How People Learn

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How People Learn
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What Have We Learned So Far About How Children
Learn?

• Children think about ideas.
• Childrens capacity to think is, in part, a
  function of their developmental level
• As children get older, they develop different
  cognitive structures which both help them acquire
  information and constrain what they do with this
  information
• Children encounter new information and have to
  assimilate that information into existing
  concepts. When the information has trouble
  fitting in well with those concepts, they need to
  change the concepts to take the new information
  into account a process of accomodation.

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More of What We Have Learned

• Children acquire factual information
• Their familiarity with these facts improves with
  practice. Certain types of reinforcement are
  effective ways to motivate them to learn these
  facts.
• Some factual knowledge gets organized into
  domain specific knowledge structures.
• Sometimes this factual knowledge must be
  reorganized into new knowledge structures. The
  reorganization of this knowledge may require
  conceptual change.
• Sometimes, although children may not on their own
  understand a concept, they can, with the help of
  a more advanced learner grasp this concept. This
  process is called scaffolding and the childs
  ability to learn these ideas which are slightly
  above his current capability represent his zone
  of proximal development.

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Other Things We Have Learned

• The context in which something is learned can
  influence a childs ability to transfer that
  knowledge to another situation.
• Sometimes a child can understand a concept and
  solve a problem in a school situation but be
  unable to solve a related problem that requires a
  concept in a context outside of school. And
  sometimes the reverse is true a child can solve
  a problem outside of school but be unable to
  solve a related problem in a school context.

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What Can We Do With This Knowledge?

• We are trying to understand these ideas to help
  us figure out why children sometimes have
  difficulty learning conceptually challenging
  mathematics.
• One idea we focus on today is whether prior
  knowledge about natural numbers stands in the way
  of understanding rational numbers.

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What Can Help You Understand This Challenge?

• If you are a teacher of mathematics, try to think
  about an instance of one of your students who had
  difficulty understanding some conceptually
  challenging mathematical idea.
• If you are not a teacher, think about an example
  of a child having difficulty learning one of the
  mathematical concepts described in an article we
  have read.
• Think for example about the difficulty a child
  had in understanding ideas about decimal
  fractions.
• Or think of the study about difficulty students
  have learning about the structure of the set of
  rational numbers.

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Discreteness Of A Set Of Numbers

• Natural numbers
• what natural number(s) fall(s) between the
  following two natural numbers?

2 discrete numbers 3, and 4
1 discrete number, 3
No other discrete number
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Density Of A Set Of Numbers

• Rational numbers
• what rational number(s) fall(s) between the
  following two rational numbers?

Between 2.5 and 5.5 there are an infinite number
of rational numbers
Between 2.5 and 4.5 there are an infinite number
of rational numbers
Between 2.5 and 3.5 there are an infinite number
of rational numbers
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How does prior knowledge influence the
acquisition of new knowledge?

• Knowing about discreteness gets in the way of
  acquiring the concept of density.

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Vosniadou Theoretical Framework for Conceptual
Change

• The knowledge acquisition process is not always a
  process of enriching existing conceptual
  structures. Sometimes the acquisition of new
  information requires the radical reorganization
  of what is already known.
• Learning that requires the reorganization of
  existing knowledge structures is more difficult
  and time-consuming than learning that can be
  accomplished through enrichment. Moreover, it is
  likely that in the process of reorganization
  students will create misconceptions.
• Many misconceptions are synthetic models that
  reveal students attempts to assimilate new
  information to their existing knowledge base.

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Goal Of Education

• Helping students develop the intellectual tools
  and learning strategies needed to acquire the
  knowledge that will allow them to think
  productively about mathematics

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Is Knowing Facts Important?

• Yes. Experts know lots of facts.
• Distinction between usable knowledge and
  disconnected facts.
• Experts knowledge is organized around concepts.

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The Role Of Prior Knowledge

• Children construct new knowledge and
  understandings based on what they already know
  and believe.
• If you want to know why a student may be getting
  something wrong, you might want to look at her
  incomplete understandings and concepts about the
  subject being learned.
• Sometimes you need to work on changing
  fundamental concepts of prior understandings in
  order to allow students to correctly understand
  new concepts.

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Metacognition

• A persons ability to monitor their understanding
  and how they are going about working at a task.

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Transfer

• The ability to extend what one has learned in one
  context to new contexts
• In situated learning, we are concerned about
  transfer. Is some type of learning so bound to a
  particular context that it is difficult to
  transfer it to different contexts?
• Question what kinds of initial learning are most
  likely to support transfer?

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Initial Learning

• You have to have a substantial body of initial
  knowledge in order to have knowledge to transfer
  to a new setting.
• Learning multiplication facts may be helpful for
  learning new material.
• Presenting multiplication facts in conceptually
  different ways may help in acquiring the needed
  body of knowledge more effectively.
• Example Everyday Math uses fact triangles rather
  than simply asking students to memorize the
  multiplication table.
• Developing expertise requires a lot of practice.
• Does this suggest a role for meaningful homework
  activities?

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The Problem Of Introducing Too Many Topics Too
Quickly

• May hinder transfer because
• Isolated facts are not organized into concepts
• Not enough facts to see the relationship between
  facts that supports organizing principles

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Contrasting Cases Can Help To Understand When,
Where, And Why To Use New Knowledge

• Helps you notice new features
• Helps you learn which features are relevant to a
  particular concept and which are not
• ConceptBIPED
• Is this a BIPED?
• Which features?
• (also helps to learn bi2 and pedfeet)

YES
NO
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Motivation To Learn

• Competence (Robert White)
• Intrinsic as well as extrinsic
• Performance orientation versus learning
  orientation (Carol Dweck)
• Performance Will I make a mistake? What grade
  will I get? Will my teacher think I did well?
  What will my mother think of my work?
• Learning this problem is a challenge. I wonder
  whether I can solve it. It will feel good to
  figure it out.