The essential act leading to a successful learning experience is the establishment, and hewing to, of coherent and important learning goals. While this statement may seem obvious, too many courses (e.g., United States History (1492 — Present) or Algebra II) are little more than a title and a grab bag of poorly connected topics. Figuring out what activities will aid students in achieving the sought after goals is an exciting and challenging task.

The habits of problem posing, creating representations, explaining connections, and testing and checking are central to the development of interesting new mathematics. Students need to see these habits as worthwhile activities. Mathematics investigations can involve multiple variables and methods of solution, appealing questions with no obvious path to understanding, and answers that vary according to the assumptions made. Encounters with such settings dispel students’ notion that the sole trademark of mathematics is the exactness and uniqueness of results. Rather, creativity and the recognition of underlying structure and abstraction become dominant features of the discipline. We must help students become comfortable with the roles that creativity, analysis, and clear communication play in active learning and discovery. Lastly, they must be curious and willing to take risks. Successful students in traditional math courses are rewarded for speed and technical accuracy. A different type of confidence is required when they begin posing problems with no immediately clear method of solution and no guarantee that a solution can be found.

Goals should be made explicit and shared with students at the start of an experience. The greater students’ initial and subsequent understanding of what they are doing and why it is worthwhile, the more readily they will connect each activity to that larger understanding and retain its lessons. A sample handout of goals for a mathematics research strand or course is in Appendix A. The point of such a handout is not to detail every learning goal (e.g., note the lack of specific proof techniques which might be studied), but to invite students into a reflection on how they have learned math in the past and what the expectations of a research experience might be. In general, goal setting is even more effective if the students themselves can generate the objectives and standards. This approach is possible when the students already have some prior experience with the task at hand (e.g., have engaged in open-ended mathematics explorations). The Building Collaborative Skills section provides an example of student generated standards.

Once the goals have been read and discussed in class, a homework assignment (see Figure 1 below) asks students to reflect on the goals and the challenges that the goals pose for them in the coming year. Because doing mathematics research is, for most students, a radical departure in both content and approach from typical math classes, it is crucial that they be aware of, and ultimately value, the changed expectations. A class should periodically discuss the meaning and purpose of the goals in order to develop this support. Some of the goals will make more sense as the activities provide students with a sense of context. For students who encounter research as a non-elective part of a standard mathematics course, approval of, or commitment to, the goals should be not be assumed and cannot be forced. It must be negotiated, encouraged, justified, and inspired.

The three quotes below (Figure 2) are excerpts from student responses from both mathematical modeling and pure mathematics research courses to the first option from the goals reflection assignment. These responses reveal several common reactions: fear, optimism, and skepticism. Many other students write about which goals they like because they already feel competent in those areas (e.g., being persistent or writing clearly). These essays are the start of a dialogue. Responses to this assignment should be positive and encouraging comments (samples in italics) which agree with, or at least acknowledge, the feelings expressed and observations made. A student citing a particular weakness (e.g., calculator use) can be offered help to reach these goals.

Few students choose the second essay option, which is a more difficult question. For either question, students sometimes give answers without providing much explanation. For example, favorite or feared goals may be noted without reason for the reaction. A student might suggest that they keep a diary in order to demonstrate their progress toward meeting the goals without detailing how the record would demonstrate improvement. As with all feedback, it is important to point out the need to elaborate on comments and to justify claims.

Several weeks into the course, the teacher can exchange written evaluations with each student on their progress. It is helpful to look back on their initial reflections and note how students have fared with the goals that particularly concerned them. Students often make the most progress in those areas that they were aware of enough to write about in the first place. If not, advice should be offered on how to start improving. Positive feedback on what progress has been made in each area should be detailed (see Class Time in the Assessment section).

An outline of the main goals for Mathematics Research Seminar Essential questions to explore:

• What is mathematics? What is a mathematical system?
• What are the processes of understanding and discovering mathematics?
• What does it mean for a mathematical statement to be true? What constitutes a proof of a claim? What is the role of proof in mathematics?

• What you learn this year will not benefit you unless you look to apply it on your own after this course. Wanting to continue studying and using mathematics throughout your life is more important than any given skill that you learn now.

• The richest mathematical experiences frequently evolve out of problems posed by the problem-solver. New mathematics is created through the modification of existent questions, statement of new definitions, or the identification of a new area of exploration.

• Actively seek to uncover symmetries, relationships, connections, and patterns in the settings that you explore.
• Attempt to generalize your observations into conjectures.

• The reading of primary source mathematics is neither a speedy nor a linear process. Strive to read both peer and professional mathematics writings carefully and patiently as you identify the assumptions, test the conditions, and verify the conclusions that you read.

• Develop logical arguments which prove or disprove the conjectures that you investigate.
• Make informed choices about which of your mathematical skills would be helpful in constructing a proof or whether new mathematics needs to be studied in order to support a claim.

• Write narrations of your explorations and problem-solving efforts.
• Abstract content does not require impenetrable prose. The motivation behind your work and your reasoning should be clearly crafted and presented. Technical language and symbols should be used only when they enhance the communication.

• Only accept conclusions if you can verify or estimate the validity of an answer.

• Take responsibility for guiding the activities of the class, for responding to each other’s ideas, and for persisting in the face of difficult challenges.

• Know how and when to use calculators and computer tools. Understand the limitations and power of each.

All of the above have these overarching objectives: That you broaden and refine your own aesthetic and intuition about pure mathematical questions. That you become mathematicians who create their own pathways for investigation.