Model-Eliciting Activities (MEAs) - The UTeach Institute

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.
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