The following passage is quoted from: Teacher Investigations of Students’ Work: The Evolution of Teachers’ Social Processes and Interpretations of Students’ Thinking, a doctoral thesis by Michelle Heger Chamberlin. It provides information on the background and design principles for Case Studies for Kids.
Another source of information can be found in Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching Chapter 13, “The Case for Cases” http://tcct.soe.purdue.edu/books_and_journals/models_and_modeling/powerpoints/ch13.swf
Case Studies for Kids were originally
designed as a result of a multi-tiered teaching experiment (Lesh & Kelly,
2000; Lesh, et al., 2000). The overall goal of the research was for middle
school teachers to collaborate with researchers in developing performance
assessment activities, accompanying teaching materials, and respective design
principles so that other teachers could create similar activities. The activities
were to focus on deeper and higher order understandings of approximately ten
major mathematical ideas at each grade level. The research was conducted over
a 15-week period in which teachers met with researchers weekly for two-hour
seminars and worked together to express, examine, compare, test, refine, and
reach consensus about activities that were shown to be excellent for performance
assessment of their students.
The main criteria for judging the activities in development was to ask, “Do the students’ solutions provide useful diagnostic information about the students’ mathematical understandings?” In other words, when the teachers observed their students working on the activities and when the teachers examined their students’ final solutions, did the observations and solutions reveal useful assessment information? To engage the teachers in cycles of expressing, testing, and revising their performance assessment activities and their associated models, the researchers involved the teachers in various iterative activities:
As the teachers and the researchers worked to define rules for designing effective performance assessment activities, they began to realize that the most promising activities were ones that required students to reveal their thinking and to continue thinking in an increasingly more productive manner while only needing minimal intervention from the teacher. Specifically, the teachers and the researchers began to realize that the activities had to be (a) self-adapting in that the students would be able to interpret the activity meaningfully from each of their different levels of mathematical ability without needing the teacher to bring them all to a mathematical starting point first, (b) self-documenting in that the students’ solutions would naturally reveal how they were thinking about the problem, and (c) self-monitoring in that the students could assess their thinking on their own and could continue thinking in productive ways without relying on the teacher or the researcher to prod them on. The teachers and the researchers then began to realize that in order for the activities to be self-documenting, the most successful mechanism was to ask the students to create complex products such as descriptions, explanations, or procedures for interpreting significant mathematical situations. By the end of the study, the teachers and the researchers had refined their three criteria into six guiding principles for designing Case Studies for Kids.
To develop Case Studies for Kids, designers rely upon six design principles that are based on the work of the teachers and the researchers described above and that have subsequently been refined by Lesh et al. (2000). Each principle is briefly described in this section and illustrated by referring to the case study, Departing On Time (See Appendix B). Departing On Time presents students with departure times for five different airlines. The client for the case study, the Ridgewood High School Spanish Club, needs help selecting an on time airline for an upcoming study abroad trip. Thus, students have to develop a procedure that the club can use to rank the five airlines from most-likely-to-depart-on-time to least-likely-to-depart-on-time. The activity is specifically designed to help students develop conceptual foundations for statistical concepts such as mean, standard deviation, spread, and frequency.
The first principle for designing a Case Study for Kids is called the Model Construction Principle. This principle ensures that the solution to the case study requires the construction of an explicit description, explanation, procedure, or justified prediction for a given mathematically significant situation. Such products externalize how the students interpret the situation and also reveal the types of mathematical quantities, relationships, operations, and patterns that they take into account. In the Departing On Time case study, students are specifically asked to “Develop a procedure for ranking the five airlines in terms of most-likely-to-depart-on-time to least-likely-to-depart-on-time.”
The second design principle is the Reality Principle. This principle could also be referred to as the meaningfulness principle, and it relates to two important characteristics of a case study. First, it requires the case study to be designed so that students can interpret the activity meaningfully from their different levels of mathematical ability and general knowledge. In the Departing On Time case study, students relate to the idea that they want to select an airline that is more likely to leave on time, and they quickly realize that lower numbers in the table indicate more punctuality. Second, this principle requires the case study to pose a problem that could happen in real life. In the Departing On Time case study, selecting an airline that is most likely to depart on time is a pragmatic problem.
The third design principle is the Self-Assessment Principle. This principle ensures that the case study contains criteria the students themselves can identify and use to test and revise their current ways of thinking. Specifically, the case study should include information that the students can use for assessing the usefulness of their alternative solutions, for judging when and how their solutions need to be improved, and for knowing when they are finished. For the Departing On Time case study, students can carry out strategies such as finding the total or average number of minutes late for each airline or counting the number of times that each airline departs on time. Then, they can return to the data in the table to self-assess whether the results of their calculations seem to reflect what they visually see in the data. As an example, students often begin this problem by finding the average number of minutes late for each airline. When they find that the averages are all nearly the same, they return to the data and begin noticing that although the averages are the same, the airlines differ in the amount of time that they are typically late when they are late and in their frequency of being late. Such reflections typically lead the students to revise their strategies for solving the case study and to try different approaches such as counting the number of on time flights for each airline or finding the average number of minutes late when the flights are late.
The fourth principle, the Model Documentation Principle, ensures that while completing the case study, the students are required to create some form of documentation that will reveal explicitly how they are thinking about the problem situation. Requiring external documentation of their thinking is beneficial for both the teacher and the students. First, the documentation is helpful for the teacher because it reveals how the students are interpreting and thinking about the given situation. Second, the documentation is beneficial for the students because when students externalize their thinking, it becomes easier for them to self-assess or to reflect on their thinking. In other words, externalizing their thinking helps students engage in metacognition. This principle is typically accomplished in two ways. First, students are working in groups; thus, they explicitly reveal their thinking when they communicate with each other to carry out processes such as planning, monitoring, and assessing. Second, as in the example case study, the problem is stated to require students to produce explanations, procedures, or descriptions as part of their solution and to explain their solutions in written letters. Together, these two requirements produce documentations that reveal how students are thinking about the given situation.
The fifth principle is the Construct Share-Ability and Re-Usability Principle, which requires students to produce share-able and re-usable solutions. By asking the students to produce products that can be used by others beyond the immediate situation, case studies require students to go beyond personal ways of thinking to developing more general ways of thinking, often resulting in more powerful mathematics. In the Departing On Time case study, the students are specifically asked to develop a ranking procedure that can be shared with the Spanish club, which encourages the students to be more detailed in their explanations of their solutions. In addition, the students are specifically asked to make their procedure more general so that the Spanish club “may use the procedure to rank additional airlines that they may identify at a later time.”
The sixth principle, the Effective Prototype Principle, ensures that the case study will be as simple as possible yet still mathematically significant. The goal is for students to develop solutions that will provide useful prototypes for interpreting other similar situations. In the Departing On Time case study, the problem scenario is clear and simple – find the airlines that are more likely to depart on time. Furthermore, students’ solutions usually provide a useful prototype for interpreting other situations. For example, some students solve the Departing On Time case study by counting the number of on time departures for each airline. Such a process can serve as a prototype for other situations in which frequency counts are appropriate.
Table 2 summarizes the six design principles by providing questions that can be used to design a Case Study for Kids.
Table 2 - Verification Questions for the Six Design Principles
|Share-ability and Re-Usability Principle
|Effective Prototype Principle