Title: Modern Learning Theories and Mathematics Education: Bidirectional Contributions, Bidirectional Challenges
1Modern 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.
2Like 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
3Another 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
4Bidirectional Contributions, Bidirectional
Challenges
- Contributions to Mathematics Education from
Applying Modern Learning Theories - Contributions to Modern Learning Theories from
Mathematics Education Applications - Challenges to Modern Learning Theories from
Mathematics Education Applications - Challenges to Mathematics Education from Modern
Learning Theories
5- Contributions to Mathematics Education from
Applying Modern Learning Theories
69. 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
7Theoretical 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
8(No Transcript)
9Progression from Log to Linear Representation
0-100 Range (Siegler Booth, 2004)
Number Presented
Number Presented
Number Presented
10Progression from Log to Linear Representation
0-1,000 Range (Siegler Opfer, 2003)
11The 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).
12Correlations 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
13Correlations 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
14Causal Evidence External Magnitude
Representations and Arithmetic Learning(Booth
Siegler, 2008)
15Issue 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)
16- 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
17Applying Theory to Educational ProblemMight
inadequate representations of numerical
magnitudes underlie low-income childrens poor
numerical performance?
18Applied 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)
19Playing 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
20Chutes and Ladders
21Key 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
22Number Board Game
23Color Board Game
24Effects 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
25Methods
- 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.
26- 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
27Numerical Magnitude Comparison
Correct
M
p lt .001
28Counting
Mean Counts Without Error
M
p lt .001
29Number Line EstimationLinearity of Individual
Childrens Estimates
Mean R2lin
M
p lt .001
30Numeral Identification
Correct
M
p lt .001
31Percent Correct Addition Answers(Siegler
Ramani, in press)
Correct
M
p lt .05
32II. Contributions to Modern Learning Theories
from Mathematics Education Applications
33Theoretical 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
34- 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)
35III. Challenges to Modern Learning Theories from
Mathematics Education Applications
36- 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.
37Conclusion 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.
38IV. Challenges to Mathematics Education from
Modern Learning Theories
39Conclusion 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.
40- 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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42Trend in NAEP Mathematics Average Scores, 1973 -
2008
Score