examining children s conceptual subitizing skill and its
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Graduate Studies The Vault: Electronic Theses and Dissertations
2016
Examining Children’s Conceptual Subitizing Skill and
its Role in Supporting Math Achievement
Goukon, Rina
Goukon, R. (2016). Examining Children’s Conceptual Subitizing Skill and its Role in Supporting
Math Achievement (Unpublished master's thesis). University of Calgary, Calgary, AB.
doi:10.11575/PRISM/26014
http://hdl.handle.net/11023/3194
master thesis
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UNIVERSITY OF CALGARY
Examining Children’s Conceptual Subitizing Skill and its Role in Supporting Math Achievement
by
Rina Goukon
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
GRADUATE PROGRAM IN APPLIED PSYCHOLOGY
CALGARY, ALBERTA
AUGUST, 2016
© Rina Goukon 2016
ii
Abstract
The current study investigates the development of the conceptual subitizing skill, the ability to
quickly and accurately recognize quantities above 4 items, and its relation to mathematics
achievement in Grade 2 students. Both reaction time (RT) and strategy use to enumerate
patterned structures of 4 to 10 dots were examined. Using cluster analysis on the collected
enumeration speeds, three distinct groups were identified that supports the acquisition of
conceptual subitizing as a developmental process. These clusters additionally aligned with
students’ self-reported enumeration strategies used during the task, and also predicted children’s
mathematics achievement. As one of the first psychophysical investigations into the conceptual
subitizing skill, this study provides preliminary evidence for the importance of focusing on
children’s speeded enumeration of larger set sizes (e.g., between 4 and 10) as a potential
influence on their mathematical abilities.
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Table of Contents
Abstract .......................................................................................................................................... ii
Table of Contents ......................................................................................................................... iii
List of Tables ................................................................................................................................. v
List of Figures ............................................................................................................................... vi
Chapter One: Introduction .......................................................................................................... 1
Statement of the Problem .................................................................................................... 4
Purpose of the Current Research ......................................................................................... 5
Chapter Two: Literature Review ................................................................................................ 7
Perceptual Subitizing: Mechanisms and Theories .............................................................. 7
Conceptual Subitizing ....................................................................................................... 12
Stages of development .......................................................................................... 13 Mechanisms and theories ...................................................................................... 15 Empirical support for conceptual subitizing: Patterns, groups, and groupitizing . 19
Conceptual Subitizing and Mathematics Achievement .................................................... 24
Current Study .................................................................................................................... 27
Research questions ................................................................................................ 31
Chapter Three: Methods ............................................................................................................ 33
Research Design ................................................................................................................ 33
Participants ........................................................................................................................ 33
Materials ........................................................................................................................... 36
Dot enumeration task ............................................................................................ 36 Participant interviews ............................................................................................ 38 Mathematics achievement ..................................................................................... 39 Teacher questionnaire ........................................................................................... 40
Procedure .......................................................................................................................... 41
Statistical Analysis ............................................................................................................ 41
RT performance and mathematics achievement ................................................... 41 Cluster analysis ..................................................................................................... 42
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Group differences in mathematics achievement ................................................... 44 Strategy differences between clusters ................................................................... 44
Chapter Four: Results ................................................................................................................ 45
Student Exposure to Conceptual Subitizing Activities ..................................................... 45
Preliminary Inspection and Descriptive Analysis. ............................................................ 46
Accuracy and RT .................................................................................................. 46 Slope characteristics .............................................................................................. 50
Research Question 1: Is RT to determine post-subitizing range of numerosity (4-10) related to mathematics achievement? ............................................................................... 53
Research Question 2: Can RT Profiles In Different Display Conditions Show Differences in Subitizing Stages? ......................................................................................................... 55
Research Question 3: Do children in Different Clusters Show Difference in Math Achievement? ................................................................................................................... 58
Research Question 4: Does the Self-Reports on Strategies Differ Between the Clusters?59
Chapter Five: Discussion ............................................................................................................ 63
Student Exposure to and Performance on Conceptual Subitizing Activities .................... 63
Speeded Performance and Mathematics Achievement ..................................................... 67
Evidence for Conceptual Subitizing as a Developmental Process .................................... 69
Importance of Considering Children’s Enumeration Strategy .......................................... 73
Conceptual Subitizing Stages and Mathematics Achievement ......................................... 74
Limitations ........................................................................................................................ 76
Implications ....................................................................................................................... 77
Future Directions .............................................................................................................. 79
Conclusions ....................................................................................................................... 80
References .................................................................................................................................... 82
Appendix A – Dot Enumeration Task Arrays .......................................................................... 94
Appendix B – Teacher Questionnaire ....................................................................................... 96
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List of Tables
Table 1: Expected RT Performance for Different Subitizing Stages ............................................ 29
Table 2: Demographic Characteristic of Child Participants ......................................................... 35
Table 3: Mean Percent Accuracy .................................................................................................. 47
Table 4: Mean Reaction Time in Seconds .................................................................................... 47
Table 5: Average Reaction Time Slopes and R2 for Different Display Conditions for Numerosities 4-10 and the Perceptual Subitizing Range .............................................................. 51
Table 6: Average Reaction Time Slopes and R2 for Different Display Conditions for Numerosity Ranges 4-6 and 7-10 ..................................................................................................................... 53
Table 7: Pearson Correlation Between Reaction Time Slopes (Range 4-10) and Mathematics Achievement ................................................................................................................................. 54
Table 8: Regression Analyses Predicting Mathematics Achievement Scores From Two Reaction Time Slopes (Range 4-10) ............................................................................................................ 54
Table 9: Cluster Characteristics of Mean Reaction Times in Seconds ......................................... 57
Table 10: Number and Percentage of Strategy Use in Canonical and Grouped Display Conditions by the Three Clusters Combined ................................................................................................... 60
Table 11: Inaccurate Responses Provided in Each Cluster and Strategy ...................................... 62
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List of Figures
Figure 1: Canonical Patterns Created for the Task ....................................................................... 37
Figure 2: Mean Reaction Time for Each Display Condition in Seconds ...................................... 48
Figure 3: Scree Plot Using the Amalgamation Coefficients ......................................................... 56
Figure 4: RT Performance for Numerosities 4 to 10 in the Canonical and Grouped Display Conditions by the Three Clusters Formed by Cluster Analysis .................................................... 58
Figure 5: Percentage of Each Strategy (Part, Whole, Count on, Count All, Other) Used by Students in the Three Clusters ...................................................................................................... 61
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Chapter One: Introduction
The role of mathematics in our everyday lives is immense (Ansari, 2013; Mirra, 2004;
Patton, Cronin, Bassett, & Koppel, 1997). In addition to simple calculations performed in the
community such as comparing and adding prices at a grocery store and determining the wait time
for buses, the increase of jobs in the science and technology fields require students to acquire
strong mathematical knowledge to succeed (Mirra, 2004). The understanding of mathematics
and number not only influences an individual’s life, but it could impact his or her economic and
health status within society. For example, a longitudinal study in Britain revealed that low
numeracy skills in individuals lead to unemployment and/or lower income, physical and mental
illness, and higher rates of incarceration (Bynner & Parsons, 2005). Furthermore, a recent
analysis by the Organization for Economic Co-operation and Development (OECD) revealed
that historically an increase in an individual’s mathematics and science performance by one-half
standard deviation was related to a 0.87 percent increase in annual GDP per capita (Hanushek &
Woessmann, 2010). Mathematical knowledge therefore has great significance on many domains
of our lives. Given the importance mathematics has on later individual and economic success, it
is not surprising that mathematics learning and cognition is a considerable focus of research.
A number of core skills are argued as essential to building children’s earliest
understandings of mathematics and number, or number sense (Feigenson, Dehaene, & Spelke,
2004; Gersten & Chard, 1999). Among these is subitizing, defined as the quick recognition of
the exact amount of small (up to four items) sets of objects (Feigenson et al., 2004; Kaufman,
Lord, Reese, & Volkmann, 1949). In particular, arguments have been made that the subitizing
process supports the development of counting skills in children (Benoit, Lehalle, & Jouen, 2004;
Bermejo, Morales, & Garcia deOsuna, 2004; Starkey & Cooper, 1995). By attending to the
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magnitude of the set of visually presented items, children develop the idea of one-ness, two-ness,
and three-ness of visual objects and, over time, associate these perceptual sets with number
words (Butterworth, 2010; Gallistel & Gleman, 1992; Gelman & Gallistel, 1978). In other
words, they learn to associate the number and the magnitude it represents (Hannula & Lehtinen,
2005; Trick & Pylyshyn, 1994). In alignment with such views, researchers have found that
children’s subitizing skill strongly predicts their mathematics abilities in the early years
(Desoete, Ceulemans, Royers, & Huylebroeck, 2009; Reigosa-Crespo et al., 2013). Moreover,
children with mathematics learning disability (MLD) or dyscalculia show slower subitizing
latency (Butterworth, 2003; Landerl, 2013; Moeller et al., 2009; Schleifer & Landerl, 2011)
compared to typically developing peers. As such, subitizing appears to both influence children’s
ability to grasp the idea of numbers and later support the understanding of mathematics concepts.
More recent attention has been given primarily by educational researchers to the possible
existence of a higher-order form of subitizing, coined by Clements (1999) as conceptual
subitizing. The term describes the skill to quickly determine the quantity of collections above the
earlier detailed subitizing range of up to three or four items (which Clements refers to as
perceptual subitizing). Conceptual subitizing has been postulated to entail the rapid recognition
of the numerosity of visual representations of number (e.g., dice dot patterns, finger patterns).
This process occurs more quickly than does counting and it involves the recognition of a certain
number as a whole but also as comprised of smaller parts. According to Clements, individuals
who quickly identify the numerosity of a domino eight dot pattern using their conceptual
subitizing skill can focus on two sets of “four dots” and readily identify this as “one eight.” A
mere recognition of the domino pattern as “eight” therefore does not signify the engagement in
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conceptual subitizing; the ability to consciously see the whole as well as the parts of the whole is
the defining attribute of conceptual subitizing.
The way in which groups of objects are arranged appears to impact how quickly such
arrays can be conceptually subitizied. Structured or familiar patterns (e.g., rectangular
arrangement, symmetrical arrangement of objects) are detected more quickly than less structured
arrays (Beckwith & Restle, 1966; Sarama & Clements, 2009). With practice, children also
become more efficient with seeing the smaller groups within less structured patterns (e.g.,
random dot patterns; Clements, 1999). Children hence acquire and develop the conceptual
subitizing skill over time with practice and are able to see the numbers as a whole, while also
understand that the whole is made up of smaller parts. To be able to see numbers this way,
children require a higher-order understanding of number sequences, where each number is
composed of the previously counted numbers (or units; Steffe, Cobb, & von Glaserfeld, 1988).
The understanding that numbers are a unit of units (or parts representing the whole) is argued to
be essential for developing a deep understanding of number and arithmetic strategies (Clements,
1999). As conceptual subitizing requires more advanced skills than perceptual subitizing,
Clements claimed that children require experience with number and instructions to develop and
advance their conceptual subitizing skills.
Taking on this account, there has been increased discussion within educational contexts
of the importance of subitizing skills (e.g., Baroody, Lai, & Mix, 2006; Clements & Sarama,
2014; van der Walle, 2004). Currently, the mathematics curriculum in Alberta suggests that
students should be able to subitize objects up to five in Kindergarten and up to 10 objects in
Grade 1 (Alberta Education, 2014). Multiple book chapters, teacher conference notes, and
meeting documents available online outline various methods and activities to introduce
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structures or patterns to children that support a deeper exploration with numbers. For example,
some of the popular activities discussed by educators and educational researchers include: rolling
two dice and quickly identifying the overall number, flashing dot patterns with dot plates (cards
with common dot patterns like dice patterns displayed), use of ten-frames (rectangular cards with
two rows of five square boxes) with counters to represent numbers up to 10, and matching dot
cards with equal numerosity (Bobis, 2008; Huinker, 2011; van der Walle, 2004). While these
activities are not always directly identified as “subitizing” activities, they nevertheless help
children explore numbers in multiple ways that may support their ability to identify the
numerosity of collections quickly akin to conceptual subitizing.
Statement of the Problem
Despite the wide acceptance of conceptual subitizing within the field of education, a
limited number of empirical studies are available on this skill. While the term subitizing
generates 222 articles on one of the more prominent psychological research search engines,
PsychINFO, the term conceptual subitizing generates no research articles. A similar pattern is
also found within the education literature, where 39 articles are identified with the term
subitizing on the search engine ERIC, while none are found with the term conceptual subitizing.
A more generic search engine, Google Scholar would suggest 43 documents available online for
the term conceptual subitizing, of which only nine are published in journals. A number of these
documents are papers that outline the effect of training children in number representations using
different materials on mathematics understanding (discussed further in chapter two). The
remainder of the documents identified through the Google Scholar search are book chapters,
teachers’ conferences, and meeting documents that review the methods or outline activities to
increase children’s familiarity with numbers and patterns, such as those discussed above (e.g.,
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dot plates, ten-frames). Thus, although conceptual subitizing is a fairly commonly used term in
the educational field, empirical exploration of the concept is still limited.
Given the limited study of conceptual subitizing skills to date, virtually nothing is known
regarding the behavioural profile of children who engage in conceptual subitizing (i.e., speed and
strategies to enumerate displays). This is particularly surprising given that the speeded
recognition of numerosity is considered a hallmark feature of conceptual subitizing and what is
purported to distinguish it from counting. As such, it is debatable at this juncture whether the
teaching focus should be on improving children’s speed at which they determine the numerosity
of sets, or whether the understanding of the part-whole nature of numbers is enough to help
deepen children’s understanding of number and arithmetic. In addition, while a few training
studies on number representation that are similar to conceptual subitizing suggest an
improvement on children’s understanding of number (discussed in the next chapter), there is no
statistical investigation on the relationship between conceptual subitizing and mathematics
achievement per se. Thus, the education community has adopted the idea of conceptual
subitizing without a full understanding on the skill itself, its development, or its relationship to
mathematics. Further examination of this skill is necessary to argue the importance of including
conceptual subitizing in early education and to implement the appropriate activities to facilitate
its development.
Purpose of the Current Research
The purpose of the study is to provide initial empirical research on conceptual subitizing.
The current research examines the speed at which children enumerate objects within the post-
subitizing range (four to 10 items) and the relationship between enumeration speed and
mathematics achievement. In addition, the study provides preliminary exploration of children’s
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developmental progression in acquiring the conceptual subitizing skill based on their reaction
time on conceptual subitizing activities. A deeper exploration of the conceptual subitizing skill
could inform educators on the importance of incorporating activities that promote children’s
engagement in conceptual subitizing. Further, should a relationship between conceptual
subitizing speed and mathematics achievement be found, there is the possibility that the rapid
enumeration task used in this study could serve as a screening measure for determining students
who are facing, or will face, difficulties in other mathematics activities. In addition, practicing
conceptual subitizing skill may further be suggested as an activity for children with mathematics
learning difficulties to aid their understanding in mathematics and number. Therefore, an
investigation on the conceptual subitizing skill is essential for improving mathematics
instructions to assist children in having a deeper understanding of number concepts.
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Chapter Two: Literature Review
Conceptual subitizing is a skill that develops with age and experience with numbers
(Sarama & Clements, 2009). Although the underlying mechanisms are not yet fully determined,
it is believed to build upon the more innate perceptual subitizing skill. Given this developmental
progression, it is necessary to first consider the theoretical perspectives regarding the function
and mechanisms underlying perceptual subitizing. The following sections review the literature
on the exact enumeration processes discussed in the mathematics cognition and the education
literatures. First, an overview of perceptual subitizing and the theories on the processes behind
the phenomenon are discussed, retrieved mainly from the mathematics cognition literature.
Then, an overview of conceptual subitizing and the developmental trajectory of the skill are
examined, as well as the discussion on its relation to mathematics achievement, which is derived
mainly from the mathematics education literature. The chapter concludes by outlining the goals
of this study and the hypotheses that were examined.
Perceptual Subitizing: Mechanisms and Theories
As briefly mentioned previously, subitizing has been proposed as the mechanism by
which children and adults quickly and accurately determine the exact quantity of small
collections. As this awareness of small and exact quantity is observed in infants as young as five
months of age, subitizing is considered as one of the innate numerical processing skills available
in humans (Feigenson, Carey, & Hauster, 2002; Lipton & Spelke, 2003; Wynn, 1992; Wynn,
Bloom, & Chiang, 2002). While there is continued debate on the mechanism behind this rapid
enumeration skill (discussed in more detail later), it has been repeatedly shown that individuals
are able to subitize up to around three or four items (e.g., Dehaene & Cohen, 1994; Jensen,
Reese, & Reese, 1950). Once a visual set becomes larger than an individual’s subitizing range,
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children and adults tend to attend to each item and associate the items with number words to
determine the numerosity, the process of counting (Hannula, Räsänen, & Lehtinen, 2007,
Svenson & Sjöberg, 1983).
The distinctiveness of this skill is repeatedly observed from the rapid speed at which
people identify the numerosity of quantities up to three or four within around 40-100 ms per item
(Trick & Pylyshyn, 1993). The time it takes to detect the amounts above this range (i.e.,
engaging in counting) significantly increases, taking about 250-350 ms per item (Trick &
Pylyshyn, 1993). This change in the enumeration process (from subitizing to counting) is
evident when viewing the reaction times (RT) for different quantities when plotted on a graph.
For the range of one to three or four items, the RT is fairly stable resulting in a relatively flat RT
slope. A sudden change in the slope is visible at the point where individuals engage in counting,
as demarcated by a significant linear increase in RT per item. This point is referred as the point
of discontinuity, with the subitizing and counting ranges represented by different linear
regression line slopes (Reeve, Reynolds, Humberstone, & Butterworth, 2012).
The fundamental process involved in perceptual subitizing is still under debate.
Amongst the numerous extant theories on the mechanism underlying perceptual subitizing, the
object tracking system appears to be one of the more empirically supported. The authors of the
object tracking system theory argue that perceptual subitizing utilizes a visual mechanism in
which people place a visual marker to each object that allows individuated attention (Trick &
Pylyshyn, 1994). These markers are assigned to the items pre-attentively (i.e., without conscious
attention), and people use this pre-attentive, visual information to quickly process the number of
objects and match it with the number names. Trick and Phylyshyn thus consider perceptual
subitizing as a non-numeric, two-step process; the markers are first assigned to the items pre-
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attentively; next, people consciously assign the number name to this perceptual experience.
Trick and Phylyshyn explain that perceptual subitizing is limited to small numbers of items
because of the limited amount of visual markers people could track at a time. When the number
of items in the display exceeds the number of traceable visual markers (i.e., not subitizable),
people start to attend to each item to assign each marker, which is the process of counting.
Therefore, the object tracking system theory, in essence, argues that people pre-attentively store
small quantities in the sensory memory as a whole first and then quickly retrieve the sensory
quantity separately (as parts) to enumerate the whole display (Wender & Rothkegel, 2000). This
distinction between perceptual subitizing and counting could thus be viewed as a whole-first,
part-second (whole-part) process versus a part-first, whole-second (part-whole) process,
respectively.
In support of this theory is neural evidence of differences in the visual mechanisms
engaged during perceptual subitizing versus counting. While both skills involve brain regions
responsible for vision, counting seems to entail greater activation of certain visual regions. For
example, Piazza, Mechelli, Butterworth, and Price (2002) have found increased activation in the
occipital and intraparietal areas when counting (six to nine dots) compared to subitizing (one to
four dots). Ansari, Lyons, Eimeren, and Xu (2007) similarly found an increased activation of
occipito-parietal region in the counting range, while they also found that processing items in the
perceptual subitizing range showed an increased activation in the right temporo-parietal junction
compared to the counting range. Thus, it is evident that visual processing (prominent in the
occipital and occipito-parietal junction area) is more involved in the counting range than the
perceptual subitizing range, in line with Trick and Pylyshyn’s (1993) argument for requiring
visual attention to each item to enumerate through counting.
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Mandler and Shebo (1982) proposed an alternative theory that attributes pattern
recognition as the mechanism underlying perceptual subitizing. They argued that children and
adults build familiarity with the typical patterns that represent two (e.g., line) and three (e.g.,
triangle), which allows for a fast processing of small numbers. These typical arrangements of
dots to represent numerosity are often called canonical patterns, in which Mandler and Shebo
described them as patterns that “do not contradict cultural expectations, that are symmetrical, and
that may be discriminated from one another with relative ease” (p.15). Mandler and Shebo
observed in their study that there is no significant difference in RT or accuracy between
randomly placed dot displays and dots arranged in familiar, canonical patterns for displays up to
three. Therefore, the authors argued that any configuration of doublets or triplets could be
recognized as two and three, and thus, subitizing occurs. For numerosity above this range,
however, a significant difference between RT and accuracy was observed between randomly
arranged dots and canonical patterns. Thus, the authors argued that random generations of items
in this range often do not produce a familiar pattern because of the exponential increase in the
possible arrangement of dots as the number of items increase, and thus, children and adults
switch to counting (Logan & Zbrodoff, 2003; Mandler & Shebo, 1982).
The pattern recognition approach to perceptual subitizing has been heavily discussed by
researchers, which resulted in the theory later modified to specify that the recognition of the
patterns are done by the outline of the shapes of the dots (i.e., two often forming a line, and three
often forming a triangle shape; Dehaene, 1992; Krajcsi, Szabó, & Mórocz, 2013; Trick, 1992).
A major criticism of the modified pattern recognition theory has been that it fails to account for
how individuals would distinguish between two and three objects presented in a line. In
alignment with the modified pattern recognition theory, it would be expected that if individuals
11
are presented with three dots positioned in a line, a frequent mistake would be to identify the
numerosity of the set as “two,” given that two dots can sufficiently form the shape of a line.
However, evidence suggests this is not the case. Rather, individuals accurately and rapidly
process three dots positioned in a line (Atkinson, Campbell, & Francis, 1976; Frick, 1987).
Despite such criticism, some evidence provides support for the original pattern recognition
theory suggested by Mandler and Shebo (1982). For example, neuroimaging studies by Fink and
his colleagues (2001) found that the temporo-occipital cortex is activated in tasks involving both
shape recognition and numerosity recognition of dot arrays. Thus, in both types of tasks, visual
pattern recognition appears to play an important role.
Other studies using familiar (canonical) patterns further provide evidence for the pattern
recognition theory by showing that, when canonical presentations are used, adults can process
quantities above their perceptual subitizing range faster and more accurately compared to when
the presentation is non-canonical (Krajcsi et al., 2013; Mandler & Shebo, 1982; Piazza et al.,
2002). In fact, Krajcsi et al. (2013) found participants’ RT to canonical patterns up to six dots
were similar to the perceptual subitizing range. Such work suggests that the subitizing range can
be expanded using canonical patterns. Similar effects were recently observed in children four to
six years old. Jansen and her colleagues (2014) presented children with different dot
configurations (random, line, and dice patterned dots) up to six items and observed their
accuracy. They found that children were more accurate in the dice pattern condition than the
random or line displays, suggesting their use of pattern recognition to enumerate the displays.
From these behavioural studies, researchers that support the pattern recognition theory argue that
pattern recognition should not be dismissed as a theory behind perceptual subitizing (Krajcsi et
al., 2013; Piazza et al., 2002). Extending the earlier whole-part discussion, familiar patterns may
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allow individuals to process both small (1-3 items) and larger arrays (above 4 items) as a whole
first and then link the patterns to the number words to enumerate. While similar to the whole-
first process suggested by the object tracking theorists, the pattern recognition theory
additionally accounts for a whole-first process in the rapid recognition of familiar displays above
the perceptual subitizing range.
In summary, children and adults’ ability to rapidly and accurately enumerate small set
sizes is a widely reported and robust finding. As a skill seen in early childhood, perceptual
subitizing provides an initial step to quantifying numbers, which is a core skill for mathematics
understanding. While the underlying mechanism for perceptual subitizing is still under debate,
perceptual subitizing is argued to be a foundational skill of a more complex form of rapid
enumeration of quantity, namely, conceptual subitizing. The proposed developmental trajectory
in acquiring the conceptual subitizing skill is discussed in the next section.
Conceptual Subitizing
Clements and other educational researchers have argued that subitizing is not limited to a
“low-level” (p. 401), biological sense of quantity but rather is also inclusive of a high-level,
conceptual understanding of numbers (Clements, 1999). Clements distinguished the smaller-
number (low-level) subitizing, which he termed perceptual subitizing, and the larger-number
(high-level) subitizing, which he termed conceptual subitizing. To date, there has been limited
empirical investigation into the characteristics of or mechanisms underlying conceptual
subitizing. Rather, researchers in this area have given initial attention to detailing a theoretical
account of conceptual subitizing. Sarama and Clements (2009) proposed that the conceptual
subitizing skill follows a developmental trajectory, with origins in the low-level perceptual
subitizing ability. Specifically, children encounter four developmental stages in advancing their
13
subitizing skill: pre-attentive and quantitative, attentive perceptual subitizing, imagery-based
subitizing, and conceptual subitizing. The following sections outline the behaviours and
strategies children use to enumerate the presented displays in each stage, and discuss the
proposed mechanisms and theoretical support provided for the stages of development.
Stages of development. The first of the four developmental stages is the pre-attentive
and quantitative stage, which describes the initial quantification process seen in infants.
Considered to be the most primitive, Sarama and Clements (2009) argue that young children first
process the objects’ information and individuate the objects visually using the “object files”
system proposed by Kahneman, Treisman, and Gibbes (1992). This perceptual system allows
each item in the array to be represented as a symbol (file) without consciousness (thus defined as
a pre-attentive stage). Pulling from multiple theories on early quantification systems, Sarama
and Clements argue that these visual experiences could be given some quantitative meaning.
The authors reference two systems that could support this process: For small quantities, the
“number perception module” (Dehaene, 2011) that could process a very small quantity (one or
two) exactly, allows children to distinguish the difference between one and two items. For larger
items (above two), an estimator (like an accumulator of quantity, suggested by Meck and
Church, 1983) activates to process the approximate numerosity of the objects in the array. As
such, Sarama and Clements argue that children in this stage can distinguish small quantities of
items, as seen in infant studies (e.g., Feigenson et al., 2002; Wynn, 1992). However, as the
number perception module and the accumulator are not yet connected with the number words (as
the infants are nonverbal), children at this stage do not understand the cardinality of the array
(i.e., cannot identify the number of elements in the set).
14
Once young children have practiced and recognized that the previous sensorimotor
experiences (i.e., individuating objects) of a certain numerosity are similar to one another, they
become more attentive to the numerosity of the array (Hannula & Lehtinen, 2005; Sarama &
Clements, 2009; Steffe, 1992). With the development of number names, children then connect
these visual experiences with enumeration experiences (e.g., naming collections), and move on
to the next stage. Known as the attentive perceptual subitizing stage, it is different from a mere
sensitivity to the visual experience of the preceding stage as children can now quantify a set of
objects (Sarama & Clements, 2009). Children in this stage can quickly and accurately
individuate and process many different kinds of objects and, thus, the authors argued this stage to
be akin to the original definition of subitizing. In accordance with the definition of subitizing,
this stage allows quick enumeration of items up to three or four. Quantification of numerosity
beyond this amount requires other methods, such as counting.
In general, the two initial stages of conceptual subitizing follow a similar developmental
course to learning the number concepts in childhood. The next two stages of conceptual
subitizing development involve furthering this understanding of numbers in children.
Acquisition of the higher-level, larger-number subitizing skills (i.e., imagery-based and
conceptual subitizing) is believed to occur through repeated practice and exposure that supports
the development of internal representations (i.e., imagery) of larger numerosities (Clements and
Sarama, 2014). Clements and Sarama suggested that children require repeated opportunities to
engage in multiple activities that allow them to increase their familiarity with different
arrangements of the same numerosity. Such activities help children to readily see the numerosity
in the perceptual subitizing range (i.e., one to three or four), and also develop the internal
representation (i.e., imagery) of larger numerosities.
15
In the imagery-based subitizing stage, children use these internal representations to
quickly process larger numerosities. A particular characteristic of this stage is children’s use of
these imageries without a full account of what constructs them. As such, children may identify
the correct numerosity (e.g. six) of well-practiced patterns, but only through recall of familiar
pattern-number associations. In other words, children in this stage use the structure as a whole in
and of itself. With unfamiliar patterns or larger structures, children may not use the structure
effectively. For example, they may try to locate the known structure in the array and count the
remaining items individually. In the conceptual subitizing stage, children understand that the
practiced patterns (e.g., six) consist of smaller components (e.g., two threes). With this
understanding, children become faster at recognizing the numerosity of the presented array as a
whole, as well as recognizing smaller groups of objects within the whole. In cases when children
encounter larger sets of objects or unfamiliar arrangement of numerosity, children can more
readily use the smaller groups to recognize the numerosity of the entire array (Sarama &
Clements, 2009). For example, increasing familiarity with the ten-frame structure allows easy
visualization of numbers beyond 10. Thus, 25 can be readily identified as two tens and one five,
or five fives (Sarama & Clements, 2009). The authors argue that the skill to see the parts in the
whole is therefore the key process of conceptual subitizing. Developing and using the internal
imagery of certain numerosities is the first step (imagery-based subitizing); children then are able
to visually decompose and recompose the imagery without counting (conceptual subitizing).
The ability to see numbers visually in this manner should support children’s deep understanding
of larger numbers and arithmetic operations in the later years (Sarama & Clements, 2014).
Mechanisms and theories. To date, limited attention has been given to specifying the
mechanisms responsible for conceptual subitizing and its development. Sarama and Clements
16
(2009) have largely referenced existing theories to account for their first two proposed stages,
although even here they provided limited discussion of the underlying mechanisms. Their
specification of possible mechanisms for the latter two stages is even more limited, as is
discussion on the speed at which children can engage in conceptual subitizing. However, some
speculation of the mechanisms for the higher stages can be drawn from the perceptual subitizing
theories, which is discussed below.
For the initial two stages (i.e., pre-attentive and quantitative, attentive perceptual
subitizing), Clements and Sarama (2014) primarily referenced existing theories on early
numeracy discussed in the cognition literature. As detailed above, the pre-attentive and
quantitative stage relies on the theories for the exact magnitude representation and the
approximate magnitude systems, which are the two core systems of number discussed by
Feigenson et al. (2004). The attentive perceptual subitizing stage is explained with the original
theory on subitizing; the idea that perceptual subitizing involves a “numerical” process of
quantifying the visual information of small set sizes (Kaufman et al., 1949). The authors point
out its similarity to Kaufman’s definition of subitizing to contrast it from the heavy focus on the
visual mechanisms (e.g., tracking objects, finding patterns) in the perceptual subitizing literature.
However, Kaufman and colleagues do not specifically discuss the underlying processes in
perceptual subitizing, but rather focus on outlining the distinctiveness of the subitizing skill from
the other enumeration processes. While Sarama and Clements stress the importance of
considering subitizing as an enumeration process rather than a mere innate perceptual ability
(and thus call it the “attentive” perceptual subitizing), a lack of discussion on the cognitive
processes limits our understanding of the perceptual subitizing skill. In addition, it obscures our
17
understanding of the skills that comes later in development, as the authors argue that conceptual
subitizing develops from the perceptual subitizing skill.
For the higher-level stages (i.e., imagery-based subitizing and conceptual subitizing),
Clements and Samara point to the importance of the internalization of patterns and structures.
However, what is not clear is what mechanism supports the development of these mental
templates or possible links with the mechanisms responsible for the initial two stages. Despite
the lack of attention Clements and Samara have given to underlying mechanisms, it is possible to
argue that the perceptual subitizing theories discussed above can offer some insights to the
processes involved in the higher-level stages. In particular, an argument could be made that
conceptual subitizing skill, as described by Sarama and Clements (2009), is supported by both
the ability to use familiar and internalized patterns (suggested by Mandler and Shebo) and to
track objects as groups (suggested by Trick and Pylyshyn) to process the post-subitizing range of
numbers. Thus, conceptual subitizing could be argued as explainable with a combination of, and
an extension to, the pattern recognition theory and the object tracking theory of perceptual
subitizing.
Sarama and Clements’ (2009) focus on internalization of patterns and structures for the
higher-level subitizing skill, especially in the imagery-based subitizing stage, aligns with the
pattern recognition theory for perceptual subitizing. The pattern recognition theory argues that
the typical shapes formed by the small quantity of dot displays allow perceptual subitizing to
occur (Mandler & Shebo, 1982). Similarly, Clements and Sarama (2014) argue that children
create mental templates of post-subitizing numerosities through various experiences and
extended practice, which, in turn, allows for the rapid recognition of set sizes up to about five
(above perceptual subitizing range). In alignment with the rapid enumeration expected in higher-
18
order subitizing, the canonical pattern study outcomes provide support that familiar patterns
speed up the enumeration of larger arrays. It seems that familiar pattern are seen as a whole that
assists with the rapid identification of the numerosity of the array (whole-first process).
While pattern recognition mechanisms alone could account for how larger number
patterns above four items are readily recognized, it is possible that the object tracking system
may also support conceptual subitizing skills. In particular, conceptual subitizing involves both
the ability to quickly recognize certain “whole” patterns as well as the rapid recognition of
smaller groups in an array. Trick and Pylyshyn (1994) have postulated that as individuals
become efficient at enumeration (subitizing and counting), the visual trackers could be assigned
to groups of subitizable sizes (2-4 items) rather than single items. Individuals then add these
groups (or count in groups) to reach the total numerosity (i.e., part-first, whole-second process).
Similarly, Sarama and Clements view conceptual subitizing as involving children’s ability to
accurately identify the smaller groups in the whole. Yet, Sarama and Clements seem to argue
further that children would not just “count” the groups, but rather be fast and accurate at
“subitizing” the groups together to enumerate the whole, especially if the structure was
unfamiliar. The authors therefore seem to suggest that conceptual subitizing could decrease the
speed of the part-first, whole-second process at which children determine post-subitizing ranges.
Further, with the increased familiarity with mental templates of the post-subitizing ranges,
children could readily identify groups of larger-sized groups (above four items) than what Trick
and Pylyshyn suggests. Thus, children may be able to decrease their speed at enumerating even
larger quantities with conceptual subitizing skills. However, as Sarama and Clements have not
detailed mechanisms responsible for the latter two stages, the above discussion on the
mechanisms of conceptual subitizing is, at this point, largely speculative.
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Empirical support for conceptual subitizing: Patterns, groups, and groupitizing. As
mentioned earlier, conceptual subitizing is a fairly widely used term within the field of education
(Alberta Education, 2014), yet has had limited empirical study to date. In absence of empirical
studies specific to conceptual subitizing, findings from the cognition literature best inform our
understanding of the behavioural characteristics when engaging in conceptual subitizing.
Specifically, of particular relevance to our understanding of conceptual subitizing are studies that
have explored the perceptual subitizing skill using set sizes in the post-subitizing ranges (up to
10 items) and different display arrangements (e.g., dot patterns, grouping cues). The following
sections review such studies.
Subitizing patterned structures. A number of researchers have examined RT and
accuracy with canonical patterns of dots (i.e., patterns that are frequently encountered and
overlearned, Mandler & Shebo, 1982). Typically, this work has used quantities of up to 10 items
and a variety of canonical pattern arrangements. Common displays for quantities between “1”
and “6” include arranging items in geometric shapes (e.g., for “2”, for “3”) and
common dice patterns (e.g., for “2” and for “3”). For numerosities above seven, the
variability in displays is even wider with, for example, eight dots displayed as two rows of four
dots (Mandler & Shebo, 1982), two columns of four dots (Ashkenazi, Mark-Zigdon, & Henik,
2013), or two columns of three dots with a column of two dots in the middle (Krajcsi et al.,
2013).
Despite this variability in displays used, researchers that has examined enumeration speed
of canonical displays above the perceptual subitizing range have found that structured patterns
improve the speed in determining larger numerosity compared to counting (Jansen et al., 2014;
Krajcsi et al., 2013; Piazza et al., 2002). Further, a recent study showed that children who were
20
identified as having Developmental Dyscalculia (DD) failed to benefit from the canonical
arrangement of dots up to nine, as they showed no enumeration speed difference between
random and canonical pattern stimuli (Ashkenazi et al., 2013). Thus, the ability to use patterned
structures to improve efficiency in enumeration may be lacking in children with DD, suggesting
that conceptual subitizing may also be a challenge for them. In general, the findings from
subitizing studies using canonical patterns support the internalization of patterns (structured
arrays) as assisting children in quickly and accurately recognizing numerosities above the
perceptual subitizing range.
Grouping skills. In addition to studies that have examined the recognition and
enumeration of canonical patterns, a number of researchers have examined subitizing with
grouped sets. As conceptual subitizing involves the recognition of smaller groups within the
whole, and also utilizing those groups to enumerate the array, it is important to see if this skill
has been explored in other research. Wynn, Bloom, and Chiang (2002) examined infants’
abilities to see groups in visual stimuli. The authors showed 5-month-old infants either two sets
of three dots or four sets of three dots until they habituated to the stimuli (measured by the
decrease in time looking at the stimuli). They then measured whether infants would look longer
at subsequent trials that contained two collections of four objects versus four collections of two
objects. The authors found that infants who habituated to two collections looked longer at four
collections during the trial, while those who habituated to four collections looked longer at two
collections. Infants as young as 5 months could therefore track a collection of items as an
individual entity, as well as detect the parts of the array that were separated by space.
Wender and Rothkegel (2000) investigated the role of spatial distinctions in cuing adults
to group visual stimuli and add those groups to process the numerosity of the whole array. The
21
authors presented dot displays that consisted of spatially separated subgroups of dots, where
numerosities 3 to 6 items were presented using two subgroups, 7 to 9 items presented using three
subgroups, and 10 items presented using four subgroups. Each subgroup was kept within the
perceptual subitizing range (up to three items), with items arranged to form canonical patterns.
Wender and Rothkegel (2000) found that when adults were provided with spatially grouped
stimuli, they were able to process the numerosity of the whole array faster than when the stimuli
were randomly arranged. Further, the RT increased as the number of subgroup increased (i.e.,
RT significantly increased between numerosities 6 shown with two subgroups and 7 shown with
three subgroups, and 9 shown with three subgroups and 10 shown with four subgroups),
suggesting that adults where conceptually combining the parts.
A recent paper by Starkey and McCandliss (2014) discussed a similar concept to
conceptual subitizing, which they called groupitizing (McCandliss et al., 2010). They defined
groupitizing as the speeded enumeration process by using grouping cues (i.e., spaces between the
dots) to create subgroups within the perceptual subitizing range. The authors presented children
in kindergarten to Grade 3 with stimuli that contained five to seven dots; half of the stimuli were
unstructured, where the dots were randomly placed on the display, while the other half was
grouped into three subgroups (e.g., five was always presented with two subgroups of two dots
and one more dot, six was presented with three subgroups of two dots, and seven as two groups
of three dots and one more dot). The subgroups were arranged so that the dots would not
produce a canonical pattern. Starkey and McCandliss found that the stimuli with subgroups of
dots allowed children in all age groups to enumerate the grouped arrays faster than the
unstructured arrays, but the older children (in Grades 2 and 3) benefited more from this
arrangement compared to the younger children (in kindergarten and Grade 1). The authors
22
explained that the older children’s experience with counting on and the understanding that
numbers are made up of smaller numbers supported their groupitizing skill, leading to the
benefits in speed.
Discussion of related studies. Empirical studies from the cognition literature on speeded
processing of dot structures thus provide some initial support for conceptual subitizing. The
canonical pattern studies show that familiar images allow speedy enumeration of post-subitizing
numerosity, and the studies using stimuli with subgroups suggest that spatially grouped
structures assist in the process of quickly combining smaller parts to enumerate the entire array.
In addition, the groupitizing work specifically discusses the developmental nature of a similar
skill to conceptual subitizing, in that it requires experience with numbers to engage in part-whole
processing of the visual number representations (dots).
While such studies hold relevance to increasing our understanding of conceptual
subitizing, there are several areas in which the above outlined areas of study are limited in terms
of increasing our understanding of conceptual subitizing skills. First, these studies do not
examine “how” children are viewing patterned structures. According to the previously detailed
developmental theory of conceptual subitizing, it would be expected that children at different
developmental stages would focus on different aspects of the displays. For example, children in
the imagery-based subitizing stage may be able to process the familiar, canonical patterns fast,
but may not understand that the patterns consist of smaller parts to create the whole as is required
in the conceptual subitizing stage. Similarly, the studies using grouped display studies do not
explore whether children can flexibility see both the parts of the whole and the whole as
consisting of parts without any cues for groupings. Thus, it is still unclear whether children fully
understand the structure of presented patterns (i.e., the patterns are consisting of smaller parts),
23
which is important for the conceptual subitizing skill. In other words, findings from subitizing
research using canonical patterns appear to be supportive of the existence of the imagery-based
subitizing stage (i.e., quick processing of internalized representation of numbers), but have not
been designed to assess for the more advanced conceptual subitizing stage. Further investigation
is therefore necessary to see if there are differences in enumeration latency between children who
are seeing the structured patterns as just familiar patterns as a whole (i.e., in the imagery-based
subitizing stage) and those that comprehend the patterns as constructed of smaller parts (i.e., in
the conceptual subitizing stage).
Second, the subitizing studies using grouped structures have been limited to subgroup
sizes within the perceptual range (for both canonical and non-canonical structures). There have
not yet been studies of grouping skills in which the subgroups are comprised of items beyond the
perceptual subitizing range. Sarama and Clements (2009) encourage children to learn the visual
representations of numbers that are beyond their perceptual subitizing range, such as four and
five (done in imagery-based subitizing stage), in order to develop their efficiency with
conceptual subitizing. Therefore, it is still unknown whether children would be faster with
subgroups with larger numbers compared to random arrangements. Further exploration of
children’s performance on larger subgroups becomes necessary to see if, in fact, children do
develop internalized structures that are above their perceptual subitizing stage, and use them
effectively to quickly enumerate the entire array.
Third, while the improvement in the speed of enumeration using different structures is
discussed in the cognition literature, little is known about whether this is important for
mathematics understanding. The groupitizing study by Starkey and McCandliss (2014) provides
some evidence for the importance of the speeded enumeration of grouped items on mathematics
24
achievement. The authors showed that groupitizing speed uniquely predicted young children’s
mathematics fluency (addition and subtraction) independently from perceptual subitizing skills,
counting, and magnitude comparison tasks (identifying which of the two arrays has more dots).
Yet, the difference between conceptual subitizing and groupitizing discussed above limits our
full exploration on the level of influence conceptual subitizing speed has on children’s
mathematics understanding, and whether it is important over and above the other enumeration
strategies (i.e., perceptual subitizing and counting). The importance of conceptual subitizing in
general is discussed within the educational literature. Thus, while the cognition literature is
limited in linking the speeded enumeration of post-subitizing quantities and mathematics
achievement, the educational literature offers some attention to the connection between
conceptual subitizing and mathematics understanding, as discussed in the following section.
Conceptual Subitizing and Mathematics Achievement
Aside from a need to better understand the basic properties of and processes supporting
conceptual subitizing, further examination is needed of the link between conceptual subitizing
and mathematics achievement. To date, research in this area appears limited to two studies.
While further study in this area is warranted, these initial studies support interventions that target
the development of conceptual subitizing skills as impacting children’s mathematics abilities.
Clements and Sarama (2003) founded a project called Building Blocks, which entails a
number of activities to develop young children’s (Pre-K to Grade 2) understanding of the
concept of number and geometry. Specific to the number concept component of the project, the
authors created computer games intended to facilitate children’s conceptual subitizing skills.
One of these games is called Snapshots, which presents an array of dots (random and structured
presentations) for two seconds and children are asked to choose the corresponding number
25
(displayed in Arabic numbers) to the numerosity of the dots. Another game is called the
Dinosaur Shop, in which children are asked to determine the number of dinosaurs in a 2 × 5 (i.e.,
ten-frame) box as fast as possible. Clements and Sarama (2007) have shown that children who
played these games showed an increase in their conceptual subitizing skill (i.e., their accuracy on
identifying magnitudes five to 10) compared to a control group, suggesting the importance of
experience in improving larger number subitizing skills (Clements & Sarama, 2007).
Furthermore, children who were at-risk of developing mathematics difficulties (i.e., children
from lower SES with little exposure and experience with number) improved their overall
understanding of numbers and geometry by using the Building Blocks program (in its entirety),
as measured by Sarama and Clements’ own assessment system (The Building Blocks
Assessment of Early Mathematics, Sarama & Clements, in press). Similarly, Jung, Hartman,
Smith, and Wallace (2013) showed that preschool children who played the conceptual subitizing
games from the Building Blocks project improved their early mathematics ability (measured by
the TEMA – Third Edition; TEMA-3) more than those who received regular instructions. These
studies suggest that children who are trained in conceptual subitizing skills indeed improve their
mathematics achievement scores.
Similar to intervention studies that have directly targeted conceptual subitizing skills,
there are a number of studies in which emphasis has been given to developing children’s internal
representation of numbers in order to influence mathematics achievement. For example,
Mulligan, Mitchelmore, and Prescott (2006) have argued the importance of an understanding of
patterns and structure in children’s development of mathematical concepts. They define pattern
as a regularity in the environment, including space and numbers, and structure as the relationship
between patterns. They argue that this awareness of mathematical patterns and structures, such
26
as equal groups, units, rows and columns, numerical patterns and geometric patterns, is
important for mathematics achievement (Mulligan & Mitchelmore, 2012). Moreover, Mulligan
and her colleagues (2003, 2006) have found that children who are low-achievers in mathematics
tend to show a lack of understanding in structural representation of numbers. For example, when
children where shown a structured pattern of dots or boxes (e.g., dots arranged in a triangle, five
boxes in two rows) and asked to replicate the amount of dots and patterns they saw, low-
achievers tended to either reproduce the structure (but with the wrong quantity) or the quantity
(but with a dissimilar structure). In contrast, high-achievers tended to accurately reproduce the
shown structure with the correct quantity and a similar structure. The authors therefore
suggested that using structured representations of numbers in instructions is important to
facilitate children’s development of number concepts.
The importance of children developing internal representation of number is also
supported by a number of intervention studies using different manipulatives. For example,
Tournaki, Bae, and Kerekes (2008) trained Grade 1 teachers to use rekenreks (a manipulative
that consists of 20 beads in two rows (10 each), in which beads are colour-coded into sets of
fives) to teach addition and subtraction in numbers one to 20. Rekenreks utilize the five-
structure representation of numbers, allowing children to easily see and use doubles
addition/subtraction, making tens, borrowing and carry-overs instead of resorting to counting
(Fosnot & Dolk, 2001; McClain & Cobb, 1999). The participants were identified as having both
reading and mathematics disabilities based on New York state criteria (i.e., average IQ but
minimum two-year delay in reading and mathematics achievement scores). Children with
learning disabilities who received 30-minute of daily instruction using rekenreks for a three week
period significantly improved their addition and subtraction scores for numbers 0 to 20 compared
27
with those who did not receive daily instruction. The authors argued that the rekenrek helped to
facilitate the use of five- and 10-structure in arithmetic, and children developed their own mental
configurations of numbers that they could utilize (Tournaki et al., 2008). Similar structures such
as five- and ten-frames (one or two rows of five boxes) also has proved similar improvement in
children’s understanding of number concepts such as counting in preschool children (Jung, 2011;
McGuire, Kinzie, & Berch, 2012), further supporting the importance of using structured
representation of numbers.
In general, the mathematics education literature suggests that children can improve
certain aspect of mathematics understanding through improving their internal representations of
number using different tasks. These internal representations in turn seem to facilitate a faster
recognition of magnitudes above the perceptual subitizing range. Primary emphasis has been
given within the above outlined studies to improvements in children’s accurate reproduction or
use of structured groupings as it relates to mathematics achievement. However, less attention
has been given within the education literature to children’s speed on conceptual subitizing (and
related) tasks. If conceptual subitizing is related to perceptual subitizing, it is reasonable to
conclude that the speed at which children enumerate grouped sets is also important. Starkey and
McCandliss’s (2014) groupitizing study discussed above offers some insight into this
relationship of speed and math achievement. However, further evidence is needed to support this
view.
Current Study
Educational practices around instructing number concepts commonly incorporate
conceptual subitizing activities. However, there is, at present, limited empirical support to
suggest that children’s skill in conceptual subitizing is linked to improved mathematics
28
achievement. What is evident from the cognitive psychology literature is that structured arrays
facilitate children’s accurate and rapid enumeration of up to around 10 items. There is also
initial preliminary evidence to suggest that children’s skill to use the small parts to enumerate the
entre array predicts their mathematics fluency skills (Starkey & McCandliss, 2014). However,
such work largely focused on this as a low-level skill. Less is known, or has been studied, with
respect to conceptual subitizing as a higher-order skill that entails the conceptual understanding
of the array as a whole that consists of smaller parts, while utilizing subgroups that are larger
than their perceptual subitizing range (1-3 items), and its relation to mathematics achievement.
The current study attempted to address this gap in the literature by providing a preliminary
investigation into Clements and Sarama’s developmental stages of conceptual subitizing.
Emphasis was given to the last two stages (imagery-based subitizing and conceptual subitizing)
and exploring their unique behavioural profiles (i.e., RT, accuracy, and strategy use by stage).
Part of the difficulty in studying conceptual subitizing is that the developmental stages
have not yet been operationalized, particularly as it relates to expected RT performance specific
to the four stages. I aimed to address this in my study by proposing that children’s performance
on different types of numerosity representation may reveal their fundamental understanding of
number (i.e., part-whole and whole-part processing) and thus, reveal their developmental stages
of subitizing. Specifically, canonical patterns assess children’s ability to process structures as a
whole, whereas grouped patterns (consisting of subgroups) assess children’s ability to process
parts to represent the whole. Based on Clements and Sarama’s descriptions of the developmental
stages, the hypothesized RT on canonical and grouped patterns by developmental stage is
detailed in Table 1.
29
Table 1 Expected RT Performance for Different Subitizing Stages
RT on grouped pattern condition
RT on canonical condition Fast Slow
Fast
Conceptual subitizing stage Student fluid in viewing displays as “parts” and/or “wholes.”
Imagery-based subitizing stage: Type 1 Student more readily processes displays as “whole” than as “parts.”
Slow
Imagery-based subitizing stage: Type 2 Student more readily processes displays as “parts” than as “whole.”
Perceptual subitizing and counting stage (counters) Student counting the entire array regardless of the structure.
Essentially, children who are in the conceptual subitizing stage are expected to be fluent
at seeing both the whole and the part of number displays. As such, they are expected to be fast at
identifying the numerosity for familiar patterns as a whole in the canonical condition and in
using spatial cues of the grouped condition to quickly find the smaller groups (parts) of dots in
the array, and determine the numerosity (whole).
Children who are in the imagery-based subitizing stage are those that show less fluidity in
viewing numerosities as both wholes and parts. As Sarama and Clements (2009) stress the use
of the internalized imagery in this stage, it is generally expected that they would be quick at
recognizing familiar, canonical patterns as they view the pattern as a whole, but struggle with the
grouped display condition, as they are inexperienced at enumerating the whole array using
multiple parts. Children at this level are designated in Table 1 as being in the “imagery-based
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subitizing stage type 1.” However, as the hallmark of this stage is a lack of fluidity in moving
between “whole” and “part,” it is possible that some children at this stage could conversely find
it easier to enumerate the provided groups (parts) in the grouped arrays but are slower at
quantifying the whole canonical patterns (“imagery-based subitizing stage type 2” in Table 1).
Lastly, children who have not yet developed the higher-order subitizing skills may not
utilize the structure of the patterns to enumerate these displays (i.e., not fluid at seeing numbers
as a whole or as consisted of smaller parts). Children who are still counting individual items in
the display are marked by slow performance in both the canonical and grouped conditions.
Using the above proposed framework, Grade 2 students were asked to quickly enumerate
three types of display conditions for numerosities 4 to 10: (1) random, (2) canonical patterns, and
(3) spatially separated groups (grouped array). RT performances across the three display
conditions were explored to see whether the participants improved their enumeration speed using
the dot structures, and whether this performance could predict mathematics achievement. In
addition, the RT performance on the canonical patterns and the grouped arrays were compared to
determine if there was evidence for distinct stages in the development of conceptual subitizing
skills (expected performance is shown in Table 1). In addition, random arrangements of 1-3 dots
were included to examine children’s perceptual subitizing skills, and to compare the performance
with the higher-level subitizing performance. Much of earlier work in the cognitive literature
includes four items as the upper range for perceptual subitizing. The current study, however,
placed the upper perceptual subitizing range at three items, as there is evidence for children’s
subitizing range as limited to three that develops to four over time (Maylor, Watson, & Hartley,
2011; Reeve, Reynolds, Humberstone, & Butterworth, 2012; Svenson & Sjoberg, 1983).
31
The current study also extends previous work by examining the strategies children
reported using while enumerating the various arrays. Specifically, the Grade 2 students were
additionally interviewed and asked how they determined the total number of dots in the array.
The way in which they arrived at the total number of dots in the array is important to consider as
children in the different developmental stages of conceptual subitizing are expected to focus on
different aspects of the structure in the arrays (Sarama & Clements, 2009). As such, observing
the behavioural difference (i.e., RT) between the developmental stages of conceptual subitizing
and exploring the students’ self-report on their enumeration strategies are both crucial for a full
understanding of the conceptual subitizing skill. Explanations of their enumeration strategy on
the canonical and grouped array conditions were analyzed and compared against the
developmental stages determined from the RT performance.
Research questions. The four research questions explored in this study are outlined
below.
Question one: Does the RT to enumerate the post-subitizing quantity of dots (4-10)
relate to mathematics achievement? As there is some evidence to suggest that the rate at which
children enumerate using subgroups of dots predicts mathematics fluency (Starkey &
McCandliss, 2014), it is hypothesized that children who are faster at enumerating larger
quantities would be better at math achievement. While it is unknown whether RT performance
on canonical and grouped array conditions would show difference in the strength of association
with mathematics achievement, it is expected that both conditions would show some
predictability of math achievement scores. This prediction was based on Starkey and
McCandliss’ findings, as well as Ashkenazi et al.’s (2013) findings that children with DD failed
to benefit from the canonical patterns.
32
Question two: Can the RT profiles on the different display conditions (i.e., canonical
and grouped patterns) distinguish groups that are at different developmental stages of
conceptual subitizing? Based on the hypothesized difference in the RT outcome between the
developmental stages (see Table 1), children’s RT for numerosity 4 to 10 on the two conditions
were analyzed to see if different clusters could be created based on the RT performance. It is
expected that RT on the two conditions could distinguish the four different groups that represent
the developmental stages of subitizing, as outlined in Table 1.
Question three: Do children in different clusters show differences in their mathematics
achievement scores? It is expected that students with more advanced conceptual subitizing
skills also demonstrate higher mathematics achievement. Specifically, it is expected that those
who are in the conceptual subitizing cluster would attain higher scores in the mathematics task
compared to those that are in the imagery-based and counting clusters.
Question four: Do children’s reports on how they saw the quantity differ between the
clusters generated earlier? Comparing children’s self-reports on their enumeration strategies on
canonical and grouped display conditions between the groups created with question two above
could also inform the difference between children in different subitizing stages. It was expected
that those who were categorized as conceptual subitizers would show more flexibility in seeing
the numbers as both a whole and as consisting of smaller parts than those who were categorized
as imagery-based subitizers (i.e., similar with outlined response behaviours in Table 1). Those
who were categorized as counters in question two are also expected to report using the structure
of the arrays less, explaining that they counted out the dots to determine the quantity.
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Chapter Three: Methods
This chapter provides a description of the research design, participants, tasks and
materials used to investigate children’s enumeration skills and mathematics achievement, and the
procedures taken in conducting the current research. Consideration is also given to the statistical
methods applied to address the previously outlined research questions.
Research Design
The current study took a mixed method approach in investigating children’s conceptual
subitizing skill. Examination of the participant’s RT in enumerating the dot displays adopted a
within-subject design, in which all participants were presented with different display conditions.
Investigation on the participant’s strategies in enumerating the displays used a descriptive survey
approach, using an interview to solicit student feedback on how they enumerated the dots across
display conditions.
Participants
Thirty Grade 2 students (mean age 7.42 years, SD = 0.34) were recruited from two school
districts in Alberta for the current study. Grade 2 children were selected as it is expected that
they have already been provided with and should have the ability to subitize arrays up to 10
items. Specifically, the Alberta Grade 1 mathematics curriculum outlines that children should be
able to “subitize (recognize at a glance) and name familiar arrangement of 1 to 10 objects or
dots” (Alberta Education, 2014, p.13). In addition, Starkey and McCandliss found that children
in more advanced grades (Grades 2 and 3) were able to utilize the grouped structures better to
improve their efficiency in enumeration. One student was omitted from the study due to low
accuracy in correctly identifying the numerosity of displays and resulting inability to calculate
the required RT data. As such, data from 29 participant were included in the analysis.
34
Information on the students’ gender, age, first language and other languages spoken at
home, students’ length of time lived in Canada, and the family SES were collected from the
parents. Language and length of time in Canada would indicate the students’ familiarity with
English, as language skills are important in understanding the mathematics questions and
concepts (Ellerton & Clarkson, 1996). Family SES is often reported as having an impact on
child academic achievement (Sirin, 2005; White, 1982), and thus is important to be mindful of.
While various measures of SES are available, the SES information was determined based on the
level of parental education. This information is the most common method to collect information
within the educational literature (Sirin, 2005; White, 1982), and has been considered as a good
indicator of income in Canada, since the attainment of post-secondary education significantly
decreases the likelihood of unemployment (Berger & Parkin, 2009).
The demographics of the student participants are shown in Table 2. The ethnicity of the
sample was mostly Caucasian, and all participating students were fluent in English. The parental
education level showed that 18% of the parents of the student sample did not attain post-
secondary education. Berger and Parkin (2009) show that the median income level of those who
had not attained a high school diploma and those who received a high school diploma is at
$32,029 and $37,403 respectively. According to Statistics Canada’s (2013) data, the low-income
cutoffs for the districts under the population of 30,000 for families with 2-4 persons is $22,714-
$33,905 (before tax). As such, 18% of the sample for the current study may meet the low-
income cutoff. In addition, one student was identified by the teacher as receiving a diagnosis for
Attention-Deficit Hyperactivity Disorder (ADHD).
35
Table 2 Demographic Characteristic of Child Participants
Demographic characteristic n Percentage (%)
Gender
Male 16 55
Female 13 45
Ethnicity
Caucasian 27 93
Other (South Asian, Mixed)
2 7
English as First Language 29 100
Highest Parental Education
Some High School 1 4
High School Diploma or Equivalent
4 14
Certificate (after High School)
6 21
University/College Degree (2 years)
5 18
University/College Degree (4 years)
7 25
Master’s Degree 5 18
The current teachers of participating students were also asked to provide more
information about the types of conceptual subitizing activities they used, and the frequency they
introduced these activities within their classrooms. As experience with number representation is
important to develop the conceptual subitizing skill, the teacher reports were collected to
determine the extent to which participating students had prior exposure to conceptual subitizing
activities. Four teachers from three different classrooms responded to the questionnaires
regarding the mathematics activities. One class was a Grade 1 and 2 split class, and one class
36
had two teachers co-instructing the same class. The teachers had four to 24 years of teaching
experience, all within the elementary school grades.
Materials
Dot enumeration task. The following sections outline the details of the dot enumeration
task.
Apparatus. The experiment ran on a Macintosh MacBook Air laptop with a 13-inch
display (1440x900 pixels). The task was programed using the PsychoPy software (Peirce, 2007,
2009), which is an open-source, free software that runs on any platform. The RT and accuracy
of the participants’ performance were recorded using this software and an Insignia® one-
directional microphone.
Task conditions. Black dots (3mm diameter, 0.29° visual angle) appeared on a grey
background, to minimize eyestrains caused by the computer screen. Dot size was chosen
following Starkey and McCandliss’ (2014) stimuli. The visual angle size of the dots was set
using common perimeters outlined in the subitizing literature, ranging from 0.20° (Krajcsi et al.,
2013) to 0.35° (Schleifer & Landerl, 2011). Three types of dot display conditions were included
in the task: random, canonical, and grouped. The specific arrays used in the three dot display
conditions are outlined in Appendix A. The random condition displayed 1-10 dots in a random
manner (i.e., did not form a familiar pattern).
The canonical condition consisted of 4-10 dots displayed in a symmetrical fashion.
While there are a number of displays used as “canonical patterns” within the literature, the
patterns for the current study were taken from the original canonical pattern study (i.e., Mandler
& Shebo, 1983), with an exception of the numerosity 8 that was derived from Ashkenazi et al.’s
(2013) study. Ashkenazi and her colleagues’ pattern was chosen, as this is one of the few studies
37
that used canonical patterns on child participants. Numerosity 8 was taken specifically from
Ashkenazi et al.’s design because Mandler and Shebo’s design for numerosity 8 consisted of four
rows of two dots, which is much wider than the other patterns. As such, Ashkenazi et al.’s
design of two rows of four dots for numerosity 8 was used to maintain the consistency of the
widths of the canonical patterns.
The grouped condition displayed arrays of 4-10 dots in two subgroups (e.g., 10 presented
as six and four). Each subgroup formed a familiar, canonical pattern up to six items. Previous
studies using grouped arrays (i.e.,Starkey & McCandliss,2014; Wender & Rothkegel, 2000) have
limited the subgroups to the perceptual subitizing range. The current study utilized subgroups up
to six in alignment with Sarama and Clements’ (2009) view that children develop mental
templates for numbers above the perceptual subitizing range.
Stimulus design. The canonical patterns were created within a 2 × 2 table (2 cm × 2 cm
size, or 1 cm × 1 cm per cell, visual angle of 1.9° in total). Each dot was positioned just within
the 2 × 2 table in order to create a pattern with consistent size (see Figure 1). The visual angle of
the canonical images was appropriate for viewing the entire image within the foveal visual field
(i.e., centre of the retina of the eye; Reinagel & Zador, 1999), which is normally around 2°.
Figure 1. Canonical patterns created for the task. The outline of the table would not appear in the actual computer task.
38
Dot arrays in the random condition were also created within this 2 cm × 2 cm space so
that the total display area was constant across the random and canonical pattern conditions. For
the grouped condition, two canonical patterns were used as the subgroups, which were linearly
(horizontally) placed on the screen. Both combinations of the two canonical patterns in the array
(e.g., three and four and four and three) were included in the task in order to minimize order
effect. A constant distance was maintained between the centre point of the subgroups (i.e. 4 cm),
resulting in 5.27° visual angle. This size is appropriate for viewing the images within the
stationary visual field, which is argued to be around 6° visual angle (Sanders, 1970). The entire
image within the stationary visual field could be viewed within peripheral vision.
Task design. The task started with a 500 ms fixation point in the middle of the screen,
which disappeared for 10 ms, and the stimuli then appeared in the middle of the screen. Each
stimuli was presented until a verbal response was given. RT was measured from the stimuli
presentation until the start of a verbal response.
The tasks presented arrays of 1-10 dots in the random condition and 4-10 dots in the
canonical and grouped conditions, totaling 24 types of displays across the three conditions. Each
display was presented eight times, resulting in 192 trials, divided into two tasks (i.e., 96 trials per
task). The number of data points required for each display was determined based on past
subitizing studies with children at this age range (i.e., Ashkenazi et al., 2013; Starkey &
McCandliss, 2014). The arrays were presented in a random order (determined prior to task
creation), with three display conditions intermixed.
Participant interviews. After completion of the dot enumeration task, the students were
interviewed about how they determined the numerosity of the displays. Students were shown
four canonical patterns and four grouped patterns used in the computer task with arrays of 7-10
39
items. Each display was shown one at a time and each participant was asked, “How many dots
are there?” and, “How did you know?” These questions were retrieved from Clements and
Sarama’s (2014) suggestion on the activities to develop children’s conceptual understanding of
number. Vague and incoherent answers were probed for further explanations. Each of the
responses were recorded and categorized by two raters into five strategy categories. Answers
that implied seeing the smaller parts to enumerate the whole (e.g., “I saw two fours, so eight”; “I
know this is five, and three more, so eight”) were categorized as part strategy. If the student
explained that they just knew (e.g., “Because it looks like the right seven,” (Benz, 2012), “I’ve
seen it before”), the answer was categorized as whole strategy. If they saw one part of the shape
but did not identify the number of the rest of the dots (e.g., “There’s five, and counted the rest,”
“Five, six, seven, eight”), the answers were categorized as counting on strategy. If the students
explained that they counted (e.g., “Because one, two, three, four, …”), their answers were
categorized as counting all strategy. Finally, answers that did not belong to the above strategies,
or were not articulated well, were categorized as other strategy.
Mathematics achievement. Participants’ mathematics ability was examined using the
two mathematics subtests from the Wechsler Individual Achievement Test – Third Edition
(WIAT-III; Wechsler, 2009). WIAT-III is a standardized measure that helps to identify
children’s strengths and weaknesses in multiple domains of academic skills including oral
language, reading, writing, and mathematics. The mathematics subtests assess Math Problem
Solving skills (e.g., word problems) and Numerical Operations skills (i.e., computation). In
addition, the measure provides Canadian norms, which was more appropriate for the sample for
the current study.
40
The WIAT-III is reported to have high reliability and validity scores (Wechsler, 2009).
The reliability coefficients reported using the split-half method for the mathematics subtests for
Grade 2 students are .95 and .93 for Math Problem Solving subtest in the fall and spring terms
respectively, and .90 for Numerical Operation subtest for both the fall and spring terms. The
overall Mathematics composite score was reported at .96 for the fall term and .95 for the spring
term, suggesting the Mathematics composite to be a highly reliable measure. Construct validity
of the measure was tested by assessing the intercorrelations between the subtests and the
composite scores. The technical manual reports the correlation coefficients of Math Problem
Solving and Numerical Operation subtests to be at .74, and the correlation between the subtests
and the Mathematics composite score to be .93. As such, the WIAT-III Math subtests and the
composites assess the students’ mathematics abilities in a reliable and valid manner.
Teacher questionnaire. As conceptual subitizing is considered as a learned skill, the
amount of instruction participants were receiving using conceptual subitizing-related tasks was
also examined. The Grade 2 teachers of participating students were provided with a short
questionnaire that asked about their teaching activities and materials specific to number
representation and perceptual and conceptual subitizing. Specifically, the questionnaire asked
whether they had used specific display formats to represent number in their instructions (e.g.,
dice dots, ten-frames), and required the teachers to identify the top three most used display
formats in their activities (see Appendix B for the full questionnaire). The choices in the
questionnaire were chosen based on the suggested activities in the educational literature (e.g.,
van der Walle, 2004, Young-Loveridge, 2002). Further, the questionnaire collected some
information regarding the teachers’ familiarity with the term subitizing and conceptual
41
subitizing. This was included to explore the level of acceptance of the idea and the term of
conceptual subitizing between the educators in Alberta.
Procedure
The students were tested on an individual basis across two testing sessions, with the
WIAT-III Math tasks and the dot enumeration (computer) task administered on separate days.
On the dot enumeration task, students were instructed to sit 60 cm away from the computer
screen in order to maintain the visual angle size constant across participants. After the
completion of the dot enumeration task, students were interviewed as to how they enumerated
the quantity of the array (i.e., “ How many dots did you see?” and “How did you know?”).
The teachers were provided with the questionnaire at the start of data collection from
their classrooms, and were instructed to return the questionnaires within one week of
distribution.
Statistical Analysis
RT performance and mathematics achievement. In order to see whether there is a
general RT difference between counting and enumerating structured dot patterns, a repeated
measures ANOVA analysis was conducted as a preliminary exploration of the RT on the three
display conditions. The RT analysis was conducted on only those trials for which participants
had produced a correct response. RT data was cleaned by winzorizing the outliers and
transforming the data using square-root transformation so that the distribution of the RT on each
numerosity in each condition approached normality.
Then, the RT slope was explored within each display condition. Enumeration
performance in different conditions (numerosity range and/or display types) is commonly
compared using the best fitting linear regression lines for each participant in each condition
42
(display types and/or numerosity ranges; e.g., Gray & Reeve, 2014; Starkey & McCandliss,
2014). The slope provides information on the efficiency at which children enumerate each
additional item in the display (Landerl & Kölle, 2009; Schleifer & Landerl, 2011). The smaller
slope value indicates that children are faster at enumerating each additional item in the display,
suggesting their fluency or efficiency at enumeration. As the slope takes away the individual
differences in the processing speed (or RT) of the overall display, it is easier to see and compare
the differences in how fast they could process the items in different conditions (Starkey &
McCandliss, 2014). The current study focused on the slope of the regression lines, as the main
objective was to see if the increased efficiency in enumerating post-subitizing ranges is related to
mathematics achievement. The regression lines were determined from the means of the RT to
each numerosity (4-10 items) in each condition. RT was winzorized so the skew of the data was
minimized. Transformation of the data was not conducted in order to see the actual values of the
slopes that would indicate the increase of RT per item in seconds.
The regression slopes were then used to explore the relationship between RT
performance and mathematics achievement using regression analysis. WIAT scores were
winzorized to increase normality in the data distribution.
Cluster analysis. In order to see if students’ performance on the canonical condition and
grouped condition could be used to categorize them into developmental stages for subitizing, RT
for numerosities 4-10 on the two display conditions (grouped, canonical) were explored. More
specifically, an agglomerative hierarchical cluster analysis was conducted to form clusters based
on participants’ RT performance on the two conditions. This method of clustering starts with
each case forming individual clusters, and at each step merges similar clusters into one (Norušis,
2011). Winzorized data were used for this analysis.
43
The entered variables were not transformed for the analysis as the same unit of measure
(i.e., RT in seconds) was used in all variables. The Euclidean distances of the cases were
analyzed to form the clusters, using the complete linkage method, also known as the furthest
neighbor method. This method uses the distance between the furthest points to define the
clusters (i.e., cluster distance is defined from the distance between the furthest elements).
Complete linkage was selected amongst other methods due to its ability to find compact clusters
(Everitt, Landau, & Leese, 2004), which may be important as the variability of the RT are small,
especially for the smaller numerosities. It also avoids the chaining phenomenon seen in the
single linkage (nearest neighbor) method that determines the cluster difference using the closest
difference between the clusters. The chaining effect could occur when the distance of one of the
elements is close (which defines the clusters) but the other elements are still distant from each
other. As the RT performance characteristics on larger numerosities may differ from the smaller
numerosities, this dataset may be prone to produce a chaining phenomenon. Therefore, the
complete linkage method was utilized to avoid this phenomenon as much as possible. The final
clusters were determined based on the amalgamation coefficients, creating a scree plot to
visually see the change of the coefficient values. The data was analyzed using the IBM-SPSS
version 20.0 for Macintosh systems.
There is no rule of thumb for sample size requirement to conduct cluster analysis
(Dolnicar, 2002; Mooi & Sarstedt, 2011). Literature on a similar statistical analysis method
called the latent class analysis offers some recommendation on the sample size, using at least 2m
participants, where m equals the number of variables included in the analysis (Formann, 1984).
Using this as a guideline, the use of cluster analysis in the present study with 14 variables would
require 16384 participants. However, there is some trend in the literature that utilizes cluster
44
analysis with small sample sizes and a large number of variables (i.e., high dimensional space;
see Dolnicar, 2002). Dolnicar conducted a systematic review on 243 studies that used cluster
analysis and found that 22% of the studies used sample sizes of less than 100. As such, the use
of small sample sizes in cluster analysis is not uncommon or without precedence.
Group differences in mathematics achievement. The non-parametric, Kruscal-Wallis
test was conducted on the mathematics achievement scores to explore whether the three clusters
showed differences in mathematics understanding. The data that were winzorized for the
previous analyses were used. Non-parametric test was utilized to compare the means of the three
groups, as the number of participants in each cluster differed.
Strategy differences between clusters. In the final step, two graduate students
categorized the interview data into five different strategy groups, as detailed earlier. Inter-rater
reliability of the categorization was tested. Inconsistencies on the ratings were resolved by
verbal discussions on the reasons for the rating, and the final rating that both raters agreed upon
was assigned to the answer. The interview answers were compared between the clusters formed
from the analysis conducted above to see if there are any differences in the reported strategies to
enumerate the different displays. As this question is exploratory, percentages of answers that
were identified as using parts, whole, counting on, counting all, and other strategies was reported
as a first step into discovering the methods children use to determine the quantity when they are
in different stages of subitizing.
45
Chapter Four: Results
This chapter describes the results of the current study. First, the details of the conceptual
subitizing activities the teachers reported they used in the classroom are described. Then,
preliminary explorations of the RT to the dot enumeration task, along with the slope of the RT to
the different display conditions are discussed. The results of the statistical analyses are then
discussed in regards to the four research questions explained in chapter two.
Student Exposure to Conceptual Subitizing Activities
The teacher questionnaire asked about the teachers’ understanding of subitizing, and the
use of conceptual subitizing activities in the classroom. All four teachers were familiar with the
term and the definition of “subitizing.” All the teachers expected their students to be able to
“subitize” up to at least 10 items, but the definition of “conceptual subitizing” was not familiar to
half of the teachers.
All four teachers reported that they incorporate on a weekly basis multiple activities that
target the rapid identification of numbers. Some of the activities included representing the
“number of the day” with ten frames, money, and tally marks, matching the rolled dice patterns
to the domino blocks, discussing how large numbers (up to 100) are constructed using ten frames,
and comparing the values on the ten frames to choose the larger value. One of the classrooms
limited the presentation time (i.e., hid the images after five to seven seconds) for some of the
structured images, and the other two classrooms were planning on using this method later in the
year. Thus, the students were exposed to activities that would encourage them to visually
represent numbers. However, speed was not reported as a main focus of these activities.
The most commonly used visual for these activities were ten frames (see Appendix B
question five), ranked as the most often used display in the instruction in two of the classrooms
46
(ranked second in one classroom, after tally marks). Ten frames were reported to be used to
develop student understanding of the structure of numbers up to 10, as well as to conceptualize
larger numbers up to 100. Dice dot patterns up to six were incorporated in the math activities in
all classrooms, but it was ranked as the third most commonly used patterns in two classrooms,
and fourth (out of the four displays chosen) in one classroom. Finger patterns were ranked
higher than dice patterns. Nevertheless, the student participants were exposed to the dice dots in
the classroom setting. Domino patterns above six that were incorporated in the current dot
enumeration task were only explored by one of the classrooms. Dot patterns separated by space
(i.e., similar to the grouped condition included in the dot enumeration task) were used in two of
the classrooms. Random patterns of dots were never used in the Grade 2 classrooms.
Preliminary Inspection and Descriptive Analysis.
Accuracy and RT. On the dot enumeration task, participants demonstrated a high
degree of accuracy on each number and display condition, showing that students were able to
identify the amount of dots from each display at above chance levels (see Table 3). The average
RT (on correctly responded trials) for each numerosity and display condition is shown in Table 4
and visually represented in Figure 2.
47
Table 3 Mean Percent Accuracy
Display Condition
Numerosity Random (SD) Canonical (SD) Grouped (SD)
1 100 --- ---
2 100 --- ---
3 100 --- ---
4 97.4 (0.08) 100 99.1 (0.05)
5 94.8 (0.12) 98.3 (0.06) 94.0 (0.11)
6 93.1 (0.13) 96.6 (0.11) 95.7 (0.15)
7 86.2 (0.18) 87.1 (0.22) 92.2 (0.17)
8 75.0 (0.26) 86.2 (0.25) 94.8 (0.10)
9 83.6 (0.21) 88.8 (0.20) 91.4 (0.15)
10 82.8 (0.24) 89.7 (0.20) 90.5 (0.16)
Table 4 Mean Reaction Time in Seconds
Display Condition
Numerosity Random (SD) Canonical (SD) Grouped (SD)
1 0.90 (0.11) --- ---
2 1.02 (0.28) --- ---
3 1.18 (0.41) --- ---
4 2.24 (0.73) 1.25 (0.43) 2.06 (0.61)
5 2.99 (0.92) 1.57 (0.65) 3.20 (0.85)
6 3.53 (1.02) 1.90 (1.32) 3.26 (0.93)
7 4.36 (1.02) 3.54 (1.39) 4.07 (1.21)
8 4.59 (1.39) 3.14 (1.64) 3.67 (1.10)
9 5.16 (1.43) 3.41 (1.87) 4.20 (1.74)
10 5.66 (1.55) 4.42 (1.66) 4.78 (1.48)
48
Figure 2. Mean Reaction Time for each display condition in seconds. The RT within the perceptual subitizing range (1-3 items) was faster than the RT for the
numerosities above this range in all three conditions, which is in line with previous research on
perceptual subitizing (Trick & Pylyshyn, 1993). RT difference within the counting range (4-10
items) between the random, canonical, and grouped conditions was also observed, suggesting
that children used different processes to enumerate this range of numerosity in different
conditions. Canonical patterns allowed fastest RT in the post-subitizing range quantities
compared to the other two display conditions, consistent with previous research on canonical
patterns. Numerosity 4 in the canonical condition also shows a very similar RT to numerosity 3.
This suggests that the canonical pattern of numerosity 4 may have been perceptually subitized by
the students. This is in alignment with the literature that suggests that canonical patterns could
extend the perceptual subitizing range beyond the original perceptual subitizing range (Krajcsi et
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10
RT(s)
Numerosity
Random
Canonical
Grouped
49
al., 2013). The sharp increase in RT between numerosity 3 and 4 in the random condition
suggest that the participants in the current study were, as expected, perceptually subitizing up to
three items, and engaged in counting from sets above three items.
It is also interesting to note that there seems to be a difference in performance between
the numerosities 4-6 range and the 7-10 range. The canonical display condition showed a
notable increase in RT between numerosities 6 and 7. The grouped display condition showed
similar performance patterns to the random display condition between the 4-6 range, while
showing similar performance patterns to the canonical display condition between the 7-10 range.
As such, caution is necessary to interpret the slope of the RT discussed in the next section.
To compare the RT across the three display conditions, a 3 (display condition) × 7
(numerosity 4-10) repeated measures ANOVA on a square-root transformed data was conducted.
The performance data for numerosities 1 to 3 was omitted for the ANOVA analysis, as the
canonical and grouped conditions did not contain these numerosities. The repeated measures
ANOVA revealed a main effect of the display condition, F(2, 56) = 74.23, p < .001, ηp2 = .73,
and a main effect of numerosity, F(4.08, 114.32) = 114.42, p < .001, ηp2 = .80 (utilizing
Greenhouse-Geisser correction due to non-sphericity). A post hoc comparison using Bonferroni
correction showed that the RT differed between all display conditions; random condition took
significantly longer than canonical and grouped conditions (both p < .001), and the canonical
patterns allowed faster RT than the grouped patterns (p < .001). A post hoc pairwise comparison
of numerosity revealed a difference in RT for all numerosities (all p < .001), except between
numerosities 5 and 6, and numerosities 7 to 9. The significant difference in numerosity show
that, as expected, students took longer to enumerate the larger numerosities. The interaction
between the two factors was also significant, F(12, 336) = 4.72, p < .001, ηp2 = .14, which
50
reflects the slower processing of numerosity 5 on grouped condition compared to the random
condition, as shown in Figure 2. However, examining the margins of error for the average RT
for numerosity 5 in the random condition (2.07-3.91 seconds) and grouped condition (2.35-4.05
seconds) reveals that the range of RT that lie within the 95% confidence interval of error
overlaps.
Slope characteristics. The RT slopes for each participant were generated by calculating
the best fitted linear regression lines for each condition. Table 5 shows the average of these
individual RT slopes for the numerosities 4 to 10 on each display condition, as well as for the
perceptual subitizing (PS) range (i.e., random display condition for numerosities 1 to 3).
Repeated measures ANOVA for the four slopes was initially conducted to explore
whether the efficiency in processing the post subitizing range (4-10) is, in general, statistically
significantly different from the PS range and from each other in different conditions. The
analysis showed that there was a statistically significant effect of display condition on the RT
slopes, F(3, 84) = 22.06, p < .001, ηp2 = .44. A post-hoc comparison using Bonferroni correction
showed that the PS slope was significantly different from the slopes for the numerosity range of
4 to 10 (all p < .001). Statistically significant difference was also observed between the grouped
condition slope and the random condition slope (p < .001). No statistical difference was
observed between the random condition and the canonical condition slopes. The difference
between the grouped and the canonical condition slopes approached significance (p = .06).
Therefore, the students benefitted from the spatial grouping cues in the grouped condition to
improve efficiency in determining the numerosity of the post-subitizing range (i.e., little addition
of RT was seen as the numerosity of the dots increased). This effect was not observed from the
slope in the canonical condition, although caution is required to conclude so as there is a
51
significant jump in the RT between numerosity 6 and 7, affecting the slope. Thus, the slopes in
the numerosity 4-6 range and the 7-10 range were also explored separately to understand the
outcome further.
Table 5 Average Reaction Time Slopes and R2 for Different Display Conditions for Numerosities 4-10 and the Perceptual Subitizing Range
Display Condition Slope (SD) R2
Random 0.58 (0.52) .99
Canonical 0.51 (0.05) .89
Grouped 0.38 (0.04) .86
PS range 0.14 (0.21) .99
Note. PS = Perceptual subitizing. Table 6 shows the different RT slopes for each display condition in different numerosity
ranges. A 2 (ranges) × 3 (display condition) repeated measures ANOVA revealed a significant
effect of display condition, F(2, 56) = 3.95, p = 0.03, ηp2 = .12, and ranges, F(1, 28) = 4.96, p =
.01, ηp2 = .15. No significant interaction was observed. Post hoc test with Bonferroni correction
on the display condition identified that canonical RT slopes are significantly flatter than the
random RT slopes (p = .03), and the post hoc test on the numerosity range suggested that the
upper ranges of numerosity showed a flatter slope on average compared to the lower numerosity
range (p = .03). Repeated measures ANOVA run separately for each numerosity range showed a
significant effect of display condition on the slopes of the 4-6 range, F(1.70, 47.60) = 4.81, p =
.02, ηp2 = .15, with Greenhouse-Geisser correction, but the post-hoc test did not show statistically
significant differences between the slopes. No significant effect of display condition was seen in
the 7-10 numerosity range. A paired-sample t test was conducted to further test the changes in
52
slopes within each display conditions, which revealed a statistically significant change of slope
in the grouped display condition, t(28) = 3.37, p = .002.
In order to see if the efficiency of enumerating each additional dot (shown by RT slope)
were similar to the PS range performance, the three display condition slopes of the two ranges
were also compared with the PS slope using repeated measures ANOVA separately by
numerosity range. The repeated measures ANOVA for the 4-6 range revealed a significant
effect of display condition on RT slopes, F(3, 84) = 10.42, p < .001, ηp2 = .27. Post-hoc test with
Bonferroni correction showed a significant difference between PS slope and random and grouped
slopes, but no significant difference between the PS and canonical slopes, indicating that the
canonical condition allowed a fast enumeration of each additional dot. Repeated measures
ANOVA did not reveal a significant effect of display condition on RT for numerosity range 7-
10. As this indicates that some comparisons were similar to each other, a post-hoc, paired-
sample t test was conducted. The t test revealed a statistical significance only between PS and
random slopes, t(28) = -3.45, p = .002. The comparison between the PS slope and canonical and
grouped slope in this range showed no statistical difference, suggesting that the canonical and
grouped displays allowed a quick enumeration of each additional dot in this range as well.
53
Table 6 Average Reaction Time Slopes and R2 for Different Display Conditions for Numerosity Ranges 4-6 and 7-10
Numerosity Range
4-6 7-10
Display Condition Slope (SD) R2 Slope (SD) R2
Random 0.65 (0.50) .99 0.45 (0.44) .97
Canonical 0.33 (0.57) .99 0.29 (0.55) .60
Grouped 0.61 (0.47) .80 0.27 (0.47) .56
Research Question 1: Is RT to determine post-subitizing range of numerosity (4-10) related
to mathematics achievement?
Pearson’s correlation between the three slopes for the whole numerosity range and the
WIAT-III Math score (M = 98.27, SD = 13.47) was first calculated to see if the fluency at which
participants determined the numerosity above the perceptual subitizing range was related to
mathematics achievement. Pearson correlations revealed a significant relationship between
mathematics scores and canonical and grouped condition performance. The negative correlation
suggests that the more fluent the students are at enumerating the dots, the better they were with
mathematics (see Table 7).
Multiple regression analysis was planned to determine how predictive the enumeration
fluency of the canonical and grouped conditions are on mathematics achievement. However,
multicollinearity was observed between the two independent variables (i.e., canonical and
grouped condition RT slopes were significantly correlated), which violates the assumption of
multiple regression. Therefore, regression was run individually to see how predictive the RT
54
slopes of each display condition are for the mathematics achievement scores. The outcome is
shown in Table 8.
Table 7 Pearson Correlation Between Reaction Time Slopes (Range 4-10) and Mathematics Achievement
Measure 1 2 3 4
1. WIAT-III ---
2. Random .06 ---
3. Canonical -.42* .07 ---
4. Grouped -.48** .24 .38* ---
Note. WIAT-III = Wechsler Individual Achievement Test – Third Edition (Mathematics composite score). * p < .05. ** p < .01. Table 8 Regression Analyses Predicting Mathematics Achievement Scores From Two Reaction Time Slopes (Range 4-10)
Slopes Β β t R2
Canonical -20.89 -.42 -2.37* .17*
Grouped -30.31 -.48 -2.84** .23**
* p < .05. ** p < .01.
The results indicate that both the canonical condition RT slope and the grouped condition
RT slope significantly predict mathematics achievement; the canonical condition RT slope
predict 17% of the variance of mathematics achievement, F(1, 27) = 5.63, p = .03, and grouped
condition RT slope predict 23% of the variance, F(1, 27) = 8.08, p = .008. As such, speeded
performance on both types of display conditions seem to be important for mathematics
55
achievement, while the grouped condition had a slightly stronger influence on the WIAT-III
score change.
When the influence of RT slope for the different ranges (4-6 and 7-10) of numerosity on
the WIAT-III Math scores were explored, only the canonical 4-6 range RT slope was a
significant predictor of math achievement (R2 = .16, F(1, 27) = 5.20, p = .03, t = -2.28, p = .03).
Both ranges of the grouped condition did not present any significant results individually (R2 =
.09, F(1, 27) = 2.72, p = .11, t = -1.67, p = .11 for 4-6 range; R2 = .001, F(1, 27) = 0.07, p = .80,
t = -0.25, p = .80 for 7-10 range), possibly due to the lower average fit of the regression line,
suggested by R2.
Research Question 2: Can RT Profiles In Different Display Conditions Show Differences in
Subitizing Stages?
An agglomerative hierarchical cluster analysis using RT performance on canonical and
grouped display conditions was conducted to form clusters that were hypothesized to distinguish
the subitizing stages suggested by Sarama and Clements (2009).
One student was excluded from the cluster analysis since the cluster analysis resulted in
allocating the student into its own cluster using any method of exploration (single linkage,
complete linkage, Ward’s method, etc.). Thus, 28 participants were included in further analyses.
While it was expected that the student performance could be distinguished into four
groups (shown in Table 1), the hierarchical cluster analysis method resulted in creating three
clusters, suggested by the amalgamation coefficients. Figure 3 shows the scree plot generated
from the coefficients. The graph shows a noticeable increase in the coefficient at the three-
cluster solution, suggesting the best fitting solution. The amalgamation coefficient, however,
shows a very small change between three-cluster and four-cluster solutions. This may suggest
56
that with more participants, the four cluster solutions may be informative. For the current
sample, however, three-cluster solution may be the most ideal outcome.
Figure 3. Scree plot shown using the amalgamation coefficients. Table 9 shows the characteristics of the three clusters formed. Most students were
categorized into cluster 2 (n = 18), while clusters 1 and 3 were each comprised of five students
categorized in each cluster. Based on visual inspection of the RT (Figure 4), both display
conditions present differences between the clusters more readily starting from numerosity 6.
Differences in RT between cluster 1 and 3 appear large, showing a two-second difference
starting from numerosity 6 in the canonical condition, and numerosity 7 in the grouped
condition. Differences between cluster 2 and 3 are more visible starting numerosity 8 in the
canonical condition, and the difference become larger as the numerosity becomes larger. In the
grouped condition, cluster 3 shows a stable fast RT performance across the numerosity, while
cluster 2 shows a slight but steady increase in the RT across the numerosity.
0
2
4
6
8
10
12
14
272625242322212019181716151413121110 9 8 7 6 5 4 3 2 1
AmalgamationCoef7icients
NumberofClusters
57
In general, the cluster analysis identified three distinct groups based on RT performance.
Importantly, the ranked performance of the three clusters were stable across the two display
conditions and across all numerosities; cluster 3 were consistently ranked the fastest in all
numerosities in both display conditions, cluster 1 the slowest in all numerosities for both
conditions, and cluster 2 performing in the middle.
Table 9 Cluster Characteristics of Mean Reaction Times in Seconds
Cluster 1 (n = 5)
Cluster 2 (n = 18)
Cluster 3 (n = 5)
Numerosity Canonical
(SD) Grouped
(SD) Canonical
(SD) Grouped
(SD) Canonical
(SD) Grouped
(SD)
4 1.77 (0.57)
2.22 (0.74)
1.13 (0.30)
2.06 (0.54)
1.06 (0.24)
1.62 (0.26)
5 1.93 (0.59)
3.44 (0.43)
1.52 (0.66)
3.29 (0.80)
1.22 (0.39)
2.33 (0.62)
6 3.16 (1.06)
3.80 (0.81)
1.57 (0.88)
3.18 (0.60)
1.01 (0.18)
2.38 (0.42)
7 5.06 (1.07)
4.90 (1.47)
3.09 (1.05)
4.19 (0.85)
2.94 (0.68)
2.48 (0.61)
8 4.82 (1.34)
4.90 (0.50)
2.71 (0.95)
3.54 (0.87)
2.00 (0.98)
2.51 (0.54)
9 5.64 (1.26)
5.93 (1.27)
3.02 (1.18)
4.15 (1.15)
1.62 (0.50)
1.92 (0.26)
10 4.90 (1.30)
6.35 (1.21)
4.54 (1.57)
4.67 (1.09)
2.81 (0.83)
3.05 (0.32)
58
Figure 4. RT performance for numerosities 4 to 10 in the canonical and grouped display conditions by the three clusters formed by cluster analysis. Research Question 3: Do children in Different Clusters Show Difference in Math
Achievement?
A non-parametric, Kruskal-Wallis test was conducted to compare the differences between
clusters in mean WIAT-III Math scores, indicative of their mathematics achievement. The
results revealed a significant difference in WIAT-III Math scores between groups, χ2 (2) = 7.30,
p = .03. The difference in the mean WIAT-III Math score between cluster 1 (M = 88.40, SD =
7.47) and cluster 3 (M = 111.92, SD = 18.97) was 23.52 points, which is about 1.5 SD difference
in a WIAT-III score distribution (Wechsler, 2009). In addition, the score difference between
clusters 2 (M = 97.22, SD = 10.28) and cluster 3 was 14.69 points, which is close to 1 SD
difference in the WIAT-III score distribution. Thus, the current results show a trend in the
differences in mathematics achievement scores between the three clusters. However, it is noted
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
4 5 6 7 8 9 10 4 5 6 7 8 9 10
RT(s)
Numerosity
Cluster1
Cluster2
Cluster3
CanonicalPatterns GroupedPatterns
59
that the average WIAT-III Math scores of the three clusters are all within 1 SD of the mean score
in the WIAT-III distribution.
Research Question 4: Does the Self-Reports on Strategies Differ Between the Clusters?
The primary investigator and one other graduate student independently categorized the
eight interview responses per participant (224 responses in total) into part strategy, whole
strategy, counting on strategy, counting all strategy, and other strategy. The inter-rater reliability
for this categorization resulted in a Cronbach’s alpha of .92. Answers that were categorized as
“other” included those that were vague and non-descriptive even with probe for further
explanation, and those that the strategy was not derivable from the answers (e.g., “one more, one
less”).
The total number and percentage of strategies used across all clusters and interviewed
displays are shown in Table 10. Overall, students reported using the parts strategy most
frequently. The counting on strategy, which requires identification of one of the parts, was seen
more in the grouped condition. The whole strategy was only seen in the canonical condition,
which is expected due to the structure of the image (canonical pattern forming one structure
instead of two separate structures in the grouped condition). Counting all strategy was seen
equally in both conditions, suggesting that the display format did not matter when the items were
counted.
60
Table 10
Number and Percentage of Strategy Use in Canonical and Grouped Display Conditions by the
Three Clusters Combined
Display Conditions
Canonical Grouped
Strategy N % N %
Parts 77 69 65 58
Whole 7 6 0 0
Counting On 5 4 23 21
Counting All 21 19 21 19
Other 2 2 3 3
Total 112 100 112 100
The use of different strategies in each display condition between the three clusters is
shown in Figure 5. While there is some variability in the strategies children used in the three
clusters, there seems to be a trend in the type of strategies they used most often. In cluster 1,
counting strategy was the most frequently used strategy in both display conditions. On the other
hand, students in clusters 2 and 3 used the parts strategy in the majority of displays. The
difference between the two clusters was that the variability in the type of strategies students used
in cluster 3 was wider compared to cluster 2, using the whole strategy 20% of the time in the
canonical condition versus 0% in cluster 2. This suggests that students in cluster 3 showed
higher flexibility in viewing numbers both as whole and as parts. The ratio of the strategies used
to enumerate the grouped condition did not differ significantly between the clusters 2 and 3.
61
Cluster 1
Cluster 2
Cluster 3
Figure 5. Percentage of each strategy (part, whole, count on, count all, other) used by students in the three clusters. Response accuracy on the interview questions (i.e., answer to “How many dots are
there?”) was also compared among the clusters. Table 11 shows the number of inaccurate
20% 15%0%
60%
5%0%20%40%60%80%100%
Parts WholeCountOn
CountAll
Other
Canonical
15%0%
15%
60%
10%
0%20%40%60%80%100%
Parts Whole CountOn
CountAll
Other
Grouped
83%
0% 7% 8% 1%0%20%40%60%80%100%
Parts WholeCountOn
CountAll
Other
Canonical
69%
0%21%
8% 1%0%20%40%60%80%100%
Parts Whole CountOn
CountAll
Other
Grouped
65%
20%0%
15%0%
0%20%40%60%80%100%
Parts WholeCountOn
CountAll
Other
Canonical
60%
0%
25%15%
0%0%20%40%60%80%100%
Parts Whole CountOn
CountAll
Other
Grouped
62
answers given by students in each cluster, as well as the reported strategy the students used to
arrive at their inaccurate answers. The number of inaccurate responses given by students in
cluster 1 (n = 6, 15% of total answers within the cluster) was much higher than the other two
clusters (4% for cluster 2, 3% for cluster 3). Students in cluster 1 often showed inaccuracy in
enumerating using the counting all strategy (n = 4, 10%), suggesting that they may have not yet
mastered the counting skills. The inaccurate answers in cluster 2 were mostly from the parts
strategy (n = 3, 2%) and the counting on strategy (n = 2, 1%). This may indicate that they have
not yet shown the full fluidity and accuracy in using the parts of the displays.
Table 11 Inaccurate Responses Provided in Each Cluster and Strategy
Strategy Total
Cluster Parts Whole Count
On Count
All Other N %
Within Cluster
Cluster 1 2 0 0 4 0 6 15
Cluster 2 3 0 2 1 0 6 4
Cluster 3 0 1 0 0 0 1 3
Total 5 1 2 5 0 13 6
Note. The column “% within cluster” shows the percentage of the total inaccurate responses in each cluster. For the last row, the percentage of the inaccurate responses across three clusters is reported.
63
Chapter Five: Discussion
The objective of the current study was threefold: (1) to further explore children’s skills to
quickly enumerate larger numbers, (2) to provide empirical support for the developmental stages
for subitizing skills, and (3) to investigate the importance of speeded enumeration skills on
mathematics understanding. This study is one of the first in exploring children’s conceptual
subitizing skill more empirically, and thus considered exploratory. This initial exploratory study
provides some promising results in capturing the conceptual subitizing skill and its
developmental trajectory. However, given the small sample size, the conclusions reached are
tentative at this point and further study in this area is required. The following sections discuss
the findings of this study through five themes; on children’s experience with conceptual
subitizing activities, the relation between RT on structured patterns and mathematics
achievement, conceptual subitizing development observed from the RT profiles and strategy use,
importance of exploring children’s enumeration strategy, and the difference in mathematics
performance across the different developmental stages. In addition, the implications of the
study, the limitations to the current study, and some suggestions for future research on
conceptual subitizing are discussed.
Student Exposure to and Performance on Conceptual Subitizing Activities
The teacher questionnaires were conducted in order to learn about the teachers’
understanding of conceptual subitizing in general, as well as to investigate the amount of
experience and exposure grade two children have with conceptual subitizing activities. The term
conceptual subitizing itself was only known to half of the teachers, while all the teachers
expected children to be able to “subitize” above the perceptual subitizing range. This outcome is
in line with the general trend that the educational field considers “subitizing” to include the
64
higher-level, conceptual processing of numbers compared to those in the cognitive research field
that consider the skill as innate and limited to a smaller range of numbers (Berch, 2005).
The teacher reports on use of subitizing activities in the classroom revealed that Grade 2
students are involved, on a weekly basis, in activities that represent numbers in non-symbolic
visual forms (e.g., using dots, blocks, etc.). The visuals were mainly used in various matching
and numerosity comparison games, with the focus being discussion of how numbers are made up
of smaller numbers. The presentation styles were mostly consistent across classrooms; ten-
frames were most popular, followed by finger patterns and dice dot patterns. Importantly, the
speed at which children identify the numerosity was not a large focus in the classrooms. Only
one teacher mentioned activities using flash cards (i.e., quick processing of displays), and only
one classroom limited the time children were allowed to see the displays. Rather than focus on
speeded identification, the activities mainly focused on instructing the number concepts and
allowing children to explore multiple number representations.
Information about the classroom subitizing activities is of relevance to the tasks and
observed performance in this study. The reported discussions on number concepts in the
classroom may have helped children to be aware of the idea that numbers could be represented
with smaller numbers, indirectly helping them with the current study’s task. However, given that
the participants likely had limited formal experience with the speeded recognition of displays,
most of the performance on the dot enumeration task may likely be related to children’s informal
experiences with number (e.g., play, parental instruction, or other encounters with number) and
their ability to enumerate unfamiliar patterns using their prior knowledge. Moreover, the dot
enumeration task in the current study was also built on the assumption that the “canonical”
patterns would be familiar to Grade 2 students and, thus, relatively easy to for them to
65
enumerate. However, the teacher questionnaire revealed that the domino dot patterns, which are
similar to the canonical patterns used in the dot enumeration task, were only explored in one
classroom. As such, there is some likelihood that the canonical patterns 7 to 10 shown in the
current task were new to the student participants, which may have influenced their performance
on these displays.
With respect to the grouped display condition, two teachers reported incorporating
similar types of displays (i.e., two patterned dots separated by space) in their classrooms.
Therefore, some of the participants in this study may have been more familiar with the
enumeration tasks, especially for those students whose teacher mentioned incorporating both the
domino dot patterns and the grouped patterns in the class activities. Performance differences
between the classrooms was not examined, as different number of students were recruited from
the three classrooms, and the main objective of the study was to investigate children’s conceptual
subitizing skill overall. Future studies, however, should attempt to test the difference in RT
across the different classroom settings to fully investigate the effect of classroom experience in
children’s performance.
Despite the lack of experience in enumeration activities similar to the dot enumeration
task in the current study, the participants seemed to show a better efficiency in enumerating the
patterned displays compared to the random displays. In general, the canonical patterns were
processed the fastest, followed by grouped displays, and then random displays. The random
arrangements were expected to take the longest to enumerate, as the definition of counting is a
slow and error prone process (Piazza et al., 2002; Vuokko, Niemivirta, & Helenius, 2012). It is
also not surprising that the canonical patterns were processed the fastest, as the pattern displays a
single structure that allows for the processing of the pattern as a whole easily (Krajsci et al.,
66
2013). In contrast, the grouped displays were expected to take longer than the canonical
patterns, as it requires children to identify the two distinct patterns, and consider the combination
of the two patterns.
The slope characteristics also distinguished the performance in the three display
conditions. As discussed earlier, slope suggests the speed at which children enumerate each
additional dot in the array, and perceptual subitizing performance is known to show a relatively
flat slope compared to the counting range, which suggests that processing of each dot is much
faster for perceptual subitizing (Trick & Pylyshyn, 1994). The RT slope of the canonical
condition for the numerosity ranges 4-6 and 7-10 in this study did not show a significant
difference from the PS slope, which further supports the notion that canonical patterns helps
students to more efficiently enumerate displays. However, a similar performance pattern was not
observed in the grouped condition. The slope in the 4-6 range was similar to the performance in
the random condition in this range (i.e., similar to counting), while the slope in the 7-10 range
showed similarity to the PS slope. This pattern of performance is particularly interesting because
even though the grouped condition displays for 4 and 5 contained subgroups within the
perceptual subitizing range, the RT performance and response slopes suggest that Grade 2
students are counting all the dots in such displays rather than picking up on and utilizing the
structure of the subgroups. This finding is contradictory to Starkey and McCandliss’ (2014)
results where the students efficiently enumerated the dots using subgroups that were within the
perceptual subitizing range. The difference may have resulted from the fact that Starkey and
McCandliss used at least two of the equal-sized subgroups (e.g., two groups of 2 and one dot for
numerosity 5, three groups of 2 for 6), which may have allowed easier access to the number
facts, compared to the current study that used two unequal-sized subgroups. With two equal-
67
sized subgroups, it may allow children to utilize their knowledge from skip counting to retrieve
the number facts.
Interestingly, children improved their efficiency in enumerating each additional dot when
the subgroups were above their perceptual subitizing range. As shown in Appendix A, the
grouped display condition for range 7-10 consisted of subgroups with the canonical patterns 4, 5,
and 6. Therefore, it may be likely that the students did not utilize the structure of the displays
until the subgroups were beyond their perceptual subitizing range in the grouped condition. The
overall RT in determining the numerosity of the 7-10 range, however, was still slower than the 4-
6 range. This may indicate that while the children were able to identify the numerosity of the
parts (subgroups), they took some time to retrieve the mathematical facts on the number
combination. Yet, the smaller increase in RT with each additional dot in the higher range of
numbers suggests that children were better at utilizing the parts in this range. Thus, the current
study provided evidence that children are able to improve their efficiency in enumerating
additional items even when the subgroups are above their perceptual subitizing range.
Speeded Performance and Mathematics Achievement
The first research question explored the importance of fluency to enumerate post-
subitizing ranges of numbers on mathematics achievement. As expected, the faster students
were on enumerating the overall post-subitizing ranges in both canonical and grouped
conditions, the better they were with mathematics achievement. The correlation between the
slopes of the canonical and grouped pattern conditions also suggests that children who were fast
at enumerating the canonical condition were also fast at enumerating the grouped condition.
These findings are in line with Starkey and McCandliss’ (2014) findings, which suggested that
68
the increased efficiency of enumerating numbers in the counting range (numerosity 5, 6, and 7 in
their study) was a strong predictor of mathematics fluency scores.
This relationship between the overall slope and mathematics achievement revealed in the
current study, however, needs caution to interpret. First, there was a significant change in the RT
between numerosity 6 and 7 in the canonical condition. This could affect the overall slope of the
canonical condition. However, the analysis revealed that on average, children’s RT slope did not
change between the 4-6 range and the 7-10 range, suggesting that although children’s speed to
enumerate the two ranges differed, the efficiency of enumerating one additional item did not
change between the two ranges. As such, the relationship between the overall canonical slope
and mathematics achievement may suggest that continued efficiency on enumerating each item
in the canonical display arrays is related to better mathematics performance.
Second, as discussed earlier, there is a significant change in the RT slope in the grouped
condition between the two ranges (i.e., 4-6 range and 7-10 range). This may suggest that
children were utilizing a different strategy to enumerate the dots in the grouped display
condition, specifically, counting in the numerosity 4-6 range and then conceptually subitizing in
the 7-10 range. This change in their performance results in the flatter slope in the overall slope
for the grouped display condition. Yet, the significant relationship between the mathematics
score and the overall grouped condition slope may suggest that those who were able to change
their strategy to enumerate the 7-10 range from the 4-6 range (resulting in flatter slope for the
overall grouped condition) did better at mathematics.
These outcomes combined suggest that the speeded enumeration performance on the
post-subitizing range of numbers is related to understanding of mathematics. As such, it would
be important to examine the speeded performance in the range of four to 10 in Grade 2 children.
69
In addition, the efficiency in enumerating each additional dot in the canonical and grouped
display conditions suggests that these structured displays could successfully promote students to
efficiently process these larger sets of numbers, indicating its usefulness in the classroom
activities.
Evidence for Conceptual Subitizing as a Developmental Process
The second research question was exploratory in nature; to see if children’s performance
on different types of displays could distinguish those who are in the different developmental
stages of subitizing as proposed by Sarama and Clements (2009). Sarama and Clements only
discussed children’s subitizing stages based on children’s verbal responses of their strategies
(e.g., conceptual subitizers explaining, “I saw two rows of three, so six,” versus imagery-based
subitizers explaining, “Because that’s the shape of six”). Thus, the current thesis proposed a
possible RT performance profile to distinguish the developmental stages (see Table 1) and, thus,
allowing for a psychophysical investigation of the area. The cluster analysis of the RT was
generally consistent with the proposed RT framework in that three distinct groups were
identified. Specifically, a three-cluster solution was suggested that aligned with the proposed
counters (cluster 1), imagery-based subitizers (cluster 2), and conceptual subitizers (cluster 3)
groupings. As proposed, children in cluster 1 showed the slowest RT in both canonical and
grouped display conditions, showing a steady increase in the RT as the numerosity increased.
Children in cluster 2 performed faster than cluster 1 in both display conditions, but they also
showed a steady increase in RT as numerosity increased in the grouped condition, suggesting
their inefficiency with putting together the parts to enumerate the whole. Children in cluster 3
performed the fastest of the three clusters in both display conditions.
70
One area in which the cluster analysis results deviated from the proposed RT framework
was with respect to the two hypothesized types of imagery-based subitizers; the distinction of the
two types was not supported by the cluster analysis, as the cluster solution supported a three-
cluster solution instead of four. The distinction between the two types may not have been
detected, as performance on the canonical condition was, in general, faster than the grouped
condition, and thus, did not find any group that may have showed fast performance in the
grouped patterns but slower performance in the canonical patterns (imagery-based subitizers type
2). However, it could be argued that children in cluster 2 are displaying what could be expected
in the imagery-based subitizing type 1, with fast RT in the canonical condition and slow RT in
the grouped condition. This could be explained from the comparison between cluster 2 and
cluster 3 performances on the grouped condition. As the grouped displays were expected to
force children to use the parts (subgroups) of the whole, the biggest differences were expected in
the grouped display. As predicted, children in cluster 2 were consistently slower in the grouped
condition compared to those in cluster 3, and it displayed a consistent increase in the RT as the
numerosity increased, suggesting that children in cluster 2 did not improve their efficiency in
enumerating extra items like those in cluster 3. Thus, it could be argued that children in cluster 2
were struggling to use the parts to enumerate the whole, in line with the imagery-based
subitizer’s characteristic (Sarama & Clements, 2009).
While these results are promising with respect to elucidating the developmental stages of
conceptual subitizing, the findings are still tentative given the relatively small sample size used
in this study. Promising, however, is that the analysis of the interview responses in the fourth
research question provides additional support for the outcome of the cluster analysis, in
particular, strategy use aligning with the hypothesized subitizing stages. To review, strategy use
71
was classified using a similar framework to the RT-based categories. Namely, strategies were
classified based on whether the focus was on the parts of the structure, the whole of the structure,
counting on from one structure (seeing some items as units but not for all items), counting all of
the individual items, and other methods that do not fit with the above.
Differences were identified between the three clusters in the types of strategies children
reported using that aligned, to some extent, with the three clusters formed from the RT profiles.
As expected, counting all was the most common strategy reported by students in cluster 1. It is
likely that their slow RT performance was due to them counting each item individually. When
canonical patterns are examined, it is found that cluster 3 students used the whole strategy along
with the part strategy in enumerating the canonical patterns. This may indicate that children in
cluster 3 were flexible with seeing the numbers as both whole and as parts, which is a quality
expected in conceptual subitizers (Clements, 1999). The high accuracy of these children’s
performance may also suggest that students in Cluster 3 were able to select the most efficient and
accurate method in enumerating the displays.
Cluster 2 students predominately reported using the parts strategy. As Sarama and
Clements (2009) argue that imagery-based subitizers are those who have not yet reached the
flexible view of numbers as a whole and as parts, their lack of flexibility in strategy use allows
us to classify them as our imagery-based subitizers. Again, it is important to mention that this
classification is contradictory to Sarama and Clements’ proposal that imagery-based subitizers
focus on the image as a whole instead of the parts. However, their lack of flexibility in using
both the whole and the parts of images could suggest that children in this cluster are not yet fully
developed in the conceptual subitizing skill, and are in the imagery-based subitizing stage. Lack
of flexibility may, in part, account for overall slower response to the canonical patterns compared
72
to children in cluster 3. As children over-focused on the parts of the patterns, they may not have
used the whole structure of the pattern itself.
The results of this study, both analysis of the strategy use and RT results, thus provide
some initial suggestion that imagery-based subitizing, a stage before the most advanced stage of
subitizing development (conceptual subitizing), was observed in the participating children.
Contrary to expectation, however, children’s inflexibility with strategy in enumeration did not
stem from their viewing of the displays as a whole, but rather from them focusing on the parts of
the whole. While this is surprising, it could be argued that the current study may provide a
different perspective to the developmental trajectory of the conceptual subitizing skill. As
children acquire the skill to quickly and flexibly see numbers as a whole that consist of smaller
numbers, children may need to consciously follow each step to first remember that numbers
consist of parts, and then identify those parts before enumerating the two parts together to reach
the numerosity of the entire array. Thus, it may be important to consider that the conceptual
subitizing stage may consist of those who are still consciously engaging in identification of the
parts, and those who are fluent with the steps that they are able to quickly and flexibly use the
whole and the parts in different display formats and numerosity. As such, cluster 2 of the current
study may have captured those that have acquired the conceptual subitizing skill, but are still in
the “conscious phase” of conceptual subitizing. This may make more sense when considering
the participating children’s age. As Clements and Sarama (2014) suggested that conceptual
subitizing could be performed by children at the age of 5 with instruction, it is expected that
children in Grade 2 are able to engage in conceptual subitizing. The current study may thus have
provided evidence that children continues on the developmental trajectory of conceptual
73
subitizing by improving their efficiency of the skill after acquiring the conceptual understanding
of the skill, beyond the age of 7.
Overall, the current study was able to capture the developmental stages of conceptual
subitizing using RT performance on different types of displays. The results of the interview
analysis supported the cluster analysis outcome by providing evidence that different clusters
were using different types of strategies that are indicative of the developmental stages suggested
by Clements and Sarama. The contradiction from expected strategy use in cluster 2 students
implies, however, that RT cannot be the only way to assess children’s conceptual subitizing skill,
or to identify their developmental stage of the skill. Investigation on children’s strategy in
enumerating the displays is thus essential in understanding their competence in engaging in
conceptual subitizing.
Importance of Considering Children’s Enumeration Strategy
As discussed in the above section, children’s strategy on enumerating arrays supported
the outcomes of the RT analysis, and provided further insight into the developmental theory of
conceptual subitizing. This highlights the fact that in order to understand children’s conceptual
subitizing skill, RT and strategy use cannot individually be the sole method in exploring
children’s conceptual subitizing skill. In addition to this, exploring children’s enumeration
strategy provided some key findings worth discussing.
First, it should be noted that children in the current study appeared to be fairly accurate in
their ability to report their strategy use. This is most evident for children in Cluster 1, as they
were disproportionately more likely to report the use of counting strategies compared to those in
the other two clusters. Their use of the counting strategy seems to be supported by the overall
slower RT that characterized this group. As children are aware of their methods in arriving at
74
their answers, discussion of the peers’ enumeration strategies in the classroom may be helpful.
Increasing awareness of other strategies and internalizing them during classroom instruction may
allow children to possess a different, and more advanced tool in viewing different representations
of numbers.
Second, reported strategy use helped to highlight that display presentation methods
mattered to strategy use. This is most evident in the grouped displays, in which none of the
students reported using the whole strategy. This is in contrast to the canonical displays, where
both cluster 1 and cluster 3 students reported use of the whole strategy (15% and 20%,
respectively). As such, the way the items are displayed affects how children view the numbers.
Thus, it could be suggested that the use of both canonical and grouped displays may be important
for future study in this area, as it helps children to privilege certain aspects of the display during
enumeration. In addition, including both types of number representations allow for studying
children’s flexibility in their use of the whole and part strategies.
Focusing on children’s enumeration strategy would therefore provide many benefits as
children’s conceptual subitizing skill is explored empirically. Future research, as well as
educational practices, should consider the discussion of enumeration strategies in children to
fully understand how children approach enumeration of number representations.
Conceptual Subitizing Stages and Mathematics Achievement
The third research question intended to see if there were any differences in the
mathematics achievement scores between the three clusters. Significant group differences were
found between the slowest- and the fastest-performing clusters. This is in line with previous
research on the relation between perceptual subitizing and mathematics abilities, suggesting that
the slower the enumeration of displays, the lower their mathematics achievement (Gray &
75
Reeve, 2014; Reigosa-Crespo et al., 2013). The current research contributes to the existent
literature by providing initial evidence to suggest that this relationship continues beyond the
more typically studied perceptual subitizing range. Specifically, enumeration of the numerosity
range between four and 10 were found to be predictive of children’s understanding of
mathematics and number. The findings also suggest that teaching and promoting the use of
conceptual subitizing skills, and advancing children into the higher-level subitizing skills may
support children’s mathematics skills. In addition, it may suggest that while the understanding of
the numbers as a whole and as parts is essential, the efficiency in their ability to utilize this
understanding is important. As such, classrooms also need to incorporate activities that focus on
how quickly children are able to enumerate the numbers, as discussed by Clements and Sarama
(2014).
In review of the current results, while these results do seem to lead support for conceptual
subitizing as important for mathematics understanding, additional explanations of the obtained
results are possible. It could also be argued that children’s former experience with number, and
children’s math fact fluency, may have shaped the cluster groupings as opposed to differing part-
whole strategy use. In other words, those who may have been faster at applying their previous
knowledge of the dot representations of number and/or retrieving math facts for addition of the
parts in the displays may result in faster RT, which distinguishes the groupings rather than the
strategies they are using to enumerate the array. The interview on their strategies attempted to
overcome this limitation, which revealed that those who were fast at detecting the numerosity
were also those who showed flexibility in their strategy to enumerate the displays. However,
previous research on strategy choices suggests that children who use more mixed methods in
solving simple addition problems are better at mathematics (Geary & Brown, 1991). Thus, the
76
current evidence may merely be suggesting that the cluster groupings just revealed those who
were better at mathematics overall. Future research should attempt to control for the math fact
fluency, especially when investigating the relationship of the RT to overall mathematics
achievement.
Limitations
A number of limitations of the study should be considered in interpreting the current
outcomes, and addressed in future research. The primary limitation of the current study stems
from the limited sample size. Small sample size of the current study was mostly a result of the
time-consuming analysis procedure applied to the behavioural RT data. Extraction of the RT
information from the vocal data took a significant amount of time, which constrained the number
of data analyzable within the allocated time frame. As such, a larger sample size is required to
validate the cluster analysis outcomes from this study. A common method to attempt replication
of the cluster solution is to split the sample into half and implement the cluster analysis
separately (Pastor, 2010). However, this approach was not feasible in the current study, and
thus, the validity of the cluster solution was not tested. As such, future studies require a much
larger sample size to improve replicability of the outcomes.
Small sample size also posed some limitation to the comparison of mathematics
achievement scores between the cluster groups. With a large difference in the sample size
between the three clusters, a robust statistical test could not be conducted. Future research
should thus include a large sample to minimize the difference in the cluster sizes. However,
there is some likelihood that the difference in the sample size between the clusters was due to the
fact that Grade 2 students are mainly in the imagery-based subitizing stage. To investigate this
possibility, future research should attempt to utilize a cross-sectional design by investigating
77
children in Grade 1 and 3, in addition to Grade 2, to see if children in different age groups would
show a difference in the number of students clustered into the three developmental stages.
Another limitation to the current study is from the fact that teachers reported that they do
not use the canonical patterns in their instruction for Grade 2 students. As such, it was not
possible to consider that children were familiar with these structures, which were supposed to be
“familiar” and readily identifiable. The lack of confidence that children were familiar with these
structures affects the interpretation of the RT on canonical patterns. This outcome may be,
however, the result of asking the Grade 2 teachers who were expected to instruct on larger
numbers above 10 (Alberta Education, 2014). One option in future studies would be to ask the
Grade 1 teachers whether they used domino dot patterns in their instructions when teaching the
current Grade 2 students. The current study recruited from the Grade 1 teachers in the
participating schools, but did not receive interest from any of the teachers approached (n = 4).
Future research should attempt to attain more information regarding the type of instructions the
students have received in the current grade as well as in the earlier years.
Implications
The current study offered an initial empirical investigation into children’s quick and
accurate enumeration skills for numerosities within the post-subitizing range (4-10 items) and
provided preliminary evidence for its relation to mathematic achievement. A key implication of
this study is the potential usefulness of conceptual subitizing as a screening tool for children’s
mathematics understanding. The current study provides an initial roadmap to how Clements and
Samara’s proposed developmental framework of conceptual subitizing can be operationalized
and studied (see Table 1, p. 29). Ideally, this roadmap could be used in developing conceptual
subitizing screeners that could be used as early indicators of children’s mathematics skills.
78
One population that may particularly benefit is children with MLD. Future research may
explore conceptual subitizing skills in children with MLD to determine possible differences in
speed and strategy use from typically developing children, and determine its contribution to the
learning difficulties of this group. Assessing their skills on both canonical displays and grouped
displays may reveal what skills are lacking in children with MLD. It may further provide
suggestions to educators on the kind of conceptual subitizing activities that are necessary for
each individual to improve their understanding of number concepts. In addition, evaluation of
the lacking skills at an individual level could expand to a method of assessment for the response-
to-intervention (RTI) strategy of supporting those who are showing some early difficulty in
mathematics.
More recently, discussion on the importance of symbolic and non-symbolic number skills
on mathematics achievement at different developmental stages have proposed that symbolic
skills (using Arabic numerals) show stronger relationship to mathematics achievement as
children age (De Smedt, Noël, Gilmore, & Ansari, 2013; Merkley & Ansari, 2016), while non-
symbolic skills (using dots) are more important for mathematics skills in younger children
(before age 7) and children with mathematics difficulties (Brankaer, Ghesquière, & De Smedt,
2014; Rousselle & Noël, 2007). As conceptual subitizing is a skill that concerns non-symbolic
representation of number, it aligns with Clements and Sarama’s (2014) argument for the
importance of promoting children’s engagement and development of conceptual subitizing skill
in the early years (before age 6). While the discussion of the non-symbolic skill has lead to
focus on younger children, the current study supports the continued importance of non-symbolic
skills with Grade 2 students. Thus, it may be argued that conceptual subitizing may still be a
relevant skill to assess older children’s non-symbolic number skills. In fact, some researchers
79
have provided evidence that mapping on non-symbolic representations of number to the
symbolic numbers are important for mathematics understanding in school-aged children
(Kolkman, Koresbergen, & Leseman, 2013; Nöel & Rousselle, 2011), suggesting the importance
of children’s experience with a more concrete non-symbolic representation of number to become
fully aware of the quantity of certain numbers. In addition to symbolic skills, experience with
non-symbolic representation of number can help in advancing children’s understanding of
number principles (Kolkman et al., 2013). Thus, it may be argued that conceptual subitizing
may still be a relevant skill for children’s mathematics understanding, and therefore is important
to assess older children’s non-symbolic number skills.
The conceptual subitizing task used in this study also offered a slightly different way to
measure children’s non-symbolic skills from what is more commonly used in the current
literature. The activity to assess children’s non-symbolic skill in the current literature mainly
involves comparing the magnitude of two different dot arrays and choosing the array that has a
larger magnitude (e.g., Lyons, Ansari, & Beilock, 2012; Rousselle & Nöel, 2007). On the other
hand, conceptual subitizing requires exact enumeration of the non-symbolic number
representations. Thus, the conceptual subitizing may offer a novel approach to measure
children’s non-symbolic skills as a screener for children’s mathematics skills.
Future Directions
While the current study offers a possible way to operationalize and empirically explore
children’s conceptual subitizing skill, the framework and the methods used in this study are
exploratory. A number of recommendations for future study in this area are outlined below
based on the experience from conducting this study.
80
First, different data recording methodology is suggested for future studies that attempt to
explore children’s conceptual subitizing skills. One suggestion is to use a different tool to record
children’s RT on the dot enumeration task. As discussed earlier, the current study’s small
sample size was due to the method used to extract children’s RT. The use of the free software
resulted in a process that required a tremendous amount of time, which renders the use of a large
sample impractical. Software such as E-prime would allow more automatic retrieval of RT
information from the task performance, providing an ideal environment for larger sample
analysis. Thus, it is recommended to use software that allows easier, less labour intensive,
collection and analysis of RT data.
Future research may also benefit from observing and recording children’s behaviours
while enumerating the arrays in the computer task. Videotaping the students during the
computer task may capture some behaviours during the enumeration task, such as head nodding
to count, finger counts, pointing to the dots on the screen, which may be informative of the
children’s actual strategies to enumerate the displays. These behaviours were observed during
the current task, but were not recorded as a measure for the current research. Improvement in the
quality of software to collect RT information, as well as deeper observation of children’s
behaviours during task performance would allow future research to more easily attain richer
information on children’s conceptual subitizing skills from a larger sample.
Conclusions
The current research provided some preliminary empirical support for the existence of the
developmental stages of conceptual subitizing, which could be generated from the RT profiles on
different types of displays. It also revealed some trend in the relationship between the subitizing
stages and mathematics achievement, which is a good initial step to support the argument that
81
including conceptual subitizing skills in classroom activities in the early grades may be
important. However, as the current study is a preliminary investigation on the skill and its
relationship to mathematics achievement, further empirical examination is necessary to
determine the potential value of using conceptual subitizing activities in the classrooms. With
further investigation on the significance of the conceptual subitizing skill to mathematics
achievement, conceptual subitizing activities may serve as an effective screening tool for
children’s mathematics understanding and as a useful RTI intervention strategy for those who are
struggling in mathematics.
The overall findings from this study suggest that it is important to distinguish the term
“subitizing” as perceptual subitizing and conceptual subitizing, as it allows a deeper exploration
and understanding of the quick enumeration skill overall. Moreover, a lack of distinction
between perceptual and conceptual subitizing obscures the recognition of the more limited
evidence in support for conceptual subitizing to this date. The current research attempted to
provide some preliminary support for the conceptual subitizing skill, but further exploration is
necessary as the investigation in the current study is still exploratory in nature. However, the
findings from the current study provide some initial argument for the importance of developing
this skill to deepen children’s understanding of number concepts and mathematics achievement.
Future research should explore conceptual subitizing further to provide more empirical support
for the conceptual subitizing stages and to offer ways to incorporate conceptual subitizing
activities in the classroom.
82
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Appendix A – Dot Enumeration Task Arrays
Number Random Canonical Grouped 1
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Appendix B – Teacher Questionnaire Name:____________________________________________GradeYouCurrentlyTeach:_____________________________Inthe2013-14schoolyear:
• Whatschooldidyouteachin?__________________________________
• Whatgradedidyouteach?_____________________________________
Howmanyyearshaveyoubeenteaching?__________________________________________Listthegradesyouhavetaughttodate:____________________________________________
1. Howfamiliarareyouwiththeterm“subitizing”?
a. Iknowthetermanditsdefinition
b. Iknowthetermbutnotthedefinition
c. Iamnotfamiliarwiththeterm
2. Howfamiliarareyouwiththeterm“conceptualsubitizing”?
a. Iknowthetermanditsdefinition
b. Iknowthetermbutnotthedefinition
c. Iamnotfamiliarwiththeterm
3. Subitizingisoftendefinedasaquickandaccuraterecognitionofthequantityofitemsdisplayed.Howmanyitemsdoyouexpectyourstudentstobeabletosubitize?
4. Doyouincorporateactivitiestoencouragesubitizinginyourmathinstructions?YES/NO
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IfselectedYES,pleaseanswerthefollowingquestionsinsidethebox.IfselectedNO,pleasemoveontoquestion5onpage4:
4.1Pleasedescribethetypesofactivitiesyouusetodevelopyourstudents’abilitiestoquicklyidentifydisplayednumbersets.
4.2Howfrequentlydoyouusetheaboveactivitiesinyourclassroom?
a. Daily
b. Weekly
c. Monthly
d. UnitDependent(pleasespecify):
4.3Whatisthemaximumnumberofitemsyouaskyourstudentstoidentify?4.4Didyoulimittheamountoftimestudentscouldlookattheitems?Ifso,howlongweretheyallowedtoseetheitems?
YES/NOHowlong:_____________
98
5. Inthebelowchart,placeacheckmarkbesideeachtypeofdisplayyouhaveusedtorepresentnumbersinyourmathematicsinstruction.Inaddition,identifythetopthreedisplaysthatyouusethemostfrequently.Write1forthedisplaythatisusedthemost,2forsecondmost,and3forthirdmost.(NOTE:Imagesareexamples.Pleaseselectifyouhaveusedsimilardisplays):
Used Top3 Displays Used Top3 Displays Dicedotpatterns
Randomdots
Dominopatternslargerthan6
Fingerpatterns
Geometricshapepatterns
Tenframes
Otherstructureddotpatterns
Other(pleasedrawbelow)
6. Howdidyouusetheabovedisplaysintheclassroom?Pleasedescribebrieflybelow.
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7. Inthebelowchart,placeacheckmarkbesideeachtypeofconfigurationsofdotsyou
haveusedtorepresentnumbersinyourmathematicsinstruction.Inaddition,identifythetopthreeconfigurationsthatyouusethemostfrequently.Write1forthedisplaythatisusedthemost,2forsecondmost,and3forthirdmost.(NOTE:Imagesareexamples.Pleaseselectifyouhaveusedsimilardisplays):
Used Top3 Configurations Used Top3 Configurations Patterneddotsseparatedbyspace
Randomdotsseparatedbycolour
Patterneddotsseparatedbycolour
Other(pleasedrawbelow)
Randomdotsseparatedbyspace
Thankyouverymuchforyourtime!