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Model-Eliciting Activities (MEAs) - The UTeach Institute

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Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>-<strong>Eliciting</strong> <strong>Activities</strong> (<strong>MEAs</strong>)<br />

5 th Annual <strong>UTeach</strong> <strong>Institute</strong>-NMSI Conference<br />

Thursday, May 26, 2011<br />

Adem Ekmekci<br />

Science & Mathematics Education/<br />

<strong>The</strong> <strong>UTeach</strong> <strong>Institute</strong><br />

<strong>The</strong> University of Texas at Austin<br />

ekmekci@mail.utexas.edu<br />

&<br />

Gladys Krause<br />

Science and Mathematics Education<br />

<strong>The</strong> University of Texas at Austin<br />

gladyshrueda@gmail.com<br />

1


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>-<strong>Eliciting</strong> <strong>Activities</strong> (<strong>MEAs</strong>)<br />

Agenda<br />

• Group Work: Solve an MEA in Groups<br />

• What are <strong>MEAs</strong>?<br />

• Designing <strong>MEAs</strong><br />

• Questions and Comments<br />

2


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>-<strong>Eliciting</strong> <strong>Activities</strong> (<strong>MEAs</strong>)<br />

Group Work<br />

• Toolbox!<br />

• In groups of 3-4<br />

• Solve the problem ~30 min.<br />

• Present solutions and discuss ~10-15 min.<br />

3


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>The</strong> Team Ranking<br />

Problem<br />

Summer Jobs Problem<br />

Carmona & Greenstein (2010).<br />

Investigating the Relationship<br />

Between the Problem and the<br />

Solver: Who Decides What Math<br />

Gets Used? In R. Lesh et al. (eds.),<br />

<strong>Model</strong>ing Students’ Mathematical<br />

<strong>Model</strong>ing Competencies.<br />

Case Studies for Kids<br />

Purdue University<br />

4


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>-<strong>Eliciting</strong> <strong>Activities</strong> (<strong>MEAs</strong>)<br />

What are <strong>MEAs</strong>?<br />

• Thought-revealing activities that elicit students’<br />

mathematical and scientific knowledge (Lesh, 1999;<br />

Lesh & Doerr, 2003; and others),<br />

• Complex problem solving situation for students that<br />

incorporate the development of a mathematical or<br />

scientific model to describe a real-life situation,<br />

• Encourages students to describe, revise, and refine<br />

their mathematical and/or scientific ideas,<br />

• Encourages the use of representational media to<br />

explain and document students’ conceptual systems,<br />

5


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>-<strong>Eliciting</strong> <strong>Activities</strong> (<strong>MEAs</strong>)<br />

What are <strong>MEAs</strong>?<br />

• Designed for small groups (3-4 people),<br />

• Similar to a Case Study version for Kids,<br />

• Introduce a problem related to a real life situation,<br />

- <strong>The</strong> context can be given in an introductory<br />

newspaper article -<br />

• <strong>The</strong>ir solution is a model describing a mathematical<br />

or scientific situation,<br />

• <strong>The</strong>y allow for multiple solutions.<br />

6


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>-<strong>Eliciting</strong> <strong>Activities</strong> (<strong>MEAs</strong>)<br />

<strong>MEAs</strong> are designed to be:<br />

• Purposeful<br />

• Meaningful<br />

What are <strong>MEAs</strong>?<br />

And…<br />

• Elicit knowledge (models)<br />

7


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>-<strong>Eliciting</strong> <strong>Activities</strong> (<strong>MEAs</strong>)<br />

What are <strong>MEAs</strong>?<br />

• Students’ different approaches to solving the problem<br />

are valued and recognized<br />

• Extending the notion of what it means to do<br />

mathematics and science extends the number of<br />

students who are capable of doing math and science<br />

• <strong>The</strong> group becomes greater than the sum of its parts<br />

• Diversity is seen as an asset that engages and<br />

promotes learning<br />

8


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Designing <strong>MEAs</strong><br />

6 Principles<br />

<strong>The</strong> Reality Principle<br />

<strong>The</strong> Simple Prototype Principle<br />

<strong>The</strong> <strong>Model</strong> Construction Principle<br />

<strong>The</strong> <strong>Model</strong>-Documentation Principle<br />

<strong>The</strong> <strong>Model</strong> Generalizibility Principle<br />

<strong>The</strong> Self-Evaluation Principle<br />

9


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Designing <strong>MEAs</strong><br />

<strong>The</strong> Reality Principle<br />

(Personal Meaningful Principle )<br />

• Could this really happen in a “real life” situation?<br />

• Will students be encouraged to make sense of the situation<br />

based on extensions of their own personal knowledge and<br />

experiences?<br />

10


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Designing <strong>MEAs</strong><br />

<strong>The</strong> Simple Prototype<br />

Principle<br />

• Is the situation as simple as possible, while still<br />

creating the need for a significant model?<br />

• Will the solution provide a useful prototype (or<br />

metaphor) for interpreting a variety of other<br />

structurally similar situations?<br />

11


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Designing <strong>MEAs</strong><br />

<strong>The</strong> <strong>Model</strong> Construction<br />

Principle<br />

• Does the task create the need for a model to be<br />

constructed, modified, extended, or redefined?<br />

• Does the task involve constructing, explaining,<br />

manipulating, predicting or controlling a structurally<br />

interesting system?<br />

• Is attention focused on underlying patterns &<br />

regularities rather than on surface-level<br />

characteristics?<br />

12


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Designing <strong>MEAs</strong><br />

<strong>The</strong> <strong>Model</strong> Documentation<br />

Principle<br />

• Will the response require students to explicitly reveal<br />

how they are thinking about the situation (givens,<br />

goals, possible solution paths)?<br />

• What kind of system (mathematical objects, relations,<br />

operations, patterns, regularities) are they thinking<br />

about?<br />

13


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Designing <strong>MEAs</strong><br />

<strong>The</strong> <strong>Model</strong> Generalization<br />

Principle<br />

• How shareable and reusable is the model developed<br />

by the students? Does it only apply to the particular<br />

problem, or can it be reused in other situations?<br />

14


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Designing <strong>MEAs</strong><br />

<strong>The</strong> Self-Evaluation<br />

Principle<br />

• Will the students be able to determine the usefulness<br />

of alternative responses?<br />

• Will students be able to determine when their<br />

responses are good enough?<br />

• Why does the client need the students’ result? For<br />

what purposes? Under which circumstances?<br />

15


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Multi-tier Design of thought revealing<br />

activities that promote research,<br />

teaching, and learning<br />

(Based on Lesh & Kelly, 1999)<br />

16


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>ing Cycles in <strong>MEAs</strong><br />

Example: Historic Hotels MEA<br />

Problem:<br />

<strong>The</strong>re is a client who inherited a historic hotel and does<br />

not have management skills. He wants to determine the<br />

rate per room where he was told by the previous owner<br />

that all of 80 rooms are occupied when the daily rate is<br />

$60 per room; the rate per room increases by $1 for every<br />

vacant room; and each occupied room has a $4 cost for<br />

service and maintenance.<br />

(Based on Aliprantis & Carmona, 2003)<br />

17


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>ing Cycles in <strong>MEAs</strong><br />

Example: Historic Hotels MEA<br />

<strong>The</strong>se are the most common testing<br />

-revising cycles emerged from<br />

students’ solution process of the<br />

Historic Hotels problem.<br />

(Based on Ekmekci & Dominguez, 2007)<br />

18


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

<strong>Model</strong>-<strong>Eliciting</strong> <strong>Activities</strong> (<strong>MEAs</strong>)<br />

• QUESTIONS...<br />

• COMMENTS...<br />

Adem Ekmekci & Gladys Krause<br />

ekmekci@mail.utexas.edu<br />

gladyshrueda@gmail.com<br />

<strong>The</strong> University of Texas at Austin<br />

19


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

References<br />

Aliprantis, C.D. & Carmona, G. (2003). Introduction to an economic<br />

problem: A <strong>Model</strong>s and <strong>Model</strong>ing Perspective. In R. Lesh & H. Doerr<br />

(Eds.), Beyond constructivism: <strong>Model</strong>s and modeling perspectives on<br />

mathematics problem solving, learning, and teaching (pp. 255-264).<br />

Mahwah, NJ: Erlbaum.<br />

Carmona, G. & Greenstein, S. (2010). Investigating the Relationship<br />

Between the Problem and the Solver: Who Decides What Math Gets<br />

Used? In R. Lesh et al. (eds.), <strong>Model</strong>ing Students’ Mathematical<br />

<strong>Model</strong>ing Competencies, DOI 10.1007/978-1-4419-0561-1_21, C<br />

Springer Science+Business Media, LLC 2010.<br />

Ekmekci, A. and Dominguez, A. (2007, Oct) "College Level Students’<br />

Reasoning of an Optimization Problem: Historic Hotel MEA (<strong>Model</strong><br />

-<strong>Eliciting</strong> Activity)" Paper presented at the annual meeting of the North<br />

American Chapter of the International Group for the Psychology of<br />

Mathematics Education, University of Nevada, Reno, Reno, Nevada<br />

Retrieved from http://www.allacademic.com/met/p201387_index.html<br />

Publication Type: Poster<br />

20


Adem Ekmekci & Gladys Krause<br />

<strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

References<br />

Lesh, R. & Doerr H. M. (Eds.), 2003. Beyond Constructivism. Manwah,<br />

NJ: Lawrence Erlbaum Associates.<br />

Lesh, R., Hoover, M., Hole, B., Kelly, A. E., & Post, T. (2000). Principles<br />

for developing thought-revealing activities for students and teachers. In<br />

A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in<br />

mathematics and science education (pp. 591-646). Mahwah, NJ:<br />

Lawrence Erlbaum Associates, Inc.<br />

21


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

College Level Students’ Reasoning of an Optimization Problem:<br />

Historic Hotel MEA (<strong>Model</strong>-<strong>Eliciting</strong> Activity)<br />

Adem Ekmekci<br />

University of Texas at Austin<br />

ekmekci@mail.utexas.edu<br />

Angeles Domínguez<br />

Instituto Tecnologico de Estudios<br />

Superiores de Monterrey<br />

angeles.dominguez@itesm.mx<br />

<strong>Model</strong>-eliciting activities (<strong>MEAs</strong>) are based on real-life situations where students,<br />

working in small groups, present a mathematical model as a solution to a client’s need<br />

(Zawojewski and Carmona, 2001). <strong>The</strong> problem of Historic Hotels is an MEA in which<br />

students are asked to develop a mathematical model to maximize profit that can be<br />

calculated with a quadratic equation. In the problem, there is a client who inherited a<br />

historic hotel and does not have management skills. He wants to determine the rate per<br />

room where he was told by the previous owner that all of 80 rooms are occupied when<br />

the daily rate is $60 per room; the rate per room increases by $1 for every vacant room;<br />

and each occupied room has a $4 cost for service and maintenance. <strong>The</strong> problem can be<br />

solved by using quadratic formulas, 1 st and 2 nd derivative method, or simply looking at<br />

the profit for each value. In this study, college level students are asked to solve this<br />

optimization problem to see possible their ways of solution to the problem. Two groups<br />

of students were given this non-traditional problem. <strong>The</strong> first group was a calculus class<br />

where the majority of students were engineering major and the second one was an<br />

education course required for teaching certification where students were either science or<br />

math major. <strong>The</strong>re were 23 groups of 3 in calculus class and 11 groups of 3 in the other.<br />

After the problem was solved in groups, students were asked to present their<br />

models to their class-mates. Discussions about different solutions were done after each<br />

presentation. Authors and their colleagues observed students and took field notes while<br />

students were working on the problem and discussing their results afterwards. This<br />

particular type of MEA made students’ thinking about optimization as a mathematical<br />

concept visible to both their peers and teachers. Formative assessment (Black & William,<br />

1998) cycles that students went through appeared in the problem solving process.<br />

Students’ final works are analyzed in terms of the function they used, variables and their<br />

definitions, graphical representations, and generalizations.<br />

Analysis of solutions of both groups to the problem and observations by authors<br />

and their colleagues will be reported in the poster. In addition, math major students’ and<br />

pre-service teachers’ ways of thinking will be discussed. Following questions will also be<br />

touched upon in the poster:<br />

- How much calculus knowledge do college level students associate with solving an<br />

optimization problem?<br />

- In what ways, did college level students model the maximization of profit in the<br />

given context?<br />

- What symbols and tool did students use in solving the problem?<br />

- Were there any differences in the method used between calculus students and<br />

students in teacher certification program?<br />

1


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

References<br />

Aliprantis, C.D. & Carmona, G. (2003). Introduction to an economic problem: A <strong>Model</strong>s<br />

and <strong>Model</strong>ing Perspective. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: <strong>Model</strong>s<br />

and modeling perspectives on mathematics problem solving, learning, and teaching (pp.<br />

255-264). Mahwah, NJ: Erlbaum.<br />

Black, P. & Wiliam, D. (1998). Inside the black box: raising standards through classroom<br />

assessment. Phi Delta Kappan, October 1998, 139-148.<br />

Dominguez, A. (2007). Single solution, multiple perspectives. Proceedings of thirteenth<br />

annual meeting of <strong>The</strong> International Community of Teachers of Mathematical <strong>Model</strong>ing<br />

and Applications (ICTMA). Bloomington, IN.<br />

Lesh, R., Hoover, M., Hole, B., Kelly, E., & Post, T. (2000). Principles for developing<br />

thought revealing activities for students and teachers. In A. E. Kelly & R. A. Lesh (Eds.),<br />

Handbook of research design in mathematics and science education (pp. 591-645).<br />

Mahaway, NJ: Lawrence Erlbaum.<br />

Zawojewski, J., & Carmona, G. (2001). A developmental and social perspective on<br />

problem solving strategies. In R. Speiser & C. Walter (Eds.), Proceedings of the twentythird<br />

annual meeting of the north American chapter of the international group for the<br />

psychology of mathematicseducation.<br />

Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental<br />

Education.<br />

Citation for this proceeding:<br />

Ekmekci, A. and Dominguez, A. (2007, Oct) "College Level Students’ Reasoning of an<br />

Optimization Problem: Historic Hotel MEA (<strong>Model</strong>-<strong>Eliciting</strong> Activity)" Paper presented at the<br />

annual meeting of the North American Chapter of the International Group for the Psychology of<br />

Mathematics Education, University of Nevada, Reno, Reno, Nevada Online Retrieved<br />

2008-09-04 from http://www.allacademic.com/meta/p201387_index.html<br />

Publication Type: Poster<br />

2



<br />

Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />


<br />


<br />

College
Students’
Reasoning
in
an
<br />

Optimization
Problem
<br />

INTRODUCTION<br />

<strong>Model</strong>-eliciting activities (<strong>MEAs</strong>) present a problem based on<br />

real-life situations to be solved by students in small groups (Lesh et.<br />

al., 2000; Zawojewski & Carmona, 2001). <strong>The</strong> activity used in this<br />

study, the Historic Hotel Problem designed by Aliprantis and<br />

Carmona (2003), is a model-eliciting activity with single numerical<br />

answer with multiple solutions (Dominguez, 2007). Through <strong>MEAs</strong><br />

students’ ways of thinking, as they work on the problem, become<br />

visible to the teacher as well as to their peers (Lesh et. al., 2000). In<br />

that sense, <strong>MEAs</strong> are thought-revealing activities in which students<br />

express their initial ideas and revise them according to feedback they<br />

get from their peers.<br />

SIGNIFICANCE<br />

Classroom practices play a crucial role in efforts to improve<br />

student learning. One of the most important steps for teachers to<br />

accomplish this goal is to know what their students know. Black &<br />

William (1998) defined classroom assessment as the activities<br />

undertaken by teachers and students to provide information about<br />

how students think and what they know. <strong>MEAs</strong> can be used and be<br />

very beneficial as a non-traditional way of classroom assessment.<br />

<strong>MEAs</strong> usually start with a newspaper article that warms student to<br />

problem statement. Readiness questions related to the article make<br />

us sure that all students are ready to the problem. Finally, problem<br />

statement emphasizes the need of a model rather than just a numeric<br />

answer. As the output of <strong>MEAs</strong>, attention should be paid to the<br />

model students develop rather than the numerical answer. That is,<br />

descriptions, explanations, constructions, and the math student use<br />

(Lesh et. al., 2000) in their models are more important than their<br />

single-numerical answer to the problem.<br />

PARTICIPANTS & THE ACTIVITY<br />

This study is an extended version of an earlier study conducted by<br />

Dominguez (2007) with an addition of 30 pre-service science and<br />

math teachers (10 groups of 3) to 94 calculus students (23 groups of<br />

4 or 5) at a university in central Texas. <strong>The</strong> activity was done in two<br />

episodes, 50-60 minutes each. In the first episode, students read the<br />

newspaper article and answered the readiness questions as a class.<br />

<strong>The</strong>n, they worked on the problem in groups.<br />

In the second episode, students completed their letters and each<br />

group presented their models to other groups and discussed their<br />

ways of solving.<br />

PROBLEM STATEMENT<br />

Mr. Frank Graham has just inherited a historic hotel with 80<br />

rooms. He is told that all of the rooms were occupied when the<br />

daily rate was $60 per room and, for every dollar increase in the<br />

daily rate, one less room is rented (if the daily rate was $61, 79<br />

rooms would be occupied). Each occupied room has a $4 cost for<br />

service and maintenance per day. Mr. Graham would like to know<br />

how much he should charge per room in order to maximize his<br />

profit, what his profit would be at that rate, and a procedure for<br />

finding the daily rate that would maximize his profit in the future<br />

even if the hotel prices and the maintenance costs change.<br />

THE PURPOSE OF THE STUDY<br />

We focused on the following questions:<br />

• What different models and optimization strategies were there?<br />

• How many college level students incorporated calculus<br />

knowledge and at what level?<br />

• Were there any difference in solution ways of calculus<br />

students and pre-service math and science teachers?<br />

• Were there any particular testing and revising cycles students<br />

went through?<br />

DATA COLLECTION<br />

All of students’ written work was analyzed and field notes were<br />

taken by researchers during the solution process.<br />

Strategy
<br />

SINGLE ANSWER – MULTIPLE<br />

PERSPECTIVES<br />

Calculus
groups
<br />

(n=23)
<br />

Adem
Ekmekci
<br />

Department
of
Science
&
Mathematics
Education
<br />

University
of
Texas
at
Austin
<br />

Pre‐service
Teacher
<br />

Groups
(n=10)
<br />

Derivative
 24
 5
<br />

Table
 1
 1
<br />

Graph
 1
 1
<br />

Vertex
formula
 1
 1
<br />

Historic
Hotel
MEA
(<strong>Model</strong>‐<strong>Eliciting</strong>
Activity)
<br />

Total
 27*
 8**
<br />

*3 groups gave 2 solutions and 1 group used a different strategy to verify.<br />

**<strong>The</strong> optimization strategy of 2 groups could not be identified.<br />

All calculus groups used derivative strategy, whereas 5 out of 8<br />

of pre-service teacher groups favored that method.<br />

n
<br />

x
<br />

y
<br />

r
<br />

&
<br />

Austin, TX / May 24 – 26, 2011<br />

Angeles
Domínguez
<br />

Instituto
Tecnologico
de
Estudios
<br />

Superiores
de
Monterrey
<br />

Functions and Variables in Students’ Solutions<br />

Definition
of
 
<br />

variables
<br />

<strong>Model</strong>ing
Function
<br />

#
of
vacant
 
 P(n)
=
(80
‐
n)(56
+
n)
<br />

rooms
 
 P(r)
=
(80
‐
r)(56
+
r)
<br />

#
of
occupied
 
 P
=
x
y
‐
4
x

&

y
=
140
‐
x
<br />

rooms
 
 P(x)
=
x
(140
‐
x)
‐
4
x
<br />

Daily
rate

 
 P
=
x
y
‐
4
x
&
x
=
140
‐
y
<br />

per
room
 
 P(y)
=
y
(140
‐
y)
‐
4
(140
‐
y)
<br />

Amount
raise

 
 P(y)
=
(80
‐
(y
‐
60))
(y
–
4)
<br />

in
daily
rate

 
 P(y)
=
(y
–
4)
(140
‐
y)
<br />

Calculus
<br />

Students
<br />


<br />

Pre‐service
<br />

Teachers
<br />

20
 5
<br />

2
 4
<br />

2
 2
<br />

2
 0
<br />


 
 
 Total
 26*
 11**
<br />

*Three
teams
gave
two
solutions
<br />

**
One
team
gave
two
solutions
<br />


<br />

Test-Revise Cycles<br />

<strong>The</strong>se are the most common testingrevising<br />

cycles emerged from students’<br />

solution process of the problem. <strong>The</strong> most<br />

common pattern was I – II – III. A few<br />

groups went through all cycles and very<br />

few groups did not use III at all: they went<br />

through I – II – IV. <strong>The</strong>se were all preservice<br />

teachers<br />

CONCLUSIONS<br />

Although it seems like there is one single-numerical answer to the<br />

Historic Hotel Problem, students used 4 different strategies for<br />

optimization. Moreover, the modeling functions students developed can<br />

be classified into four groups. From students’ written work, it was<br />

possible to see that there are multiple pathways to find a singlenumerical<br />

answer. Observing students during the modeling process<br />

made it possible to see how they went through various modeling cycles<br />

(Lesh et. al., 2000) to develop their models. Calculus students were<br />

more likely to use derivative strategy than pre-service teachers, as<br />

expected.<br />

REFERENCES:<br />

Aliprantis, C.D. & Carmona, G. (2003). Introduction to an economic problem: A <strong>Model</strong>s and <strong>Model</strong>ing Perspective. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: <strong>Model</strong>s and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 255-264). Mahwah, NJ: Erlbaum.<br />

Black, P. & Wiliam, D. (1998). Inside the black box: raising standards through classroom assessment. Phi Delta Kappan, October 1998, 139-148.<br />

Dominguez, A. (2007). Single solution, multiple perspectives. Proceedings of thirteenth annual meeting of <strong>The</strong> International Community of Teachers of Mathematical <strong>Model</strong>ing and Applications (ICTMA). Bloomington, IN.<br />

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.<br />

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.<br />

Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.<br />

3


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Summer Jobs Case Study 1<br />

Newspaper Article: Heat Is on for Summertime Jobs.<br />

Saturday is the Meers Middle School’s annual job fair, organized by the Youth Action<br />

Club. <strong>The</strong> fair gives students an early start to plan how they will earn money during the<br />

summer.<br />

It also honors innovative work efforts. This year’s Earnest Earner award went to<br />

seventh-grader Tyrone Wakes. His Kids on the Run service provided mail pickup, errand<br />

running, and pet walking for residents in his apartment complex.<br />

“Summer jobs are big business,” claims Randy Tye, President of Youth Action. “Our<br />

fair gives ideas about the kinds of jobs kids can get and the money they can earn. We<br />

will hand out information on pay rates for different jobs and advice on ‘talking money’<br />

with employers.”<br />

<strong>The</strong> fair will show how to track down hard-to-get jobs. Student Greg Maby reports,<br />

“<strong>The</strong> basic idea is to go after the jobs. Don’t wait for them to come to you. It’s up to<br />

kids to show what they can do. We’ll give money-making tips, like how to win babysitting<br />

jobs by getting to know young children in the neighborhood.”<br />

Students Jack Elliot drums up yard work by mowing his own lawn early on Saturday<br />

mornings. “Wake up and smell the grass clippings! Is my motto. When people see the<br />

job I do on my parents’ lawn, they often ask me to do theirs.” Showing off your skills is<br />

a good way to attract business.<br />

Visitors to the fair can take a lesson on washing cars for fun and profit. Prices for<br />

commercial car washes are high enough hat young people can compete for the business.<br />

Tye says, “Most adults don’t care who washes their car. If you provide services as good<br />

as others but at a better price, you’ll get the job.”<br />

A profit-making venture often overlooked is services for the elderly. As life<br />

expectancy increases, people live longer, and they can use different kinds of help. Senior<br />

citizens often need help shopping, clearning hourse, or doing yard work. Households<br />

with infirm elderly may need someone to visit with then, so that family members can take<br />

a break.<br />

Jenny Crisp says, “Believe it or not, my baby-sitting job is with my next-door<br />

neighbor’s grandmother. She has Alzheimer’s disease. We read stories together.<br />

Sometimes I play my flute. We have a good time and I also get paid. I’m going into<br />

medical research when I grow up. I’d like to find a cure for Alzheimer’s.”<br />

Today’s students use brochures to get summer jobs. Teacher Helen Soleski says, “Our<br />

computer lab has software for designing flyers that advertise job skills and experience.<br />

Some students design resumes giving potential employers information about their<br />

training and previous work experience.”<br />

Eighth-grader Justin Shar says, “My resume tells about the baby-sitting I’ve done in<br />

the past, my class in first aid, and my advanced swimmer’s certificate. I listed references<br />

with names and phone numbers of people I worked for. <strong>The</strong>y gave me such good<br />

recommendations that I got five job offers.”<br />

Sixth-grader Marsha Turi advises, “Don’t give up. Last year I made flyers, went doorto-door,<br />

and advertised on bulletin boards. I must have asked a hundred people for work.<br />

I got ninety-nine no’s and one very good job that lasted all summer. By September, I had<br />

put $350 in the bank. I did such a good job that I expect to get the job again next<br />

summer.”<br />

4


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Summer Jobs Case Study 2<br />

Typical Pay for Student Summer Jobs<br />

Washing Cars<br />

Own Parents<br />

Others<br />

$2/car MEDIAN $4/car<br />

$2 to $5 LOW/HIGH $3 to $6<br />

Lawn Mowing<br />

Own Parents<br />

Others<br />

$5/lawn MEDIAN $10/car<br />

$3 to $10 LOW/HIGH $5 to $15<br />

Paper Routes<br />

MEDIAN<br />

$12.50/week<br />

LOW/HIGH $7 to $25<br />

5


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Summer Jobs Case Study 3<br />

READINESS QUESTIONS<br />

1. About how much money might a student earn mowing his or her parent's lawn<br />

once a week for a month? Be sure to show your calculations below.<br />

2. About how much might a student earn mowing 10 average-size lawns once a<br />

week for a month? Show your calculations below.<br />

3. About how much might a student earn in a month mowing 5 small-size lawns<br />

once a week and 5 average-size lawns once every two weeks? Show your<br />

calculations below.<br />

4. Next summer Jack Elliot wants to do more than just yard work. He plans to mow<br />

10 yards, get a newspaper route, and wash cars on Saturdays. Jack wants to know<br />

about how much money he can realistically expect to make during the summer,<br />

but this depends on many things, such as how often he mows a lawn or how big<br />

the lawn is.<br />

Estimate what his earnings might be and list the assumptions you make. Give at<br />

least 5 of your assumptions.<br />

6


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Summer Jobs Case Study 4<br />

Last summer Maya started a concession business at Wild Days Amusement Park. Her<br />

vendors carry popcorn and drinks around the park, selling wherever they can find<br />

customers. Maya needs your help deciding which workers to rehire next summer.<br />

Last year Maya had nine vendors. This summer, she can have only six – three full-time<br />

and three half-time. She wants to rehire the vendors who will make the most money for<br />

her. She doesn’t know how to compare them because they worked different numbers of<br />

hours. Also, when they worked makes a big difference. After all, it is easier to sell more<br />

on a crowded Friday night than a rainy afternoon.<br />

Maya reviewed her records from last year. For each vendor, she totaled the number of<br />

hours worked and the money collected – when business in the park was busy (high<br />

attendance), steady (medium attendance), and slow (low attendance). (See the tables.)<br />

Please evaluate how well the different vendors did last year for the business and decide<br />

which three she should rehire full-time and which three she should rehire half-time.<br />

Write a letter to Maya giving your results. In your letter describe how you evaluated the<br />

vendors. Give details so Maya can check your work, and give a clear explanation so she<br />

can decide whether your method is a good one for her to use.<br />

7


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

Summer Jobs Case Study 5<br />

HOURS WORKED LAST SUMMER<br />

June July August<br />

Busy Steady Slow Busy Steady Slow Busy Steady Slow<br />

MARIA 12.5 15 9 10 14 17.5 12.5 33.5 35<br />

KIM 5.5 22 15.5 53.5 40 15.5 50 14 23.5<br />

TERRY 12 17 14.5 20 25 21.5 19.5 20.5 24.5<br />

JOSE 19.5 30.5 34 20 31 14 22 19.5 36<br />

CHAD 19.5 26 0 36 15.5 27 30 24 4.5<br />

CHERI 13 4.5 12 33.5 37.5 6.5 16 24 16.5<br />

ROBIN 26.5 43.5 27 67 26 3 41.5 58 5.5<br />

TONY 7.5 16 25 16 45.5 51 7.5 42 84<br />

WILLY 0 3 4.5 38 17.5 39 37 22 12<br />

MONEY COLLECTED LAST SUMMER<br />

June July August<br />

Busy Steady Slow Busy Steady Slow Busy Steady Slow<br />

MARIA 690 780 452 699 758 835 788 1732 1462<br />

KIM 474 874 406 4612 2032 477 4500 834 712<br />

TERRY 1047 667 284 1389 804 450 1062 806 491<br />

JOSE 1263 1188 765 1584 1668 449 1822 1276 1358<br />

CHAD 1264 1172 0 2477 681 548 1923 1130 89<br />

CHERI 1115 278 574 2972 2399 231 1322 1594 577<br />

ROBIN 2253 1702 610 4470 993 75 2754 2327 87<br />

TONY 550 903 928 1296 2360 2610 615 2184 2518<br />

WILLY 0 125 64 3073 767 768 3005 1253 253<br />

Figures are given for times when park attendance was high (busy), medium (steady), and low (slow).<br />

8


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

THE TEAM RANKING PROBLEM<br />

Each point in the figure below represents the win-loss record of<br />

each of the twelve elementary school soccer teams in Flat Mountain<br />

School District.<br />

READINESS QUESTIONS<br />

1. Which team(s) won the most games? How do you know?<br />

2. Which team(s) lost the most games? How do you know?<br />

3. Did all teams play the same number of games? How do you know?<br />

9


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

THE PROBLEM<br />

Your task is to develop a method for determining which of the<br />

twelve teams wins the first, second, third, fourth, and fifth place trophies.<br />

Make sure you explain very clearly why the teams should be ranked this<br />

way, so that the kids on all the teams will understand that your system is<br />

fair.<br />

<strong>The</strong>re are twelve teams in the Flat Mountain School District, but<br />

many, many more in the whole state! At the end of the season, the state<br />

soccer officials will need to find the top five teams for the state. Write<br />

a letter to the soccer officials where you explain how your system works<br />

and why it is fair. Make sure the officials will understand how to use your<br />

ranking system for any number of teams.<br />

10


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

11


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

12


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

13


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

14


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

15


Adem Ekmekci & Gladys Krause HANDOUTS <strong>UTeach</strong> <strong>Institute</strong> - NMSI Annual Conference<br />

Austin, TX / May 24 – 26, 2011<br />

16

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