Characteristics of Interesting Mathematics Lessons

AERA 2014 (Philadelphia, PA) 1

Characteristics of Interesting Mathematics Lessons

Leslie Dietiker Boston University

What if mathematics educators knew how to design lessons as stimulating as a chapter

from Harry Potter, where students eagerly await the next installment? Literature is designed to hold the attention of readers, but why not mathematical sequences? Despite some efforts to design math experiences to resemble the work of mathematicians, the typical daily march through content still often sends a meta-message similar to study this math content because you need it to study related content coming later that you also will likely have little interest in or realize is coming (Chazan, 2000). Research indicates that when a student’s curiosity or interest is activated (what Dewey refers to as “catch”), the student is more likely to engage with the material and look forward to more, which, in turn, increases potential learning benefits (Csikszentmihalyi, 1990; Dewey, 1913; Guthrie, Hoa, Wigfield, Tonks, & Perencevich, 2005; Wong, 2007). This suggests that ideally, educative experiences should attend to the aesthetic as well as the rational and logical (Wong, 2007). However, studying the aesthetic dimension of mathematics learning experiences has been largely ignored in mathematics education research and is an area of need (Sierpinska, 2002).

The TIMSS video study suggests that the mathematical structure of a math lesson may also explain the poor performance on math assessments in the U.S. (Stigler & Hiebert, 1999). Comparing the curriculum of U.S. classrooms with those of high-performing countries, Stigler and Hiebert use the metaphor of story to describe key differences:

Imagine the lesson as a story. Well-formed stories consist of a sequence of events that fit together to reach the final conclusion. Ill-formed stories are scattered sets of events that don't seem to connect. As readers know, … well-formed stories are like coherent lessons. They offer the students greater opportunities to make sense of what is going on. (p. 61) This study similarly uses the notion of “mathematical story” to explore the qualities of

enacted mathematics lessons, this time focusing on the aesthetic qualities to inform curriculum design on what stimulates student interest.

Theoretical Framework

In order to analyze how the temporal unfolding of mathematical content builds across a sequence, a mathematical story framework is used (Dietiker, 2012). Similar to a literary story, a mathematical story is the ordered sequence of connected mathematical events connecting a beginning with its end. In this conceptualization, mathematical characters are the mathematical objects brought into existence through reference in the story, such as a linear function. Mathematical action describes the changes to the mathematical ideas or objects of study, such as rotating a function about an axis. Mathematical characters and actions are brought into being in a constructed “space” such as a white board or a coordinate plane, referred to as its mathematical setting.

Another aspect of a mathematical story is its mathematical plot, which describes the way it holds the interest of its audience. When a mathematical story hints of a future revelation, it may spur the formulation and pursuit of questions (“What if the lines are parallel?”) similar to


how a reader of a literary story might wonder how the story will progress (i.e., “Will Romeo and Juliet live happily ever after?”). Thus, the mathematical plot describes the dynamically changing tension between what is already known and desired to be known by a reader as the story progresses (Dietiker, 2012). It enables the description of how the mathematical sequence can generate suspense (by setting up anticipation for a result) and surprise (by revealing a different result than the one anticipated). The transition from asking to answering a question forms a story arc. Since a mathematical story may involve answering multiple questions at any point along a sequence, multiple story arcs may overlap.

Although stories that interest readers vary widely in design, this framework assumes that those that hold a reader’s interest do so by sustaining one or more questions throughout the story while resolving more minor questions of a shorter span at multiple points of the story. A reader may become frustrated when too many questions remain unanswered at a point (which can feel as though the story may never resolve any of them) or when the story repeatedly answers questions immediately after they are raised.

The present study uses a selection of videotaped algebra lessons to examine if and how the structure of the mathematical plots, as coded by knowledgeable mathematics teachers, reveal potential differences that can be explored further in a later study. This study assumes that aesthetic mathematical experiences vary by individual (Sinclair, 2009). It also assumes that the meaning developed throughout a lesson is in part affected by a reader’s goals, experience, and purpose, and thus is individual in nature (Rosenblatt, 1994). Since the audience of an enacted mathematical story in a school is arguably its students, the students’ evaluation of the experience is an appropriate measure of its aesthetic value.

The goal of this study is to examine the structural qualities of mathematical plots found in algebra lessons found in classes with high student interest measures and not found in classes with low student interest measures. It starts to address the question: When math lessons are interpreted as stories with mathematical plots, what plot features (if any) distinguish lessons with high interest measures from those that have low interest measures?

Methods

This exploratory study is a qualitative analysis of enacted 9th grade algebra lessons from the Measures of Effective Teaching (“MET”) video database in order to learn more about the qualities of high and low interest lessons using the mathematical story framework. The availability of the student perception data offers a rare opportunity to compare mathematical lessons found in classes which students describe as interesting with those for which they do not. Data selection

The videos analyzed in this study were carefully selected from the MET video database. While not a random sample, this database is large (over 5000 lessons) and sufficient in its representation of classrooms from different parts of the country to enable interesting differences to emerge. As part of a large study, a Student Perception Survey (“SPS,” developed by the Tripod Project) of more than 80 questions was also administered for a sample of students in the algebra class being taped. In addition to questions that measured a variety of attitudes and dispositions, such as whether the teacher appeared to care for the students, four questions on the SPS were designed to measure the degree to which a student was captivated by the mathematical content and the way it was taught. These 5-point Likert-scaled questions asked students whether they agreed or disagreed with the statements: “This class does not keep my attention – I get


bored,” “My teacher makes learning enjoyable,” “My teacher makes lessons interesting,” and “I like the ways we learn in this class.” The scores for these questions were aggregated for students in the same class to represent a captivate measure for each algebra classroom for the year. It is important to note that the SPS was only administered once and is therefore reflective of a sample of student responses at one point of the in the school year in which the lesson was taught. Therefore, the captivate measure does not indicate how a student felt about a particular mathematical lesson; it is only a measure of how a subset of students felt about the mathematical teaching and learning throughout the school year in that class.

Of all the algebra lessons in the MET video database, those that had fewer than 7 student responses to the SPS were excluded. In addition, to control for whether the students perceived the class as not captivating because they perceive the teacher does not care about their success and wellbeing, those lessons with negative aggregate care measures were eliminated. From the remaining pool of videotapes, the lessons were sorted based on their aggregate captivate measures. Five videotaped algebra lessons of 9th grade sections with captivate aggregate measures that are in the top 10% of all 9th grade math sections and three lessons in the bottom 10% were identified and analyzed for a total of 8 lesson videos. The selected lessons represent a range of typical algebra topics such as solving systems of equations (typically with a given strategy, such as graphing), rewriting variable expressions using the laws of exponents, and factoring. Analyzing the mathematical plots

Each lesson video was transcribed and independently analyzed by three researchers (the lead author and two research assistants) before coding differences were collectively resolved. Each researcher is an experienced middle and/or high school teacher and has taught an algebra course previously. Throughout the coding process, no one on the research team knew which videos came from classes with high or low captivate measures.

After an algebra lesson was transcribed, its coding took three key stages: identifying acts, identifying mathematical questions under pursuit in the lesson, and coding the mathematical story for the progress of these question.

Identifying acts. First, a sequence of mathematical events (“acts”) was identified, representing the points in a story for which we could identify changes to the mathematical characters, actions, settings, or questions being pursued in the story. These acts generally represent a segment of a lesson that has a focus that can be distinguished from what came before and what follows. An act might include all or part of a task. For example, when a teacher introduces and solves a task on the board, this segment, regardless if whether it lasted 10 seconds or 3 minutes, would likely constitute a single mathematical event (since the mathematical question being pursued had changed and likely was followed by something that either introduced a new question or changed the mathematical character, action, or setting). However, sometimes a task can span multiple acts. For example, when a teacher assigns a task, has students work together on it, then pulls the focus back to the board and proceeds to discuss the task, this would be coded as two acts: the first being the portion of time in which the students were able to make progress on the mathematical question, the second being the portion of time in which the mathematical work shifts from the student papers to the front board (representing a shift in mathematical setting). Thus, the acts do not represent equal portions of time, but rather represent steps forward in the development of the content of the mathematical story.


Identifying mathematical questions. Once the mathematical acts were identified, each researcher re-watched the lesson, noted the mathematical questions that were either explicitly or implicitly driving the content of the lesson, and recorded the time in the lesson at which these questions emerged. Mathematical questions may be explicitly raised by the teacher, explicitly raised by curricular materials (such as on a worksheet), or explicitly raised publically by one or more students during a whole class discussion or during work in small groups. In addition, implicit questions, such as those that are not stated in question form but instead are represented in teacher or student statements of goals or are identified through statements of confusion by students, were included. The mathematical questions identified by the individual researchers were then collectively shared and negotiated until a consensus was found.

Coding the progress of the mathematical questions. Once the sequence of mathematical questions were set for a mathematical story, each act of the mathematical story was coded for progress of inquiry (if any), using sub-codes adapted from Barthes (1974) to describe progress throughout a mystery (see Table 1). Table 1 Sub-codes used to describe the mathematical plot, adapted from Barthes (1974). Sub-code Description Mathematical Story Example

Proposal (PRP)

The recognition of a possible mystery, suggesting there is something to learn.

Evidence of a pattern or relationship that might lead to inquiry.

Formulation (FQ)

A question raised by the participant or narrator for pursuit.

A question such as “How can a shape be folded in half?”

Promise (PRM)

An indication that the question will eventually be answered.

A goal such as “We will learn how to halve a shape”

Snare (SN)

An attempt to mislead the reader with a falsehood.

A false statement, such as “All shapes have symmetry”

Equivocation (EQ)

An ambiguity that may lead the participant to a false answer.

A restriction of shapes that allows an incorrect assumption to emerge.

Jamming (JM)

The perception that a question may be unanswerable.

The acknowledgement that it may not be possible to find a line of symmetry.

Suspended Answer (SP)

The delay of an answer by turning attention to a different question.

Turning from an exploration of symmetry to multiplying fractions.

Partial Answer (PA)

Progress toward answering a question, which leaves some part of the question still open.

Such as recognizing the answer for some shapes (such as a circle) but not others (such as a square).

Disclosure (DS)

The explicit answering of a question.

An answer to a question such as “Not all shapes have symmetry.”

Thus, once a mathematical question was perceived, the remainder of the mathematical

story was analyzed for its progress in answering that question (or lack thereof).


Story arcs and their coding. Barthes’ sub-codes offers an opportunity to recognize how a mathematical story provokes curiosity, sustains inquiry, or frustrates the reader to be with the development of mathematical content. For example, when there is an indication of an impending mathematical question, and yet there is no particular question raised yet, this is coded as a proposal, which signals the opening of a mathematical mystery. To demonstrate what a proposal looks like in a mathematical story, consider how a teacher might hint at a deepening complexity without directly addressing it. For example, after defining the greatest common factor, a teacher can ask “What two numbers multiply together to give me 12?” and receives a response of “one and 12.” If the teacher responds with, “Well, we do not want to use one. Four and three, right?”, this teacher indicates that something is different about one. Mathematically, 1 and 12 do multiply to make 12 and yet, as the teacher’s statement suggests, 1 does not qualify to be part of this discussion. Although it is not explicit, this phrase indicates that one might not count as a factor, which hints that there is more to learn about what does and does not count as a factor. This important mathematical question, though, has not been explicitly raised and it appears only as a hint at this point.

To demonstrate how a story arc is then coded, it is then analyzed when this question (if at all) is raised and addressed. A few acts later, 1.5 is offered repeatedly by a student as a potential factor. By stating it multiple times, the student appears to want the issue raised and addressed. Although it is not yet addressed (it is ignored until much later), the formulated question is implied at this point. A few acts later, this question is finally addressed. When a student says the factors of five are “1 and 5,” the teacher states, “right just one times five. There is nothing else that is going to multiply to give me five. Whole numbers. So two point five times two, yes that does work, but that is not what we are talking about. Just integers.” With this, the teacher offers a partial answer that only whole numbers are factors in this case. This is an example of a partial answer since the teacher did not broadly resolve the conditions of what counts as a factor.

Story arcs may include all of Barthes’ sub-codes or only some, and sub-codes may repeat and occur in any order. Each act may also have more than one sub-code since it may simultaneously offer information for the students to answer one mathematical question under pursuit while stimulating the formulation of another. The coding of story arcs using these sub-codes enable the distinction between mathematical stories in which inquiry is sustained through long portions of the lesson and those in which mathematical questions are quickly addressed, leaving little for the student to anticipate. The letters T, S, or R were placed at the end of each code to indicate whether the progress was made by the teacher, student, or researchers (in the case of implicit questions), respectively. For example, ‘PAT’ represents a partial answer provided by the teacher.

The mathematical plot codes per act were organized in a spreadsheet, such as the one shown in Figure 1.


Figure 1. Partial coding of the mathematical plot for a mathematical story, with acts numbered #1-16 represented in columns and formulated questions #1 through 9 (of 20), represented in rows.

In this table, each row represents a story arc, explaining how the formulated question is raised and addressed throughout the mathematical story. The example story arc described earlier can be found in question 6 of this table. In this particular table, both explicit and implicit questions are found, and these questions are either raised by the teacher (such as question #8 in Act 3), one or more students (such as in Act 5 in questions #2 and #8), or the researcher based on the discourse of the class (such as in Act 5 for question #6).

The negotiation of coding. It is important to reiterate that the mathematical story framework does not assume there is a perfect interpretation of a mathematical story. In short, the consensus does not bring with it an assumption that the resulting codes are “the” mathematical story. Instead, it should be recognized as a reasonable interpretation of the mathematical story as negotiated by three knowledgeable mathematics educators. Thus, we accept that there are other interpretations of the mathematical stories found in these videotaped lessons. At the same time, we argue that, while rejecting a postitivist stance of a “true” reading of mathematical stories, these negotiated interpretations based on thorough analysis offer insight into the potential mathematical structures represented in these lessons, that differences in interpretation would likely be minor rather than significant, and that differences between mathematical stories would likely persist. To guard against weak interpretations, the negotiation process included the thoughtful consideration of opposing interpretations (“Perhaps this is not an equivocation. How else could we interpret it?”) and resolution through the analysis of the transcript and videotaped lesson.

Determining the scope of a mathematical story. For this study, the entire length of the lesson that was available on videotape was analyzed as a single mathematical story with the exception of one lesson involving linear functions and word problems. Due to a change in thematization roughly halfway through this lesson, the content was subdivided into two disparate parts. Namely, the first half of the lesson consisted of having the students solve review problems involving graphs of linear functions as part of a “warm-up.” After this, the focus shifted to new

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material involving representing and solving word problems where the teacher presented material, provided examples, and assigned some problems for the students to try. This lesson was not the only lesson with a warm-up; however, because of the significant length of this beginning portion of the lesson (22 minutes), it represented nearly half of the 48-minute lesson. Although considering both parts together as a mathematical story raises important questions for a later study, it was decided to use portion representing new material in this study for consistency and comparison with other lessons in the dataset.

Potential measures of interesting mathematical stories. We identified two measures that literary theory suggests could distinguish potential aesthetic differences in mathematical stories. These are the average density of the mathematical stories (that is, the average number of story arcs that are coded per act of a mathematical story) and the average length of story arcs (that is, the average number of acts for which story arcs are sustained throughout a mathematical story). These measures offer potential because it literary theory suggests that when multiple questions are being pursued, the story might be rich with complexity and thus more intriguing for the reader. In addition, longer story arcs in a mathematical story would suggest that the mathematical lesson might sustain and build interest throughout the story. We therefore predicted that lessons in high-captivate classrooms would have a higher average density of story arcs in each act and a higher average length of story arcs.

Findings

When examining the coded structures of the mathematical plots of the algebra lessons included in this study, some interesting and unexpected patterns emerged. To explain, this section first starts at a top level to present some general findings across all eight mathematical stories. Then, to describe mathematical plot structures that potentially distinguish algebra lessons, the next part compares the structures and codes of two lessons that represent extremes to highlight potential qualities that may make a mathematical plot more appealing or interesting. Top Level Findings

This section describes some of the qualities of the mathematical plots that may point to aesthetic differences in the algebra lessons included in our dataset. Across the mathematical stories, we found a wide variance in most measures of the mathematical plots. While there were some differences between the measures of mathematical plots in high-captivate and low-captivate classrooms, the potential measures of interest did not distinguish these groups of mathematical stories as we had expected. This section starts with a description of the acts and story arcs found in these mathematical stories. It then presents the observations regarding the potential measures of interest of these mathematical stories. Note, however, that the small number of lessons in this study does not allow for generalization.

Findings regarding acts and story arcs. The eight selected mathematical stories had between 7 and 20 events (“acts”) and the median was 15 acts per story. The mathematical stories found in high-captivate classrooms tended to have more acts than those in low-captivate classrooms (with a median of 16 acts and 12 acts, respectively). Since the videos had consistent length (approximately 45 minutes), this indicates that the low-captivate lessons tend to spend more time on average per act, and thus take longer to make mathematical progress.

The number of story arcs in the mathematical stories also varied. In our data, the smallest number of story arcs found was 11, the highest was 22, and the median was 18. Interestingly, the lessons found in high-captivate classrooms on average had fewer story arcs overall than those in


low-captivate classrooms (with 16 and 20 story arcs, respectively). Thus, based on this small set of data, it suggests that lessons found in high-captivate classrooms may tackle fewer mathematical questions overall but may make more points of progress while answering them.

An interesting feature of every mathematical story in this study was the existence of story arcs that last for only one act. This happened when a mathematical question is raised and answered within the same act (referred to as momentary story arcs). Although momentary story arcs were found in every lesson, there was one mathematical story that had only a single occurrence (found in a high-captivate classroom), while another had 14 momentary story arcs (found in a low-captivate classroom). However, when considering the median of the proportion of momentary story arcs to the total number of story arcs in each lesson, both high-captivate and low-captivate lessons had, on average, one-third of story arcs within a single one act.

Comparing potential measures of interesting mathematical stories. The measures average density and average length of story arcs were found to have wide variance as well. The smallest average density of the eight mathematical stories was 1.42 story arcs per act while the highest average density was 5.81 story arcs per act. Interestingly, both of these extremes were found in lessons from high-captivate classrooms. Also, within our limited dataset, we were surprised to note that the mathematical stories from classrooms with low-captivate measures generally had higher density than those in high-captivate classrooms. Possible explanations are offered in the discussion section.

Likewise, the average length of story arcs measure did little to distinguish between the mathematical stories found in high-captivate and low-captivate classrooms. While the lowest average length of story arcs measure (2 acts) was found in a low-captivate classroom and the highest (4.65) was found in a high-captivate classroom, the median average length of story arcs measures for the two groups were virtually identical, at approximately 3.6 acts per story arc. Again, possible explanations are offered in the discussion section.

To consider how these measures distinguish mathematical stories, the lessons with extremes for average density and average length of story arcs measures are listed in Table 2. Table 2 Ranking of the highest and lowest lessons along potential measures of interesting stories. Lowest Next lowest Next highest Highest

Which lessons appear at extremes in regards to the average density?

Exponents (1.42)

Solving system by graphing (3.71)

Linear word problems (5.71)

Factoring (5.81)

Which lessons appear at extremes in regards to the average length of story arcs?

Linear word problems (2.00)

Exponents (2.45) Solving system by graphing (4.33)

Factoring (4.65)

The results of this table are very surprising. Not only did the extremes represent only four of the eight lessons, two of the lessons (“Solving systems by graphing” and “Linear word problems”) appeared at both the low and high ends of the scales. Therefore, these potential measures of interesting mathematical stories do not appear to make the same distinction between lessons. In addition, the “Factoring” lesson had the greatest measure for both interest measures while the “Exponents” lesson was ranked with either the lowest or second lowest interest measures. Based


on our assumptions stated earlier, it might be reasonable to expect the “Factoring” lesson to be found in a high-captivate classroom and the “Exponents” lesson to be found in a low-captivate classroom. Instead, surprisingly, both lessons came from classrooms with high-captivate measures.

Distinguishing Qualities of Mathematical Stories

Regardless of the fact that the “Factoring” and “Exponents” lessons were both found in classrooms with high-captivate measures, we argue that they represent mathematical stories with considerably different structures of content with potentially different aesthetic dimensions. This section analyzes the contrast in these two mathematical plots to offer information about how the mathematical plot framework might reveal potential aesthetic differences. This section will compare these lessons along three dimensions: the overall structural differences, the forms of story arcs, and the differences in the sub-codes found in these lessons.

Overall structural differences. In order to analyze the structural differences of the content in these mathematical stories, the sub-codes in Table 1 were assigned different colors without regard to whether the sub-code was attributed to the teacher, students, or researchers. The cells were then colored to represent the sub-codes contained in the cell. The resulting diagrams of the Factoring and Exponents lessons are displayed in Figure 2. If there were two different sub-codes, such as “formulated question by the teacher (FQT)” and “partial answer by the student (PAS),” then these two colors (red and light green, respectively) were represented in the cell so that the first sub-code found in the transcript appeared on the upper left and the second appeared on the bottom right. When three different sub-codes from Table 1 appeared in the same cell, in each case, we noted that the “partial answer” sub-code was present. Since the partial answer sub-code was found in most cells with sub-codes and was by far the most prevalent sub-code, this color was dropped to allow the less-common sub-codes to be brought into focus. No cells contained more than three types of sub-codes.

The visual differences between these two mathematical plots in Figure 2 are striking. While the Factoring lesson appears busy, overlapping, and complicated, the Exponents lesson appears thin and disconnected. The higher density of the Factoring lesson is evident when looking vertically for any particular act. For example, in Act 7, the class is entertaining 10 different story arcs at the time. In contrast, the greatest density of the Exponents lesson is 3 story arcs at one time in Act 6. Beyond the “thicker” appearance of the Factoring lesson, however, the difference in density of these two mathematical stories can be also recognized in the graph in Figure 3, which displays the number of story arcs in play in each act across the lesson. This graph represents the shape of mathematical inquiry throughout the lesson. While the Exponent lesson appears nearly flat across the lesson, the graph of Factoring lesson, in contrast, shows an interesting rapid increase of mathematical questions in Acts 1 and 2, a leveling off for Acts 3, 4, and 5, and yet another increase in Acts 6 and 7. Although some of these story arcs are finding closure during this portion of the story, this increase indicates that for nearly the first half of the Factoring lesson, there are more mathematical questions being raised than answered.


Figure 2. The mathematical plots of two algebra lessons from the MET video database. The top lesson is focused on factoring while the bottom lesson is on the laws of exponents. Color codes: purple (proposal), red (formulated question), light green (partial answer), yellow (promise of an answer), dark green (disclosure), and pink (equivocations).

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Figure 3. The density of story arcs per act for the Factoring lesson (green) and Exponents lesson (red). Note that the Factoring lesson has 16 acts while the Exponent lesson has 19 acts.

Forms of story arcs. There are several interesting observations that can be made about

these mathematical plots by contrasting the story arcs of algebra lessons. Both the Factoring and Exponents lessons contain story arcs that end with partial answers, which occur when the research team compares the ending conclusions of the story arc with the mathematical question and considers the answer incomplete in a mathematical sense. Admittedly, these decisions were based on our advanced knowledge as mathematics teachers and are not indicative of failures of these mathematics teachers. In many cases, it means that there are additional ways to make sense of the mathematical question under pursuit or that the question can continue into more complex content. For example, the question “how do we factor?” was a question that likely guides an entire unit of study in this particular algebra classroom. Given that we saw a lesson near the very beginning of the study of factoring (starting with greatest common factors of whole numbers), this question likely remains open for a much broader mathematical story (indeed, even beyond a single course).

Unlike the Exponents lesson, however, the Factoring Lesson has two story arcs that remain open at the end of the lesson. This occurs for questions in which, during the progress of reaching an answer, there is an indication by the teacher that there is more to be learned and that the students will learn more about the topic in a subsequent lesson. The explicit indication that it will continue to develop is important. This “promise” builds anticipation and interest in what is to come.

Both lessons also contain story arcs that are sustained for more than one act (referred to as sustained story arcs). However, the Factoring Lesson had nearly double the percentage of sustained story arcs (14 of 20, or 70%) than the Exponents Lesson (4 of 11, or 36%). Both lessons had at least one story arc that was sustained for at least half the lesson (which I will refer to as a dominant story arc), but while the Exponent lesson had only one dominant story arc, the Factoring lesson has at least six dominant story arcs and there is at least one dominant story arc in play for every act.

Finally, another important difference involves how answers are positioned in mathematical stories relative to the question. In the Exponents lesson, only one story arc is asked (“Is there a law for when the bases aren’t the same?” in Act 3) for which its progress is


delayed (until Acts 6 and 7). Every other question is followed immediately by some development toward an answer either immediately within the same act, in the next act, or both. In contrast, the Factoring lesson has many story arcs that move through a cycle of progress and no progress. To represent when the story arc fails to make progress in an act although the focus of the story remains relevant to the question at hand in the diagrams in Figure 2, an arrow is used between codes. For example, in question #1 of the Factoring Lesson, no progress is made between Acts 3, 4, or 5 even though the lesson still is focused on factoring. When a story shifts attention and appears to a reader that it will not address a particular question, no arrow is given. This only occurred one time in these lessons, in Act 10 for question 13 of the Factoring lesson (“What is 5x3 + 25x2 written in factored form?”). Note that all of the dominant story arcs contained in the Factoring lesson contain at least two periods of time during which no progress is made.

Types of sub-codes found in these stories. In terms of Barthes’ (1974) sub-codes, a few stand out in the way they may indicate suspense and/or anticipation by a reader. In addition to the proposal (PRP) sub-code, which was discussed earlier, the promise of an answer (PM) sub-code indicates that an answer is forthcoming, and thus gives a reader something to anticipate. These sub-codes were both found multiple times in the Factoring lesson. There were four story arcs that began with proposals and three with promises of an answer. Interestingly, they are mostly found in longer story arcs, possibly to enable interest to be sustained. In contrast, the Exponents lesson only had one promise of answer at the start of the mathematical story in the form of a goal statement indicating that the lesson would focus on the laws of exponents.

Another sub-code that offers the potential for curiosity and intrigue in mathematics is equivocation. This occurs whenever an ambiguity is raised in the pursuit of an answer for which the reader is able to make a mistaken assumption. When this happens in a story, there opens an opportunity for the reader to be surprised by the outcome by learning that the assumption was incorrect. In the Exponent lesson, no equivocations were found. In the Factoring lessons, five equivocations arise: once in question 2 (“How do we factor?” in Act 12), once in question 8 (“How can I figure out what they have in common?” in Act 3), and three times in question 9 (“Does the GCF have to be one of the numbers?” in Acts 3, 5, and 8). In the case of question 2, this equivocation arises when the teacher addresses a student observation about the need for parentheses. The teacher explains that you need parentheses, “because you are pulling something out. And you will end up distributing something back through.” This statement was recognized an equivocation because it was ambiguous. One interpretation would assume that factoring then include multiplying the factors as well. While the teacher may not have meant this, the possibility remains that a student may have interpreted this statement that way. In this case, the equivocation is never addressed.

It is worth mentioning that the equivocation sub-code is similar to another, snare, which also involves misleading a reader. However, in the case of a snare, the story explicitly and unequivocally misleads the reader. In contrast, equivocations never explicitly mislead but instead offer open interpretations and ambiguous references that enable a reader to be misled. Neither the Exponents nor Factoring lessons contained a snare, although it should be noted that some other mathematical stories in our analysis did contain one or more snares.

Discussion

This exploratory study compared the mathematical plots of algebra lessons found in classes with high-captivate measures to those in classes with low-captivate measures.


Throughout the analysis, the coders did not know which lessons were which. When all of the mathematical stories were analyzed, we were surprised to learn which were from high-captivate classrooms. As the findings described in the previous section, the mathematical stories in high-captivate classrooms were found to contain striking differences in structure. Many of the algebra lessons, such as the Laws of Exponents lesson, did not contain any plot features that would signal high interest, such as high density, long story arcs, or special codes, such as equivocation, jamming, and snare. This result makes us question whether the aggregate captivate measure of the Student Perception Survey, which were gathered on a single day of a course, can reflect any quality of a particular lesson from that course. Throughout a year, the level of interest of the students and the interesting quality of the lessons may change. This study suggests that future analysis would benefit by attributing captivate measures directly to the lesson of study rather than a group of lessons for a course.

Another question that was raised by this surprising result was along the lines of what the captivate measure measures. That is, when students respond to the questions “This class does not keep my attention – I get bored,” “My teacher makes learning enjoyable,” “My teacher makes lessons interesting,” and “I like the ways we learn in this class,” we wonder if they are signaling their overall attitude towards mathematics and not how the content is taught and/or learning in their class. While we recognize ways the questions direct attention to the class and the teacher specifically, we also could interpret these questions to reflect some broader statements about mathematics. Because of this, the context beyond the classroom, which includes social support for enjoying mathematics, could be a large factor in these measures. In fact, we suspect that some of the high-interest mathematical plot features that we found in lessons with low-captivate measures could represent an effort by teachers to improve the mathematical disposition of their students.

One expected difference that was not supported by the data was the particular sub-codes found in the lessons. In particular, snares and equivocations, sub-codes that represent opportunities for surprise, were found in both high- and low-interesting lessons (although not in the example in Figure 2). In fact, the most snares occurred in a low-interest lesson due to a high number of errors that went uncorrected for at least 10 seconds of time.

Although this study did not identify key differences between the sets of algebra lessons in high-captivate classrooms and low-captivate classrooms, it did offer some insight into the structure of algebra mathematical stories. First of all, as shown with the Factoring and Exponents lessons, the structure of content in algebraic mathematical stories can vary dramatically. This study demonstrates that algebra lessons differ in how questions are raised, how many questions are entertained at any point in the lesson, and answered as well as how the story arcs overlap or not. In addition, some lessons contain supports that can offer anticipation or sustain attention to a question under pursuit, while others do not. It would be interesting to learn if this variance is also found in other content strands of mathematical stories. Also, what other structures are found in our mathematics classrooms? And which of these structures are viewed favorably in terms of student interest?

Limitations

While some interesting dimensions of the structure of mathematical plots of algebra lessons were identified, this study has several factors that limit its usefulness. For example, whenever a lesson is interpreted separately from what comes before and comes later in the broader mathematical story, it is certainly possible that the story arcs were started before the


lesson of focus and that questions formulated but not disclosed during a mathematical lesson may reach closure in a subsequent lesson. Due to the nature of the data accessed in this study, the prior and subsequent lessons were not available. This is particularly problematic for lessons that appear to be review of previous topics - like a retelling, but as a new audience member, the research does not know upon what the statements of the lesson are built.

It is also important to point out that this framework does not capture any non-mathematical factors that may influence student motivation. For example, this analysis would not identify if, for example, the use of technology or the teacher’s personality were important factors that students used as a basis for rating a math course as interesting. In addition, since the SPS measures were not reflective of the particular lesson analyzed and instead reflects broad student attitudes at a single point of the school year, there may be no relationship between the SPS measures and the mathematical plot as represented by our analysis. Therefore, we cannot associate our findings with the increased interest, nor that any particular mathematical plot can lead to increasing interest in students. Therefore, future studies of the mathematical plots of lessons that interest students should be identified using student measures that reflect the particular lesson and should measure potential other aspects of the classroom environment that may influence a students’ motivation.

Unfortunately, due to the nature of the camera used to film the lessons, many student utterances were not recorded. While they often could be inferred by a teacher’s response, such as “2? Right”, this severely limited the recognition of student mathematical questions and other contributions for the mathematical story. It became clear to us that future research in this area will need to carefully record audio so that each student’s utterances can be included for analysis.

Conclusion

By studying the coherence and aesthetic in enacted algebra lessons, this study begins a line of research that eventually can lead to improving mathematical stories and enabling students to learn more content and develop productive dispositions toward mathematics. Our analysis shows some mathematical plot differences emerging that may offer aesthetic differences for students. These patterns of mathematical structure can offer insights into how sequences of mathematical activity within lessons can temporally build to stimulate and sustain student inquiry. However, this study does not account for other motivators that may also account for student interest measures. Future research should examine differences in lessons students found interesting compared to those the same students judged as not interesting.

Just as a literary story can appeal to some readers and not to others, this study assumes that mathematical stories may be interesting to some students and not others. However, as is evidenced by the enthusiasm shown for Harry Potter, there are some stories that have wider appeal than others. If the mathematical plots that have wide student appeal are studied, what is learned can be shared with teachers and other curriculum designers to improve mathematics lessons for students.

Acknowledgements: I would like to thank the MET New Scholar Award (funded by the William T. Grant Foundation) for funding this study and enabling my access to the database of videos. In addition, I very much appreciate the assistance of Katie Jenks and Samantha Rabinowitz throughout this study.


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Characteristics of Interesting Mathematics Lessons

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