1
Introduction
A Teacher Handbook
for
Implementing “Case Studies for Kids” and Understanding Students’ Solutions
Spring 2002
Table of Contents
Introduction 3
Description of Case Studies for Kids 3
Description of Students’ Thinking Sheets 4
Implementation Suggestions for Case Studies for Kids 7
Observation Form11
Presentation Form13
Student Debriefing Form15
Departing On-Time Case Study17
Short Description17
Teacher Comments17
Students’ Perception of Case Study17
Case Study18
Students’ Thinking Sheet22
Historic Hotels Case Study24
Short Description24
Teacher Comments24
Students’ Perception of Case Study24
Case Study26
Students’ Thinking Sheet29
Phone Plans Case Study32
Short Description32
Teacher Comments32
Students’ Perception of Case Study33
Case Study34
Students’ Thinking Sheet40
Summer Jobs Case Study42
Short Description42
Teacher Comments42
Case Study43
Students’ Thinking Sheet48
Volleyball Case Study50
Short Description50
Teacher Comments50
Students’ Perception of Case Study50
Case Study51
Students’ Thinking Sheet56
1
Introduction
Introduction
The “Teacher Handbook for Implementing ‘Case Studies for Kids’ and Understanding Students’ Solutions” was developed as part of the professional development schools collaboration between a mid-western university and a local middle school. For one school year, six teachers from the local middle school collaborated with me, a graduate student from the university, to implement five Case Studies for Kids in their own classrooms and to participate in a series of teacher workshops centered around the case studies. The overarching purpose during these workshops was to a) examine and capture students’ mathematical thinking as revealed by their work on the case studies and b) to produce a handbook for fellow teachers that would provide suggestions on how to implement Case Studies for Kids and explanations of how students typically think about and solve the five case studies that the teachers implemented.
For each of the case studies that the teachers implemented, the teachers participated in an introductory workshop and a follow-up workshop. During the introductory workshop, the teachers completed the associated case study, discussed the mathematics within the case study, addressed how they felt their students would solve the activity, and shared implementation ideas. After the introductory workshop, each teacher implemented the case study within his/her own classroom. After implementing the case study, the teachers met again for a follow-up workshop where they discussed the students’ various solution strategies. Over the course of implementing these five case studies and as the teachers refined their methods for implementing the case studies, this handbook was developed.
What are Case Studies for Kids? Case Studies for Kids are explicitly designed to help middle school students develop conceptual foundations for deeper and higher order ideas in pre-college mathematics. Each task asks students to mathematically interpret a complex real-world situation and requires the formation of a mathematical description, procedure, or method for the purpose of making a decision for a realistic client. Because groups of students are producing a description, procedure, or method (instead of a one-word or one-number answer), students’ solutions to the task reveal explicitly how they are thinking about the given situation.
Each case study consists of four components:
1)Newspaper article: Students read the newspaper article to become familiar with the context of the problem.
2)Readiness questions: Students answer these reading comprehension questions about the newspaper article to become even more familiar with the context.
3)Problem statement: In groups of three, students work on the problem statement for 60 – 90 minutes.
4)Process of sharing solutions: Each group writes their solution in a letter to the client. Then, each group presents their solution to the class. Whole class discussion is intermingled with these presentations to discuss the different solutions, the mathematics involved, and the effectiveness of the different solutions in meeting the needs of the client.
In totality, each case study takes approximately three class periods to implement.
In this handbook, you will find several sections including overall classroom implementation suggestions for Case Studies for Kids, an Observation Form that can be used to make notes about your students’ mathematical thinking and group functioning as they work on a case study, a Students’ Debriefing Form that can be used to conclude a case study, five example Case Studies for Kids with corresponding teacher comments, and five “Students’ Thinking Sheets” for each of the five case studies. These Students’ Thinking Sheets outline the different ways that students tend to think about and solve each of the case studies. They include excerpts of students’ work, descriptions of the mathematics involved in the students’ different ways of thinking, and evaluations of the students’ different ways of thinking in terms of meeting the needs of the client. You may find the sheets helpful for any of the following reasons:
- Knowing in advance how students tend to think about and solve the associated case study,
- Providing feedback to students about their work on the case study,
- Evaluating the students’ final solutions,
- Guiding a classroom discussion about the case study,
- Describing to parents at conferences the case studies and the work of their children,
- Identifying the mathematics inherent in the case study (you can share this with the students), and
- Prompting questions to ask of the students during their group presentations.
Before you begin examining the remainder of this handbook, the teachers and I would like to offer three comments. First, the ideas and suggestions contained in this Teacher Handbook should be viewed as just that – suggestions. We have gathered information about what we have learned from implementing Case Studies for Kids, and we wanted to share this information with you in the event that it may be helpful. However, please do not view our suggestions as rules or fixed guidelines. Feel free to adapt our suggestions to your own classroom needs.
Second, Case Studies for Kids do not typically have one “right answer”; however, some solutions are better than others. Thus, it is more helpful to ask, “How well does this particular solution meet the needs of the client?” instead of asking, “Is this the right answer?” when assessing your students’ work or providing feedback to your students. You will probably find it helpful to share this perspective with your students before they begin working on their first case study.
And finally, although the case studies within this handbook emphasize mathematical concepts, teachers of other subjects, such as language arts, science, and social studies, have also found implementing case studies in their classrooms to be productive. One possibility that we have found productive is for teachers to team together. For example, it may be fruitful for the language arts teacher, the science teacher, and the math teacher to team together and implement a case study. Then, the teachers can work together to each stress the particular aspects of their subject that are related to the case study.
We hope that this handbook will prove helpful for you in implementing Case Studies for Kids. If you ever have any questions, please do not hesitate to contact us.
Shanna BradfordSusie BurtonMichelle Chamberlin
Tim FolandMarie FureyRaymond Gee
Jon PostmaLisa RoetkerSchool Mathematics
d ScienceCenter
(765) 494-5889
1
Case Study Implementation
Classroom Implementation Suggestions for
“Case Studies for Kids”
Recommended supplies (depending on the actual activity implemented):
- Overhead transparencies and transparency markers/pens
- Calculators
- Rulers, scissors, tape
- Markers, colored pencils, pencils
- Construction paper, graph paper, lined paper
- Paper towels or kleenex (for cleaning transparencies)
- Manila folders or paper clips for collecting the students’ work
- Optional: Computers with programs such as Microsoft Word and Excel
Recommended Progression of the Activity:
Newspaper Article and Readiness Questions:
The purpose of the newspaper article and the readiness questions is to introduce the students to the context of the problem. Depending on the grade level and/or your instructional purposes, you may want to use a more teacher-directed format or a more student-directed format for going through the article and the questions. Some possibilities include:
- More teacher-directed (½ hour): Read the article to the students and give them class time to complete the readiness questions individually. Then, discuss as a class the answers to the readiness questions before beginning work on the problem statement. This approach also works well when you can team with a language arts teacher, and they can go through the article in their class.
- More student-directed (10-15 minutes): Give the article and the questions to the students the day before for homework. If you wish, you may provide some class time for the students to complete the article and questions. Then, on the day of the case study, discuss as a class the answers to the readiness questions before beginning work on the problem statement.
- More student-directed (10-15 minutes): Give the article and the questions to the students in their groups right before the students begin working on the problem statement. The students answer the questions as a group and then proceed to work on the problem statement.
Working on the Problem Statement (45-75 minutes):
Place the students in groups of three. If classroom management is an issue, the teacher may form the groups. If classroom management is not an issue, the students may form their own groups. Encourage (but don’t require or assign) the students to select roles such as timer, collector of supplies, writer of letter, etc. Remind the students that they should share the work of solving the problem.
Present the students with the problem statement. Depending on the students’ grade level and previous experience with case studies, you may want to read the problem statement to the students and then identify as a class: a) the client that the students are working for and b) the product that the students are being asked to produce.
Allow the students to work on the problem statement. As they work, your role should be one of a facilitator and observer. Avoid questions or comments that steer the students toward a particular solution. Also during this time, try to get a sense of how the students are solving the problem so that you can ask them questions about their solutions during their presentations.
Presentations of Solutions (30-45 minutes):
The groups present their solutions to the class. Each presentation typically takes 3 – 5 minutes. You may want to limit the number of presentations to five or six or limit the number of presentations to the number of original solutions to the case study.
Before beginning the presentations, encourage the other students to not only listen to the other groups’ presentations but to also a) try to understand the other groups’ solutions and b) consider how well these other solutions meet the needs of the client. You may want to offer points to students that ask ‘good’ questions of the other groups, or you may want students to complete a debriefing page in which they explain how they would revise their solution after hearing about the other solutions.
As students offer their presentations and ask questions, whole class discussions should be intermixed with the presentations in order to address conflicts or differences in solutions. When the presentations are over, collect the students’ work.
Assessment of Students Work:
You can decide if you wish to evaluate the students’ work. If you decide to do so, you may find the following Assessment Guide[1] helpful:
Performance Level
/ Effectiveness:Does the solution meet the client’s needs?
Requires redirection / The product is on the wrong track. Working longer or harder with this approach will not work. The students may need additional feedback from the teacher.
Requires major extensions or refinements / The product is a good start toward meeting the client’s needs, but a lot more work is needed to respond to all of the issues.
Requires only minor editing / The product is nearly ready for the client to use. It still needs a few small modifications, additions, or refinements.
Useful for this specific situation / No changes are necessary to meet the client’s immediate needs.
Share-able or re-usable / The tool not only works for the immediate solution, but it would be easy for others to modify and use in similar situations.
OR
The solution goes above and beyond meeting the immediate needs of the client.
Note: If you use this Assessment Guide for grading purposes, please keep in mind that a performance level of “requires only minor editing” or higher indicates a satisfactory solution. For example, you may want to assign a grade of B for “requires only minor editing”, while assigning an A for the next two higher levels.
If you give a written score or letter grade after assessing the students’ work, we encourage you to provide the students with an explanation (i.e. written comments) as to why they received that score and/or how their solution could be improved. In particular, we found it helpful to phrase the feedback as if it was coming from the client of the problem. So for example, in the volleyball problem, the client is a team of program directors that is trying to devise a system for creating fair and equal volleyball teams, and feedback to the students could include statements such as the following:
"We understand how you formed your three teams of six members each from the eighteen members, but we need more information from you about how we are going to apply your procedure when we have over 200 campers."
Implementing a Case Study with Students for the First Time:
You may want to let students know the following about case studies:
- Case studies are longer problems; there are no immediate answers. Instead, students should expect to work on the case study and gradually revise their solution over a period of 45 minutes to an hour.
- Case studies often have more than one solution or one way of thinking about the problem.
- Let the students know ahead of time that they will be presenting their solutions to the class. Tell them to prepare for a 3-5 minute presentation, and that they may use overhead transparencies or other visuals during their presentation.
- Let the students know that you won’t be answering questions such as “Is this the right way to do it?” or “Are we done yet?”. You can tell them that you will answer clarification questions, but that you will not guide them through the case study.
- Remind students to make sure that they have returned to the problem statement to verify that they have fully answered the question.
- If students struggle with writing the letter, encourage them to read the letter out loud to each other. This usually helps them identify omissions and errors.
Observing Students as They Work on a Case Study
You may find the Observation Form on the following page useful for making notes about one or more of your groups of students as they work on a case study. We found that the form could be filled out during “real-time” as you observe the students working or sometime shortly after you observe the students. The form can be used to record observations about what mathematical concepts the students are using, how they are interacting as a group, how they are organizing the data, what tools they use, what revisions to their solutions they may make, and any other miscellaneous comments.
1
Case Study Implementation
Observation Form
Observer: ______Date: ______
Group: ______Activity: ______
Math Concepts Used: What mathematical concepts and skills did the students use to solve the problem? / Group Interactions: How did the students interact within their group or share insights with each other?Data Organization & Problem Perspective: How did the students organize the problem data? How did the students interpret the task? What perspective did they take? / Tools: What tools did the students use? How did they use these tools?
Miscellaneous Comments: About the group functionality or the problem. / Cycles of Assessment & Justification: How did the students question their problem-solving processes and their results? How did they justify their assumptions and results? What cycles did they go through?
Presentation Form
As the groups of students present their solutions to the class, you may find it helpful to have students complete the following presentation form. This form asks students to evaluate and provide feedback about the solutions of at least two groups. It also asks students to consider how they would revise their own solution to the case study after hearing of the other groups’ solutions.