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A longstanding debate concerns the use of concrete versus abstract instructional materials, particularly in domains such as mathematics and science. Although decades of research have focused on the advantages and disadvantages of concrete and abstract materials considered independently, we argue for an approach that moves beyond this dichotomy and combines their advantages. Specifically, we recommend beginning with concrete materials and then explicitly and gradually fading to the more abstract. Theoretical benefits of this “concreteness fading” technique for mathematics and science instruction include (1) helping learners interpret ambiguous or opaque abstract symbols in terms of well-understood concrete objects, (2) providing embodied perceptual and physical experiences that can ground abstract thinking, (3) enabling learners to build up a store of memorable images that can be used when abstract symbols lose meaning, and (4) guiding learners to strip away extraneous concrete properties and distill the generic, generalizable properties. In these ways, concreteness fading provides advantages that go beyond the sum of the benefits of concrete and abstract materials.
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Educational Psychology Review
ISSN 1040-726X
Volume 26
Number 1
Educ Psychol Rev (2014) 26:9-25
DOI 10.1007/s10648-014-9249-3
Concreteness Fading in Mathematics and
Science Instruction: a Systematic Review
Emily R.Fyfe, Nicole M.McNeil, Ji
Y.Son & Robert L.Goldstone
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Concreteness Fading in Mathematics and Science
Instruction: a Systematic Review
Emily R. Fyfe &Nicole M. McNeil &Ji Y. Son &
Robert L. Goldstone
Published online: 14 January 2014
#Springer Science+Business Media New York 2014
Abstract A longstanding debate concerns the use of concrete versus abstract instructional
materials, particularly in domains such as mathematics and science. Although decades of
research have focused on the advantages and disadvantages of concrete and abstract materials
considered independently, we argue for an approach that moves beyond this dichotomy and
combines their advantages. Specifically, we recommend beginning with concrete materials and
then explicitly and gradually fading to the more abstract. Theoretical benefits of this con-
creteness fadingtechnique for mathematics and science instruction include (1) helping
learners interpret ambiguous or opaque abstract symbols in terms of well-understood concrete
objects, (2) providing embodied perceptual and physical experiences that can ground abstract
thinking, (3) enabling learners to build up a store of memorable images that can be used when
abstract symbols lose meaning, and (4) guiding learners to strip away extraneous concrete
properties and distill the generic, generalizable properties. In these ways, concreteness fading
provides advantages that go beyond the sum of the benefits of concrete and abstract materials.
Keywords Concrete manipulatives .Abstract symbols .Learning and instruction
A longstanding controversy concerns the use of concrete versus abstract materials during
mathematics and science instruction. Concrete materials connect with learnersprior knowl-
edge, are grounded in perceptual and/or motor experiences, and have identifiable correspon-
dences between their form and referents. In contrast, abstract materials eliminate extraneous
perceptual properties, represent structure efficiently, and are more arbitrarily linked to their
Educ Psychol Rev (2014) 26:925
DOI 10.1007/s10648-014-9249-3
E. R. Fyfe (*)
Department of Psychology and Human Development, Vanderbilt University, 230 Appleton Place,
Peabody #552, Nashville, TN 37203, USA
N. M. McNeil
Department of Psychology, University of Notre Dame, Notre Dame, IN, USA
J. Y. Son
Psychology Department, California State University, Los Angeles, Los Angeles, CA, USA
R. L. Goldstone
Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN, USA
Author's personal copy
referents. Although the concreteness of materials varies on a continuum, researchers often refer
only to the extremes, pitting concrete and abstract materials against each other.
Here, we argue that concreteness fading is a promising instructional technique that moves
beyond the concrete versus abstract debate and exploits the advantages and minimizes the
disadvantages of both. Concreteness fading refers specifically to a three-step progression by
which the concrete, physical instantiation of a concept becomes increasingly abstract over
time. This fading technique offers a set of unique advantages that surpass the benefits of
concrete or abstract materials considered in isolation. In this review, we focus on the theoretical
benefits of this concreteness fading technique, discuss supporting evidence in mathematics and
science instruction, and lay out unanswered questions and directions for expansion.
Concrete Versus Abstract Materials
Concrete materials, which include physical, virtual, and pictorial objects, are widely used in
Western classrooms (Bryan et al. 2007), and this practice has support in both psychology and
education (e.g., Bruner 1966; Piaget 1970). There are at least four potential benefits to using
concrete materials. First, they provide a practical context that can activate real-world knowl-
edge during learning (Schliemann and Carraher 2002). Second, they can induce physical or
imagined action, which has been shown to enhance memory and understanding (Glenberg
et al. 2004). Third, they enable learners to construct their own knowledge of abstract concepts
(Brown et al. 2009). Fourth, they recruit brain regions associated with perceptual processing,
and it is estimated that 2540 % of the human cortex is dedicated to visual information
processing (Evans-Martin 2005). Despite these benefits, there are several reasons to caution
against the use of concrete materials during learning. Specifically, they often contain extrane-
ous perceptual details, which can distract the learner from relevant information (e.g., Belenky
and Schalk 2014; Kaminski et al. 2008), draw attention to themselves rather than their
referents (e.g., Uttal et al. 1997), and constrain transfer of knowledge to novel problems
(e.g., Goldstone and Sakamoto 2003; Sloutsky et al. 2005).
A number of researchers recommend avoiding concrete materials in favor of abstract
materials, which eliminate extraneous perceptual details. Abstract materials offer increased
portability and generalizability to multiple contexts (Kaminski et al. 2009; Son et al. 2008).
They also focus learnersattention on structure and representational aspects, rather than on
surface features (Kaminski et al. 2009; Uttal et al. 2009). However, abstract materials are not
without shortcomings. For example, solving problems in abstract form often leads to ineffi-
cient solution strategies (Koedinger and Nathan 2004), inflexible application of learned
procedures (McNeil and Alibali 2005), and illogical errors (Carraher and Schliemann 1985;
Stigler et al. 2010). In general, abstract materials run the risk of leading learners to manipulate
meaningless symbols without conceptual understanding (Nathan 2012).
Concreteness Fading
Given that both concrete and abstract materials have advantages and disadvantages, we
propose a solution that combines their advantages and mitigates their disadvantages.
Specifically, we argue for an approach that begins with concrete materials and gradually and
explicitly fades toward more abstract ones. This concreteness fading technique exploits the
continuum from concreteness to abstractness and allows learners to initially benefit from the
grounded, concrete context while still encouraging them to generalize beyond it.
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Concreteness fading was originally recommended by Bruner (1966). He proposed that new
concepts and procedures should be presented in three progressive forms: (1) an enactive form,
which is a physical, concrete model of the concept; (2) an iconic form, which is a graphic or
pictorial model; and finally (3) a symbolic form, which is an abstract model of the concept. For
example, in mathematics, the quantity twocould first be represented by two physical apples,
next by a picture of two dots representing those apples, and finally by the Arabic numeral 2.
The idea is to start with a concrete, recognizable form and gradually strip away irrelevant
details to end with the most economic, abstract form. We use the term concreteness fading to
refer to the three-step progression by which the physical instantiation of a concept becomes
increasingly abstract over time. Since Bruners time, several researchers have adopted similar
approaches to the concrete versus abstract debate, advocating the use of concrete materials that
are eventually decontextualized or faded to more abstract materials (Goldstone and Son 2005;
Gravemeijer 2002; Lehrer and Schauble 2002;Lesh1979).
Concreteness fading offers a unique solution to linking multiple instantiations of a concept
that distinguishes it from alternative approaches to the concrete versus abstract debate. For
example, it is widely assumed that providing multiple external representations will benefit
learners relative to providing a single representation in isolation (e.g., Ainsworth 1999;Gick
and Holyoak 1983). Although some studies have found support for multiple representations,
others have found limited benefits (see Ainsworth 2006). For example, when multiple
representations are presented simultaneously, novices often experience cognitive overload
due to the burden on limited cognitive resources (e.g., Chandler and Sweller 1992). Further,
learners may struggle to understand how the representations are related to one another and fail
to extract key concepts (e.g., Ainsworth et al. 2002). Thus, learners need support in relating or
integrating the different representations (e.g., Berthold and Renkl 2009;Schwonkeetal.
2009). Concreteness fading involves the presentation of multiple examples but overcomes
the limitations of simply presenting them simultaneously by varying the concreteness in a
specific progression. That is, the examples are clearly aligned on the dimension of concrete-
ness, and there is a specified directionality in that the concrete examples are the referents for
the iconic and abstract examples.
A related approach to presenting multiple representations is to ask learners to explicitly
compare them (e.g., Kotovsky and Gentner 1996). Comparison involves the simultaneous
presentation of two examples and the mapping of elements across examples to highlight
similarities or differences. For example, in one study, preschoolers were tested on their ability
to detect patterns (i.e., ABA) by asking them to match one pattern (e.g., small square, large
square, small square) with one of two other patterns (e.g., diamond, circle, diamond versus
circle, diamond, diamond; Son et al. 2011). Children who were trained to align and compare
the elements in each set were better able to detect and generalize the pattern than children who
did not make this direct comparison. Although concreteness fading may support spontaneous
comparison, the focus is not on mapping similarities and differences, but on linking the
examples as mutual referents. Further, concreteness fading does not require that two repre-
sentations be held in mind simultaneously; it only requires the presentation of one example at a
time. Concreteness fading is a unique solution to linking concrete and abstract examples that
provides advantages over a general presentation and comparison of multiple representations.
Theoretical Benefits of Concreteness Fading
On the surface, the benefits of concrete and abstract materials seem to be in opposition. It
appears we have to choose grounded, contextualized knowledge or portable, abstract
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knowledge. However, we argue that both goals can be achieved simultaneously. Specifically,
we suggest that concreteness fading allows concepts to be both grounded in easily understood
concrete contexts and also generalized in a manner that promotes transfer. In this way,
concreteness fading offers the advantages of concrete and abstract materials considered
separately, the advantages of presenting concrete and abstract materials together, and finally,
the advantages of presenting them together in a specific gradual sequence from concrete to
abstract. Here, we explicate the unique theoretical benefits of this progressive fading sequence
for both learning and transfer. See Fig. 1for a schematic theoretical model of concreteness
fading and its potential benefits.
Benefits of concreteness fading emerge because it starts with a concrete format and only
later explicitly links to the more difficult abstract symbols. Concrete materials are advanta-
geous initially because they allow the concept to be grounded in familiar, meaningful scenarios
(Baranes et al. 1989;Carraheretal.1985). The presentation of this initial concrete stage gives
rise to at least three specific advantages.
First, it helps learners interpret ambiguous abstract representations in terms of well-
understood concrete objects (Goldstone and Son 2005; Son et al. 2012). Abstract symbols
are devoid of context and often difficult to interpret. This can lead to the manipulation of
meaningless symbols without understanding. However, people interpret ambiguous objects in
terms of unambiguous familiar objects (e.g., Medin et al. 1993). For example, when an
ambiguous man-rat drawing is preceded by a drawing of a man, it is spontaneously interpreted
as a man. But when it is preceded by a drawing of a rat, it is interpreted as a rat (Leeper 1935).
Thus, if the concrete materials precede the abstract materials, the learner can successfully
interpret the ambiguous abstract materials in terms of the already understood concrete context.
This process may underlie childrens improved performance on symbolic equations (e.g., 5+
2= 4 +__) when they are preceded by equations constructed from concrete manipulatives
Fig. 1 Theoretical model of concreteness fading
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(Sherman and Bisanz 2009). Children can interpret arbitrary symbols, such as the equal sign,
in terms of familiar ideas, such as balancing or sharing. If teachers present materials in this
fading sequence, learners can link the concrete and abstract materials as mutual referents, thus
mitigating the disadvantage of ambiguity in the abstract materials.
Second, the initial concrete stage takes advantage of embodied cognition by giving learners
experience with perceptual and physical processes that are constrained to give rise to proper
inferences. According to an embodied theory of cognition, high-level cognitive processes (e.g.,
language, mathematics) stem from action and perception (Barsalou 2003; Lakoff and Nunez
2000). Thus, comprehension of abstract symbols requires mapping those symbols onto bodily
experiences or representations of those experiences. For example, having children act out a
sentence with toys helps them connect words (i.e., abstract symbols) to particular objects and
actions and enhances comprehension of the text (e.g., Glenberg et al. 2004). The concrete, or
enactive, stage engages learners in physical and perceptual processes thereby providing the
necessary bodily experience. Further, these embodied experiences are linked to the abstract
symbols, thus facilitating the mapping between abstract concepts and perceptual processes.
Third, the concrete stage enables learners to acquire a store of images that can be used when
abstract symbols are forgotten or disconnected from the underlying concept. According to
Bruner (1966), once an understanding of the abstract concept has been achieved, learners do
not give up their imagery, but rather rely on this stock of representations as a means of relating
new problems to those already mastered. The stored images provide learners with an acces-
sible, back-up representation that can be used when the abstract symbols are detached from
their referent. For example, elementary school students who initially learned about the equal
sign in the context of balancing a seesaw were presented with challenging open-ended
sentences later in the school year (e.g., 5+6=__ +2). Students who were in doubt about their
solutions were encouraged to rely on their knowledge of a seesaw and check if their solution
made sense (Mann 2004). Concreteness fading not only encourages teachers to focus on both
concrete and abstract understanding but also provides learners with abstractions explicitly
linked to a stock of images. Each of these images can act as a salient retrieval cue for
connecting an otherwise reclusive abstraction.
The concrete, enactive stage provides numerous benefits for the learning of a concept.
Equally important, however, are the two subsequent stages: the iconic and symbolic stages of
Bruners(1966) sequence. If given only concrete materials (or concrete and abstract materials
in isolation), it is likely that the learners knowledge would remain too tied to the concrete
context and would not transfer to dissimilar situations. Indeed, Kaminski et al. (2008)found
that students who learned a math concept only through an abstract example outperformed
those who learned through a concrete example followed by an abstract example. However, the
examples in this study were presented one after another in isolation, rather than in a fading
progression. It is through the gradual and explicit fading that learners are able to strip the
concept of extraneous, concrete properties and grasp the more portable, abstract properties
(Bruner 1966).
Stripping away extraneous features facilitates transfer by highlighting structural information
and reducing the salience of concrete information. Indeed, the problem of transferring knowl-
edge to novel contexts has often been attributed to learnersinability to detect the underlying
structure of the problem at hand (e.g., Ross 1987). Further, drawing attention to structure is one
of the central tenets of several theories of learning, including the preparation for future learning
account (e.g., Schwartz et al. 2011) and schema theory (e.g., Chi et al. 1981). By slowly
decontextualizing the concrete materials, concreteness fading hones in on the pertinent,
structural features and results in a faded representation that can be a useful stand-in for a
variety of specific contexts. This mitigates the disadvantageous context specificity of concrete
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materials and allows learners to more easily extend learned material to new and superficially
dissimilar spheres of application.
In sum, concreteness fading can benefit learning and transfer by starting with a concrete,
recognizable format then gradually and explicitly removing context-specific elements to
generate a more abstract representation. Resulting knowledge is not only grounded and
meaningful but also abstract and portable. The fading process allows learners to more
explicitly link the concrete materials and abstract symbols as mutual referents. Importantly,
the result is a rich, grounded understanding of the underlying concept that is connected to
conventional, abstract symbols.
Support for Concreteness Fading
Although empirical support for concreteness fading is increasing, most evidence in favor of
this method is admittedly indirect and restricted to the domains of mathematics and science.
Various features of concreteness fading have been found to be helpful in learning contexts. For
example, concreteness fading involves presenting the same concept in three different instan-
tiations and linking their common structures. Several studies have found benefits of presenting
multiple instantiations of a concept, as it affords the possibility of extracting commonalities
(e.g., Gick and Holyoak 1983). In addition, concreteness fading reduces representational
support as domain knowledge increases, which may have analogous benefits to fading out
instructional support more generally (e.g., Kalyuga 2007; Wecker and Fischer 2011). Learners
with low domain knowledge tend to benefit from methods with heavy instructional support,
whereas higher-knowledge learners benefit from methods with little to no instructional support
(see Kalyuga 2007). Further, concreteness fading maximizes smooth transitions from concrete
to abstract as opposed to abrupt shifts. Recent studies suggest smooth transitions adapted to
learners knowledge level are particularly beneficial (e.g., Renkl et al. 2002).
More relevant indirect evidence comes from a variety of studies demonstrating benefits of a
concrete-to-abstract sequence. For example, Koedinger and Anderson (1998) found high
learning gains from a cognitive tutor that presented concrete, algebra word problems first,
followed by an intermediate step, followed by the symbolic expression. In more recent
research, worked examples accompanied by animations that faded from concrete to abstract
were better at facilitating transfer performance than were worked examples that were not
accompanied by any animations (Scheiter et al. 2010). However, it remains unclear whether
the benefits were due to the fading or to the mere presence of animations. Similarly, physics
simulation environments that contained concrete images that transitioned into more abstract
images promoted better transfer than simulations that contained only concrete images
(Jaakkola et al. 2009). Further, transitioning from physical experimentation with actual
concrete models to a simulation environment promoted better conceptual understanding than
physical experimentation alone (Zacharia 2007). Unfortunately, without an abstract or
simulation-only condition, it is difficult to determine whether concrete models should have
been included at all.
There has been some direct, experimental evidence in favor of concreteness fading, but
more is needed. Goldstone and Son (2005) had undergraduates learn a scientific principle (i.e.,
competitive specialization) via computer simulations that varied in perceptual richness. The
concrete elements appeared as colorful ants foraging for fruit, whereas the abstract elements
appeared as black dots and green shapes. Students exhibited higher transfer when the elements
in the display switched from concrete to abstract than when the elements remained concrete,
remained abstract, or switched from abstract to concrete. However, the simulations only used
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two steps in the fading progression as opposed to the three steps recommended by Bruner
(1966). Additionally, the concrete and abstract elements were both fairly concrete as they both
represented ants and foodthe abstract elements were simply stripped of perceptual detail.
Also, the abstract elements were not arbitrarily linked to their referents, as most abstract
symbols are.
Braithwaite and Goldstone (2013) also examined a two-step fading progression, but in the
mathematical domain of combinatorics. The researchers used outcome listing (i.e., listing all
the possible outcomes) and numerical calculation (i.e., using a formula to calculate the possible
outcomes) as examples of concrete and abstract materials, respectively. Students were assigned
to one of four conditions: concrete (outcome listing only), abstract (numerical calculation
only), concrete first (listing followed by calculation), or abstract first (calculation followed by
listing). Concrete first led to higher transfer than concrete or abstract first and to similar transfer
as abstract. These results suggest that when both concrete and abstract materials are employed,
concrete materials should precede the abstract. However, the combination was not necessarily
more effective than using abstract materials alone. Again, only two-step fading was used as
opposed to the three-step fading. Also, in this study, the concrete and abstract materials were
both fairly abstract (i.e., graphic notation written on paper). Perhaps including the initial,
enactive stage would have enhanced the effectiveness of this form of concreteness fading.
There is some direct empirical support for the full three-step concreteness fading progres-
sion. For example, McNeil and Fyfe (2012) had undergraduates learn modular arithmetic in
one of three conditions: concrete, in which the concept was presented using meaningful
images; abstract, in which the concept was presented using arbitrary, abstract symbols; or
concreteness fading, in which the concept was presented using meaningful images that were
faded into abstract symbols. The concreteness fading progression included an intermediate,
fadedstage that retained the identifiable correspondence between the form and referent, but
was stripped of all extraneous perceptual detail. This allowed the concrete and abstract
elements to be explicitly linked as mutual referents. Students completed a transfer test
immediately, 1 week later, and 3 weeks later. Importantly, students in the concreteness fading
condition exhibited the best transfer performance at all three time points.
Two additional studies provide direct support for concreteness fading but are currently
unpublished. In the first experiment, children with low knowledge of math equivalence
received instruction on math equivalence problems (e.g., 3+ 4=3+ __) in one of four condi-
tions (Fyfe and McNeil 2009). In the concrete condition, problems were presented using
physical, concrete objects (e.g., toy bears on a balance scale). In the abstract condition,
problems were presented as symbolic equations on paper. In the concreteness fading condition,
problems were presented in three progressive formats: first with the physical objects, next with
fading worksheets that used pictures to represent the objects, and finally with symbolic
equations (see Fig. 2). In the concreteness introduction condition, problems were presented
in the reverse progression such that it began with abstract materials and the concrete materials
were gradually introduced. Following instruction, children solved five transfer problems.
Children in the concreteness fading condition solved more transfer problems correctly than
children in the other conditions.
A follow-up experiment tested the generalizability of the results by extending the target
population to children who had high prior knowledge in the domain (Fyfe and McNeil 2013).
In this study, high-knowledge children were taught an advanced procedure for solving
equivalence problems that none of the children had used on a screening measure.
Specifically, they were taught to cancelequivalent addends on opposite sides of the equal
sign (e.g., canceling the 3s in 3+4=3 +__). Children were assigned to the same four conditions
described above. Following instruction, children solved transfer problems, including one
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challengeproblem (258+ 29+173= 29 + __+258) designed to elicit the cancel strategy.
Children in all conditions had similar overall transfer performance, but the challenge problem
revealed meaningful differences. Children in the concreteness fading condition were more
likely than children in the other conditions to use the cancel strategy and to solve the challenge
problem correctly.
Importantly, in both of these experiments, children in the concreteness introduction condi-
tion did not perform as well as children in the concreteness fading condition, ruling out a
number of alternative explanations for why concreteness fading is effective. For example,
children in both the concreteness fading and concreteness introduction conditions solved
problems in three formats (i.e., concrete, worksheet, abstract), manipulated physical objects,
and were encouraged to map concrete and abstract examples. Thus, these results suggest it was
the specific progression from concrete to abstract that promoted childrens mathematical
Although evidence in favor of concreteness fading continues to increase, it is worth
acknowledging situations in which a concrete-to-abstract sequence did not optimize learning
outcomes. As mentioned previously, both Kaminski et al. (2008) and Braithwaite and
Goldstone (2013) found that the concreteabstract combination was not more effective than
using abstract materials alone. In another study (Tapola et al. 2013), concrete materials alone
faired better than the combination: A simulation on electricity in which the elements remained
concrete promoted higher learning than one in which the elements switched from concrete to
abstract. Finally, a recent experiment found benefits for a reverse abstract-to-concrete se-
quence. Middle school students who studied electrical circuits using abstract diagrams that
transitioned to concrete illustrations exhibited higher transfer than those who studied concrete
illustrations that transitioned to abstract diagrams (Johnson et al. 2014). There are several
reasons why concreteness fading may not have optimized outcomes in these situations. First,
concreteness fading was not ideally implemented. In all of these studies, only two stages were
employed as opposed to the three stages recommended by Bruner (1966), the elements did not
span the full concreteness continuum, and the stages were rarely linked in an explicit manner.
Second, learner characteristics may have played a role. Concreteness fading assumes that
learners can easily comprehend the concrete materials, are unfamiliar with the abstract
materials, and have a certain level of readinessto learn. If learners lack sufficient prior
Fig. 2 Progression of materials used during instruction in the concreteness fading condition in Fyfe and McNeil
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knowledge to understand the concrete materials, then a fading progression may be too
premature. On the other hand, if learners already have a sophisticated understanding of the
abstract material, then they may benefit from working directly with the symbolic
Although these examples suggest that there may be boundary conditions to the efficacy of
concreteness fading, the number of studies supporting the technique continues to grow. As
shown here, there is a variety of support in favor of concreteness fading, ranging from relevant
existing literatures to direct, experimental evidence. Clearly, research suggests that concrete-
ness fading can be an effective instructional technique, though the number of studies that
instantiate the complete three-step transition is admittedly small. More research is needed to
confirm the effectiveness of concreteness fading across tasks, settings, and populations.
Applications of Concreteness Fading
Despite the need for more direct support, concreteness fading has already been incorporated
into several mathematics curricula and programs. For example, MathVIDS is a supplementary
resource for teachers with struggling mathematics learners (Allsopp et al. 2006). One of the
core recommended strategies is to teach through CRA, a concreterepresentationalabstract
sequence. With CRA, each math concept is first modeled with concrete materials, then with
pictures that represent the concrete materials, and finally with abstract numbers and symbols.
The website provides example materials for each stage of the CRA sequence as well as how to
use them effectively. The CRA sequence is primarily studied and recommended by special
education researchers (e.g., Butler et al. 2003;Petersonetal.1988), though some suggest CRA
may be a beneficial approach for all learners (Berkas and Pattison 2007).
At least four widely used curricula also use a form of concreteness fading in their lessons.
Singapore Math is a teaching method and curriculum based on the national math practices of
Singapore (Thomas and Thomas 2011). The overarching goal is to facilitate mastery of each
basic concept before introducing new material. This is accomplished by covering fewer topics
in greater depth and by employing a three-step process that progresses from concrete to
pictorial to abstract (Wang-Iverson et al. 2010). As with concreteness fading, this sequence
starts with physical, hands-on activity; moves to pictorial diagrams or models; and eventually
transitions to conventional abstract symbols. Singapore Math continues to grow in popularity
and is used by an increasing number of schools in the USA. However, despite reports of high
math achievement in Singapore, there is little evidence on the effectiveness of the Singapore
Math curricula as it is used in the USA (WWC 2009).
Mathematics in Context is a comprehensive middle school curriculum (Romberg and
Shafer 2004) that supports mathematics knowledge via progressive formalization
(Freudenthal 1991). The goal is to use studentsprior, informal representations to support
the development of more formal mathematics. In practice, this means the mathematical model
is first instantiated in a context-specific, concrete situation and only later generalized over
situations as a more abstract model. This instructional approach is based heavily on
Gravemeijers(2002)bottom-upprogression, in which he recommends starting from situ-
ational mathematics and progressing to more formal mathematics. Longitudinal evidence
suggests that Mathematics in Context leads to achievement gains (Romberg et al. 2005);
however, its overall effectiveness remains unclear due to a lack of studies meeting sufficient
evidence standards (WWC 2008).
Similarly, Everyday Mathematics is a research-based and field-tested elementary school
curriculum that emphasizes the use of concrete, real-life examples as introductions to key
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concepts (Carroll and Issacs 2003). In this way, formal concepts and procedures are not
presented in isolation, but are linked to informal, concrete situations. The goal is to help
learners think carefully about mathematics by relating the abstract symbols to actions and
referents that are familiar to young students. The curriculum also emphasizes facility with
multiple representations, especially the ability to translate among representations, which is also
a component of concreteness fading. Everyday Mathematics was found to have potentially
positive effects on math achievement for elementary school students (WWC 2010).
Finally, Building Blocks is a supplemental mathematics curricula designed to develop
preschool childrens early math knowledge through the use of concrete manipulatives and
print material (Clements and Sarama 2007). It embeds math learning in daily activities with the
goal of relating childrens informal math knowledge to more formal mathematics concepts.
The general goal is to build upon childrens early, situational experiences with mathematics to
help them develop a strong foundation for more abstract mathematical thinking and reasoning.
Although the use of explicit and gradual fading is not necessarily a formal component of the
Building Blocks curricula, it is naturally built in as concepts are initially grounded in concrete
experience and eventually connected to more conventional mathematics symbols and num-
bers. Importantly, Building Blocks has been evaluated by the What Works Clearinghouse and
was found to have a positive effect on mathematics knowledge (WWC 2007).
Although most of these approaches are intended for teachers in classrooms, concreteness
fading is also an important design component in technology-driven, individualized instruction.
Tablet-based or web-based interventions, though newer and not as fully evaluated, also
progress from concrete to abstract. Initial results of a randomized field trial show that one of
these interactive programs, Spatiotemporal (ST)Math, a game-based intervention for touch
tablets, shows signs of early promise (Rutherford et al. 2010). In ST Math, elementary students
interact with simple, dynamic representations of visuospatial puzzles. These exercises increase
in difficulty and can only be solved when a pattern or concept has been extrapolated. Only
after students understand the concept visually are they introduced to corresponding symbolic
representations and procedures. Examples of this sequence from pure visual to more symbolic
can be viewed at the MIND Research website,
A final example of interactive instruction that implements the three-stage fading sequence is
the DragonBox system for teaching algebra, illustrated in Fig. 3. Learners begin playing a
game with monstersthat follow certain rules (i.e., rules of algebra): (a) day and night
monsters annihilate each other in a vortex, (b) monsters above and below a line can be
converted into a special monster that can be eliminated when positioned next to any other
Enactive Iconic S
Fig. 3 The enactive, iconic, and symbolic stages of concreteness fading as implemented by the DragonBox
tutoring system for teaching algebra
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monster, and (c) a monster can be added to one side of a screen as long as it is also added to the
other side. The goal of the game is to isolate a special star boxon one side of the screen.
Passing through an iconic stage, in which monsters are replaced with simple dice patterns,
learners finally reach a stage in which algebraic symbols are used. The students learn that the
rulesthey had learned before correspond to algebraic axioms (e.g., A+[A]=0, the additive
inverse property). Learners implicitly internalize the constraints of algebraic transformations
during the concrete, enactive stage, and these constraints are progressively made more explicit
and formalized.
Questions for Future Research
In general, concreteness fading has been incorporated into a variety of programs and curricula,
all of which have shown potential for success. This is promising for the practical application of
concreteness fading in real-world learning contexts, though admittedly more empirical support
is needed. Once the efficacy of concreteness fading is established, there are several potential
directions for future research. These include, but are not limited to, (1) explicating the
underlying mechanisms, (2) optimizing the outcomes, and (3) specifying key moderators.
Here, we briefly outline and discuss several issues related to these future directions.
What underlying mechanisms explain the benefits of concreteness fading? There are a
number of possibilities, not all of which are mutually exclusive. One potential mechanism by
which concreteness fading causes better learning outcomes is by supporting a persistent
interpretation of the materials. The fading process may allow learners to maintain their already
understood, concrete representation of a concept even as the materials become more abstract.
Abstract symbols are arbitrarily linked to their referents and often difficult to interpret.
However, people spontaneously interpret ambiguous elements in terms of things with which
they are familiar (e.g., Leeper 1935). With concreteness fading, a familiar concrete scenario
precedes the abstract scenario. One possibility is that learners maintain this concrete interpre-
tation of the abstract symbols, even after the concrete image is no longer present (Bruner
1966). That is, the abstract symbols carry meaning because of the cognitive availability of the
concrete context.
Another potential mechanism is by encouraging structural recognition and alignment (e.g.,
Kaminski et al. 2009). The fading method strips away extraneous, superficial details thereby
reducing opportunities for these details to compete with, or be confused for, the deeper relational
structure. Previous research has shown that when situations are presented only in a perceptually
rich manner, the structure is not likely to be noticed (e.g., Gentner and Medina 1998;Markman
and Gentner 1993). For example, when perceptually rich objects are used in a relational task,
children often notice the concreteness of the objects (e.g., both sets contain a toy car), rather than
the structural relation (e.g., both sets have objects that increase in size). Concreteness fading can
overcome these limitations by narrowing in on the relational structure, rather than surface
elements. Also, by highlighting the structure in multiple examples that vary in concreteness,
the fading progression likely increases the learners ability to recognize that structure in diverse
contexts. Indeed, several studies have shown that providing at least two examples of a concept,
rather than only one, greatly improves learnersability to identify a novel instance of the concept
(e.g., Christie and Gentner 2010;Grahametal.2010; Kurtz et al. 2013).
Concreteness fading may also improve learning by grounding a concept in an intuitive
manner (e.g., Baranes et al. 1989). This allows learners to gain confidence in easily graspable
ideas before that grounding is systematically removed. This mechanism may also explain the
success of a related instructional technique, which we refer to as prototype fading. A common
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approach is to present a simple, prototypical example of a concept and introduce increasingly
distant examples. For instance, a math teacher might begin a unit on multiplication by first
describing repeated addition, a prototypical example that works well for multiplying positive
integers. Eventually, studentsconcept of multiplication extends further and repeated addition
falls by the wayside as the basis for understanding multiplication (Devlin 2011). Similarly, a
physics teacher might begin a unit on waves by first describing water waves, then sound
waves, and finally light waves (Hofstadter and Sander 2013). Analogous to fading out concrete
aspects of learning materials, prototype fading works by grounding a concept in an intuitive
manner and then systematically removing aspects of that grounding to promote an under-
standing that is transportable to previously unimagined areas. The most striking cases of
prototype fading are powered by the development of symbol systems that are built to express
general patterns. One of the best ways to avoid being limited by concrete, superficial
appearances is to translate situations into formal symbol systems that are unfettered by the
constraints of enactive or iconic representations. Science as a whole historically progresses by
the development of such formal understandings (Quine 1977), as algebraic notation, DNA, and
the periodic table all attest.
Although the cognitive sciences offer several viable mechanisms, errorless learning might
also provide potential mechanisms underlying the success of concreteness fading. Errorless
learning was originally investigated in animal learning; pigeons were able to learn difficult
discriminations by first presenting them with easily discriminable stimuli superimposed on the
difficult stimuli (Terrace 1963). Although pigeons would first respond based on the easily
discriminable stimuli, gradually those would be faded out so that only the difficult stimuli
remained. Learning the more difficult information is incidental, but still powerful. Errorless
learning may also explain why native Japanese speakers learn the distinction between Rand
Lsounds most effectively when they are first trained on an exaggerated /r//l/ discrimination
that accentuates the differences between the two sounds and then slowly progressed to a more
typical /r//l/ discrimination (McClelland et al. 2002). The exaggeration helps learners focus on
the relevant discriminating aspects, and then the gradual reduction of the exaggeration gives
learners experience with naturalistic speech. Concreteness fading may be driven in a similar
manner. Learners with no prior intention of learning abstractions may find themselves acquiring
this information simply by responding to easily grasped concrete materials, which perfectly co-
occur with relevant abstract characteristics. With concreteness fading, these concrete scaffolds
eventually disappear, allowing learners to lean more on abstract information.
In addition to explicating the mechanisms by which concreteness fading works, future
research should also work to optimize the technique. One way to optimize the technique is to
test whether all types of concreteness are beneficial during the enactive stage. So far we have
been positing a single concrete-to-abstract continuum, but it is likely more productive to
distinguish several such continua. For example, imagine teaching the distance formula by
describing a car ride from Chicago to New York. The teacher can vary perceptual concreteness
(e.g., colorful car model versus line drawing of a car), narrative concreteness (e.g., provide a
contextualized story versus provide only the relevant variables), familiarity concreteness (e.g.,
story involving the student versus story involving random person), or representational con-
creteness (e.g., sketch of the trip versus line graph of distance traveled). These different types
of concreteness are unlikely to be cognitively equivalent (Son and Goldstone 2009), and they
may interact to impact learning outcomes. For example, preschoolers exhibited better counting
when they used perceptually rich, but unfamiliar objects than when they used objects that were
perceptually rich and familiar, or perceptually bland (Petersen and McNeil 2013). Future work
should continue investigating different types of abstractness and whether they can be com-
bined to facilitate learning and transfer.
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A related issue for optimizing concreteness fading is to manipulate components of the
sequence. This would allow researchers to determine if it is possible, for example, to bypass
the enactive stage. As currently conceptualized, concreteness fading ought to begin with actual
physical objects, which provide perceptual and physical experiences. At least one study
suggests that the enactive stage is necessary. Butler et al. (2003) compared a full CRA
sequence to an RA sequence for teaching students with math disabilities about fractions.
The CRA group used concrete manipulatives for the first few lessons, while the RA group
used representational drawings. Students in the CRA group exhibited higher posttest achieve-
ment than students in the RA group. However, it is not clear if physical manipulation is always
feasible or even necessary. For example, although McNeil and Fyfe (2012) employed a three-
step fading progression, all three representations were graphic in nature and did not afford
physical manipulation. Indeed, perceptually grounded computer simulations of scientific
phenomena are often as effective, if not more, than enactive, physical instantiations of the
phenomena (de Jong et al. 2013). One key difference between these studies is the knowledge
level of the participants. Indeed, Bruner (1966) suggested that the enactive stage may not be
necessary for learners with high prior knowledge.
This point highlights that individual differences play a large role in the learning process,
often determining whether an instructional technique will be effective or not (Cronbach and
Snow 1977). Thus, it will be important to determine which factors serve as potential moder-
ators of concreteness fading. A large body of research indicates that learnersprior domain
knowledge may be particularly important (see Kalyuga 2007). A relevant study by Goldstone
and Sakamoto (2003) suggests that learners with low knowledge are more adversely affected
by distracting concrete materials relative to learners with high knowledge. However, an
experiment by Homer and Plass (2009) offers different conclusions. They found that concrete
visualizations in a science tutorial were more effective for students with low knowledge than
for students with high knowledge. These studies suggest that interactions between concrete-
ness and individual differences may be complicated. Future work is needed to tease apart the
distinctions between these studies and to consider the relationship between concreteness fading
and knowledge levels.
Another potential moderator to consider is the level of direct instructional guidance.
Although many influential learning theorists advocate discovery learning (e.g., Bruner 1961;
Piaget 1973), recent evidence suggests that including some explicit instructional elements may
be more beneficial than pure discovery alone (e.g., Alfieri et al. 2011;Mayer2004). However,
the optimal amount and type of instructional guidance remains unclear and likely depends on
the task, learner, and instructional technique. One relevant study suggests the level of direct
instruction influences learning from concrete materials. Kaminski and Sloutsky (2009) exam-
ined kindergartenersability to recognize common proportions across different instantiations.
When explicit training and examples were provided, children who learned with either concrete
or abstract materials successfully transferred their knowledge. However, when no explicit
instruction was provided, only children who learned with abstract materials exhibited transfer.
Thus, one possibility is that concreteness fading is more effective when there is a high level of
direct instructional guidance. Another possibility is that concreteness fading is equivalent to
direct instruction and may circumvent the need for more explicit training and instruction.
In sum, concreteness fading represents a promising instructional technique that moves beyond
the concrete versus abstract debate and exploits advantages of multiple examples across the
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concreteness continuum. It refers specifically to the three-step progression by which the
physical instantiation of a concept becomes increasingly abstract over time. This fading
technique offers unique advantages that surpass the benefits of concrete or abstract materials
considered in isolation. Additionally, it has some support in the research literature and is
widely used in existing mathematics curricula, though more direct experimental evidence is
Given the widespread use and endorsement of concrete materials by researchers and
teachers alike, it is pertinent that we find optimal ways to use these materials to facilitate both
learning and transfer. This will require considering the types of concrete materials to use, when
to use them, and most importantly how to connect them to conventional, abstract symbols.
Indeed, as Brown et al. (2009)note,linking nonsymbolic conceptual understanding to more
abstract, symbolic representations may be one of the most significant challenges teachers face
today(p. 162). We propose concreteness fading as a solution to that challenge, as it offers the
best of both concrete and abstract instruction.
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... It thus seems realistic to identify, for any type of problem, which problem statement as well as which sequence of training problems might be the most beneficial to help learners abstract a representation as close to the deep structure as possible. A congruence fading process akin to concreteness fading (Fyfe, McNeil, Son, & Goldstone, 2014) could thus help learners abstract the deep structure of the problems by resorting to increasingly incongruent examples. An interesting venue to capitalize on such effects would be to alternatively present problems attached to different world semantics congruent with different representations, in order to help learners switch from an initial representation to another one, more efficient with regard to the resolution of the problem. ...
... We mentioned earlier how the use of increasingly semantically incongruent examples may complement a learning strategy based on concreteness fading (Fyfe et al., 2014), to guide learners from a concrete grasp of a problem to a more abstract understanding of its solution principle. It may be possible to develop a similar strategy in arithmetic learning, by progressively varying the semantic congruence between the concrete situations presented to the learners and the arithmetic notions to be taught. ...
... This is especially problematic since mathematics education does not usually control for content effects Lee, DeWolf, Bassok, & Holyoak, 2016), which is partly due to mathematics being primarily considered the realm of abstraction (Son & Goldstone, 2009;Goldstone & Sakamoto, 2003;Day, Motz & Goldstone, 2015). Similarly to how concreteness fading is proposed as a way to improve transfer by resorting to increasingly abstract examples (Fyfe, McNeil, Son, & Goldstone, 2014), it may be a promising route to develop a semantic congruence fading process using increasingly incongruent examples. In the case of the problems used in the current study, starting with teaching the 1-step algorithm on ordinal problems, then moving to hybrid problems and then to concrete problems may be a way to help learners acquire a better understanding of this algorithm, and consequently learn to use it in any situation, regardless of the semantics conveyed by the problem statement. ...
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With its context-independent rules valid in any setting, mathematics is considered to be the champion of abstraction, and for a long time human mathematical reasoning was thought to follow nothing but the laws of logic. However, the idea that mathematics is grounded in nature has gained traction over the past decades, and the context-independency of mathematical reasoning has come to be questioned. The thesis we defend concerns the role played by general, non-mathematical knowledge on individuals' understanding of numerical situations. We propose that what we count has a crucial impact on how we count, in the sense that human's representation of numerical information is dependent on the semantic context in which it is embedded. More specifically, we argue that general, non-mathematical knowledge about the entities described in a mathematical word problem can shape its interpretation and foster one of two representations: either a cardinal encoding, or an ordinal encoding. After introducing a new framework of arithmetic word problem solving accounting for the interactions between mathematical knowledge and world knowledge in the encoding, recoding and solving of arithmetic word problems, we present a series of 16 experiments assessing how world knowledge about specific quantities can promote one of two problem representations. Using isomorphic arithmetic word problems involving either cardinal quantities (weights, prices, collections) or ordinal quantities (durations, heights, number of floors), we investigate the pervasiveness of the cardinal-ordinal distinction in a wide range of activities, including problem categorization, problem comparison, algorithm selection, problem solvability assessment, problem recall, sentence recognition, drawing production and transfer of strategies. We gather data using behavioral measures (success rates, algorithm use, response times) as well as eye tracking (fixation times, saccades, pupil dilation), to show that the difference between problems meant to foster either a cardinal or an ordinal encoding has a far-reaching influence on participants from diverse populations (N = 2180), ranging from 2nd graders and 5th graders to lay adults, expert mathematicians and math teachers. We discuss the general educational implications of these effects of semantic (in)congruence, and we propose new directions for future research on this crucial issue. We conclude that these findings illustrate the extent to which human reasoning is constrained by the content on which it operates, even in domains where abstraction is praised and trained.
... Studies have generally demonstrated that instruction with manipulatives has better outcomes than instruction without manipulatives (Gürbüz, 2010;Lane 2010;Suh & Moyer, 2007). However, research has also shown that the use of manipulatives does not always produce better learning outcomes (Brown et al., 2009;Fyfe et al., 2014;Stein & Bovalino, 2001). Awareness of this latter fact led Baroody (1989) to write, "Perhaps manipulatives should carry the following warning label: The Secretary of Education [or other ...
... 396). Fyfe et al. (2014) concluded, Given the widespread use and endorsement of concrete materials by researchers and teachers alike, it is pertinent that we find optimal ways to use these materials to facilitate both learning and transfer. This will require considering the types of concrete materials to use, when to use them, and most importantly how to connect them to conventional, abstract symbols. ...
... I use this explanation to address how abstract concepts can be promoted on the basis of students' activity with manipulatives. Although the theories of Piaget and Bruner have been invoked in justifying the use of manipulatives (e.g., Liggett, 2017;Fyfe et al., 2014), learning theories have generally not been used to provide insight into how manipulative use is implicated in the mechanism of conceptual learning. ...
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Elements of the Learning Through Activity (LTA) research program's theoretical framework are used to provide a learning theoretical explanation of the role of manipulatives in the learning of mathematical concepts. In particular, the LTA elaboration of reflective abstraction is used to explain how manipulatives can be used to elicit particular activity that can serve as the basis for intended concepts. The constructs discussed provide a basis for instructional design using manipulatives and for modifying ineffective lessons involving manipulatives. The constructs also serve to explain one of the major shortcomings of many lessons involving manipulatives, students' apparent disconnect between their activity with manipulatives and abstract mathematics. Mental-run tasks, one of the constructs discussed, is explained as a specific antidote to this shortcoming. The article also focuses on how the constructs provide a framework for the design and use of virtual manipulatives.
... Concreteness Fading is a progression by which the physical instantiation of a concept becomes increasingly abstract over time (e.g. Bruner, 1966;Fyfe et al., 2014;Goldstone & Son, 2005). Bruner (1966) originally proposed that new concepts should be presented in three progressive forms: (1) an enactive form, a concrete model of a concept; (2) an iconic form, a graphic or pictorial model; and (3) a symbolic form, an abstract model of the concept. ...
... This comparison process makes it possible for students to learn to adopt a different point of view on the problems they encounter, flexibly categorizing the situation to identify the optimal way to broach and solve it. In this perspective, a sequence of comparisons between increasingly dissimilar problems sharing the same solution could result in a deeper understanding of the notions at hand, similarly to how concreteness fading (Fyfe, McNeil, Son, & Goldstone, 2014) uses examples of increasing abstraction to promote transfer. ...
Conference Paper
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Because of its importance in academic achievement, especially in mathematics, training cognitive flexibility at school is a major issue. The present research investigates the effectiveness of a school-based intervention to improve proportion arithmetic problem solving. The study was conducted with 5th graders of 10 classes from 5 high-priority education schools in the Paris region. Students of the control and experimental groups took part in 8 learning sessions about proportion problem solving. The experimental group's training focused on comparing and flexibly categorizing the problems in the hopes to help students achieve a deeper understanding of proportion problems. Results show that training flexible categorization allowed the experimental group to progress more than the control group, in both categorization and solving tasks. The educational implications of our results are discussed.
... In addition, we may have seen no differences across training because determining how to support representational competence through models, actions and imagined action is not straightforward. For example, Fyfe and colleagues' work (Fyfe et al., 2014;Fyfe et al., 2015) suggests that starting with manipulatives (models) and moving to symbolic representations (called concreteness fading) is particularly helpful. It may be that gesture would work better only after learners have had an opportunity to use models. ...
Many undergraduate chemistry students struggle to understand the concept of stereoisomers, molecules that have the same molecular formula and sequence of bonded atoms but are different in how their atoms are oriented in space. Our goal in this study is to improve stereoisomer instruction by getting participants actively involved in the lesson. Using a pretestinstruction-posttest design, we instructed participants to enact molecule rotation in three ways: (1) by imagining the molecules’ movements, (2) by physically moving models of the molecules, or (3) by gesturing the molecules’ movements. Because gender differences have been found in students’ performance in chemistry (Moss-Racusin et al., 2018), we also disaggregated our effects by gender and examined how men and women responded to each of our 3 types of instruction. Undergraduate students took a pretest on stereoisomers, were randomly assigned to one of the 3 types of instruction in stereoisomers, and then took a posttest. We found that, controlling for pretest performance, both women and men participants made robust improvements after instruction. We end with a discussion of how these findings might inform stereoisomer instruction.
Conference Paper
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Loin d’être une compétence élitiste, la compréhension mathématique est mobilisée dans la vie de chacun pour les calculs de la vie quotidienne, la mesure des longueurs et des quantités, la lecture de graphiques, l’interprétation de toute donnée chiffrée, etc. La nouvelle synthèse du CSEN a pour objectif de favoriser l’appropriation par les élèves de notions essentielles et néanmoins difficiles, que sont les multiplications, les divisions et les fractions, dont une certaine maîtrise est essentielle tant pour les compétences générales évoquées ci-dessus que pour les apprentissages mathématiques ultérieurs.
In this survey paper we focus on mathematics learning in Chinese contexts, as a way to contribute to broader discussions about mathematical learning. We first review the features of Chinese students’ mathematical learning depicted in the literature, followed by a review of student mathematical learning in recent Chinese research journals. This leads to an introduction of the papers on Chinese students’ learning in this issue. For Chinese students’ learning contexts, we discuss four aspects, namely, classroom instruction, teachers’ professional learning, curriculum materials, and learning outside of school. For each context, we review the literature findings on the identified features, introduce emerged practices and most recent policies under the reformed era, and discuss the relevant papers in this special issue. Whenever possible, we connect findings on Chinese students’ learning with the associated contexts and relate these findings in the Chinese contexts to findings in the broader world context. We conclude this survey paper with possible lessons learned from Chinese students’ learning features and from the varied Chinese contexts. In particular, we discuss these aspects from culturally contextualized and semantically decontextualized dimensions, which is expected to facilitate broad international discourse centering on the three questions proposed at the end of this paper.
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Introductory statistics students struggle to understand randomness as a data generating process, and especially its application to the practice of data analysis. Although modern computational techniques for data analysis such as simulation, randomization, and bootstrapping have the potential to make the idea of randomness more concrete, representing such random processes with R code is not as easy for students to understand as is something like a coin-flip, which is both concrete and embodied. In this study, in the context of multimedia learning, we designed and tested the efficacy of an instructional sequence that preceded computational simulations with embodied demonstrations. We investigated the role that embodied hands-on movement might play in facilitating students’ understanding of the shuffle function in R. Our findings showed that students who watched a video of hands shuffling data written on pieces of paper learned more from a subsequent live-coding demonstration of randomization using R than did students only introduced to the concept using R. Although others have found an advantage of students themselves engaging in hands-on activities, this study showed that merely watching someone else engage can benefit learning. Implications for online and remote instruction are discussed.
Im Seminar „Elementare Differentialgeometrie zum Anfassen“ erarbeiten sich Lehramtsstudierende (Gymnasiallehramt Mathematik) durch forschungsähnliches Lernen das fortgeschrittene mathematische Themengebiet der Elementaren Differentialgeometrie, also der Theorie der gekrümmten Kurven und Flächen. Sie arbeiten dazu in Gruppen zunächst mit Hands-on-Materialien und bauen ihre dabei entstandenen Ideen dann zu mathematisch präzisen Herleitungen aus. Dabei erweitern sie ihr fachliches Wissen im Bereich Differentialgeometrie und festigen durch Wiederholung und Anwendung ihr fachliches Wissen aus den Grundvorlesungen. Gleichzeitig erkennen sie Verbindungen zum Schulstoff. Quasi nebenbei erfahren sie einen Zugang zu Mathematik, der im Studium sonst nicht vorkommt und der sich auch für den Einsatz in der Schule eignet. Das Seminar wurde von den Autorinnen bisher zweimal mit insgesamt 31 Studierenden (siebtes Semester oder höher) an der Universität Tübingen durchgeführt.
This study investigates the emergence of empathic framing in a small group of university students’ discussions of equity-oriented concepts in a service-learning course. Empathic framing refers to the making of emotional connections that enable one to experience the world from another’s perspective, particularly when they are from different cultures, means of socialization, and life experiences. The study used collaborative coding for both concepts and empathic framing in six discussions of three scholarly books devoted to different equity concerns focused on the phenomenon of teacher-student reciprocal burnout, the differential experiences of affiliative or ‘jock’ students and disaffiliative or ‘burnout’ students, and African American speech and its political consequences. The findings identify examples of empathic framing in the six discussions, with most instances occurring in the two books that include narrative accounts of people experiencing oppression and inequity; the final volume, centered on textuality more than human action, produced a single instance of empathic framing recruited from outside the book’s contents. The study suggests that empathy can serve as a beginning point to concept development toward more equitable teaching and school culture, and can be available for formal academic learning when it is combined with worldly experience such as that available in service-learning courses.
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Previous studies have demonstrated that children use oral calculation procedures not taught in school. The present study provided evidence for situational variables that strongly influence the tendency to use such procedures. It also provided a qualitative analysis of the oral mathematics used by Brazilian third graders. Concrete problem situations were powerful elicitors of oral computation procedures, whereas computation exercises tended to elicit school-learned computation algorithms. Oral computation procedures involved the use of two reliably identifiable routines, decomposition and repeated grouping, that revealed the children's solid understanding of the decimal system. In general, the children were far more successful in using oral mathematics than written mathematics. An understanding of children's oral procedures may be useful in developing more successful programs for elementary mathematics instruction.
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The terms concreteness fading and progressive formalization have been used to describe instructional approaches to science and mathematics that use grounded representations to introduce concepts and later transition to more formal representations of the same concepts. There are both theoretical and empirical reasons to believe that such an approach may improve learning outcomes relative to instruction employing only grounded or only formal representations (Freudenthal, 1991; Goldstone & Son, 2005; McNeil & Fyfe, 2012; but see Kaminski, Sloutsky, & Heckler, 2008). Two experiments tested the effectiveness of this approach to instruction in the mathematical domain of combinatorics, using outcome listing and numerical calculation as examples of grounded and formal representations, respectively. The study employed a pretest-training, posttest design. Transfer performance, that is, participants’ improve- ment from pretest to posttest, was used to assess the effectiveness of instruction received during training. In Experiment 1, transfer performance was compared for 4 types of instruction, which differed only in the types of representation they employed: pure listing (i.e., listing only), pure formalism (i.e., numerical calculation only), list fading (i.e., listing followed by numerical calculation), and formalism-first (i.e., listing introduced after numerical calculation). List fading instruction led to transfer performance on par with pure formalism instruction and higher than formalism-first and pure listing instruction. In Experi- ment 2, an enhanced version of list fading training was again compared to pure formalism. However, no difference in transfer performance due to training was found. The results suggest that combining grounded and formal representations can be an effective approach to combinatorics instruction but is not necessarily preferable to using formal representations alone. If both grounded and formal representations are employed, the former should precede rather than follow the latter in the instructional sequence.
Research has shown that it is effective to combine example study and problem solving in the initial acquisition of cognitive skills. Present methods for combining these learning modes are static, however, and do not support a transition from example study in early stages of skill acquisition to later problem solving. Against this background, the authors proposed a successive integration of problem-solving elements into example study until the learners solved problems on their own (i.e., complete example --> increasingly more incomplete examples --> problem to-be-solved). The authors tested the effectiveness of such a fading procedure against the traditional method of using example-problem pairs. In a field experiment and in 2 more controlled laboratory experiments, the authors found that (a) the fading procedure fosters learning, at least when near transfer performance is considered; (b) the number of problem-solving errors during learning plays a role in mediating this effect; and (c) it is more favorable to fade out worked-out solution steps in a backward manner (omitting the last solution steps first) as compared with a forward manner (omitting the first solution steps first).
The fading of instructional scripts can be regarded as necessary for allowing learners to take over control of their cognitive activities during the acquisition of skills such as argumentation. There is, however, the danger that learners might relapse into novice strategies after script prompts are faded. One possible solution could be monitoring by a peer with respect to the performance of the strategy to be learned. We conducted a 2×2-factorial experiment with 126 participants with fading and peer monitoring as between-subjects factors to test the assumptions that (1) the combination of a faded script and peer monitoring has a positive effect on strategy knowledge compared to only one or none of the two types of support; and (2) this effect is due to a greater amount of self-regulated performance of the strategy after the fading of the script when peer monitoring takes place. The findings support these assumptions. (
Fifty Brazilian children between the ages of 7 and 13, all number conservers, were individually given seven addition and four subtraction exercises. The children were asked to write down and do each exercise and then explain how the answer was reached. The results showed that: (a) counting was the preferred procedure; (b) the use of school-taught algorithms was rather limited; (c) some children decomposed numbers into tens and units and then worked at both levels, combining results subsequently; and (d) children rarely referred to previously obtained results when doing related exercises. Mathematics educators could profit from a knowledge of the procedures used naturally by children and the specific difficulties related to natural and to school-prescribed routines.