SYLLABUS -- MATH 279
Mathematical Problem-Solving Strategies I
Instructor: Dr. Ron
Wenger
304 Ewing Hall
831-8664
wenger@math.udel.edu
http://www.math.udel.edu/~wenger/index.htm
Meeting Time: Mondays
4:00pm – 4:50pm
Meeting Place: Ewing
Hall 210
Office Hours: Mondays 11:15-12:30pm, Tuesdays 12:30-1:30pm, Wednesdays 1:30-2:30 pm, Thursdays 1-2 pm, Fridays 11:15-2:30 pm, (or any other time by appointment)
Problem solving is a process that provides students with opportunities
to experience the power of mathematics in the world around them. It is also an
instructional approach that provides a consistent context for students to learn
and apply mathematics. Thus, problem solving has special importance in the
study of mathematics. A primary goal of mathematics teaching and learning is to
develop students' ability to solve a wide variety of complex mathematical
problems. The National Council of Teachers of Mathematics Curriculum Standards
indicates that: In grades 9-12, the mathematics curriculum should include the
refinement and extension of methods of mathematical problem solving so that all
students can:
*use, with
increasing confidence, problem-solving approaches to investigate and understand
mathematical content;
*apply integrated
mathematical problem-solving strategies to solve problems from within and
outside mathematics;
*recognize
and formulate problems from situations within and outside mathematics;
*apply the
process of mathematical modeling to real-world problem situations.
In order to implement this goal in the classroom,
mathematics teachers must learn to be competent problem solvers. This course is designed to study and
experience various problem-solving strategies. Most importantly, it is designed
to explore problem solving as an instructional goal (teaching problem solving).
MATH 379 will emphasize the use of problem solving as an effective approach to
teach mathematics (teaching mathematics through problem solving). In
particular, the objectives of the instructor are to structure this course in
such a way that students will be able to:
a) become
competent and confident problem solvers;
b)
construct knowledge about nature and strategies of problem solving;
c) experience
aspects of problem-solving processes as a learner, such as group problem
solving, justification and communication of solutions, and models of problem
solving;
d) become
familiar with some useful instructional materials for use in your own teaching.
To achieve these objectives you must become quite
introspective about your own thinking as you grapple with problems. Documenting
such introspection in a journal will be an important catalyst to this process.
Methods of
Instruction
This course is based on problem-solving activities using
nonroutine problems designed or selected by the instructor. Group work,
classroom discussion, and individual problem solving together with your
thoughtful introspection about your thinking are the teaching methods for this
course. The working groups will also
collaborate on the construction of a Problem Solving Handbook throughout the
semester. That handbook should be a useful instructional document as you
prepare to teach your own students in the schools in the future.
The following books have been placed on reserve in the
Morris Library:
Johnson,
K., &Herr, T. (2001 & 2004). Problem solving strategies: Crossing
the river with dogs. Key Curriculum Press.
Polya, G. (1973). How to solve it. Princeton University.
Schoenfeld, A. (1985). Mathematical problem solving. Academic Press.
Problem
Solving in school mathematics: 1980 NCTM Yearbook. The
National Council of Teachers of Mathematics.
In addition to these references, relevant sections of
Principles and Standards for School Mathematics (2000) (National Council of
Teachers of Mathematics) will be read.
The final grade will be based on the following
components. Approximate weightings for
each component are provided.
Introspective Journal (10%): You must
keep a contemporaneous journal of your thinking as you grapple with the
problems assigned in the course. This
journal must effectively document your
thinking and attitudes with dated and timed entries in a written form including
references to copies of scratch work. It should be in a form that would allow a
reader to effectively reconstruct your thinking while exploring the problems
assigned. Think of this document as “data” if you were doing research on how a
person grapples with and adjusts to engaging challenging, non-standard
problems. Think of this journal as having two other related purposes: as
baseline information for gauging the development of your own learning and as a
catalyst for becoming more perceptive about the learning of others (e.g., your
own students).
The instructor will request to see your journal at least
twice: at least once during the semester (so bring it to class each week) and
at the end of the semester. He may elect to copy a journal for informal
(anonymous) use in the design of future versions of MATH279.
Problems (25%):
Each week you will be given one or more problems to solve. These
problems have several instructional purposes. They will help you review or
learn some new mathematical ideas and strategies. They are a primary catalyst for you to think about your own
thinking and attitudes. Your reflection on your work on them should lead to
your identification of problem solving strategies that have value beyond the
particular problem for which you constructed them (see the content of the Problem
Solving Handbook below). Always attempt to determine more than one solution
for a problem.
You should work alone on the problems assigned for a day or
so and record that experience in your jounal before working on them with your
group. Also record in your journal your experiences in working in your group.
Each week the solutions to the problems for that week will
be collected from each of the students in a few of the groups. So you must
bring your individual solutions and any group solutions to each class in case
they are collect that week. Such work will be collected at least twice for the
students in each group during the semester. They will be evaluated by the
instructor using the Scoring Rubric
for Problem Solving attached.
That rubric and the Scoring Rubric for Affect were developed by Dr. J.
Cai. The latter is provided for your
use—perhaps with your own students when you teach.
Problem Solving Handbook (20%): Each week a different group from the class
will be responsible for using the references in the library (and other sources
if you wish) to explore a different problem solving strategy or heuristic. That
group is required to prepare a section for the handbook that focuses on that
strategy. Although the labels used for the strategies in the outline below are
usually chapter headings in the Johnson-Herr book, all the books on reserve,
except Schoenfeld’s, must be examined as part of this assignment. Note that
different authors use somewhat different labels for similar strategies. You are
urged to skim the books on reference to become familiar with their content and
uses beyond the particular strategy assigned to your group.
You should keep the future teaching needs of your fellow
preservice teachers in mind as you develop your section because the value of
this handbook to your future teaching will depend on the quality of your
group’s work. A section should have at least the following features:
Your group must do its work on its assigned section early
enough that it is given to the instructor by late Thursday of the week your
section is due. He will then add it to the Problem Solving Handbook web page.
This will permit students in MATH279 to see and use an updated version each
week.
Mid-term Test (20%):
There will be a mid-term test in class. Prior to that test you will each
be assigned a chapter from the 1980 NCTM Yearbook on reserve. I will give each
of you your assignment early in the semester so don’t procrastinate on
reading/outlining it for yourself. The structure of that test will be
essentially the following:
Final Exam( 25%): There will be an in-class final exam. That exam will have a
structure similar to that of the mid-term but the question concerning the
chapter in the 1980 NCTM Yearbook will be replaced by one concerning readings
in Schoenfeld’s book. You are expected to read pages xi-xiii, pp. 1-45, of that
book sometime during the semester. The question(s) on this reading will be
essentially: List and briefly discuss the four components of Schoenfeld’s
framework for analyzing problem solving. Be prepared to provide as part of that
discussion a brief example to illustrate each of the four components. You may also be asked to describe your own
personal experience and introspections from problem solving during the semester
in terms of these components. You are
advised to select a time early in the semester to prepare yourself for these
final exam questions.
Our
class time will be devoted to: A brief report from the group whose section of
the Handbook was posted the preceding Friday; groups reporting on their
solutions to that week’s problems; and solving new problems. Note the deadlines for the reports from
groups for the Problem Solving Handbook.
The group assigned the strategy for that week is listed. That group
must get their report to the instructor by Thursday, 4:30 p.m., of that week. Please
look ahead and don’t wait until that week to start your group’s work!
2/9 Introduction to the course: What is a problem?
What
is problem solving?
What is a problem-solving
strategy?
2/16 “Draw a Diagram” [GROUP 1 – Your section report is due
Thursday, February 19]
2/23 “Systematic Lists/Organizing
Data” [GROUP 2; due Thursday, February
26]
3/1 “Look for a Pattern” [GROUP 3; due
Thursday, March 4]
3/8 “Guess and Check” [GROUP 4; due Thursday, March 11]
3/15 “Subproblems” [GROUP 5; due Thursday, March 18]
Do random drawing of group numbers
to determine which groups are responsible for the strategies listed below for
4/19 – 5/10.
3/22 Spring Break
3/29 Mid-term Test
4/5 “Solve
and Easier Problem” [GROUP 6; due
Thursday, April 8]
4/12 “Work
Backwards” [GROUP 7; due Thursday,
April 15]
4/19 “Algebra” [GROUP ??; due Thursday, April 22]
4/26 “Finite
Differences” [GROUP ??; due Thursday,
April 29]
5/3 “Finite differences” [GROUP
??; due Thursday, May 6]
5/10 “Other
Ways to Change Focus” [GROUP ??; due
Thursday, May 13]
5/17 General discussion about the
role of problem solving in teaching secondary students
Scoring Rubrics for Problem Solving
Excellent: 4
* problem is
solved with a clear, complete strategy and strong supporting explanation
*
appropriate issues and questions are understood
* the reader finds nothing missing;
the problem is completely and correctly solved and the answer is interpreted
correctly
* may show
originality or nonstandard procedures
* solution
may go beyond what is asked – including providing alternative solutions
* checks for
accuracy and precision
Adequate: 3
* problem is
solved and there is clear evidence of the strategy
* solution process is complete or nearly complete and
correct as far as it goes
* shows understanding of the problem’s elements,
relationships, issues, and questions, with only minor omissions
* argumentation is complete or nearly complete and may
contain only minor flaws
* checks
for accuracy
O.K.: 2
* shows
partial understanding of what the problem is about
* shows only some understanding of
the relationships between important problem elements
* explanations
are muddled, incomplete or missing
* shows some indication of a solution plan but makes
errors in carrying it out
Minimal: 1
* begins
but fails to give evidence that some strategy is operating
* shows a
limited understanding of the problem situation
* may attempt to use irrelevant information or put too
much emphasis on unimportant elements
* no
attempt
* illegible
scratching
* combines numbers or applies rules without evidence
of understanding
Scoring Rubric for
Affect
Excellent: 5
* shows persistence and makes effort to solve problems
* shows a good, confident attitude toward mathematical
problem-solving processes
* no unreasonable complaints
* does not blame failure on someone else
* takes personal responsibility for problem solving,
learning, improving, and other course related activities
Adequate: 4
* nearly always maintains a good, confident attitude toward
problem solving
* may become slightly discouraged, but keeps trying
* very little complaining
* most of the time, takes personal responsibility for
problem solving, learning, improving, and other course related activities
O.K.: 3
* may complain, but is willing to continue to meet the
course requirements
* may claim to be stupid or not good at problem solving, but
at least believes there is a chance to improve
* may tend to blame past or present teachers, but admits
some need for personal exertion order to be successful
Minimal: 2
* shows passive attitudes toward mathematical problem
solving
* may not complain if an idea for solving arises right away,
but complains when the solution path is not immediately obvious
* blames past or present teachers or others for difficulties
* uses never having been good at math as an excuse not to
try
* claims that the problems are too hard
* claims to be trying when there is little or no evidence of
that happening
Confused: 1
* shows very negative attitudes toward math and teaching of
math
* either fails to get started or gives up on a problem
before doing anything
* shows no belief that personal responsibility is needed for
learning
[1] Parts of this section of the syllabus and the two
rubrics at the end were designed by Dr. J. Cai.