Conceptual Change and Mathematics Learning

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Are children fundamentally different kinds of
thinkers and learners than adults?
Yes, at the level of representational format
data structures for representing information
and/or processes for manipulating those
structures (e.g., Piaget Vygotsky) VS. Basic
mental machinery is the same but children and
adults differ in other ways related to
differences in knowledge
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Representational format changes
Piaget shifts between his major periods
(especially sensorimotor to preoperational and
preoperational to concrete operational)
interpreted as changes in representational
formats and processes that operate on
them Vygotsky children have different kinds of
concepts from adults dont have true concepts
until puberty
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Stimuli 2 sizes of horizontal surface (large
small) 5 colors 6 shapes 2 heights (tall flat)
This is a mur. Pick out all the other ones that
might be the same kind.
Target concepts lag tall large bik flat
large mur tall small cev flat small
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Vygotskys views of conceptual development

• Childrens concepts are different in structure
  and meaning from adults
• Development proceeds through 3 basic phases
• 1. Congeries/heaps (only connections are some
  chance subjective impression in childs mind)
• 2. Thinking in complexes (bonds actually exist
  between the objects child puts together).
  Maturest form is pseudoconcepts which
  correspond more closely to adult meanings and
  serve as a bridge to the development of true
  concepts.
• 3. Only at puberty do children become capable
  of abstract concepts, which are true concepts.

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Both Piaget and Vygotsky would argue for
fundamental domain general changes in mental
machinery from childhood to adulthood (Bruners
theory hypothesizing shifts from iconic to
symbolic concepts is also of this type)
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• Thus most major theories of cognitive development
  through the 70s argued that children are
  fundamentally different in the kinds of concepts
  they can represent and operate on.
• There have been two important changes since then
• We have more data from children in a greater
  variety of situations
• There have been significant changes in what is
  accepted as viable models of ADULTS concepts
  many now view adults concepts as embedded in
  theories rather than as having stand-alone
  definitions consisting of necessary and
  sufficient features

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What does Carey mean by domain?

• Involves a set of real-world phenomena
• Uses a set of concepts to represent those
  phenomena
• Draws on a set of explanatory mechanisms and
  principles that constitute an understanding of
  that set of phenomena

Carey uses domain as roughly synonymous with
something one would have a theory about. When a
new theory emerges, it carves out a set of
phenomena that it applies to, represents them in
characteristic ways, and has an explanatory
framework to account for them. (e.g., historical
emergence of theories Piagets theory of
genetic epistemology group theory in math)
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How might children and adults differ in knowledge?
Metaconceptual knowledge adults but not
children can think about their own mental
processes. Note that Carey treats metaconceptual
knowledge as a form of domain specific knowledge
that has implications for other domains of
learning Foundational concepts children lack
certain concepts that apply widely across other
domains e.g., causality Tools of wide
application children lack specific tools such
as math or literacy that apply to many other
domains Domain specific knowledge children
have different theories about the world children
are universal novices
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Carey treats all four of these as contrasting
with the first claim that there are developmental
changes at the level of representational
structures and processes. She examines each of
them as an example of ways in which children
could differ from adults by having different
knowledge bases.
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Metaconceptual knowledge, Foundational Concepts,
and Tools of Wide Application are all instances
of kinds of knowledge that have an impact across
many other domains of thinking and learning. The
last one, differences in domain specific
knowledge, argues that children use a different
set of theories to organize the world. This is
where conceptual change comes in.
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Cases Carey considers Classes Class
Inclusion Hypothesis Formation Testing Mental
Lexicon Causal Explanatory Notions Appearance/
Reality Distinction Number Measurement
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Careys position is that most of the
developmental action for all of these cases falls
under her last category of changes in domain
specific knowledge . . . The acquisition and
reorganization of strictly domain-specific
knowledge (e.g., of the physical, biological, and
social worlds) probably accounts for most of the
cognitive differences between 3-year-olds and
adults. (p. 512) This is a particularly strong
claim. It is not a necessary assumption of
theories of conceptual change, but it is a
provocative argument to consider.
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But its critical to note that knowing more
does not simply mean accumulating more facts or
bits of knowledge. Carey argues that whats
interesting to look at in development is
knowledge reorganization. When the reorganization
is substantial, conceptual change is involved.
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What is conceptual change?

• A kind of learning in which knowledge must be
  radically restructured and reorganized from prior
  conceptual framework.
• Weak restructuring additive, enriches and
  extends previous knowledge, old concepts are
  compatible with new ones
• Strong restructuring old and new concepts and
  explanatory frameworks are incommensurate

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Conceptual Change theorists argue that it is a
kind of learning that involves different learning
mechanisms. The learning is not additive.
Vosniadou et al. believe that using additive
instructional mechanisms when conceptual change
is involved can lead to misconceptions. Argue
instead for developing learners who have the
metacognitive skills to recognize difficult
learning situations and who can use strategies of
intentional learning.
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Vosniadou

• Also focuses on changes in concepts and
  explanatory frameworks as hallmarks of conceptual
  change.
• Adds that misconceptions may arise as learners
  attempt to bring new ideas into prior conceptual
  structures. Refers to these as synthetic models.

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The earth is round
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No, round like a ball
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Conceptual Change Rational Numbers

• Proposition that transition from natural numbers
  to rational numbers requires conceptual change
• Changes in concept of what numbers are and what
  they are used for
• Changes in explanatory frameworks

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Comparison of Natural and Rational Numbers

• Discreteness of natural numbers For every
  number a successor is defined and no two numbers
  have the same successor
• In contrast, rational numbers are characterized
  by the property of infinite densitythere is an
  infinite number of numbers between any two
  numbers. There is no unique successor to any
  given number.

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Discreteness of Natural Numbers

• Natural numbers (1, 2, 3, etc.) are for counting
  entities.
• Numbers can be used to count entities that can be
  individually differentiated. Typically we count
  natural objects, but we can also count events
  (e.g., when jumping rope), fictitious things,
  absent things, etc. Note that the constraint on
  counting, even for children, is discreteness, not
  concreteness.

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Childrens Counting Experience

• Using numbers to count small sets of discrete
  objects appears very early in development
  hypothesized to be based on innate learning
  mechanisms.
• Children gain much everyday practice with
  counting.
• First formal math instruction reinforces
  counting.

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Preschool Counting

• Preschoolers honor the counting principles before
  formal instruction in math
• Initially count only small set sizes accurately
• Knowledge of the extended counting list grows
  gradually (e.g., base 10 structure)
• Children become better at coordinating task
  demands (e.g., keeping track of which items were
  already counted)

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Discreteness vs. Density

• Because natural numbers have unique successors
  but rational numbers do not, the rules for
  ordering and comparing them are different.
• Operations on natural and rational numbers also
  yield different results.

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Generalizations for Natural Numbers that Do Not
Apply to Rational Numbers

• Addition and multiplication make bigger
• Subtraction and division make smaller
• (Teaching multiplication as repeated addition and
  division as repeated subtraction can play into
  these ideas as extended from counting numbers)

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Operations

• Students think that principles that apply to
  counting numbers are general principles that
  apply to all numbers.
• Their overgeneralizations lead to errors. For
  many children, this is when math stops making
  sense.

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Important reminder from Vosniadou (2004)

• Presuppositions that constrain learning are
  not under the conscious control of the learner.
  It is important to create learning environments
  that allow students to express and elaborate
  their opinions, so that they become aware of
  their beliefs.

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Infinity and Density

• Children can engage in productive, abstract
  thinking within their understanding of natural
  numbers. E.g., Can use the successor principle to
  induce the concept of infinity for positive
  integers (can always add one more)
• Difficulty with density of rational numbers
  doesnt seem to be problem with understanding
  infinity per se. Seems instead to be constrained
  by assumption of discreteness.

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• Stafylidou Vosniadou Understanding of
  numerical value of fractions.
• Shifts in explanatory frameworks
• Fraction as two independent natural numbers
• Fraction is part of a whole of a natural object
• Relation between numerator/denominator (with
  without infinity)

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Michelene Chi et al.

• Conceptual change involves ontological
  reclassification of key concepts
• What does this mean? What are ontological
  categories?

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Ontological Categories

• Fundamental backbone to peoples categories of
  what kinds of things exist physical objects
  (living things, non-living natural kinds,
  artifacts) events processes ideas
• Diagnosed by category error when applying
  predicates
• The book weighs 3 pounds. Statement may be
  true or false, but either way it is sensible.
• The idea weighs 3 pounds. This is a category
  error it is not just that its false, but that
  it is nonsensical to assign attribute of weight
  to an idea.

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• Chi et al. argue that learning that involves
  ontological reclassifications will be especially
  difficult.