Traditional paper-pencil assessments (subset of problem types
shown below):
Christine Massey,1 Philip J. Kellman,2 Zipora Roth,1 Ji Y. Son,2 Timothy Burke 2 & Warren Longmire1
1University of Pennsylvania & 2University of California, Los Angeles
Naming shaded pieces/Shading pieces
Judging Fraction Magnitude
Perceptual learning (PL): experience-induced
changes in the ability to extract information.
Declarative
•  Experts selectively perceive problemknowledge
Procedural
(e.g.,
knowledge
relevant features as well as higher-level
concepts,
(e.g., how to)
relationships.
facts)
•  Even when explicitly instructed about
Perceptual
Learning
higher-level relationships, novices often ignore
(e.g., pattern
them or cannot recognize them in new cases.
recognition
& structural
•  PL is crucial in the development of expertise
intuitions)
but largely neglected in K-12 instruction.
The learning focus: creating and iterating unit fractions to produce
new quantities (integers, fractions, and mixed numbers with the
same denominator).
Current Research Questions:
(1)  Can PL be systematically accelerated in mathematics – in
particular, in the domains of fractions and measurement?
(2)  Can PL interventions (short varying interactions w/ critical relational
structures) improve performance on traditional assessments?
Variation in Trial Difficulty
1. Parent/offspring value:
may be integer (1 or >1) or
fraction (<1, improper, or
mixed fraction)
2. # of conceptual steps:
•  One-step: No chunking to
slice or clone.
•  Two-steps: The cloning
Students have limited interpretations of fractions: e.g., the
common part-whole interpretation (3 out of 5 slices of a pizza) is
hard for students to generalize to improper and equivalent
fractions, and to operations with fractions.
Iterable
1/5 unit
(but only in
relation to the
whole 1 unit)
1 unit
5/5 units
7/7 or 7/5
units?
Each trial involves
one parent quantity
(given).
and one offspring quantity
(to be created).
User controls interactive
“slicing” and “cloning” tools.
of the offspring requires
chunking to create one
whole out of 1/5 units;
then adding on another
three 1/5 units.
•  Control participants (7th graders) completed the most
relevant fraction/measurement units taught in their curriculum
before testing (7 weeks)
•  PLM participants (6th graders) received all treatment & tests
except last delayed posttest prior to these curriculum units.
Group
Spring 08
Spring 08
6th graders,
Intervention
group
(n=53 at start,
40 at end)
Completed
S&C1 PLM
Baseline/pretest
Immediate
on S&C1 and
S&C2 subscales Posttest on both
subscales
(Posttest 1)
7th graders,
Uninstructed
control (n=21)
Baseline on
S&C1 and
S&C2 subscales
Fall 08
After 5 month
delay from PLM,
Delayed
Posttest on both
subscales
(Posttest 2)
Fall 08
Naming shaded pieces/Shading pieces
Circle pairs representing equal amounts.
Word Problems
Short Number Sentences
Word Problems
Criteria for completion of S&C 2 PLM
•  8 categories (5 active slice/clone categories; 3 passive slice/active
clone categories); 8/10 correct in each category
•  70% learned to criterion in the 8 categories of SC2
•  Average # of categories learned to criterion: 6.8
5 month delay
one
Write another fraction that represents
the same amount.
5.5 month delay
2 conceptual steps to clone
The learning focus:
equivalent fractions involving
different denominators
Each trial involves
one parent quantity (given)
and
two different offspring
quantities (to be created).
Incorrect Slice Response
Correct Slice Response
Shown feedback;
given another slice opportunity
before going on to clone.
One offspring is shown cloned;
subjects must clone to create the
remaining offspring.
S&C1
Intervention
Effect sizes
Posttest vs. Pretest: 2.03
Posttest vs. Control: 3.45
S&C2
Intervention
Delayed Posttest vs. Pretest: 1.75
Delayed Posttest vs. Control: 2.79
S&C1
Intervention
Effect sizes
Posttest vs. Pretest: .84
Posttest vs. Control: 1.14
S&C2
Intervention
Delayed Posttest vs. Pretest: 1.03
Delayed Posttest vs. Control: 1.40
•  PL technology can facilitate structure discovery and structure mapping, offering an important complement to declarative and procedural
instruction.
•  PL technology may accelerate fluency in processing key structures, reducing cognitive load and facilitating more advanced problem
solving.
Spring 09
Completed
S&C2 PLM
Immediate
Posttest on both
subscales
(Posttest 3)
Short Number Sentences
Criteria for completion of S&C 1 PLM
•  11 categories (one-step/low difficulty: 4 categories; two-step/high
difficulty: 7 categories); 8/10 correct in each category
•  100% learned to criterion in the 11 categories of SC1
•  Average # of categories learned to criterion: 11
Example S&C 2 Trial
The two modules presented here (Slice&Clone 1 and 2) are
designed to help students come to understand fractions as
representing quantities, in which the denominator of the fraction
defines an iterable unit and the numerator of the fraction specifies
a given number of iterations of that unit.
Division/Multiplication of fractions
(symbolic notation)
Traditional paper-pencil assessments (subset of problem types
shown below):
Final Delayed
Posttest on only
S&C2 subscale
•  PL techniques show great promise even in high-level, symbolic domains such as mathematics −− including
notoriously difficult areas such as fractions and measurement.
Acknowledgements
The researchers would like to thank Hye Young Cheong (Upenn) and Joel Zucker (UCLA) for assistance on
this project. Special thanks go to the Feltonville School of Arts and Sciences. This research was supported by the Institute of Education
Sciences, US Department of Education through Grant R305H06070 to UCLA. The opinions expressed are those of the authors and do not
represent views of the US Department of Education.