Model-Eliciting Activities (MEAs) - The UTeach Institute

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  Adem Ekmekci & Gladys Krause HANDOUTS UTeach Institute - NMSI Annual Conference     College Students’ Reasoning in an  Optimization Problem  INTRODUCTION Model-eliciting activities (MEAs) present a problem based on real-life situations to be solved by students in small groups (Lesh et. al., 2000; Zawojewski & Carmona, 2001). The activity used in this study, the Historic Hotel Problem designed by Aliprantis and Carmona (2003), is a model-eliciting activity with single numerical answer with multiple solutions (Dominguez, 2007). Through MEAs students’ ways of thinking, as they work on the problem, become visible to the teacher as well as to their peers (Lesh et. al., 2000). In that sense, MEAs are thought-revealing activities in which students express their initial ideas and revise them according to feedback they get from their peers. SIGNIFICANCE Classroom practices play a crucial role in efforts to improve student learning. One of the most important steps for teachers to accomplish this goal is to know what their students know. Black & William (1998) defined classroom assessment as the activities undertaken by teachers and students to provide information about how students think and what they know. MEAs can be used and be very beneficial as a non-traditional way of classroom assessment. MEAs usually start with a newspaper article that warms student to problem statement. Readiness questions related to the article make us sure that all students are ready to the problem. Finally, problem statement emphasizes the need of a model rather than just a numeric answer. As the output of MEAs, attention should be paid to the model students develop rather than the numerical answer. That is, descriptions, explanations, constructions, and the math student use (Lesh et. al., 2000) in their models are more important than their single-numerical answer to the problem. PARTICIPANTS & THE ACTIVITY This study is an extended version of an earlier study conducted by Dominguez (2007) with an addition of 30 pre-service science and math teachers (10 groups of 3) to 94 calculus students (23 groups of 4 or 5) at a university in central Texas. The activity was done in two episodes, 50-60 minutes each. In the first episode, students read the newspaper article and answered the readiness questions as a class. Then, they worked on the problem in groups. In the second episode, students completed their letters and each group presented their models to other groups and discussed their ways of solving. PROBLEM STATEMENT Mr. Frank Graham has just inherited a historic hotel with 80 rooms. He is told that all of the rooms were occupied when the daily rate was $60 per room and, for every dollar increase in the daily rate, one less room is rented (if the daily rate was $61, 79 rooms would be occupied). Each occupied room has a $4 cost for service and maintenance per day. Mr. Graham would like to know how much he should charge per room in order to maximize his profit, what his profit would be at that rate, and a procedure for finding the daily rate that would maximize his profit in the future even if the hotel prices and the maintenance costs change. THE PURPOSE OF THE STUDY We focused on the following questions: • What different models and optimization strategies were there? • How many college level students incorporated calculus knowledge and at what level? • Were there any difference in solution ways of calculus students and pre-service math and science teachers? • Were there any particular testing and revising cycles students went through? DATA COLLECTION All of students’ written work was analyzed and field notes were taken by researchers during the solution process. Strategy  SINGLE ANSWER – MULTIPLE PERSPECTIVES Calculus groups  (n=23)  Adem Ekmekci  Department of Science & Mathematics Education  University of Texas at Austin  Pre‐service Teacher  Groups (n=10)  Derivative  24  5  Table  1  1  Graph  1  1  Vertex formula  1  1  Historic Hotel MEA (Model‐Eliciting Activity)  Total  27*  8**  *3 groups gave 2 solutions and 1 group used a different strategy to verify. **The optimization strategy of 2 groups could not be identified. All calculus groups used derivative strategy, whereas 5 out of 8 of pre-service teacher groups favored that method. n  x  y  r  &  Austin, TX / May 24 – 26, 2011 Angeles Domínguez  Instituto Tecnologico de Estudios  Superiores de Monterrey  Functions and Variables in Students’ Solutions Definition of    variables  Modeling Function  # of vacant    P(n) = (80 ‐ n)(56 + n)  rooms    P(r) = (80 ‐ r)(56 + r)  # of occupied    P = x y ‐ 4 x  &  y = 140 ‐ x  rooms    P(x) = x (140 ‐ x) ‐ 4 x  Daily rate     P = x y ‐ 4 x & x = 140 ‐ y  per room    P(y) = y (140 ‐ y) ‐ 4 (140 ‐ y)  Amount raise     P(y) = (80 ‐ (y ‐ 60)) (y – 4)  in daily rate     P(y) = (y – 4) (140 ‐ y)  Calculus  Students    Pre‐service  Teachers  20  5  2  4  2  2  2  0        Total  26*  11**  *Three teams gave two solutions  ** One team gave two solutions    Test-Revise Cycles These are the most common testingrevising cycles emerged from students’ solution process of the problem. The most common pattern was I – II – III. A few groups went through all cycles and very few groups did not use III at all: they went through I – II – IV. These were all preservice teachers CONCLUSIONS Although it seems like there is one single-numerical answer to the Historic Hotel Problem, students used 4 different strategies for optimization. Moreover, the modeling functions students developed can be classified into four groups. From students’ written work, it was possible to see that there are multiple pathways to find a singlenumerical answer. Observing students during the modeling process made it possible to see how they went through various modeling cycles (Lesh et. al., 2000) to develop their models. Calculus students were more likely to use derivative strategy than pre-service teachers, as expected. REFERENCES: Aliprantis, C.D. & Carmona, G. (2003). Introduction to an economic problem: A Models and Modeling Perspective. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 255-264). Mahwah, NJ: Erlbaum. Black, P. & Wiliam, D. (1998). Inside the black box: raising standards through classroom assessment. Phi Delta Kappan, October 1998, 139-148. Dominguez, A. (2007). Single solution, multiple perspectives. Proceedings of thirteenth annual meeting of The International Community of Teachers of Mathematical Modeling and Applications (ICTMA). Bloomington, IN. Lesh, R., Hoover, M., Hole, B., Kelly, E., & Post, T. (2000). Principles for developing thought revealing activities for students and teachers. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 591-645). Mahaway, NJ: Lawrence Erlbaum. Zawojewski, J., & Carmona, G. (2001). A developmental and social perspective on problem solving strategies. In R. Speiser & C. Walter (Eds.), Proceedings of the twenty-third annual meeting of the north American chapter of the international group for the psychology of mathematicseducation. Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. 3
Adem Ekmekci & Gladys Krause HANDOUTS UTeach Institute - NMSI Annual Conference Austin, TX / May 24 – 26, 2011 Summer Jobs Case Study 1 Newspaper Article: Heat Is on for Summertime Jobs. Saturday is the Meers Middle School’s annual job fair, organized by the Youth Action Club. The fair gives students an early start to plan how they will earn money during the summer. It also honors innovative work efforts. This year’s Earnest Earner award went to seventh-grader Tyrone Wakes. His Kids on the Run service provided mail pickup, errand running, and pet walking for residents in his apartment complex. “Summer jobs are big business,” claims Randy Tye, President of Youth Action. “Our fair gives ideas about the kinds of jobs kids can get and the money they can earn. We will hand out information on pay rates for different jobs and advice on ‘talking money’ with employers.” The fair will show how to track down hard-to-get jobs. Student Greg Maby reports, “The basic idea is to go after the jobs. Don’t wait for them to come to you. It’s up to kids to show what they can do. We’ll give money-making tips, like how to win babysitting jobs by getting to know young children in the neighborhood.” Students Jack Elliot drums up yard work by mowing his own lawn early on Saturday mornings. “Wake up and smell the grass clippings! Is my motto. When people see the job I do on my parents’ lawn, they often ask me to do theirs.” Showing off your skills is a good way to attract business. Visitors to the fair can take a lesson on washing cars for fun and profit. Prices for commercial car washes are high enough hat young people can compete for the business. Tye says, “Most adults don’t care who washes their car. If you provide services as good as others but at a better price, you’ll get the job.” A profit-making venture often overlooked is services for the elderly. As life expectancy increases, people live longer, and they can use different kinds of help. Senior citizens often need help shopping, clearning hourse, or doing yard work. Households with infirm elderly may need someone to visit with then, so that family members can take a break. Jenny Crisp says, “Believe it or not, my baby-sitting job is with my next-door neighbor’s grandmother. She has Alzheimer’s disease. We read stories together. Sometimes I play my flute. We have a good time and I also get paid. I’m going into medical research when I grow up. I’d like to find a cure for Alzheimer’s.” Today’s students use brochures to get summer jobs. Teacher Helen Soleski says, “Our computer lab has software for designing flyers that advertise job skills and experience. Some students design resumes giving potential employers information about their training and previous work experience.” Eighth-grader Justin Shar says, “My resume tells about the baby-sitting I’ve done in the past, my class in first aid, and my advanced swimmer’s certificate. I listed references with names and phone numbers of people I worked for. They gave me such good recommendations that I got five job offers.” Sixth-grader Marsha Turi advises, “Don’t give up. Last year I made flyers, went doorto-door, and advertised on bulletin boards. I must have asked a hundred people for work. I got ninety-nine no’s and one very good job that lasted all summer. By September, I had put $350 in the bank. I did such a good job that I expect to get the job again next summer.” 4
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