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Stabilizing the Concept of Teaching Mathematics for (Conceptual) Understanding
Matthew LeachSED RS 654Spring, 2012
Dr. Alan K. GaynorMarch 18, 2012
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Introduction
Mathematics teaching and learning over the course of recent years has undergone much
change and reform. Between the Principles and Standards of School Mathematics (NCTM,
2000) and recent studies and various reports from the Trends in Mathematics and Science
Studies (TIMMS) there has been a push in the United States for more mathematics instruction
that aims at developing understanding. In an effort to better understand what it means to teach
mathematics for understanding, it is necessary to compare and contrast the two main ways of
teaching mathematics: namely teaching for procedural efficiency and teaching for conceptual
understanding. Both of these types of understanding represent a constant debate in the field of
mathematics education, with respect to traditional instruction (lecture) versus reform instruction
(discussion, inquiry).
In this paper, I will examine the two main types of learning goals for mathematics: procedural
efficiency and teaching for conceptual understanding. These two main goals in mathematics
education, while somewhat opposing, are also complementary to one another, which is why it is
necessary to consider both concepts to better understand what is meant by conceptual
understanding. I will provide an example of each in hopes that the reader will gain an
understanding as to what exactly constitutes teaching for procedural understanding and what
constitutes teaching for conceptual understanding. Secondly, I will identify and justify the
elements of mathematics instruction that aim for conceptual understanding as well as procedural
efficiency, based on many of the ideas proposed over time. After all, knowledge of
mathematical procedures are a necessary aspect of mathematics knowledge for teaching and
learning, although the conceptual depth will augment the power and understanding of
mathematical procedures, concepts, and theorems. Finally, I will examine the potential role that
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a balance between the two types of understanding will play in my proposed study of what
mathematics knowledge is gained or strengthened when teachers introduce basic
theory/composition techniques in select mathematics lessons at the high school level.
Procedural efficiency vs. Conceptual understanding
Two contrasting learning goals of mathematics include teaching for skill efficiency and
teaching for conceptual understanding (Resnick & Ford, 1981, as cited by Hiebert & Grouws,
2007). With respect to procedural efficiency, sometimes referred to as “procedural fluency” (e.g.
Kilpatrick, Swafford, & Findell, 2001) or “skill efficiency” (Gagne, 1985, as cited by Hiebert &
Grouws, 2007), there is a strong emphasis on correctly carrying out mathematical procedures and
algorithms. Kilpatrick, Swafford & Findell (2001) refer to procedure fluency as one of the five
main strands of mathematical proficiency. Procedural fluency refers to “the knowledge of
procedures, knowledge of when and how to use them appropriately, and skill in performing them
flexibly, effectively, and accurately” (p. 121). Hiebert & Grouws (2007) do not consider
procedural (skill) efficiency to include the flexible use of skills or adapting skills in different
mathematical situations, which is contrary to the view offered by Kilpatrick, Swafford & Findell
(2001).
When considering conceptual understanding, there is a lot to take in. Kilpatrick, Swafford &
Findell (2001) refer to conceptual understanding as “an integrated and functional grasp of
mathematical ideas” (p. 118). When considering the students’ learning with conceptual
understanding, students understand the “why” behind a mathematical idea as well as what types
of contexts the idea is useful. Conceptual understanding is a way for knowledge to be organized
in to a “coherent whole,” which gives the learner the ability to make connections to prior
knowledge as they gain new knowledge. These connections are explored by Hiebert &
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Carpenter (1992) as well, when considering what it means to learn mathematics with
understanding. States Hiebert & Carpenter (1992): “A mathematical idea, procedure, or fact is
understood thoroughly if it is linked to existing networks with stronger or more numerous
connections” (p. 67). While there is not an explicit mention of conceptual understanding, there is
definitely a requirement of connections needed between mathematical ideas, which is one facet
of conceptual understanding.
Teaching for procedural fluency: an example
We shall now consider an example from high school geometry where the emphasis is on
procedural fluency. Below is an excerpt from a lesson about applying the distance formula to
circles on the coordinate plane (Focus in High School Mathematics: Reasoning and Sense
Making, NCTM, 2009).
Teacher: Today’s lesson requires that we calculate the distance between the center of a circle and a point on the circle in order to determine the circle’s radius. Who remembers how to find the distance between two points?
Student 1: Isn’t there a formula for that?
Student 2: I think it’s x1 plus x2 squared, or something like that.
Student 1: Oh yeah, I remember – there’s a great big square root sign, but I don’t remember what goes under it.
Student 3: I know! It’s x1 plus x2 all over 2, isn’t it?
Student 4: No, that’s the midpoint formula.
(Eventually, the teacher reminds the students about the correct formula).
While there is a connection made with respect to a formula being used to calculate the radius
of a circle, there is not much more depth given to the concept of circles in the coordinate plane.
Perhaps the prior day’s lesson might have looked something like this.
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Teacher: Today, we are going to learn the distance formula to calculate the distance between two points on the coordinate plane. The distance formula, given two ordered pairs, ( x1 , y1 ) and
( x2 , y2 ), is D=√(x2−x1)2+( y2− y1 )2. Copy this into your notebooks.
(students copy the formula into their notes and label it)
Teacher: OK, so now let’s calculate the distance between (-2, -5) and (-3, 7). I will let ( x1 , y1 )=(−2 ,−5) and ( x2 , y2 )=(−3 , 7). So I will plug into the distance formula like this
√(−3−(−2 ))2+(7−(−5 ))2=√(−1)2+122=√1+144=√145∨about 12
(students copy the formula in the same procedure as the teacher demonstrates in their notes)
After another example or two, some students might be called on to answer questions such as
“what do we plug in for x1…-5 – (-8) is what…what’s the next step…” Then the students do
some on their own and start their homework in class, practicing the procedure in calculating the
distance between two points. There is little, if any, reference to other connections such as the
Pythagorean Theorem or how the horizontal and vertical distances are related to the actual
distance between the points.
When analyzing the above excerpt with respect to procedural efficiency, there is an emphasis
on using a formula. Formulas in mathematics are methods for making calculations efficiently.
In classrooms where there is an emphasis on procedural efficiency, the teacher typically gives
the formula to the students without much explanation, other than to tell them what it is and when
to use it. In this case, the distance formula is the formula that was given to the students by the
teacher and given again by the teacher, when a reminder was needed. There is more of a sense
of rote learning rather than meaningful mathematics in a lesson like the one described above (e.g.
Davis, 1984).
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Teaching for conceptual understanding: an example
We shall now consider a different version of the above lesson on the distance formula, where
there is more of a need for students to reason and make connections, which are foundations for
conceptual understanding in mathematics. In this alternative situation, we see more reasoning
about mathematics and making connections to other aspects of mathematics (Focus in High
School Mathematics: Reasoning and Sense Making, NCTM, 2009).
Teacher: Let’s take a look at a situation in which we need to find the distance between two locations on a map. Suppose this map shows your school; your house, which is located two blocks west and five blocks north of the school; and your best friend’s house, which is located eight blocks east and one block south. If the city had a system of evenly spaced perpendicular streets, how many blocks would we have to drive to get from your house to your friend’s house?
Student 1: Well, we would have to drive ten blocks to the east and six blocks to the south so I guess it would be sixteen blocks, right?
Teacher: Now what if you could use a helicopter to fly straight to your friend’s house? How could we find the distance “as the crow flies”? Work with your partners to establish a coordinate-axis system and show the path you’d have to drive to get to your friend’s house. Next, work on calculating the direct distance between the houses if you could fly.
Student 1: What if we use the school as the origin? Then wouldn’t my house be at (-2, 5) and my friend’s house at (8, -1)?
Student 2: Yeah, that sounds right. Here, let’s draw the path on the streets connecting the two houses and then draw a line segment connecting the two houses.
Student 1: Maybe we could measure the length of a block and find the distance with a ruler?
(students draw and label a diagram on a coordinate plane)
Student 3: Wait a minute – you just drew a right triangle, because the streets are perpendicular.
Student 4: So that means we could use the Pythagorean theorem: 102+62=c2, so c=√136.
Student 2: But how many blocks would that be?
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Student 3: Shouldn’t the distance be between eleven and twelve blocks, since 121 < 136 < 144? Actually, it’s probably closer to twelve blocks, since 136 is much closer to 144.
(The teacher then extends the discussion to consider other examples and finally develop a general formula.)
Considering the dialog presented previously, we can see “an integrated and functional grasp
of mathematical ideas” (Kilpatrick, Swafford & Findell, 2001). The ideas that students are
integrating include the Pythagorean Theorem, right triangles, perpendicular and parallel lines,
measurement of distance using a coordinate system, and the gradual construction of the direct
distance, relating the horizontal and vertical distances. It is through these ideas that the
knowledge needed for the distance formula is organized into a coherent whole. By working out
the problem, students are using their prior knowledge to develop and reason about the distance
formula. As such, they experience the "why" of the distance formula (why is it true, where does
it come from) in addition to the "how" of the distance formula (plug in two ordered pairs to
calculate the distance between them). Furthermore, by applying the mathematical concept in a
useful situation (e.g. finding the most direct distance between two locations), both the teacher
and student can experience "meaningful mathematics" rather than simply rote learning (e.g.
Davis, 1984).
Let us now consider how understanding of mathematics has developed its definitions over
time. There are a number of recurrent themes in understanding of mathematics, such as linking
prior knowledge to current knowledge and relating many integrated important ideas to a main
mathematical idea. While the term conceptual understanding was not always used, there have
been some other terms used which convey the same ideas as conceptual understanding.
Likewise with procedural efficiency, there have been some other terms that have been linked to
the ideas conveyed by procedural efficiency. There have been some notable debates as well,
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particularly when considering conceptual understanding from a psychological viewpoint versus a
social viewpoint.
Towards a definition of conceptual understanding:
As early as the 1930s, there has been a desire for teaching and learning mathematics with
understanding, although the modifier conceptual was not yet used. It is certainly noteworthy that
many of the ideas seen in history are much more prevalent today, especially during the call for
reform teaching and learning in mathematics. Three theories of arithmetic learning were
proposed (Brownell, 1935 as cited by Baroody & Dowker, 2003). The first was drill learning,
which essentially focused on rote memorization and computational skills. Four assumptions
were noted for drill learning. First of all, children must learn to imitate the skills and knowledge
of adults. Secondly, what is learned are associations or bonds between otherwise unrelated
stimuli. Thirdly, forming such bonds does not require understanding. Finally, the most efficient
way to accomplish bond formation is through direct instruction or drill (Brownell, 1935 as cited
by Baroody & Dowker, 2003).
Brownell was not satisfied with drill theory and proposed meaning theory. Here, there is a
desire for the “meaningful memorization of skills” (Brownell, 1935 as cited by Baroody &
Dowker, 2003). States Brownell:
“The ‘meaning’ theory conceives arithmetic as a closely knit system of understandable ideas, principles, and processes. According to this theory, the test of learning is not mere mechanical facility in ‘figuring.’ The true test is an intelligent grasp upon number relations and the ability to deal with arithmetical situations with proper comprehension of their mathematical significance as well as their practical significance.” (p. 19). We can see such key words as "grasp" and "comprehension." Brownell is using this theory to
go beyond "figuring," which is what is typically common in rote and procedural work. Brownell
seems to be implying the importance of seeking relations and being able to handle arithmetic
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situations from both the mathematical/theoretical and practical point of view. If Brownell were
alive today, perhaps he would be a major proponent of the process standards outlined in the
Principles and Standards for School Mathematics (NCTM, 2000), such as connections, reasoning
and proof and problem solving.
Brownell continues to discuss the meaning theory with respect to teachers and teaching.
There are three interrelated factors that are needed to promote the learning of arithmetic with
understanding. The first is called the complexity of arithmetic learning. In the beginning,
teachers might allow students to use basic strategies such as counting. As students develop and
become ready, however, teachers are encouraged to encourage students to adopt more advanced
strategies, such as reasoning (e.g. transforming an unfamiliar problem into a familiar one) when
they are ready.
The second factor that is important for teachers is the pace of instruction. Learning
arithmetic, according to Brownell, is viewed as a “slow, protracted process.” As such, the
meaning approach requires teachers to allow time for the students to construct an understanding
of arithmetic ideas, especially with basic number combinations and multi-digit procedures, and
to discover and rediscover regularities and patterns in arithmetic before practice makes basic
number combinations automatic. This, in turn, leads to knowledge that is more easily transferred
(Brownell, 1935 as cited by Baroody & Dowker, 2003).
The third factor that is important for teachers is the emphasis on relations. Addition and
subtraction patterns such as 5 + 4 = 9 and 9 – 5 = 4 are not facts but generalizations (Brownell,
1935 as cited by Baroody & Dowker, 2003). Teachers need to make students discover a
regularity such as 5 + 4 = 9 many times in many different contexts. More importantly, teachers
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need to guide students in seeing relations among combinations so that they (students) can see the
basic combinations as a system of knowledge.
The final of the three main theories of arithmetic teaching and learning is that of incidental-
learning theory. Like meaning theory, this was in reaction to drill theory. Much of the ideas
behind this theory require that children explore the world around them, identify regularities and
patterns, and actively construct their own understandings and patterns. This theory seems to be
in line with many of Dewey’s ideas on progressive education as well as Jean-Jacques Rousseau’s
fictitious student, Emile, where the student’s learning is determined by their own natural
curiosity. There does seem to be a number of limitations with this theory, particularly with the
time required as well as the lack of teacher expertise to effectively carry out this particular
practice.
What is of interest in Brownell’s work is that we see a number of the same debates back in the
1930s as we have seen develop over time and are in existence today. Brownell seems to take a
middle approach in valuing a combination of drill approaches and incidental-learning (students
making their own discoveries). Brownell does not seem to value computation over conceptual
understanding or vice versa (Baroody & Dowker, 2003). Brownell in short advocated for both
the “what” and “why” in mathematics learning, but during that time, he left unclear what
teachers need to do to achieve that balance. There is, however, a call for students creating a
network of knowledge with guidance from teachers of mathematics. This network of knowledge
is now what has become a major component of conceptual understanding.
Skemp (1976) discusses the ideas of instrumental mathematics and relational mathematics.
Skemp (1976) basically describes instrumental mathematics as easier to understand because of
easily remember rules (e.g. a negative times a negative equals a positive). He also cites some
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advantages such as more immediate rewards, such as a test page of all correct answers and the
quickness of getting at the right answer. Much of procedural and skill efficiency would fall into
this category deemed instrumental mathematics.
Skemp (1976) proceeds to discuss relational mathematics, which would fall into much of
today's ideas of conceptual understanding. Some advantages cited by Skemp (1976) include the
following: relational mathematics is more adaptable to new tasks, it is easier to remember,
despite being harder to learn, which suggests a paradox, and that relational knowledge can be a
goal in itself (the process of problem solving versus the product). Skemp (1976) discusses how
many teachers might lose the benefits of teaching topics as topics are taught as separate topics
rather than fundamental concepts which create interrelations within mathematics. With respect
to adapting problem solving to different contexts, Kilpartick, Swafford, & Findell (2001) also
consider the idea of adaptive reasoning as one of their five strands of mathematical proficiency.
We have also seen this argument of topics being taught to form a more coherent whole in much
of the modern research as well (e.g. Hiebert & Carpenter, 1992; Bransford et al, 2000; Hiebert &
Grouws, 2007). As we advance into the next decade, we see two similarly coined terms, which
refer to the procedural aspect and the connectedness aspect of mathematics.
Davis (1984) discusses “rote mathematics” and “meaningful mathematics.” These two
contrasting ideas parallel with the modern day procedural efficiency and conceptual
understanding, In an example of “rote mathematics,” Davis (1984) offers an example from
arithmetic where the student carries out an addition problem mechanically (e.g. add the ones,
carry a one, add the next column, carry, etc.). When speaking of “meaningful mathematics,”
Davis (1984) refers to rote mathematics as leaving out meaning and understanding. Again, his
next question is, what is meaning and understanding? Davis (1984) discusses the role of mental
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representations used in the “creation of appropriate frames.” These frames, of which Davis
(1984) speaks are no different than the construction of networks of knowledge, which is what
constitutes conceptual understanding. While Davis (1984) does not use the term conceptual
understanding, he does use the term, meaningful mathematics, which is again, pushing towards a
definition of conceptual understanding.
Davis (1984) discusses the idea of “creative mathematics.” Here, he takes his rote vs.
meaningful mathematics and parallels it with routine vs. creative. On one hand, this creativity
might appear to be in line with Brownell’s (1935) discussion of incidental-learning theory.
There does, however, seem to be less of an emphasis on nature and as technology is advancing,
the teachers of mathematics need to add more tools to their professional development. Creative
mathematics is viewed in light of computers being designed to handle many routine and
repetitious calculations. Davis (1984) cites six reasons for the necessity for creative
mathematics. The first reason pertains to the complexity of mathematics (p. 15). An example of
an quadratic equation is shown which requires ingenuity and originality. That is, students would
need to recognize the quadratic-like form to solve an easier problem and apply it to the solution
for the more difficult problem. The example given is e2 t−5e t+6=0, which looks like a variation
of the equation x2−5 x+6=0. Here, there is a strong cry for the ability to transfer among
knowledge networks (e.g. recognizing a quadratic-like pattern and factoring the equation
accordingly). Again, this component of creative mathematics proposed by Davis (1984) is seen
in much of today’s research on teaching and learning mathematics with conceptual
understanding.
The second component of creative mathematics proposed by Davis (1984) is that students do
invent original solutions. With older children, there may be a number of insightful ideas, which
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demonstrate conceptual understanding. Peter, a 17 year old, was asked to make up and solve a
percent problem. He posed and answered the following problem (p. 18).
Ten is what percent of 12?
"You’ve got one-sixth left over…so…you need to figure out how to get 6 out of 100. Six times fifteen is 90 and you’ve got 6 left over, so you’ve got 16…You’ve got 4 left over, so it’s two thirds or point six six six and so on, so that’s sixteen point six six six (forever). Now, you need the opposite [sic!] of that…that subtracted from 100 – so you’ve got eighty-three point three three three (forever) percent."
Davis (1984) concludes that this boy’s thinking was not taught to him by a teacher, but rather
this was the boy’s original solution. Furthermore, we see that the student engaging in creative
mathematics by making connections among different aspects of mathematics (fractional
thinking, division, handling remainders, subtracting a part from a whole to find the remaining
part). Again, these connections within this complex of network of knowledge demonstrates a
deep conceptual understanding of how percents work.
The third component of creative mathematics proposed by Davis (1984) is the need to
recognize originality (p. 16). Kye Hedlund, a third grade boy, was credited for an original
method used to subtract without borrowing. When Kye subtracted 64 from 28, he subtracted the
ones, saying that 4 – 8 = -4. Then he subtracted the tens 60 – 20 = 40. Combining 40 and -4,
Kye got 36, the correct answer to the problem that the teacher solved using the original
regrouping strategy. Furthermore, the teacher was not familiar with this strategy, as she taught
the common misconception of not being able to subtract 8 from 4. These inventive strategies
may evolve through cooperative learning as well as students’ own intuitions. Again, students
being able to construct strategies for solving various problems may show evidence of solid
conceptual understanding.
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The fourth component of creative mathematics is knowing how to make classes dull (p. 19).
Davis (1984) suggests that teaching mathematics strictly for rote learning makes mathematics
learning dull. While the dullness may not completely avoidable, mathematics can certainly be
“treated for what it really is – an exploration of a rich world of possibilities and a persistent
Challenge – mathematics can be exciting, and mathematics classes can be fun.” (p. 19). The
fifth component of creative mathematics is sometimes the originality is the very thing we need to
teach (p. 20). Davis (1984) cites geometry for instance as a subject which typically has three
main goals: 1.) developing a mathematical system starting with a given set of axioms and
definitions, 2.) providing students with some analytical knowledge for one model of physical
space, and 3.) to help students learn how to make original proofs of theorems (p. 20). Far too
often, teachers may neglect the last goal as there is too much emphasis on procedure and not
developing an understanding of the why in mathematics. Conceptual understanding, as
mentioned earlier, emphasis the why in mathematics in addition to the what/how.
The final component of the creative mathematics proposed by Davis (1984) is mathematics in
the real world (p. 20). Many situations encountered in the real world may be modeled using
mathematical problem solving and modeling (e.g. evaluating financial investments, comparing
interest rates from 20 years ago to now). While conceptual understanding does not emphasize
real world connections as much as the connections formed within a network of knowledge,
perhaps the term might be adjusted in the future to incorporate more real world situations.
In summary, Davis (1984) encourages the need for meaningful mathematics. Much of what
is considered to make mathematics meaningful is the need to teach and learn it with
understanding. This understanding requires the ability to make connections to other aspects of
mathematical knowledge, whether it is procedures or prior concepts. As we progress a few years
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later, we finally see the evolution of the term conceptual understanding, although many of its
features have been called for and noted during the earlier part of the 20th century.
Hiebert & Carpenter (1992) define the term understanding with respect to the way
information is represented and structured. While they do not use the modifier conceptual, they
state the following when defining understanding:
"A mathematical idea or procedure or fact is understood if it is part of an internal network. More specifically, the mathematics is understood if its mental representation is part of a network of representations. The degree of understanding is determined by the number and the strength of the connections." (p. 67).
This theme/definition has been seen over time when considering what constitutes (conceptual)
understanding. We have seen this theme in Davis (1984) when he calls for more "meaningful
mathematics" as well as in Brownell's (1935) call for "meaningful memorization of skills."
Hiebert & Carpenter (1992) propose a framework that places the idea of understanding in a
conceptual framework. When considering the idea of understanding, Hiebert and Carpenter
(1992) consider that understanding requires the following two components: external and internal
representations and connecting representations. Understanding requires the thinking of
mathematical ideas be represented internally. That is, in our minds, we need these ideas to
operate on them. Unfortunately, the idea of internal representations is inferential at best as
mental representations are not observable (p. 66). With respect to external representations, ideas
can be represented by means of spoken language, written symbols, pictures, or physical objects.
In short, external representations are what the National Council of Teachers of Mathematics
(NCTM) considers multiple representations (e.g. table, graph, equation/rule, situation)
(Principles and Standards, 2000).
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With respect to connecting representations, Hiebert & Carpenter (1992) consider external
connections and internal connections. Connections between external representations of
mathematical information are often constructed when looking at structures of similarity and
differences (p 66). Connections of an external nature might be constructed by the learner using
different representation forms of the same mathematical idea. For example, connections can be
made between the graph, equation, and table that represent the same linear equation, showing
different features (e.g. y-intercept, slope, sequences of constant differences, ratios of constant
differences). Also, connections for related ideas within the same representational form can be
constructed (e.g. comparing the graphs of linear quadratic cubic, quartic, etc. functions).
With respect to internal connections, Hiebert & Carpenter (1992) propose that when
relationships between internal representations of ideas are constructed, there are developments of
networks of knowledge. When speaking of these networks, Hiebert & Carpenter (1992) consider
two metaphors. The first metaphor is a vertical hierarchy, where special cases of knowledge and
ideas are stored but in an overarching manner, generalizations are created. These generalizations
are none other than the "coherent wholes" that were mentioned earlier whereas the knowledge
components were special cases, which organized into the coherent whole. In the second
metaphor, knowledge is viewed as being constructed like a spider's web, where the junctures
(nodes) represent the pieces of represented information. The threads that connect the nodes are
thought of as connections or relationships between the knowledge. Again, when we consider the
two metaphors proposed by Hiebert & Carpenter (1992), we see the idea of various mathematical
ideas, processes, and concepts being connected and depending upon the strength of the internal
connections, students' learning of mathematics can truly be experienced with conceptual
understanding by means of this complex network. Carpenter & Lehrer (1999) also identify five
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components needed to develop understanding: constructing relationships, extending and applying
mathematical knowledge, reflecting about experiences, articulating what one knows, and making
mathematical knowledge one's own.
Boix-Mansilla & Gardner (1997) consider the concept of understanding with respect to two
dimensions. The first dimension considers understanding of domain-specific knowledge, which
in this case, would be mathematics and mathematical topics. The second dimension of
understanding takes into consideration the “disciplinary modes of thinking embodied in the
methods by which knowledge is constructed, the forms in which knowledge is made public, and
the purposes that drive inquiry in the domain (p. 382).” States Boix-Mansilla & Gardner (1997):
“Students demonstrate their understanding when they are able to go beyond accumulating
information and engage in performances that are valued by the community in which they live (p.
382).” There seems to be a social aspect to understanding which shall be discussed later when
debates are reviewed about understanding with respect to psychological foundations and
understanding with respect to social foundations. For now, however, we shall consider primarily
the psychological foundations of conceptual understanding, which seem to be much more rooted
in psychology and the mind.
Some examples of these disciplinary modes of thinking are considered by Bransford et al
(2000) in How People Learn: Brain, Mind, Experience and School. With respect to some of
these disciplinary modes of thinking, Bransford et al (2000) propose three main findings: 1.)
Students come to the classroom with preconceptions about how the world works. If their initial
understanding is not engaged, they may fail to grasp the new concepts and information they are
taught, or they may learn them for purposes of a test but revert to their preconceptions outside
the classroom (pp. 14, 15). 2.) To develop competence in an inquiry, students must a.) have a
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deep foundation of factual knowledge, b.) understand facts and ideas in the context of a
conceptual framework, and c.) organize knowledge in ways that facilitate retrieval and
application (p. 16). 3.) A “metacognitive” approach to instruction can help students to learn and
take control of their own learning by defining learning goals and monitoring their progress in
achieving them (p. 18). In short, knowledge is connected to other knowledge, even though the
other knowledge may not be accurate but through correcting misconceptions (which is a dual
responsibility for both teachers and students), students can create and be guided toward creating
a conceptual framework (see also Davis, 1984 in his discussion of frames).
Conceptual understanding in mathematics goes beyond learning a mathematical fact,
generalization or procedure. Conceptual understanding in mathematics requires knowledge
about the “why” in mathematics topics in addition to the “how” or “what” in mathematical
topics. Conceptual understanding requires a mental capacity to create many different pieces of
knowledge into a coherent whole such that retrieval is facilitated when different problem solving
contexts arise. As seen throughout the 20th century and early 21st century, conceptual
understanding has held fairly constant threads, such as the need to connect and organize
knowledge. We shall now turn our attention to some of the debates that have been fueled in
learning mathematics with conceptual understanding.
Controversy over emphasis on internal or mental representations and lack of consideration for social/cultural factors.
During the 1980s and 1990s, there has been some criticism over the emphasis of learning
from a psychological standpoint and a lack of consideration of social/cultural elements of
learning with understanding (e.g. Lave, 1988, Wenger, 1998). In an effort to consider
social/cultural factors, Lave (1988) created the Adult Math Project. In the Adult Math Project,
there are many observational and experimental investigations of everyday arithmetic practices in
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different settings, such as the supermarket or the kitchen. Two questions are posed for the
teachers and students using the Adult Math Program. How does arithmetic unfold in action for
everyday settings? Are there differences between school situations (e.g. tests) and situations far
removed from school (in the kitchen or the supermarket such as buying apples for four children,
what considerations are needed)? These questions seem to align with Davis (1984) and his idea
of real world situations as a major element of creative mathematics. This practical element of
mathematics would also be in line with many of Dewey’s ideals with respect to progressive
education.
Wenger (1998) argues that much of our institutionalized teaching and training is perceived as
would-be learners as irrelevant and most of us come out of this treatment feeling that learning is
boring and arduous, and that we are not really cut out for it. This argument resonates with Davis
(1984) as well when he cautions educators about making classes dull. Wenger (1998) proposes
that instead, an alternate perspective is needed for learning and teaching with learning with
understanding. Learning should be treated as a human activity, like eating or sleeping. Again,
we see the social element when considering the humanness of learning.
A call for a balance between social and psychological factors for conceptual understanding.
Over recent years, there has been a call for a balance that considers both psychological and
social/cultural factors when considering the meaning and elements of conceptual understanding
in mathematics (e.g. Cobb, 1994; Sfard, 1998). Cobb (1994) considers mathematical learning
with understanding from both the constructivist (psychological) vs. sociocultural viewpoints.
Students actively construct their mathematical ways of knowing as they aim for effectiveness by
creating coherence to their own personal worlds of experience. This experience mostly considers
how students learn and how they go about learning, from a psychological/constructivist point of
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view. Cobb (1994) also considers the idea that understanding requires a certain degree of
activity. According to Cobb (1994), students’ arithmetical activity is profoundly influenced by
completing worksheets, shopping in the supermarket, selling candy, etc., which are examples of
conditions for learning possibilities. Hence, there is a desire for a balance between the
sociological (conditions for possibility of learning) and constructivist (what students learn and
how they go about learning) when considering teaching and learning mathematics for
understanding.
Sfard (1998) proposes two metaphors for learning with conceptual understanding. The first
metaphor, called the acquisition metaphor (AM), regards concepts as basic units of knowledge.
When considering the language of knowledge acquisition, minds as viewed as things to fill (p.
5). Sfard (1998) lists a number of terms associated with AM: knowledge, concept, conception,
idea, notion, misconception, meaning, sense, schema, fact, representation, material, contents (p.
5). We have seen such terms as meaning and representation used in the historical review of
conceptual understanding. A number of terms are also proposed by Sfard (1994) for making
entities one’s own: reception, acquisition, construction, internalization, appropriation,
transmission, attainment, development, accumulation, grasp (p. 5). The teacher may help student
to reach his or her learning goal(s) by delivering, conveying, facilitating, mediating, etc. (p. 5)
Once acquired, the knowledge, like any other commodity, may now be applied, transferred (to a
different context) and shared with others (p. 6). The sharing of knowledge is one of the first
social/cultural factors we see in Sfard’s (1998) acquisition metaphor. States Sfard (1998): “If
we can only become cognizant of something by recognizing it on the basis of the knowledge we
already possess, then nothing that does not yet belong to the assortment of the things we know
can ever become one of them (p. 7)”
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The second metaphor for learning that Sfard (1998) proposes is the participation metaphor
(PM), where learning is viewed as a process of becoming a member of a community. Learners
are considered newcomers and potential reformers of the practice while the teachers are the
preservers of its continuity (p. 6). There seems to also be a constructivist/psychological element
here as well. The PM for learning with understanding requires that the whole and the parts affect
and inform each other. We see the coherent whole and many units of knowledge needed for
conceptual understanding although it is the mind that is doing the “participating” in this case.
We also see the social element needed for learning with understanding. According to Sfard
(1998), there is “interaction in learning…’people in action’” (p. 6).
Taking into consideration some of the debates that have occurred over the years in
mathematics education, it seems that there is a need for students and teachers of mathematics to
experience mathematics through conceptual understanding in addition to procedural efficiency.
Taking away some of the key ideas that constitute conceptual understanding, it seems that
conceptual understanding in mathematics requires students to create many connections, both
internal and external, among many mathematical ideas to form a more coherent whole.
Achieving conceptual understanding has a number of factors embedded within it that are of a
psychological/constructivist nature (e.g. what students learn and how do they learn) and
social/cultural factors (sharing knowledge with others and actively engaging in mathematics as a
human activity). Conceptual understanding calls for connections both made within the field of
mathematics as well as everyday life outside of mathematics. Connections within mathematics
include a grasp of how and why various procedures/algorithms work, why mathematical
theorems and ideas are true, and how many ideas seemingly unrelated can be linked together to
form a more coherent whole. We now shall turn our attention to some key features of teaching
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that promote conceptual understanding, as it is the teachers who are the managers and facilitators
of mathematical knowledge.
What key features of teaching promote conceptual understanding?
Despite the fact that educators have always been interested in promoting conceptual
understanding, there has been in recent years a burst of interest in the topic. Hiebert & Grouws
(2007) consider two features of teaching that promote conceptual understanding. First of all,
teachers and students need to attend explicitly to concepts (Hiebert & Grouws, 2007, p. 383).
Secondly, students need to struggle with important mathematics (p. 387).
With respect to the first key feature, students can acquire conceptual understanding if
teaching attends explicitly to concepts. These concepts are defined as connections among
mathematical facts, procedures, and ideas (Hiebert & Grouws, 2007). Teachers attend to
concepts when they treat mathematical concepts in an explicit and public way (Hiebert &
Grouws, 2007). Additional suggestions and guidelines are offered for teachers. Brophy (1999)
suggests that students learn best when their classroom learning climate is caring and cohesive,
which seems to suggest some of the social elements discussed earlier with regard to learning
with understanding. Brophy also speaks of coherent content, which applies greatly to creating
opportunities for students to learn mathematics with conceptual understanding. States Brophy
(1999): “To facilitate meaningful learning and retention, content is explained clearly and
developed with emphasis on its structure and connections” (p. 17). Some additional thoughts
proposed by Brophy (1999), which teachers need to be aware is the creation and management of
thoughtful discourse, practice and application activities, and scaffolding students’ task
engagement. With regard to the last idea, the teacher’s role is to provide assistance and
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facilitate, where appropriate to steer his/her students in a general direction of learning with
understanding
With respect to the second key feature of teaching that promotes conceptual understanding,
students need to struggle with important mathematics. Hiebert & Grouws (2007) use the word
struggle to refer to students’ expending of effort to make sense of mathematics, figuring out
something that may not be immediately apparent (p. 387). Couco et al (1996) discusses five
habits of mind which students need to develop when doing mathematics. These habits include
1.) Process (the way one works through problems), 2.) Visualization (how one “pictures
problems), 3.) Representation (what one writes down), 4.) Patterns (what one finds), and 5.)
Relationships (what one finds or uses). Some of these habits of mind may be developed through
mathematical discourse or cooperative learning situations (see also Brophy, 1999).
We shall now turn our attention to some of the research that has focused on conceptual
understanding in mathematics teaching and learning. There have been studies have focused
comparing and contrasting teaching methods that emphasize skills versus understanding. Other
studies have also focused on supporting the claim that some aspects of teaching can facilitate
students’ opportunities to struggle (see prior definition of struggle) with mathematical ideas.
A review of research on teaching for (conceptual) understanding.
There is some research that have focused on comparing and contrasting teaching methods that
emphasize skills versus understanding (e.g. Cobb et al, 1991, Heid, 1988, Hiebert & Wearne,
1993, Boston & Smith, 2009). Cobb et al (1991) conducted an experimental study looking at ten
second grade classes participating in a project that focused on instruction that was
socioconstructivist in nature. Eight second grade classes were in the control group and did not
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receive the alternative instruction. Conceptual understanding was measured in a qualitative
manner by comparing different student algorithms used to solve problems across both groups.
Heid (1988) examined the effects of re-sequencing skills and concepts in a college calculus
class. Like Cobb et al (1991), patterns of understanding were measured in a qualitative manner.
Students in this study used computer programs that aided in the graphical and symbolic
manipulations of various calculus concepts. Heid (1988) used field notes, transcripts, student
notes and test results to measure the level of understanding students achieved. It was found that
students who focused on skill development after concept development, their test results on the
final examinations were similar to those who spent the entire semester focusing on skill
development.
Hiebert & Wearne (1993) compared and contrasted six second-grade classes with respect to
traditional versus alternative instruction on place value and addition and subtraction of whole
numbers. There was an initial split with respect to ability (two higher achieving classes, one of
which received the alternate instruction; four lower classrooms, one of which receiving the
alternate instruction). Like in the previous studies, the conceptual understanding gained was
observed and measured qualitatively, using audiotapes and observation notes. Two areas of
focus in observations were discourse (e.g. Brophy, 1999) and the tasks or problems in which
students engaged. Hiebert & Wearne (1993) looked in particular at the questioning strategies
used by the teacher. Some noteworthy areas including coding questions of recall (e.g. factual
information, procedural information, prior work), questions of strategy (e.g. describe strategy,
describe alternative strategy), questions that require students to generate a problem or a story,
was worked a certain way, analyze how a problem is different from one before) (p. 402). When
24
considering the last type of question, we see one way a teacher can help students create
connections to different ideas, as they develop conceptual understanding in mathematics.
The results indicated that in the classrooms receiving the alternate instruction featured a wider
array of representations used than those classes receiving the traditional instruction (e.g. stories,
stories and materials, physical materials, pictures of objects, symbols). It is also noted that in the
alternate instruction classes, the teachers seemed to ask less questions regarding procedure/fact
recall and more questions requiring students to explain and analyze. Much of the explanation
asked of students tends to aim more toward the why of mathematical ideas and procedures, not
just the how and what of mathematical ideas and procedures. Recall from earlier discussions that
in conceptual understanding requires the ability to make many connections and analyze problems
in different contexts as well as to have multiple (external) representation systems. Furthermore,
there is a sense of the why as well as the what/how in mathematics learning.
Teachers need to be educated and trained in their professional development to assist them in
promoting conceptual understanding in their own classrooms. Also, models of teaching that
focus on student conceptual development might be of use for teachers and education researchers.
Some recent research has focused on the importance of task selection and how teachers examine
their students’ conceptual development (e.g. Doerr et al, 2006; Boston & Smith, 2009). Doerr
(2006) conducted a qualitative case study on the link between the examination of students' work
and the teachers' actions in the classroom, particularly at the secondary level. Doerr (2006)
wished to determine the ways in which teachers examine their students' conceptual development
of exponential growth in the context of their own classrooms. A well-known modeling task was
administered to the students. Again, methods such as interviews, transcriptions and coding were
employed.
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It should be noted that Doerr (2006) did not measure student concept development in this
study, but rather made inferences to teachers’ ways in which they examine their students’
concept development and growth. Doerr (2006) found that the teacher had six main
responsibilities, which would allow them to examine and encourage the growth of their students’
conceptual development. First of all, the teacher needs to set expectations for student thinking
(e.g. explicitly encourage students to "work hard” and sometimes "give students a little prod" so
that they could continue working productively on the task) (p. 12). Secondly, the teacher needs
to focus the task (recognize the central mathematical difficulty with the task and press the
students to engage with it). In this case, the task was finding the equation) (p. 12). Thirdly, the
teacher needs to listen to students' ways of thinking about the task. Here, the teacher was
observed to have pressed the students to continue to follow their way of thinking (e.g. patterns of
perfect squares, slopes and linear patterns) and to investigate why their current claims may be
true (p. 14). Fourthly, the teacher needs to ask for student descriptions, explanations, and
justifications. By asking a student to describe a solution, the teacher supported the student in
revising his own solution, rather than guiding him along a learning trajectory that she might have
had in mind. In this way, the descriptions and explanation served not only to help the teacher in
understanding the student's current way of thinking, but also to help the student in evaluating and
revising his current way of thinking about the problem (p. 16). Fifthly, the teacher needs to have
students share and compare solutions. The teacher had intentionally let the students "work on it
[the task] until they had two different solutions," which enabled the teacher to further her goal of
engaging students in describing and/or explaining solutions. However, the teacher also shifted
the task beyond a description and explanation of the solutions. The teacher had created an
opportunity for students to engage in a discussion of mathematical equivalency of the two
26
solutions. (e.g. students were now in a position where they were the ones evaluating the validity
of their work.) (pp. 17, 18). Lastly, the teacher needs to recognize mathematical connections.
The teacher in this case study appeared to have recognized the mathematical connections
between the students' responses to this task and later tasks within the overall sequence of tasks
(p. 18).
Taking a look at the above six codes, we can see many themes which have been prevalent in
conceptual understanding. The last code in particular, makes a reference to mathematical
connections, which is considered essential in conceptual understanding (e.g. There is also the
social/cultural element of sharing and comparing solutions where mathematics is viewed as a
human activity (see also Wenger, 1998). We also see the need for students to provide
descriptions, explanations and justifications, which we have seen in prior research (e.g. Hiebert
& Wearne, 1993). These explanations require that students go beyond the what/how in
mathematics and think about the why with respect to procedures, patterns and problem solving.
Doerr (2006) also is considering the teacher’s need to set expectations for students’ thinking and
listen to students’ thinking, which is an essential component of students struggling with
important mathematics (see Hiebert & Grouws, 2007). There are elements of students’ being
guided toward certain structural elements of the task as well as connections to prior knowledge
(e.g. Brophy, 1999).
We shall now look at a recent example of research with respect to the development of
teachers’ ability in selecting tasks appropriate for students’ development of conceptual
understanding. Boston & Smith (2009) examined the impact of a professional development
program, Enhancing Secondary Mathematics Teacher Preparation (ESP), on the instructional
practices of the secondary mathematics teacher participants. The program was specifically
27
focused on the selection and implementation of cognitively challenging tasks. Boston & Smith
(2009) concluded that he changes in teachers’ instructional practices throughout their
experiences in ESP have the potential to improve students’ opportunities for learning
mathematics. While there was no measurement in this study to determine the impact of task
selection of student understanding, Boston and Smith (2009) have suggested that increased
involvement and extended engagement with high-level cognitively challenging tasks increase
students’ learning of mathematics (see also Hiebert & Wearne, 1993).
Conceptual understanding as a variable
Taking into consideration the idea of conceptual understanding in mathematics and reflecting
upon some of the studies reviewed in this section, a number of ideas come to light. When
considering conceptual understanding as a variable, it seems to be treated as a dependent
variable. When considering some of the other variables involved in examining conceptual
understanding, some of the independent variables seem to include the teacher, the textbook, tasks
implemented, and pedagogy in general. Many of the studies reviewed thus far have largely been
qualitative, making use of methodologies and techniques including interviews, field notes, and
video/audio transcriptions. What seems to be lacking is more quantitative methodologies when
looking at conceptual understanding. Perhaps conceptual understanding is understood better in a
qualitative light.
Conceptual understanding has been an idea that has existed for many years, although in recent
years, especially after recent calls for mathematics education reform (e.g. NCTM, 2000).
Hiebert & Grouws (2007) offer some directions with regard to future research on teaching and
learning in general. Two ideas that seem to apply toward further research on conceptual
understanding is to build and use theories and to correlate features of teaching with student
28
learning. When considering conceptual understanding with respect to some aspects of teaching,
much of the research has been qualitative. While qualitative methods provide many rich details
and categories, generalizability is an issue. Some of the studies have used some averages, but
there seems to be a lack of statistical significance considered and most of the studies used more
of a counting strategy.
Considerations for my own research
Given that much of research in mathematics is aimed toward developing conceptual
understanding and making connections, I am thinking about examining the effects of applied
mathematics from an interdisciplinary perspective on students' conceptual understanding in
mathematics. In particular, I am interested in how some music theory and composition
fundamentals might be implemented from a mathematical perspective in select topics in the high
school mathematics curriculum. With respect to potential methodology, I am seeking a mixed
methods approach. With respect to qualitative methods, I am considering making observations,
taking field notes, and audio taping of student and teacher discourse. Transcriptions of student
discourse and teacher actions will help to determine the level of connections students make,
which is a major element of conceptual understanding. With respect to quantitative methods, I
am still in the process of thinking about the best analysis, using student test scores as a source of
quantitative measurement. I foresee a potential for an experimental approach, where one similar
class receives the treatment of the music applications where the control class receives no special
treatments.
When considering potential variables, I am looking to have one or two teachers who teach
two of the same level class (e.g. Algebra 2 college prep) in order to control for the teacher
influence but using more than one teacher might provide a better picture of the effects of the
29
treatment. Some other considerations that I have in mind include the size of the classes involved
as well as the size of the student body. The textbook and curriculum sequence will also be taken
into consideration as certain applications will lend themselves to topics nicely. Examining
smaller schools might make it less likely that the teacher teaches two of the same level classes
that I would be looking for. Selection of materials will require some thought as well, as I can't
be guaranteed that all teachers will have a deep background in music although perhaps careful
selection of music theory and composition applications and appropriate teacher
training/professional development will ease matters for the participating teachers.
Closing and summary
There is a strong push in mathematics education for teaching and learning with understanding
(e.g. Hiebert & Grouws, 2007, Bransford et al, 2000, Hiebert, J. & Carpenter, T.P., 1992).
Furthermore, the Connections Standard (NCTM Principles and Standards, 2000) requires
students (and teachers) to experience mathematics as a connected set of ideas both within and
outside of mathematics. Teaching and learning mathematics with understanding has been
considered over the course of almost a century (e.g. Brownell, 1935 as cited by Baroody &
Dowker, 2003; Davis, 1984, Hiebert, J. & Carpenter, T.P., 1992, Bransford et al, 2000, Hiebert
& Grouws, 2007). Given many of the recent Trends in International Mathematics and Science
Studies (TIMSS) as well as the Principles and Standards for School Mathematics (NCTM, 1989,
2000), there is a much more greater call for the teaching and learning of mathematics with
understanding. While procedural and skill efficiency is still a necessary part of teaching and
learning mathematics for understanding, there is definitely a need for a balance between skill
efficiency and conceptual understanding as one seems to inform the other.
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