Illinois State University Mathematics Department

 MAT 326: Technology Tools for Secondary School Mathematics Dr. Roger Day (day@ilstu.edu)

### Extended Mathematics Investigation (EMI)

One of two options for a long-term assignment is to carry out an extended mathematics investigation (EMI) for which technology plays some role. The information here is intended to help you understand what constitutes an EMI, to provide suggestions for resources to explore in choosing an EMI, to clarify the submission requirements for the EMI report you submit, and to state the criteria by which your EMI report will be evaluated.

• A statement describing your EMI is due October 29, 2012.
• The final written version of your EMI report is at due  the end of the semester

You are encouraged to consult with others in completing an EMI, and your final submission may be made with up to two other classmates. That is, you may work individually, in pairs, or as a triple to complete this assignment.

In addition to your written report, you must present a brief summary of your EMI to your classmates during one of our last class meetings of the semester.

• What is an Extended Mathematics Investigation?

An Extended Mathematics Investigation (EMI) is a significant exploration into some mathematical problem, topic, idea, or context. The problem, topic, idea, or context may be entirely abstract, primarily at the level of an application, or somewhere in between. At its most fundamental level, it represents doing mathematics.

Here is one of a million examples.

 Consider the Fibonacci sequence, the first few terms of which are shown to the right. It appears, at least from this short list, that every 5th term is divisible by 5. Is this indeed the case? If so, what sequence results when every 5th term is divided by 5? For "the rest of the story" that includes a connection to Pascal's Triangle, turn to the March 1998 Mathematics Teacher and read the article by Corey Andreasen! term number value 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987

Technology played an important role in Corey's EMI. Among other things, he first used a graphing calculator and then a computer spreadsheet to generate and manipulate elements of the Fibonacci sequence. The spreadsheet, in particular, gave him the opportunity to explore "What if . . . ?" questions that arose as a result of his continued investigation. It is intended that technology play some role in your EMI.

In oprder to get a sense for what an EMI is, here are some suggestions for what an EMI is not:

• An EMI is not primarily a report on some historical aspect of mathematics, although you may learn some history along the way. An historical report suggests researching something that has been done or that has occurred. Your EMI ought to be about you doing mathematics.
• An EMI is not a nice write-up of your response to a homework or test question from this class or some other course. It may be that some problem you've encountered in a class may motivate you to extend the problem to go beyond its original intent, or to go off in some direction that was motivated by a required problem; these would constitute legitimate EMIs. It is not the intent of the assignment that you simply recycle work you've already done.
• An EMI is not a summary or report of someone else's work. As you explore mathematics and search for EMI problems, you will come across reports and other write-ups that could indeed stand as an EMI report for this course. The intent is that an EMI represents your efforts in doing mathematics, not a report of what someone else has accomplished.
• An EMI is not a three-sentence write-up to a problem you found who-knows-where. This type of response would not likely be in the spirit of an extended investigation nor a thoughtful and complete report of your efforts and results.
The EMI option is part of this course to help you continue, or perhaps begin, investigating mathematics outside the traditional textbook and course assignments that have likely dominated your career in mathematics.
• Sources and Resources

There are many places to look for potential topics for EMIs.

• First and foremost ought to be your interests and capabilities. What do you enjoy about mathematics? If geometry is one of your favorite mathematics subjects, perhaps you should look there for a problem, topic, idea, or context that you can investigate. Reflect on what you enjoy about mathematics and consider those areas for EMIs.
• Talk with your classmates about possible problems, topics, ideas, or contexts that you could investigate. Conversation with others may spark an idea or two for you.
• Tap into the resources of Dr. Jim Wilson, long-time mathematician, mathematics educator, and problem solver from the University of Georgia. Dr. Wilson's students are likely to be exploring various problem situations that could be fodder for an EMI of yours. Look at these courses of his in particular:
• Browse through or conduct searches at internationally known web sites for mathematics education:
• Do some of the problems in one of the many "contest books" in STV 334, or in the middle pages of each issue of the Mathematics Teacher. While a particular problem may not constitute an EMI in and of itself, it may spark you to carry out an extended investigation.
• Talk to your mathematics instructors!

Submission Guidelines (with thanks to Dr. Wilson, UGa, for the ideas and requirements he uses with his students)

An EMI report represents your synthesis and presentation of a mathematics investigation you have carried out. Fundamental to your report is that it convincingly communicates what you have found to be important from the investigation.

Who are you writing for? You might consider your audience to be your students (present and future), your classmates, or classroom mathematics teachers in general. Mathematics Teacher articles, like the one describing Corey's EMI, might serve as a good model. Remember that such published work often has been limited in length and scope in order to present a topic in a reasonable amount of space. Such constraint may limit your use of non-text or interactive materials, or descriptions of your explorations into and through alleys, roadblocks, freeways, and bumpy roads, all of which may have been a part of your EMI.

Keep that in mind as you generate your EMI report. It should communicate the essential material you have synthesized from your investigation. It could be entirely a word-processed document, or in HTML format, where you can put together text, graphics and figures, and even software applications in a dynamic document.

Your EMI Presentation

As part of this project requirement, you must present your EMI report to your classmates. The intent here is for you to plan and present a verbal report that helps your classmates to know what you investigated and what you have learned. While your presentation must convey the significance of your EMI, it need not be lengthy nor should it be verbose and filled with detailed minutia.

A successful presentation will:

• provide a brief introduction to the mathematical problem, topic, idea, or context you have investigated,
• share the highlights of what you have learned,
• mention extensions, unsolved aspects of the investigation, or new investigations that have materialized based on your EMI, and
• share some of your interest in and motivation for the EMI.

A one-page handout to be distributed when you begin your presentation may help listeners better know what you're reporting on and to focus on the important points you intend to make. I'm happy to print this for you if you give me a bit of lead time.

EMI Evaluation

An wholistic grading scheme will be used to evaluate your work, based on the submission guidelines stated above. Components I will look for include a statement of the problem, a description of your solution approaches, a statement of your results, a discussion of conclusions and extensions, and a list of resources used.