Bob’s Productive Task Engagement in Fractional Reasoning: A Case Study

Research Question 1: How Bob’s Engagement Becomes Evident/Patterns Across Tasks

We began this process by coding for instances of each of the three kinds of engagement. First, to assess Bob’s behavioral engagement, we used simple counts of utterances during Bob’s problem-solving and in his discussion of his solution with the researcher-teacher, coded as on- or off-task language (e.g., whether the talk was about the math task or something else). We also documented Bob’s effort in terms of how long he worked on a problem, whether he stopped working at any point during the session, and whether he returned to problem-solving or not. Second, for Bob’s cognitive engagement, we coded his words, utterances, and gestures for evidence of his thought processes and attention. Specifically, we coded for his entry points and solution paths in problems, how he used resources (prior knowledge, monitoring, self-talk, and the researcher-teacher) during difficulty with a problem, and the length and time spent explaining his solutions and written work. Finally, for Bob’s emotional engagement, we looked for evidence in the videos of Bob’s emotions that occurred as part of completing a task and coded for whether these emotions were positive and produced excitement rather than negative.

Next, by analyzing field notes, student work, video data and transcripts, and our codes, we constructed a thematic understanding of Bob’s patterns of engagement, both within and across sessions, with regard to objects of his engagement (i.e., tasks), developing claims and counterclaims around specific elements that we adapted items from elements of Goldin and colleagues’ (2011) nine engagement structures to four main structures: (a) Bob’s goal or motivating desire to engage with the task or teacher object (i.e., his fulfilling a need within the situation); (b) Bob’s patterns of behavior in response to the goal or motivating desire; (c) Bob’s affective pathways and expressions during the pattern of behavior; and (d) aspects of the tasks or teacher interactions that mediated Bob’s engagement. Analysis of Bob’s patterns of engagement was concurrent with data collection so that claims could be either supported or refuted during later stages of data collection. The first author led the writing of “Bob’s patterns of engagement memo” which was reviewed by the team during collaborative work (Brantlinger et al., 2005).

The analytic process stressed building claims from multiple data sources, as well as exploring counterexamples. For example, in a field note during Phase 1 of the study, a researcher noted “Bob is persistent to engage in problems that are familiar and where he can show what he knows.” Another field note described the richness of Bob’s explanations during this time. Because Bob’s engagement in problem-solving and the tasks that support it is a major focus of our study, we made an initial claim that when the task was accessible yet sufficiently challenging to Bob’s prior knowledge, Bob better utilized cognitive resources to engage with the problem. In later data collection, we kept careful track of what Bob did, analyzing the transcripts and accompanying video footage in depth. We found that, when the task was too easy (i.e., engaged Bob’s prior knowledge yet not in a way that challenged it), Bob consistently offered quick explanations and engaged in off-task behaviors. Claims are supported by multiple data sources and are analyzed concurrently with data collection to increase credibility; disconfirming evidence was included for every claim (Brantlinger et al., 2005). For each claim about Bob’s patterns of engagement, we actively sought counterevidence for the claim. When no counter evidence could be found, we included the claim in our findings.

Research Question 2: Bob’s Engagement Related to His Understanding of Fractions

To answer our second research question, we created “Bob’s fraction learning memo” which included every instance of data pertaining to the student’s evolving notions of fractions. All coding and analyses for this initial stage were completed by the first author to ensure consistency, and the coding and analyses were presented in research meetings for collaborative work (feedback; analysis) from the full research team (Brantlinger et al., 2005).

To begin, we used ongoing analysis of video data and written work immediately following each session. The focus was on documenting Bob’s problem-solving strategies and generating hypotheses of how problem-solving coincided with patterns of engagement. These hypotheses guided later data collection and retrospective analysis. After the first phase of data collection, we noted that Bob not only engaged very little in tasks that addressed unit fractions but also evidenced much higher levels of cognitive and emotional engagement when tasks linked whole numbers and fractions together multiplicatively (e.g., make ⅙ from ⅓). We added the consideration of the function of the task, such as those that were accessible to prior knowledge versus those that challenged it, and designed an ongoing analysis tool for analyzing Bob’s interactions with the tasks and teacher interactions, using video recordings for all rounds.

Next, the first author coded all Phase 2 tutoring sessions, with a focus on the interaction between Bob, the task, and the teacher. This coding process entailed watching the video and stopping in 30-second increments to describe Bob’s problem-solving strategies, gestures, and utterances. Student work and field notes were incorporated into the analysis to provide a thick depiction of Bob’s operations and units coordination within each task and tutoring episode (Geertz, 1988). We used this coding to retrospectively identify broad indicators of conceptual growth across the sessions (Leech & Onwuegbuzie, 2007). We concluded with fine-grained analysis to consider how Bob advanced his thinking from one stage to the next (Tzur, 1999).

Comparing Engagement and Fraction Understandings Over Time

Finally, we compared “Bob’s fraction learning memo” against “Bob’s patterns of engagement memo.” Assertions were made about how shifts in knowledge occurred within patterns of engagement. We looked for multiple explanations for these shifts, which were then reviewed and confirmed by all members of the research team. Throughout our process, we worked to ensure the trustworthiness, credibility, transferability, and dependability of our analyses (Brantlinger et al., 2005). First, the trustworthiness of the data was established through: (a) thick description, (b) ongoing and iterative data analysis and sharing, (c) multiple data sources for triangulation, and (d) collective research team analysis. The prolonged engagement with the focal student, the persistent observation of engagement and fraction learning, and triangulation (i.e., three coders, three sources of data, three methods to collect data: video, student work, field notes) address the credibility of our data. Transferability is supported by describing not just the observable behavior but also the context of Bob’s engagement and fraction learning. We also report examples of our coding processes and rationale for what codes were clustered together to form the basis of a claim to ensure dependability. Finally, we attended to rival explanations (Merriam & Tisdell, 2015). When doing analyses of the sessions, we documented claims and counterclaims, providing evidence that each of our claims was not contradicted in our data set. We constructed assertions based on evidence. For each assertion, we sought plausible rival explanations. In the Discussion section, we unpack rival explanations.

“Get the Job Done.”

The first theme that became apparent in our tutoring session data was “GTJD.” This theme became apparent during work where Bob seemed to complete the mathematical tasks out of a sense of obligation to complete the work assigned or to follow instructions. This need was often observable through Bob’s requests for clarification of the task, his work toward fast or straightforward completion of the work, often with behavioral distractors, lower overall cognitive engagement (e.g., less self-talk, less employed solution pathways), and emotional satisfaction of task completion (e.g., lower level emotional engagement, except upon task completion, at which point the student indicated happiness), suggesting an arguably less productive form of mathematical engagement. Get the Job Done was coded in 16 out of 30 of the tasks Bob worked on, which is consistent with Goldin and colleagues (2011) who note GTJD as the most commonly observed structure in mathematics classrooms.

An example of Bob’s interaction with a task in Session 4 illustrates the GTJD engagement cluster. In this task, Bob was asked to make a whole length given a unit fraction of that length. Specifically, Bob was asked to show the length of 1 m given the length that a ladybug (⅙ of a meter) and an ant (⅓ of a meter) traveled in a given time period:

Bob’s behavior in this task is consistent with a student who wishes to quickly arrive at completion (understanding directions, nodding head to indicate completion) and who exhibits a lower overall amount of cognitive engagement (e.g., little to no self-talk). We concluded that the motivating goal, the patterns of behavior in response to the goal, and Bob’s affective pathways and expressions during the pattern of behavior were most consistent with GTJD engagement.

“I’m Really Into This.”

The second theme that became apparent in our work with Bob was “IRIT.” Here, Bob’s apparent motivating desire is “flow,” or to experience the actions involved in addressing the task. Unlike the previous theme, Bob was interested, curious, and intrigued by the actions, processes, and the mathematics of solving the problem, at times so much that he tuned out environmental distractors. Developing connections and a desire to understand are often connected with this engagement cluster; in our data, higher counts of behavioral (increased time in the task; more math), cognitive (more self-talk; employed solution pathways; work to find entry points into the problem), and emotional (positivity; curiosity) engagement were consistent with this theme. Goldin et al. (2011) note “a sense of accomplishment may be derived from achieving mathematical understanding, from solving a difficult problem, or simply from the experience of fascination during active involvement” (p. 551) in this engagement cluster, suggesting an overall more productive form of mathematical engagement. In our data, IRIT was coded in 14 out of 30 of Bob’s tasks across the sessions.

An example of IRIT engagement occurred during Session 6. In this task, Bob was asked to make a whole length given a unit fraction of that length. Specifically, Bob was asked to share three cookie dough logs among five bakers and describe the resulting quantity.

Patterns of Themes Across Sessions and Tasks

The first research question also asks about the patterns of engagement that are evident across tutoring sessions and mathematical task objects. These patterns are illuminated in Figure 1. Overall, Bob’s patterns of engagement appear to be somewhat erratic when looking at engagement as a function of tutoring sessions. For example, Bob displayed GTJD engagement throughout the sessions, with the highest levels within the first five sessions and then again sporadically in the eighth session. Similarly, IRIT engagement clusters emerge in the 5th session and become more consistent in the 7th, 9th, and 10th tutoring sessions. Yet, when we examine engagement by task, a different pattern emerges. Specifically, the task types that consistently supported engagement that supported developing connections and a desire to understand (i.e., IRIT) for this student were: (a) tasks that involved sharing multiple whole items among multiple sharers and (b) tasks that take a unit or non-unit fraction of a unit or non-unit fraction. Arguably, the alignment with this form of engagement with tasks that require at least two levels of units speaks to the mathematical knowledge that Bob was bringing into the tasks as well as the productive challenge he experienced attempting to build a third level of units within his problem-solving actions. We unpack these ideas more in the next section.

OPEN IN VIEWER

Figure 1.

Engagement Patterns by Session and by Task Type.

Pre-Interview to Session 4: Getting the Job Done With Two Levels of Units

Data from both Bob’s pre-interview and his work in the first four sessions confirm that mathematically he was working with two levels of units. First, in the pre-interview, Bob was given tasks that asked him to draw one out of five and three out of four equal parts of a bar, respectively. In each task, Bob created the required number of equal parts in each bar and indicated the correct number of parts. When given tasks that asked him to create the whole bar from a part (e.g. make the length of the whole pan of brownies if this part represents 1/7 of the length), Bob correctly iterated the given part seven times to create the length of the whole. Bob was also presented with a long strip of paper that we called a “French fry”; the teacher folded the fry into three parts and then folded one of the three parts into three more parts. Bob called this part 1/6 of the fry. Losing the three-level unit relationship (one-third within one-third within one whole) suggests that Bob was coordinating two levels of units with fractions but not three.

Bob’s work in the first four sessions was consistent with a student that worked with two levels of units with respect to fractions. For example, we engaged Bob in sharing one French fry among multiple numbers of sharers (i.e., 2, then 3, 4, and 5, then 10) in the first two sessions. Bob accurately estimated the length of one of the shares in each situation and was able to adjust his estimate correctly if it proved to be too short or too long. As discussed earlier in the article, Bob was also asked to re-create the whole from a length that represented one-third and one-sixth during the fourth tutoring session. In these tasks, Bob readily iterated, or repeated, each unit fraction the correct number of times to reproduce the whole in each situation.

We argue that Bob’s coordination of two levels of units was related to his GTJD engagement. His quick solutions, interest in other objects (e.g., binder clips), and lower overall cognitive (i.e., low self-talk, one solution path) and emotional (i.e., lower level emotional engagement, except upon task completion) engagement suggest that the tasks, while accessible to his prior knowledge, did not support more productive forms of engagement or give Bob opportunities to advance his reasoning.

“Really Into” Sharing a Unit Fraction

We noted that Bob was unable to reason about three-level unit relationships in a problem about taking a part of a unit fraction (take one-third of one-third of a French fry) he was given during the pre-interview. Yet, during Session 6, Bob was productively engaged in sharing one-thirteenth of a churro among two friends. To begin the problem, the teacher asked Bob to create a bar (as opposed to creating one for him or giving him a bar to act upon) in the computer program Javabars and split it into 13 equal parts. The teacher then asked Bob to quantify the created part before posing the next part of the task:

We note three features of the task that seemed to support Bob’s engagement as well as his ability to coordinate three levels of units in his problem-solving actions. First, Bob was given a problem with an open representation—one where a representation was not given to him to use to think with. Instead, Bob was asked to create a bar, partition it to create thirteenths, and partition again to create equal shares of each one-thirteenth. Second, the use of two as the number of sharers supported Bob to access his skip counting, a form of iteration, by twos to relate two parts inside each of the thirteenths he created to the original whole churro. Finally, the language and context of sharing seemed to bring forward Bob’s partitioning actions. As opposed to using formal language (i.e., “take one-half of one-thirteenth”), the context of sharing along with the visible three-unit structure (e.g., 2 parts inside each of 13 parts inside of a whole) supported Bob to relate his notion of “two” to the actions of partitioning to create one-half or one-thirteenth.

Two of the task features—“sharing” and open representation—were used in a similar task in Session 8. This time, the teacher extended the range of number values and challenged Bob to share one-seventh of a cookie dough log equally among three bakers at a baking party. After Bob made a bar to represent one whole cookie dough log, he partitioned the bar into seven equal parts. Bob and the teacher agreed that one out of those parts represented one-seventh of the whole, at which point the teacher presented the task:

As noted above, the problem began with a sharing context and an open representation; Bob created a bar that he used to think about the problem and partitioned one of his seven parts of the cookie dough log into three parts. Interestingly, Bob does not partition the remaining six-sevenths into three parts each this time. Instead, he connects his skip counting to the problem, counting by threes to “iterate” the three parts into each of the remaining six-sevenths. However, upon doing this, Bob loses track of the first one-seventh that he partitioned, calling the created quantity “18.” This suggests that, even though Bob was “Really Into” the actions involved in the problems, he was not yet coordinating the units at three levels in his actions. Yet, with the teacher questioning and positioning Bob as competent by encouraging and talking through his efforts in the task, Bob was able to coordinate the units and reason about the problem.

Reaching Into Three Levels of Units: Infusing Language and Verbalizations Into Engagement

In previous sessions, we presented tasks to Bob using accessible task features (e.g., sharing contexts, open representations, number values he was familiar with). We did this to promote Bob’s engagement in creating fractions of unit fractions through his actions and his entrance into coordinating three levels of units. Later, we began to increase the difficulty of the numerical values used in the problems (e.g., shares of 3 or 5 as opposed to 7 and 13 parts of a whole as opposed to 2 or 4 parts of a whole). In the following excerpt, we continue to use these elements of task design within problems involving taking a part of a unit fraction. Yet, we also began to infuse fractional language and explicate the action of taking parts out of the whole. Later, we also infused student verbalizations of imagined actions as a part of the task design. To begin, we started to pose the problems as taking “a unit fraction of a unit fraction” (Hackenberg et al., 2016). In Session 9, we posed such a task to Bob within the context of eating a part of a cookie dough log that a child received for her birthday:

Bob continued to evidence IRIT engagement as he displayed finite differences in his ability to coordinate three levels of units in his problem-solving actions. We relate these differences to the design moves of the tasks. We explicated the action of taking parts out of the whole by requiring Bob to show the part separately from the whole as a part of the task. The computer program facilitated Bob’s ability to remove the part from the whole without destroying it (Biddlecomb, 1994). Bob used his skip counting to iterate and relate five parts to each of the three parts of the whole cookie dough log. In addition, Bob was given a problem with fraction language that he related to his whole number multiplication (“If you break them all into five parts it would be like five times three”). Bob evidenced similar reasoning when we engaged him in a problem about taking one-half of one-sixth of a sandwich (Hackenberg et al., 2016). This time, we challenged Bob to imagine and explain the actions he would take to solve the problem:

Bob: [Constructs a bar and partitions it into six parts, then takes the first part out and cuts it into two parts. Re-explains the iteration of the two parts inside each of the six parts and relates that to six times two as 12].

Bob continuously showed IRIT engagement and changed in his ability to coordinate three levels of units by using or verbalizing imagined problem-solving actions. The specificity of the language (i.e., “taking 1/n of 1/n”) and the request for verbalization of enacted or imagined reasoning and justification (“Explain to me how you are imagining this problem.” “[nodding and smiling] How do you know?”) supported Bob to relate his actions (And then you, um, you take, one out. Then cut that part you take out into halves and take out one of the halves) to his whole number multiplication (“like two times six”) and quantify the fraction quantity as units of units of a unit (i.e., one-half of one-sixth of a whole).