LEARNERS' DIFFICULTIES WITH QUANTITATIVE UNITS IN
ALGEBRAIC WORD PROBLEMS AND THE TEACHER'S
INTERPRETATION OF THOSE DIFFICULTIES
Abstract
This study examines 8th grade students' coordination of quantitative units arising from word
problems that can be solved via a set of equations that are reducible to a single equation with a
single unknown. Along with Unit-Coordination, Quantitative Unit Conservation also emerges as
a necessary construct in dealing with such problems. We base our analysis within a framework
of quantitative reasoning (Thompson, 1988, 1989, 1993, 1995) and a theory of children’s unitscoordination with different levels of units (Steffe, 1994) that both encompass and are extended by
these two constructs. Our data consist of videotaped classroom lessons, student interviews and
teacher interviews. On-going analyses of these data were conducted during the teaching
sequence. A retrospective analysis, using constant comparison methodology, was then
undertaken during which the classroom video, related student interviews and teacher interviews
were revisited many times in order to generate a thematic analysis. Our results indicate that the
identification and coordination of the units involved in the problem situation are critical aspects
of quantitative reasoning and need to be emphasized in the teaching-learning process. We also
concluded that unit coordination and unit conservation are cognitive prerequisites for
constructing a meaningful algebraic equation when reasoning quantitatively about a situation.
Introduction
The research presented in this paper indicates that the ability to coordinate different units
in a quantitative situation is an important skill for students to develop in order to be successful in
both representing and solving algebraic word problems. Whether this be coordinating different
levels of units in a whole number multiplicative situation (e.g. Steffe, 1994) or in a fraction
situation (e.g. Lamon, 1994; Olive, 1999; Olive and Steffe, 2002; Steffe, 2002) or in dealing with
intensive (e.g., miles per hour) as well as extensive (e.g., number of hours) quantities (Schwartz,
1988) the crucial point is to understand what is being done with the varying quantities in these
situations and how the units involved can be related (Thompson, 1988, 1989, 1993, 1995).
Theoretical Framework
This study is part of Project CoSTAR (Coordinating Students’ and Teachers’ Algebraic
Reasoning)1 that has as its main purpose the coordination of research on students’ understandings
and teachers’ practices and interpretations of students’ actions relative to algebraic reasoning.
This particular study is informed by recent research on students’ understanding of algebraic
symbols (Kieran and Sfard, 1999), and students’ construction and coordination of quantitative
units (Lamon, 1994; Olive, 1999; Smith and Thompson, (in press); Steffe, 1994, 2002;
Thompson, 1988, 1989, 1993, 1995).
The main theoretical framework for this study is based on the construct of quantitative
reasoning. Thompson (1995) states that “Quantitative reasoning is not reasoning about numbers,
it is about reasoning about objects and their measurements (i.e., quantities) and relationships
among quantities” (p. 206). Thompson (1988) defined a quantity as follows: “A quantity is a
measurable quality of something. A magnitude of a quantity is the quantity’s measure in some
unit.” The important distinction that Thompson makes in this definition is between the quality of
something and the magnitude of that same thing. A quantity has to be both named (by its
quality) and measured by some identified unit. We concur with Thompson on this necessity for
forming or identifying a unit by which the named quality may be measured or quantified. We
need to delve further into the nature of these quantities in order to uncover the units associated
with them. An ordering of the form (name, unit) is helpful for the sake of proper coordination.
For instance, the coordination (dime, number of dimes) is not the same as (dime, value of a
dime) or (dimes, value per dime). Smith and Thompson (in press) formalize this “name—unit”
coordination in their representation of quantitative relationships throughout their article. They
“use ovals to represent quantities and place inside those ovals all relevant information about
those quantities—their ‘name,’ their units of measure, and any numerical value or expression
that is given or can be inferred.” (footnote 4 on p. 19)
Schwartz (1988) used the term referent in a way similar to how we are using identified
unit of measure and called such quantities adjectival quantities. (p. 41) He stated that all
quantities have referents and that the “composing of two mathematical quantities to yield a third
derived quantity can take either of two forms, referent preserving composition or referent
transforming composition.” (p. 41) The referent transforming composition, Schwartz claims,
forces us to distinguish between two different kinds of quantity: extensive quantity and intensive
quantity. An extensive quantity can be counted or measured directly, whereas an intensive
quantity is derived from the multiplicative combination of two like or unlike quantities, and is
usually recognized by the use of “per” in its referent unit (e.g., miles per hour, price per pound).
Schwartz (1988) also distinguished the name of the quantity from its referent unit. For instance,
the intensive quantity speed could have referent miles per hour or feet per second.
In word problems involving extensive and intensive quantities, one further step is needed,
beyond coordination of each one of those quantities. We somehow would need to reconcile all
these quantities, each of which can be coordinated in the form (name of quantity, unit of the
quantity). In other words, we not only look at each coordinated quantity separately, but also look
at all these quantities together as a whole. This coherence of the whole requires that we
meaningfully combine each coordinated quantity: A coordination of coordinated quantities. We
refer to this second level of coordination as “quantitative unit conservation”.
Introducing a second level of coordination of quantitative units necessitates a view of
different levels of quantitative units. Behr, Harel, Post, & Lesh, (1994) conducted a conceptual
analysis of different levels of quantitative units. Steffe (1994) provided a psychological analysis
of children’s construction of composite units at different levels of composition (a singleton unit,
a unit of units, and a unit of units of units) involved in multiplicative reasoning. Olive (1999),
Olive and Steffe (2002), and Steffe (2002) extended the idea of multiple levels of units to
children’s reasoning with fractional quantities. We shall show, in the analysis of the coins
problem in this study, that students need to operate with three levels of units in order to
successfully make a coordination of coordinated quantities necessary for quantitative unit
conservation.
Thompson (1988, 1995) provides another way of thinking about this second level of
coordination through his description of quantitative operations and relationships. Thompson
(1995) states: “quantitative operations and numerical operations should not be thought as being
the same.” (p. 212) In his 1988 paper he describes four types of quantitative operations:
combining quantities either additively or multiplicatively, and comparing quantities either
additively or multiplicatively. He goes on to state that:
Complex quantitative reasoning entails relating groups of quantitative mental operations, such as in
forming a multiplicative comparison of an additive comparison and an additive combination (i.e.,
“How many times bigger is this difference than is this combination?”). Quantitative reasoning also
entails reasoning relationally about quantitative structures, entails the constitutive mental operations
for comprehending a quantity situationally, and entails the constitutive mental operations which
allow one to recognize a quantity as one whose value varies or can vary. (p. 165)
We consider this elaboration to be supportive of our construct of quantitative unit
conservation. With this construct, we are opening one more theoretical perspective that covers a
range of mathematical practices associated with solving word problems. Such mathematical
practices include, but are not limited to, taking care of priority of operations, using parentheses
appropriately (in order to combine quantities), and substituting literal expressions for other literal
symbols. All these mathematical practices serve one crucial idea, and that is to maintain the
equality of expressions on both sides of an equation (Chazan and Yerulshalmy, 2003), while
being aware of what's happening on both sides: Things we are adding or subtracting have to be
like terms while those we multiply or divide do not necessarily have to be so. The simplified
expressions on both sides of the equation must be “like-terms” in the sense that they both have
the same “simplified” unit. The simplified unit throughout the process of obtaining equivalent
equations must be conserved.
In this paper we explore the units coordination arising from situations that can be
represented by linear equations involving more than one unknown or variable but that can be
reduced to an equation in a single variable; that is, a system of linear equations that can be solved
by substitution. We consider these situations to be ones that require complex quantitative
reasoning (Thompson, 1988) in that they entail relating groups of quantitative relationships; they
also involve the use of algebraic notation that adds another layer of symbolic complexity to
students’ quantitative reasoning. Students often associate the algebraic symbol with the name of
the quantity rather than its magnitude, as Thompson (1995) pointed out:
When we reasoned symbolically, we needed to remind ourselves continually that W stood for the number of
women and that M stood for the number of men. When students fail to keep in mind that letters represent
numerical values, they will think of an expression like W=8/9M as saying “one woman is eight-ninths of a
man” instead of thinking “the number of women is eight-ninths the number of men.” Also, students will often
read the (equivalent) equation 9W=8M as “There are 9 women for every 8 men” instead of as “9 times the
number of women equals 8 times the number of men.” Students’ thinking of letters as standing for objects is
well researched, and it has pernicious consequences for students’ understanding of algebra... (p. 209)
Several students in this study made similar associations between the letters in an algebraic
expression as standing for objects (names of quantities) rather than numbers (magnitudes of the
quantities). Through our analysis of the classroom discussions, students’ explanations and
responses to interview tasks, along with interviews with the classroom teacher, we have come to
realize that the identification and coordination of the units involved in the problem situation (the
name-unit coordination) are critical aspects of quantitative reasoning that need to be emphasized
in the teaching-learning process.
Context and Methodology
This study took place in an 8th-grade classroom in a rural middle school in the
southeastern United States. The 24 students were between 13 and 14 years old and had been
placed in the algebra class based on their success in 7th-grade mathematics. The students were
racially, socially and economically diverse, with an approximately equal distribution of gender.
All eight class lessons on a unit that focused on writing and solving algebraic equations from
word problems were videotaped using two cameras, one focused on the teacher and the other on
the students. Four students were interviewed twice in pairs (a pair of girls and a pair of boys)
during the three weeks of the study. The classroom teacher was also interviewed twice during
the three weeks. All interviews were videotaped. Excerpts from the classroom videotapes were
used during both student and teacher interviews to initiate discussion of the learning that was
taking place in the classroom. Excerpts from the videotapes of student interviews were also used
in the teacher interviews. The first author conducted all of the interviews.
This paper focuses on problems arising from a particular contextual situation (the Coins
Problem). The data for the Coins Problem were collected during the two class lessons and
subsequent student and teacher interviews that dealt with the following word problem from
UNIT 4 of College Preparatory Mathematics (CPM) Algebra 1, 2nd edition (2002):
Mrs. Speedy keeps coins for paying the toll crossing on her commute to and from work.
She presently has three more dimes than nickels and two fewer quarters than nickels.
The total value of the coins is $5.40. Find the number of each type of coin that she has.
Students were first asked to create a “guess and check” table to find possible solutions to the
problem. They were then challenged to write an equation to represent the problem. This Coins
Problem gave rise to student difficulties that can be explained in terms of unit identification and
coordination.
Analysis Process
Each day the classroom video data from the two cameras were viewed and digitally
mixed using a picture-in-picture technology. A written summary of the lesson with time-stamps
for video reference was created from the mixed video. This written summary also contained
comments about any significant events and screen shots from the video when needed for
clarification or highlight. These written “lesson graphs” were then used to select excerpts from
the classroom video to be used in the student or teacher interviews, and to plan questions and
related problems to pose to the interviewees in an effort to understand how the students (and
teacher) had interpreted the problem and the classroom discussions that followed from different
students’ attempts to address the problem.
CPM ALGEBRA I UNIT 4, Choosing a Phone Plan (CP)
CPM UNIT 4
CLASS
CP 0,1
10/25/04
CP 1,4-8
10/26/04
CP 15,16
10/27/04
Students P&M
Students B&G
TEACHER
10/28/04
10/29/04
CP 17,18
11/01/04
CP 27, 28, 38
11/02/04
CP 45, 40, 48(a),
39(c), 65(b)
11/03/04
11/03/04
11/03/04
11/05/04
CP 92
11/08/04
CP 92
11/09/04
11/10/04
Figure 1: Connections among class lessons, student interviews & teacher interviews
After the end of the three weeks of data collection, the corpus of classroom video data was
reviewed, along with the associated lesson graphs to generate possible themes for a more
detailed analysis. All student and teacher interviews were transcribed from audio files created
from the videotapes of the interviews. A chart of relationships among class lessons, student
interviews and teacher interviews was then created. This chart indicated which class lessons
(including the specific activities from the CPM unit) were used or referenced in which student
and teacher interviews, and which student interviews were used or referenced in the teacher
interviews (see figure 1). A retrospective analysis, using constant comparison methodology, was
then undertaken during which the classroom video, related student interviews and teacher
interviews were revisited many times in order to generate a thematic analysis from which the
results emerged.
Results of the Analysis
The protocols we are presenting below came from important events in class lessons, student
and teacher interviews. By important events, we mean critical instances that are meaningfully
related to our two theoretical constructs (quantitative unit coordination and conservation). For
the coins problem, we started with a detailed description of the situation from the related class
lesson. We then looked for where else the coins problem was used among class videos, student
interviews, and teacher interviews. We revisited the transcripts of these video-episodes many
times, and refined them further. The protocols presented below do not necessarily follow a
chronological order; rather, critical incidents that occurred during class lessons are followed up
through excerpts from both student and teacher interviews before returning to the protocols
excerpted from the class lessons. This sequence of evidence follows the thematic analysis that
emerged from the retrospective analysis of the total set of video data.
The Coins Problem
Mrs. Speedy keeps coins for paying the toll crossing on her commute to and from work. She
presently has three more dimes than nickels and two fewer quarters than nickels. The total value
is $5.40. Find the number of each type of coin she has. (from CPM Algebra 1, UNIT 4, CP-16,
2002)
This problem was introduced during the third class period in Unit 4 on 10/27/04 (see Figure
1). Problem CP-3 in the Unit was very similar, also dealing with combinations of nickels, dimes
and quarters, but the teacher had skipped over that problem in a previous class period, thus
problem CP-16 was the first one of this type that the students had encountered. In this problem
situation, when trying to calculate the total value of all coins, the monetary values of specific
coins are intensive quantities (they are the values per coin) and the numbers of each coin and
total value are extensive quantities. Distinguishing between these two different types of
quantities surfaced as a problem during the classroom discussions. Associating appropriate units
with the different quantities and combining unknown quantities emerged as further problems
during the student interviews.
A major confusion arose during the class lesson on 10/27/04 in naming the quantities in the
situation. Students had chosen the letter N to represent the nickels in the problem, however, it
became apparent from the discussion that, while N stood for the number of nickels for the
teacher and for some of the students, for others it either represented the value of all the nickels
together or just stood for the coin (a nickel). When the teacher, Ms. Jennings2 asked the students
“What are we gonna call dimes?” (immediately after writing “n=nickels” on the classroom
board) some students answered “two N”, and this could be a corroboration that those students
saw N as the value, and not the number of coins under consideration. The following dialogue
between Ms. Jennings and a student, Cathy, taken from the classroom video illustrates the
confusion:
Protocol I: Student's confusion about naming coins (from classroom video on 10/27/04)
Ms. Jennings: We are not done... We are just naming our variables right now. We haven't begun to
make an equation yet. We have to know what we are naming, before we put in an equation.
Cathy: So why can't we just put them all with their first letter? Like N equals nickels, just keep doing,
D dimes, Q quarters.
Ms. Jennings: Let me ask you this question and see if you can solve it: “N plus D plus Q equals 5
dollars and forty cents. How many of each one do I have?”
By challenging Cathy with the statement “N plus D plus Q equals five dollars and forty
cents. How many of each one do I have?” in the above dialogue, Ms. Jennings may have added
to the confusion (over what the letters represented in the situation). Ms. Jennings actually wrote
on the white-board during the lesson: “n=nickels, d=dimes, q=quarters” following Cathy’s
suggestion. In her interview with the first author a few days later, Ms. Jennings commented that
students name a coin by its first letter to make it easy to identify in the equation but that later
confuses them.
Protocol II: Students' confusion (from Teacher interview on 11/03/04)
Ms. Jennings: But I also know that they will not be able to solve that equation with N and D and Q
because they have no value at that point.
Interviewer: Okay.
Ms. Jennings: And even though they know that they’re adding 3 different kinds of coins together to
make an amount of money, without consolidating that variable in some way, they won’t figure
out the number of each one that they need to find.
The source of this confusion partly comes from what Ms. Jennings had written on the board:
“n=nickels, d=dimes, q=quarters”. Ms. Jennings’ pedagogical approach in the classroom is to
accept students’ suggestions without evaluation from her, with the intent of having her students
evaluate and discuss what is said during the lesson. This approach leads to rich discussions and
productive arguments among the students but can also leave some students confused as to what
is mathematically acceptable and what is not. Ms. Jennings’ introductory question “What are we
gonna call nickels, dimes, quarters?” could have been misleading (as she did not specifically say
number of nickels, dimes and quarters). In the interview with students Pam and Maria, on the
morning following the classroom lesson, the interviewer (first author) showed the classroom
video episode from Protocol I above. While Pam and Maria appear to have understood the
situation and did not appear to have been mislead by the confusion evidenced in the classroom
episode, this was not the case with the two boys who were interviewed later in the week. We
begin with Pam and Maria’s interview as their responses to the interviewer’s questions framed
the tasks for the later interview with the two boys. Protocol III begins after the interviewer has
shown Pam and Maria the video of the classroom episode:
Protocol III: Students' interpretation of Cathy's remark (from student interview on 10/28/04)
Interviewer: Okay. What do you think Cathy means by N for nickels, D for dimes, and Q for
quarters?
Pam: That represents how much you have, that’s what she’s talking about.
Interviewer: How much?
Maria: No, she was thinking about the value of each one.
Pam: The value, yeah.
Interviewer: Oh, rather than what?
Maria: The number of coins.
Later in the same interview, Maria distinguished the differences among three different types
of quantities: the value of a coin, the number of that coin and the total value of all the coins of
that type. She was then able to combine her total values for each type of coin to produce the
total of all coins ($5.40).
Protocol IV: Unitizing quantities (from student interview on 10/28/04)
Maria: Yes. Okay. Okay, this is the value of the nickel, so it would be…
Interviewer: What is “this?”
Maria: .05. And, in any number, let’s say 5, so it’ll be .05 times 5 will give the amount of nickels and
you do…
Interviewer: The amount of nickels?
Maria: No, the value of the whole nickels that you have.
Interviewer: Does that make sense?
Maria: Yeah. And, then you do the same for D and Q and it comes out to $5.40.
Interviewer: Do the same for D and Q for me.
Maria: Okay, so it will be, let’s say, 10 dimes. So, it’ll be 10 times .1 will give you the value and the
same for Q. If you do times any number, so Q… the letters mean any number you can think
of.
Interviewer: Well, what in the terms of the problem what do those letters stand for?
Maria: The number of coins you need to get $5.40.
Maria’s statement “the letters mean any number you can think of” is evidence that she knows
she is dealing with the quantity “number of a coin”. Moreover, she separately calculates the total
value for each coin, and this could be seen as her coordination of units before adding them
together. In fact, during this interview, by explaining this unit coordination, Maria makes sure
that the “terms” she is adding are like terms, and then she concludes the addition and writes the
first equation 0.05n+0.1d+0.25q=5.40; and after substitution, the second equation
0.05n+0.1(n+3)+0.25(n-2)=5.40. Both Maria and Pam correctly write and explain these
equations and they agree it makes sense, as the following protocol indicates:
Protocol V: Student's Explanation of How the Second Equation comes from the First One (from
interview on 10/28/04)
Interviewer: Yes. Now how does the second one come from that first one.
Pam: Because it said that we had 3 more dimes than nickels, right? So, that’s why it says N plus 3.
Interviewer: And, what’s the N plus 3 instead of?
Pam: The…
Maria: D.
Pam: The T?
Interviewer: D.
Maria: D.
Pam: D, oh. The D. That’s right ‘cause it gets you the dime… the value of the dime and how much
coins and how much dimes we have.
Interviewer: How many dimes we have. Okay.
Pam: And, then nickels, the N minus 2 gives you how much quarters you have.
Interviewer: Why?
Pam: Because it said you have 2 fewer quarters than you do nickels.
Interviewer: Okay. Does this make sense now?
Maria: Mm-hmm.
Interviewer: Is that how you would explain it, Maria? Okay. Okay, good
During the process of obtaining her equivalent form 0.05n+0.1(n+3)+0.25(n-2)=5.40, Maria
performed several notable mathematical practices. First was her correct substitution of
expressions for literal symbols, as in this case, n+3 for d, and n-2 for q. Second, her placing of
parentheses around those expressions appropriately. This way of writing the expressions had a
purpose. Each product on the left hand side represented a composed quantity, and had to possess
a unit inherent in its structure. Moreover, each product, having the same unit: value, was
connected meaningfully via the addition operation. This was when she identified these products
as monetary values. In this whole process of obtaining the equivalent form, there is another
meaningful mathematical practice, which we call quantitative unit conservation: Not only did
each product on the left hand side of the equation have the same unit as the quantity on the right
hand side, but their combination in the form of a sum – they could be combined because they
were like terms – had the same unit as the quantity on the right hand side of the equation. In this
way, there is this notion of coherence between each term on the left with the term on the right, as
well as the coherence of the combined expression on the left with the expression on the right. In
Thompson’s (1988, 1993) terms, Maria and Pam had constructed a quantitative structure – a
network of quantitative relationships that were embodied in this second equation.
We can also analyze Maria’s solution in terms of different levels of units: a single coin is the
first level, the value of the single coin and the number of those single coins are units established
at a second level (a composite unit of units), whereas establishing the value of all the coins
requires a third level of units (a composed unit of units of units). The ease with which Maria
established the second equation, with correct parentheses, indicates that she had these three
levels of units available to her prior to operating.
Even though these two students eventually wrote both equations correctly, Ms. Jennings had
some doubts about other students’ realization that they would not be able to solve the first
equation in terms of the three different variables:
Protocol VI: Ms. Jennings’ Comments on Students' Setting up the Equations (from Teacher
Interview on 11/03/04)
Ms. Jennings: I know that they understand what the N, D and Q stand for and that they do know the
value of each coin, obviously, but I’m not, I’m not sure how they will reach a conclusion that
they can’t solve the first equation. I don’t know how they will decide they can’t do it.
Interviewer: Yeah.
Ms. Jennings: But if they can’t do it, then I guess my job is to re-route them into naming a dime in
terms of a nickel by the information from the problem. I mean I really liked the first
equation.
Interviewer: Which?
Ms. Jennings: The first one that both of them wrote.
Interviewer: Oh. In terms of N, D and Q?
Ms. Jennings: In terms of just making sense of how many you have and the value of each coin and
then turning it more into each variable in terms of the first ‘cause the N + 3 and the N – 2
were implied in the problem's information, but they weren’t getting that the first time or the
first couple of times.
In this last remark, Ms. Jennings is referring to many students “not getting that” during the
class lesson rather than Pam and Maria during the interview. Her reflection on her role as
teacher indicates that she is aware of students’ needs to first represent the situation in terms of
nickels, dimes and quarters and it is then up to her to help them use the relations among the
numbers of the different coins to create an equation in one unknown that they can solve.
In contrast to Maria’s success in setting up both equations in Protocol V above, the two boys
who were interviewed (Ben and Greg), while able to make the unit coordination to produce the
first equation similarly to Maria, were not able to produce the second equation through
substitution. In searching for a possible explanation for this behavior, we found that in class, in
the process of naming quantities, when the teacher challenged students by her statement “I have
three more dimes than nickels”, Greg had commented “n plus 6”. In the interview, when he was
questioned about this, he said “I was thinking about the value... because it takes six nickels to get
three dimes”. Therefore, in class, it appears that Greg was aware of the two units concerning the
coin – its number and its value – his answer, however, did not reflect an appropriate coordination
between these two units. Greg’s partner, Ben, had given a very clear explanation in class for
coordinating the value of each coin with the number of each coin. The interviewer asked him to
repeat what he had said in class. Protocol VII begins with Ben’s explanation.
Protocol VII: Ben's Explanation for Setting up the First Equation (from student interview on
10/29/04)
Ben: Okay. Before you find the answer, I said whatever the values are for the numbers, like, a nickel
is .05 and you have to multiply that times the number of nickels and that gives you the value
and then the dimes, .1, I think, yeah, .1 times the number of dimes that you have and that’ll
give you the value of that one and then the quarters… then .25 (inaudible) you have to
multiply that times the number of quarters you have.
Interviewer: And, what will that give you?
Ben: It should give you $5.40.
Ben appears to have reasoned with unit coordination, like Maria. He knew that when he
multiplied the value of a coin by the number of that coin, the result was a monetary value.
Following this explanation, the interviewer asked Ben to write down an equation based on his
explanation, using the symbols N for number of nickels, D for number of dimes and Q for
number of quarters. Ben eventually ended up with the following equation:
(.05N)+(.1D)+(.25Q)=$5.40. The interviewer then asked the boys what they knew about D and
Q. Greg responded that you have to have the nickels to find D and Q. The interviewer asked
him to write down what D equals in terms of “that number of nickels” (pointing to the N). Greg
wrote the expression: (.05n+3). Protocol VIII picks up at this point in the interview:
Protocol VIII: Finding D in terms of N (from student interview on 10/29/04)
Interviewer: Do you agree, Ben?
Ben:
Yeah.
Interviewer: That’s… what about… what is that about the dimes? Is that the value of the dimes, the
number of the dimes, the picture of the dimes?
Ben:
I think it’s the value.
Interviewer: The color of the dimes?
Ben: I think it’s the value.
Interviewer: You think that’s the value of the dimes.
Ben: Because the… ‘cause the… .05 is 5 cents and that’s a nickel and then the N is the number of
nickels, so that gives you the value, plus 3 should give you another value.
Ben's interpretation of Greg’s expression suggests that he was aware of the different types of
quantities involved (value and number of coins) but may have had problems coordinating the
quantitative units meaningfully. Ben was aware that 0.05n and 3 both must have the same unit,
value, in order to be added. There is also the possibility that Ben interpreted the statement in the
problem “three more dimes than nickels” to mean that the value of the dimes was three more
than the value of the nickels. This would explain the acceptance of 0.05 (5 cents) as the unit
value used to find the value of the dimes.
In her interview Ms. Jennings interpreted Ben’s problem as stemming from the use of an
unknown value for N in the expression, as indicated in the following protocol:
Protocol IX: Ms. Jennings’ Interpretation of Ben's Explanation (from Teacher Interview on
11/03/04)
Ms. Jennings: I think he really knows what he means, but he’s really having trouble with interpreting
how to write it as an expression. I think, I think that if he knew a number of nickels and he
knows the order of operations, if he knew the number of nickels and multiplied by the 5 cents
and then added a 3, he would see his mistake, but I don’t think he would do it without
knowing a number of coins.
Interviewer: Okay. So what does that imply about his understanding of the role of N there?
Ms. Jennings: I don’t think he sees N + 3 as one number. I think he only sees the N as a number and
whatever it is, you’re gonna add 3 to it.
In the continuation of the student interview, the interviewer also believed that Ben would
realize his mistake if he was to work with an actual value for N. He suggested a little experiment.
He asked the two boys to assume that they have 2 nickels, and evaluate their conjecture for this
value of N. The students accepted that if they have 2 nickels, they must have 5 dimes and the
value of 5 dimes is 50 cents. When they were asked to evaluate the expression 0.05n+3 for n=2,
they realized that it would exceed three whole dollars, and realized that their conjecture must be
false. The interviewer then shifted the focus back to what D was in terms of N. Protocol X
begins at this point in the interview:
Protocol X: Creating Expressions for D and Q (from student interview on 10/29/04)
Interviewer: But, I don’t want the value, I want D. What’s D in terms of N?
Greg: N plus 3.
Interviewer: Can you write that down? Put D equals. [Greg writes: D=N+3] Now, what do you
know about Q?
Ben:
Q is 2 less then N.
Interviewer: Yeah. So, write down the equation for me.
Ben:
N minus 2. [Greg writes Q=N-2]
Interviewer: Okay. So, now can you rewrite this equation [pointing to the first equation: (0.05 n) +
(0.1 d) + (0.25 q) = 5.40] just using N? Okay, can you do that for me?
Greg: To get the value?
Greg started by writing .05n, then put a plus sign, and then put n+3 and stopped. He hesitated
for a while in this step, and he asked himself (audibly) “How can you get the value?” He erased
the n+3, and replaced it with .1 n + 3, without parentheses. He did the same thing for the last
expression and wrote .25 n – 2 . His complete expression for the second equation was
0.05n+0.1n+3+0.25n–2=5.40. Upon the interviewer's question whether he agrees, Ben said that
0.05n was correct. They both hesitated for the remaining terms on the left hand side. They tried
to compare this expression, namely their second equation with their first equation (0.05 n) + (0.1
d) + (0.25 q) = 5.40 (note that Ben had placed parentheses around each of the expressions that
indicated the value of each set of coins in this first equation). Protocol XI continues from this
point:
Protocol XI: Finding the Second Equation by Substituting for D and Q (from student interview
on 10/29/04)
Interviewer: Let’s look at this equation that you all agreed was correct [points to first equation].
Ben:
5 cents times the number of nickels plus 10 cents…
Greg: (inaudible), but you’re trying to get D equals N plus 3?
Interviewer: Yeah. Read what this says.
Ben:
0.1 times D or times the number of dimes.
Interviewer: Keep that. 0.1 times the number of dimes. Is that what this says [pointing to the second
part of Greg’s equation: 0.1n+3]?
Ben:
This says, .1 times the number of nickels. So…
Greg: Plus 3.
Interviewer: Plus 3.
Greg: It gives you the number of dimes.
Ben:
Oh, if you say it like that then that means the nickels are the same as … the two N’s are the
same?
Interviewer: Yeah, the two N’s are going to be the same. Once you’ve picked the number for the
number of nickels that stays the same throughout the equation. Okay. This says, .1 times the
number of dimes [pointing to (.10D) in the first equation]. Does this say .1 times the number
of dimes [pointing to.10n+3 in the second equation]? What did it say?
Ben:
.1 times the number of nickels…
Interviewer: Plus 3. [Greg draws parentheses around the (n+3)] Okay… what about the other
one…[Greg draws parentheses around the (n-2)] Now, does it say, .1 times the number of
dimes?
Greg: Yes.
Ben:
Yeah.
Interviewer: What does this say [pointing to .25(n-2)]?
Greg: That says, .25 times (overtalking)
Ben:
Times the number of nickels.
Greg: Quarters.
Ben:
Yeah, quarters.
The big surprise in the above protocol is Ben’s question about the nickels – the two N’s –
having to be the same in the second equation. His question and surprise indicate that Ben (like
several other students in the class) had not realized what we, as adults, take for granted when
reading an algebraic expression: that the same literal symbols stand for the same values
throughout the expression.
Greg’s question after writing n+3: “How can you get the value?” indicates that he was
making a distinction between the two different quantitative units (number of coins and value of
coins). His lack of use of parentheses when writing the expression 0.1n+3 caused a perturbation
for Ben, whereas Greg knew that the n+3 gave the number of coins. Greg eventually realized
that he needed to add parentheses to distinguish this quantity (number of coins) from the value of
the coin (0.1), and produced the second equation: .05n+.10(n+3)+.25(n-2)=5.40. The lack of
appropriate parentheses in Greg’s first expression for D (.05n+3) and in his first attempt at the
second equation led to Ben and Greg’s difficulties in coordinating units within their quantities
and seeing the coherence of the whole equation, namely that each product on the left hand side
must be consistent in units with the term on the right hand side, and that they could then be
added because they were in terms of the same unit: monetary value.
In her interview, Ms. Jennings commented about the boys' mistakes concerning
parentheses:
Protocol XII: Ms. Jennings’ comment on boys' omitting parentheses (from Teacher Interview on
11/03/04)
Ms. Jennings: I wonder how they know that parentheses make that difference?
Interviewer: It helps unitize that expression (N + 3) is my interpretation.
Ms. Jennings: And I know that and you know that, but I’m not sure how they realize that.
Interviewer: Yeah. For the girls, it was there immediately. For Pam and Maria, they wrote their
equations with the parentheses.
Ms. Jennings: And this took quite a while.
In our opinion, the use of parentheses is one demonstration of how much a student is able to
understand and to see the relation between the quantities and their units, and the different levels
of units that are available to them prior to operating. At the beginning, both in the classroom
video and students’ interviews, students hesitated a lot about whether they were dealing with
values or numbers of coins. For the time being, this problem has not been resolved entirely, and
it seems to us that the boys' omitting parentheses is in a way related to their misunderstanding
and misinterpreting the quantities arising from the problem. In contrast to the girls, Ben and
Greg were not able to construct a meaningful quantitative structure using the literal symbols in
these equations. They were able to work with units at the second level (number of coins and
value of a coin), but did not have available a third level of units that would enable them to
envision the quantitative structure required to meaningfully combine and find the value of all the
coins. They can construct this third level structure through their activity but it was not available
to them prior to activity (as it was for Maria). As stated in the above protocol, the use of
parentheses helped the boys to unitize the unknown quantities (numbers of each coin) so that
they could then combine them with the known quantities (values of each coin) to produce a
value, but this only required operating with two levels of units at one time. The third level of unit
(value of combined coins) only emerged after their struggles with these two levels of units.
The teacher interview on 11/03/04 started with asking Ms. Jennings her perception of what
was going on with the students. Her initial comments are worth noting:
Well, my first perception is probably that I jumped in too fast and that they weren’t ready to
think about three different variables in terms of one. And, also, the fact that with coins we
have decimals to play with didn’t make the problem any easier for them to think about. Even
though they know what value is represented by coins, to multiply that value to a variable that
they don’t understand yet was a leap.
Ms. Jennings recognizes the complexity in the coins problem but focuses on there being
more than one “variable” and the use of decimals rather than the difficulty we perceived as
distinguishing between names of coins and the different quantitative units, and levels of units.
Throughout the class lesson Ms. Jennings used the name of the coin, (nickel, dime, quarter) to
stand for the number of coins. There are several instances from the interview where this issue
becomes apparent. For instance, her comments “I guess my job is to re-route them into naming a
dime in terms of a nickel by the information from the problem.” could imply that for the teacher,
a dime is a variable that can be expressed in terms of the other variable, nickel. She probably
used the names of coins assuming that number of the coins was to be understood. We believe
that this practice became ambiguous for students; they were unsure as to whether the number or
the value of a dime was the referent, when using its name (a dime). In fact, at the beginning of
the class session, we believe that it was this communication problem that lead students in an
unproductive direction.
Ms. Jennings commented about the boys' construction and confusion over the expression
(0.05n+3); she said that Ben really knew what he meant, but he was having trouble with how to
write it as an expression. She thought that if he knew the number of nickels and multiplied by the
5 cents and then added a 3, he would see his mistake. Ms. Jennings commented, however, that
Ben would not realize his error without knowing a specific number of coins. In other words,
when focused on the nature of these coins, Ms. Jennings realized that the coin must be associated
with the units inherent in its structure. When questioned about Ben's understanding of the role of
n in (0.05n+3), she said she does not think Ben sees n + 3 as one number. She added “I think he
only sees the n as a number and whatever it is, you’re gonna add 3 to it”. Ms. Jennings is aware
of students’ difficulties with interpreting algebraic expressions as a “process” to be carried out
rather than as an “object” on which to operate (Tall et al., 2001). In another instance from her
interview, Ms. Jennings was asked to compare the behavior of the boys with that of Maria. She
commented that Maria understood that the “n + 3” was an expression meaning d and that Maria
had an understanding of what an expression means: just to rename one thing by another name.
The boys, however, did not really have that understanding. According to Ms. Jennings, Ben and
Greg understood how to name the new unknown quantity, but they didn’t know how to use it to
coordinate an amount of money with an amount of coins. In other words, they could not make
an appropriate “name-unit” coordination.
Conclusions & Discussion
With this study, we have attempted to extend Thompson’s (1988, 1989, 1993, 1995)
theoretical framework that encompasses a way to look at word problems that can be solved via a
set of equations that are reducible to a single equation with a single unknown. We claim that
through this lens, one could interpret all the variables arising from word problems as not simply
ordinary quantities named in the problem, but, rather, as mathematical objects with names,
values and associated units. We tried to rely on students' judgments essentially on this
coordination of different quantities’ names and their units. Hence the theoretical construct
“quantitative unit coordination” is suggested as the main extension to Thompson’s theoretical
model of quantitative reasoning that we want to bring forth through this study.
In applying a quantitative unit coordination construct to the coins problem, a coin was not to
be understood simply as its name (e.g. “dime”). Rather we needed to associate or coordinate a
coin with a unit present in its monetary nature. The Coins Problem helped us focus on each coin
in the problem as an object with a name and the unit associated with the specific coin. A dime is
not a quantity; dime is only the name of the coin stated in the problem. Once a dime is associated
with its value, for instance, that's when a quantity is born. Therefore, with this quantitative unit
coordination, “value of a dime” or “value per dime” becomes an intensive quantity. Similarly, as
our analysis also shows, neither n, nor d, nor q stands for the names of the coins: nickel, dime,
and quarter, respectively. It was essential to coordinate once again the names of these coins with
their “number”, hence to make of them extensive quantities with referent units “number of
nickels”, “number of dimes”, and “number of quarters”. We needed not only to coordinate each
mathematical object with its name and unit, but also to do a coordination of the big picture that
consisted of six different values (three of which were unknown) enabling us to write the linear
equation 0.05n+0.1d+0.25q=5.4. Such complex coordinations are necessary to form what
Thompson (1993) calls a quantitative structure – a network of quantitative relationships.
We relied on evidence of students' judgments concerning this coordination of the quantitative
referents and their units. Maria’s successful coordination of the quantitative referent and its unit,
and the difficulties that Ben and Greg experienced in interpreting the various quantitative
referents and combining them with appropriate units to produce quantities that conserved units
within an expression (the quantitative structure), support our theoretical conjectures. From an
analysis of units point of view (Behr, Harel, Post and Lesh, 1994; Olive, 1999; Olive & Steffe,
2002; Steffe, 1994, 2002) the boys appeared to be limited to operating with only two levels of
units (singletons and a composite unit of units), whereas the girls appeared to be able to operate
with three levels of units (a composite of composite units). In order to construct a network of
quantitative relationships, the students need to be able to form a composite unit of the separate
quantitative relationships, each of which is a composite unit (a coordination of different unit
quantities). Thus, the ability to work with three levels of units is necessary to form a quantitative
structure of this complexity.
Another key issue arising from the above analysis is the “consistency” of quantities on the
left and right hand sides of an equation. In the coins problem and various other problems,
students looked for “likeness” of terms on each side of an equation. This desire for consistency
emphasizes another crucial issue, namely the conservation of quantities. The reduced equation
for the coins problem 0.05n+0.1d+0.25q=5.4 had to have a consistency, therefore students
wanted to look at this equation as representing “value” on each side. As our analysis shows,
transforming this first equation in three unknowns to the second equation in only one unknown,
0.05n+0.1(n+3)+0.25(n-2)=5.4 was not very easy for the students. Some students had trouble
with parentheses and they overcame this difficulty by relying on quantitative unit conservation:
They wanted to have “value” on each side. Therefore here, we hypothesize that quantitative unit
coordination necessitates quantitative unit conservation.
In Thompson’s 1988 study he found cognitive obstacles to students’ quantitative reasoning
that he believed constituted obstacles to algebraic reasoning. One major obstacle was that
students’ “failure to distinguish between a quantity and its measure hindered their ability to
explicate relationships.” (p. 168) Another major obstacle was that “Multiplicative quantities of
any sort (products, ratios, rates) were commonly mis-identified or given an inappropriate unit.”
(p. 168) Our study suggests that unit coordination and unit conservation are necessary constructs
for overcoming these cognitive obstacles when reasoning quantitatively about a situation. Smith
and Thompson (in press) also emphasize a necessity to coordinate between two levels of
reasoning:
Quantities that result from quantitative operations exist in two different senses, as quantities in their own
right and as relationships between the two quantities. It can be conceptually demanding to reason and
communicate about such quantities because we must distinguish and coordinate these two senses, and,
when necessary, shift between them. (p. 23)
This necessity to “shift between them” is what we have referred to as our second level of
coordination: a coordination of coordinated units that has as its goal unit conservation. The
current study suggests that such a coordination, however, requires operating a priori with three
levels of units.
The particular teacher in this study had developed a pedagogy based on constructivist
principles of how students learn. It was her intension to have students discuss and debate their
own interpretations of a particular problem. Her awareness and interpretation of her students’
difficulties in solving the coins problem, led her to question her role in the learning process and
helped her to realize that there are appropriate times when she needs to lead students in a more
productive direction, and appropriate times to make productive use of a student’s (insightful)
interpretation. While realizing that her use of her students’ convenient but ambiguous naming of
quantities (without clarification) could contribute to her students’ confusions, she also became
aware that her goal to have students simplify algebraic expressions could lead to a miss-match
(for her students) between the original situation and the simplified equation. This rush to
simplify equations can be understood when the goal throughout solving quantitative problems
arithmetically has been to get an answer. Teachers’ and students’ goals and processes need to
shift, however, for the development of quantity-based algebraic reasoning. As Thompson (1989)
has suggested, the shift in goals is from the goal of “getting an answer” to the goal of “laying out
a pattern of reasoning.” The shift in cognitive processes is from immediate evaluation of
expressions to the postponement of calculations so that one constructs numeric expressions that
capture a “history” of a quantity’s value. These shifts in goals and processes amount to students
developing the intention of constructing “numeric formulas” for calculating the values (or
magnitudes) of quantities. We conclude from the results of this study that unit coordination and
unit conservation are key aspects of such numeric formulas, and their construction requires
operating with three levels of units.
NOTES:
1.
CoSTAR is supported by a grant from the National Science Foundation, grant # REC 0231879. The
opinions expressed in this paper are those of the authors and do not necessarily reflect the views of NSF.
2.
All names are pseudonyms.
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