Phase 4 was completed, the codes of the refined CMIAS for each unit of data were interpreted as the students’ narratives for solving the open number sentence in Phase 5. Some of these counts and percentages calculated of the students’ uses that were generated during Phase 5 are also provided in Chapter IV. These are provided as a way to make sense of the qualitative data to highlight the students’ use of the CMIAS. The percentages were also utilized to help illustrate the differences in CMIAS use that different students use, as well as, the use in different types of problems: contextual and symbolic. Then, the percentages of use for each CMIAS across each of the Individual Sessions for each student were examined, to make sense of the students’ use of the CMIAS. Although all the percentages for all three students are provided in Chapter IV, Alice was selected as a student to provide further insight into what these uses look like. A qualitative account with examples of her work is provided for a focused and in-depth account of her use of the CMIAS.
Analysis of Individual Session Data for Jace
Capturing development, or learning, as a change in discourse (Sfard, 2008) over a significant amount of time requires a way to productively manage and reduce the data in order to make sense of the data to describe learning. To address the second part of the research question, about the ways students use conceptual models of the integers as they learn about integer addition and subtraction, I analyzed the changes exhibited in Jace’s thinking across the Individual Sessions. I selected Jace because he had attended the most sessions throughout the teaching experiment, missing only one of the Group Sessions. Also, Jace was the only student in the study that eventually used all of the CMIAS at some point in the study (see Chapter IV).
Drawing upon Sfards’ (2008) definition of learning as a change in discourse, I analyzed the data with regard to changes in word use, visual mediators, narratives, and routines. Although learning is not defined as an “acquisition” of getting the correct answer, the identification of correct and incorrect answers helped narrow the data. For example, as Jace solved open number sentences during his Individual Open Number Sentence Sessions he solved some problems correctly across the four sessions. Other open number sentences, he solved incorrectly across the four sessions. And, other open number sentences he solved incorrect across the sessions and eventually got them correct. A single open number sentence type (see Table 13) from each of these categories was selected for examination. Three open number sentence types were selected: -a + ☐ = b (a, b > 0 and b > a), -a – b = ☐ (a, b > 0 and a > b), and -a – ☐ = -b (a, b > 0 and b > a). I identified and described Jace’s word use, visual mediators, narratives, and routines for each of these three open number sentences. That is, each of these tenets across the four
sessions were placed in a chart next to each other. I then examined and described the changes in each of the tenets by looking across the sessions for similarities and
differences for each tenet of commognition (i.e., word use, visual mediators, narratives, routines) for each of the open number sentences. Descriptions of these changes are provided, by each commognitive tenet, in Chapter V.
Table 15 below highlights how each of the two parts of the research question is related to the data sources used and the data analysis.
Table 15
Summary of Research Question, Data Sources, and Data Analysis
Research Question Data Sources Data Analysis
In what ways do fifth- grade students use conceptual models of the integers (e.g.,
Bookkeeping, Counterbalance, Translation, Relativity, Rule) as they
(a) attempt to make sense of the integers
Transcripts Drawings
Word use and visual mediators were used to determine the students’ narratives by examining routines, or repetition of thinking, with constant comparative methods (Merriam, 1998). And (b) learn about integer
addition and subtraction?
Transcripts Drawings
Teacher-researcher journal
Identifying changes in word use, visual
mediators, narratives, and routines with a qualitative description of changes of over time of Jace solving three different open number sentences.
CHAPTER IV
REFINEMENT OF THE CONCEPTUAL MODELS FOR INTEGER ADDITION AND SUBTRACTION & THE STUDENTS’ USE OF THE REFINED CONCEPTUAL
MODELS
This chapter begins with describing the refined Conceptual Models for Integer Addition and Subtraction (CMIAS) and how they were refined. Descriptions of the refined CMIAS are purposefully mathematical and focus on the roles of the integers within each of the conceptual models. The decisions that were made how the models were refined are provided with the supporting data. After the refined models are described, this chapter transitions to the ways that Alice, Jace, and Kim utilized the CMIAS in the Individual Sessions. The purpose of the descriptions of these uses of the three students is to provide broad descriptions of how the three Grade 5 students used the CMIAS. Then, this chapter concludes with an in depth illustration of how one student, Alice, used the CMIAS throughout the Individual Sessions of the teaching experiment.
Refinement of CMIAS
The CMIAS are ways of reasoning, thinking about, and mathematical uses of adding and subtracting the integers both symbolically and within contexts. The original CMIAS were Bookkeeping, Counterbalance, Translation, Relativity, and Rule. After data analysis, the CMIASs have been modified to seven different conceptual models:
Bookkeeping, Counterbalance, Translation, Relativity, Analogies, Algebraic Reasoning, and Proceduralization. Each of these seven modified CMIASs and how they were refined is described next, with the coded data from Phase 2 being reported on as well as the refinements and refinement decisions of Phases 3 and 4. For all the CMIAS reported on, the emphasis is that these are ways of thinking and using the integers and being right or wrong is not important.
Each of the descriptions of the CMIAS were first modified by examining the units of data coded in Phase 2 from Alice, Kim, and Jace from the Individual Open Number Sentence Sessions and the Individual Context Sessions. However, the units of data from the Individual Context Sessions did not provide insight into the refinement of these models. That was to be expected given that the initial CMIAS descriptions were initially developed from examining the stories that student generated when they posed stories for integer open number sentences. After those modifications were made using the units of data from the Individual Open Number Sentence Sessions, the descriptions were further refined to include clear and consistent descriptions that defined role of zero and the role of the integers in addition and subtraction. This was done from my mathematical
perspective, rather than the units of data. The mathematical descriptions of the CMIAS as well as this process of the refinement are provided next.
Bookkeeping
Approximately 13% of the students’ responses to the Individual Open Number Sentences (each coded unit of data) exhibited use of the Bookkeeping model of thinking. However, all of these units of data came from Alice. Of Alice’s 93 units of data from the Individual Open Number Sentence Sessions, there were 36 units coded as Bookkeeping.
Although only Alice’s units were used in this refinement, the original description did not fully capture the mathematical use in the students’ responses. Based on my analysis of the data, I made one modification to the definition (see Table 16). That change involved the clarification of wording to help distinguish it from other models.
Table 16
Original and Refined Bookkeeping CMIAS
Original CMIAS Description from Wessman-Enzinger & Mooney (2014)
Refined CMIAS Description
A conceptual model of Bookkeeping model involves using integers to describe gains and losses. The use of zero in this model represents a status of neither loss nor gain. The bookkeeping model can represent a gain and loss of any item and is not
necessarily limited to the context of money. For example, gains and losses can be conceptualized through such scenarios as owing and acquiring candy bars or wanting and receiving baseball cards. (p. 203)
Bookkeeping is utilized when the integers
are used in a way to describe losses and gains of quantities. The zero in the
bookkeeping conceptual model represents having neither a specific gain nor loss. In Bookkeeping, the positive and negative integers represent a gain or loss of something applied to a singular quantity. That is, the addition of a positive integer may be treated as gain to the singular quantity and the addition of a negative integer may be treated as a loss to the singular quantity. Or, the subtraction of a positive integer may be treated as a loss to the singular quantity and the subtraction of a negative integer may be treated as a gain to the singular quantity.
Bookkeeping points to thinking that is mostly quantitative reasoning without explicit reference to order and without reference neutralization. Although the positive integers are treated as an increase in the quantity and the negative integers are treated as loss in quantities, the addition of positive integers to a negative quantity may also be considered a loss of negative quantity. A distinguishing element of Bookkeeping compared to other models is that the gain or loss is being applied to a singular quantity.
Clarification of wording. The clarification of wording included defining integer
addition and subtraction as a gain or loss to a singular quantity. The refinement of recognizing gains and losses applied to a singular quantity was made from all of Alice’s responses. All of her responses (the 36 units of data) utilized discrete quantities without order and without neutralization. All of the responses included a gain, adding discrete quantities to an already existing quantities, or removing quantities from an existing
quantity. For example, when Alice solved -18 + 12 = ☐, she stated:
I did eighteen lines with (points at the green tallies with finger), which that was the eighteen negative. I crossed off twelve (points at the right with the pink). And, I still had six negative left (points at the tallies that were on the left that had not been crossed off).
In this example, Alice drew one singular quantity (i.e., a set of tallies) and changed this singular quantity by crossing off tallies (i.e., a loss applied to the singular quantity). This type of thinking seemed distinct from Counterbalance because positive and negatives were not being compared. Also, this did not seem like Translation because there was not movement, shift, or continuous distance incorporated. Although these types of encounters happened with Alice frequently, they didn’t occur for Kim or Jace. Although none of Kim’s of Jace’s units of data influenced the refinement of this definition, all of Alice’s responses included her verbally discussing a gain or loss to a singular quantity and applying a gain or loss to that set of drawn tallies.
Counterbalance
Approximately 4% of the students’ responses from the Open Number Sentence Sessions (each coded as a unit of data) was coded Counterbalance. That is, only 10 units
of the 279 units of data were coded as Counterbalance. All 10 of those units were from Alice’s Open Number Sentence Sessions. Based on my analysis the data, there were two refinements made to counterbalance (see Table 17).
Table 17
Original and Refined Counterbalance CMIAS Original CMIAS Description from
Wessman-Enzinger & Mooney (2014)
Refined CMIAS Description
A conceptual model of Counterbalance involves using positive and negative integers to balance, or cancel, each other out. This model is similar to the
“Balanced Metric,” cancellation, and chip modeling concepts found in the literature (e.g., Battista 1983). The zero in this counterbalance model indicates
neutralization. The distinguishing element of this model is that the quantities always remain, even when neutralized. (p. 203)
Counterbalance is utilized when the negative and positive integers are treated as two separate quantities in a way that balances each other out. The zero in the counterbalance conceptual model represents a status of neutralization. Positive and negative numbers in
Counterbalance are not just opposites, but opposites that neutralize. Addition of integers, whether positive or negative, represents joining quantities where the equal number of positive and negative integers are neutralized and the sum is the integers not neutralized. Subtraction of integers, whether positive or negative, represents removing quantities, which may entail removing quantities from a status of neutralization.
Similar to Bookkeeping,
Counterbalance points to thinking that is mostly quantitative. A distinguishing feature of Counterbalance from other models, and specifically Bookkeeping, is that there are two quantities that always remain present in the Counterbalance, even when neutralized. The absolute value, or magnitude, of these two different quantities may be compared to solve addition and subtraction problems.
One change involved clarifying wording the wording of the description. The other change addressed ways the students’ ways of thinking about the integers that were not adequately described in the original definition. This change pertained to the mathematical use of absolve value, or absolute magnitude, comparisons.
Clarification of wording. All 10 of the units of data included a comparison of
two different discrete quantities. The original definition included two quantities without explicitly stating it, when it included, “using positive and negative integers to balance, or cancel, each other out.” However, within these units of data, sometimes only two positive numbers were compared, rather than a positive and negative number. For example, when
Alice solved 12 – 18 = ☐, she drew 12 tallies above 18 tallies. She compared these two
quantities, and determined the answer was -6:
Well, I did twelve (points at the green tallies.) as the twelve (points at 12 in the number sentence). And then, I did eighteen (points at the pink tallies). And this is twelve (points at the left side of the pink tallies). So I (takes hand and covers green and pink tallies) knew that the six extra ones (still covering both sets of twelve tallies, uses right hand to point at the uncovered pink tallies) were the answer.
Alice, using her hand to cover up the tallies, illustrated implicit neutralization. Alice used Counterbalance, but she did not do with positive and negative numbers. Alice used Counterbalance with two quantities, even when both were positive, like 12 and 18. Also, within these 10 units of data, there was a unit of data that was accidentally coded as Bookkeeping. For this reason, there was a need for clarity in wording. The wording that a
comparison of “two quantities” was added to account for the use of two quantities and to help provide clarity in the distinction from Bookkeeping.
Absolute value or magnitude. These 10 units of data were also related to the
units that were coded as Other. Approximately 6% of the students’ responses (17 of the 279 units of data) from the Open Number Sentences were coded as Other. All of the codes of Other came from Jace’s Individual Sessions. Noticing the commonality between the units of data coded Counterbalance and Other, Counterbalance was refined to include a discussion about this commonality—absolute value or magnitude comparison.
The inclusion of comparisons of absolute magnitude, or absolute value, still fit within the definition because it is still a comparison of two different quantities. Notably, magnitude is distinguished from directed magnitudes, where a quantity is defined by distance and direction. The refined definition uses magnitude that is absolute magnitude. That is, magnitude refers to the use of the absolute value of a quantity, with no distinct
direction. For example, when Alice solved 15 + -24 = ☐, she drew two quantities and
used implicit neutralization:
(Draws 15 tallies with green marker. Draws 24 tallies with pink maker below the 15 green tallies. Then, counts the tallies that do not have green tallies above them. She uses the pink marker to count each of the tallies one by one. Writes -9 in the box.) I did fifteen (points at the green tallies). And then twenty-four negatives (points at the pink tallies). And I had nine negative left.
Alice also used absolute magnitude of comparing 15 and 24. Similarly, the responses from Jace (the units originally coded Other) included similar reasoning to Alice, except his responses used only verbal descriptions and he did not include drawings. For
example, when Jace solved 12 + -16 = ☐, he stated, “Sixteen is greater than twelve. And,
sixteen’s the negative number. So sixteen minus twelve equals four. And, it’s got to be negative four because the negative number’s bigger than the regular number.” Here, Jace used the absolute magnitude when he compared 16 and 12 by stating that, “Sixteen is greater than twelve.” He then recognized that this was the absolute magnitude and not the actual number, when he stated that, “the negative number’s bigger than the regular
number.” This type of magnitude comparison use was present in all of the units of data coded as Other and was similar to units coded as Counterbalance.
These units of data from Jace that were coded as Other were influential in the refinement of Counterbalance. They were not coded as Counterbalance before because it was not clear that he was employing neutralization of two quantities at first. Because the use of neutralization was implicit from both the students and they were each using two different quantities, it was important to refine this definition. Both Alice and Jace the compared the magnitude, or absolute value, of two different quantities, in all of these units of data and this influenced the refinement of Counterbalance.
Relativity
There were no students’ responses (coded units of data) from the Individual Open Number Sentence Sessions that were coded as Relativity. The only responses that were coded as Relativity came from Jace’s Individual Context Sessions. Of the 28 units of data from Jace’s Individual Context Sessions, there were only 3 units that provided use of integers in relative positions as defined in the original definition of Relativity. Because there were so few units of data, refinement to this model had to be strictly just
defining the role of addition and subtraction and emphasizing the meaning relative numbers.
Table 18
Original and Refined Relativity CMIAS Original CMIAS Description from
Wessman-Enzinger & Mooney (2014)
Refined CMIAS Description
A conceptual model of Relativity involves using integers in relative positions and as a comparison to a referent. With relativity, the zero is not actually zero but treated as a referent, or point of reference, for comparison. (p. 203)
Relativity is utilized if the integers are used as comparative numbers, otherwise known mathematically as relative
numbers. That is, the integers are utilized in a way to describe relative, or arbitrary, positions. With Relativity, the zero represents the point of reference, which may be intentionally or arbitrarily selected. Distinctively, the zero does not represent a quantity of nothing, but is treated as a referent, or point of reference, for comparison. What distinguishes the Relativity from other models is that the actual cardinality of the numbers, or the quantities involved is not necessary. Using the integers as relative numbers in comparison to an unknown referent is the distinguishing feature of Relativity. And, the use of order and integers as relative numbers are a unique feature to Relativity.
Clarification of wording. The first clarification in wording was to define zero as
a referent, or point of reference. This was already in the definition, but it was refined to emphasize that the integers are used in comparison to this reference and that this zero as well as the integers do not have cardinality. This was refined by my mathematical perspective and reflection about treating integers as relative numbers. This refinement, although made from my mathematical perspective, was supported by Jace’s responses.