Connecting concrete objects and abstract symbols promotes children’s place value knowledge

Abstract

Children often learn abstract mathematics concepts with concrete
manipulatives. The current study compared different ways of
using specific manipulatives – base-ten blocks – to support children’s
place value knowledge. Children (N¼112, M age ¼
6.88 years) engaged in place value learning activities in one of
four randomly assigned conditions in a one-on-one tutoring setting:
Concrete Only, Two-Step Fading, Three-Step Fading, and
Fading-with-Comparison. Performance on a posttest measure was
higher in the Two-Step and Three-Step Fading conditions relative
to the Fading-with-Comparison condition, suggesting potential
benefits of gradual, sequential transitions from concrete objects
to written numerals. Children in the Concrete Only condition also
exhibited high place value knowledge on the posttest. Finally,
across conditions, children who exhibited knowledge of the connections
between the base-ten-blocks and written number symbols
had higher posttest and transfer test scores relative to
children who did not exhibit knowledge of these connections.

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Connecting concrete objects and abstract symbols
promotes children’s place value knowledge
Andrea Marquardt Donovan & Emily R. Fyfe
To cite this article: Andrea Marquardt Donovan & Emily R. Fyfe (2022): Connecting concrete
objects and abstract symbols promotes children’s place value knowledge, Educational Psychology,
DOI: 10.1080/01443410.2022.2077915
To link to this article: https://doi.org/10.1080/01443410.2022.2077915
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Connecting concrete objects and abstract symbols
promotes children’s place value knowledge
Andrea Marquardt Donovan
a
and Emily R. Fyfe
b
a
Psychology Department, University of Wisconsin-Madison, Madison, WI, USA;
b
Department of
Psychological and Brain Sciences, Indiana University, Bloomington, IN, USA
ABSTRACT
Children often learn abstract mathematics concepts with concrete
manipulatives. The current study compared different ways of
using specific manipulatives –base-ten blocks –to support child-
ren’s place value knowledge. Children (N¼112, Mage ¼
6.88 years) engaged in place value learning activities in one of
four randomly assigned conditions in a one-on-one tutoring set-
ting: Concrete Only, Two-Step Fading, Three-Step Fading, and
Fading-with-Comparison. Performance on a posttest measure was
higher in the Two-Step and Three-Step Fading conditions relative
to the Fading-with-Comparison condition, suggesting potential
benefits of gradual, sequential transitions from concrete objects
to written numerals. Children in the Concrete Only condition also
exhibited high place value knowledge on the posttest. Finally,
across conditions, children who exhibited knowledge of the con-
nections between the base-ten-blocks and written number sym-
bols had higher posttest and transfer test scores relative to
children who did not exhibit knowledge of these connections.
ARTICLE HISTORY
Received 3 February 2021
Accepted 11 May 2022
KEYWORDS
Manipulatives; place value;
concreteness fading;
comparison
Introduction
One common practice used to support children’s understanding of symbolic numbers is
to incorporate learning activities with concrete manipulatives such as blocks, counters, and
balance scales. A consistent recommendation is to help learners build explicit connections
between manipulatives and corresponding symbols (e.g. McNeil & Uttal, 2009). The goal
of the current study was to experimentally compare different ways of using manipulatives
to support children’s knowledge of place value from a targeted learning experience.
We also tested whether differences in children’sknowledgeoftheconnections between
the manipulatives and symbols predicted their place value knowledge at posttest.
Maths manipulatives
Researchers and educators often support the use of manipulatives to aid children’s
learning. For example, the National Council for Teachers of Mathematics lists physical
CONTACT Emily R. Fyfe efyfe@indiana.edu Department of Psychological and Brain Sciences, Indiana
University, 1101 E. 10th Street, Bloomington, IN 47405, USA
Supplemental data for this article can be accessed online at https://doi.org/10.1080/01443410.2022.2077915
ß2022 Informa UK Limited, trading as Taylor & Francis Group
EDUCATIONAL PSYCHOLOGY
https://doi.org/10.1080/01443410.2022.2077915

materials as part of the curriculum standards for mathematics (National Council of
Teachers of Mathematics, 2000). Theoretical support for manipulatives stems from
developmental psychology and the belief that children’s thinking is inherently con-
crete (e.g. Piaget, 1970). Other support for manipulatives stems from theories of
embodied cognition, emphasising action and perception (e.g. Glenberg et al., 2007;
Martin & Schwartz, 2005). Still other support is from socio-constructivist perspectives
that highlight building meaning-making with these objects in a classroom community
(Cobb et al., 1992).
Despite support for manipulatives, the empirical evidence regarding their effective-
ness is mixed. Meta-analytic data showed instruction with manipulatives as beneficial
for children’s learning (Carbonneau et al., 2013), with effects depending on several fac-
tors. But just using manipulatives does not guarantee positive outcomes (see McNeil &
Jarvin, 2007). In fact, there are situations in which symbols alone yield benefits over
conditions that incorporate concrete materials (e.g. McNeil et al., 2009; Mix et al.,
2017). Mixed evidence does not mean concrete manipulatives should be abandoned.
In fact, some research citing advantages of abstract symbols also identify ways in
which concrete materials are beneficial (e.g. McNeil et al., 2009; Mix et al., 2017).
Further, teachers continue to value and use concrete materials in their classrooms (e.g.
Kaminski & Sloutsky, 2020; Puchner et al., 2008).
Given the use of manipulatives in many learning experiences, research is needed to
compare different ways of using manipulatives. There are a variety of suggestions for
enhancing the effectiveness of manipulatives, and here we focus on the recommenda-
tion to draw explicit connections between concrete objects and the symbols they are
intended to represent. For example, Morin and Samelson (2015) suggest teachers
need to ‘achieve greater congruence between the numerical concepts and procedures
they are teaching and the manipulative displays they are using to represent them’
(p. 362). Similarly, McNeil and Uttal (2009) suggest ‘drawing linkages between concrete
and abstract representations of mathematical concepts may do far more to advance
students’understanding than working on either in isolation’(p. 139).
This recommendation suggests how manipulatives are used can be just as import-
ant as whether they are used. One cannot provide children with manipulatives and
assume that successful learning will occur. Rather, manipulatives may be helpful to
the extent that they support children’sknowledge of the connections between the
objects and symbols (e.g. Brown et al., 2009; Moyer, 2001). The goal of the current
study was to compare different ways of using concrete manipulatives with a focus on
a technique referred to as Concreteness Fading.
Concreteness fading
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 pre-
sented in three progressive forms: (1) an enactive form, a concrete model of a con-
cept; (2) an iconic form, a graphic or pictorial model; and (3) a symbolic form, an
abstract model of the concept. The theoretical benefits of Concreteness Fading include
2 A. M. DONOVAN AND E. R. FYFE

helping children interpret ambiguous symbols in terms of well-understood concrete
objects and guiding children to strip away extraneous concrete properties (e.g. Fyfe &
Nathan, 2019).
There has been some experimental evidence in favour of Concreteness Fading with
children. In one study, children ages 7-to-9 received one-on-one instruction on math-
ematical equivalence in one of four conditions (Fyfe et al., 2015). Problems were pre-
sented: (1) using concrete objects, (2) using abstract equations, (3) using Three-Step
Fading (objects, then pictures, then equations), or (4) using a reverse progression
(equations, then pictures, then objects). Children in the Three-Step Fading condition
exhibited better transfer to symbolic problems than children in the other conditions.
Similarly, Osana et al. (2017) had 7-year-olds learn about place value with base-ten
blocks presented before or after written symbols. Children in the Two-Step Fading
condition (i.e. blocks then symbols) gained more place value knowledge from the les-
son when instructional guidance was low.
However, empirical evidence for Concreteness Fading with children is limited, and
it is unclear whether certain implementations of the fading sequence are optimal.
Here, we tested three forms of Concreteness Fading. Two forms included step-wise,
sequential transitions from concrete to abstract whereby each representation was pre-
sented one at a time; one had a Two-Step progression (i.e. blocks then symbols) and
one had a Three-Step progression (i.e. blocks, pictures, then symbols). Some hypothe-
sise that a Three-Step progression is more effective than a Two-Step progression
because the intermediate, iconic stage maintains some correspondence to the
manipulative, but also starts to strip away the extraneous perceptual details, which
potentially helps learners see the connection between the concrete manipulative and
the written symbol (Bruner, 1966; Fyfe & Nathan, 2019). However, empirical tests of
this hypothesis are lacking.
Direct comparison
The third form of Fading we investigated started with concrete materials, ended with
abstract symbols, but included an intermediate stage in which the two representations
were directly compared. Comparison is a powerful tool that aids learning in a variety
of domains (Rittle-Johnson & Star, 2011). By placing objects and symbols side by side,
comparison allows children to notice the similarities and differences between the rep-
resentations. Indeed, comparison is thought to support learning by helping learners
abstract the key structural features of each representation so their knowledge is not
tied to narrow problem features (e.g. Gentner, 1983).
There is some evidence that comparing representations simultaneously is more
effective for learning than viewing the same representations one at a time (e.g. Son
et al., 2011). For example, preschoolers completed a category learning task, and per-
formance was optimised when they compared two target pictures simultaneously
than when they saw the same pictures sequentially (Christie & Gentner, 2010).
However, evidence supporting the comparison of manipulatives and symbols is sparse.
There is evidence to suggest that teachers engage in this type of comparison. Alibali
and Nathan (2007) recorded a sixth-grade mathematics teacher explaining algebraic
EDUCATIONAL PSYCHOLOGY 3

relations. The teacher frequently compared a picture of a pan balance manipulative
with the corresponding symbols in an equation. More recently, Mix and Colleagues
(2017) conducted an experiment to teach 7-year-olds about place value with (a) sym-
bols alone or (b) symbols and base-ten blocks. The training with base-ten blocks
included comparison to the symbols and resulted in better understanding of base-ten
structure. Although promising, the training included multiple components and more
evidence is needed on the benefits of comparing manipulatives and symbols.
On the one hand, including comparison within the fading sequence may capitalise
on the advantages of both techniques and result in better understanding of the
underlying concept. On the other hand, including comparison may overwhelm learn-
ers’limited cognitive resources as it requires them to consider both representations
simultaneously while tracking the fading process.
The current study
Weaimedtocontributetothisliteratureintwokeyways.Thefirstgoalwastoexperi-
mentally compare different ways of using manipulatives to support children’sknowledge
of place value. To do so, children engaged in place value activities in one of four condi-
tions: (1) Concrete Only, with physical, base-ten blocks, (2) Two-Step Fading, with base-
ten blocks that transitioned to symbols, (3) Three-Step Fading, with base-ten blocks that
transitioned to pictures and ended with symbols, and (4) Fading-with-Comparison, with
base-ten blocks, comparison of base-ten blocks and symbols, and ending with symbols.
This is one of the first randomised experiments to contrast three forms of concreteness
fading with children. We hypothesised that (H1) the three fading conditions would be
more effective than the Concrete Only condition, and (H2) the Three-Step Fading condi-
tion would be more effective than the Two-Step Fading condition. Differences between
the Fading-with-Comparison condition and the other two Fading conditions were also
examined, though no explicit hypotheses were made, given that both techniques are
intended to help learners draw connections between representations.
The second goal was to test whether children’s knowledge of the connections
between the manipulatives and symbols during the activities predicted their overall
knowledge of place value. This represents one of the only studies to include a stand-
alone measure that assesses individual differences in this type of connection
knowledge. We hypothesised (H3) that children who exhibited knowledge of the
connections between the objects and symbols during the activities would have higher
scores on the posttest and transfer test than children who did not. We addressed our
questions in the context of children engaging in place value activities as several recent
studies suggest that training with both base-ten blocks and symbols hold promise for
supporting children’s place value knowledge (Mix et al., 2017; Osana et al., 2017).
Method
Participants
Prior to launching this study, the research team decided to target a sample size of
about 140 children, so that we could account for any potential exclusions and have a
4 A. M. DONOVAN AND E. R. FYFE

final analytic sample of approximately 120 children (which provides 80% power to
detect a medium effect size with four conditions). By the conclusion of the study, par-
ent consent and child assent had been obtained for 146 children who attended a sin-
gle session in one of two laboratories. Children were recruited from a population of
working- and middle-class families from two databases in Midwestern cities that con-
tain public universities (51% from Bloomington, IN and 49% from Madison, WI). Eight
children were excluded from analyses for experimenter error (n¼3), withdrawing
(n¼2), or off-task behaviour (n¼3). Of the remaining 138 children, 26 had missing
data on a key covariate (i.e. self-report of their prior experience with the blocks). Thus,
the final analytic sample included 112 children (M
age
¼6.88 years; SD ¼0.58), with 25
kindergarteners (M
age
¼6.25 years; SD ¼0.22), 64 first-graders (M
age
¼6.84, SD ¼0.39)
and 23 second-graders (M
age
¼7.66, SD ¼0.31). Based on parent report, 12% of chil-
dren were ethnic minorities and 51% were female. Analyses on the full sample that
did not include the prior experience covariate (n¼138) produced similar findings and
are in the supplemental material.
Design and procedure
Children participated in a single one-on-one session with a pretest-activities-posttest
design. We acknowledge that comprehensive place value instruction takes place over
many sessions and that other studies include longer training intervals (e.g. Fuson &
Briars, 1990; Mix et al., 2017; Peterson et al., 1988). We opted to use a single one-
on-one training session based on the scope of the lesson content. Specifically, our
goal was to focus on one specific aspect of place value knowledge –identifying the
value of each digit in a three-digit number –and to inform decisions about designing
a lesson on this topic with base-ten blocks. This methodological design is similar to
other experimental studies in educational psychology that target specific topics during
a single session to gain insight into students’cognitive processes (e.g. Alibali et al.,
2018; Cook et al., 2008; Rittle-Johnson, 2006). Using between-subjects random assign-
ment, children were placed in one of four conditions: Concrete Only (n¼31), Fading-
with-Comparison (n¼29), Two-Step Fading (n¼28), or Three-Step Fading (n¼24). As
shown in Table 1, there were no significant condition differences in age, proportion of
females, or proportion of ethnic minorities, ps>.05. The session took place with a
trained experimenter and lasted 30–40 min. The experimenter first administered a brief
paper-and-pencil pretest to assess children’s prior knowledge of place value, and then
proceeded to the learning phase.
Learning phase
All children completed place value activities focussed on identifying the value of each
digit in three-digit numbers (e.g. the value of 2 in the number 524 is 20). The child
was asked to be a detective and to crack ‘secret codes’by identifying the value of
each number in the code. The experimenter said, ‘we’ll think about whether the num-
ber is in the ones place, the tens place, or the hundreds place’and then demonstrated
the meaning of a ones-block, a tens-block, and a hundreds-block. The experimenter
then placed the blocks into separate piles in front of the child (30 ones-blocks, 30
EDUCATIONAL PSYCHOLOGY 5

tens-blocks, and 10 hundreds-blocks), and the child completed six practice problems
with guidance and corrective feedback. The materials provided on the practice prob-
lems differed by condition, but the procedure and content were the same for
all children.
On three trials (Problems 1, 3, 5), the experimenter provided the value of each digit
individually, and the child was asked to represent that digit using the available materi-
als. For example, on a trial where blocks were provided, the experimenter started with:
‘The number six is in the ones place. How can we show six ones with our blocks? What
is the value of six ones blocks?’. Similarly, on a trial where paper and pencil were pro-
vided, the experimenter started with: ‘The number six is in the ones place. How can we
show six ones on our paper? What is the value of six ones?’. On the other three trials
(Problems 2, 4, 6), the experimenter named the entire numeral, the child was then
asked to represent the total numeral with the available materials, and the experimenter
discussed the value of each digit. For example, on a trial where blocks were provided,
the experimenter started with: ‘The number is four hundred seventy-five. Can you
show me that number with your blocks?’. Similarly, on a trial where paper was pro-
vided, the experimenter started with: ‘The number is four hundred seventy-five. Can
you write the number on your paper?’. Across all trials, if the child answered incor-
rectly, the experimenter showed them how to use the target materials to represent the
number and explained the value of each digit. Throughout the learning phase, children
were guided through 15 questions across six problems. We calculated a general meas-
ure of learning by summing children’s correct responses to those 15 questions.
For the learning phase, conditions differed in terms of the format of the problems
and the materials provided (see Figure 1). In the Concrete Only condition, all six prob-
lems were presented solely with base-ten blocks. In the Fading-with-Comparison condi-
tion, the first two problems were presented with base-ten blocks. The middle two
problems were presented with blocks and written numerals on paper simultaneously.
The paper included three blank lines representing the ones, tens, and hundreds place.
The experimenter mapped the similarities across the materials (e.g. ‘Look, the value of
two ones-blocks is two and the value of a written 2 in the ones place is also two’),
and the child completed the problems using both formats side-by-side. The last two
problems were presented with written numerals on paper. In the Two-Step Fading con-
dition, the first two problems were presented with blocks, and the last four problems
were presented with written numerals on paper. In the Three-Step Fading condition,
Table 1. Raw descriptive statistics by condition.
Concrete Only
Fading-with-
Comparison Two-Step Fading Three-Step Fading Total
(n¼31) (n¼29) (n¼28) (n¼24) (n¼112)
Pretest Score (out of 12) 5.94 (3.07) 6.83 (3.10) 6.79 (2.64) 8.17 (2.73) 6.86 (2.97)
Posttest Score (out of 12) 7.10 (3.12) 7.10 (3.30) 8.11 (2.56) 9.13 (2.92) 7.79 (3.07)
Transfer Score (out of 6) 3.23 (1.69) 3.34 (1.86) 3.79 (1.79) 4.42 (1.53) 3.65 (1.76)
Age in Years 6.85 (0.53) 6.96 (0.58) 6.68 (0.64) 7.05 (0.51) 6.88 (0.58)
% Female 50.12 58.62 50.00 45.83 50.89
% White 80.65 86.21 96.43 91.67 88.39
% Reporting Prior
Experience with Blocks
67.74 86.21 53.57 83.33 72.32
Note. Raw means are reported with standard deviations in parentheses.
6 A. M. DONOVAN AND E. R. FYFE

the first two problems were presented with blocks. The middle two problems were
presented with ‘faded’worksheets, which displayed a line drawing of the blocks along
with three blank lines. The last two problems were presented with numerals on paper.
In all Fading conditions, the experimenter verbally supported the transition to new for-
mats (e.g. ‘We’re going to play the same game, but this time on paper. First, let me
show you how it works on the paper …’).
Test phase
After the learning phase, children were given a brief break. Then, they completed the
Block Task, which assessed their knowledge of the connections between the base-ten
blocks and the symbols. Next, the blocks were removed from sight and children com-
pleted the posttest and transfer test taking as much time as needed. Feedback about
performance was not provided. The posttest was used to assess general learning. The
transfer test was used to determine whether children could apply their knowledge to
novel problems (i.e. numbers in the thousands). At the end of the session, the experi-
menter asked the child if they had ever used the base-ten blocks before. This was to
obtain an informal measure of children’s prior experience with the materials. We
acknowledge this is a somewhat crude measure of prior experience; however, as noted
in the results, it reliably related to children’s pretest scores, suggesting it may capture
a relevant factor.
Measures and scoring
Pretest and posttest
The pretest was a paper and pencil, 12-item multiple-choice measure designed to
assess children’s knowledge of symbolic place value with three-digit numbers
(Cronbach’sa¼.75). The posttest was isomorphic to the pretest; it included the same
12 item types but with different numerals (Cronbach’sa¼.79). Items were adapted
from previous assessments used with this age range (Mix et al., 2017). Several items
tapped numeral identification knowledge (e.g. ‘How is two hundred six written?’).
Other items tapped numeral magnitude knowledge (e.g. ‘What is the value of the 5 in
Concrete Only Fading-with-
Comparison
Two-Step Fading Three-Step Fading
Problems 1 & 2
Problems 3 & 4
Problems 5 & 6
Figure 1. Schematic of materials used across conditions during the learning phase.
EDUCATIONAL PSYCHOLOGY 7

this number [points to 526]?’) or the value of three-digit numerals relative to others
(e.g. ‘Which number could be between 134 and 197?’). Children were assigned a score
from 0 to 12 on each assessment based on their correct multiple-choice selections.
Transfer test
The paper-and-pencil transfer test included six items (Cronbach’s alpha ¼.76). These
items were multiple-choice items that were similar to the posttest items, but the
numerals were four-digits, which were not discussed during the learning phase.
Children were assigned a score from 0 to 6 based on their correct multiple-
choice selections.
Block task
The block task was a three-item measure assessing children’sknowledge of the connec-
tions between base-ten blocks and symbols adapted from prior work (Laski et al., 2014).
Piles of ones-blocks, tens-blocks, and hundreds-blocks were placed in front of the child.
The experimenter showed a laminated card with ‘7’printed on it, selected the correct
number of blocks, and said, ‘Look, this shows seven [points to card] and this shows
seven [points to blocks]’. The child then completed three test items with numerals 12,
28, and 134. On each trial, they were shown a card with the number and asked to
show the number using blocks. Responses were scored as correct if the quantity of
blocks matched the numeral (e.g. representing the numeral 12 with one tens-block and
two ones-blocks). Children were assigned a score from 0 to 3.
Results
Pretest results
The average score on the pretest was 6.86 (out of 12; SD ¼2.97), and scores ranged
from 1 to 12. As shown in Table 1, there were descriptive differences in pretest scores
as a function of condition, and the effect of condition on pretest scores approached
conventional levels of significance, F(3, 108) ¼2.68, p¼.051, g
p2
¼.07. We took sev-
eral steps to ensure these initial differences did not explain away any conclusions.
First, we included pretest scores as a covariate in subsequent analyses. Second, we re-
ran all subsequent analyses excluding extreme scores on the pretest (i.e. scoring
higher than 10 or scoring lower than 3), and the results remain unchanged. Third, we
controlled for other variables that were significantly related to pretest scores. Pretest
scores were significantly higher for children who reported prior experience using the
base-ten blocks (M¼7.37, SD ¼2.85) relative to children who did not (M¼5.52,
SD ¼2.90), F(1, 110) ¼9.42, p¼.003, g
p2
¼.08. Also, pretest scores were positively
correlated with children’s age, r(110) ¼.51, p<.001. Thus, prior block experience and
age were also included as covariates in subsequent analyses. Preliminary analyses indi-
cated no reliable interactions between these three background variables and condi-
tion, so no interactions with age, pretest score, or prior block experience were
included in the final models.
8 A. M. DONOVAN AND E. R. FYFE

RQ1: Condition differences predicting posttest and transfer
The first set of planned analyses focussed on the effects of condition on the posttest
and transfer test. The average score on the posttest was 7.79 (out of 12, SD ¼3.07,
Skewness ¼.31). Given the isomorphic nature of the posttest and pretest, we exam-
ined improvement across these two measures (see Table 1). Posttest scores were sig-
nificantly higher than pretest scores, t(111) ¼5.83, p<.001, suggesting that children
learned from the place value activities. When split by condition, there was significant
improvement in the Concrete Only, t(30) ¼4.35, p¼.001, Two-Step Fading, t(27) ¼
3.59, p¼.001, and Three-Step Fading conditions, t(23) ¼3.15, p¼.004. There was
not significant improvement in Fading-with-Comparison, t(28) ¼0.89, p¼.380.
To formally analyse condition differences, a linear regression model was used with
condition as the independent variable and posttest scores (out of 12) as the depend-
ent variable. Pretest scores, age, and prior block experience were included as covari-
ates. Three Helmert Contrasts were used to represent the four levels of condition.
Helmert Contrasts compare the first condition to all subsequent conditions (second/
third/fourth), then compare the second condition to all subsequent conditions (third/
fourth), and so on. We selected Helmert Contrasts as they aligned with our hypotheses
and allowed us to test the contrasts of interest: (1) Concrete Only versus the three fad-
ing conditions, (2) Fading-with-Comparison versus the step-wise Fading conditions,
and (3) Two-Step Fading versus Three-Step Fading.
The model predicting posttest scores accounted for 74% of the variance. Figure 2
displays the adjusted means by condition. There was a significant, positive effect of
pretest score, ß¼.78, p<.001, but not an effect of age, ß¼.09, p¼.136, or prior
block experience, ß¼.05, p¼.378. The first condition contrast was not significant, ß
¼–.02, p¼.772; the three fading conditions had an average adjusted score of 7.77
(SE
adj
¼0.18) and the Concrete Only condition had an average adjusted score of 7.87
(SE
adj
¼0.28). The second condition contrast was statistically significant, ß¼.15, p¼
0
1
2
3
4
5
6
7
8
9
10
Concrete Only Fading-with-Comparison Two-Step Fading Three-Step Fading
Number Correct (out of 12)
Adjusted Posttest Scores
Figure 2. Adjusted posttest scores by condition. Note. Scores are adjusted means and error bars
represent adjusted standard errors after controlling for pretest scores, age, and prior experience
with blocks.
EDUCATIONAL PSYCHOLOGY 9

.005; the step-wise Fading conditions scored significantly higher (M
adj
¼8.13, SE
adj
¼
0.23) than Fading-with-Comparison (M
adj
¼7.04, SE
adj
¼0.29). The third condition con-
trast was not significant, ß¼–.04, p¼.416; the Three-Step Fading condition had an
average adjusted score of 7.94 (SE
adj
¼0.32) and the Two-Step Fading condition had
an average adjusted score of 8.32 (SE
adj
¼0.30).
On the transfer test, the average score was 3.65 (out of 6, SD ¼1.76, Skewness ¼
.21). The same linear regression model was used with transfer test scores (out of 6)
as the dependent variable, the three Helmert Contrasts as the independent variables,
and pretest scores, age, and prior block experience included as covariates. The model
predicting transfer scores accounted for 52% of the variance. Figure 3 displays the
adjusted means by condition. There was a significant, positive effect of pretest score,
ß¼.63, p<.001, but not an effect of age, ß¼.14, p¼.111, or prior block experi-
ence, ß¼.04, p¼.611. None of the condition contrasts were significant. The first con-
dition contrast was not significant, ß¼.03, p¼.664; the three fading conditions had
an average adjusted score of 3.69 (SE
adj
¼0.14) and the Concrete Only condition had
an average adjusted score of 3.58 (SE
adj
¼0.23). The second condition contrast was
marginal, ß¼.12, p¼.084; the step-wise Fading conditions had an average adjusted
score of 3.86 (SE
adj
¼0.18) and the Fading-with-Comparison condition had an average
adjusted score of 3.34 (SE
adj
¼0.23). The third condition contrast was not significant,
ß¼.00, p¼.980; the Three-Step Fading condition had an average adjusted score of
3.87 (SE
adj
¼0.25) and the Two-Step Fading condition had an average adjusted score
of 3.86 (SE
adj
¼0.24).
To summarise, there were limited condition differences on the posttest. Our first
hypothesis was not supported; children in the Concrete Only condition performed well
and were not significantly different from children in the other three conditions. Our
second hypothesis was not supported; children in the Three-Step Fading condition
0
1
2
3
4
5
Concrete Only Fading-with-Comparison Two-Step Fading Three-Step Fading
Number Correct (out of 6)
Adjusted Transfer Test Scores
Figure 3. Adjusted transfer scores by condition. Note. Scores are adjusted means and error bars
represent adjusted standard errors after controlling for pretest scores, age, and prior experience
with blocks.
10 A. M. DONOVAN AND E. R. FYFE

performed similarly to children in the Two-Step Fading condition. However, children in
the step-wise Fading conditions performed significantly better than children in the
Fading-with-Comparison condition. Condition effects were not found on the transfer
test, though patterns were in the same direction.
RQ2: Connection knowledge predicting posttest and transfer
The second set of planned analyses focussed on children’s knowledge of the connec-
tions between the objects and the symbols, as assessed on the Block Task. On aver-
age, children did well on the Block Task with an average score of 2.51 (out of 3,
SD ¼0.79). The distribution was skewed (Skewness ¼1.62), with 66% of children
solving all three items correctly and demonstrating mastery in their knowledge of the
connections between the symbols and the blocks. Block Task scores were correlated
with pretest scores, r(110) ¼.57, p<.001, but only moderately so suggesting that
Block Task scores assessed knowledge that was unique relative to children’s prior
knowledge of symbolic place value.
A linear regression model was used to predict children’s posttest scores from their
Block Task score. Given the skewed distribution of Block Task scores, a dichotomised
variable was created: children at mastery (solved all three items correctly, n¼74) and
children not at mastery (n¼38). Pretest scores, age, and prior block experience were
included as covariates. The model accounted for 75% of the variance in posttest
scores. Block Task Mastery significantly predicted posttest, ß¼.24, p<.001. Children
at mastery on the Block Task had significantly higher posttest scores (M
adj
¼8.31,
SE
adj
¼0.19) than children not at mastery (M
adj
¼6.75, SE
adj
¼0.29). In this
model, pretest scores also predicted posttest scores, ß¼.66, p<.001, but age did
not, ß¼.08, p¼.189, and prior block experience did not, ß¼.00, p¼.982.
Similar results were found with transfer test scores. The model accounted for 53%
of the variance in transfer scores. Block Task Mastery significantly predicted transfer
test scores, ß¼.23, p<.001, even after controlling for pretest scores, age, and prior
block experience. Children at mastery on the Block Task had significantly higher trans-
fer test scores (M
adj
¼3.95, SE
adj
¼0.16) than children not at mastery (M
adj
¼3.07,
SE
adj
¼0.24). In this model, pretest scores also predicted transfer test scores, ß¼.51,
p<.001, but age did not, ß¼.12, p¼.128, and prior block experience did not,
ß¼.07, p¼.296.
Exploratory analyses on connection knowledge
These results provide empirical support for the theoretical notion that children’s
knowledge of the connections between objects and symbols is critical for learning
from manipulatives. It was the children who had mastered the connections on the
Block Task that exhibited the highest place value knowledge following the lesson.
However, there is an alternative explanation; the associations between the Block
Task and posttest assessments could have nothing to do with children’s knowledge
of the connections –instead, the Block Task could be capturing any learning or
knowledge from the session, and this knowledge could be related to the posttest
scores. To rule out this possibility, we conducted several unplanned exploratory
EDUCATIONAL PSYCHOLOGY 11

analyses using a different measure of general learning from the lesson. If there is
something unique about children’s knowledge of the connections,thenBlockTask
performance should still predict posttest scores even after accounting for this other
measure of generic learning.
As noted in the method, children were guided through 15 questions during the
learning activities. We used their performance on these questions as a measure of
generic learning. On average, children answered 74% of these questions correctly
(SD ¼28%). However, the distribution was skewed (Skewness ¼1.00), with over half
of them scoring 85% or higher. Given this distribution, we split children into low
learner status (<median, n¼51) and high learner status groups (median, n¼61).
Then, we used two linear regression models to predict children’s posttest scores and
children’s transfer scores with both Block Task Mastery and General Learning Status as
predictors. Pretest scores, age, and prior block experience were included as covariates.
The models were significant and accounted for 75% of the variance in posttest scores
and 53% of the variance in transfer scores. Figure 4 displays the adjusted means. In
these models, Block Task Mastery remained a significant predictor of posttest scores,
ß¼.22, p<.001, and transfer scores, ß¼.24, p¼.007. Further, in these models,
General Learning Status was not a significant predictor of posttest scores, ß¼.06,
p¼.435, or transfer scores, ß¼.00, p¼.932. Thus, children exhibited higher posttest
and transfer scores to the extent that they had knowledge of the connections between
the objects and symbols.
We also explored Block Task performance by condition. A logistic regression was
used with Block Task Mastery as the dependent variable and the three Helmert
Contrasts as the independent variables. Pretest scores, age, and prior block experience
0
1
2
3
4
5
6
7
8
9
10
Transfer TestPosttest
Number Correct
Adjusted Posttest and Transfer Test Scores
Block Task Mastery
Block Task Non-Mastery
Figure 4. Adjusted posttest and transfer scores as a function of Block Task Mastery. Note. Scores
are adjusted means and error bars represent adjusted standard errors after controlling for General
Learning Status, pretest scores, age, and prior experience with blocks.
12 A. M. DONOVAN AND E. R. FYFE

were included as covariates. None of the condition contrasts were statistically signifi-
cant: the first contrast, ß¼.22, p¼.705, OR ¼1.25; the second contrast, ß¼.19, p¼
.779, OR ¼1.21; and the third contrast, ß¼.51, p¼.571, OR ¼1.66. Descriptively, the
percentage of children at mastery on the Block Task was highest in the Three-Step
Fading condition (83%), followed by Fading-with-Comparison (66%), then Two-Step
Fading (64%), and lowest in Concrete Only (55%).
There was also no indication that Block Task performance mediated the effect of
condition on posttest scores. When the condition contrasts and Block Task Mastery
were entered as simultaneous predictors of posttest scores, both were unique predic-
tors. Specifically, the second condition contrast (step-wise Fading conditions relative to
Fading-with-Comparison) remained significant, ß¼.14, p¼.004, and Block Task
Mastery was significant, ß¼.24, p<.001. As before, the first condition contrast was
not significant, ß¼.02, p¼.641, and the third condition contrast was not significant,
ß¼–.05, p¼.308. This suggests that condition effects and Block Task performance
explained unique variance in children’s posttest scores.
Discussion
The current study compared different ways of using base-ten blocks to support child-
ren’s knowledge of place value. We hypothesised (H1) the three fading conditions
would outperform the Concrete Only condition, and (H2) the Three-Step Fading condi-
tion would outperform the Two-Step Fading condition. Neither hypothesis was con-
firmed; children in the Concrete Only, Two-Step Fading, and Three-Step Fading
conditions all made significant gains from pre-test to posttest and exhibited similarly
high scores at posttest and transfer test. However, children in the step-wise Fading
conditions had higher posttest scores than children in the Fading-with-Comparison
condition. We also hypothesised (H3) that children who exhibited knowledge of the
connections between the objects and symbols would have higher scores on the postt-
est and transfer test. This hypothesis was supported; children demonstrating mastery
on the Block Task had higher posttest and transfer scores relative to children who did
not, even after accounting for general learning from the lesson. Implications are
discussed as well as limitations.
Concrete manipulatives
The current findings indicate that children can benefit from activities that include
manipulatives (e.g. Carbonneau et al., 2013), even when learning is assessed with
abstract symbols. Children in three of the four conditions demonstrated significant
improvements from pre-test to posttest after working through six targeted problems
with explanations and feedback. Further, children in the Concrete Only condition,
who were only exposed to base-ten blocks, exhibited similar posttest and transfer
performance as children in the other conditions. Of course, children in one of the
conditions did not make improvements; thus, engaging with physical objects can, but
does not always, lead to improvements in children’s symbolic number knowledge.
Further, many, if not all the children likely had some exposure to place value concepts
EDUCATIONAL PSYCHOLOGY 13

prior to participation, so the learning activities may have primarily served to activate
prior knowledge rather than to expose children to entirely novel concepts. This
knowledge activation may help explain why the Concrete Only condition led to
improvements as children had a wealth of existing symbolic knowledge from which
to draw.
The current conclusions about whether manipulatives are effective are based on
children’s improvements over the session, not on comparisons to a no-manipulatives
control. The literature in this area continues to find mixed effects, with some research
showing benefits of no-manipulative conditions in some contexts (e.g. McNeil et al.,
2009; Mix et al., 2017). But manipulatives are pervasive in classrooms, and that is not
likely to change (e.g. Kaminski & Sloutsky, 2020). Thus, determining under what condi-
tions manipulatives work is a clear priority.
Concreteness fading
The current results contribute to a growing body of literature on the potential benefits
of Concreteness Fading. Several studies have demonstrated that Concreteness Fading
within a targeted lesson can improve learning for older children and adults relative to
a variety of control groups (e.g. Goldstone & Son, 2005; McNeil & Fyfe, 2012; Ottmar &
Landy, 2017). The current study is one of few to experimentally investigate the bene-
fits of Concreteness Fading for younger children (Fyfe et al., 2015; Osana et al., 2017;
Trory, Howland, & Good, 2018). Children in the step-wise Fading conditions exhibited
the highest scores on the place value posttest and significantly outperformed children
in the Fading-with-Comparison condition.
Contrary to our hypothesis, the Three-Step Fading condition was not more effective
than the Two-Step Fading condition. It has been suggested that a Three-Step progres-
sion is most effective because the intermediate stage serves to explicitly connect the
concrete representation and the symbolic representation via gradual decontextualiza-
tion (e.g. McNeil & Fyfe, 2012). Research suggests that may be true in some cases. For
example, Butler et al. (2003) had middle school students with learning disabilities
(ages 11–15) learn about fractions with manipulatives, pictures, then symbolic numer-
als or with just pictures then symbolic numerals. The condition with three representa-
tions performed best; but, it is unclear whether the benefit comes from having three
stages or from having physical manipulatives at all.
One possibility is that Two-Step Fading is sufficient when the initial representation
is from Bruner’s(1966) enactive stage (e.g. with physical objects) rather than the iconic
stage (e.g. with pictures) and when there is verbal support to connect the two stages,
as was done in the current study. Another possibility is that Two-Step Fading is suffi-
cient for typically developing learners or those with some background knowledge. A
third possibility is that Three-Step Fading is more effective than Two-Step Fading, but
it was not detected in the current study due to the nature of the design. It is possible
that the advantages of the intermediate iconic stage take time to develop and the
brevity of the study prevented us from detecting effects. More work is needed to
experimentally contrast Two-Step and Three-Step Fading sequences.
14 A. M. DONOVAN AND E. R. FYFE

Fading with direct comparison
The current study suggests that embedding comparison within the fading sequence
may not support children’s learning. Children who received the comparison activities
did not make significant gains from pre-test to posttest, and they scored significantly
lower than the step-wise Fading conditions at posttest. Direct comparison has been
shown to improve learning in a variety of domains for children and adults (e.g.
Gentner et al., 2003; Rittle-Johnson et al., 2009; Son et al., 2011). The current findings
do not contradict this work; rather, they suggest that comparing objects and symbols
using the current implementation may be ineffective.
One possibility is that limited instances of comparison may be insufficient to pro-
mote learning. The current study utilised a limited number of comparison trials. This
was done to maintain some similarities with the other conditions (e.g. they all solved
six problems total); however, most studies demonstrating benefits of comparison pro-
vide learners with many trials.
A second possibility is that the Fading-with-Comparison condition was cognitively
overwhelming. Presenting objects and symbols side-by-side and having children
attempt to represent place value using both representations simultaneously may have
resulted in cognitive overload (e.g. Sweller et al., 1998). Additionally, including
comparison within the fading sequence may have resulted in conflicting cognitive
processes confusing children and overwhelming their cognitive system. With true
step-wise fading, each representation is presented sequentially to help the learner
think of the representations as one and the same, as if one were truly morphing into
the next via gradual decontextualization (Fyfe & Nathan, 2019). However, with com-
parison, representations are presented simultaneously which can reinforce their dis-
tinctiveness (Rittle-Johnson & Star, 2011), and this may be at odds with a cognitive
fading process. Future work is needed to examine different types and amounts of
comparison with manipulatives –both within the fading sequence and as a stand-
alone technique.
More broadly, future research is needed to examine the benefits of a variety of
Concreteness Fading sequences. The number of studies demonstrating positive effects
of Concreteness Fading is increasing (Fyfe & Nathan, 2019), but it is far from a panacea
(e.g. Jaakkola & Veermans, 2018; Osana et al., 2017). For example, the benefits of step-
wise Fading in the current study were small, only significant on the posttest, and only
relative to Fading-with-Comparison. Research is needed to explore the factors that
lead to optimal Fading sequences.
Knowledge of connections
Finally, the current study provides empirical support for a theoretical claim that know-
ledge of connections between the manipulatives and symbols is a key factor when learn-
ing from manipulatives. Children who mastered the connections between the blocks
and symbols had higher posttest and transfer scores relative to children who did not.
Because we did not include a pretest measure of children’s Block Task performance,
these results are compatible with at least two interpretations. It could be that children
who learned the connections during the place value activities achieved higher place
EDUCATIONAL PSYCHOLOGY 15

value knowledge than children who did not. It could also be that children who started
with knowledge of the connections were primed to learn from activities and ended up
with higher place value knowledge relative to children who did not.
A third potential interpretation is that these associations are simply a function of
some other aspect of prior knowledge and not specific to knowledge of connections,
though we do not think it likely. It is possible that children with higher Maths achieve-
ment may perform well on the block task and on the posttest and transfer measures.
However, our results suggest that the Block Task is measuring something unique, over
children’s pretest scores, their general learning from the lesson, their age, and their
exposure to base-ten blocks. Thus, our results are consistent with the interpretation
that some aspect of Block Task performance was due to their connection knowledge,
and it was this connection knowledge that supported their place value learning.
Importantly, performance on the Block Task did not differ as a function of condition
and did not mediate the effect of condition. One possibility is that this is due to the
measures in the current study; perhaps step-wise Fading helped children make con-
nections between objects and symbols in ways that were not captured by the Block
Task. Another possibility is that the step-wise Fading conditions benefitted children
relative to Fading-with-Comparison via a different mechanism. Perhaps the step-wise
Fading progression changed children’s memory of the objects in a way that promoted
posttest performance (e.g. better encoding of the relative sizes of the blocks).
Additional research is needed to verify the mechanism by which learning with manip-
ulatives promotes learning, and whether individual differences in knowledge of con-
nections matters more than instructional manipulations of concrete objects.
Limitations and future directions
Limitations of the current research provide additional avenues for future research. The
current study examined children’s learning from a single, brief one-on-one session pre-
sented by an experimenter. However, it is more normative in the area of place value
to include multiple lessons and learning sessions as comprehensive instruction and
training in place value goes well beyond a single targeted topic (e.g. Fuson & Briars,
1990; Mix et al., 2017; Osana et al., 2017). Also, the benefits of manipulatives may take
longer to emerge or may differ in group contexts (e.g. Cobb et al., 1992), which the
current results cannot attest to. Thus, future research should continue to examine
these questions using multiple-session studies in various contexts.
Perhaps the biggest limitation of the current study was the lack of a no-manipulatives
control condition. Because all four conditions in the current study included exposure to
base-ten blocks, we cannot make any conclusions about the general effectiveness of les-
sons with manipulatives relative to lessons without them. Even though we found
improvements in children’s knowledge in three of the conditions here, it is possible that a
no-manipulatives condition would have resulted in even greater improvements.
Additional research should continue to examine the effective use of manipulatives, but
with stronger no-manipulatives control conditions. Several additional aspects of the
research design and measures prevent more precise conclusions about the effects –
including the lack of a pretest Block Task measure, the limited number of training items,
16 A. M. DONOVAN AND E. R. FYFE

and the limited number of Block Task items. Finally, despite using random assignment,
there were some condition descriptive differences at pretest. These initial differences are
unlikely to account for the results because we controlled for pretest differences in our
models. Yet, replicating these results across variants in those design issues will
be important.
Despite these limitations, the present research provides novel insights into the use of
manipulatives for developing children’s symbolic mathematics understanding. It addresses
a call to investigate different ways of combining concrete manipulatives and abstract
symbols. The findings suggest general benefits of using manipulatives for children’smath-
ematics learning and specific benefits of step-wise Concreteness Fading relative to
Fading-with-Comparison. More robustly, the findings indicate that children’sknowledge
of the connections between objects and symbols relates to place value knowledge.
Acknowledgments
Portions of this work were completed when Fyfe was supported by U. S. Department of
Education, training Grant R305B130007 as part of the Wisconsin Centre for Education Research
Postdoctoral Training Program. The authors thank Kathryn Anderson, Gregory Bond, Prabhjot
Chahal, Mariah Ferri, Aleksa Kresovic, Amanda Nafe, and Bret Schreckenghaust for their help
with data collection and data coding.
Disclosure statement
The authors have no conflicts of interest to declare. Data reported in the current manuscript
can be obtained by emailing the corresponding author.
Funding
Funded by the U.S. Department of Education via Institute of Education Sciences (IES) training
grant R305B130007.
ORCID
Andrea Marquardt Donovan http://orcid.org/0000-0002-6418-3171
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