By Jonathan Brendefur, Ph'D'

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Mathematical Development

• By Jonathan Brendefur, Ph.D.
• Boise State University

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• Most students enter school confident in their own
  abilities, and they are curious and eager to
  learn more about numbers and mathematical
  objects. They make sense of the world by
  reasoning and problem solving, and teachers must
  recognize that young students can think in
  sophisticated ways.

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• Young students are active, resourceful
  individuals who can construct, modify, and
  integrate ideas by interacting with the physical
  world and with peers and adults. They make
  connections that clarify and extend their
  knowledge, thus adding new meaning to past
  experiences. They learn by talking about what
  they are thinking and doing and by collaborating
  and sharing their ideas.
• Principles and Standards for School Mathematics

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• What are Piagets four Stages of Cognitive
  Development?

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Stage 1 Sensorimotor (Infancy)

• At this stage, cognitive development focuses on
  motor and reflex actions. The child learns about
  herself and her environment through sensation and
  movement. They have a great interest in the faces
  and voices of their caretakers.

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Stage 2 Preoperation (Toddler and Early
Childhood)

• At this stage, the main focus of the child's
  intellectual development is language and using
  symbols (e.g., pictures and words) to represent
  ideas and objects. The child does these
  intuitively. The child at this stage has an
  active imagination and vivid fantasies. It is not
  uncommon for him to personify objects.

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Stage 3 Concrete Operation (Elementary and
Early Adolescence)

• At this stage, children begin to process abstract
  concepts such as numbers and relationships but
  they need concrete examples to understand these
  concepts. If before they could manipulate objects
  only physically, now they can do so mentally.

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Stage 4 Formal Operation(Adolescence and
Adulthood)

• At this stage, the child begins to reason
  logically and analytically without requiring
  references to concrete applications. This
  symbolizes the reaching of the final form of
  intelligence. At this stage, the child is also
  capable of hypothetical and deductive reasoning.

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The Nature of Number

• Match up with a partner. Choose who will be the
  teacher and who will be the student.
• Put out seven chips of one color (e.g., red).
• Teacher Place all seven chips in a row and ask
  the student to put out the same number of (e.g.,
  blue) chips as you have (red) chips.

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Equality

• Teacher Arrange both sets of chips in two equal
  rows and ask whether the two rows have the same
  amount. If the student says yes, ask How many
  are there? If the student says no, ask How many
  more chips are needed? Or How many extra chips
  are there?

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Conservation

• Teacher Spread out out the chips in one of the
  rows and ask
• 1. Are there as many (the same number of) red
  ones as blue ones or are there more her (red) or
  more here (blue)?
• 2. If correct answer ask, Look how long this
  line is. Another student said there are more
  counters in it because this row is longer. Who is
  right, you or the other child?

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• 3. If incorrect response remind the student of
  the initial equality. Another student said that
  there is the same number of red and blue ones
  now. Who do you think is right, you or the other
  student?

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Quotity

• Teacher Hide the blue counters with your hand or
  arm and ask, How many blue chips do you think
  there are? Can you guess without counting them?
  How do you know?
• Question
• What do each of these ideas mean for teaching
  mathematics?

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Hierarchical Order of Development
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Physical Knowledge

• Physical properties
• External
• Empirical abstractions
• Things known by observation
• How many chips are below?
• What are the differences?
• What are the similarities?

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Logico-Mathematical Knowledge

• Relationships
• Internal
• Reflective or constructive abstraction
• The difference is a relationship created
  mentally by the individual who puts the two
  objects into a relationship. The difference is
  neither in one chip nor in the other. If the
  person did not put the objects into the
  relationship, the difference would not exist.

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• Place the following statements into
  logico-mathematical knowledge or physical
  knowledge
• They weigh the same.
• They are chips.
• There are two chips.
• One chip is red and the other blue.

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• How many raindrops below?

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• How many raindrops below?

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Empirical and Reflective Abstractions

• Empirical abstractions
• Child focuses on one specific property and
  ignores the others
• E.g., color, weight, material/substance
• Reflective abstractions
• The relationship exists only in the mind
• E.g., number
• Why is this knowledge important?

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Why is it important to understand physical and
logico-mathematical knowledge?

• It implies that the child must put all kinds of
  content (objects, events, actions) into all kinds
  of relationships to construct number.
• Numbers are learned by reflective abstraction as
  the child constructs relationships from
  physical knowledge.
• What is 3, 5, or 1,000,002?
• Have you ever seen the actual quantity of the
  above numbers?

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Order vs. Hierarchical Inclusion

• Count the following chips
• How many chips are there?
• How do you know?
• What does the number mean?
• (2 possible ideas)

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nine
nine
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• What do you see?
• How many animals?
• How many dogs?
• How many cats?
• Are there more dogs
• or more animals?

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Reversibility

• By age 7 or 8 childrens thought becomes mobile
  enough to be reversible.
• ability to mentally do opposite actions
  simultaneously
• 3 7 10 and 7 3 10
• Importance Children must put all kinds of
  content (objects, events, actions) into all kinds
  of relationships. When children are in these
  situations their thought becomes more mobile and
  they build logico-mathematical structures or
  knowledge.

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Social Knowledge

• Conventions worked out by people
• Christmas is on December 25th
• Tree means tree
• Three means
• is (3 2 5)
• Children can be taught to say 3 2 5, BUT
  they cannot be taught directly the relationships
  underlying this addition.

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Activity

• Ask a partner to solve the following problems in
  her/his head and to write down how s/he solved
  it.
• 3 4
• 8 11
• 47 45
• 54 17

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Decomposing Numbers

• What are different ways to solve these numbers
  that use number relationships?
• 3 4 8 11
• 47 45 54 17
• What importance is the use of decomposition?

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Norms of Promoting Conceptual Understanding

• Explanations consist of mathematical arguments,
  not simply procedural summaries of the steps
  taken to solve the problem.
• Errors offer opportunities to reconceptualize a
  problem and explore contradictions and
  alternative strategies.
• Mathematical thinking involves understanding
  relations among multiple strategies.
• Collaborative work involves individual
  accountability and reaching consensus through
  mathematical argumentation.

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Questions and Prompts for Promoting Conceptual
Understanding

• Understanding students solutions
• How did you figure it out?
• How did you solve that?
• Describe how you used numbers/pictures/words to
  figure out this.
• Why did you decide to add/subtract/multiply/divide
  ?
• What do these numbers/tallies/cubes stand for?
• How did that help you? 

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• Involving other students
• Do you agree or disagree? Why?
• Who did it in a similar/different way?
• Who thinks they know what the presenter is
  going to do next? Why?
• Who can explain what the presenter said?

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• Extending thinking
• Can you think of another way to solve this
  problem using . . . ?
• What other strategy/problem does this
  strategy/problem remind you of?
• How are these strategies alike? Different?
• How could you write this strategy with
  numbers/words?
• Can you solve this problem another way?
• Would this strategy always work? Explain.
• How do you know you have found all the possible
  ways?
• How could you make this into a more challenging
  problem for yourself?

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• Using mistakes as learning sites
• What makes this a hard problem? Why is this a
  hard problem?
• How could you make this problem easier?
• I have seen other children make that mistake. Why
  do you think they make that mistake?
• How can you avoid that mistake next time you are
  solving a problem like that?