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2008-11-25
The State of Balance Between Procedural Knowledge and The State of Balance Between Procedural Knowledge and
Conceptual Understanding in Mathematics Teacher Education Conceptual Understanding in Mathematics Teacher Education
Damon L. Bahr
Michael J. Bossé
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Original Publication Citation Original Publication Citation
Bossé, M.J., & Bahr, D.L. (28). The state of balance between procedural knowledge and
conceptual understanding in mathematics teacher education. International Journal of
Mathematics Teaching and Learning, 25 Nov 28 found at http://www.cimt.plymouth.ac.uk/
journal/default.htm.
BYU ScholarsArchive Citation BYU ScholarsArchive Citation
Bahr, Damon L. and Bossé, Michael J., "The State of Balance Between Procedural Knowledge and
Conceptual Understanding in Mathematics Teacher Education" (2008).
Faculty Publications
. 924.
https://scholarsarchive.byu.edu/facpub/924
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Running head: Procedural and Conceptual Balance
The State of Balance Between Procedural Knowledge and Conceptual Understanding
in Mathematics Teacher Education
October, 2008
Michael J. Bossé
Department of Mathematics and Science Education
East Carolina University
Greenville, NC 27858
Phone (252) 328-9367
FAX (252) 328-9371
Damon L. Bahr
Department of Teacher Education
McKay School of Education
Brigham Young University
201-E MCKB
Provo, UT 84604
Phone (801) 422-6114
Key Words:
Balance
Conceptual Understanding
Procedural Knowledge
Teacher Education
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The State of Balance Between Procedural Knowledge and Conceptual Understanding
in Mathematics Teacher Education
ABSTRACT
The NCTM Principles and Standards for School Mathematics calls for a balance between
conceptual understanding and procedural knowledge. This study reports the results of a survey
distributed to AMTE members in order to discover the opinions and practices of mathematics
teacher educators regarding this balance. The authors conclude that there is wide disparity of
views regarding the meaning of the terms "conceptual" and "procedural" as well as the meaning
"balance" between the two, in terms of what constitutes mathematics, the learning and teaching
of mathematics, and the assessment of mathematical proficiency.
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Running head: Procedural and Conceptual Balance
The State of Balance Between Procedural Knowledge and Conceptual Understanding in
Mathematics Teacher Education
The word "balance" has become popular in today's educational climate as educators have
become weary of pendulum swings between extreme pedagogical perspectives. Historically,
traditional mathematics instruction has been characterized by an extreme commitment to the rote
memorization of procedures with little concern for the associated concepts that underlie them.
NCTM's Principles and Standards for School Mathematics [PSSM] (2000) states that balance
ought to exist between conceptual and procedural learning in mathematics classrooms.
"Developing fluency requires a balance and connection between conceptual understanding and
computational proficiency." (p. 35)
On the one hand, computational methods that are over-practiced without
understanding are often forgotten or remembered incorrectly ... On the other hand,
understanding without fluency can inhibit the problem-solving process ... The
point is that students must become fluent in arithmetic computation-they must
have efficient and accurate methods that are supported by an understanding of
numbers and operations. (p. 35)
Balancing the acquisition of conceptual understanding and procedural proficiency is far
from being strictly an American concern. Standards documents and curriculum frameworks
from around the world, such as the Australian (Leonelli & Schmitt, 2001) and British
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frameworks (ATM, 2006) are quite pronounced in their calls for such a balance.
In this introduction, we will demonstrate support for examining the effects of the
conceptual/procedural balance upon four concerns: the type of mathematics that should be
learned in school, preservice teacher preparation, instructional conceptualization and design, and
assessment.
The PSSM consistently connects "learning mathematics with understanding" (p. 20) with
calls for balance between conceptual and procedural learning and the "... ability to use
knowledge flexibly, applying what is learned in one setting appropriately in another." (p. 20)
Propounding that mathematics proficiency is dependent upon learning both concepts and
procedures, the PSSM states,
One of the most robust findings of research is that conceptual understanding is an
important component of proficiency, along with factual knowledge and
procedural facility. The alliance of factual knowledge, procedural proficiency,
and conceptual understanding makes all three components usable in powerful
ways. Students who memorize facts or procedures without understanding often
are not sure when or how to use what they know, and such learning is often quite
fragile. (p. 20)
Other mathematics educators call for a similar balance in the learning of concepts and
procedures. Ma (1999) describes the development of a "profound understanding of fundamental
mathematics" as a well-organized mental package of highly-connected concepts and procedures.
Regarding further discussions of an algorithm, Ma states, "'Know how, and also know why.' ...
Arithmetic contains various algorithms ... [and] one should also know why the sequence of steps
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in the computation makes sense." (p. 108)
The PSSM contends that a conceptual-procedural balance is fundamental to all of school
mathematics, and envisions that throughout their K-12 mathematical experiences "students will
reach certain levels of conceptual understanding and procedural fluency by certain points in the
curriculum." (p. 7; see also p. 30) Such balance is understood as fundamental, leading to the
understanding that "Learning ... the mathematics outlined in [all grade bands] requires
understanding and being able to apply procedures, concepts, and processes." (p. 20)
The conceptual-procedural balance provides a context for investigating the preparation of
preservice teachers. Ma (1999) views teacher preparation as the vehicle to break the "vicious
cycle formed by low-quality mathematics education and low-quality teacher knowledge of
school mathematics." (p. 149), yet both she and Fosnot and Dolk (2001) decry the ever-widening
gap between the needed and actual content understanding of teachers. Simon (1993) investigated
gaps in prospective teachers' knowledge of division by examining the connectedness within and
between procedural and conceptual knowledge and suggests conceptual areas of emphasis for the
mathematical preparation of elementary teachers. Ambrose, Clement, Philipp, and Chauvot
(2003) describe seven beliefs they hope to engender in their preservice students. The first four
are based on the conceptual-procedural balance: (1) Mathematics, including school mathematics,
is a web of interrelated concepts and procedures. (2) One's knowledge of how to apply
mathematical procedures does not necessarily go with understanding of the underlying concepts.
(3) Understanding mathematical concepts is more powerful and more generative than
remembering mathematical procedures. (4) If students learn mathematical concepts before they
learn procedures, they are more likely to understand the procedures when they learn them. If
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they learn the procedures first, they are less likely ever to learn the concepts.
Revised views in what it means to learn mathematics have caused a careful examination
of the effects of traditional pedagogical paradigms. Marilyn Burns (1999) indicates that this
examination has yet to achieve widespread reform. Frequently, a direct instruction model is
linked to traditional teaching methodologies (Confrey, 1990) although some have reexamined the
role of telling in light of both procedural and conceptual learning (Lobato, Clarke & Ellis, 2005).
Many scholars (e.g., Carpenter, Ansell & Levi, 2001; Cobb, Wood & Yackel, 1990;
Davis, 1990; Wood, 1999; Wood, 2001) argue from a constructivist perspective for an
instructional sequence that begins with the presentation of a meaningful task or problem and
continues with an invitation to solve that problem in multiple ways, which leads to the sharing,
justifying, and discussing of those problem solving strategies in small or large group discourses.
The roles assumed by teacher and student, as well as the environment, associated with this type
of teaching are radically different from tradition and are thoroughly described in the Professional
Standards for Teaching Mathematics (1991). The PSSM suggests that this constructivist-oriented
sequence can be used to promote both conceptual and procedural learning: "Moreover, in such
settings, procedural fluency and conceptual understanding can be developed through problem
solving, reasoning, and argumentation." (p. 21)
With calls for pedagogical change have come calls for change in assessment practice.
Traditional assessment processes tend to reward speed and accuracy in remembering pre-existing
facts (Bell, 1995) and communicate a message about the meaning of mathematics that fails to
represent its complexity (Galbraith, 1993). The PSSM states that assessment tasks communicate
what type of mathematical knowledge and performance are valued (p. 22) and that assessment
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should be aligned with instructional goals (p. 23). Its predecessor, the Curriculum and
Evaluation Standards (NCTM, 1989) promotes a similar view: "As the curriculum changes, so
must the tests. Tests also must change because they are one way of communicating what is
important for students to know." (pp. 189-190) All of these statements suggest that if there is to
be a balance between conceptual and procedural learning, there should also be a balance in the
types of assessment used to capture that learning.
Suggesting that the conceptual-procedural dichotomy is incomplete in defining the
meaning of mathematical proficiency, Adding It Up (Kilpatrick, Stafford & Findell, 2001)
describes five interrelated strands of mathematical proficiency of which procedural fluency and
conceptual understanding are but two. The other three are strategic competence, adaptive
reasoning, and productive disposition. Knowing Mathematics for Teaching Algebra Project and
Teachers for a New Era projects at Michigan State University, and the University of Chicago
School Mathematics Project additionally indicate that the dichotomy is incomplete. However, it
is possible that this characterization of the incompleteness of the dichotomy may be more a
reflection of curricular perspective than a reflection of the nature of mathematics. For example, a
careful reading of Adding It Up reveals that the first two strands, procedural fluency and
conceptual understanding, repeat the conceptual-procedural balance suggested by the PSSM
Content Standards. The next two strands, strategic competence and adaptive reasoning, are a
reiteration of the PSSM Process Standards, namely, problem solving, reasoning and proof,
communication, connections, and representation. The last strand, productive disposition,
reiterates the PSSM consistent reference to the affective side of mathematical performance. Thus,
while these and other sources may opine that the dichotomy investigated in this project and paper
8
is incomplete, they do not refute the important central question at hand. Indeed, these sources
may indicate that the dichotomy between procedural learning and conceptual understanding is
more complex than often recognized and warrants both additional research and refinement of the
fundamental understanding of the issue.
Altogether, therefore, the literature suggests that the investigation of the opinions and
practice of mathematics teacher educators regarding the concept-procedure balance warrants
further investigation.
2. Background of the Survey and Respondents
The membership of the Association of Mathematics Teacher Educators (AMTE) primarily
consists of preservice and/or inservice mathematics teacher educators employed by institutions
of higher education. They typically teach early childhood through secondary mathematics
content and/or methods, may work with educators in schools, and may hold various local, state
and national leadership positions in the field. Many are engaged in research about the teaching
and learning of mathematics and/or mathematics teaching. Thus, the membership of AMTE is,
herein, altogether considered expert and its opinions are highly valued within the field.
In preparation for a presentation at the Ninth Annual AMTE Conference in Dallas in
January, 2005, the researchers in this study identified some issues related to the balance of
conceptual and procedural learning and developed an initial survey to be used at the conference.
Subsequent conference discussions and initial survey responses allowed for the refinement of
questions associated with those issues and lead to the production of the current survey
instrument. This refined survey, which attempts to discern opinions and practice among AMTE
9
members, was developed and distributed via the membership email list. Voluntary responses
were returned (N=40) to the researchers for analysis and synthesis into a report. The survey
consisted of multiple choice and short answer questions and appears in the appendix. Qualitative
and quantitative methods were used to develop the report.
The expertise provided by AMTE member respondents makes generalization of the
findings of this study to mathematics education in toto tenuous. AMTE member opinions are not
considered the general state of mathematics education in the U.S.; rather, their opinions are
generally considered among the most well versed opinions in the field. Others in the field may
not share some of the opinions of AMTE members. In addition, it is recognized that a small
sample can hardly be considered representative of the entire AMTE organization, particularly
because the sample was formed strictly from those whole voluntarily completed the survey.
However, the significant variance in the professional experience of the respondents (gender; age;
current employment (university and/or City & State); number of years at this position; school
and year at which highest degree was earned) provides an increased level of confidence that
some of the responses can be cautiously generalized to a larger population.
3. Researchers' Presuppositions
The researchers in this study believe that the debate between conceptual understanding and
procedural knowledge in mathematics learning can be compared with learning to play a musical
instrument. When a child begins to learn to play the trumpet, she may begin with musical theory
but must develop the instrumental skills to continue to learn the instrument. Both the
development of musical theory and instrumental skill go hand in hand. Conversely, both are
10
limited by the other. Only in the rarest instance can a musician of limited instrumental skill
become an excellent theorist or composer. Redirecting this concern to mathematics education,
the authors believe that this is the type of conceptual-procedural balance recommended in the
NCTM PSSM. The authors believe that mathematical skills are learned through procedural
knowledge and novel, connected, extended, and applied mathematical ideas are developed
through conceptual understanding.
The researchers in this study define the following: a problem is a scenario in which, upon
initiation, neither the result nor a method for solution is known; an exercise is a scenario in
which the result is unknown but a method for solution is known. Notably, what may be an
exercise for one student may be a problem for another. Furthermore, when students do not know
a method to solve a scenario, even though they should, it is a problem until they learn a method
for solution; then it becomes an exercise. Within any group of supposed exercises (for instance at
the end of sections and chapters in texts), as concepts are further investigated, selected examples
may indeed be problems to some students. This is often the case when examples look alike and
yet methods for solutions change due to alterations among the examples. Therefore, factoring
90x
2
− 27x − 40 may be an exercise to a precalculus student and a problem to an algebra I or II
student who lacks the necessary foundation. The authors believe that mathematical exercises are
solved using procedural knowledge and problems are solved using conceptual understanding.
The conceptual understanding required for the latter may lead to, or incorporate, associated
procedural knowledge.
4. Initial Relevant Findings
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Approximately 63% of respondents believe that students naturally learn mathematics in a
sequence that begins with conceptual understanding and leads to procedural knowledge and
another 37% believed that students learn from either conceptual understanding or procedural
knowledge to the other. No respondents state that procedural knowledge leads to conceptual
understanding.
Respondents categorize the focus of the instruction which they provide in undergraduate
content courses for preservice K-12 teachers as well blended (50%), mostly conceptual (46%),
and all conceptual (4%). Virtually all responses agree with the need to emphasize, or even over
emphasize, conceptual understanding over procedural knowledge in order to overcome the
perennial focus on procedurally-based instruction in K-12 mathematics education.
Almost all respondents state that they use the NCTM PSSM process standards for
developing student conceptual understanding, which suggests a high regard for the constructivist
instructional sequences discussed previously. Although most respondents claim to use Bloom's
Taxonomy to develop conceptual understanding, a significant number of respondents state that
since they work in mathematics departments at their universities, Bloom's Taxonomy is not
deeply considered. Three respondents admit that they are completely unfamiliar with the
taxonomy.
Respondents categorize the skills/knowledge which their mathematics education students
demonstrate as they enter their course/program as mostly procedural (90%) and all procedural
(10%) and as well blended (66%), mostly procedural (24%), and mostly conceptual (10%) as
they exit. Most respondents indicate that they saw a greater need to teach conceptual
understanding to preservice teachers than to mathematics students not planning to enter the
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teaching profession.
Respondents categorize the focus of the assessments which they use in their
undergraduate content courses for preservice K-12 teachers as well blended (59%), mostly
conceptual (37%) and mostly procedural (4%). They categorize the focus of the assessments
which they use in their undergraduate pedagogy courses for preservice K-12 teachers as mostly
conceptual (52%), well blended (40%), and mostly procedural (8%). Nearly half of the
respondents do not discuss how they taught their undergraduate preservice K-12 teachers to
assess their future K-12 students for conceptual understanding.
Approximately 25% of respondents state that they realized that they needed to return to
the issue to reconsider their teaching practices. Of these, many note that their self-recognized
difficulty in defining and distinguishing between procedural knowledge and conceptual
understanding led them to realize their own need to more thoroughly investigate and understand
the issues and reinvestigate their teaching and assessment strategies in light of these findings.
5. Digging Deeper
The survey for this study melded multiple choice questions with open ended, short answer
questions. Therefore, much more rewarding analysis was possible than could have been
accomplished through multiple choice questions alone. The following findings and discussions
are supported by these responses. Respondents provided the following descriptors and
characteristics for conceptual understanding and procedural knowledge. The numbers in the
parentheses indicate the frequency with which the descriptors were found in the responses. It was
quite common for a response to include multiple facets.
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Respondents opine that conceptual understanding: is adaptable, adjustable, transferable
and applicable to other situations (13); is knowing "why"(9) or "how"(4) something works; is
born from and develops connections (9); makes math (and the world) more sensible or
meaningful (6); is a flexible foundation for long-term retention (5) and understanding; is the
essence of mathematical thinking and the only true kind of understanding (3); leads students to
see the bigger picture (2); does not rely on memory (2); assists the reconstruction of the idea if
details are forgotten (2); is at the heart of "knowing" a topics (2); links facts and procedures (1);
leads to new learning (1); and is continually growing and developing (1).
Respondents state that procedural knowledge: produces algorithmic efficiency and
accuracy (17); is a set of skills (6); is primarily "how to" do (perform) some operations and
employ some properties (4); is learned by rote, repetition, and drill (4); makes "meaning"
unnecessary (2); is only used, remembered, and applicable in the short-term (3); only applies to
context in which it was first learned (3); saves from rediscovering the wheel each time (2); is
worthless by itself (2); is connected to a lack of, or incorrect, memory (1); does not emphasize
making connections (1); is a step-by-step sequence (1) automating the routine (1); preserves
"harder thinking" for less routine things (1); is "just doing what you are told" (1); allows quick
calculations of things that one no longer needs to worry about how it works (1); does not require
an understanding of what one is doing (1) and requires no need to think (1); depends on
memorization (1) and is easily forgotten (1); is merely superficial learning (1); is useful to
communicate concepts (1); is a discrete set of factoids (1); is a stagnant set of rules (1); and is
obtained with less personal involvement with the mathematics (1).
Connecting the two concerns, respondents claim that conceptual understanding and
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procedural knowledge: work together (5) and are both necessary (5); complement each other (2);
form a continuum (1); are both important (1); develop interactively (1) and should not be
separated (1); and can both result in identical answers to a question (1).
These responses in toto paint a more comprehensive picture of respondents' opinions
regarding conceptual understanding and procedural knowledge. This paper reports aspects which
are more deeply embedded in the responses.
All characteristics of conceptual understanding provided by responses can be
unanimously seen as positive. Procedural knowledge, however, while seen as valuable in
producing algorithmic efficiency and accuracy, is much maligned in many ways. The negative
tone and context of the majority of the responses overwhelmingly suppresses the minority
position, which values procedural knowledge as complementary to conceptual understanding.
Only four responses are considered positive characterizations of procedural knowledge: produces
algorithmic efficiency and accuracy; saves from rediscovering the wheel each time; automating
the routine; is useful to communicate concepts. While another two characterizations are
considered neutral, twenty-one other responses are interpreted as negative perceptions of
procedural knowledge.
Issues of memory, memorization, and retention deeply connect many comments regarding
conceptual understanding and procedural knowledge. Numerous responses denigratingly
associate procedural knowledge with a mere exercise of memory and rote practice. Memory is
recognized as being in opposition to understanding. Conversely, many responses indicate that
conceptual understanding, via connections and sense-making, leads to greater retention of salient
concepts; in such, memory is no longer maligned, but rather applauded. The majority of
15
responses imply that, in respect to learning, understanding should lead to memory, but memory
can not lead to understanding. Some responses go further and imply that ideas which are
constructed and integrated into a student's schema are independent of memory - that is, if
something is learned, then it is not memorized. Conversely, others argue that if something is
memorized then it is not learned. Therefore, together, a false dichotomy is established and
promoted from both sides of the argument.
While many respondents associate conceptual understanding with deeper thinking, many
responses indicate that procedural knowledge is completely bereft of any real thinking
whatsoever, makes "meaning" unnecessary, and does not require an understanding of what one is
doing. One respondent insinuates that using procedural knowledge allows for the conservation of
thought energy which can later be directed to situations which require "harder thinking".
Although these more irregular opinions are less commonplace, their existence further
demonstrates the degree to which procedural knowledge is repudiated among some of the
respondents. This is particularly deleterious in that it blurs the meaning of thinking. Thinking,
understanding, and learning are consistently associated with a student's introduction to novel
ideas. Respondents never use the phrase "learn a skill."
Respondents recognize the purpose of procedural knowledge in expediting calculations,
as a tool which allows students to avoid continually rediscovering the wheel. While devaluing
procedural knowledge, respondents highly value the expediency it brings to routine operations.
Respondents declare that procedural knowledge cannot be transferable and applicable to other
situations and that procedural knowledge is inefficient as a problem-solving tool.
Responses indicate that a contextualized purpose for procedural knowledge is
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inconsistently understood. The purposes for conceptual understanding and procedural knowledge
are only occasionally seen complementarily and only once mentioned contextually. The context
of the use of procedural knowledge is continually clouded by the notion of problem-solving.
Except in one unique case, respondents indicate no distinction between problems and exercises.
Thus, most respondents do not express value for procedural knowledge to facilitate the solution
of exercises. Nor do they indicate any value for the mathematical practice afforded by exercises
and the role of practice and exercises for the reinforcement of memory and understanding after
concepts are developed.
6. Discussing Balance
A small number of respondents fail to provide cogent definitions which articulate both
similarities and difference between procedural knowledge and conceptual understanding. This
may have been due to the open-ended nature of the survey question and the respondents
unwillingness to write lengthy responses. Many of the descriptors among different respondents
for procedural knowledge and conceptual understanding are inconsistent and contradictory. The
responses often repeat the notion that procedural knowledge is tantamount to no understanding at
all. While a small number of responses delineate distinctions between the two, most respondents
devalue procedural knowledge and highly value conceptual understanding.
The imbalance of the relative values attached to conceptual understanding and procedural
knowledge within the responses may be best recognized when it is understood that 100% of
responses demonstrate a preference that, as students both enter and exit their course/program,
students understand mathematical concepts well but lack procedural skills. While this one-sided
17
opinion may be due to the fact that the only other seemingly unsatisfactory option to the question
was that respondents preferred students to have strong procedural skills, this result demonstrates
the existing imbalance.
As previously noted, the NCTM PSSM calls repeatedly for balance between procedural
knowledge and conceptual understanding. Most respondents claim to either present a balance of
conceptual and procedural instruction and assessment or provide emphasis on mostly conceptual
instruction and assessment. Through additional comments on the surveys, those who denote that
they use instruction and assessment which are "mostly conceptual" do so primarily for two
significant reasons. First, virtually all recognize the need to emphasize, or even over emphasize,
conceptual understanding over procedural knowledge in order to overcome the perennial focus
on procedurally based instruction in K-12 mathematics education. Second, a number of
respondents indicate that overemphasis of conceptual understanding within the mathematics
education courses within their program is necessary to counterbalance the overemphasis of
procedurally based instruction and assessment in required mathematics courses held in the
respective mathematics departments. Thus, respondents often perceive that an overemphasis in
respect to conceptual learning is necessary in order to provide both students and the program as a
whole a balance between procedural knowledge and conceptual understanding. Unfortunately,
even with a purposive imbalance in favor of conceptual instruction, assessment, and
understanding, 24% of respondents admit that their students continue to exit these programs with
skills and knowledge which are mostly procedural.
Interesting inconsistencies can be found among groups of responses and within individual
responses regarding the notion of balance between conceptual understanding and procedural
18
knowledge. As previously noted, the majority of respondents categorized the focus of the
instruction and assessment which they provide in undergraduate content courses for preservice
K-12 teachers as well blended and 40% categorize the focus of the assessments which they use in
their undergraduate pedagogy courses for preservice K-12 teachers as well blended. However,
this purported claim toward balance may be philosophically inconsistent with the negative
perception which the majority of responses provided regarding procedural knowledge. This
naturally leads to questions regarding how so many respondents simultaneously denigrate
procedural knowledge and promote balance. If procedural knowledge has as little value as the
majority of responses indicate, then it could be argued that it should receive less attention in
preservice classrooms. This philosophic inconsistency cannot be answered through the responses
provided on this survey and warrants further investigation.
A relatively small number of responses argue that the conceptual/procedural debate poses
a false dichotomy. Some responses indicate that the issues reviewed in the survey are outdated,
and that these issues have been thoroughly discussed for more than two decades. Unfortunately,
the wide diversity of opinion by respondents argue more forcefully that this discussion is far
from resolved among the community of professional mathematics educators. Far too much
variety of often contradictory opinion resides within responses to indicate common grounding,
understanding, and focus on this concern. Thus, the opinions of the AMTE members who were
surveyed seem to remain fragmented regarding the balance recommended by NCTM.
7. Connecting to Demographics
The researchers were particularly interested in investigating whether strong correlation could be
19
discovered among various demographic aspects and respondents’ opinions regarding the balance
between procedural knowledge and conceptual understanding. For instance, among other
questions, it was initially wondered if the respondents’ number of years in the profession or
institution from which they received their last degree could be correlated to one particular
opinion over another. A number of factors made this dimension of the investigation of less value.
First, as previously mentioned, although great disparity existed among definitions for both
conceptual understanding and procedural knowledge among responses, agreement
overwhelmingly existed regarding a simultaneous denigration of procedural knowledge and a
unified valuation of conceptual understanding. Second, a number of doctoral candidates were
respondents. It is not clear to what extent these respondents either mirrored the opinions of their
faculty advisors or cautiously responded according to what they believed was the correct
response. Third, the numerous responses previously provided within this discussion seem well
distributed among all respondents. Fourth, the previously mentioned philosophic inconsistency
discovered within responses regarding the notions of the need and application of balance
between conceptual understanding and procedural knowledge again permeated all demographic
distinctions. Altogether, therefore, since all responses shared similarities, the comparison of
demographic data to various differing positions was determined to be of little value.
8. Conclusion and Suggestions
While the members of AMTE who participated in this study demonstrate significant
inconsistencies in their opinions and practice concerning the balance of procedural knowledge
and conceptual understanding in respect to mathematics teacher education, it is assumed that
20
beyond these members the entire field of mathematics and mathematics teacher education would
prove to be even more fragmented. Only additional dialog will remedy this fracture.
It is feared that balance between procedural knowledge and conceptual understanding
will not be fully met within mathematics education until both are valued and seen as necessarily
complementary, albeit each with their primary contextualized focus. Educators must understand
that learning new concepts and practicing the skills associated with further unfolding or applying
those concepts are interconnected. Learning procedures must be recognized as situated within the
process of learning.
Memory must also be seen as a necessary component of learning. Retention of concepts
along with the procedures which apply to, and can be employed in expanding upon, those
concepts is vital to learning. Memory and retention must not be seen as in opposition to learning;
they must be recognized as a necessary and valuable component of learning. One of the problems
associated with isolated procedural learning is that learning is quite fragile (NCTM, 2000;
Bransford, Brown, & Cocking, 1999), meaning that those procedures are often forgotten quickly,
or remembered inappropriately. A balance of learning both concepts and procedures, particularly
if the connections between them are made explicit, has been shown to enhance the long term
retention of both (Schoenfeld, 1988).
The survey and responses opened many more areas of investigation which this brief
paper was unable to address. Many dimensions yet remain fodder for future research, not the
least of which is the role of the conceptual-procedural balance in light of No Child Left Behind
and high stakes testing. In addition, further investigations into what is meant by “procedural
knowledge” such as those discussed by Star (2005) could also prove fruitful. It is hoped that this
21
study will elicit further consideration of this concern.
It seems obvious to us that if the mathematical reforms championed by AMTE, NCTM,
MAA, and the like are to achieve widespread implementation, then a greater degree of
correspondence should exist among those who champion them particularly with regards to issues
as fundamental as the balance between conceptual understanding and procedural knowledge. We
believe that the disparity in views held by mathematics teacher educators regarding the meanings
of conceptual and procedural learning – and thereby the resulting inconsistent classroom
pedagogical methodologies – might, if not rectified, spell trouble for the current mathematics
education reform effort.
The New Math Movement of the 1950’s and 60’s, which had its roots in the same
psychological and epistemological perspectives of current reform efforts, suffered from similar
fractured pedagogical classroom delivery (Bossé, 1995, 1999; Davis, 1990). While a core of
curriculum projects (e.g., P.S.S.C. SCIS, the Madison Project) focused upon reasoning,
creativity, understanding, discovery, big ideas, and real world application, other
contemporaneous efforts had incongruous foci.
There was no one “New Math.” There were many different ideas offered; some
were fundamentally and foundationally different from others. The entire
movement was very diverse. The only thing in common within the entire
movement was [the perception] that school mathematics education was
insufficient as it stood. Therefore, the most historically and intellectually honest
definition of the New Math Movement might actually be: All educational
movements during the 1950s and 1960s that had an aim of reforming, repairing,
22
or enhancing mathematics education on the K- I2 level (Bossé, 1995, p. 173).
While many of the New Math reform efforts based on constructivist epistemologies
demonstrated significant success, virtually all efforts of that era were collected under the
pejorative moniker “New Math”, and dubbed as failures. Therefore, it may be valuable for
current mathematics educators to learn from the past, increase dialog regarding epistemological
and pedagogical concerns (conceptual understanding versus procedural knowledge), and
sufficiently unite in beliefs and practice so as to avoid repeating the mistakes of the past.
23
Appendix
AMTE Member Voluntary Survey – April, 2005
What is the Meaning of the Term "Balance" as Used by the Principles and Standards Document in
Relation to Conceptual and Procedural Learning?
Demographics (Optional):
Gender: Age (or approximate):
Where you currently work/teach (university and/or City & State):
Number of years at this position:
School at which you earned your highest degree: Year:
DIRECTIONS: For questions which you select a response, either highlight (electronic) or circle
(hardcopy) the appropriate response. For open-ended questions, please limit your responses to no more than
2-3 sentences. These responses can be typed into the electronic document or sent on separate sheets of paper
by mail.
PART 1: Curriculum and Instruction.
1. What is the purpose/value of conceptual understanding?
2. What is the purpose/value of procedural knowledge?
3. What are the similarities between procedural knowledge and conceptual understanding?
4. What distinguishes procedural knowledge from conceptual understanding?
5. As students naturally learn mathematics, what it the natural flow of understanding? (Highlight or circle one
choice.)
conceptual understanding leads to procedural knowledge
procedural knowledge leads to conceptual understanding
either leads to either
6. How would you catagorize most K-12 mathematics curricula? (Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
7. How would you categorize most “reformed” K-12 mathematics curricula? (Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
8. How would you catagorize the focus of the training which you received in the program of your highest
degree? (Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
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9. How would you categorize the focus of the instruction which you provide in your undergraduate content
courses for preservice K-12 teachers? (Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
10. How would you categorize the focus of the instruction which you use in your undergraduate pedagogy
courses for preservice K-12 teachers? (Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
11. Many mathematics teacher educators have opined regarding the need to emphasize, or even over
emphasize, conceptual understanding over procedural knowledge in order to overcome the perennial focus on
procedurally based instruction in K-12 mathematics education. If you responded “mostly conceptual” or “all
conceptual” on questions 9 and/or 10 and your reason for doing such is different from the opinion just
mentioned, please indicate why you made your selection(s).
12. How would you categorize the textbooks (if applicable) which you most frequently use in your classes?
(Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
13. Rate the following NCTM process standards in rank order of 1 (highest) to 5 (lowest) in how significantly
they can assist students to gain conceptual understanding.
Problem Solving Reasoning & Proof Communication Connections Representations
_____ _____ _____ _____ _____
PART 2: Student Understanding.
14. How would you categorize the skills/knowledge which your mathematics educations students demonstrate
as they enter into your course/program? (Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
15. How would you categorize the skills/knowledge which your mathematics education students demonstrate
as they exit your course/program? (Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
16. As a student enters your course/program, if you were REQUIRED to choose ONLY ONE of the following,
would you prefer that a student (Highlight or circle one choice.)
can perform mathematics procedures well but lacks deep conceptual understanding of the mathematics
understands mathematical concepts well but lacks procedural skills
17. As a student exits your course/program, if you were REQUIRED to choose ONLY ONE of the following,
would you prefer that a student (Highlight or circle one choice.)
can perform mathematics procedures well but lacks deep conceptual understanding of the mathematics
understands mathematical concepts well but lacks procedural skills
18. Please explain your rationale for your selections in Questions 15 and 16. Would your responses be
different if you were considering mathematics students who are not planning to become teachers and those who
are planning to teach? Why?
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19. Could a student succeed through your program if s/he possessed excellent procedural skills, but had limited
conceptual understanding? (Highlight or circle one choice.)
Yes Unsure No
20. How have you employed the NCTM process standards to deepen conceptual understanding?
21. How have you employed Bloom’s taxonomy of the cognitive domain to deepen conceptual understanding?
PART 3: Assessment.
22. How would you categorize the focus of the assessments which you use in your undergraduate content
courses for preservice K-12 teachers? (Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
23. How would you categorize the focus of the assessments which you use in your undergraduate pedagogy
courses for preservice K-12 teachers? (Highlight or circle one choice.)
All Procedural Mostly Procedural Well Blended Mostly Conceptual All Conceptual
24. How do you assess undergraduate preservice K-12 teachers for conceptual understanding in the content
courses you teach? Give some examples. Tell if these evaluations for conceptual understanding are graded.
25. How do you assess undergraduate preservice K-12 teachers for conceptual understanding in the pedagogy
courses you teach? Give some examples. Tell if these evaluations for conceptual understanding are graded.
26. How do you teach your undergraduate preservice K-12 teachers to assess their future K-12 students for
conceptual understanding? Give some examples.
27. How have you employed the NCTM process standards to assess conceptual understanding in your
undergraduate preservice K-12 teaching courses (content and pedagogy)?
28. How have you employed Bloom's taxonomy of the cognitive domain to assess conceptual understanding in
your undergraduate preservice K-12 teaching courses (content and pedagogy)?
PART 4: Change in Practice.
29. How has this survey affected your view of your teaching/assessment for conceptual understanding?
30. Which question posed above made you most reflect upon your/your school’s/your students’ focus on this
issue and why?
31. If you completed a previous version of this survey, how did it affect your educational view and practice on
this issue?
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