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Doing Math Case Study

“What do we learn about when we learn about the processes of doing mathematics and science? 

Introduction

This case describes pre-service math and science teachers’ doing mathematics in an advanced course called Perspectives and Research in Teaching Mathematics and Science: A Focus on Modeling. For teachers and teacher candidates, we hope this case provides a vicarious experience the of doing and reflecting that now occurs more often in our program as a result of this capacity building project. The case work was motivated by an interest in understanding the pedagogy of doing applied in teacher education, and exploring the claim that a purposeful experience in doing can have impact on future teachers' teaching mathematics practices (see Common Core Mathematics Standards).

The mathematics problems selected for the doing module that this case is based on will be recognized by some as Split-a-Pile problems. The Split a Pile Problem Set the class worked on was organized as a sequence of games directed by Joshua Zucker for the Julia Robinson Math Festival. The problems/games range from trivial counting problems to more complex problems with no known general solution. Dr. D, a math professor and Dr. C, and associate professor of education, selected this problem set after: (a) considering the goals for the course, (b) discussing the nature of modeling in mathematics, (c) brainstorming tasks with potential for eliciting active engagement in modeling practices that did not entail the traditional application of algebraic models to word problems, (d) taking into account Dr. D's interest and expertise in Combinatorial Group Theory, and (e) considering what would be engaging mathematics content accessible to a class of teacher candidates of varied math backgrounds.

The class under investigation was made up of a group of five students, three learning to teach science and two learning to teach math. They met weekly in the evenings for two and one-half hours during a regular semester. The stated aim of course was to develop a deeper understanding of the nature of the practices of mathematics, with an emphasis on modeling. In addition to the doing math module, the other course experiences included an examination of historical examples of doing math and science and an exploration of practices as described in today’s curriculum standards, both the Common Core Math Standards and Next Generation Science Standards.

Five sessions were dedicated to engagement in the doing math module and  Dr. D, and Dr. C worked on the problems together with the class. The group sat in a semi-circle or around a table in front of a whiteboard. The whiteboard was in the corner of a rather large room which gave meetings the feel of a study group rather than a typical class. Board-work with markers was common during the doing math class sessions. Often someone was at the board illustrating some aspect of the conversation, whereas powerpoint presentations, web documents and teaching videos were the focus of the "regular" sessions.

Several interlaced activities were identified from our video case analysis as potential instructional landmarks in their doing. These activities are presented below in roughly the order they emerged, acknowledging that there is much overlap.  The activities presented are:

Getting Started is a period of getting to know the problem set and negotiating the expectations for solutions.

Testing Strategies represents a series of working sessions focused is on winning or solving each game in the set.

Using Models shows how representations are used to model games and solutions sets.

Building Understanding Together is an example of the kind of collaborative discourse that emerged over time as the doing together unfolded.

Reflecting presents some evidence that the experience provided an anchor for discussion of practices and opportunities to think about teaching.

The Central Problem

“Which player has a winning strategy?”

The games involve two players (Alice and Bob) and a pile of objects (matches, pebbles, paper-clips, beads, etc). Players take turns splitting an original pile and the resulting piles until one player can’t make a move. Each game has a unique set of rules (e.g., in Game 2, piles of 2 can’t be split). The winning player prevents his opponent from making a move.

Solving the problem of each game entails determining a winning strategy for either Alice or Bob. Initial problems are simple cases employing mostly low-level arithmetic and counting that give way to more complex problems with non-trivial solutions, thus covering a range in cognitive demand (Stein, Smith, Henningsen, and Silver, 2000). For a first-hand account of the problems and a discussion on how they can be used in the classroom see Zucker and Davis (2007)

Chapter 1: Getting Started

After several trials with Game 1 the class determines that the outcome of the game is determined by the starting number of matches and the "solution" is split into two cases (odd and even starting piles). As one student said, "The initial game and its solution are largely straight forward and simple. The game is a simple countdown, where players only remove matches . . . and the game is decided simply by who takes the last match."

However, attempts to formalize their observations and intuition into an acceptable solution in general terms resulted in a range of  proof methods and an unplanned discussion of the nature of proof. Through this a collective sense of what was expected for presenting solutions emerged.

As you watch the clip think about ...

When can a diagram, representation, or model of a solution serve as a proof?

Chapter 2: Testing Strategies 

After getting familiar with the nature of the problems and how to play the games the class often acted out game play. Two class members would play a game to demonstrate their strategy using words and pictures as a form of think-aloud while others followed along and made observations. This naturally occurring mathematics fishbowl emerged as a common way to informally present and test a strategy on new problem cases. These moments are interpreted as opportunities to try ideas and explore solutions. Notice how everyone is responding to and contributing to S3's strategy. These collaborative sessions differed from a more formal presentation of a strategy where one student would "present" their idea and the class would work to understand and critique it. One student reflecting on this reoccurring activity in our doing found it inefficient noting that more time (perhaps too much time) was spent playing the games rather then actually learning how to solve them.  However, the phase of activity seemed necessary step, especially in early phases of doing.

As you watch the clip think about ...

Are there instructional advantages or drawbacks to having students do problems in real time, as compared to presented prepared answers to questions?

Chapter 3: Using Models

Working the problems almost always resulted in producing a game tree like the one below to model a particular game or strategy.

slide3

The trees became the groups shared/accepted method for modeling a complete game to "see" a strategy. Applying the tree strategy resulted in several trees being required for each case. With small starting piles, drawing every possible tree showing the desired result  became an accepted solution method. However, as the games be came longer (involving more branches and requiring more trees than could be drawn easily) the tree models became cumbersome. The table and discussion in the clip below reveal a concomitant shift from tree-based reasoning to analyzing the numbers that are part of solutions and the possibility in strategies that control the numbers in the outcome (e.g., for N=22, if Bob can keep a 6 in the solution he wins). Notice the table in clip shows the outcome space for 4 cases (1 win for Alice, 3 for Bob) revealing the pattern in the occurrence of 6. The table-model was initiated by Dr. C as a conscious attempt to use a representation to model the problem in a way different than  the trees.

As you watch this clip think about ...

How different models can reveal different facets of a problem

Note about the table model. What is on the board is a table of game outcomes or cases with Alice winning in column #1 and then Bob winning in columns 3-5.  The table is not big enough so the column entries are incomplete. 

Chapter 4:  Building Understanding Together 

Over time collaborative problem solving moments became more normal rather than the exceptional. That is, the entire group focused more on a common problem or developing a shared strategy rather than working independently on their own ideas while others presented (as was more typical early on). In the example below the group is building a new strategy for a more complex problem by building from the prior strategy of controlling outcomes. The new idea about the possible role of triangular numbers has been taken up by the entire group. A table-model is again used to show a pattern that highlights the control strategy and triangular numbers. The clip shows that everyone is involved in the discourse as they build an understanding of the strategy and there is a sense of real possibility that it works. Interestingly, this class lead to motivation for a special presentation in the following class by Dr. D. on computational methods that could ease the burden of on working through the cases. 

As you watch this clip think about ...

What is the role of a model in supporting the discourse of reasoning and argumentation?

Chapter 5: Reflecting

The module concluded with a planned reflection on the process of doing mathematics. The teacher candidates were explicitly asked to use their experience in doing as a basis for explaining their ideas about doing. The clip below shows S1 summarizing from his point of view the steps we went through in doing mathematics. For him, the doing involved (1) exploring, (2) pattern identification, (3) hypothesis generation, testing and searching for counter examples, and finally (4) formulating a solution with proof.

As you watch this clip think about ...

Your model of the process of doing mathematics.

Following the candidate summaries of doing Dr. C pressed the group to specifically discuss the role of models and modeling. One student's description of the process  was still on the board clearly showing that creating representations [trees] was definitely part of the process for her. Interestingly, she positioned modeling as an act of representing [diagraming] an outcome that comes after the exploring phase [of game play] and after considering prior knowledge. This suggests for her that modeling is a personal representational act that comes after rather than during problem solving.  Interestingly, as this conversation continued, and after everyone weighed-in, Dr D proposed an alternative model of modeling that captured some of the other group member's ideas. He proposed that modeling entails: (1) transforming the initial problem into a new problem (i.e. P1 => P2), (2) the new problem is then solved (P2 => S2) because it is known or a simpler case (i.e., provides a link to prior knowledge), which leads to (3) inferences about the solution to the original problem ( S2 =>S1).

A noteworthy outcome of this reflective discourse was that all participants were actively engaged in the reflection and referred to the module activities, and they all could articulate their own model of doing.

The students' ideas about doing and teaching and modeling are, however, best captured in their reflecting through a final writing assignment. Below are some examples of written evidence that the experience can serve as an anchor for thinking not just about doing but about pedagogy as well.

Written Reflection 1 - The importance of content in doing

This student's comments reveal an emphasis on evaluating the content relevance of the doing tasks [with a view of content distinct from that of practice]. He contrasts the doing [to learn content] with his valuing of the direct instruction segment on curriculum standards for learning about modeling.  In his own words,

"The activities involving splitting piles of pebbles, while fun, did not have much effect on modeling for me. It was useful experience in analyzing a new problem and assessing my peers and my own understanding. The problems themselves were not all that engaging since they were fairly easy for me to figure out and I couldn’t figure a way to tie these into a situation that high school students would be solving them or similar ones. Learning about the current and past focuses on modeling in high school curriculums was the most engaging aspect of the course for me." 

Written Reflection 2 - Teacher learning about modeling

For this student the doing activity along with other aspects of the course raised her awareness of the role of modeling in doing math, and perhaps more importantly, in teaching math.  She wrote,

"The experience in this class has resulted in me creating a definition of modeling that I have not considered.  Before this class, I felt that a model is a minimized object of something.  My definition has now evolved to modeling being a visual representation of something that we cannot see in real form.   Modeling provides us with a visual method of seeing something as a whole instead of a small part of it and I feel is a crucial aspect in teaching middle and high school students.  Without modeling, students would have a difficult time visualizing certain content that is outside of the classroom."

Written Reflection 3 - Thinking about teaching modeling

This student in reflecting on her experience of the open-ended and responsive pedagogy enacted in the doing module expressed a personal planning dilemma.

"If I am trying to teach students how to model, I would first need to make an important decision: am I going to first provide them with the background knowledge and and skills necessary to model, examine, and solve problems; or am I going to set students free to examine the problem in any way they see fit. Each option provides its own benefits and drawbacks."

Conclusions and Recommendations for Future Work

What do we know about the nature of the doing module at this point in time?

The problem set and collaborative teaching strategy did stimulate authentic doing of mathematics, it allowed time for content mini-lessons to emerged authentically in the process, and afforded the instructors opportunities to be responsive to the directions an interests of the students ideas. In example, the keen insight of Dr D. supported working the more complex games based on students' ideas and easily diverged into relevant investigations of proof, the study of related problems (e.g., NIM, Chomp, etc ), and an exploration of computational strategies used at higher-levels.  Dr. C noted that this support could not have happened immediately in context of problem solving had he been teaching alone.

At a finer grain size, our analyses of the video dat show evidence that ideas and strategies are threads that run through the module and that they seem to provide coherence to micro segments of activity, and that there are identifiable shifts in the groups activity initiated by both the instructors and the students. With a more nuanced focus on the idea and strategy threads we are also able to confirm that the doing supported general principles of constructivist pedagogy in that the candidates: (1) actively constructed personal knowledge of problems and possible solution strategies, (2) listened closely to ideas and perspectives of others, (3) developed and assessed alternative ideas and strategies, (4) made personally meaningful choices about where to go next on the basis of the ideas already on the table, and (5) reflected on their experience.

What did these teacher candidates learn about the processes of doing mathematics by engaging in doing mathematics?

  • There is some evidence that they now have personal ideas about doing involving steps or stages that reasonably map to the CCMS and NGSS practices.
  • Some developed new ideas about the nature of modeling as part of the doing process.
  • Some developed ideas for planning doing as pedagogy, citing issues around task content, open-endedness, prerequisite knowledge, and instructional time.
  • And in combination with the other class activities some developed deeper understanding of the nature of practices as explicitly presented in the CCMS and NGSS.

Future considerations

First, we suspect that a deeper analysis of the impact of the collaboration between Dr. D, the math faculty member, and Dr. C, the math educator, will show that each had a particular influence on the shifts in activity and contributed to a productive problem solving experience in ways that would not be achieved otherwise.

Second, the students had some specific critiques concerning their experience that seems counter to the stated course goals and instructors' experiences, and these deserve more attention. Specifically noted in journal entries and the final prompt to reflect on their experience where these points:

  1. The doing for them lacked content learning goals (i.e., no math learned)
  2. They were uncomfortable with the doing at some points, describing a desire to have goals for the doing more clearly articulated each session
  3. Working on a problem with no known solution was frustrating and they felt deceived when one problem was left unresolved, because they expected to get to a solution.
  4. They did not like engaging in the process with out a defined endpoint

Third, it is not clear that five class sessions is the optimal dose or amount of doing relative to the goal of doing to learn practice. It seems that this is a worthwhile beginning, but we have questions about the foundation that has been created and where to go from here.

Finally, there are may who have proposed actively involving future teachers in authentic disciplinary work for many purposes, most reason that you can't teach what you have not experienced and/or learned, and that many candidates are deficient in such disciplinary experiences. With that as a backdrop we hoped to learn something about what teacher candidates actually might learn from authentic doing with a spot light on new ideas about practices. We remain positive about the merit in this approach in combination with other instructional experiences, however we are even more aware of the need for clear descriptions of what comprises a framework for doing that has pedagogical relevance. Will the personally meaningfuly and experienced-based models of doing these teachers left this experience with help them teaching practices?