Concrete, Representational, Abstract 1
THE EFFECTS OF CONCRETE, REPRESENTATIONAL, ABSTRACT
INSTRUCTIONAL SEQUENCE ON COMPUTATIONAL FLUENCY IN THE
MATHEMATICS CLASSROOM
Julianna Pasetto
School of Education, Liberty University
Concrete, Representational, Abstract 2
Introduction
Research has shown that the concrete, representational, and abstract methods of teaching
math is the most effective way to introduce new topics to students, particularly those with
disabilities (Milton et al., 2019). In addition to this approach, there are also various iterations of
this procedure, such as virtual, representational, and abstract, as well as concrete,
representational, abstract, and integrated (Bouck et al., 2018; Flores & Hinton, 2021). According
to Lafay et al., concrete, representational, and abstract provide students with a physical
manipulative before moving to a pictorial phase and then the final stage: the abstract or number
phase (2019).
Students are introduced to the concrete portion of this procedure using math
manipulatives (Flores, 2009). Once they have mastered the concrete phase, they transition to a
two-dimensional or pictorial depiction in what is referred to as the representational phase (Flores,
2009). When students have mastered both stages, they move to the abstract stage by solving
problems containing numerals as representations (Flores, 2009). Although this process is time-
consuming, the findings from several studies show promising results, especially for students with
learning disabilities (Bouck et al., 2018; Flores, 2009; Flores et al., 2018; Hinton & Flores, 2019;
Lafay et al., 2019; Milton et al., 2019; Park et al., 2020).
The lesson's concrete portion can include physical or virtual manipulatives, such as
counters or ten frames. Jerome Bruner, a cognitive psychologist, create a model which states that
learners should progress from the enactive phase by manipulating concrete materials through to
the iconic stage, which is creating drawings to represent the concrete materials, and finally: the
symbolic stage, which is when one uses mathematical symbols to represent the objects in the
situation (Bakhurst & Shanker, 2001). These stages coincide with Bruner's engagement levels:
enactive, iconic, and symbolic (Bruner, 1966).
Importance of Mathematics
The National Council for the Teachers of Mathematics, otherwise known as NCTM,
states in its Principles to Actions: ensuring mathematical success for all that effective teaching
of mathematics builds fluency with procedures on a foundation of conceptual understanding so
that students, as instruction progresses, become more proficient in utilizing a variety of
procedures as they solve contextual and mathematical problems (NCTM, 2014). These
procedures are taught best after a student has mastered mathematics and actively participated in
their learning (NCTM, 2014). They also emphasize the importance of effective mathematics
teaching that engages students in making connections, using mathematical representations to
deepen understanding of mathematics concepts and procedures, and as tools for problem-solving
(NCTM, 2014).
Mathematics is essential to daily life; we need it to determine how much we pay for gas,
coffee, or tips for servers. As a result, students need to learn how to be efficient, flexible,
accurate, and automatic with their computation skills (Hannula et al., 2010). True computational
fluency, according to Bay-Williams, is defined as "answering within three seconds, either
Concrete, Representational, Abstract 3
through recall or automatic strategy application" (Bay-Williams & Kling, 2019, p. 3). More
specifically, according to Hannula et al., academic success is predicted by understanding
numbers and performing simple operations (Hannula et al., 2010).
Students must regularly engage in problem-solving and reasoning tasks to develop high-
level thinking skills (NCTM, 2014). Multiple entry points, including different representations
and tools and diverse approaches to problem-solving, are used in these tasks to foster reasoning
and access to mathematics. However, learning and applying procedures are in the curriculum and
necessary for developing fluency (NCTM, 2014).
Mathematical representations play a crucial role in effective mathematics teaching, and,
as the NCTM Process Standard for Representation highlights, these also play an important role
in teaching and learning mathematics (NCTM, 2014). Diagrams and words illustrate and explain
fundamental mathematical concepts and actions, such as fractions, ratios, and multiplication
(NCTM, 2014). It is also pertinent to note that physical objects can be considered representations
(NCTM, 2014).
Students who become fluent in math learn number concepts, informal reasoning
strategies, and eventually general methods for solving problems based on the properties and
meanings of operations. A fluent student can choose methods and strategies flexibly, explain
their approaches, and deliver accurate answers rapidly (NCTM, 2014). Computational fluency
and number sense are closely related, and the relationship encompasses much more than had
been previously realized. The development of computational fluency extends far beyond
memorizing facts and following steps unrelated to understanding (NCTM, 2014).
As tools for communication, thought, and calculation, mathematical ideas are
externalized, shared, and archived through physical representations (National Research Council,
2001). Clarifying ideas helps to build mathematical understanding and supports logical
reasoning. Additionally, these representations facilitate algorithm development. Mathematics
relies heavily on representations. Mathematical ideas are only accessible through their
representations because mathematics is abstract (National Research Council, 2001). Therefore,
the representations aid children in understanding mathematics.
Multiplication
In most states, the Common Core State Standards for Mathematics emphasize students'
conceptual understanding of operations and numbers (2010). Procedural fluency in algorithms
depends on understanding and reasoning about multiplication (Milton et al., 2018). Developing
multiplicative reasoning skills in younger students will also be crucial to their future success in
advanced mathematics courses (Milton et al., 2018). A particular focus has been placed on
mathematics instruction, as it directly correlates to college and career readiness (Milton et al.,
2018). Harel and Sowder state, "Students develop the foundations for advanced mathematical
thinking in elementary grades, and one particularly critical foundation is multiplicative
reasoning” (Harel & Sowder, 2005, p. 29).
Addition
According to the Common Core State Standards, fluency is knowing sums from memory
Concrete, Representational, Abstract 4
(2010). Students must understand which strategies to use and under what circumstances to use
them, and calculating math facts occupies working memory resources. Math facts retrieval frees
up working memory resources for other tasks, such as understanding the relevant problem in a
text question (Gliksman et al., 2022). In order to succeed at math, students need efficient
strategies or to have progressed through the learning continuum (Flores & Hinton, 2021). It is
common for students who struggle with math to get stuck at the first strategy, counting every
number in the problem to add up. For students to become efficient in math, teachers must model
these strategies with manipulatives.
Definition of Key Terms
Concrete, Representational, Abstract Model
The lesson's concrete portion can include physical or virtual manipulatives, such as
counters or ten frames. These stages coincide with the levels of engagement outlined by Bruner
(1966); enactive, iconic, and symbolic. Bruner's model states that "learners should progress from
the enactive (manipulating concrete materials) through to the iconic stage (creating drawings to
represent the concrete materials) and finally the symbolic stage (using mathematical symbols to
represent the objects in the situation)" (Bruner, 1966, p. 2). This instructional strategy coincides
with current commercial curriculum products such as Go Math and enVision 2.0. Students are
first introduced to a topic with manipulatives in the initial lesson by the teacher and then recreate
a similar picture of the manipulative in the guided practice. Finally, the independent practice
portions are only numerical representations (Dixon et al., 2016; Berry et al., 2018).
Students move from the concrete to the representational as they gain more experience
with the given manipulative. In the concrete phase, students can conceptualize numbers in their
terms rather than simply seeing them as numbers (Flores & Hinton, 2019). In addition to
providing students with opportunities to manipulate physical objects, this stage also helps them
develop visual images and fades visual supports explicitly (Flores & Hinton, 2019). The student
will eventually stop using representational tools. The child's attention increasingly shifts to
focusing on relationships as they gain experience working with representational tools.
Eventually, the representation becomes a referential basis for the following formal mathematical
reasoning (Ramsingh, 2020). The learner will no longer need to think about representational
tools to understand math (Ramsingh, 2020). Students can rely on something other than the
manipulative for the entire process but only for the concrete aspects. The manipulative, such as
counters, will lessen the cognitive load at first and allow the student to concentrate on the
quantity represented by the object(s) (Ramsingh, 2020). According to Milton et al., students
should actively participate in manipulating objects, pictures, drawings, and other representations.
As students learn operations, multiple representations support their cognitive processing and
connections (Milton et al., 2019). This type of learning is particularly effective for students with
learning disabilities (Bouck et al., 2018; Flores, 2009; Flores et al., 2018; Hinton & Flores, 2019;
Lafay et al., 2019; Milton et al., 2019; Park et al., 2020).
Concrete, Representational, Abstract 5
Manipulatives
According to the National Research Council (2001), "Mathematics requires
representations., because of the abstract nature of mathematics, people have access to
mathematical ideas only through the representations of those ideas" (p. 96). Students naturally
develop these ideas in their heads as representations. Manipulatives are teacher-provided,
structured supports that scaffold the mathematics curriculum. "'Manipulatives' are concrete or
virtual objects intended to reify central concepts in the mathematics curriculum" (Lafay et al.,
2019, p. 1). Children may find these materials distracting initially, and it takes time to develop
mathematical meanings for them and use them effectively (National Research Council, 2001).
Willingham associated manipulatives with Piaget's concrete operational stage, stating, "the child
uses concrete objects (during this stage) to support logical reasoning, whereas, in the formal
operations stage (age 12 to adulthood), the child can think of using pure abstraction"
(Willingham, 2017, p. 26).
As an aid in learning, manipulatives require teacher guidance. Students are unlikely to
learn the target concepts by simply providing them with materials and allowing them to do with
them as they see fit. The use of arrays and the area model are instrumental in third through fifth-
grade instruction for multiplication and its inverse operation: division. While the use of the "ten
frames" is commonly seen in early childhood levels, such as kindergarten through first or second
grade, with the focus on ten and the ability to subitize, or "intuitively see," a small quantity of
objects. "Subitizing is an early numeracy mathematical skill where students can state the quantity
of something without counting" (Jimenez & Saunders, 2018, p. 23). At the early elementary
grade levels, one-to-one correspondence is a focus, and students often use bears or counters.
Providing multiple learning environments, such as manipulatives, offers remembrance,
materialization, saving time, and facilitating understanding for learners (Boz et al., 2020). When
students use manipulatives, they can reallocate their cognitive resources to facilitate observation,
reflection, and connection-based activities (Suh & Moyer-Packenham, 2007).
Concrete, Representational, Abstract 6
Virtual Manipulatives
Another option for the concrete stage is to use virtual manipulatives, such as the National
Library for Virtual Manipulatives (http://nlvm.usu.edu/en/nav/vlibrary.html). “Virtual
manipulatives are digital versions of concrete manipulatives” (Moyer et al., 2002, p. 372). In
Moyer-Packenham’s International Perspectives on Teaching and Learning Math with Virtual
Manipulatives, the author defines virtual manipulatives as "an interactive, technology-enabled
visual representation of a dynamic mathematical object, including all of the programmable
features that allow it to be manipulated, that present opportunities for constructing mathematical
knowledge" (p. 13). For instance, students can engage on a device with base ten blocks and
manipulate them to show the process of regrouping to subtract. They can also overlap fractions
to add unlike denominators and manipulate those denominators to arrive at the sum. Virtual
manipulatives are another option for the "concrete" phase of concrete, representational, and
abstract methods even though students are not physically touching the manipulative. Another
popular virtual manipulative site, besides the National Library for Virtual Manipulatives
mentioned earlier, is BrainingCamp.com (https://www.brainingcamp.com/manipulatives). In
both sites, virtual manipulatives are available for various grade levels and skills.
Strategic Instruction Model
Several research studies also included using the Strategic Instruction Model (SIM).
Once students show understanding of concepts such as operations, properties, places
value, and so forth, instruction involves a procedural strategy, often a mnemonic, to
remember the procedures required to complete an abstract problem. The strategy serves
as a bridge between the conceptual understanding of the numbers and operations to
computational fluency. Finally, at the abstract level, the instructional focus is the
Concrete, Representational, Abstract 7
computation of abstract problems with the assistance of a mnemonic strategy" (Flores et
al., 2020, p. 166).
This mnemonic device provides students with a lasting strategy for conceptual understanding.
These same devices can also transfer to other mathematical thinking (Flores et al., 2020).
At the same time, the SIM model focuses on using mnemonics during the abstract portion
of learning. This is especially helpful during problem-solving when students have more than one
step and remembering a procedure becomes an issue (Flores & Hinton, 2019). During Flores and
Hinton's 2019 research, students were taught the mnemonic strategy, RENAME, which is more
specific to regrouping while multiplying ((a) read the problem, (b) examine the ones, (c) note the
notes, (d) address the tens, (e) mark the tens, (f) examine the hundreds, and exit with a check)
(Flores & Hinton, 2019). All three students improved their computation performance and
achieved fluency in the multiplication computation with regrouping. Ultimately, students
demonstrated at least 90% of the correct digits after these interventions (Flores & Hinton, 2019).
During the final stage of CRA-I, this mnemonic instruction becomes an important strategy.
“Mnemonics use the principle of association, which is the information to be remembered
associated with other information that is easy to remember” (Ni & Hassan, 2019, p. 93).
Computational Fluency
Computational fluency is the ability to efficiently and accurately solve mathematical
problems using mental math (Flores et al., 2019). It is a critical skill that students need to
develop in elementary school, as it forms the foundation for success in higher math courses
(Flores et al., 2019). In fact, according to Flores et al., the importance of a student developing an
understanding of multiplicative reasoning directly correlates to that student's success in advanced
mathematics courses (Flores et al., 2019). This understanding is similar to the concept of
fractions. According to Flores et al. in "Teaching Fraction Concepts Using the Concrete-
Representational-Abstract Sequence," if a student does not have an established understanding of
fractions, they will struggle with more sophisticated mathematics in years to come (Flores et al.,
2018).
According to the "Principles to Actions" from the National Council of Teachers of
Mathematics (2014), a mathematically fluent student must be able to choose from various
Concrete, Representational, Abstract 8
computational methods in an accurate and efficient amount of time. This idea is far from simple
since there need to be several options for the child to choose from in their mathematical
"toolbox."
The math facts consist of problems involving operands from 0 to 10 and the four basic
arithmetic operations: addition, subtraction, multiplication, and division from the multiplication
table (Gliksman et al., 2022). The ability to solve math facts accurately and quickly is known as
math fluency. One of the essential fluencies a child can have is additive since it is the
foundational skill for all other fluencies (Gliksman et al., 2022). Children begin with the
"counting all" strategy, using a concrete item to count each number in the equation. The
subsequent strategy children use is "counting on," where children “count on” from one of the
operators in the problem, preferably the larger. Afterward, children use related or "known" facts
such as doubles or doubles plus one, frequently decomposing numbers into parts. Finally, the
facts have been stored in long-term memory and can be retrieved fluently and efficiently
(Gliksman et al., 2022).
Learning thinking strategies helps students discover, label, and internalize relationships -
processes that are fundamental to mastering the basics. It is also possible that internalized rules,
procedures, and principles become routinized, contributing to the efficient production of number
combinations in adults and children (Baroody, 1985). A stored procedure, rule, or principle can
easily construct a range of combinations and is cognitively more efficient than relying solely on
a network of individual facts (Baroody, 1985). Applying rules, procedures, and principles reduce
cognitive workload (Baroody, 1985). A special needs child is faced with memorizing many
isolated facts because he or she lacks a rich network of rules and principles to rely upon
(Baroody, 1985). However, the drill is also essential. According to Baroody, in addition to
fostering the formation of specific numerical associations, drills may also aid in the routinization
of rules, procedures, and principles (Baroody, 1985). Once students have a conceptual
understanding of the numerical facts, the drill becomes appropriate.
Conceptual Knowledge
Every year a student's mathematics instruction builds upon the prior year. Students must
enter the next grade level with the necessary fluencies and conceptual understandings of the year
Concrete, Representational, Abstract 9
prior. “Conceptual knowledge is a multifaceted construct that includes knowledge of categories,
relationships, principles, and representations” (Braithwaite & Sprague, 2021, p. 2). For instance,
in kindergarten, students work with quantities to ten, including one-to-one correspondence and
the beginning stages of additive quantities. Whereas in first grade, they must know about
kindergarten strategies to be successful. In first grade, students are working with quantities to
one hundred, including more advanced place value methods and compensation (Common Core
State Standards for Mathematics 2010). At this point, students must have had access to the
concrete, representation, and abstract method of instruction. A rekenrek is one essential concrete
tool students can use to see the pattern in the numbers and be able to manipulate them fluently.
This manipulative aids in the conceptual knowledge of numbers, allowing students to build in
groups of five and ten, using the count-on strategy, doubling and halving strategy, and solving
from a known fact (Todenvold, 2019).
Students must have a solid understanding of procedures before proceeding with
conceptual knowledge (Flores et al., 2019). It is crucial that conceptual understanding is
complemented by procedural understanding so that students can "make sense" of the math and
ensure that it is not simply procedures applied to numbers. The concept of place value and
number concepts is necessary for a student to understand the regrouping procedure when adding
two-digit numbers (Flores et al., 2019). In addition, students must understand the concept of
equal groups in order to succeed with division (Milton et al., 2018). In order to learn
mathematics, it is necessary to have a conceptual and procedural understanding (Braithwaite &
Sprague, 2021).
Conceptual knowledge may compensate for a problem's weak procedural knowledge
(Braithwaite & Sprague, 2021).
Based on Braithwaite and Sprague (2021), conceptual knowledge becomes important
when procedural knowledge fails, such as when a student does not know or is uncertain about an
appropriate procedure or when he or she commits an error if the procedure is incorrectly used or
executed (Braithwaite & Sprague, 2021). Conversely, a student who is confident in retrieving
and correctly executing a given procedure may do so without using conceptual knowledge
(Braithwaite & Sprague, 2021). This research solidifies the fact that students need both
conceptual and procedural knowledge to be successful in mathematics.
Concrete, Representational, Abstract 10
Related Literature
Practices, Policies, and Procedures that Relate to the Topic
The Common Core State Standards for Mathematical Practice lend themselves to the
concrete, representational, and abstract method of instruction. The second Standard for
Mathematical Practice states, "Reason abstractly and quantitatively" (Common Core State
Standards for Mathematics, 2010, p. 6). While the fourth standard states, "Model with
mathematics" (Common Core State Standards for Mathematics, 2010, p. 7). These standards
emphasize the importance of concrete and pictorial referents throughout the mathematical
learning continuum. Furthermore, the standards at the kindergarten level include "counting out a
given number of objects; comparing sets or numerals; and modeling simple joining and
separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2
= 5" (Common Core State Standards for Mathematics, 2010, p. 11). This numeracy continuum
continues into first grade when "They use a variety of models, including discrete objects and
length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-
together, take-apart, and compare situations to develop meaning for the operations of addition
and subtraction, and to develop strategies to solve arithmetic problems with these operations"
(Common Core State Standards for Mathematics, 2010, p. 15). Concrete models return to the
Common Core standards in third grade when the focus is on multiplication, "Students develop an
understanding of the meanings of multiplication and division of whole numbers through
activities and problems involving equal-sized groups, arrays, and area model" (Common Core
State Standards for Mathematics, 2010, p. 23).
Further, multiplication assimilates into division in fourth grade. Models once again
become necessary, "Students apply their understanding of models for division, place value,
Concrete, Representational, Abstract 11
properties of operations, and the relationship of division to multiplication as they develop,
discuss, and use efficient, accurate, and generalizable procedures to find quotients involving
multi-digit dividends" (Common Core State Standards for Mathematics, 2010, p. 29).
Overall, concrete models are a powerful tool for teaching and learning mathematics. By
engaging with these models, students can develop a deep and intuitive understanding of
mathematical concepts that will serve them throughout the mathematics continuum.
Steps That Have Been Taken to Address the Topic
As addressed throughout this paper, the CRA approach has been studied extensively with
general and special education students. One such study, "The Effects of a CRA-I Intervention
on Students' Number Sense and Understanding of Addition'' clearly showed a marked increase in
students' achievement over time (Flores & Hinton, 2021). In fact, the results showed that
"students increased their understanding of the operation, numbers, increased their flexibility, and
showed automaticity" (Flores & Hinton, 2021, p. 192). All students wrote 30 correct digits
within the time frame. Students maintained this learning for at least two weeks after the
intervention, with some continuing this mastery four weeks later (Flores & Hinton, 2021).
Another study completed by Flores, "Using the Concrete-Representational-Abstract Sequence to
Teach Subtraction With Regrouping to Students at Risk for Failure" (Flores, 2009), addressed
vital deficits in the area of subtraction with regrouping to special education students already at
risk of failing mathematics. Their results were so promising that not only did six weeks post-
instruction, four of the six students continued their progress in computing subtraction with
regrouping fluently, but teachers also noted increased student participation (Flores, 2009). In
fact, "All of the students met criteria in the two regrouping phases-the tens place and the tens and
hundreds place- writing at least 20 digits correctly" (Flores, 2009, p. 204).
Concrete, Representational, Abstract 12
Hinton and Flores also studied the CRA sequence for students at risk for mathematics
failure. The effects of CRA on special education elementary students at risk of mathematics
failure were explored in Flores and Hinton's 2019 study, "The Effects of Concrete-
Representational-Abstract Sequence for Students at Risk of Mathematics Failure." Students in
this study effectively demonstrated mastery of subtraction according to a set probe with 100%
accuracy, providing all 25 digits correctly in two minutes (Flores & Hinton, 2021). A lack of
conceptual understanding of numbers (rounding, regrouping, fraction comparisons) leads to
several conceptual needs (rounding, regrouping, fraction comparisons). One can attribute that to
a greater sense of the student's conceptual understanding.
Learning Theory Association
As a key figure in cognitive psychology, Jerome Bruner (1915-2016) made significant
contributions to instructional theory and practice. As a psychologist, he encouraged educators to
introduce problem-solving and intellectual development into the curriculum (Stapleton &
Stefaniak, 2018). In his works, he advocates a cognitive constructivist view of learning in which
knowledge is constructed through exploration of the world or prior knowledge (Stapleton &
Stefaniak, 2018).
Bruner (1966) explored different ways of representing and organizing knowledge
throughout his work. In his constructivist theory, new material can be understood best by
following a progression from an enactive to an iconic to a symbolic representation; this also
applies to adults. Furthermore, Bruner's work suggests that even young children can learn any
material as long as they are appropriately instructed, contrary to the beliefs of Piaget and other
stage theorists (Bruner, 1966).
Constructing mathematical knowledge based on meaningful experiences for students is a
part of mathematical thinking (Díaz, 2017). According to Bruner, children construct their
knowledge in three ways: enactive (actions, actual words), iconic (images and pictures), and
symbolic (words and symbols) (Bruner, 1966). According to Bruner's stages of representation
(enactive, iconic, symbolic), manipulations by using concrete or manipulative media are then
Concrete, Representational, Abstract 13
represented graphically (drawn) and carried out symbolically (Zuliana et al., 2019). Bruner states
these stages are a cognitive human learning process (Bakhurst & Shanker, 2001).
Gaps in Research
Currently, most of the research includes students with disabilities or those needing tier-
three intervention. The research focuses on students with specific intellectual disabilities. There
was limited research done on general education students. Most research, not including special
education students, focused on tier three instruction according to the Multi-Tiered System of
Support (MTSS). MTSS is a framework schools follow to provide targeted instruction for
students on a tiered system. 80% or more of students are effectively reached by differentiated
core instruction at tier one. Supporting the remaining 80% with scaffolding is part of tier two,
while intensive instruction is for tier three students with the highest needs (Jimenez & Barron,
2022). While tier three does not necessarily mean special education for MTSS purposes, it has
been identified as the targeted tier in the research provided.
The research used concrete, representational, and abstract method of instruction with
students. In the research from Hinton and Flores (2019) on the concrete, representational, and
abstract method, fraction comparison was taught during this tier three (Multi-Tiered System of
Support) intervention time, explicitly using the area and length models. As a result of this
instruction, students in this study could effectively demonstrate mastery of subtraction, rounding,
and fractions using a set of probes with 89%-100% accuracy. While in the research of Milton et
al. (2018), students with special education services could write 30 correct digits in a minute on
three consecutive probes with 100% accuracy. However, during the maintenance phase, students
were not as accurate, with an average of 22 correct digits in a minute on similar probes with
100% accuracy (Todenvold, 2019).
Concrete, Representational, Abstract 14
Another area for improvement in the research is the variety of manipulatives. Much of
the research utilized fraction blocks (Flores et al., 2018), base ten blocks (Flores et al., 2019;
Gibbs et al., 2018), or "homemade" manipulatives (such as food on plates, base ten blocks, etc.)
(Boz et al., 2020; Milton et al., 2018). While those are key manipulatives at the elementary level,
there are many more. One such manipulative is the rekenrek. An arithmetic rack or calculating
frame, the rekenrek, was created in the Netherlands and translated into English as an arithmetic
rack (Todenvold, 2019). Rekenrek is a unique tool that teaches number sense to children at their
own pace. Through the built-in 5 & 10 structure, rekenreks help children strengthen their number
relationships and develop more advanced strategies (Todenvold, 2019). In the rekenrek, children
can model and solve problems as they would with Unifix cubes, but they can also develop more
advanced strategies due to the rekenrek's built-in structure (Todenvold, 2019).
Biblical Worldview
The Bible guides teachers toward understanding perseverance. Romans 5:3-4 says, "And
not only that but we also glory in tribulations, knowing that tribulation produces perseverance;
and perseverance, character; and character, hope" (King James Bible Online, 2023, Romans 5:3-
4). By teaching students to persevere through math, using various strategies, teachers can bring
forth this characteristic.
Also, James 1:2-3 says, “My brethren, count it all joy when you fall into various trials,
knowing that the testing of your faith produces patience” (King James Bible Online, 2023, James
1:2-3). To be an effective teacher, one should possess patience. Teachers frequently have to
provide instruction in various manners on the same topic. As a result, students may need help
grasping the concept. This requires patience. As stated in Ephesians 4:2-3, “Always be humble
and gentle. Be patient, allowing for each other's faults because of your love. Make every effort to
Concrete, Representational, Abstract 15
keep yourselves united in the Spirit, binding yourselves together with peace" (King James Bible
Online, 2023, Ephesians 4:22-3). Teachers are God's vessels for imparting knowledge through
patience and love.
Conclusion
The benefits of mathematical learning include fostering students' curiosity, developing
critical thinking, and developing problem-solving abilities (Park et al., 2021). As students with
learning disabilities progress through higher grades without establishing a fundamental
understanding of mathematics, their struggle to learn mathematics intensifies each year (Park et
al., 2021). Examining previous research to identify effective instructional strategies for teaching
mathematics to students with these difficulties is crucial. One evidence-based practice is the
concrete, representational, and abstract method (Park et al., 2021). The concrete,
representational, and abstract (CRA) approach is a method of teaching where the teacher
provides students with various ways of visualizing mathematical concepts that increase the
difficulty level from concrete to pictorial to abstract in three stages (Park et al., 2021). Students
can participate in three different modes of representations: "enactive" (physical manipulatives),
"iconic" (pictorial representation), and "symbolic" (abstract mathematical symbols)
representations (Bruner, 1966).
According to the What Works Clearinghouse, teachers should “Use a well-chosen set of
concrete and semi-concrete representations to support students’ learning of mathematical
concepts and procedures” (Assisting Students Struggling with Mathematics: Intervention in the
Elementary Grades, Recommendation 3, WWC: What Works Clearinghouse 2021). Also
according to their Recommendation Number Three for Assisting Students Struggling with
Mathematics: Intervention in the Elementary Grades, What Works Clearinghouse identifies the
Concrete, Representational, Abstract 16
use of the concrete, representational, and abstract method as an instructional practice for students
at tiers two and three (Assisting Students Struggling with Mathematics: Intervention in the
Elementary Grades, Recommendation 3, WWC: What Works Clearinghouse, 2021).
Teachers make choices every day. They choose the most effective research-based
strategies to ensure students have the best outcomes. It is necessary that, as teachers and leaders,
we make sure that effective interventions are in place. Concrete, representational, and abstract is
a key judgments that teachers can make as a consideration to combine procedural knowledge and
conceptual knowledge in the teaching of mathematics. The concrete, representational, and
abstract method is an effective intervention for tier three and special education students bridging
the gap from using manipulatives to drawing pictorial representations to, finally, the abstract
representation of a concept.
Bibliography
Bakhurst, D., & Shanker, S. (2001). Jerome Bruner: Language, culture, self.
https://doi.org/10.4135/9781446217634
Baroody, A. J. (1985). Mastery of basic number combinations: Internalization of relationships or
facts? Journal for Research in Mathematics Education, 16(2), 83–98.
https://doi.org/10.5951/jresematheduc.16.2.0083
Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An Alternative Reconceptualization of
Procedural and Conceptual Knowledge. Journal for Research in Mathematics Education,
38(2), 115–131.
Bay-Williams, J. M., & Kling, G. (2019). Math fact fluency: 60+ games and assessment tools to
support learning and retention. ASCD.
Berry, R. Q., Champagne, Z., Milou, E., Schielack, J. F., Wray, J. A., Charles, R. I., & Fennell,
F. (2018). EnVisionMath 2.0. Pearson.
Bouck, E. C., Park, J., Shurr, J., Bassette, L., & Whorley, A. (2018). Using the virtual–
representational–abstract approach to support students with intellectual disability in
Mathematics. Focus on Autism and Other Developmental Disabilities, 33(4), 237–248.
https://doi.org/10.1177/1088357618755696
Concrete, Representational, Abstract 17
Bouck, E. C., Satsangi, R., & Park, J. (2018). The concrete–representational–abstract approach
for students with learning disabilities: An evidence-based practice synthesis. Remedial
and Special Education, 39(4), 211–228. https://doi.org/10.1177/0741932517721712
Boz, M., Uludag, G., & Erdogan, S. (2020). The Effect of The Manipulative Material on The
Early Mathematical Skills. Bartin University Journal of Education, 9(3), 492–500.
Braining Camp. Brainingcamp. (n.d.). Retrieved March 2, 2023, from
https://www.brainingcamp.com/manipulatives
Braithwaite, D. W., & Sprague, L. (2021). Conceptual knowledge, procedural knowledge, and
metacognition in routine and nonroutine problem solving. Cognitive Science, 45(10).
https://doi.org/10.1111/cogs.13048
Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.
Dixon, J. K., Burger, E. B., Leinwand, S., Larson, M. R., Sandoval-Martinez, M. E., Petrello, R.,
& JimTnez Elizabeth. (2016). Go math! Houghton Mifflin Harcourt Publishing Company.
Díaz, L. D. (2017). The teaching and learning process of mathematics in the primary education
stage: A Constructivist proposal within the framework of key competencies.
International Electronic Journal of Mathematics Education, 12(3), 709–713.
https://doi.org/10.29333/iejme/643
Flores, M. M. (2009). Using the concrete-representational-abstract sequence to teach subtraction
with regrouping to students at risk for failure. Remedial and Special Education, 31(3),
195–207. https://doi.org/10.1177/0741932508327467
Flores, M. M., & Hinton, V. M. (2019). Improvement in elementary students' multiplication
skills and understanding after learning through the combination of the concrete-
representational-abstract sequence and strategic instruction. Education and Treatment of
Children, 42(1), 73–99. https://doi.org/10.1353/etc.2019.0004
Flores, M. M., & Hinton, V. M. (2021). The effects of a CRA-I intervention on students’ number
sense and understanding of addition. Remedial and Special Education, 43(3), 183–194.
https://doi.org/10.1177/07419325211038009
Flores, M. M., Hinton, V. M., & Meyer, J. M. (2018). Teaching fraction concepts using the
concrete-representational-abstract sequence. Remedial and Special Education, 41(3),
165–175. https://doi.org/10.1177/0741932518795477
Flores, M. M., Moore, A. J., & Meyer, J. M. (2020). Teaching the partial products algorithm with
the concrete representational abstract sequence and the strategic instruction model.
Psychology in the Schools, 57(6), 946–958. https://doi.org/10.1002/pits.22335
Gibbs, A. S., Hinton, V. M., & Flores, M. M. (2018). A case study using CRA to teach students
with disabilities to count using flexible numbers: Applying skip counting to
Concrete, Representational, Abstract 18
multiplication. Preventing School Failure: Alternative Education for Children and Youth,
62(1), 49–57. https://doi.org/10.1080/1045988x.2017.1342218
Gliksman, Y., Berebbi, S., & Henik, A. (2022). Math fluency during Primary School. Brain
Sciences, 12(3), 371. https://doi.org/10.3390/brainsci12030371
Hannula, M. M., Lepola, J., & Lehtinen, E. (2010). Spontaneous focusing on numerosity as a
domain-specific predictor of arithmetical skills. Journal of Experimental Child
Psychology, 107(4), 394–406. https://doi.org/10.1016/j.jecp.2010.06.004
Harel, G., & Sowder, L. (2005). Advanced mathematical-thinking at any age: Its nature and its
development. Mathematical Thinking and Learning, 7(1), 27–50.
https://doi.org/10.1207/s15327833mtl0701_3
Hinton, V. M., & Flores, M. M. (2019). The effects of the concrete-representational-abstract
sequence for students at risk for mathematics failure. Journal of Behavioral Education,
28(4), 493–516. https://doi.org/10.1007/s10864-018-09316-3
Hurrell, D. (2021). Conceptual knowledge or procedural knowledge or conceptual knowledge
and procedural knowledge: Why the conjunction is important to teachers. Australian
Journal of Teacher Education, 46(2), 57–71. https://doi.org/10.14221/ajte.2021v46n2.4
Institute of Education Sciences. (2021). WWC: What Works Clearinghouse. WWC | Find What
Works! Retrieved March 4, 2023, from https://ies.ed.gov/ncee/wwc/PracticeGuide/26
Jimenez, B. A., & Barron, T. (2022). Specially designed instruction of early numeracy in the
Inclusive Elementary Classroom for students with extensive support needs. Inclusion,
10(3), 168–182. https://doi.org/10.1352/2326-6988-10.3.168
Jimenez, B., & Saunders, A. (2018). Increasing efficiency in mathematics: Teaching subitizing to
students with moderate intellectual disability. Journal of Developmental and Physical
Disabilities, 31(1), 23–37. https://doi.org/10.1007/s10882-018-9624-y
King James Bible Online. (2023). Retrieved February 20, 2023, from
https://www.kingjamesbibleonline.org/.
Lafay, A., Osana, H. P., & Valat, M. (2019). Effects of interventions with manipulatives on
immediate learning, maintenance, and transfer in children with Mathematics learning
disabilities: A systematic review. Education Research International, 2019, 1–21.
https://doi.org/10.1155/2019/2142948
Milton, J. H., Flores, M. M., Moore, A. J., Taylor, J. L. J., & Burton, M. E. (2018). Using the
concrete–representational–abstract sequence to teach conceptual understanding of basic
multiplication and division. Learning Disability Quarterly, 42(1), 32–45.
https://doi.org/10.1177/0731948718790089
Moyer, P. S., Bolyard, J. J., & Spikell, M. A. (2002). What are virtual manipulatives? Teaching
Children Mathematics, 8(6), 372–377. https://doi.org/10.5951/tcm.8.6.0372
Concrete, Representational, Abstract 19
Moyer-Packenham, P. (2016). International Perspectives on teaching and learning mathematics
with virtual manipulatives. Springer.
Moyer-Packenham, P. S., & Suh, J. M. (2012). Learning Mathematics with Technology: The
Influence of Virtual Manipulatives on Different Achievement Groups. Journal of
Computers in Mathematics and Science Teaching, 31(1), 39–59.
National Governors Association Center for Best Practices & Council of Chief State School
Officers. (2010). Common Core State Standards for Mathematics. Washington, DC:
Authors.
National Library of Virtual manipulatives. National Library of Virtual Manipulatives. (n.d.).
Retrieved February 25, 2023, from http://nlvm.usu.edu/
National Research Council. (2001). Adding it up: Helping Children Learn Mathematics. National
Academy Press.
NCTM, National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring
mathematical success for all.
Newton, N., Record, A. E., & Mello, A. J. (2020). Fluency doesn't just happen with addition and
subtraction: Strategies and models for teaching the basic facts. Routledge.
Ni, L. B., & Hasan, N. A. (2019). The Use of Mnemonic and Mathematical Mnemonic Method
in Improving Historical Understanding. International Journal of Educational and
Pedagogical Sciences, 13(2).
Park, J., Bouck, E. C., & Fisher, M. H. (2020). Using the virtual–representational–abstract with
overlearning instructional sequence to students with disabilities in Mathematics. The
Journal of Special Education, 54(4), 228–238.
https://doi.org/10.1177/0022466920912527
Park, J., Bryant, D. P., & Shin, M. (2021). Effects of interventions using virtual manipulatives
for students with learning disabilities: A synthesis of single-case research. Journal of
Learning Disabilities, 55(4), 325–337. https://doi.org/10.1177/00222194211006336
Ramsingh, V. (2020). Conceptual and representational confusion: An analysis of three
foundation phase teachers’ descriptions of how they teach division. South African
Journal of Childhood Education, 10(1). https://doi.org/10.4102/sajce.v10i1.796
Stapleton, L., & Stefaniak, J. (2018). Cognitive constructivism: Revisiting Jerome Bruner’s
influence on instructional design practices. TechTrends, 63(1), 4–5.
https://doi.org/10.1007/s11528-018-0356-8
Suh, J. M., & Moyer-Packenham, P. S. (2007). The Application of Dual Coding Theory in Multi-
Representational Virtual Mathematics Environments. Proceedings of the 31st Conference
of the International Group for the Psychology of Mathematics Education, 4, 209–216.
Concrete, Representational, Abstract 20
Todenvold, C. (2019). What the heck is a rekenrek? Therecoveringtraditionalist.com. Retrieved
February 22, 2023, from https://www.therecoveringtraditionalist.com/heck-rekenrek/
Uribe-Flórez, L. J., & Wilkins, J. L. (2016). Manipulative use and elementary school students’
mathematics learning. International Journal of Science and Mathematics Education,
15(8), 1541–1557. https://doi.org/10.1007/s10763-016-9757-3
Willingham, D. T. (2017). Ask the Cognitive Scientist: Do Manipulatives Help Children Learn?
American Educator, Fall 2017, 25–31.
Yakubova, G., Hughes, E. M., & Baer, B. L. (2019). Supporting students with ASD in
mathematics learning using video-based concrete-representational-abstract sequencing
instruction. Preventing School Failure: Alternative Education for Children and Youth,
64(1), 12–18. https://doi.org/10.1080/1045988x.2019.1627999
Zuliana, E., Retnowati, E., & Widjajanti, D. B. (2019). How should elementary school students
construct their knowledge in mathematics based on Bruner’s theory? Journal of Physics:
Conference Series, 1318(1), 012019. https://doi.org/10.1088/1742-6596/1318/1/012019
