Introduction To Research In the Classroom
Answers To Frequently Asked Questions
Ken Levasseur, UMass Lowell
This is an abridged version of a longer FAQ list compiled by Joshua Abrams for Making Mathematics, an NFS - funded project based at Center for Mathematics Eductaion of Education Development Center (http://www2.edc.org/cme/).
Mathematics research is the long-term, open-ended exploration of a set of related mathematics questions whose answers connect to and build upon each other. Problems are open-ended because students continually come up with new questions to ask based on their observations. Additional characteristics of student research include:
Mathematics research influences student learning in a number of ways:
This question is usually more bluntly framed as "Can kids really do this?!" The experience of teachers in all types of school settings is that all children can successfully engage in mathematics research. In Making Mathematics, a recently completed project based at Education Development Center, teachers undertook research with urban, rural, and suburban students from grades 4 through 12. They guided at-risk, honors, and English as a Second Language (ESL) classes through projects lasting from a few weeks up to a year. Students in math clubs, individual students, and home-schooled students carried out successful investigations. One teacher first introduced research to her honors seventh graders. Once she was confident in her own experience, she tried the same project with two low-tracked eighth-grade sections. The quality of the questions, experimenting, reasoning, and writing was excellent in all three sections and indistinguishable between the honors and non-honors students. Research drew upon a richer array of student abilities than were assessed for tracking purposes.
Research can thrive in a heterogeneous class of students if you pick a project that does not require a lot of background to get started but which also inspires sophisticated questions. Students will pose problems at a level that is both challenging and appropriate for them.
Making Mathematics teachers have been most comfortable trying research for the first time with one of their "stronger than average" sections. Some teachers have begun work with one or more interested students as part of a mathematics club or independent seminar. The purpose of these first excursions has been for the students to become familiar with the research process and for the teacher to see how students respond to lengthy, open-ended problem-solving.
You should commit at least three consecutive class periods at the start of a first investigation in order to maintain the momentum of the experience. You want students to appreciate that the questions are not typical quick exercises, so it is important that they get to wade into the work. Interruptions also make it harder for them to maintain a line of thinking. After the initial burst, you can sustain a project through weekly discussions of work done at home. If a problem is working well, do not be afraid to let kids pursue it for a long period of time.
Many teachers independently introduce research into a class. Your work will have greater impact on students if they encounter research in all of their mathematics classes. Both for that reason and in order to feel less isolated as you experiment, it is helpful to recruit one or more colleagues to try out research along with you. Share ideas and observations and even visit each others classes on days when the students are doing research. Talk with your department head or supervisor to garner support for your efforts.
Mathematicians in the Focus on Mathematics program would all be eager to serve as a mentor for you and your students. The rest of this paragraph describes how you might search for a mentor if independently of FOM. If you want an advisor for yourself or an outside audience for the work that your students do, you can contact the mathematics or mathematics education department at a local college and ask if any of the professors would be willing to serve as a mentor (either via email, phone, or in person) for you and your class. We have also found good mentors contacting corporations that employ scientists and mathematicians. Your mentor may just communicate with you or she may be willing to read updates or reports from the students and provide responses. You should make these exchanges via your email accountparental consent is required by law for direct internet communication. Be sure to let any prospective mentor know what your goals and expectations are for the students and for their involvement.
Mentors can help in a number of ways. They can:
The teaching notes accompanying the Making Mathematics projects (http://www2.edc.org/makingmath/) can serve as models that you can adapt to other projects. As noted earlier, it is best if you can introduce research with a burst that permits a coherent presentation of the research process before separating discussions with several days of non-research studies.
Once research is underway, each student or group of students may work on different, but related, questions. During whole-class discussion, classmates should describe the different problems that they are exploring. Students should report back on their progress (new questions, conjectures, proofs, etc.) periodically.
At the end of a class session devoted to research, each group should give themselves a homework assignment in a logbooks. You can check these recorded tasks to make sure that the assignments were meaningful and check the subsequent entry in the logbook to make sure that the student made reasonable progress with the tasks. Typical homework challenges include:
Students can think about where they are in the research process (see below for one model for the process) in order to decide what step to attempt next. Their work should have some narrative explanations ("I did this because "). Students can work on their homework for a few days, but groups will also need regular class time to catch up on each others thinking, to work together, and to then coordinate next steps before their next stretch of independent work.
Although some projects, such as the one in Making Mathematics include teaching notes of some kind that suggest what to do on the first day, the second day, and so forth, you will need to pace the phases of a particular investigation according to the length of your class periods and the timing of a given classs particular questions and discoveries. Here are some other decisions that you should be alert to as work proceeds:
As a class works thorough its early research experiences, be sure to document for them as much of their work as possible. Posters listing the students conjectures, questions, and theorems help students grasp the cyclical nature of the research process. They see how their different questions connect and build upon each other and learn which research methods are most helpful at which stages of an investigation. After these beginning projects, students are ready to work more independently and should be encouraged to pose their own questions for research.
A project can end when a student or group has resolved some central question. Often, there are many questions and, after good progress with some of them, students enthusiasm for the others may wane. You may have established certain goals for students: to create a proof, to generate a few clear conjectures, to pose a new problem and make progress with it. Each of these possibilities is a reasonable time for work on a project to end. Students can come to a satisfying sense of closure even with a project that leaves many unanswered questions. That feeling can be enhanced if they write a final report that summarizes their main questions and work and that concludes with a list of possible extensions worth exploring. The FOM mathematician can help you with ideas about formal write-ups for students who have engaged research project.
Ultimately, research is no more demanding on your time than teaching that is more traditional. In some cases, it shifts the balance so that you spend less time preparing lessons and more time responding to student work. If you have not taught research before, there will be an initial need to think through the different issues that will arise in class. This work will prepare you to take advantage of any "teachable moments" (student comments that can lead the class to new understandings). The Making Mathematics teacher handbook is a valuable resource as you develop experience doing research with students.
One strategy for managing the demands of teaching research is to keep good notes on your observations during class. Thorough ongoing documentation will facilitate the comments that you need to make when you collect work because you will have a good sense of the entire research process that an individual or group has gone through. The more often you can read and respond to students entries in a their logbooks, the better, but you do not have to collect everyones work all at once. You can sample a few each night. Lastly, having each group submit a single final report reduces the number of papers that you need to study to a manageable number.
The above exchange between a Making Mathematics teacher and her mentor is typical of the most common and emotional question with which teachers interested in research have grappled. Many have expressed stress at feeling trapped by competing demands. In some cases, the answer is simple: if there is a major state test next week and you need to cover five topics, it is definitely a bad time to start research. But, if you are months away and you consider how often students forget what they have studied, now is a good time to introduce your students to mathematics investigations.
The content versus research question reflects a false dichotomy. We know how fruitless it is to teach disconnected topics. If you do not use knowledge in active ways that allow you to make meaning of what you have learned, you do not retain that learning. Why do students seem to forget so much of what they study? Sometimes, they still have the skills but are only able to apply them when prompted (e.g., "I am doing a chapter four problem" or "I was told to use triangle trigonometry techniques"). Sometimes, the learning experience was not memorable (consider what you have remembered and forgotten from high school and try to identify why). The more research work becomes a strand throughout a course and a schools curriculum, the better the interconnections between, and mastery of, technical content will be.
The NCTM Standards include many important goals (e.g., being able to conjecture, show persistence in problem solving, develop mathematical models, etc.) that we are supposed to "cover" that do not fit well in the framework of timed tests.
So, how do we combine research and technical content goals and what are some of the challenges that we face in our efforts? We can choose a research problem that will reinforce technical skills that a class has already studied. Alternatively, we can pick a problem that will introduce our students to and help them develop an understanding of a new topic. For example, we could use the Game of Set research project in place of or after a textbook introduction on combinatorics.
One problem that arises when using a research experience as a way to develop or reinforce a particular technical skill is that students questions and methods may not head in the direction that you expected. If you tell students to use a particular technique, then you short-circuit the research process. You are also risking turning the effort into a planned discovery activity, which usually lacks the motivational and intellectual power of true research.
You can address this problem in a few ways. A careful choice of project or framing of the question can often make certain skills inevitable. For example, a high school class proving theorems about Pythagorean Triples would be hard pressed to avoid using algebraic expressions or thinking about factors. You can also add your own questions to the classs list. This makes you a participant in the process and assures that the class will spend some time on the issues that you want considered. Alternatively, you can let the students work take them where it will knowing that some other important area of mathematics is being developed or reinforced that you will not have to spend as much time on in the future. Then, after the research is over, you can return to the topic that you originally had in mind.
When students do get to follow their own intellectual muse, they are more likely to experience a wide range of mathematics topics. For example, in a class of fifth graders working on the Connect the Dots project, one student asked what would happen if each jump was chosen randomly. The shapes were no longer as attractive, but the question of whether they would ever close led to the idea of expected value. An independent research project on randomness in DNA led a student to study matrices and Markov processes. Students will teach themselves a chapter of content from a textbook if they think it will help them on a task about which they care.