7 2 0 2 3 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

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Continuum of Understanding

Relational Understanding

Instrumental 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

Oral language

Real-world situations

Manipulative models

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 “answer-getters.” 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