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After attending the On Course I Workshop this past summer, I wrote a case study (see below) that I used in my math class for elementary education students.  The purpose of this case study is to strengthen students’ problem-solving abilities, and the experience went extremely well.

First, I had students read the case study aloud. Then I had them respond to the questions on their own. After 10 minutes, I had the students get into groups according to the way that they numbered the people in the study. I had the students in each group explain to each other why they’d numbered the people in the study the way that they had, and then we had a lively discussion among the groups about their choices.

Next, I asked students to come up with their own problem-solving strategies. I put them into seven groups of four students each. Two of the groups created identical strategies while the remaining five groups came up with different strategies. The groups all used a pattern approach to solving. Student groups shared their solutions with the whole class. They were excited to see that there was not just one way to solve a problem. Many of them had always been taught that, in math, there is only one way to get to a solution.

Students did very well on the subsequent unit exam. The class average on the unit exam was 78 with a standard deviation of 9.2. Below the case study, you will find the student results on the exam questions that correlated with the problem-strategies of the case study. I also thought you might be interested to read some of the student responses to an evaluation regarding the value of doing the case study.

Here is the case study:

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PROBLEM SOLVING

Professor Penny thought her math class needed to spend more time problem solving. In an effort to enhance her students’ problem solving ability, she decided to put students into groups and give each group a different problem to solve. Professor Penny assigned each group the following tasks:

1. solve the problem.

2. explain the strategy or strategies the group used to arrive at a solution.

One group of students--Linda, Tom, Tony, and Cindy--were given the following problem to solve:  Find the sum of the first 100 counting numbers.

                                    1 + 2 + 3 + . . . + 100

LINDA, anxious to get the work done, said that the group should break up the numbers into parts and each person should be responsible to add a set of numbers.

TONY did not like the fact that Linda was already blurting out what they were going to do to solve the problem. He wanted time to think about all of the possible ways to solve the problem. “I think we should take the problem home and make a list of all of the possibilities.”

CINDY did not like Tony’s idea. Cindy said, “This is exactly why I hate math! What does adding a bunch of numbers have to do with real-life? When am I ever going to use the sum of the first 100 numbers again?”

TOM, an easygoing guy, felt that everyone was making too much of a fuss about the problem. He suggested getting a calculator and adding the numbers from 1 to 100 inclusive. That seemed like the best idea to him and it was easy to do.

Linda then blurted out that if Tom wanted to do that he could but she was only going to add the first 25 numbers. She then told Tony that he would need to add the numbers from 26 to 50, Cindy would need to add the numbers from 51 to 75 and Tom would have to add the numbers from 76 to 100. She then proceeded to tell everyone that they needed to meet 1 hour before class started to put they results together so that they would have their answer ready for the class.

Tony then told the group that he did not feel that they were completing the tasks that Professor Penny gave them. Linda told Tony that everything was under control. Tony could not seem to get his point across to the group. Tony then went to Professor Penny and explained the situation to her. She told him that part of becoming a good problem solver was to work through the group dynamic problem in addition to solving the given problem. Tony was furious with the professor’s response. He decided to solve the problem his own way. He came up with a list of strategies that he would use to solve the math problem.

The next day, the group met 1 hour before the start of class. Three of the members of the group added the numbers that Linda had assigned, however, Tony did not add the set of numbers Linda assigned him because he did not think Linda’s strategy was a viable one. Linda said that she would handle running the presentation. Linda explained to the class the breaking up strategy that she came up with, however they were unable to arrive at a solution because Tony refused to participate. Tony then told the class that he did not think that Linda’s solution was a viable one and that he felt it best to look for patterns when trying to calculate a sum of numbers. Cindy hated the assignment and let the class know that this was a stupid assignment, which had no bearing on her life. Tom told the class to get out their calculators and add the numbers from 1 to 100 inclusive and they would arrive at the correct solution.

Now, imagine that you were a member of this group. What would you have done to contribute to the group task?

Do you identify with anyone in the group?

Which student in the group completed the task at hand? Rank the students from 1-4. Note: A "1" indicates the student you feel did the best job of completing the tasks and a "4" indicates the student you feel did the worst job of completing the tasks. Be prepared to explain your answer.

Linda  ____    Tony   ____    Cindy ____    Tom   ____

Diving Deeper: Do you think Professor Penny should have helped the group after Tony came to her for help?

In your groups: It is now your turn to come up with a strategy or a set of strategies to solve this problem.

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Now, here’s an evaluation of the impact of doing the Case Study:

The unit exam was designed to make students use the problem solving skills they had practiced in the case study.  While many of the exam problems expected students to use their problem-solving strategies, problems 4, 5, and 6 were specifically designed to test this skill. The chart below summarizes the results.

Students could earn up to 5 points for each problem. Students were awarded 3 points for their strategy (explanation needed to be clearly stated), 1 point for their conclusion (again needed to be clearly stated), and 1 point for the correct answer. The good news is that each student tried the problems. In the past, I have had students leave problems blank when they did not know how to do a problem. The students’ responses were well thought out. I did note that 16 students scored 3 points on problem 6. Part of the problem required the students to think through a seating arrangement, but they had to follow a specific diagram. Every student in the 3-point category missed one instruction in the problem, thus leading them to an incorrect conclusion and final answer.

I also sought feedback from the students about their experience with the case study.  Here are the responses to my two questions:

A. Do you think that the activity “A Case Study” helped you to become a better problem solver? If yes, explain how. If no, explain why it did not.  (Responses – 7 No’s, 18 Yes’s)

1. I don’t think it necessarily helped me in problem solving. It did, however, explain how people can work in a group together. It showed how to work good in a group and what and what not to do.

2. No, because I approached it as a math problem, something to be solved. I did not associate it with a lesson on problem solving, just something that needed to be done. I solve problems daily.

3. I don’t think it really helped me. It was a great activity and made me aware about my actions when it came to problem solving. But I still perform the way I always do.

4. No – what has helped me become a better problem solver has been getting in study groups or even in groups while in class. “A Case Study” showed 4 people with 4 different views on how to do a problem and I basically eliminated the group. I think a group should work as a team without a leader.

5. I think “A Case Study” did not help me to be a better problem solver. In the end it did not explain the right way to solve the problem. It basically showed to me how they interacted with each other more than the problem.

6. No, I do not think it helped because I could not apply the knowledge on the test except for the one problem with A, B, C, D . . .

7. I don’t think it really impacted me that much. I guess for some it might. I believe problems like that and maybe more would help.

8. Yes, because it did give me the opportunity to see different styles of learning and approaching problems.

9. Yes, it showed how not to function while problem solving in groups. It showed how different ways of problem solving exist. It made me want to not be like anyone in the group (in the assignment). The question got me to consider the answers and answer wisely.

10. Yes, I do think that “A Case Study” helped me become a better problem solver. It helped me look at different ways of solving the problem, and it helped me solve the problem faster.

11. Yes, because it made me look at problem solving in different ways instead of just my own way.

12. I remember the way that the problems was solved and I used it do pattern problems. It is easier to do pattern problems because I can see more than one solution method.

13. Yes, there are a wide variety of patterns one can implement to reach a conclusion. Knowing these multiple paths are all valid, releases the pressure of “one-way” thinking – (i.e. “I forgot the formula, so now I’ve failed the test!)

14. Yes, I think it did help me to become a better problem solver. It also helped me to see how people have different views and attitudes. It made me see how everybody needs to work together to accomplish a goal.

15. Yes, because now I know different methods for solving the same problems.

16. Yes, I think the activity helped me because it showed different ways to do a problem and it showed that there isn’t only one way to solve a problem.

17. Sort of. It gave me the idea and option of adding all the options up – doing it the simple or literal way.

18. Yes, because it helped me see that there is always more than one way to solve a problem.

19. The activity “A Case Study” helped me out a little bit. At the time of the study, I understood how to problem solve a little better, but when I tried to problem solve in the test and on other problems, it really didn’t help me.

20. I think it made me take a look at how I problem solve in a group. It really makes me be more aware of how I can get pushy instead of being open to new ideas.

21. Really the exercise and the discussion that took place afterward was helpful. I am the sort who approaches a problem far more scatterbrained than you might imagine, so hearing the way everyone else in the class handled the problem was very helpful.

22. Yes, because it made me realize that there are many ways to solve a problem. Sometimes there is not just one answer.

23. Yes, it showed different ways to approach and solve a problem.

24. Yes, because it showed me to take each person and develop problem solving by thinking.

25. It was a good example. For me it was an exercise I have done many times trying to get 30 different people with different backgrounds who spoke different languages to make 1000 plates of salad the same way. The order to make the salad doesn’t matter, only the final product must look the same.

 B.  Did the activity help you to work more effectively in a group setting? Please explain your answer.  (Responses – 2 No’s, 23 Yes’s)

1. Yes, as in number one, it showed different types of people and how they work together in a group. It showed me what type of  attitude and what actions I need to do when I work in a group with others. It also helped to get the opinions of classmates on how to act in a group and how you will be perceived in group work.

2. N/A

3. Like I said before, the activity made me aware of my problem solving style. But I have always worked great in groups.

4. No, I’ve always been able to work well in groups.

5. The activity did help me to work in a group setting. It helped me to see which character I am portrayed most closely too and how it affects the group as a whole. I learned what not to do in a group setting.

6. I think it allowed me to get to know the people I sat next to, and the way that they think. By getting to know them, we have developed a study group. (It allows you to open your mind to the ways others think about how the problem should be solved.)

7. No, because I’m one of those people who work well with others, so therefore it was not big deal.

8. Yes, in a group setting it is important to understand everyone’s point of view.

9. Yes, the example in the problem really did show how ineffectively some people work when in groups. In my group, it went well so I guess we showed how to work well in groups.

10. Yes, because you got everyone’s opinion on how to do the problem.

11. Yes, because now I know that people are going to think differently in group problems.

12. It helped you understand that people see problems differently and it helps you deal with the problems that take place within a group.

13. I always work well with others.

14. Yes, it showed me that people can’t always have their way and that they should say what they think, but also cooperate with others. It also showed me you should always put forth all of your effort and do what you say you will, otherwise, you will let down your group and may not finish the project.

15. Yes, because it helped me to see how my actions affect everyone in the group. I am the kind of person who just wants to get a job done so I would not take other peoples ideas in to make mine better.

16. Yes, because we got to discuss how we would solve the problem and see how others might do it differently.

17. It helped me realize that unity was the most important part of our group and listening to everyone’s opinion and ideas were vital.

18. Probably not. I am pretty much the same as I always was but I didn’t think I was bad at working in groups. I am pretty much a passive person.

19. The activity did help me work more effectively in a group setting. Now I recognize what type of worker I am, so I can either change that or still act as I always have in a group.

20. I think yes, it did because now I take the time to listen to others ideas before trying to come up with a solution all on my own.

21. Well honestly I’ve never been very pigheaded. But all the same, I think every group effort improves those skills. Whenever I work in a group my first response to a negative situation is to ask what I am doing wrong.

22. Yes it makes me sit back and listen to others’ ideas without rejecting them right away because I think my answer was the correct one. There may be more than one way to skin a cat.

23. Yes, because it showed how important it is to have everyone participate and cooperate in a group or nothing will be accomplished.

24. Yes, I learn very well while in groups and I think that our group works very effectively together.

25. Same answer as above. To get things done in a group, there has to be give and take. The role of a teacher in many cases is to guide the groups to the answer, how each get there can be different.

Jenny Penniman, Faculty, Mathematics, Howard Community College (MD), jpenniman@howardcc.edu

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