Math 105:
Problem Solving in Mathematics
Course Description
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This course introduces students to the true nature
mathematics, what mathematicians really do, how they
think, and what they try to accomplish.
• The focus is on using quantitative reasoning and intuitive
logical thought techniques to solve problems rather than
formal rigid processes.
General Education Goals
• Goal 3: Scientific and quantitative reasoning, including the ability to
apply generally accepted processes of mathematical and scientific
inquiry in addressing multidimensional problems/issues
• Goal 4: Ability to address problems/issues that require the
integration of ideas from multiple sources and disciplines
• Goal 5: Abilities for reading, oral and written communication,
qualitative reasoning, technological applications, critical thinking and
effective information retrieval, evaluation and use, all of which are
essential for effective learning.
General Education Goals
• Goal 3: Scientific and quantitative reasoning, including the ability to
apply generally accepted processes of mathematical and scientific
inquiry in addressing multidimensional problems/issues
• Goal 4: Ability to address problems/issues that require the
integration of ideas from multiple sources and disciplines
• Goal 5: Abilities for reading, oral and written communication,
qualitative reasoning, technological applications, critical
thinking and effective information retrieval, evaluation and use,
all of which are essential for effective learning.
Course Objectives
The student will be able to:
• Be able to recognize and produce a precise and formal statement of
a problem.
• Explore various parts of a problem, including any necessary
background information, basic examples, what sort of solution is
required, and what techniques might help to solve it.
• Demonstrate a logical reasoning process in solving problems.
• Be able to precisely present their ideas to others.
• Demonstrate an ability to understand and critique non-technical
scientific writing.
Why Solve Problems?
• What major world problems need solving?
Why Solve Problems?
• What major world problems need solving?
• Are all of these problems going to be solved by scientists?
Why Solve Problems?
• What major world problems need solving?
• Are all of these problems going to be solved by scientists?
• What major problems will you need to solve in your major field of
study?
Why Solve Problems?
• What major world problems need solving?
• Are all of these problems going to be solved by scientists?
• What major problems will you need to solve in your major field of
study?
• Will your solutions be scientific?
Why Solve Problems?
• What major world problems need solving?
• Are all of these problems going to be solved by scientists?
• What major problems will you need to solve in your major field of
study?
• Will your solutions be scientific?
• Even if your solutions are not scientific, they may require
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Formal reasoning
Critical thought
Careful re-evaluation of your solution
The careful formation of an argument to convince others
The evaluation of other people’s arguments and reasoning.
Why Solve Problems?
• What major world problems need solving?
• Are all of these problems going to be solved by scientists?
• What major problems will you need to solve in your major field of
study?
• Will your solutions be scientific?
• Even if your solutions are not scientific, they may require
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Formal reasoning
Critical thought
Careful re-evaluation of your solution
The careful formation of an argument to convince others
The evaluation of other people’s arguments and reasoning.
• We will focus on these aspects of your solutions.
The point is this…
• Use any resource available to solve a problem and
understand the solution.
• Begin by trying to solve easier versions of the problem.
• Take solutions to other problems and generalize them or
apply them to your problem.
• Create new problems. Ask questions.
Our goal
• The solution of a problem is not really our goal.
• Understanding the solution is the goal.
– If you understand the solution, then you may be able to apply it
to other situations,
– or you will know why you cannot apply it.
• Measure your understanding by how well you can
explain it to someone.
Example
You arrive in the afterlife and find two guardians, each one
standing before a door. One of the doors leads to a
place of eternal happiness and the other leads to a place
of eternal anguish.
A sign indicates that one of the guardians always tells the
truth, the other one always lies, and you are allowed to
ask one of them a single question.
What should you do?
How will your learning be evaluated?
Your grade will be determined by
– Your group’s projects. (65%)
– Your Problem-Solving Journal. (10%)
– Your weekly Reading Reflection Papers. (25%)
How will your learning be evaluated?
Your grade will be determined by
– Your group’s projects. (65%)
• Problem solving abilities
– Your Problem-Solving Journal. (10%)
• Reflection on your problem solving abilities
– Your weekly Reading Reflection Papers. (25%)
• Argument and reasoning abilities
How will your learning be evaluated?
Projects:
• Written report for each project.
• Some of these will be presented to the class.
This will be a
random selection, so you should always be prepared to present your project. The
presentation score will be added to the project score.
• Your project score will be based on:
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the organization and write-up,
problem solving strategies,
argument and reasoning abilities,
appropriate use and citation of sources,
possible new ideas generated for future research.
• Your project score will not be based on:
– the length of the project
– whether or not you solved the problem.
How will your learning be evaluated?
Problem-Solving Journal:
• Discuss all your problem solving experiences throughout
the semester.
• In each journal entry you should:
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describe the situation and the problem to be solved,
discuss any attempts to solve the problem and the merits and
shortcomings of each,
if you were able to solve the problem, describe your solution
and what techniques brought you to this solution,
discuss whether you believe this is the best solution to the
problem, and if not, find a better one.
How will your learning be evaluated?
Problem-Solving Journal:
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You may write about any problem you encounter in any aspect of
your life.
Problems of a trivial nature (like tying your shoes) are
unacceptable.
The point is to get you to think about non-trivial problems. For
example,
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You are going out to run several errands. Is there an optimal route
that will minimize your driving distance and therefore save you some
gas?
Your team captain is out with an injury and your team is disorganized
and morale is down. Can you motivate them to work together?
How will your learning be evaluated?
Reading Reflection Papers:
• The formation and evaluation of arguments.
• Weekly readings of non-technical science writing.
• Reflection paper: No more than 4 pages.
– Summarize the argument(s).
– Was the argument organized, well-written and effective?
– Was there sufficient evidence? Are you convinced?
– Ask 3 interesting questions about the article (they need not be
answered).
– How can these ideas be applied to your other classes/major
field?
Example
Which list has more numbers in it?
The even numbers:
2,4,6,8,…
The odd numbers:
1,3,5,7,…
Example
Which list has more numbers in it?
The natural numbers:
1,2,3,4,…
The even numbers:
2,4,6,8,…
Example
Which list has more numbers in it?
The natural numbers:
1,2,3,4,5,…
The natural numbers
and zero:
0,1,2,3,4,5,…
Example
Which list has more numbers in it?
The natural numbers
The integers
1,2,3,4,…
…-4,-3,-2,-1,0,1,2,3,4,…
What is expected of you
• Work towards an understanding of the Mathematical
process of problem solving.
• Contribute ideas to your group.
• Present a project that you are proud of.
• Learn some Mathematics along the way.
• You are responsible for your learning.
What should not happen in this class:
• When are we ever gonna use this stuff?
• Will this be on the test?
• You expect us to come up with that on our own?
• Do we have to know that?
• I hate Math. I’m terrible at Math.
What should not happen in this class:
• When are we ever gonna use this stuff?
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Whenever you solve problems you may be able to apply some of these techniques.
• Will this be on the test?
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No, there aren’t any tests in this class. Are you here to learn or take tests?
• You expect us to come up with that on our own?
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Now I do. Learning involves applying reasoning abilities and skills to new situations.
• Do we have to know that?
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You don’t HAVE to do anything. The fact that you’re in college tells me that you want to
learn.
• I hate Math. I’m terrible at Math.
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You probably hate the Mathematics that has been presented to you so far: numbers &
equations. This is not what Mathematics is all about.
First Assignment
• What is Mathematics?
• Describe what you think Mathematicians do.
• How do you feel about Math? Describe your experience
thus far in previous Math classes.
First Individual Project
What is Mathematics?
In a 4-page paper:
• Give a formal definition of what Mathematics is.
• Describe what Mathematicians do (they don’t just solve equations).
• Discuss how you think you will use Mathematics (beyond arithmetic) in your
intended field of study.
• Give a brief history of a famous Mathematician from before 1900 and a
summary of his/her accomplishments (i.e. what problems did they solve?).
• Give a brief history of a modern Mathematician (1900 or later) and a
summary of his/her accomplishments.
Include a bibliography of all sources cited. Visit the Writing Center with your
final draft. Submit via email.
Due Monday Sept 8 (section A).
Tuesday Sept 9 (section B).