Developing Understanding in
Mathematics
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“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
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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
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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
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Continuum of Understanding
Information from Van de Walle (2004)
Jamar Pickreign, Ph.D. 2005
Developing Understanding in Mathematics:
Benefits of Relational Understanding
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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
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Conceptual Knowledge
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Relationships or logical ideas
Procedural Knowledge
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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
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Mathematics Concepts are abstract
Models are ways of representing concepts.
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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
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Models are “thinker” toys
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Help children develop new concepts
Help children make connections between
concepts and symbols
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“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
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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
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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
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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
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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