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

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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

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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

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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

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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

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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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