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Running Head: THREE LEVELS OF UNITS: NECESSARY, BUT INSUFFICIENT 1

Three Levels of Units: Necessary for Intensive Quantity, but Insufficient

David R. Liss II

The University of Georgia

THREE LEVELS OF UNITS: NECESSARY, BUT INSUFFICIENT 2

Abstract

Many forms of advanced quantitative reasoning require constructing and operating with

increasingly complex unit structures. In particular, the ability to take three levels of units, which

is a composite unit containing a sequence of units of units, as given in operating is necessary for

many fraction operations and forms of multiplicative reasoning with quantities (Hackenberg,

2010, in press; Steffe, Liss II, & Lee, in press; Steffe & Olive, 2010). However, while the

availability of three levels of units makes several powerful ways of reasoning possible for

students, significant differences were observed in how two students constructed advanced

quantitative operations which leveraged their three levels of units. Interview data will be

presented to demonstrate these differences. Comparisons of the students’ activity revealed three

factors which impacted the students’ abilities to reason with proportional relationships and

intensive quantities: a) whether or not the students had constructed a reversible distributive

partitioning scheme; b) the degree to which the students could operate hypothetically and enact

their quantitative operations mentally; and c) the range of situations which triggered the students’

available ways of operating.


Constructing increasingly complex unit structures is a critical factor in students’

mathematical development (Steffe & Olive, 2010). In particular, the ability to coordinate three

levels of units signifies a conceptual milestone which makes new ways of reasoning possible. In

terms of fractional reasoning, three levels of units are necessary for constructing reversible

splitting, distributive partitioning, fraction composition, and iterative fraction schemes (Steffe &

Olive, 2010). These are all required to produce fractions as multiplicative objects in the sense

described by Thompson & Saldanha (2003). Further, constructing a proportionality scheme also

requires reasoning with three levels of units (Steffe, Liss II, & Lee, in press). Thus, the types of

reasoning required for making sense of algebra, such as engaging in covariational reasoning

(Carlson, Jacobs, Coe, Larsen, & Hsu, 2002) and reciprocal reasoning with inverse relationships

(Hackenberg, 2010, in press), rest in part upon structuring quantities using three levels of units.

Although crucial, the construction of three levels of units is far from the culminating

point for students’ mathematical development. The myriad of ways of operating that become

possible for students who have constructed three levels of units should not be taken as givens.

Rather, new possibilities enter students’ zones of potential construction (Steffe & Thompson,

2000) as they learn to reason with three levels of units in increasingly sophisticated manners.

Yet, the actual construction of these new ways of operating varies among students in significant

ways (Steffe & Olive, 2010). This report explores these variations and addresses the following:

1. How do students that have abstracted three levels of units differ in their efforts to

construct schemes and operations for reasoning with quantities?

2. How do these variations impact students’ abilities to construct intensive quantity?

Theoretical Perspectives

I regard mathematics in terms of the mathematics of students (Steffe, 2007) and attempt


to use conceptual schemes and operations (von Glasersfeld, 1995) to explain students’ ways of

operating. In considering quantitative reasoning, this work aligns with Thompson’s (1994)

definition that a quantity is a scheme consisting of an object concept, a measureable property of

that object concept, and an appropriate unit and process for assigning a numeric value to the

property. He continued to define quantitative operations as mental operations which account for

the construction of new quantities in relation to other, already conceived quantities.

Further, quantities can be separated into two types — extensive and intensive (Jahnke,

1983; Schwartz, 1988). Length, mass, and height are examples of the former while density and

speed are examples of the latter. Additionally, extensive quantities combine additively while

intensive quantities do not. For instance, combining two piles of carbon changes the mass of the

pile while the density of the pile remains constant. Given this distinction, the data included in

this report is part of a larger study which explored the quantitative operations which students use

in constructing and operating with proportional relationships and intensive quantities.

This report incorporates the concept of levels of units which involves the types of unit

structures that one can take as given. Students that reason with two levels of units, a unit of units,

take composite units as given in operating. In contrast, students who have interiorized three

levels of units have constructed a composite unit containing a sequence of units of units

(Hackenberg, 2010; Steffe & Olive, 2010). Thus, to a student operating with two levels of units,

seven represents a unit which also contains a sequence of seven individual unit items. However,

a student operating with three levels of units can assimilate a composite unit of 28 as a sequence

of seven composite units, each containing four individual unit items. This report focuses on

investigating the ways in which students leverage this ability to coordinate three levels of units to

construct new schemes and operations for reasoning with quantities.


Methods

A teaching experiment (Steffe & Thompson, 2000) was conducted with six algebra

students during 2013–2014 at a rural high school in the southeastern US. Participants completed

an initial interview with a pre-determined set of tasks to allow the researchers to specify their

partitioning operations and levels of units at the beginning of the study. This report incorporates

data from two students who reasoned with three levels of units and demonstrated use of equi-

partitioning, distributive partitioning, and recursive partitioning (Steffe & Olive, 2010). Jill

participated in 13, 25-minute teaching sessions while Jack participated in 22 teaching sessions.

Each session was videotaped using two views and all written work was collected. A basic goal of

the experiment was to engender specific advancements in the students’ extensive quantitative

schemes and operations as basis for supporting their construction of intensive quantities.

During the experiment researchers engaged in on-going analysis of the interactions to

build experiential models of the students’ ways of reasoning. These models served as the source

of hypotheses tested in future teaching sessions. Retrospective analysis involved re-analyzing the

completed teaching sessions in order to refine these models of the students’ thinking and explore

the implications of differences between the students (Steffe & Thompson, 2000).

Results and Analysis

While exploring quantitative relationships, several differences emerged in the ways that

the students reasoned with three levels of units. Data presented in this report will exemplify these

differences in the schemes and operations the students constructed for reasoning with quantities.

Differences in the Students’ Distributive Partitioning Schemes

We observed persistent differences in Jack and Jill’s distributive partitioning schemes. To

get a sense of the distributive reasoning characteristic of Jill’s activity, consider the following.


Protocol I. The affordances of Jill’s distributive reasoning.

I: We’ve got three people in the room here. Let’s pretend that these are our three

plates. So this would be my plate, that would be June’s plate, and this one could be

your plate. And we want to share these two Twix bars among three people.

Jill: Um hmm.

I: So how would you share those?

Jill: You would half that into threes [points to the first bar] and half that into threes

[points to the second bar] so everybody would get like [taps finger on the plates in

succession]…kind of like two-thirds. Everybody would get like two-thirds.

We see in this protocol that Jill was able to construct a fair share for each person by distributing

the partitioning activity across both of the candy bars. Significantly, Jill was also able to

structure the results of her partitioning activity into three equal shares and interpret this result in

terms of one bar to decide that there would be two-thirds of a candy bar for each person.

I claim that the ability to interpret one person’s share with respect to the unit of one

candy bar in this fashion requires the use of three levels of units. Since three levels of units

allows one to take a sequence of composite units as given, Jill was able to treat the two bars, and

the three pieces in each, as identical. She also transformed all the candy into a composite unit

containing three shares, each with two pieces. She did this while maintaining an awareness of the

original structure to conclude that each share was two-thirds of a candy bar. Thus, Jill’s three

levels of units supported the activity of her distributive partitioning scheme in several ways.

However, constraints in Jill’s distributive partitioning scheme became apparent when she

attempted to interpret one share in terms of all of the candy. Immediately prior to the following

protocol, Jill found a fair share of two different sized cakes by mentally partitioning each cake

into three parts and describing the fair share as one piece from each cake.

Protocol II. The constraints of Jill’s distributive reasoning.

I: How much of the cake would one person get?


Jill: Two-thirds.

I: Why do you say…

Jill: Well, two-sixths cause all together it’s six [Jill moves her hand across the cakes to

pantomime cutting each cake into three pieces]…A person gets one of this

[pointing to the larger cake] and one of this [pointing to the small cake], so it’s six.

…

[The interviewer and Jill discuss whether or not each piece represents one-sixth of

the amount of cake. Jill maintains that each piece is one-sixth because it is one out

of the six pieces.]

…

I: [Cuts off one piece of each cake and places them on a plate to represent the share

that Jill had described.] How many people could get a plate like this?

Jill: Three.

I: So, then what amount of the cake would I be holding?

Jill: Two-sixths…or…one-third.

I: Why?

Jill: It’s simplified.

I: What about the one-third – why does that make sense?

Jill: It’s just simplifying two-sixths.

I: Okay. Would it make sense to think of this [referring to one share] as one-third?

Jill: Ummm…yeah. It could be.

I: Besides, besides simplifying. Why, why does it make sense to you?

Jill: I don’t know. I just simplify it. That’s the only way it made sense to me. I was like,

“Maybe that sounds better.” I don’t know.

I: How many people could get plates like that? [The interviewer places the plate with

one person’s share in front of Jill.]

Jill: Two…wait three. Sorry. Cause you get another half of this and another half of this

[Jill indicates cutting up the remaining cake]. So yeah.

I: So you have three plates just like this [pointing to the one share that had been cut].

Jill: Um hmm.

I: So this would be how much then of the whole cake?


Jill: …Two-sixths.

I: Two-sixths.

Jill: Yeah.

I: Okay. Two-sixths of all the pieces.

Jill: Yes.

Throughout this protocol Jill maintained her view that one share is two-sixths of all the cake.

However, from an observer’s perspective this is problematic in that the pieces were different

sizes and neither represented one-sixth of the entire amount of cake. Yet, Jill did not see this as a

concern since she focused on the number of pieces of cake rather than the size of each piece in

relation to the total amount of cake. Thus, Jill derived the meaning for her fractional comparisons

from the part-whole relationships amongst the number of pieces.

In addition, Jill’s responses indicate that she had yet to reorganize her quantitative

operations in such a way that would allow her to interpret one share as one-third of all the cake.

For example, when explaining why it made sense to think of a share as one-third of all the cake

Jill stated, “I just simplify it. That’s the only what it made sense to me.” Thus, while she could

use a procedure for reducing fractions, Jill was unable to enact quantitative operations that would

allow her to justify that one-third of each cake is also one-third of all the cake. Jill’s ways of

operating in these protocols are consistent with the construction of a distributive partitioning

scheme (DPS) and were characteristic of her distributive reasoning throughout the study.

In contrast, Jack constructed quantitative operations which alleviated these constraints. In

the following protocol Jack is reasoning about finding one-third of two identical cakes while

using the JavaBars microworld. Immediately prior to the protocol, he had partitioned each of the

cakes (represented by rectangular bars) into three parts and moved the second cake underneath

the first on his screen so that the partitions of each were aligned vertically (see Figure 1).


Figure 1: Jack's arrangement of the two bars where each row represented one cake partitioned into three parts.

Protocol III. The affordances of Jack’s distributive reasoning.

Jack: This is three [moves cursor over the top cake]. That’s three and that’s three [moves

cursor across both rows]. It still equals six. But I guess we can imagine that line

[the horizontal lines separating the two cakes] as not there. And so these two

pieces [the left-most piece from each cake] would be…or this one piece [referring

to the left-most pieces of each cake as one piece] would be a third out of these.

I: Alright. Sounds great. Alright, so how much is that of one cake?

Jack: This of one cake? [Moves cursor over the two pieces the interviewer indicated]

I: Yeah.

Jack: It’s, um, two-thirds of a cake.

I: And how much is it of all the cakes?

Jack: Of all the cakes it’s, umm, one-third. No it’s – if we’re doing all the cakes now?

I: Yeah.

Jack: It’s still one-third.

Protocol III demonstrates that Jack interpreted two pieces of cake as both two-thirds of one cake

and one-third of all the cake. His comment, “But I guess we can imagine that line as not there”

indicates he conceptually united one-third of each cake into a composite unit of two-thirds of a

cake. Further, Jack’s awareness that this produced a share which was one-third of all the cake

suggests that he mentally restructured the total amount of cake in terms of this new composite

unit. Jack’s way of reasoning in this protocol demonstrates the characteristic way of operating

available to students that have constructed a reversible distributive partitioning scheme (RDPS).

Differences in the Students’ Fraction Composition Schemes

Both students had constructed a unit fraction composition scheme (Steffe & Olive, 2010),

but differences became apparent when they reasoned with non-unit fraction compositions. Prior


to the following protocol, Jill used JavaBars to create a rectangular unit and used it to construct

another bar which was three-fifths of that unit. When asked to find one-seventh of three-fifths,

Jill responded uncertainly saying, “It’s confusing with three boxes. Wait, can we make the lines

go away and make a seventh out of it?” Since the fraction question didn’t trigger her distributive

partitioning, the interviewer asked how she would share one-fifth of a unit among seven people.

Protocol IV. Jill’s attempt to find one-seventh of three-fifths.

Jill: Oh, you just put it into sevenths!

I: Split it into seven, right. And then what – take how many?

Jill: One

I: So what if I asked you to take one seventh of [circles the three-fifths on the screen

with the mouse].

Jill: Oh, ah…three twenty-sevenths, right? Three twenty-fourths? Oh, how do you say

that – three over twenty-one.

I: So show me what you’re thinking. How did you get that?

Jill: [Jill partitions each of the three one-fifths into seven parts]. You take one out of

each one of these [referring to the re-partitioned three-fifths].

I: And you’re saying this is how much out of the whole bar [points to a piece that is

one-seventh of one-fifth]?

Jill: One-seventh, one-seventh, and one-seventh. That would be three out of twenty-

one.

The interviewer’s redirection to sharing one-fifth among seven people seems to have had the

intended effect as Jill enacted the operations of her DPS. However, in concluding three twenty-

firsts she effectively found one-seventh of three-thirds and did not use recursive partitioning

operations. Finding one-seventh of three-fifths would require also re-partitioning the five-fifths

from the original unit into seven parts. After additional prompting Jill understood the result of

three thirty-fifths; however, she did not produce this result independently.


In contrast, Jack reorganized his available operations into an abstracted composition

scheme for proper fractions. Consider the following protocol in which Jack was asked to find

one-seventh of five-ninths. Prior to this protocol Jack had constructed a unit bar in JavaBars.

Protocol V. Jack finding one-seventh of five-ninths.

Jack: Like…you could split the bar into ninths…take out five of those. And I would say

if you combined all five of the five-ninths it would still end up being five-ninths.

And then you split that into seven…into sevenths, and you took one part of that

sevenths out and you’d have one seventh – one-seventh of five-ninths.

I: It would be one-seventh of five-ninths. Sure. Do you know what part that would be

of the whole – of the original bar?

Jack: Of the original bar? It would be…umm… [He utters a few phrases aloud as he

thinks intently for approximately 30 seconds. He then shakes his head side to side.]

I: Not sure right now?

Jack: [Shakes his head side to side again.] Like seven. Seven times nine is… [Puts his

head down as he continues to think. Then he quickly lifts his head back up.] Sixty

three. Sixty three. One – sixty three.

I: Where did you come up with the sixty three? What were you thinking?

Jack: Because you multiply…um, seven times nine because each one of the things were

split into sevenths. So seven times nine is sixty three.

I: And then how many sixty-thirds would you have then if you did a seventh of five-

ninths?

Jack: A seventh of five-ninths…you’d have five sixty-threes.

Interestingly, both students initially described splitting the composite proper fraction into seven

parts and taking one part as one-seventh. However, to determine the amount that this seventh

would be in relation to the unit, Jack used his RDPS to establish one-seventh of five-ninths as

five-sevenths of one-ninth. He also used recursive partitioning operations to re-partition each

part of the nine-ninths unit bar into seven parts. While this task was not trivial for Jack,

significantly, he was able to operate with hypothetical bars, mentally coordinate two three-levels

of units structures, and take the results of his operations as input for further operating.


Differences in the Students’ Attempts to Construct and Reason with Intensive Quantities

We also observed several differences in how the students reasoned proportionally in a

lemonade mixture context. The students mixed together three cups of water and two tablespoons

of lemonade powder. Their goal in each question was to produce a mixture with the same taste.

Jill’s responses to the questions indicated that she had constructed an awareness of

proportionality, but not a proportionality scheme (Steffe et al., in press). She reasoned that one

tablespoon of powder would need one-half of a cup of water and pointed to a place on the pitcher

that divided the mixture into two parts. This indicates an intuitive sense of proportionality in that

one-half of the lemonade would contain one tablespoon of powder. However, she did not use her

DPS to determine that one-half of the lemonade would require three-halves cups of water. Jill

also made successful coordinations for six and 15 cups using whole number operations. Yet,

whenever the comparison required fractional operations, Jill responded intuitively to maintain

the ratio but did not utilize operations that would allow her to determine the unknown quantity.

In contrast, Jack made several proportional comparisons which required fractional

reasoning. While Jack also reasoned that using one tablespoon required partitioning the mixture

in half, he used his quantitative operations to determine that one cup of water would require two-

thirds of a tablespoon of powder. His strategy involved the operations of a reversible fraction

scheme and his RDPS as he mentally coordinated partitions on each quantity. Similar to Protocol

V, Jack operated hypothetically and carried out mental operations to preserve the given ratio1.

The students also varied in their abilities to construct unit ratios in situations involving

co-varying quantities. For instance, in one session Jill reasoned about a water pump that filled a

1 Jack and Jill’s proportional reasoning abilities have been analyzed and presented elsewhere. However, summaries

of the student’s activity in the lemonade mixtures context are included here because their responses are relevant to

the current discussion. For a more thorough analysis of their responses to these tasks and the conceptual schemes

and operations which supported their understandings, see (Steffe et al., in press).


pool at a constant rate such that the water rose three inches in five minutes. One goal was to see

if Jill could construct a unit ratio as a measure of the quantity inches per minute. The interviewer

asked how much deeper the water level would get if the water pump ran for one more minute.

Protocol VI. Jill’s struggle to construct a unit ratio of three-fifths inches per minute.

Jill: Umm…it would go up half an inch.

I: Why do you say half an inch?

Jill: Umm, because it grew – it like filled up the pool three inches deep in five

minutes…And if you do like…three divided by five, or is it five divided by three?

[Jill shakes her head uncertainly.] One of those two. It’s half.

I: Can you say that (again)? I didn’t quite hear what you said there.

Jill: Umm, five divided by three I think, is one-half…Oh no. It’s three divided by five

is one-half.

I: Okay. Umm, so why – how did you know that three divided by five was one-half.

How were you thinking about that?

Jill: It’s just…like I don’t know. You just know that.

Similar to her activity in the lemonade context, Jill response of “three divided by five” shows she

had intuition about a strategy to maintain the constant rate. However, she did not use operations

which would allow her to construct this as three-fifths of an inch per minute. Instead, she viewed

three divided by five as a number fact that one just knows. Wanting to investigate the ways in

which Jill could use her result in further operating, the interviewer did not attempt to perturb

Jill’s assertion that three divided by five was one-half. However, in the context of reasoning

about the depth rising 14 inches in 28 minutes, Jill decided she had made a mistake when with

the one-half. The continuation of Protocol VI includes the continued discussion about one-half.

Protocol VI. (Continuation)

I: So why don’t you think the half is right anymore?

Jill: Because if you try to do five, wait six minutes. How many inches would that be?


I: At a half of an inch each minute?

Jill: Yeah.

I: Yeah, how much would that be?

Jill: Two minutes is one inch. Another two minutes is two inches. And then another

two minutes is three inches.

I: So at a half of an inch per minute how long would it take to get to three inches?

Jill: Six minutes. See, I messed up. It’s six minutes and then it would be like…Oh

gosh! Instead of being a half it’s something else. Oh gosh.

I: Okay. So it’s close to a half, but maybe not quite half.

Jill: Yeah.

I: So maybe let’s go back to that question. Is there a way you could think about what

fraction it would be?

Jill: I don’t know. You could do like – I’m not sure if it’s five over three or three over

five? Cause you’re trying to find…how many inches per minute.

Jill recognized a conflict between the initial condition and her initial result of one-half inch per

minute. When asked to reconsider the original question, Jill returned to the idea of dividing but

was clearly uncertain about how to carry this strategy. I suggested a diagram of three inch long

segments and Jill spent approximately 12 minutes reasoning with this and other diagrams to

make sense of the situation. After several additional prompts, Jill discovered a way of operating

with the diagrams to figure out that the water level was rising three-fifths of an inch per minute.

This achievement is significant as it indicates that the operations needed to find unit ratios such

as this lie in Jill’s zone of potential construction. Unfortunately, this was Jill’s final teaching

episode and so I was unable to explore the extent to which Jill had abstracted this strategy.

For comparison, consider Jack’s efforts to construct a unit ratio. In the following protocol

Jack was told that a brownie recipe used two sticks of butter for every three pans. The questions

centered around varying the numbers of sticks of butter and pans of brownies and coordinating


changes in the quantities in such a way that the results still adhered to the recipe. One goal of this

session was to see if Jack could construct a unit ratio as a measure of the quantity sticks per pan.

Jack constructed a diagram of two stick of butter and three pans using JavaBars (see figure 2).

Figure 2: Jack's JavaBars representation of the situation.

Protocol VII. Jack’s construction of the unit ratio two-thirds sticks of butter per each pan.

I: How much butter would just be in one pan of brownies?

Jack: Well you have two bars that go into three.

I: Um hmm, yeah. Same situation.

Jack: Two…two divided by three is…two-thirds. So two-thirds of one bar would go into

one pan of brownies.

I: Okay. Can you show me how you were thinking about that – how you decided it

was two-thirds?

Jack: [To show how he decided, Jack carried out the following activity using JavaBars

functions (in CAPS): a) Used PARTS to partition one stick of butter into three

parts; b) Used PULLOUT to disembed one of these parts from the bar; c) Used

COPY to copy this disembedded part to produce two-thirds of one stick of butter;

d) Used JOIN to unite the two sticks of butter and used CLEAR to erase the line

separating the two sticks of butter; and e) Used PARTS to partition this composite

unit into three parts. Then Jack showed that the two-thirds he had created in step c)

was equivalent to one of the three parts of the composite two sticks of butter.]

Jack: So two-thirds of one bar would go into each one of these (pans of brownies).

I: Sure. And what fraction is that of the two bars?

Jack: Of the two-bars?

I: Um hmm.

Jack: Of the two bars it is one-third.

I: So one-third of two bars goes into each pan or you’re saying two-thirds of one bar?

Jack: Two-thirds of one bar or one-third of two bars would go into each pan.


Jack’s first two responses indicate that he quickly assimilated the situation and implemented his

quantitative operations mentally to find that each pan needed two-thirds of a stick of butter.

Further, his JavaBars activity and explanations suggest that he assimilated the question as a

situation of his RDPS to construct this awareness.

Discussion

The case of Jack and Jill demonstrates that even amongst students that reason with three

levels of units, significant differences exist in the ways that they construct quantitative schemes

and operations which leverage those units coordination capabilities. In particular, Protocols I – V

exemplify several differences in how Jack and Jill with three levels of units to construct schemes

for distributive partitioning and fraction composition. While Jill could interpret solutions like

Jack’s, she never independently produced these types of explanations. Thus, despite using

similar schemes and operations during their initial interviews, by the end of the teaching

experiment Jill had yet to abstract her activity in the same ways that Jack had.

However, the contrasts between Jack and Jill are not meant to indicate that Jill never

succeeded during the teaching experiment. In fact, in tasks involving recursive partitioning or

unit fraction composition, Jill answered with relative ease. Further, she could provide detailed

justifications of the relationships among quantities in a way that was more advanced than several

other participants. Also, the fact that Jill eventually implemented her quantitative operations in

the pool context and built upon the interviewer’s suggestions construct a unit ratio is rather

significant. The participants who reasoned with only one or two levels of units could not even

carry out these operations in activity while working with the interviewer. Thus, while she had not

abstracted her operations to the same degree as Jack, Jill’s activity does indicate that these ways

of reasoning quantitatively do lie in her zone of potential construction.


Instead, the contrasts are presented to highlight these differences because of the

implications they raise for teaching and learning. First, comparing Jack’s success with reasoning

proportionally and finding unit ratios to Jill’s constraints highlights the importance of helping

students to construct units coordination, fraction composition, and RDPSs which leverage the

power of three levels of units. However, the significance of these schemes does not so much lie

in being able to multiply fractions and share units. Rather, their importance became apparent

when observing Jack use the operations of these schemes to reason proportionally and to begin to

make sense of rate contexts. Thus, helping students to construct powerful fractional reasoning is

crucial for providing students with the conceptual tools for constructing the types of algebraic

reasoning expected of students in secondary school mathematics.

Secondly, while being able to carry out the types of operations described above is

important, the degree to which one is able to abstract his/her ways of operating is ultimately just

as crucial. For example, in Protocol IV Jill constructed meaning for one-seventh of three-fifths in

the course of her interaction with the interviewer and while carrying out her operations using

JavaBars. Similarly, in the remainder of the teaching session following Protocol VI Jill carried

out her operations using the suggested diagrams to construct meaning for three divided by five.

Both examples show that Jill frequently developed meaning for her actions in the course of

enacting her operations in the JavaBars program or on drawn diagrams. Thus, while Jill had

constructed powerful ways of operating, she had not yet abstracted them an extent which would

support reasoning hypothetically or using results of her mental operations in further reasoning.

In contrast, when Jack carried out operations in JavaBars or used diagrams, he typically

did so as a way to explain his thinking. For example, in Protocol VII Jack did enact operations in

JavaBars. However, by that point he had already carried out the activity mentally to produce the


unit ratio. Further, Jack’s ability to reason hypothetically suggests that he could produce and

operate on figurative material to carry out his operations. In contrast, Jill’s need to carry out the

activity to construct meaning for her operations suggests that she required perceptual material

upon which to operate. This difference helps to explain the constraints that Jill faced in reasoning

proportionally and in producing a unit ratio in Protocol VI. In the absence of a perceptual unit or

diagram upon which to operate, Jill’s quantitative operations were not triggered.

Lastly, in reading Jack’s protocols you might have found yourself thinking that Jack was

simply using the same operations and that his mental activity didn’t vary much across the tasks.

If so, you would be right. But that realization is quite noteworthy in comparison to Jill’s activity.

While Jack used similar quantitative operations in each protocol, the contexts and the questions

asked in each were not the same. The contexts involved questions about sharing candy, fraction

composition, and co-varying the numbers of pans of brownies and sticks of butter. Significantly,

Jack assimilated the questions in each context as situations of his quantitative operations. In

contrast, while sharing units triggered Jill’s DPS, she did not assimilate fraction composition,

proportional situations, or unit ratio contexts as situations of her quantitative operations.

The differences in how the students assimilated the various contexts highlight the

importance of helping students to expand the range of situations which trigger their quantitative

operations. In attempting to find unit ratios in protocols VI and VII, both Jack and Jill referenced

using the numerical operation of division. To those with adult conceptions of rational numbers,

division and fractions are likely one and the same in that three divided by five implies the three-

fifths and vice versa. However, this was clearly not the case for either Jack or Jill as both needed

to engage in quantitative reasoning to construct this relationship. Jill assimilated three-divided by

five as a number fact to recall and only began to utilize her DPS in the course of interaction with


the interviewer. Jack’s reasoning demonstrates the how a RDPS can be used conceptually

understand the equality of two divided by three and two-thirds of one.

A student’s construction of three levels of units can be considered a type of learning

which opens the way for new constructive possibilities that were not available to the student

when operating with two levels of units. However, the case of Jack and Jill serves to demonstrate

that constructing three levels of units is critical, but insufficient for engaging in advanced

quantitative reasoning. Rather, the ways in which one reorganizes his/her operations with three

levels of units, the degree to which one can abstract those operations to operate hypothetically,

and the range of situations which can trigger those operations all play crucial roles in supporting

the construction of proportional reasoning and intensive quantities.

References

Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning

while modeling dynamic events: A framework and a study. Journal for Research in

Mathematics Education, 33(5), 352–378.

Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships.

Cognition and Instruction, 28(4), 383–432.

Hackenberg, A. J. (in press). Musings on three epistemic algebraic students. In K. C. Moore, L.

P. Steffe & L. L. Hatfield (Eds.), Epistemic algebraic students (Vol. 4). Laramie, WY:

University of Wyoming Press.

Jahnke, H. N. (1983). The relevance of philosophy and history of science and mathematics for

mathematical education. In M. Zweng (Ed.), Proceedings of the fourth international

congress on mathematics education (pp. 444–449). Boston, MA: Birkhäuser.


Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J.

Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp.

41–52). Reston, VA: The National Council of Teachers of Mathematics.

Steffe, L. P. (2007). Radical constructivism and "school mathematics". In M. Larochelle (Ed.),

Key Works in Radical Constructivism (pp. 279–290). Rotterdam, The Netherlands: Sense.

Steffe, L. P., Liss II, D. R., & Lee, H. Y. (in press). On the operations that generate intensive

quantity. In K. C. Moore, L. P. Steffe & L. L. Hatfield (Eds.), Epistemic algebraic

students (Vol. 4). Laramie, WY: University of Wyoming Press.

Steffe, L. P., & Olive, J. (2010). Children's fractional knowledge. New York: Springer.

Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying

principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Reserach design in

mathematics and science education (pp. 267–307). Hillsdale, NJ: Erlbaum.

Thompson, P. W. (1994). The development of the concept of speed and its relationship to

concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative

reasoning in the learning of mathematics (pp. 181–234). Albany, NY: SUNY Press.

Thompson, P. W., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J.

Kilpatrick, G. Martin & D. Schifter (Eds.), Research companion to the Principles and

Standards for School Mathematics (pp. 95–114). Reston, VA: National Council of

Teachers of Mathematics.

von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. New York,

NY: Routledge/Farmer.

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