Project Guide for Math 319, 419, 592, and 560

Project Overview Guide for

Math 319, 419, 592, and 560

Prof. Andrew Ross

Abstract

This project guide introduces you to why you are doing projects, gives guidance on selecting projects, and discusses the main things that are expected: ambition and explanations. It finishes with the grading rubric I use. Details on the specifics of writing a paper are left to another document.

Goals

Why am I requiring you to do a project? I want you to

• Get real-world modeling experience,
• Train for a potential future job,
• Exercise what we’ve learned in class,
• Go deeper than what we’ve learned in class,
• Actually improve the real world,
• Think about how you might teach a topic (for education majors),
• Learn and practice good academic writing,
• Prepare you for the Math Contest in Modeling (MCM),
• Perhaps submit your project to a peer-reviewed journal,
• Have fun, and
• Show me that you’ve done most of that.

Good and Bad Projects

What makes for a good project? Many people draw from their job experience. If you’ve ever cursed or grumbled about some aspect of your job that seemed suboptimal, that might be a good topic for a project. One student based his project on his job as a pizza delivery driver, optimizing routing and dispatch decisions. Other students have worked on demand prediction and staffing at restaurants where they were servers or managers. Your project could also focus on a topic related to a job you want to have, like being an actuary, K-12 teacher, stock market analyst, etc.

What makes for a bad project? Sadly, some projects turn into exercises in basic accounting, adding up a collection of various simple cost terms. This does not reflect the skill required of someone in a 300-level (or above) college math class. I try to catch these projects at or before the proposal stage, and suggest some appropriately advanced analysis to bring them up to par. While some of the mini-projects we do at the start of the semester might be like this, we do them to focus on the form of the report, while keeping the math deliberately simplistic.

I encourage you to base your projects on real problems, often involving real data. You will learn things from analyzing large sets of real data that I could never summarize for you in class. In the 500-level courses, though, it is sometimes appropriate to work on extending an existing model without having real data. I will try to guide you to find the right balance.

The Proposal

To help you along the way, I require more than just a project report. I require a project proposal (just as many agencies that make grants require a proposal), to get you thinking and exploring some topics before it’s too late. I also require a presentation, as many research projects are disseminated via conference talks long before they appear in the academic literature. We will discuss each item (proposal, report, and presentation) in more detail later.

As you are organizing your thoughts to write a proposal, it is often very helpful to sketch the graphs you expect to include in your final report. They don’t have to be precise or even match what you actually get, but they will help you formulate the questions you want to ask. For example, you might want to see how cost varies when you vary the number of servers—does it go down initially and then back up? Or does it just go up? Is it concave up or down?

Project Scope

How big should your project be? First, let’s look at some quantitative measurements:

• I don’t count pages, but most good projects are at least 6 pages, and often more like 12, including graphs, tables, and appendices.
• This is a 3-credit class, and you will devote at least two weeks’ worth of homework time (at 3 or 4 homework hours per credit) to doing and writing up the project. Add to that the time spent on the proposal and the presentation.

More important, though, is what you do in your project. Excellent projects involve either doing a substantial extension of existing mathematical models or analyzing a real (and moderately large) data set. By moderately large, I really mean large enough to have outliers, and usually with multiple related variables. In practice, this means that the data set fills more than one page. Most of the time, students can get a good data set electronically. I am not asking you to take data by hand with a stopwatch and a clipboard, for example. Sometimes, you might need to type in data that you have found in printed form (for example, a printout from a restaurant computer system that can’t export electronically). If you get a data file in a strange format, I can often help decode it in an automated way so you don’t have to retype it, but let me know early on in the semester.

In some cases, you might not be able to get the data you want. One way around this is to simulate data yourself (with my help), then analyze it as if you didn’t know how it was generated. This is also a possibility even when you have a real data set; it can help you figure out how your analysis works when you know exactly what is going on in the data set. Of course, you include in your report how you generated the data set.

You are given two scores for ambition: one for ambition in your topic selection, and another for what you accomplish. The best papers solve the problem they pose in more than one way, and compare the results. For example, a project on staffing at a restaurant might start by assuming that the future demand is known precisely, and find the optimal schedule. Then, the project would include some measure of randomness in the future demand, re-optimize, and compare the resulting schedule and bottom-line cost to the simpler model. Good-but-not-great papers solve the problem in only one way, and interpret the results rather than simply presenting the final numbers and ending. Below-average papers impose unreasonable assumptions that make the problem too simple (sometimes they do this unwittingly). Often, the line that divides “too simple” from “just simple enough” is not as clear to students as it is to professors, so check with me often to make sure you are on the right track.

Explanations

It is important to be able to do good mathematical work; it is just as important to be able to explain it well. If you can’t explain it well, nobody will know about the good work you did. One of the first rules of writing is to know your audience. I want you to write as if your reader is someone in a similar class but in a different U.S. state. That is, they have covered the same sort of material we have, but probably not the specific examples that we have done in class. Also, they have not heard the discussions that you and I had in office hours about your project. This has two effects: anything you explained to me in person should be written again in your project, and anything I explained to you or asked you to do should be explained again in your project. For example, one student did a project involving ride-on lawnmowers. I said that for a rough estimate, we could use the gas mileage of a subcompact car as the mileage of the mower; that student should think about why I suggested that, and write the reasoning into the paper.

Many times, students are not very clear on how they did their computations. A collection of equations doesn’t always show which ones are used first, which are second, etc. It might help to create a flowchart that highlights what your input values are, how you process them, and what your output values are. For example, here is how the computations go for an M/M/c queueing system:

Lesson Plan Projects

Secondary-Education majors may do a project that consists of creating a set of lesson plans that could be used in a future job. I encourage this. The danger is that, since the lessons are oriented toward pre-college students, it takes an extra effort for you to show that you have learned the material at the college level. Here are my requirements for lesson-plan projects: you should have

• A list of learning outcomes,
• Alignment with standards (where possible),
• Newexamples:
• Not just taken from another source
• Entirely worked out (show work, not just answers)
• Practical examples, with sensible values,
• Worksheets or homework set(s) with an answer key,
• Samples of forecasted student responses:
• Excellent responses,
• Good responses,
• Common errors, and suggestions for helping students overcome them,
• Material that goes beyond the lesson, in case your original material runs short, or in case a smart student asks a probing question.

If two projects are required in a semester, then at least one of them must be an ordinary modeling project rather than a set of lesson plans. This is because I want secondary-ed teachers to have experience in doing real applied mathematics, so they can share that experience with their students in the future.

Grading Rubric

Roughly speaking, scores near 4 are like an A grade, scores around 3 are like a B grade, etc. When the bottom end of a scale is not specified, it is still possible to achieve those low scores.

Proposal:

4 points for describing the system,

4 points for clarity of objectives,

4 points for references(should have roughly 3 references, mostly academic rather than Wikipedia-like)

Report:

Ambition in Topic Selection, weight = 1

4: real-life project w/large data set, or substantial extension of other's work

3: minor extension of other's work, or very good explanation w/ new examples.

2: re-implementation of other's work

1: situation is somewhat too simple for this class

The professor may choose to give a higher topic ambition score that normal for projects that he particularly wants to see done.

Ambition in output, weight = 1

4: multiple solutions based upon different assumptions about or interpretations of the task(s), usually with comparisons between solutions.

3: one good solution and interpretation of those results

2: solution is too simple for the situation

1: no actual solution achieved

Correctness (not including explanation of work), weight = 1

4: correct in almost every detail

3: a few minor details are wrong

2: one major error

1: two or more major errors

Explanation: weight = 1

4: Understandable by anyone with course prerequisites

3: clear at first reading to most people in our class

2: clear at first reading to the professor, with no gaps left.

1: professor understands after several readings, or: at first reading, but with some gaps in explanation.

0: professor is unable to understand it

Organization, weight = 1

4: sections with clear/big headings, appropriate material in each section, logical flow.

3: sections with harder-to-see headings, appropriate material in each section, mostly logical flow

2: few section divisions, inappropriate placement of material, mixed-up flow

1: little or no sectioning, no logical flow.

Grammar/Spelling/Sentence style, weight= 0.5

Style refers to intricacy of sentences, passive voice, etc., apart from professional language.

4: One or two grammar or spelling errors, good style

3: half a dozen (or so) grammar or spelling errors, acceptable style

2: a dozen errors or so, style is hard to read

1: more errors, style inhibits comprehension

Professional Language, weight = 0.5

(avoiding most contractions, informal words, slang, etc.; using We where appropriate)

4: no changes needed

3: a few changes needed

2: half a dozen (or so) changes needed

Short abstract (< 50 words), weight = 0.25

An abstract is a concise summary of the whole paper, not just the conclusions, and is understandable without reference to the rest of the paper. It should contain no citation to other published work. It should contain little or no math notation. It should include key words that someone might be searching for using Google, etc. (but no need to repeat what’s already in the title).

The abstract is listed last here because it has a low grading weight, but it comes before the paper (just after the title/author).

4: summarizes the problem and solution, proper style

3: summarizes the problem and solution, style is unsatisfactory

2: fails to describe the problem or fails to describe the solution, but style is satisfactory

1: like 2, but the style is unsatisfactory

Medium abstract (< 200 words), weight = 0.25

(same grading as Short Abstract)

Here is a sample weighted average:

3: Ambition in Topic Selection,

4: Ambition in output,

4: Correctness

3: Explanation

4: Organization

3: Grammar/Spelling/Sentence style,

2: Professional Language,

3: Short abstract (< 50 words)

4: Medium abstract (< 200 words)

Total score: 3*1+4*1+4*1+3*1+4*1+3*0.5+2*0.5+3*0.25+4*0.25 = 22.25; the sum of the weights is 6.5, so 22.25/6.5 gives 3.42, which is roughly a B+ or A-.

References

Krantz, Steven G. Primer of mathematical writing,1997

Higham, Nicholas J. Handbook of writing for the mathematical sciences, 1998

Knuth, Donald E., Tracy Larrabee, and Paul M. Roberts, Mathematical writing,1989