Reflections of a Math Professor and Project Leader
My Role in the Math Images Project
What I Did and Why
Stephen B Maurer
1. How I got interested.
I have long been interested in the special nature of mathematical writing. In addition to all the usual rules of good writing there are many special conventions in mathematics and extra attention needed to the presentation on a sentence by sentence level, since mathematical reasoning is so precise and dense. I have published various articles on this subject and put further material online.
However, I had always worked with traditional formats, articles and books. They might be created electronically, but the resulting format was traditional.
Gene Klotz approached me one day and asked if I would be a writing coach for students working in his math images project, which he pointed out was being developed on a wiki.
This intrigued me for several reasons. One of them was the same aspect that attracted Gene: the ability to insert images, and even dynamic images (i.e., short movies, or images where the reader can control the motion.). But actually I was more intrigued by other aspects. One was the collaborative nature of wiki writing, or perhaps I should say seriatim nature of wiki documents, since anyone can modify them, not just people the original author intentionally collaborates with.
But most of all I was intrigued by the following. A traditional paper or book must be presented in a linear order. While on first thought this seems perfect for mathematics, since the deductive nature of mathematical proof is linear, in fact the best way to learn a piece of mathematics in not usually linear. For instance, to do a proof it may be best to get the overall structure first and then fill in the next level, and then the next level of detail. To get an intuitive understanding it may be best to go off on digressions about examples, but how many digressions depends on the reader. Even in the traditional linear presentation one has to refer backwards often, which in traditional writing is done by section numbering and display labeling, e.g., refer back to Equation (4). An online html document doesn’t have to give names for cross references; it can just provide links.
In short, online documents make it much easier to jump around than before, and to include many more “figures” than before, and it might be that the conventions of mathematical writing should be changed to accommodate this.
By joining the Math Images Project, I would have the chance to think about this, along with the students and other faculty, as pages got written.
2. Progression of my involvement
Initially I was to be just a writing coach. However, with Gene’s retirement from the faculty of Swarthmore, someone else was needed to be the principal investigator at Swarthmore for grant purposes, including for internal grants of summer stipends to students. I agreed to do this. There were enough little responsibilities that fell to me once I agreed to be the nominal head (e.g., some faculty member had to order summer keys for students, some faculty member had to reserve a room for meetings, some faculty member had to sign their forms for obtaining early housing, etc.) that the simplest solution was to accept that I was head and proactively organize things.
As it happened, throughout the years with Math Images I had another 4-week commitment, roughly spanning July, which took me out of state and kept me very busy – a math camp. So whatever help I was going to give, it needed to be done by my departure day in June, and other people had to be prepared to take over that role after I left. This was an incentive to be organized each year from the beginning, and to ensure that other people working on the project were sufficiently prepared to continue it in my absence.
3. Value of the program
a) The original idea was that the Math Images website would fill an important function not fulfilled by other math websites such as Wikipedia, Wolfram MathWorld, or the pages intended for the public at the websites of mathematics societies. Namely, 1) Math Images would emphasize images, as hooks into learning mathematics. Each page would be based on an image with which the page starts. Second, it would be at a somewhat lower sophistication level that Wikipedia or MathWorld, which are generally read by professional mathematicians or advanced students (at least college students), where Math Images should resonate with high school students or younger. 2) Being a wiki, it would nonetheless hope to attract enough talent that it would grow on its own eventually. That is, initial funding would lead to enough high quality work that quality writers would be attracted to contribute.
b) Value to the students.
The long-term goal of most funding for student projects in STEM fields is to increase the number of students who go into STEM fields and who become very effective STEM professionals. This takes longitudinal studies, which are underway.
It is an old observation that the best way to truly learn something is to teach it to someone else, and by writing article that’s what the Math Imagers are doing. So we expected the summer to deepen their own understanding of some pieces of mathematics and thus give them more confidence. But I for one did not initially see several other reasons why working for Math Images would be an attractive option.
Foremost, Math Images was the only math summer project that hired students after their first year, and one of the few projects in any discipline to hire such students. To do traditional research in mathematics you usually need to know a lot more than rising sophomores know. But first-year college students, indeed high school students, can, by selecting an appropriate image, find something they can explain at their own level.
Second, Imagers got to learn, or learn better, several major pieces of software. Not only did they need to learn wiki markup language, but to insert math they needed to learn TeX, and to learn how to make good figures all sorts of software, from Adobe Illustrator to Mathematica to MATLAB to Geometer’s Sketchpad; and to make dynamic illustrations still other software. Furthermore, they got paid to take time to learn this software. What could be better than that, to be paid to take time to learn some useful software?
Third, Imagers got the thrill of publishing. When an article was in good enough shape, it would go live on the web. The younger the student, the more they were excited by this milestone of mathematical maturity.
Once word got out that working for Math Images had all these advantages, we got plenty of applicants.
c) The value of participation in Math Images for a faculty member may be less clear, but they were also compelling to me once I understood them.
The first benefit is that students teach you math you didn’t know. Yes, they are likely to pick elementary topics, but there are plenty of elementary topics that I don’t know much about, and even when I do know a lot about them, a student often finds something I didn’t know.
The second benefit I mentioned before: you get to think about what aspects of web documents, such as links and the capacity to hide and unhide material, might lead to very new ways of organizing mathematical writing.
Third, you get to see what your students find hard to express. One learns a version of his just by reading and commenting on traditional papers as well, but wiki papers add a new dimension. Furthermore, if the students are working on the wiki as their sole summer project, you can go over their work again and again, which is hard to do in a regular course. You get a deeper understanding off difficulties when you repeatedly work on the same paper. And students take a deeper interest in how they are doing because the final result, if good enough, will be published.
The Round of the Year
At Swarthmore, students start thinking about their summer plans in December (the end of the fall semester) or in January (the start of the spring semester.) Departments often run Show and Tell sessions in one of these months, where faculty talk about projects for which they seek students and students talk about their experiences last summer. Whether or not your department has such an event, you need to publicize your project. We do this at least 3 ways:
- Create a flyer that can be sent to individuals and widely displayed. See Ad-Flyer Math ImageSummer13.docx.
- Send an email to all potential students (say, students who have taken certain math courses). The most recent email is copied below.
- Display the poster our group made last year for the fair of research projects. See 2012 Math Images Poster_Final.pdf
Subject line: Swarthmore Summer Stipends for the Math Images Project
The Math Images Project expects to hire again this summer with Swarthmore Research Fellowships, especially first- and second-year students and underrepresented groups. Please read the attachments to see what Math Images is all about, but basically, you extend your knowledge of mathematics and then figure out (with help) how, starting with an image, to explain what you've learned with all the resources of a wiki. It's an interesting form of modern electronic writing.
This message follows up the presentation about the Math Images Project at the Math/Stat summer jobs colloquium the other week, but you don't need to have attended that.
If you are interested, please get back to me right away. There is a *pre*application to submit to us in a few days and then the application to the College is due 5pm Friday. If you write back, I will immediately send you information about the preapplication. Information on the College application process is at http://www.swarthmore.edu/x8586.xml
Steve Maurer Math/Stat Dept
Our announcement email above refers to a preapplication. At Swarthmore the application itself is a standard form for all students applying for summer stipends. Because of the large number of potential applicants for Math Images, we ran our own screening process (the pre-application), through which we limited main applications so that those who put in the main application had our backing and were likely to be accepted. Below is our preapplication, with certain local details omitted.
Your preapplication consists of an email (equivalent to a page or two on paper), in which you address the following items:
1) State your name and Swarthmore class, and that you are applying to work for the Math Images Project this summer. Women and minorities are especially welcome to apply. There is extra funding from HHMI for first- and second-year students who are women, first generation college students, and underrepresented minorities. That is, you are more likely to be funded to work with us if you are in one of these groups.
2) Tell us your possible majors. Any major is ok; it need not be math. We are just curious and hope to get students with a range of interests.
3) Explain why this project interests you.
4) Tell us what software you have worked with, if any, that you think is relevant to work on this project, and how proficient you are with it.
5) Part of the work this summer will be to produce one or more pages on grade 7-12 math topics that are hard to teach. In your mind create your own list of school topics hard to teach (or so some of your friends said). Tell us about one of these topics that you would be particularly interested in trying to develop for our wiki and any initial ideas about how you would do it. Note: We do not expect you to have very sophisticated ideas yet about how to do it. That comes with experience writing wiki pages. Rather, we are interested in knowing your initial reactions.
6) Go to http://mathforum.org/mathimages/index.php/Existing_Pages_Needing_Work . This is a page that links to Math Images pages that former summer students have worked on, but which for one reason or another are not ready yet for prime time. Tell us about one of these pages that you would be particularly interested in trying to complete and any initial ideas about how you would do it. (Caution: this webpage was last updated just before the 2012 students went to work, so some of the pages identified as needing work *have* been worked on.)
Note: Before submitting this application, you should browse the Math Images website, both the public parts accessible through http://mathforum.org/mathimages and the private pages such as the one in 6), and other pages you can get to from it. Another good private page to look at is http://mathforum.org/mathimages/index.php/S12 , which was the master coordination page for the Summer 2012 Images group. You can also go to several useful pages from the S12 page, including the pages in the navigation bar to the left
I look forward to hearing from you. To follow up your preapplication, we may ask you to come talk to us
Reading Pre-applications and Interviewing
From the written pre-applications, we looked to see how thoughtfully they had looked at current Math Images pages, how reasonable were their proposals for modifications and new pages (for instance, many applicants wanted to write about fractals, but you can’t get too far into fractals without knowing a lot of real analysis), whether they had any past software experience that might be worth sharing, and most important, any evidence that they worked well in a group. There may have been a flaw with the application announcement in terms of whether we asked good questions about group experience. Still, we could get a sense of group-work-readiness in interviews by getting a sense of how friendly and flexible the candidate seemed, and by asking directly whether they had worked on a group project and if so how did it go. We generally held interviews as well, to help further distinguish candidates. Partly it was informational, a chance for them to get clarification about what was involved. But partly it was to see how much they were in person like what they wrote in their pre-application.
We did not try to ascertain how well a candidate could work in an open-ended environment, where they had to pick what page to write and judge how much to say, all the while scheduling their time effectively. For most first-year college students, there wouldn’t be many occasions where they were given the freedom of an open-ended situation already. But it might be good to inquire, because students who thrive in open-ended situations do better as Imagers, so at least they do better without careful intervention by the staff.
Here is a memo I wrote to my interview colleagues as to what I thought we should look for.
The main thing to find out is how well you think the interviewee would work in a group (while at the same time being a self starter), and how much they have thought about how to use images and wiki writing.
If you have any concerns based on their written application, I would start with that. For instance, for those who mainly spoke in terms of calculus, ask what would they do for younger readers? For those who spoke mostly in terms of algebra, what would they do to use the visual aspects of the page?
Since they were all supposed to look at several MI pages, a question you could ask all of them is: what page did they like best? (Or name a page they really liked) This is a question I did not ask them for the written application (I asked what they wanted to extend) so I think you might learn something valuable from their answer - including that maybe they didn't look very much.
Finally, I would ask how they see their MI work as fitting in with, or possibly modifying, their major and career plans. Only a few thought of being math majors or minors, according to their written applications. I would like to be assured they are open to moving in this direction, or at least using math more.
At the same time that we were culling the new applicants, we were trying to find former Imagers who we would invite to come back as mentors. We knew whom we would particularly like; the question was whether we could get them, since as older students they now usually had several other options for summer work. Sometimes we had to settle on getting mentors only in evenings, or one day a week, or for one week, and so on.
Hiring at least one alumni mentor each summer was crucial to the success of the projects. They tended to give advice on different issues than faculty and were easier for current Imagers to relate too. The mentors were around almost all the time, whereas I was around only some of the time (though usually available on email). And finally, since the mentors had done all the work of creating pages, including inserting still images and dynamic images, whereas the faculty had not, the mentor could answer “How Do You” questions on the spot that faculty could probably not answer at all. For those who are starting this up and have no project alums, I was involved in at least one where there was no mentor who had been a “worker” in a previous year. Other project leaders had confidence in these new mentors’ ability to mentor from previous work with them on earlier math projects.
I’ve left out discussion of the Swarthmore regular summer stipend application because that is particular to Swarthmore.
Getting the Summer Started
The first thing was to arrange an information meeting that former Imagers could come to as well. This meant the meeting had to be while Swarthmore was still in session but not at a time when students were preoccupied with exams. We usually met during reading period, at a time chosen through a Doodle poll.
At this meeting, we worked through the outline in the document below. Or rather, this was the particular version for 2013.
Keep in mind that a major effort was addressed to making sure that the students always had something useful to do and were not left on their own except so far as they wanted to be. Without some interventions, the Imagers would find themselves faced on Day 1 with
- A very large number of unfinished pages
- A large gallery of images waiting for a page
- An infinite set of additional topics they might choose to write about
- The complexities of wiki writing and the two math markup languages needed to put math on their pages (one markup language being a subset of TeX and the other being an xml type markup), all of this with several starter pages for learning part of it.
All of this is daunting without suggestions for where to start. And then later, when one is in the middle of writing and rewriting a page, you can only work on a page so long without needing a break. What you are employed 9 to 5, so you need to take a break by doing something else useful for the project. We had to make sure there were other things to do. Some of the activities that provided variety during the day were
- Work on a page that was already started
- Work on a new page
- Revise one of your pages based on comments others had put in the document or on its discussion page
- Read another Imager’s page and put comments on it in the document or in its discussion page
- Have a group meeting about one of your pages
- Have a group meeting about another worker’s page
- Work for Science for Kids
- Go to one of the social events the College set up for all summer research students
- Browse through the collection of math images earlier years had collected but not written pages about
- Browse the web for images that could be used in a page you want to write
- Have a meeting with one of the professors about revising one of your pages
- Learn more about some of the software you want to use. Try some constructions and see if they work.
- Clean up something on our website, for instance, update the page that lists unfinished pages and what they need.
- Actually go to the library and look at some of the expository books on mathematics. Often, especially if they are coffee table books, they have wonderful images that might give you ideas.
Agenda, Math Images 13 Orientation Meeting
0. Hand out printout of agenda and Diana's goals and schedule for the summer (the next document) [I believe this is the next document, but I am not sure.] 1. Introductions (of people), urge all to ask questions and interrupt 2. Hand out room keys, test them 3. Everyone log on to our wiki (without creating an account yet) 4. Steve controlling the screen, first brief intro to key information pages (S13, Partner Swarthmore, wiki help) 5. daily and weekly schedule in broad outline, use of the building after hours, keeping time 6. Ethical research online course - what it is and when you take it.
Diana [our one alumni mentor that year; sometimes we had more] takes over
7. Diana and Iris [another Image alumna] personal experiences with Math Images, as a first answer to "What in the world did I sign up for" 8. Diana walks us through the rest of her handout (goals, what goes on Weds and Fri's, how feedback works, how 4Masterman [the high school we partnered with, our Imagers acting as mentors to their students who were writing pages.]works 9. Role of workshops/demonstrations, of SFK [Science For Kids, another summer program at Swarthmore, for disadvantaged kids from nearby Chester PA; all summer stipend holders were encouraged to spend some time and expertise with SFK] 10. Diana goes back to S13 and talks at more depth about the advice there.
11. Questions? (but people will be encouraged to interrupt earlier)
12. Everyone sets up an account and creates their user page include more than one way to contact you where you are living - is housing ok? [an easy first use of wiki scripting, because very little formatting is needed] 13. Link your user page to S13 and indicate any special information about availability Someone do other edits to eradicate S12 info from S13 Someone get the expectations document into Expectations12 14. Create a toy math page to practice some math writing as well as section writing, lists, etc. Can hold off on graphics for now. 15. Start you real work (reviewing current partial pages; you need to be working on one for first Review Meeting Wednesday)[we set up a lot of groups , group meetings, and review meetings for the whole project, in part to break up the day]
Math Images, 2013, Overview of Working Arrangements
Goals For the Summer:
Complete 5 pages, including: • 1 that has already been started [easiest way to start as formatting is already there] • 1 with a Masterman student • 1 helper page • 2 or more original image pages • Look for pages you can do collaboratively
Before you start each page, submit a 1-paragraph informal proposal to Steve Maurer and then arrange to go talk to him. In the second half of the summer, Steve will be gone and you will pass your proposals by Gene Klotz or Chris Taranta.
Schedule for the Summer:
Generally 9-5 with national holidays
Visits to Masterman Wednesdays (and Tuesdays) 5/22, 5/29, 6/5, 6/11, 6/12, 6/18, and 6/19
Science for Kids Science for Kids, the summer program for the participants in the Chester Children’s Choir, begins the week of June 24 (after the trips to Masterman), 10:30-11:30, Mondays–Thursdays through the last week of Math Images.
Jason is required to participate by the terms of his stipend [he had a special stipend from the Howard Hughes Medical Institute – HHMI], but as we said in our funding letters, we want all of you to participate. We are waiting to hear whether SFK yet has a good way to make use of you.
Workshops: Sketchpad Tutorial - 5/23 Photoshop - 5/27 Sketchpad Redux - 6/6 Other workshops as requested [Mathematica, MATLAB, TexShop were often asked for]
Pizza Dinners and Feedback (Diana will set up groups, they will include former Math Image participants, Math Forum staff members, and project faculty) Wednesday Nights: 5:00-5:45 Feedback groups 1 & 2 5:45-6:15 Dinner 6:15-7:00 Feedback groups 3 & 4
Come to feedback sessions for your page with an idea of questions you’d like answered or issues you’d like to discuss – you get to run your meeting.
Friday Mornings: Individual timeline meetings with Diana
Details of how a page is developed and critiqued
Notice we asked students to write a one page prospectus for each page they thought to do and then wait until I’ve discussed it with them before going ahead. The reason for this is that students at this stage often don’t have a good idea what will be doable and what won’t. Just like a social science advisor helps students formulate answerable questions, I tried to help Imagers formulate doable outlines – doable with what they knew or could reasonably learn. I also wanted to make sure that their topic had associated images which were interesting and helpful.
This brings up an interesting point. Gene’s original idea was that a student would always start with an image and produce a page to fit the image. But I found it often worked better for a student to pick some mathematics they found interesting and want to know more of and then look for an image or two that could be used as a hook to the math. If they could show me that they found some good images, I let them run with their idea. Sometimes I could contribute a suggestion for an image. Once an Imager wanted to do a page on similar figures. I mentioned the scene in The Third Man where at night one doesn’t see the person but sees a huge shadow of the person again a building. The Imager couldn’t find a photo of this scene, but found a similar one from another movie.
I let the students let me know when they wanted me to review their work – usually when they got to a natural stopping point or decision point and wanted a break from writing new things.
I like to talk about local and global aspects of mathematical writing. Global aspects are mostly not that different from global aspects of other writing – is the overall flow clear and do different sections come in a reasonable order. But I contend that on the local level there are many conventions of mathematical writing which are special. I often needed to point out to Imagers that key terms they used had not been defined, or were defined incorrectly; that their writing would have been simpler if they had defined some notation or some words; that equations should have been numbered and referred to; that some standard words had been used incorrectly (this vertex has 3 degrees instead of degree 3), and so forth.
And then there is an even broader level on which one can critique the writing: will the approach grab the intended reader? Since professional mathematicians are not the typical intended reader I was not the right person to judge this. That’s why it is essential to have peer reviewers and student mentors also look at drafts and critique them. This is why Ann Renninger’s research on what ways of wiki writing work with various readers is so important.
(There is yet another issue for which the Imagers sought me out in particular: was a proof they gave right? Or, they didn’t understand a proof they had read and wanted to include. Could I explain it to them?)
How should one convey editing critiques to the author? One way is a 1-on-1 meeting with a printed copy of the current draft with corrections marked but reasons left mostly for oral discussion. This is a good method, because the student can ask for clarification, but one should not let such meetings go on too long, say more than half an hour, because a long sequence of detailed oral explanations soon gets jumbled in the listeners mind. A possible workaround is to have a sequence of 30-minute meetings, letting the student update the paragraphs covered in one meeting before holding the next.
The other two options I used were comments on the discussion page and edits right in the main page.
I found the discussion page useful for global suggestions and their rationale. I did not find the discussion page useful for local remarks because you had to spend so much effort to identify the phrase and sentence you were referring to. Making edits right in the main page was the best way to do this. But not perfect. The usual main page edit replaces the old with the new, whereas you want the author to see both. (Also: two people cannot edit a page at the same time, so it’s necessary to be careful not to overwrite edits.) One does see both in the history page, but with only snippets of the original document (new or old version) shown. So I forced the main page to look more like what one sees with the Word revision tool: the old and new side by side with a color distinction. But it took effort, say, to color insertions in an alternate color, and it took effort by the writer to eventually delete all the comments and accept or reject all the proposed changes. The wiki software we had was not an ideal tool for doing detailed edits. Later on in the process, when Diana was working externally and the students were crunching to revise all of the pages before the end of the summer, she would download the pages as Word documents and add comments in them using the Word Revision tool
What to do about a student who is not thriving.
Occasionally a student didn’t work well. Perhaps they had grand ideas about what topic they would write about but then, when they got down to writing something, they had writers block, or they wrote a lot but it was muddled; or they never could settle on one topic.
It was because of students like this that we developed several of our protocols - start by creating your personal page; your first project should be the completion of an already started page. While I tried to let students work on some (perhaps simplified) version of a project that appealed to them, occasionally it was necessary to tell a student to do something specific or give deadlines.
Example of my work with Greg on his finishing the Taylor Series page.
Advice to Others Starting Out
I have concentrated on what I did, because that’s the part of the record that I know best. Other people writing will describe their own roles. My description of what I did includes a lot of advice, which I won’t summarize here. Let me summarize instead a few points that I didn’t talk about because it involves other people.
1. Don’t try to run such a program with yourself as the sole faculty member. You need several sorts of expertise. 2. Partnering with at least one school teacher is an excellent idea. High school students get an even bigger kick out of writing a page that might be published than college students do. Moreover, the college students will learn a lot by mentoring the high school students. 3. Include some students who are studying education as well as mathematics. They are more attuned to nuances about how the presentation of material affects its comprehension
Doing an Images Project during the Regular School Year.
There are at least 2 ways to do this: First, as a project in a course – instead of a traditional paper, students might be allowed to or required to write a images type wiki paper. Second, as an independent study project.
Ralph will try the first method this fall, but I am a little reluctant about it. There is a pretty steep learning curve to learn to write in the wiki language we had, especially if you have to learn how to do mathematics in it. At most I would make it optional to write a wiki paper instead of a traditional paper. Also, with papers from a class it is hard to give revising advice more than once per paper, and to be publishable more reviews are needed.
Doing the second method might work nicely, but you would need several students doing the independent study project together, because everyone needs feedback. The possible advantage, assuming that the instructor/facilitator has time for it, is that students won’t be trying to do this project 8 hours a day 5 days a week. There will automatically be activities (e.g., graded courses) that will break up the day from too much wiki writing. One would have to schedule pretty carefully, and make students stick to it, to make sure some useful result is obtained by the end of the semester (or year, if it is a year project). But in this regard it is not much different from running a successful senior conference (our name for the capstone math course for course majors, resulting in a reasonably long paper).