Discourse is defined in this study by four main elements: word use, visual mediators, routines, and narratives. In this chapter, I present the analyzed data grouped primarily by element of discourse, further partitioned into thematic code groups and the distinct codes used to analyze the interview transcripts for each element. Thus, within each element, the data is broken down by thematic groups with each code then discussed and examples of its use provided.
Word Use
Seventeen distinct codes were used to capture the words used by the participants during their discourses about integers. These codes were applied a total of 1,479 times over the course of 455 turns. A ‘turn’ is defined as a participant’s entire response to an interview prompt. These codes were divided thematically into three groups: sets of
numbers (five codes assigned 123 total times), sign of number read or introduced (three
codes assigned 720 total times), and role of sign verbalized or introduced (nine codes assigned 633 total times), each of which will be considered below. An overview of the codes for this element is provided by participant (Appendix G) and by question number (Appendix H).
Sets of numbers. The sets of numbers thematic group contains the codes:
Positive integer, Negative integer, Whole number, Integer, and Other names for sets of numbers. The application of each of these codes by participant and by question is shown
in Table 7. These codes capture the participants stating the names of these sets of numbers at any point in a turn.
Table 7 Code Use for Sets of Numbers by PST and by Question
‘Positive integer’. The code ‘positive integer’ occurred during 33 turns. Further analysis showed these occurrences could be divided into four types: definition of this type of number (n=13), to describe a general category of numbers (n=10), the cardinality of the number of chips associated with the magnitude of the positive integer (n=4), and the direction and magnitude of a positive integer when modeled on a number line (n=6).
The use of this phrase varied widely between participant, with Rebecca (n=9) and Ashley (n=6) using the phrase most frequently, and Christina (n=0) never using it. The full breakdown of the participants’ use of this phrase is shown in Table 7. The most frequent application of the phrase ‘positive integer’ occurred in response to Question 1 (n=13); this represented the only time the phrase was used for Amanda, Bailey, Brooke, Courtney, Jacqueline, and Monique. As Question 1 directed the participants to compare positive integers and whole numbers, the way each of these participants employ this
Code by PST Am anda A sh le y Ba ile y Br o oke Ch ris tin a Co u rtn ey Ja cque li ne Ja m ie Jor da n Ka y la M o n ique N ic o le Re b ecca T ae Sets of numbers ‘Positive integer’ 1 6 1 1 1 1 4 2 2 1 2 9 2 ‘Negative integer’ 1 7 2 1 2 1 2 2 1 6 5 ‘Whole number’ 1 1 1 3 1 1 5 1 4 1 2 1 3 1 Integer 1 2 1 1 2 2 2 4 1
Other names for sets of numbers 1 2 2 1 1 1 2 2 2 1 3 Code by Question Q1 Q2 Q3 Q4 Q5 Q6 A Q6 B Q6 C Q6 D Q7 A Q7 B Q7 C Q7 D T o ta l Sets of numbers ‘Positive integer’ 13 2 1 2 3 2 3 2 1 4 33 ‘Negative integer’ 2 5 4 2 2 1 2 1 3 3 3 2 30 ‘Whole number’ 14 4 1 1 2 1 2 1 26 Integer 2 3 4 1 2 1 1 1 1 16
Other names for sets of numbers
phrase will be explored in the first section of Chapter 5. In addition to Question 1, Jordan, Kayla, Nicole, and Tae used the phrase in answering one other question. Jordan, Kayla, and Nicole used ‘positive integer’ to describe a general category of numbers, while Tae used it in association with chip color. This usage suggests that these four PSTs would make some spontaneous use of the phrase during their discourses about integers, which shows a deeper command of the discourse, when compared to those who never spontaneously employed the phrase. Three of the participants, Jamie (n=4), Ashley (n=6), and Rebecca (n=9), used the phrase ‘positive integer’ freely. These participants show an increasing range and depth of inclusion of this phrase in their discourse about integers. Overall, this limited use of the phrase (P = 33/448 = 7.37% for turns coded for Word Use) suggest that these participants did not make overarching statements about the nature and behavior of positive integers, except where specifically asked, and not always, even when prompted.
The code positive integer was used 12 times to name a set of numbers which the participants were asked to define in Question 1. In defining ‘positive integer’ Amanda, Jacqueline, and Tae each made similar statements that a “positive integer’s anything like, on a number line, it’s anything after zero” (Amanda), and “a positive integer would be anything that is over, above, zero” (Jacqueline). These examples show that these participants understand positive integers to be numbers greater than, however, these statements do not include the fact that positive integers only exist in contrast to negative integers. In a superficially similar statement by Monique, “positive integers are, numbers that are from zero to a hundred and beyond” we see her incorrectly including zero as one
of the positive integers. Bailey and Brooke independently provided this pair of responses: “positive integers are like rational numbers I think, that word, irrational numbers” (Bailey) and “positive integers can have a decimal or a fraction of a whole number” (Brooke). These responses suggest that these PSTs understand positive integers to be the same as rational numbers or decimals, in contrast to whole numbers which are not rational numbers or decimals. While the definitions presented by Bailey and Brooke contain aspects which are accurate for real numbers, they are entirely inaccurate for ‘positive integers’ since rational numbers are defined in terms of integers. Courtney’s statement that with whole numbers you “could still do a fraction, so it would be like twelve out of twelve or like ten out of ten, where positive integers is just like seven” misrepresents both ‘whole numbers’ and ‘positive integer’. Courtney misrepresents whole numbers by indicating that they must be in fractional form, while with positive integers she does not indicate that they exist in association with negative integers. These quotes show the variation in how these participants define ‘positive integer’, each of which deviates from the way it is defined in the section about Mathematical Discourse About Integers in Chapter 2.
The structure of Questions 2 through 5 explored the PSTs’ discourse related to the properties of integers. Within this context, five of the participants (Ashley, Jamie, Jordan, Nicole, and Rebecca) employed the phrase ‘positive integer’ to describe a general
category of numbers, instead of referencing one specific value such as positive seven, being used in an expression. In response to the prompt in Question 2 about how ‘addition makes larger’ applies to integers, Jordan’s statement included:
when you add one and one together you’re making a larger number, but if you add two negatives, you’re making a lesser number, so I guess you’re not always making a larger number with integers, since integers include positives and negatives.
In this statement, Jordan used the number one as a specific positive integer and concluded with a reference to all integers greater than zero. Nicole uses ‘positive integer’ similarly when she said “Yes, because no matter if you’re adding negative integers or positive integers you’re always going to a higher number.” In Nicole’s response, she indicated that the sum of two negative integers is a larger number when it is in reality a smaller value, though larger absolute value, due to the structure of the integers. Similarly, in her response to Question 3, concerning subtraction with integers, Rebecca used ‘positive integer’ in this way, “if you’re using two positive integers, you can get a positive or a negative number depending on how large the integers are.” Jordan’s and Rebecca’s responses highlight the fact that the type, or magnitude, of the numbers included in an expression affect whether the sum or difference is larger than the original terms in the expression or not, while Nicole’s statement indicates that this does not matter. When considered together, these responses suggest that these PSTs are capable of employing ‘positive integers’ to describe a general category of numbers, while describing properties of numbers or operations while engaged in discourse about integers, without exclusively resorting to specific examples.
A more specific use of the label positive integers, to refer to specific numbers, rather than the general category as provided by Kayla and Rebecca when responding to
the question regarding additive inverse using the values -7 and 7. They made the statements: “it means that it’s the same integer but one’s positive and one’s negative” (Kayla, Q5) and “this is a positive integer of seven” (Rebecca, Q5A). These statements by Kayla and Rebecca suggest that numbers of the same magnitude exist both above and below zero. This contrasts with the idea described in the previous paragraph, since in these situations, the participants referenced individual values.
Questions 6 and 7 asked the participants to evaluate and compare two sets of four expressions. While addressing these questions, the participants continued to use ‘positive integer’ to describe a general category of numbers. While considering her initial
approach to ‘-4 – 2’ Ashley stated “four negative integers. One, two, three, four. And I would be subtracting two, and two, yeah, two is a positive integer,” identifying the contrasting nature of the two values in the expression. When discussing the expression ‘- 7 + 1’, Jamie stated that she noticed “when you add a positive integer and a negative integer it becomes a smaller negative number.” Jamie’s statement is only true when the negative integer has a larger absolute value than the positive integer, zero and positive integers are possible when the relative absolute values of the positive and negative integers are equal or reversed. Similarly, while considering the expressions ‘2 – 8’ and ‘- 7 + 1’, respectively, Rebecca pointed out that, “this [‘2 – 8’] only has positive integers involved whereas this [‘-7 + 1’ has] a negative integer,” while neglecting to point out the presence of a positive integer in the second expression. A description of the different ways in which positive and negative integers behave in an expression was provided by Ashley after evaluating the expression ‘-4 – 2’ when she said, “sometimes you would
have to do the opposite of what the symbol is asking you to do … depending on if the number you are adding, or subtracting is a negative or a positive integer.” In each of these instances, the participant was modeling the process of determining the equivalent value to expressions which were provided. In responding to each prompt, these
participants opted to verbally identified the presence of a ‘positive integer’ among the terms. This verbalization indicates that these PSTs were calling attention to effects that occur any time a positive integer is present in an expression with that operation.
The final two ways that the phrase ‘positive integer’ was used both involve visual mediators, specifically chips and number lines. In four cases, three different individuals associated a chip color as representing positive integers. While preparing to evaluate ‘-1 + 9’ using the colored chip model, Tae described sorting the brown and beige chips in the following manner, “I am using this to represent negative integers and this to represent positive integers.” Similarly, while addressing ‘6 + 2’ with the colored chip model, Ashley described her actions in this way: “I would just add two of the brown chips to get eight brown chips or eight positive integers.” These statements suggest that the PSTs used the phrase ‘positive integer’ in a way that voices their assignment of specific chip colors to the cardinality of either positive or negative integers.
Two participants described movement to the right along the number line a distance by stating the phrase ‘positive integer’ six times each. This is captured in Rebecca’s statement “because it’s a plus I’ll be moving and a positive integer I’ll be moving to the right one” while working to evaluate ‘-7 + 1’ on a number line. Similarly, Jamie approached this same expression by stating, “So we’re at negative seven right here
and we’re adding a positive integer to it so it’s going to make the number bigger and closer to zero this one, so we are going to add a positive one which will move it to the right.” These responses, when considered along with the frequency with which Jamie and Rebecca used ‘positive integer’, indicate that their discourse about integers incorporates identification of the presence of ‘positive integers’ and their associated behavior, such as a shift to the right on the number line, in expressions and models in a way that is consistent with the discourse described in Chapter 2, as well as employing the term more often than their peers.
The number of different ways that ‘positive integer’ was employed by the participants, combined with the wide range of number of uses of this phrase by
participants, from zero to nine such uses, makes it impossible to make a single general statement describing all of the PSTs in this study. However, half of the participants (n=7) only used the expression at most one time, while four others made a second reference to the phrase. This implies that only three of the participants, Ashley, Jamie, and Rebecca, have integrated the phrase ‘positive integer’ into their discourse about integers in a spontaneous and rich manner. When considering their use of the phrase ‘positive integer’ throughout the interview, different levels of consistency with established discourse practices were observed. However, these levels of consistency represent observations based on different numbers of occurrences. Ashley, Nicole, Rebecca, and Tae employed ‘positive integer’ in a way that was consistent with common practice. Most of the uses of ‘positive integer’ by Jamie, Jordan, and Kayla were determined to be consistent with common practice, after providing definitions which were partially consistent. Amanda
and Jacqueline were also partially consistent, while Bailey, Brooke, and Courtney defined ‘positive integer’ in a manner which is inconsistent with the exemplar discourse
described in Chapter 2. Christina did not employ the term and was characterized as not having exhibited this component of a mathematical discourse about integers.
Negative integer. The structure of the interview did not prompt the students to use the phrase ‘negative integer,’ however 11 of them did, for a total of 30 turns (see Table 7). As with positive integer, Ashley (n=7), Rebecca (n=6), and Tae (n=5) used the phrase ‘negative integer’ most often. Four of the participants, Brooke, Jamie, Kayla, and Monique used the phrase twice, while another four, Amanda, Jacqueline, Jordan, and Nicole only used it once. Bailey, Christina, and Courtney never used the phrase.
The code negative integer, when subjected to further analysis, can be broken down into the same uses as were noted for positive integer. ‘Negative integer’ was used as part of a definition of a set of numbers (n=2), to describe a general category of
numbers (n=20), the cardinality of the number of chips associated with the magnitude of the negative integer (n=7), and the direction and magnitude of a negative value when modeled on a number line (n=1). These uses of ‘negative integer’ complement those of ‘positive integer’ examined above, except in the additional use of negative integer in Question 4. While answering Question 4, about the role ‘ - ’ can take, Amanda and Monique employed ‘negative integer’ to describe a general category of numbers in the following ways, “It can be in front of a number and it can make the number negative so it can be a negative integer.” and “It could just signify a negative integer.” These quotes
show that these participants grasp the relationship between the symbol and the set of negative integers.
When considering their use of the phrase ‘negative integer’ throughout the interview, different levels of consistency with established discourse practices were observed. As with ‘positive integers’, these levels of consistency represent observations based on different numbers of occurrences. Amanda, Ashley, Brooke, Jacqueline, Jamie, Jordan, Kayla, Monique, Nicole, and Tae employed ‘negative integer’ in a way that was consistent with common practice, any time they used the phrase when only word use was considered. Five of Rebecca’s six uses of ‘negative integer’ were determined to be consistent with common practice, with her one partially consistent use involved her claim that both terms in an expression were ‘negative integers’ when only one actually was. Bailey, Christina, and Courtney did not employ the term, and will be characterized as not having exhibited this component of a mathematical discourse about integers.
Whole numbers. The phrase ‘whole number’ was used during 26 turns in four different ways. The most frequent way was in defining the term ‘whole number’, which was done by all 14 participants. It was also employed to describe a general category of numbers (n=10), the cardinality of the number of chips associated with the magnitude of a whole number (n=1), and the direction and magnitude of a positive integer when modeled on a number line (n=1). With five participants (Brooke, Jacqueline, Jordan, Monique, and Rebecca) using the phrase in one or more of the other ways. This includes Jordan’s use of it with the chip model, and Jacqueline’s use of it while modeling with a number line (see Table 7).
The use of the phrase ‘whole number’ as part of a definition can be seen in Tae’s statement, “I would say that whole numbers include all the numbers on the number line, but positive integers are only to the right of zero on the number line,” in which she is comparing the properties of whole numbers and positive integers. This statement can be misleading based on what is meant by the two uses of number line, since, for example, if they both refer to an integer number line, her definition of ‘whole number’ is incorrect. The other participants used the phrase in a widely varying manner as part of their definitions of this set of numbers, which I have described in more detail in Chapter 5.
Another use of this phrase is to identify the presence of a general category of the cardinal numbers greater than or equal to zero in a computation. Rebecca did so in the first part of her response to the prompt regarding addition making larger, which is shown in this passage:
addition makes larger is misleading when we’re working with integers. For instance, if we are working with simply whole numbers, whole numbers need to be positive so if we’re working with four plus three we see that they’re both positive numbers and they’re going to equal seven.
This passage includes ‘whole numbers’, to describe a general category of numbers greater than or equal to zero, which are then translated into numbers within an example
expression. This use of the phrase was distributed over responses to seven different prompts. The use of the code ‘whole number’ in this manner occurred 10 times during the 14 interviews. This shows that these participants did not employ a discourse which frequently includes a statement about ‘whole numbers’ being included in the expression,
which would be considered as consistent with a discourse about integers, if the phrase ‘positive integer’ was used more frequently for this purpose.
The other two uses of ‘whole number’ were seen while participants employed visual mediators. As Jordan was modeling ‘-1 + 9’ using colored chips, she stated “I add this whole number from the nine, well, one, it takes place of the negative, and then I only have, since I used that one, I have eight.” Jordan represented the negative one with a beige chip, while representing the whole numbers with brown chips. While making this statement, she swapped a brown chip for the beige chip, which canceled them out,