To M225 Students

Your M225 Text

Members of the EWS Mathematics Department have created the material in this book and would like to give credit to the Phillips Exeter Academy Mathematics Department for many of the original problems. As you work through it, you will discover that algebra, geometry, and trigonometry have been integrated into a mathematical whole. There is no Chapter 5, nor is there a distinct section on right triangles. The curriculum is problem-based, rather than chapter-oriented.

Problem solving techniques, new concepts and theorems will become apparent as you work through the problems, and it is your classroom community’s responsibility to make these conclusions together. Your responsibility is to keep appropriate notes for your records — there are no boxes containing important theorems. There is no index as such, but the glossary at the end of the problems should help you recall the meanings of key words that are defined in the problems (where they usually appear italicized).

Problem Solving as Homework:

You should approach each problem as an exploration. You are not expected to come to class every day with every problem completely solved.

• Reading each question carefully and repeatedly is essential, especially since definitions, highlighted in italics, are routinely inserted into the problem texts. Check the reference section regularly.
• It is important to make accurate diagrams whenever appropriate.
• Useful strategies to keep in mind are create an easier problem, guess and check, work backwards, make use of prior knowledge and recall a similar problem.
• It is important that you work on each problem when assigned, since the questions you may have about a problem will likely motivate class discussion the next day. In other words, doing homework to get ahead is not a good idea since class discussion will help you prepare for future problems.
• Try to justify each step you do – ask why not just how.
• Problem-solving requires persistence as much as it requires ingenuity. When you get stuck, or solve a problem incorrectly, back up and start over. Keep in mind that you’re probably not the only one who is stuck, and that may even include your teacher.
• If you have taken the time to think about a problem, you should bring to class a written record of your efforts, not just a blank space in your notebook. There should be a diagram, equation, or reference to similar problem.

The methods that you use to solve a problem, the corrections that you make in your approach, the means by which you test the validity of your solutions, and your ability to communicate ideas are just as important as getting the correct answer. You are not to spend more than the allotted time for that night’s homework on any one nightly assignment, so please manage your study hall time carefully!

About technology

Many of the problems in this book require the use of technology (graphing calculators or computer software) in order to solve them. Moreover, you are encouraged to use technology to explore, and to formulate and test conjectures. Keep the following guidelines in mind:

• write before you calculate, so that you will have a clear record of what you have done
• store intermediate answers in your calculator for later use in your solution
• pay attention to the degree of accuracy requested
• be prepared to explain your method to your classmates, including bringing your laptop to class with the file on it in order to project your solution to the class

Keeping a Mathematics Journal

As part of this curriculum you will be asked to write about your problem-solving processes on a regular basis. This will help you to organize your thoughts around not only problem solving, but the content of the course. When you write your journal entries you should keep in mind a few things:

• Write in complete sentences as if you were explaining to yourself or to another student how to do the problem
• Justify the steps of your process and explain to yourself why you chose the methods you used in the problem
• Make connections between why you chose a certain step in the process and ideas that have been discussed in class
• Make connections between problems – see if patterns emerge in how the problems are laid out in the curriculum
• Draw diagrams that help you to understand the problem better, even if a student used that diagram in class and explain why it helped your understanding

At any time during the year, if you have questions about journal writing or want more feedback, do not hesitate to speak with your instructor, or see your instructor’s grading rubric for journal entries.

Transitioning to Problem-Based Learning

Learning in a PBL classroom is very different for most students for many different reasons. What is valued in the PBL classroom and what is considered successful takes time to understand, so most importantly you should come with an open mind and be ready to openly communicate. Be sure to communicate your learning needs to your teacher throughout the year. Here are some comments from past students:

About presenting homework solutions:

• “The fact that we have to get up in front of the class helped in my learning”
• “The accumulative mixture of problems the book had really helped me see the connections between concepts”
• “I got more comfortable with taking mathematical risks”
• “This curriculum has made me a better problem solver”
• “It helped challenge me and taught me even if I didn’t think I was learning”

About writing journal entries:

• “Keeping a journal has really helped make reviewing and preparing for tests very easy”
• “Journals totally helped, although having them on the test is useless. Once you’ve done a journal you know the subject.”
• “Although I never fully bought into keeping a journal, it gave me a good resource for studying.”

About communication in class:

• “I loved being able to discuss issues with classmates”
• “It helps when the teacher summarizes what we learn”
• “I liked finding more than one way to do something”

About getting support:

• “Meeting with my teacher really helped”
• “Make use of the dorm affiliates”
• “I liked how it was focused on yourself figuring out the problem – though that was hard for me to adjust to – however it’s made me much more independent math-wise”

Becoming a better independent problem solver is not an easy journey, but it does need your whole-hearted curiosity and effort. The mathematics department is here to support you through this year so please make use of the support systems that are available if you feel you need them.