Abstract or concrete examples
in learning math?
A replication and elaboration
of Kaminski et al.’s study
Dirk De Bock, Johan Deprez,
Wim Van Dooren, Michel Roelens,
Lieven Verschaffel
Introduction
Abstract
mathematics learns
better than practical
examples
Is mathematics
about moving trains,
…, sowing farmers?
Or about abstract
equations with x and
y and fractions and
squares? And which
of both works best?
Introduction
Introduction
Les exemples sont mauvais
pour l’apprentissage des
mathématiques
(25 April 2008)
Examples are bad for learning math
Introduction
newspaper articles are based on
• doctoral dissertation
Kaminski, J. A. (2006). The effects of
concreteness on learning, transfer, and
representation of mathematical concepts.
• series of papers
…
Kaminski, J. A., Sloutsky, V. M., &
Heckler, A. F. (2008). The advantage of
abstract examples in learning math.
Science, 320, 454–455.
…
Kaminski et al.
• address the widespread belief in ‘from concrete to abstract’
“Instantiating an abstract concept in concrete contexts places the
additional demand on the learner of ignoring irrelevant, salient
superficial information, making the process of abstracting common
structure more difficult than if a generic instantiation were considered”
(Kaminski, 2006, p. 114)
• set up a series of controlled experiments
mainly with undergraduate students in psychology
but also one experiment with school children
Kaminski et al.
• some conclusions (Kaminski et al., 2008, p. 455)
 “If the goal of teaching mathematics is to produce knowledge that
students can apply to multiple situations, then representing
mathematical concepts through generic instantiations, such as
traditional symbolic notation, may be more effective than a series of
“good examples”.”
 “Moreover, because the concept used in this research involved basic
mathematical principles and test questions both novel and complex,
these findings could likely be generalized to other areas of
mathematics. For example, solution strategies may be less likely to
transfer from problems involving moving trains of changing water
levels than from problems involving only variables and numbers.”
Critical reactions from researchers
• in Educational Forum and e-letters in Science:




Cutrona, 2008
Mourrat, 2008
Podolefsky & Finkelstein, 2008
…
• research commentary of Jones in JRME (2009)
• informal reactions
 McCallum, 2008
 Deprez, 2008
In this seminar
1.
2.
3.
4.
Introduction
A taste of mathematics: commutative group of order 3
The study of Kaminski et al.
Two main elements of critique
1. unfair comparison
2. what did students actually learn?
5. An empirical study by De Bock et al.
6. General discussion
A taste of mathematics:
commutative group of order 3
Commutative group of 3 elements
• a set G of 3 elements …
for example
 {0,1,2}
 {r120°, r240°, r0°} , where for example r120° denotes rotation
 {a, b, c} where a, b and c are not specified
• with an operation * defined on the elements …
 {0,1,2}: addition modulo 3, for example: 2+2=1
 {r120°, r240°, r0°}: apply rotations successively, for example: first r120°,
then r240° gives r0°
 {a, b, c} : the operation can be given by a 3 by 3 table
• satisfying the following properties:
Commutative group of 3 elements
• a set G of 3 elements …
• with an operation * defined on the elements …
• satisfying the following properties:
 commutativity: x*y=y*x for all x and y in G
 associativity: (x*y)*z=x*(y*z) for all x, y and z in G
 existence of identitiy: G contains an element n for
which x*n=x=n*x for all x in G
 existence of inverses: for every element x in G there
is an element x’ for which x*x’=n=x’*x
0
the two examples are isomorphic groups
all groups of order 3 are isomorphic
2
1
The study of Kaminski et al.
The central experiment in Kaminski et al.
(80 undergraduate students)
Phase 1:
Learning domain
study + test
Phase 2:
Transfer domain
presentation + test
A: Tablets of an
archeological dig
C1: Liquid containers
C2: Liquid containers +
Pizza’s
C3: Liquid containers +
Pizza’s + Tennis balls
T: Children’s game
Phase 1
• study:
 introduction
 explicit presentation of
the rules using
examples
 questions with
feedback
 complex examples
 summary of the rules
• learning test:
24 multiple choice
questions
Phase 2
• presentation
 introduction to the game
 “The rules of the system you learned are like the rules of this game.”
 12 examples of combinations
• transfer test
24 multiple choice questions
Results
• learning test: A = C1 = C2 = C3
• transfer test: A > C1 = C2 = C3
Two main elements of critique
1. Unfair comparison
• Kaminski controlled for superficial similarity
undergraduate students read descriptions of T-A or T-C, but
received no training of the rules
low similarity ratings
no differences in similarity ratings T-A vs T-C
• critics: unfair comparison due to deep level similarity
between T and A
1. role of prior knowledge
2. (implicit) central mathematical concept
3. structure
(McCallum, 2008; Cutrona, 2009; Deprez, 2008; Jones, 2009a,
2009b; Mourrat, 2008, Podolefsky & Finkelstein, 2009)
A
C
T
1. Unfair comparison
1. prior knowledge
A and T:
 arbitray symbols
 operations governed by formal rules
 ignore prior knowledge!
C: physical/numerical referent
 physical/numerical referent for the symbols
 physical/numerical referent for the operations
 prior knowledge is useful!
A
C
T
1. Unfair comparison
2. (implicit) central mathematical concept
A and T: commutative group
(commutativity, associativity, existence of identity element,
existence of inverse elements)
C: explicitly communicated (commutative group)
vs. implicit (modular addition)
both are meaningful mathematical concepts
… but distinct!


2 and 3 elements: group determined by modular addition is
the only group
n elements, n>3, not prime: also other groups than the
group determined by modular addition
A and C learn different concepts!
concept learned in A is more useful for T
A
C
T
1. Unfair comparison
3. structure
A : identity elt. n and two symmetric elts. a and b




{n,a,b},
(1.1) a+a=b,
(1.2) b+b=a
(1.3) a+b=b+a=n
A
C: symmetry is broken (implicitly: 1 vs. 2)
 {n,a,b}
 (2.1) a+a=b
 (2.2) a+a+a=n
1+1=2
1+1+1=3
equivalent, but focus on different aspects
A and C learned/ignored different aspects
in T: no clues for 2nd set of basic rules
(A shows no transfer to modulo 4 addition (K., 2006))
C
T
2. What did students actually learn?
• Multiple choice test only shows final answers, not how
students arrived at their answers.
• What did the students learn?




a set of specific rules?
modular addition?
group properties (commutativity, …)?
…
• Is there any evidence for a conscious application of what is
learned?
commutativity, … are familiar from traditional number systems!
An empirical study by De Bock et al.
Method
• Subjects: 130 undergraduate students in
educational sciences
• Two phases
(1) training and testing in a learning domain
(2) testing for transfer
• Four experimental conditions (A = abstract, C = concrete)



AA, CA, AC, and CC
AA and CA: “Kaminski conditions”
AC and CC: important additions by us
Method
Operationalizations of the domains
• A-learning: archaeological context
• A-transfer: children’s game
• C-learning: liquid cups
• C-transfer: pizza context
(slices of pizza that behave in the same way as the liquid cups)
Method
Method
Method
Method
Method
In all conditions:
Just before test administration in the learning
domain, a summary of key ideas was presented.
Method
Method
Tests at the end of the learning phase and transfer
test consisted of 24 ‘isomorphic’ multiple choice
questions
Method
Method
Method
Second important difference with Kaminski’s procedure:
Open question immediately after learning phase
E.g., after concrete learning phase:
?
What should come on the place of the question mark?
Explain as precisely as possible how you have found
this.
Method
Or after the abstract learning phase:
?
What should come on the place of the question mark?
Explain as precisely as possible how you have found this.
Training + testing




individual
two phases immediately followed after each other
own pace
computer
Method - Analysis
• Scores on learning and transfer test: statistical analysis
(ANOVA + Tukey HSD) after removal of some outliers (using
the same procedures as Kaminski)
• Explanations on “open question”:
scoring system developed and applied to the data by two
independent raters.
Method - Analysis
Scoring system
• Unit of analysis = explanation of one participant
• Four main categories:




G (Group)
M (Modulo)
R (Rules)
N (No)
• Subcategories:
 G1 , G2 , G3 , G4
 M1, M2
• Scores: 2, 1 or 0
Method - Analysis
Scoring system
• 2 = formulation at general level
Examples
 “order doesn’t matter”
 “if you combine a flag with another symbol, you always get that other
symbol”
 “2 + 2 = 4 – 3 = 1”
• 1 = unambiguous application
• 0 = else
Results – Quantitative results
Mean and standard deviation of test
scores (Max = 24)
Condition
Learning test
Transfer test
AA (N = 23)
17.1 (3.9)
18.1 (3.8)
AC (N = 30)
15.3 (3.5)
17.4 (4.2)
CA (N = 28)
18.5 (2.9)
12.0 (4.3)
CC (N = 24)
18.3 (3.5)
20.2 (2.4)
• Learning test: AC < CA, CC
• Transfer test: CA < AA, AC, CC and AC < CC
Results – Quantitative results
• Kaminski confirmed (transfer test: AA > CA)
• Opposite holds too (transfer test: CC > AC)
• Although AC < CX (learning test), AC = AA (transfer test):
students seem to “learn” to deal with “modulo 3” arithmetic
with little or no support from (A) learning condition
Results – Qualitative results
G
Learning
Domain
A
(N = 66)
M
R
N
Score
G1
G2
G3
G4
M1
M2
2
0
6
0
0
0
0
–
–
1
16
43
0
3
0
0
62
11
0
50
17
66
63
66
66
4
55
• Literal repetition of combination rules
• Formulations of group axioms at general level are rare
(although it was explicitly asked to explain “as precisely as possible”)
Results – Qualitative results
G
Learning
Domain
C
(N = 52)
M
R
N
Score
G1
G2
G3
G4
M1
M2
2
0
0
0
0
7
0
–
–
1
13
7
0
2
22
5
5
14
0
39
45
52
50
23
47
47
38
• Application of “modulo 3” arithmetic by about half of the
students (not explicitly addressed in learning phase!)
• In some cases: even without reference to the context
• …
Results – Qualitative results
G
Learning
Domain
C
(N = 52)
M
R
N
Score
G1
G2
G3
G4
M1
M2
2
0
0
0
0
7
0
–
–
1
13
7
0
2
22
5
5
14
0
39
45
52
50
23
47
47
38
• …
• Pure repetitions of combination rules were rare
• Some spontaneous applications of group properties
(although less than in the A-learning groups)
Main conclusions
Our results confirm Kaminski’s one: transfer to a new “abstract”
domain is enhanced by an abstract, rather than by a
concrete domain
But…
• Transfer to a new “concrete” domain is also enhanced by a
concrete, rather than by an abstract learning domain
• Serious doubts about what students actually learned from
the abstract domain (group axioms vs. formal application of
combination rules)
• Some students reached a higher level of abstraction from
the concrete learning domain.
General discussion
General discussion
• Inappropriate to extrapolate Kaminski’s findings to the
broader realm of mathematics education.
• Even an extrapolation to the mathematical objects coming
next in level of abstraction is problematic (much more
specific rules are needed to determine a group of order 4 (or
higher) in Kaminski style)
• Understanding a mathematical concept also has an
epistemological meaning (where does the concept comes
from and why is it useful?).
Nor the abstract, nor the concrete instantiations of Kaminski’s
(and our) study shed any light on this issue…