Modern Learning Theories and Mathematics Education: Bidirectional Contributions, Bidirectional Challenges

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Modern Learning Theories and Mathematics
Education Bidirectional Contributions,
Bidirectional Challenges
The research reported here was supported by the
Institute of Education Sciences, U.S. Department
of Education, through Grant R305H050035 to
Carnegie Mellon University. The opinions
expressed are those of the author and do not
represent views of the Institute or the U.S.
Department of Education.
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Like many investigators funded by IES, most of my
pre-IES research was theoretical (Definition
Without any likely application)IES motivated
me to think harder about ways in which the
research could be applied to important
educational problems without sacrificing
rigorOne outcome has been my current research
applying theories of numerical cognition to
improving low-income preschoolers mathematical
understanding
A Little Personal Background
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Another outcome has been to increase my interest
in broader issues of application, i.e.,
educational policy issuesThis growing interest
in applications led me to abandon my traditional
just say no policy regarding commissions and
panels and accept appointment to the National
Mathematics Advisory Panel (NMAP). Main role was
in learning processes groupThe present talk
combines perspectives gained from doing the
applied research and from participating in the
learning processes group of NMAP
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Bidirectional Contributions, Bidirectional
Challenges

1. Contributions to Mathematics Education from
   Applying Modern Learning Theories
2. Contributions to Modern Learning Theories from
   Mathematics Education Applications
3. Challenges to Modern Learning Theories from
   Mathematics Education Applications
4. Challenges to Mathematics Education from Modern
   Learning Theories

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1. Contributions to Mathematics Education from
   Applying Modern Learning Theories

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9. Encouraging results have been obtained for a
variety of instructional programs developed to
improve the mathematical knowledge of
preschoolers and kindergartners, especially those
from low-income backgrounds. There are effective
techniques derived from scientific research on
learning that could be put to work in the
classroom today to improve childrens
mathematical knowledge.14. Childrens goals
and beliefs about learning are related to their
mathematics performance. . . When children
believe that their efforts to learn make them
smarter, they show greater persistence in
mathematics learning.
Conclusions of NMAP
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Theoretical Background The Centrality of
Numerical Magnitude Representations

• A basic issue in many modern learning theories
  involves how knowledge is represented
• In mathematical cognition, this issue involves
  the underlying representation of numerical
  magnitudes (Dehaene, 1997 Gelman Gallistel,
  2001 Case Okamoto, 1996)
• Empirical research indicates that linear
  representations linking number symbols with their
  magnitudes are crucial for a variety of important
  mathematics learning outcomes

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Progression from Log to Linear Representation
0-100 Range (Siegler Booth, 2004)
Number Presented
Number Presented
Number Presented
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Progression from Log to Linear Representation
0-1,000 Range (Siegler Opfer, 2003)
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The Centrality of Numerical Magnitude
Representations

• Linearity of magnitude representations correlates
  positively and quite strongly across varied
  estimation tasks, numerical magnitude comparison,
  arithmetic, and math achievement tests (Booth
  Siegler, 2006 2008 Geary, et al., 2007 Ramani
  Siegler, 2008 Whyte Bull, 2008).

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Correlations Among Linearity of Magnitude
Representations on Three Estimation Tasks (Booth
Siegler, 2006)
Grade Task Measurement Numerosity

2nd Number line .65 .55
Measurement .54

4th Number line .84 .70
Measurement .60
p lt .01 p lt .05
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Correlations Between Linearity of Estimation and
Math Achievement (Booth Siegler, 2006)
Estimation Task
Grade
Number Line Measurement Numerosity
2nd .53 .62 .48
4th .47 .54 .35
p lt .01 p lt .05
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Causal Evidence External Magnitude
Representations and Arithmetic Learning(Booth
Siegler, 2008)
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Issue in Mathematics Education Low-Income
Children Lag Behind in Mathematical Proficiency
Even Before They Enter School

• 1. Children vary greatly in mathematical
  knowledge when they enter school
• 2. Numerical knowledge of kindergartners from
  low-income families trails far behind that of
  peers from higher-income families (ECLS, 2001)

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• 3. Kindergartners numerical knowledge strongly
  predicts later mathematical achievement through
  elementary, middle, and high school (Duncan, et
  al., 2007 Jordan et al., 2009 Stevenson
  Newman, 1986)
• 4. Large, early, SES related differences become
  even more pronounced as children progress through
  school

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Applying Theory to Educational ProblemMight
inadequate representations of numerical
magnitudes underlie low-income childrens poor
numerical performance?
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Applied Goal Raised New Theoretical Question
What Leads Anyone to Form Initial Linear
Representation?

• Counting experience is likely helpful and
  necessary, but insufficient
• Children can count in a numerical range more than
  a year before they can generate a linear
  representation of numerical magnitudes in that
  range (Condry Spelke, 2008 LeCorre Carey,
  2007 Schaffer et al., 1974)

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Playing Board Games

• Board games might play a crucial role in forming
  linear representations of numerical magnitudes
• Designed to promote interactions between parents
  and peers
• Also provides rich experiences with numbers

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Chutes and Ladders
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Key Properties of Board Games Like Chutes and
Ladders

• The greater the number a token reaches, the
  greater the
• Distance that the child has moved the token
• Number of discrete hand movements the child has
  made
• Number of number names the child has spoken
• Time spent by the child playing the game
• Thus, playing number board games provides
  visuo-spatial, kinesthetic, auditory, and
  temporal cues to links between number symbols and
  their magnitudes

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Number Board Game
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Color Board Game
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Effects of Playing the Number Board Game(Ramani
Siegler, 2008)

• Goal was to investigate whether playing the
  number board game
• Improves a broad range of numerical skills and
  concepts
• Produces gains that remain stable over time

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Methods

• Participants 129 4- and 5-year-olds from Head
  Start classrooms (mean age 4.8), 52
  African-American
• Experimental Conditions
• Number Board Game (N 69)
• Color Board Game (N 60)
• Design Pretest, 4 training sessions, posttest, 9
  week follow-up.

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• Training Procedure
• Children play a total of 20 games over 4 sessions
  in a 2 week period, 15-20 minutes/session
• Child spins spinner, gets 1 or 2, says while
  moving token (e.g.) 5, 6 or blue, red
• Feedback and help if needed
• Measures
• 0-10 Number Line Estimation
• 1-9 Numerical Magnitude Comparison
• 1-10 Counting
• 1-10 Numeral Identification

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Numerical Magnitude Comparison


Correct
M
p lt .001
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Counting


Mean Counts Without Error
M
p lt .001
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Number Line EstimationLinearity of Individual
Childrens Estimates


Mean R2lin
M
p lt .001
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Numeral Identification


Correct
M
p lt .001
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Percent Correct Addition Answers(Siegler
Ramani, in press)

Correct
M
p lt .05
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II. Contributions to Modern Learning Theories
from Mathematics Education Applications
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Theoretical Contributions of Number Game
Application
NMAP Conclusion 10. The curriculum must
simultaneously develop conceptual understanding,
computational fluency, and problem solving
skills. . . These capabilities are mutually
supportive, each facilitating learning of the
others.

• Point to need for single theory to integrate
  learning of concepts, procedures, facts, and
  problem solving
• Demonstrate need to identify everyday experiences
  that build conceptual understanding

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• Illustrate need to identify central conceptual
  structures (Case Okamoto, 1996)
• Raise question of what other specific activities
  contribute to numerical magnitude
  representations
• Counting objects in row
• Addition via counting fingers
• Conversation about numerical properties
• Other games (e.g., war)
• Suggest that inadequate fraction magnitude
  representations partially due to lack of
  experiences that indicate correlational structure
  (1/3 1/3 2/6)

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III. Challenges to Modern Learning Theories from
Mathematics Education Applications
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• NMAP Executive Summary, p. 32
• There are many gaps in current understanding of
  how children learn algebra and the preparation
  that is needed before they enter algebra.

Considerable high quality research is available
regarding math learning in preschool and first
few grades, but far less on later math learning.
Theories and empirical studies need to address
learning of pre-algebra, algebra, and geometry.
Virtue of theory-based applications Open up
theories help avoid trap of more and more about
less and less.
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Conclusion 12 from NMAP Difficulty with
fractions (including decimals and percentages) is
pervasive and is a major obstacle to further
progress in mathematics, including algebra.

• Remarkable agreement among NMAP members and
  algebra teachers on importance of fractions for
  learning algebra. But no evidence.
• Need for robust measures of moderately general
  knowledge structures, such as understanding of
  fractions, so can investigate these relations.
• Such robust measures require better theory of
  whats central to (e.g.) understanding fractions.

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IV. Challenges to Mathematics Education from
Modern Learning Theories
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Conclusion 15 from NMAP Teachers and developers
of instructional materials sometimes assume that
children need to be a certain age to learn
certain mathematical ideas. However, a major
research finding is that what is developmentally
appropriate is largely contingent on prior
opportunities to learn. Claims that children of
particular ages cannot learn certain content
because they are too young have consistently been
shown to be wrong.

• Young students in East Asia and some European
  countries spend more time on math, encounter more
  challenging and conceptually richer curricula,
  and learn more. No reason why we cant do the
  same. Belief that young children arent ready to
  learn relatively advanced concepts contradicts
  both national and international data.

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• Conclusion 9 from NMAP
• There are effective techniques derived from
  scientific research on learning that could be
  put to work in the classroom today to improve
  childrens mathematical knowledge.

IES has generously supported research on learning
principles and on programs that implement these
principles. As always, we need more research, but
some of the research is now sufficiently advanced
for broad implementation, at least on an
experimental basis. The challenge for the field
of mathematics education is how to use the
programs and principles to improve educational
practice.
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Trend in NAEP Mathematics Average Scores, 1973 -
2008
Score