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Assessing the
Use of Technology and Using
Technology to Assess A Case Study Alex
Heidenberg Michael Huber Department of Mathematical Sciences The United States Military Academy West Point, New York The Department of Mathematical Sciences at the
United States Military Academy (USMA) is fostering an environment where
students and faculty become confident and competent problem solvers. This
assessment will reevaluate and update the math core curriculum’s program goals
to incorporate the laptop computer, enabling exploration, experimentation, and
discovery of mathematical and scientific concepts. Background and Goals: Technology has made a dramatic
impact on both education and the role of the educator. Graphing calculators and computer algebra
systems have provided the means for students to quickly and easily visualize
the mathematics that once took effort, skill, and valuable classroom time. The Calculus Reform movement sought to
improve instruction, in part, by taking advantage of these technological
resources. Mathematical solutions could
now be represented analytically, numerically, and graphically. The shift in pedagogy went from teaching
mathematics to teaching mathematical modeling, problem solving, and critical
thinking. Ideally the problem solving
experiences that students encountered in the classroom were interdisciplinary
in nature. Mathematics has truly become
the process of transforming a problem
into another form in order to gain valuable insight about the original problem. Portable notebook computers provide an even greater
technological resource that has led us to once again reexamine our goals for
education. Storage and organization
coupled with powerful graphical, analytical, and numerical capabilities allow
students to transfer their learning across time and discipline. The Department of Mathematical Sciences at USMA is
committed to providing a dynamic learning environment for both students and
faculty to develop self-confidence in their abilities to explore, discover, and
apply mathematics in their personal and professional lives. The core math program attempts to expose the
importance of mathematics, providing opportunities to solve complex problems. The program is ideally suited and committed
to employing emerging technologies to enhance the problem solving process. Since 1986, all students at USMA have been
issued desktop computers with a standard suite of software; this year the
incoming class of students (class of ’06) will be issued laptop computers with
a standard suite of software. The focus
of this assessment is to reevaluate the program goals of the math core
curriculum and update these goals to incorporate the ability of the laptop
computer to not only explore, experiment, and discover mathematical and
scientific concepts in the classroom, but provide a useful medium to build and
store a progressive library of their analytical and communicative abilities. Description The general educational goal of the United States
Military Academy is “to enable its
graduates to anticipate and to respond effectively to the uncertainties of a
changing technological, social, political, and economic world.”[1] The core math program at USMA supports this
general educational goal by stressing the need for students to think and act
creatively and by developing the skills required to understand and apply
mathematical, physical, and computer sciences to reason scientifically, solve
quantitative problems, and use technology effectively. Cadets who successfully complete the core
mathematics program should understand the fundamental principles and underlying
thought processes of discrete and continuous mathematics, linear and nonlinear
mathematics, and deterministic and stochastic mathematics. The core program consists of four semesters
of mathematics that every student must study during his/her first two years at
USMA. The first course in the core is
Discrete Dynamical Systems and an Introduction to Calculus (4.5 credit-hours). The second course is Calculus I and an
Introduction to Differential Equations (4.5 CH). The sophomore year’s first course is Calculus II (4 CH), and the
final core course is Probability and Statistics (3 CH). Five learning thread objectives have been
established for each core course. They
are: Mathematical Modeling, Mathematical Reasoning, Scientific Computing,
Communicating Mathematics, and the History of Mathematics. Each core course builds upon these threads
in a progressive yet integrated fashion. The assessment focuses on the following aspects of
our core math program. 1) Innovative curriculum, instructional, and
assessment strategies brought on by the integration of the laptop computer. 2) Student attainment of departmental goals. Innovative Curriculum and
Assessment Strategies Projects: In-class problem solving labs serve as a chance for the students
to synthesize the material covered in the course over the previous week or
two. Students use technology to
explore, discover, analyze, and understand the behavior of a mathematical model
of a real world phenomenon. Following
the classroom experience, students will be given an extension to the problem in
which they are required to adapt their model and prepare a written analysis of
the extension. Students are given
approximately seven-ten days to complete the project. For the most part, these out-of-class projects will be
accomplished in groups of two or three.
An example of a project is provided in Appendix A. Two-day Exams: Assessment of student understanding and problem-solving skills
will take place over the course of two days.
Paramount in this process is determining what concepts and/or skills we
want our students to learn in our core program. We understand that “what you test is what we you get”; therefore,
we have adapted our exams to assess these desired concepts and skills. The first day of the exam will be a traditional
in-class exam in which students do not have access to technology (calculator or
laptop computer). This exam portion
focuses on basic fundamental skills and concepts associated with the core
mathematics program. Students are also
expected to develop mathematical models of real world situations. Upon completion of this portion, students are
given a take-home scenario that outlines a real world problem. They have the opportunity to explore the
scenario on their own or in groups.
Upon arrival in the classroom the next day, the scenario is and/or
adapted to allow students to apply their problem-solving skills in a changing
environment. An example of a take-home
scenario and the adapted scenario is provided in Appendix B. Modeling and Inquiry
Problems: To
continue to develop competent and confident problem solvers, students are not
given traditional examinations in the second core mathematics course. Instead, they are assessed with Modeling and
Inquiry Problems (MIPs). Each MIP is
designed as an in-class “word problem”
scenario to engage the student for about 45 minutes in solving an applied
problem with differentiable or integral calculus or differential equation methods. The student must effectively communicate the
situation, the solution, and then discuss any follow-on scenarios, similar to
the Day Two portion outlined above, all in a report format. As an example, a MIP may involve using
differential calculus to solve a related rates problem. The Situation portion of the MIP
involves transforming the words into a mathematical model that can be solved,
by drawing a picture, defining variables with units, determining what
information is pertinent, what assumptions should be made, and most
importantly, what needs to be found.
Finally, the Situation ends with the student stating which method
(related rates in this case) will be used to solve the problem. The Solution portion involves
writing the step-by-step details of the problem and determining what is needed
to be found. Any asides or effects of
assumptions can be written in as work progresses, and this portion ends with
some numerical value, to include appropriate units. For example, “the rate at which the oil slick approaches the
shore is two meters per minute.” The MIP itself has a second
paragraph that asks follow-on questions.
“Suppose the volume of the oil slick is now doubled. How does that affect your rate?” Or what is the exact rate the moment the slick
reaches the shore?” These follow-on
questions prod the student to go back to the method and rework the problem with
new information. The final portion of the MIP
write-up is the Inquiry/Discussion section.
The MIP write-up must be coherent and logical in its flow. Students must tie together the work and
stress the solution back in the context of the problem. The Inquiry section is vital in student
understanding of the problem. Students
do not stop once they determine a numerical answer. They must continue and communicate how that answer relates to the
problem, and more importantly, if the answer passes the common sense test. As of the time of this writing, the
third core course has also incorporated MIPs, in addition to traditional
exams. The probability and statistics
course is considering the use of MIPs in future years. Electronic Portfolio: The notebook computer
provides a tremendous resource for storage and organization of
information. This resource avails the
opportunity for students to transfer learning across time and between
courses. In the novel, Harry Potter
and the Goblet of Fire, Dumbledore refers to this capability as a pensieve. “At these times,” says Dumbledore, indicating the stone basin,
“I use the Pensieve. One simply siphons
the excess thoughts from one’s mind, pours them into a basin, and examines them
at one’s leisure. It becomes easier to
spot patterns and links, you understand, when they are in this form.” [2] The
portable notebook computer provides the resource for students to create their
own pensieve. Creative exercises offer
the student exposure to mathematical concepts with the ability to explore their
properties, determining patterns and connections which facilitate the process
of constructing understanding. Thorough
understanding is feasible in either a controlled learning environment or at the
student’s leisure. Instructors will
provide early guidance to incoming students on organizational strategies and
file-naming protocol. Informal
assessments of a student’s electronic portfolio will provide information
regarding the ability to understand relationships between mathematical
concepts. Attitude and Perceptions
Survey: One
tool that will be used to assess if students are confident and competent
problem solvers in a rapidly changing world is a longitudinal attitude and
perceptions survey. Students will be
given a series of sixteen common questions upon their arrival at the Academy
and as part of a department survey at the conclusion of each of the four core
math courses. A comparison of their
confidence, attitudes, and perceptions will be made against those students who
in prior years took the core math sequence without a laptop computer. The questions used in the survey are
provided in Appendix C. Revisions Based on Initial
Experience: The assessment began in the Fall of 2002 and
will track students over a period of four semesters. A pilot study was run in the Spring of 2002 and the following
lessons were learned.
Findings Projects: Students overwhelmingly stated that the course projects helped to
integrate the material that was taught in the course. The students’ ability to incorporate the problem-solving process
(i.e., modeling) increased with each successive project. Two-Day Exams: The two-day exams provided a thorough assessment of the course
objectives. Course-end surveys revealed
that the students felt that these two-day examinations were fair assessments of
the concepts of the course. The
technology portion (Day two) magnified the separation between those who
demonstrated proficiency in solving problems using technology and those who didn’t;
there was no significant in-between group of students. Electronic Portfolios: Assessment of the electronic portfolios consisted of individual
meetings of all students with their individual instructors. The results of these meetings brought out
the point that students needed assistance in determining what material should
be retained and how it should be kept.
Students realized that material in this course would be needed in
follow-on courses, so file naming would be key. Guidance was given to students to incorporate a file management
system for later use, but no universal scheme was provided; in this manner,
students could best determine their own system. Additional Findings: Unless assessed
(tested), the students did not take the opportunity to learn how to effectively
use the computer algebra system, Mathematica. Students embraced the use of the graphing
calculator (TI-89) as the preferred problem-solving tool; they overwhelmingly
reported that the laptop computer was a hindrance to their learning. Use of the Findings: Projects: We will continue to use group projects to assess knowledge;
however, we will phase the submission of the projects to provide greater
feedback and opportunity for growth in problem-solving and communication skills. Our plan is to have students submit the
projects as each portion (Introduction, Facts and Assumptions, Analysis, and
Recommendations and Conclusions) is completed. Two-Day Exams: Content on the Day-one (non-technology) portion needs to be more
straightforward, emphasizing the concepts we want students to internalize and
understand without needing technology.
For the Day-two (technology) portion, questions should be asked to get
students to outline and explain their thought processes, identifying possible
errant methods. We need to keep in mind
that problems with syntax should not lead to severe grade penalties. Additional Use of the
Findings: We are going to introduce graded homework
sets designed to demonstrate the advantage of the computer algebra system and
the laptop as a problem-solving tool.
Use of the graphing calculator will be limited to avoid confusion and
overwhelming students with too many technology options. We plan to review course content and remove
unessential material, thus providing more lessons for exploration and
self-discovery. Next Steps and
Recommendations: The
assessment cycle will continue as we implement the changes outlined above into
the first course. The majority of
students will enter the second core course, Calculus I which will use laptops
for the first time. Six Modeling and
Inquiry Problems and one project will be used to assess the progress of our
students’ problem-solving capabilities. Acknowledgements We
would like to thank the leaders of the Supporting Assessment in Undergraduate
Mathematics (SAUM) for their guidance and support. In particular, our team leader, Bernie Madison, has been
instrumental in keeping our efforts focused. The
following Problem Solving Lab is an in-class exercise that allows the students
to model and solve a system of interacting Discrete Dynamical Systems. Humanitarian Demining Background The
country of Bosnia-Herzegovina has approximately 750,000 land mines that remain in the ground after their war ended
in November 1995. The United Nations
(UN) has decided to establish a Mine Action Center (MAC) to coordinate efforts
to remove the mines. You are serving as
a U.S. military liaison to the director of the UN-MAC. The
UN-MAC will initially have 1000
trained humanitarian deminers working in country. Each of these trained personnel can remove 65 mines per week during normal operations. Unfortunately,
there is a rebel force of about 8,000
soldiers that opposes the UN-MAC’s efforts to support the legitimate government
of Bosnia-Herzegovina. They conduct two
major activities to oppose the UN-MAC:
killing the deminers and emplacing more mines. They terrorize the deminers, killing 1 deminer for every 1,000
rebels each week. However, due to poor
training and funding, each of these soldiers can only emplace an average of 5 additional mines per week. Meanwhile,
the accidental destruction of the mines maim and kill some of both the deminers
and the rebel forces. For every 1,000,000 mines, 1 deminer is permanently disabled or killed each week. The mines have the exact same quantitative
impact on the rebel forces. Modeling and AnalysisYour current goal is to determine the outcome of the
UN-MAC’s efforts, given the current resources and operational environment. 1. Model the strength of the demining
organization, the rebels, and the number of mines in the ground. Ensure you define your variables and domain
and state any initial conditions and assumptions. 2. Write the system of equations in matrix form
A(n+1) = R * A(n). 3. If the interaction between the rebels and
deminers as well as their respective efforts to affect the minefields remain
constant, what happens during the first five years of operations? 4. Graphically display your results. Ensure you display your results for each of
the three entities you model. 5. What is the equilibrium vectors, D or Ae, for this
system? Is it realistic? 6. The General and Particular Solution for the
new system of DDS’s using eigenvalue and eigenvector decomposition. We
add a little more realism to the scenario by creating interaction between the
model’s components. The following
extension is the project that forces the students to adapt their model and
prepare a written analysis. Humanitarian Demining BETTER
ESTIMATE ON CASUALTIES DUE TO MINES We
now have more accurate data on the casualties due to mines; it may (or may
not) change part of your model.
Better estimates show that for every 100,000 mines, 2 deminers are permanently disabled or killed each week. The mines have the exact same quantitative
impact on the rebel forces. OTHER MINEFIELD LOSSES Other
factors take their toll on the number of emplaced mines as well. Weather and terrain cause some of the mines
to self-destruct, and civilians occasionally detonate mines. Approximately 1% of the mines are lost to these other factors each week. NATURAL ATTRITION OF FORCES Due
to other medical problems, infighting, and desertion, the rebel forces lose 4% of their force from one week to the
next. The deminers have a higher
attrition due to morale problems; they lose 5% of their personnel from one week to the next. RECRUITING EFFORTS
Both
the rebel forces and the deminers recruit others to help. Each week, the rebels are able to recruit an
additional 10 soldiers. Meanwhile, the UN-MAC is less
successful. They only manage to recruit
an additional 5 deminers each week. --------------------------------------------------------------------------------------------------------------------- For
the project, your report should address the following at a minimum: 1. Executive
Summary in memo format that summarizes your research. 2. The
purpose of the report. 3. Facts
bearing on the problem. 4.
Assumptions made in your model, as well as the viability of these
assumptions. 5. An
analysis detailing: a. The equilibrium
vector, D or Ae, for the
system and discuss its relevance. b.
The General and Particular Solution for the new system of DDS’s using
eigenvalue and eigenvector decomposition. c.
A description of what is happening to each of the entities being modeled
during the first five (5) years of operations. 6. The
director of the UN-MAC also wants your recommendation on the following: a.
If the demining effort is going to be successful within the first five
years, when will it succeed in eradicating all mines? If the demining effort is not going to be successful, determine the
minimum number of weekly
demining recruits needed to remove all mines within five years of operations. b. Describe at least one other strategy the
UN-MAC can employ to improve its efforts to eradicate all of the mines. Quantify this strategy within a mathematical model and show the improvement (graphically, numerically, analytically,
etc.). 7.
Discussion of the results. a.
Reflect on your assumptions and discuss what might happen if one or more
of the assumptions were not valid. b. Integrate graphs and tables into your
report, discuss them, and be sure to label them correctly. 8.
Conclusion and Recommendations. Appendix
B Example Day Two Exam Take Home Scenario While home on Spring Break, you explain
to your parents the fundamental concepts that you have learned in your Discrete
Dynamical Systems course. Following
dinner one evening you provide them a quick 10 minute presentation on how you
were able to use DDS to assist with car buying decisions. Due to your improved ability to communicate
about mathematics, your parents immediately say, “Hey, I think you might be
able to help us”. They share with you
the fact that they are negotiating the purchase of a 2003 Toyota Camry. The Wasko Federal Credit Union has agreed to
finance a vehicle loan of up to $20,000 at a yearly interest rate of 6%. Adapted Scenario
Recall from the read-ahead that your parents
have asked for your assistance to help them determine the financing option for
their purchase of a 2003 Toyota Camry.
They have successfully negotiated a price of $18,000 for the car. In order to boost slumping car sales, the
dealer has offered two financing options.
a.
In Option One the dealer has offered to finance the car at a rate of
1.9% interest for 48 months. Develop a model that predicts
the car loan balance after n months, given the loan requires 48 equal payments
of p
dollars. Define all
variables, state the domain,
and any initial conditions. b.
Determine the payment p, to the nearest cent, if your
parents bought the Camry using the 1.9% loan financed by Toyota? Explain how you used technology to obtain
this figure. Include the actual
formulas used in EXCEL or the TI-89. c. In Option Two, the dealer has offered a
$1500 rebate in lieu of the 1.9% financing.
The finance rate for this option is 5.2% interest over 48 months. Develop
a model that predicts the car loan balance after n months, given the loan
requires 48 equal payments of q dollars. Define all
variables and state the domain,
and any initial conditions. d. Should your
parents take the 1.9% financing or the $1500 rebate with a 5.2% financing
rate? Explain how you used technology
to assist in your decision. Provide
clear mathematical backing to support your decision. Include the actual formulas used in EXCEL or the TI-89. Appendix
C Questions used in Attitude and
Perceptions Survey The following questions were given to students and
were rated on a Likert-Scale from 1
(strongly disagree) to 5 (strongly agree). 1. An understanding of mathematics is useful
in my everyday life. 2. I believe that mathematics involves
exploration and experimentation. 3. I believe that mathematics involves
curiosity. 4. I can structure (model) problems
mathematically. 5. I am confident in my ability to solve
problems using mathematics. 6. Mathematics helps me to think logically. 7. There are many different ways to solve
most mathematics problems. 8. I am confident in my ability to
communicate mathematics orally. 9. I am confident in my ability to
communicate mathematics in writing. 10. I am confident in my ability to transform a
word problem into a mathematical expression. 11. I am confident in my ability to transform a
mathematical expression into my |