Mary DeYoung
Hope College, Holland Michigan
mdeyoung@hope.edu

NCTM Regional Meeting, Indianapolis
January, 2003

What will make the on-line mentoring experience a positive one for your college students? Careful preparation! Making any classroom into a place where students experience success depends upon careful preparation. That seems obvious enough, the tricky aspect is knowing exactly what and how to prepare. I believe the answer lies in three areas: the teacher's knowledge of content and pedagogy, the teacher's attitudes toward the subject and the students, and the careful arrangement of the learning environment. The on-line experience is similar in some ways to any other teaching/learning experience and preparation with your pre-service teacher mentors should address those same areas. Thinking though some pertinent questions is imperative to making the experience beneficial to the college students and to the young problem solvers. As I consider those two populations in the following pages, the college students will be referred to as "mentors" and the K-12 problem solvers as "submitters."

Consider first the nature of the mentors' mathematical preparation. Teaching without understanding the subject matter will not be profitable, so the mentors should be experienced problem solvers. If the mentors have completed some preliminary classroom observations with individual students or some tutoring with small groups, it will be helpful to their on-line mentoring experience with "virtual students."

What specific problem solving experience should the mentors have prior to mentoring?

For the mentors to have a feel for the depth of explanation expected by Math Forum, they should have struggled a bit on non-routine problems that go beyond the typical ones found in the usual textbooks. After writing in-depth explanations themselves, the mentors will be primed to follow and understand the content and flow of the solutions written by the younger submitters. The ideal mentors will be those who have completed at least one content course in mathematics and have begun some of their pedagogy coursework as well.

In my own prerequisite content course, students solve five of the dreaded "challenge problems" for each unit of study. These problems are non-routine, though related to the mathematics content at hand. Students are instructed to write a solution and provide an explanation that would make sense to one of their friends who is not enrolled in our class. Math Forum has some solution writing guidelines on the web site that will enhance the college students' own written work in preparation for mentoring younger problem solvers.

http://mathforum.com/elempow/writing.html

In general, the Math Forum website is a valuable resource for engaging those students in problem solving. Many archived problems are available on-line; these could be used as homework or class activities prior to the mentoring experience. Or, you as the instructor might require your students to submit solutions on-line just as the K-12 problem solvers do.

Ideally, the mathematical preparation of the mentors should allow them to be several levels beyond the one at which they will engage in writing replies. However, that "ideal" does not preclude mentor participation by introductory students.

The next important area to consider is the nature of the mentors' attitudes toward this teaching/learning activity and its potential as a personal growth experience. Knowing what to expect is important in aligning the activities for college students with the goals set forth by Math Forum for them as mentors and for the young problem solvers.

Exactly what will my college students learn from the mentoring experience?

As mentors, your students will easily understand how this experience will prepare them for the days when they will be the responsible teacher. They will also learn about multiple solution methods and the young minds that produced them. They will be surprised by submitters on both ends of the learning spectrum. They will encounter those whose writing skills make any conclusive interpretation difficult and others whose mastery of the vocabulary of mathematics and attention to detail will be astounding. The international nature of the experience will also allow the mentors to view our own educational system through a different lens.

Do my students have to be mathematical stars to participate in mentoring?

My own students participated in the mentoring experience as part of a course on the teaching of problem solving. Prior to that course nearly all had completed 6 credit hours of mathematics content in a preparatory course designed exclusively for pre-service elementary teachers. All mentors should possess a strong interest in mentoring the submitters. Any personal enthusiasm for problem solving is a further asset. Some mentors will be confident of their own mathematics ability, but many will not. One important benefit of the mentoring experience is that it will foster greater confidence on the part of mentors from both ends of the confidence spectrum.

Generally, my student mentors responded well to the experience. They found the contact with the submitters exciting and seemed to take a personal interest in following through the threads of individual K-12 students.

What does a well-written mentor reply look like?

This is a important topic for discussion with your students after they have studied the writing hints given at the Math Forum web site and have engaged in the prescribed on-line training activities. Your class discussion might then address such topics as: a common format, their questioning strategies, the nature of praise or constructive criticism (including the tone of the reply), and finally the "nit-picking details" such as grammar, spelling and punctuation. Deciding together with your students what constitutes a well-written response will be a time-saver for you as you seek to evaluate their written work as mentors.

How should praise and criticism be handled by the mentors?

The new scoring rubric has addressed well the nature of an excellent solution by a submitter. Your college student mentors need to understand that praise needs to be earned in order to be authentic. No one is served well if praise is handed out easily for sub-standard quality work. Phrases like "great job" should mean that someone has indeed done so. That eliminates the danger of problem solvers not reading further into the mentor's reply. The college students can be enthusiastic in other ways that are genuine. "Thanks for working on this problem" is a way to encourage students who may not have achieved a correct solution or clear explanation. Criticism needs to be somewhat specific in order to help the problem solver, with the expectation that the problem solver is capable of fixing the situation. "You started this problem in a way that makes sense to me, but I don't understand why you divided by 2 in the third step. Perhaps you could explain that in a revision." Sometimes the problem solver's work will be correct and will require a more precise explanation. At other times the math may be wrong as well. The rubric scoring should help clarify the difference for the submitters.

How should my students be evaluated for their participation as mentors?

Establishing an evaluation rubric with your college students is a natural extension to the discussion of a quality mentoring reply. Future teachers will benefit from brainstorming together those aspects of their writing experience that are measurable and then moving on to coalesce those ideas into a checklist or point system. A self-evaluation of their replies should be required before submitting the replies for approval. I strongly believe that the college students need to understand their personal accountability to the submitters to go beyond generalized statements by giving detailed and attentive mentor reply.

Is there a way for students to practice ahead of time?

In order to prepare my students for mentoring, we "rehearsed" as follows. Our selected problems were shared with some teachers in nearby schools. I promised individual responses to any youngsters who worked on them, and the public school teachers passed their students' work back to me. My college students worked in pairs to write responses to the youngsters, taking into account the guidelines we had discussed for good responses. This experience was a reality check for the college students; they were unprepared for the skimpy and incomplete nature of some solutions. They also got a taste for what it was like to interpret such incomplete thinking in order to write appropriate mentor replies.

Another preparation resource exists on the Math Forum site. A number of former problems, with sample solutions, are available. These can be cut and pasted into a practice activity for class. Or, perhaps doing that activity via an email distribution allows you to check the nature of their mentor response time and the technology communication channels.

In order to effectively use the new rubric established by Math Forum, your students will also be expected to complete the on-line training activities prior to mentoring. However, this training will be most beneficial to them if they have previously done adequate problem solving and some mentoring rehearsal activities.

Lastly, consider the learning environment. With an on-line experience, "environment" has a new meaning, since it is no longer physical. The folks at Drexel and Math Forum have worked out the programming details. They have worked hard to make the system friendly to all those involved, with good bookkeeping access for you as the professor. Thinking through the following questions will be helpful to you and your college student mentors.

How should I prepare myself for the process of directing my students as mentors and for approving their replies?

Serving as a mentor yourself will provide you with experience that you are unable to get in any other way. You should engage in the on-line training module provided my Math Forum. You will thus learn to navigate the site and will familiarize yourself with the new scoring rubric. The different paths that you travel throughout your own training and mentoring will allow you to more readily trouble-shoot when your college student begin their on-line experience.

What will be my role as the instructor?

Your responsibility will include two important areas: preparation and monitoring. As you consider preparation, that necessarily refers to the mathematical problem solving that students have previously completed and the preparation for mentoring. To help us select a problem for submission, I did the following: First, a set of problems was assigned as homework. During class the next day, the students presented different solution methods on the board for the entire group. We then discussed the pros and cons of the individual problems, deciding which ones we wished to submit for approval by the folks at Math Forum. (They had also been given a copy of our problem set. Following discussions internally among the Math Forum staff, our problem options were jointly agreed upon for problem posing.)

Who will assign the mentoring as the problems come in?

I assigned solutions for myself, but the task could be handled by a student grader or secretarial assistant. My system was simply to "deal them out" to the next person on the list. Another possible method might be to have students specify ahead of time which days would allow them more timely access. Solutions could then be assigned in small chunks to the small group of students who is responsible for each day. Students may prefer to do five in succession, rather than to write replies spread out over several days.

Having said that, I want to emphasize that the mentors must understand the importance of checking daily in at the mentoring site. They will receive three different types of related assignments: new submissions that require them to write a personal mentor reply, a directive from you to "please edit the reply before sending" for a previously written reply, or a revision from a student previously assigned to their charge.

Whoever handles the assignment should be committed to checking the site several times daily, since responses come in at all hours from around the world. Faster assignment of mentoring will mean shorter turn-around time for the problem solvers.

What is a reasonable turn-around time expectation for my students?

This is an important question to answer together with your students. Your system of solution assignment should include a time a commitment from them. In order to help them focus during the mentoring, it is best to omit other significant written work for that two-week period. Students should know that prompt replies are expected and will affect their grade for this portion of your course. I'd recommend requiring that mentor replies be written within 24-36 hours from the time the problem submission is assigned to the individual college student mentor. This will provide you (or some other trained approver) with a reasonable window to review their replies and to allow for a "timely" turn-around to the original submitter. All submitters will receive an automated confirmation email within 48 hours. (Some are almost immediate; other may take longer. The variation depends on browser and email capabilities of individual computers.)

What is the time commitment for me and for my students?

This question is difficult to answer. Some problems generate more solutions that others, and you may want to plan some options. One guideline set forth by Math Forum is that individual mentors should follow ten "threads" through the reply-approval-send-revision cycle. If more submissions arise, you should discuss ahead of time with the staff how the excess submissions will be handled. Your students need to understand the inherent unpredictability of this mentoring activity, since a more difficult problem may generate more incorrect solutions and thus more revision. You can best serve your own college students and the young problem solvers by focusing on the mentoring activity for two weeks and minimizing other course homework. Scheduling a class session to meet in a computer lab about three days after your problem is posted might facilitate the smooth running of the mentoring process for both you and your student.

You should be cognizant of the time commitment for you to serve as the "approver" of the replies written by your mentors. Hopefully, you can arrange your schedule to permit you to check the site several times daily, approving replies and assigning new submissions as they show up in your "in box." You are truly the cog that keeps the machine in motion! Plan your personal schedule with that in mind.

How did previous mentors respond to the mentoring activity?

The following comments are taken directly from the evaluation comments written by my own college students.

The POW mentoring was worthwhile because it gave me first-hand experience with correcting math problems. It made me think about responding from a teacher's perspective.

It gave me a glimpse into how kids approach problem solving and gave me a chance to try to guide them. The atmosphere was also neat…with a chance to just interact with the kids.

I liked seeing the thought processes of some of the kids.

I would recommend that lots of other colleges adopt this plan of evaluating problems.

This experience made me more aware of different thinking styles for students and also how they put their thoughts on paper.

This was our first taste at grading and responding, being both positive and reinforcing to actual students. I found that very worthwhile.

It has given me practice responding to students and giving them good feedback.

In conclusion, I hope that you can get excited about sharing this mentoring experience with your college students. In these pages, I have tried to lay-out a strategy that will allow you to go into the process with a somewhat detailed plan. The nature of any interaction with real live students necessitates some flexibility. Knowing what worked for me will allow you to be more realistic in adopting this assignment into your own course for pre-service teachers. I wish you the best of luck with this adventure!

How do I get started? Whom should I call?

Kristina Lasher at Math Forum is available to discuss possibilities with you. Email her at:
kristina@mathforum.org