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Developing Mathematics Knowledge
Bethany Rittle-Johnson
Vanderbilt University
Child Development Perspectives, 2017
doi:10.1111/cdep.12229!
Published Version available at:
http://onlinelibrary.wiley.com/doi/10.1111/cdep.12229/abstract
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Key words: conceptual knowledge, procedural knowledge, flexibility
Abstract
Developing strong knowledge about mathematics is important for success academically,
economically, and in life, but more than one children fail to become proficient in math. Research
on the developmental relations between conceptual and procedural knowledge of math provides
insights into the development of knowledge about math. First, competency in math requires
children to develop conceptual knowledge, procedural knowledge, and procedural flexibility.
Second, conceptual and procedural knowledge often develop in a bidirectional, iterative fashion,
with improvements in one type of knowledge supporting improvements in the other, as well as
procedural flexibility. Third, learning techniques such as comparing, explaining, and exploring
promote more than one type of knowledge about math, indicating that each is an important
learning process. Researchers need to develop and validate measurement tools, devise more
comprehensive theories of math development, and bridge more between research and educational
practice.
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Proficiency in mathematics is critical to success academically, economically, and in life.
Greater success in math is related to entering and completing college, earning more in adulthood,
and making more optimal decisions concerning health (1, 2). Knowledge of math begins to
develop at a young age, and this early knowledge matters: Knowledge of math at or before
school entry predicts outcomes in math and reading across primary and secondary school (3).
More than one children struggle to learn math. For example, only 40% of fourth-grade
and 33% of eighth-grade students in the United States performed at or above proficiency in math
on the 2015 National Assessment of Educational Progress, and proficiency rates were even lower
for African-American and Hispanic children and for children from low-income homes (4). More
than one students do not master challenging math content.
Thus, it is critical to understand how children develop knowledge about math and how
educators can support this process more effectively. For example, when children practice solving
math problems, does this enhance their understanding of the underlying concepts? Under what
circumstances do abstract math concepts help children invent or implement correct procedures?
How do knowledge of math concepts and procedures contribute to flexible problem solving?
These questions tap a central research topic—the developmental relations between conceptual
and procedural knowledge of math—which is the focus of this article.
Developmental Relations Between Types of Knowledge
Conceptual knowledge refers to knowledge of concepts, which are abstract and general
principles such as cardinality and numeric magnitude (5, 6). Conceptual knowledge can be
explicit or implicit, meaning some conceptual knowledge cannot be put into words. Procedural
knowledge is often defined as knowledge of procedures—what steps or actions to take to
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accomplish a goal (5, 6). This knowledge often develops through problem-solving practice, and
thus is tied to particular types of problems. Both types of knowledge promote procedural
flexibility, which is knowing more than one procedures and applying them adaptively to a range
of situations (7). For example, mathematicians know and use more procedures than novices,
appreciate efficient and elegant solutions to problems, and identify the most appropriate
procedure for a given problem based on different factors (e.g., characteristics of problems; 8).
Table 1 provides examples that represent each type of knowledge.
Historically, researchers have debated whether conceptual knowledge develops first or
procedural knowledge develops first (see 7, 9 for reviews). According to a concepts-first view,
children initially acquire conceptual knowledge by learning from adults or by innate constraints.
Then, they derive and build procedural knowledge from their conceptual knowledge through
repeated practice solving related problems. According to a procedures-first view, children
initially learn procedures by imitating adults, and then gradually derive conceptual knowledge
from implementing the procedures, abstracting the structure and principles of the problems.
More recently, I proposed an iterative view in which the causal relations are bidirectional, with
increases in conceptual knowledge leading to subsequent increases in procedural knowledge and
vice versa (6). For example, in one study (6), prior conceptual knowledge of decimals predicted
gains in procedural knowledge after a brief problem-solving intervention, which in turn predicted
subsequent gains in conceptual knowledge.
The iterative view is now the most well-accepted perspective among researchers (10, 11).
First, this view accommodates gradual improvements in each type of knowledge over time. Each
type of knowledge is multifaceted, and if knowledge is measured using continuous rather than
categorical measures, one type of knowledge is not well developed before the other emerges,
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arguing against a strict view that puts concepts or procedures first. Second, an iterative view
accommodates evidence that supports concepts-first and procedures-first views, as initial
knowledge can be conceptual or procedural, depending on environmental input and relevant
prior knowledge. For example, even if children are born with a basic ability to track and
discriminate between numerical magnitudes (12), conceptual knowledge of numerical magnitude
develops in concert with experience counting and learning the counting procedure. Third, an
iterative view recognizes the role each type of knowledge can play in developing the other.
Conceptual knowledge can help with constructing, selecting, and appropriately executing
problem-solving procedures, and practice implementing procedures may help students develop
and deepen their understanding of concepts, especially if the practice is designed to make
underlying concepts more apparent (5).
Evidence also supports an iterative view. Numerous longitudinal studies indicate
predictive, bidirectional relations between conceptual and procedural knowledge. For example,
in one study, elementary school children’s knowledge of fractions was assessed in the winter of
fourth grade and the spring of fifth grade (13). Procedural knowledge in fourth grade predicted
conceptual knowledge in fifth grade after controlling for prior conceptual knowledge and other
factors; similarly, conceptual knowledge in fourth grade predicted procedural knowledge in fifth
grade.
Similar bidirectional relations across grade levels have been found in elementary school
children’s knowledge of whole number concepts and procedures (14). Over shorter time frames,
bidirectional relations have been found in preschoolers learning about counting (e.g., 15),
elementary school children learning addition and subtraction (e.g., 5) and about decimals (6, 16),
and middle school students learning about solving equations (17; see Figure 1).
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Causal evidence for bidirectional relations comes from studies that experimentally
manipulate at least one type of knowledge and then measure both types of knowledge. For
example, in one study (18), elementary school children were given a brief lesson on a procedure
for solving problems of mathematical equivalence (e.g., 6 + 3 + 4 = 6 + __) or the concept of
mathematical equivalence, or were given no lesson. Children who received either lesson gained
greater conceptual knowledge and greater procedural knowledge than children who received no
lesson, indicating that a lesson on a procedure led to improvements in conceptual knowledge and
a lesson on a concept led to improvements in procedural knowledge.
Furthermore, studies on carefully constructed practice problems (5) suggest that
improving procedural knowledge can support improvements in conceptual knowledge. Practicing
nontraditional arithmetic problems such as __ = 3 + 5 improved second- and third-grade students’
procedural knowledge as well as their conceptual knowledge of the equal sign relative to
traditional practice formats such as 3 + 5 = ___ or no practice (e.g., 19). Overall, both
longitudinal and experimental studies indicate that procedural knowledge improves conceptual
knowledge, and vice versa, suggesting that the relations between the two types of knowledge are
bidirectional.
An iterative view further predicts that the bidirectional relations between conceptual and
procedural knowledge persist, with increases in one supporting increases in the other in an
iterative feedback loop (6). In addition, iterating between lessons on concepts and procedures on
decimals supported greater procedural knowledge and equivalent conceptual knowledge than
presenting concept lessons before lessons on procedure (16). These studies suggest that relations
between the two types of knowledge are bidirectional and iterative over time.
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This does not mean that relations between the two types of knowledge are always
symmetrical. In a recent study, the relations were symmetrical—the strength of the relationship
from prior conceptual knowledge to later procedural knowledge was the same as it was from
prior procedural knowledge to later conceptual knowledge (17). However, in other studies,
conceptual knowledge or conceptual instruction influenced procedural knowledge more strongly
than vice versa (13, 18). Furthermore, brief procedural instruction or practice-solving problems
does not always support growth in conceptual knowledge (5, 20). How much gains in procedural
knowledge support gains in conceptual knowledge is influenced by the nature of the procedural
instruction or practice (e.g., 19). Crafting procedural lessons to encourage children to notice
underlying concepts can promote a stronger link from improved procedural knowledge to gains
in conceptual knowledge (5).
Relations to Procedural Flexibility
Although it has received much less attention than conceptual and procedural knowledge,
evidence on the development of procedural flexibility has emerged recently. The development of
procedural flexibility is related to children’s conceptual and procedural knowledge (21). For
example, greater procedural flexibility for multidigit arithmetic is related to greater conceptual
and procedural knowledge of arithmetic (22) [AU: than?]. Furthermore, middle school students’
prior conceptual and procedural knowledge for solving equations each uniquely predicted their
procedural flexibility at the end of a classroom unit on solving equations (see Figure 1; 17).
Summary
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Proficiency in math requires developing conceptual knowledge, procedural knowledge,
and procedural flexibility. Evidence from a variety of math domains indicates that the
development of conceptual and procedural knowledge is often bidirectional and iterative, with
one type of knowledge supporting gains in the other. Greater conceptual and procedural
knowledge is also related to greater procedural flexibility, and evidence suggests that conceptual
and procedural knowledge support the development of procedural flexibility.
Learning Techniques for Improving Mathematics Knowledge
Given the importance of developing conceptual knowledge, procedural knowledge, and
procedural flexibility, we need to understand how learning techniques improve these types of
knowledge. Three powerful activities—comparing, self-explaining, and exploring before
instruction—can promote both conceptual and procedural knowledge, and one (comparing) also
improves procedural flexibility. This experimental research also helps validate instructional
methods for promoting knowledge of math.
Comparing
Comparing is a ubiquitous cognitive process, and comparing alternative ways to solve
problems can promote learning in math. In five studies, students looked at pairs of examples
illustrating two correct procedures for solving the same problem and were prompted to compare
them, or they studied the examples individually and were prompted to reflect on them (23). For
students who knew one of the solution procedures at pretest, comparing procedures supported
greater conceptual knowledge, procedural knowledge, and procedural flexibility. For novices,
who did not know one of the solution procedures at pretest, comparing improved procedural
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flexibility, but not conceptual or procedural knowledge. Comparing can improve all three types
of knowledge in part because comparing examples side by side promotes perceptual learning of
the structure of problems within the domain (24).
In addition, comparing incorrect procedures to correct ones can also aid conceptual and
procedural knowledge (25). For example, fourth- and fifth-grade students gained greater
conceptual and procedural knowledge when they compared examples of correct and incorrect
solution procedures rather than comparing only correct procedures (26). Another promising form
of comparison is when students compare easily confusable problem types, which helps learners
distinguish the two problem types and improves procedural knowledge (27).
Self-Explaining
Generating explanations to make sense of new information (i.e., self-explanation) is
another common and powerful learning process (28, 29). Furthermore, prompting students to
explain new information, such as examples of solutions to math problems, helps promote
learning in math. For example, prompting primary school children to explain why solutions to
problems of math equivalence were correct or incorrect supported greater conceptual and
procedural knowledge than having them solve problems without self-explanation prompts (30).
Self-explanation aids conceptual knowledge by integrating knowledge, as explanations often link
new information or link new information with prior knowledge (31). In addition, self-explaining
facilitates conceptual and procedural knowledge by guiding attention to structural features
instead of to surface features of the content to be learned, helping students notice key structural
features of exemplars and use procedures less frequently tied to particular surface features of the
exemplars (20, 30).
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Our recent meta-analysis of 26 experimental studies on prompted self-explanation with a
wide range of ages (age 4 to 22) learning math confirmed that self-explanation prompts promote
greater procedural knowledge, especially procedural transfer, as well as greater conceptual
knowledge when knowledge was assessed immediately after the intervention (32). The effect
was stronger if support for high-quality explanation was provided, such as partial explanations to
complete. Without support, children and adults sometimes have difficulty generating useful
explanations when prompted. Training on self-explanation and structured self-explanation
responses, such as selecting an explanation from a list, supported learners effectively. Overall,
prompting children to generate explanations when learning math promotes conceptual and
procedural knowledge, especially when explanations are supported.
Exploring Before Instruction
Children are intrinsically driven to explore, and exploration can help children discover
and pay attention to important information (28). At the same time, children often fail to discover
important information on their own and benefit from direct instruction (33). A productive
combination is to offer opportunities for children to explore problems before instruction (34). For
example, primary school children solved unfamiliar math problems and received a lesson on
equivalence, and the order of problem solving and the lesson was manipulated (35, 36).
Compared to children who solved the problems after the lesson, children who solved the
unfamiliar problems before the lesson gained more conceptual knowledge or procedural
knowledge. Similarly, middle school students who explored problems and invented their own
formulas for calculating density before receiving instruction on density gained deeper conceptual
and procedural knowledge of the topic than students who had the lessons first (34). Exploring
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problems followed by instruction fits with the recommendation from researchers of math
education that students have opportunities to struggle—to figure out something that is not
immediately apparent—before direct instruction (37).
Summary
Comparing, self-explaining, and exploring before instruction are learning techniques that
can improve conceptual and procedural knowledge of math. Comparing solution procedures also
improves procedural flexibility. Research confirms the causal role of each type of knowledge in
math development and validates techniques educators can use to promote such development.
Certainly, more than one other learning techniques promote math development. These
include studying worked-out examples of solution procedures (38), and discussing math ideas
with peers (39). These activities promote active thinking about math concepts and procedures,
not simply memorizing terms and solution procedures as dictated by adults. More than one
children in U.S. math classrooms spend much of their time implementing procedures
demonstrated by their teachers rather than reflecting actively on concepts and procedures (40).
Concluding Remarks
Overall, research on developmental psychology has helped illuminate how children learn
math. Competency in math requires that children develop conceptual knowledge, procedural
knowledge, and procedural flexibility. These three types of knowledge often develop
bidirectionally and iteratively, with improvements in one type of knowledge supporting
improvements in the other types. Furthermore, comparing, self-explaining, and exploring before
!
instruction promote conceptual and procedural knowledge of math, and comparing also promotes
procedural flexibility.
Despite a growing number of studies on the psychology of math development, current
research has its limits. First, researchers have not developed standardized approaches to assess
the different types of knowledge with proven validity, reliability, and objectivity (10). Rather,
they typically develop their own study-specific measures, often without evidence of convergent
or divergent validity. Some topics, such as conceptual knowledge of cardinality and numeric
magnitude, are receiving increased attention, but we lack consensus on the most effective way to
measure each construct (41). As a field, we need to invest more resources in measurement
development and validation. In Table 1, I have provided examples of types of items that lend
themselves to standardized administration and scoring. Evidence for bidirectional relations may
be driven, in part, by impure measures that each tap a mixture of types of knowledge rather than
by true bidirectional relations in the underlying constructs. Only one study has provided evidence
for bidirectional relations after establishing the divergent validity of the measures (17).
Second, we need a more comprehensive, integrative theory of how the different types of
knowledge develop and interact. Such a theory should consider how age and individual
differences affect relations between the three types of knowledge and the effectiveness of
different learning techniques. It should also identify when developmental relations and learning
process differ for different math topics, as well as the impact of affective factors such as math
anxiety (42).
Finally, we need to invest more effort in bridging research and practice (43). Instead of
trying to apply our research to practice, we need to do research that is inherently relevant to and
driven by the needs of practice. We should incorporate research topics and methods that consider
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current problems of practice (e.g., what math educators identify as their most pressing concerns).
We also need to conduct research within educational settings to ensure the method is feasible
outside the lab and the findings generalize to those settings. For example, we have capitalized on
the common educational practice of partner work. We randomly assigned pairs of students to
different conditions within classrooms, having students work on our materials with a partner
during their math class on content relevant for that course (23). Such research is often most
successful when conducted by interdisciplinary research teams that include psychologists, math
education researchers, mathematicians, and math teachers. Collaboration like this can often lead
to publishing findings in journals for practitioners (e.g., Teaching Children Mathematics), which
require a different approach to writing than journals for researchers. Interdisciplinary work also
facilitates translating research-based findings into curriculum and professional-development
materials for teachers (44). Translating psychological principles and findings into useable
practices is not straightforward. For example, psychological research often focuses on isolating
particular processes and components of knowledge, and rarely speaks to how to combine and
integrate different processes and components to address broad learning goals, but this is
necessary in practice (45). Overall, bridging research and practice benefits both, and will help
advance our understanding of how children learn math and how we can promote this learning
more effectively.
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Author’s Note
Bethany Rittle-Johnson, Department of Psychology and Human Development, Peabody
College, Vanderbilt University.
Correspondence concerning this article should be addressed to Bethany Rittle-Johnson,
Department of Psychology and Human Development, 230 Appleton Place, Peabody #552,
Vanderbilt University, Nashville, TN 37203; e-mail: bethany.rittle-johnson@vanderbilt.edu.
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Table 1
Sample Tasks and Items Used to Assess Each Type of Mathematics Knowledge
*Note: Designating problems as being in a familiar format requires some knowledge of students’
instructional history, but can often be inferred because either the researcher intentionally exposed
students to the problem type before the assessment or it is reasonable to assume from content
standards for mathematics in the particular country or state.
Sample!task!type!
Sample!Item!
Conceptual+Knowledge++
a.!Evaluate!examples!of!concept!
a.!Decide!whether!the!number!sentence!3!=!3!
makes!sense!
b.!Translate!quantities!between!
representational!systems!
b.!Place!symbolic!numbers!on!number!lines!
c.!Compare!quantities!
c.!Indicate!which!symbolic!integer!or!fraction!
is!larger!
d.!Generate!or!select!definitions!of!
concepts!
d.!Define!the!equal!sign!
Procedural+Knowledge+
a.!Solve!problems!in!a!familiar!
format*!
a.!8/10!+!6/10!=!__!
b.!Solve!problems!with!a!new!
surface!or!problem!feature!
b.!2!½!+!¼!=!__!
Procedural+Flexibility+
a.!!Generate!multiple!methods!
a.!Solve!this!equation!in!two!different!ways:!
4(x!+!2)!=!12!!
b.!Evaluate!nonconventional!
methods!
b.!Do!you!think!this!way!of!starting!this!
problem!is!(a)!a!very!good!way;!(b)!OK!to!do,!
but!not!a!very!good!way;!(c)!not!OK!to!do?!!
!
Figure 1. Regression paths of the best-fitting structural equation model of the relations
among conceptual knowledge, procedural knowledge, and procedural flexibility in Study 1 (17).
Conceptual+
Knowledge)
Time%1
Procedural*
Knowledge)
Time%1
Conceptual+
Knowledge)
Time%2
Procedural*
Knowledge)
Time%2
Procedural*
Flexibility)
Time%2
.32
!
.46
.34
.30
.28
.26
.42
.80
.87
.89
