drawings produced when posing a story or solving an open number sentence) was analyzed for each of the sub-questions of the research question by using constant comparative methods (Merriam, 1998) with the CMIAS descriptions as the initial codes (see, e.g., Wessman-Enzinger & Mooney, 2014) as well as the option to code Other. The data were organized with respect to each of the discursive tenets from commognition. Each of the transcripts, which included the students’ verbal use about integers (i.e., word use) and descriptions of drawings (i.e., visual mediators), were paired with the all of the drawings produced by the students (i.e., visual mediators). Each of these 354 units of data was examined by the witness and myself.
Constant comparative methods were selected as an analytic tool to refine the CMIAS because the CMIAS already had initial descriptions and an aim of this study was develop more robust descriptions of them and generate new possible CMIAS.
Furthermore, the act of coding with the CMIAS was viewed as a way to describe the students’ narratives (i.e., narratives) or the students’ use of the CMIAS, which also addressed the second part of the research question. Constant comparative methodology allowed refinement of the CMIAS, with the allowance of generating new CMIAS to describe the students’ narratives betters.
Each of these units of data was first coded individually by both the witness and me with the initial CMIAS or Other (Wessman-Enzinger & Mooney, 2014). The
descriptions of the initial CMIAS, which are described in both Chapter I (see, e.g., pp. 9– 13) and Chapter II, were compared to the units of analysis for solving open number sentences and posing stories for all three students. The witness and I allowed the option to code “Other” when the current descriptions of the CMIAS broke down or were not sufficient. The coding of Other was thought to highlight areas that would need to be discussed for refinement in the coding process to develop more robust descriptions of the CMIAS. Because the initial descriptions of the CMIAS were developed from contextual stories, this presented little issues in coding the units of data from the Individual Context Sessions. However, this was more challenging when coding the units of data from the Individual Open Number Sentence Sessions. For this reason, having the code of Other was important and highlighted areas to pay attention to for the refinement of the CMIAS.
For example, when Jace solved 5 + ☐ = -3, he responded, “I figured out that eight is
is the negative number.” This was coded as Other in the first pass of coding. Both the witness and I coded this as Other because it was agreed that Jace used magnitude comparisons here by comparing the size of 5 and 3. It was unclear in the first round of coding how magnitude was related to the initial descriptions of the CMIASs and so anything that had apparent magnitude ideas like this was coded as Other. This was an issue that was negotiated in Phase 3 and 4.
In this initial coding of the data, where the initial CMIAS descriptions and Other were utilized, Rule was the most utilized code by both the witness and I for all three of the students. In fact, Rule was coded for more than half of the units of data. For this reason, the category Rule was examined further in Phase 3 to investigate the differences in the units of data that were coded with Rule.
As described above, the witness and I met to discuss each of these initial sets of codes, negotiate differences, and note places of difficulty in coding weekly or bi-weekly from August 2014 to December 2014. We met to compare our codes and discuss
discrepancies on this first pass of coding. The codes, the agreement, and notes about the disagreement were recorded in a coding sheet. An example of a Phase 2 coding sheet from Kim’s Individual Open Number Sentence 1 is provided in Appendix O. In addition to discrepancies and negation, both the witness and I met to discuss challenges in using the current CMIAS descriptors to describe the students’ narratives of solving the open number sentences or posing a story. For example, the witness and I discussed that coding Rule was ambiguous and there seemed to be significant differences in some of the units of analysis that were coded with Rule. Additionally, how students made many analogies to whole numbers to solve some problems came up in discussion. For a problem like -2 +
-3, students often reasoned that because two plus three is five, negative two plus negative three must be negative five. We discussed and debated whether this should be included as Rule since this analogous reasoning seemed different than other types of reasoning. Similarly, for the problem 5 + -3, student may have used a Rule that adding a negative is the same as subtracting, and solve 5 + -3 as 5 – 3. The analogous reasoning seemed different from this implicit rule that the students were using; yet, because the analogous reasoning was based on connecting it to a whole number rule, we agreed to code this as Rule still and the other was coded as Rule as well in Phase 2. For this reason, we knew we needed to do a second pass and look at other possible emergent theme within the units of data coded Rule in Phase 3.
Another challenge that emerged during this initial pass of the data was how to code strategies that used directed distances and strategies that used distance without direction. We negotiated the meaning the Translation to be any directed distance and distance without directions during this phase. More about details about this discussion is in Chapter IV. Because this first pass of the data using the CMIAS was messy and difficult, it was important to look for commonalties and challenges across all of the units of data for all of the students, do generate better CMIAS descriptions or establish new CMIAS in Phase 3.