Math 105:Problem Solving in Mathematics

Course Description

• 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.

Course ObjectivesThe 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.

Course ObjectivesThe 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.

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.

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

– 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.

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.

• 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.

• 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.

• 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.

• 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.

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 – Group Projects. (65%)

– Reading Reaction Papers. (20%)

– Midterm (5%) & Final Individual Projects. (10%)

How will your learning be evaluated?

Your grade will be determined by – Group Projects. (65%)

• Problem solving abilities

– Reading Reaction Papers. (20%)• Argument and reasoning abilities

– Midterm (5%) & Final Individual Projects. (10%)• Problem solving, research, argument abilities

How will your learning be evaluated?

Projects (65%):• Typed 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.

• Your project score will be based on: – the organization and detail of your explanation, – 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?

Reading Reaction Papers (20%):• The formation and evaluation of arguments.• Bi-weekly readings of Mathematical writing.• 3-4 page paper:

– Summarize the main argument(s)  in the reading.

– Is the argument well-organized, well-written, and effective?

– Is sufficient evidence presented? Were you convinced? 

– What did you learn from the reading that you did not know before?

– Ask and discuss 3 interesting questions about the ideas presented in the reading.

– Discuss the reading’s relationship to other courses.

How will your learning be evaluated?

Midterm (5%) & Final Individual Projects (10%):• Independent Problem Solving.• More on this later.

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 1,2,3,4,…

The integers …-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 projects 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?– Whenever you solve problems you may be able to apply some of these techniques.

• Will this be on the test?– 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?– Now I do. Learning involves applying reasoning abilities and skills to new situations.

• Do we have to know that?– 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.– You probably hate the Mathematics that has been presented to you so far: numbers &

equations. This is not what Mathematics is all about.

Book

The Language of Mathematics by Keith Devlin

First Individual Project

What is Mathematics?

Read the Prologue of The Language of Mathematics (p.1-12) and do the following in a 3-4 page paper:

• Present and discuss a formal definition of Mathematics.• Describe at least one particular field of Mathematics.• 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. • Discuss a brief history of a famous Mathematician from before 1900 and a summary

of his/her accomplishments (i.e. what problems did they solve?).• Discuss a brief history of a modern Mathematician (1900 or later) and a summary of

his/her accomplishments.

You may need to use additional sources (not Wikipedia) and you should cite these sources using any format you are familiar with and include a bibliography. Visit the Writing Center with your final draft. Submit via email.

Due: Sunday September 4 by midnight.