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Honing Strong Habits of Mind

At Milton, students learn the concepts and habits of mind that are key to the mastery of mathematics: analytical thought, exploration, organization—quantitative and spacial perspective; understanding numbers, abstraction logic, patterns and proofs, structure, space and change. Ultimately, students learn to speak and write the language of mathematics.

Through increasingly challenging problems—geared toward helping a broad range of students learn and succeed—and through extensive use of instructional technology and mathematical software, students experiment with higher level mathematical exploration.

The program encourages collaboration; faculty know that communicating the process of mathematical problem-solving (with teachers, with classmates) further strengthens students’ understanding of the concepts and enhances students’ skills as mathematicians. Collaboration also opens doors to new ways of approaching a problem, and innovative paths toward solving it.

Faculty work to give students ownership of their learning, creating a foundational framework from which to launch creative application. From early courses in Geometry and Algebra II through advanced courses in Multivariable Calculus, Abstract Algebra and Group Theory, and Topology, connecting the concepts of applied mathematics to other disciplines, and to the world beyond the classroom, is a primary goal.

Mathematics: Teaching Standards and Goals

The Mathematics Department is committed to providing outstanding teaching but it acknowledges that excellence does not require conformity to a single model of teaching. We recognize that the on-going debate about the “student-centered” as opposed to the “teacher-centered” classroom is a fruitful one and that there are outstanding teachers on both sides of the issue. Whether the teacher’s role is to be a questioner and facilitator or to be the provider of clear explanations, our task is to actively engage our students in doing mathematics. The resourceful teacher, in fact, should be flexible in his/her methods; the ability to vary one’s teaching style to respond to student needs is essential. For example, while deductive methods require careful instruction, inductive thinking is best learned when the teacher resists the temptation to explain and instead encourages the student to experiment and invent. Both deductive and inductive reasoning are essential in mathematics, and we need to strike a balance between the two.

Because the learning process requires that students, every day, confront the limits of their knowledge and competence, it is essential that we provide them a safe and supportive environment. To that end, we should engage the students in class and we should get to know them individually, building upon their strengths and helping them to overcome their weaknesses. An excellent teacher should be able to meet the needs of students of differing abilities, providing different ways of explaining concepts or providing a variety of experiences so that all students can master the relevant skills. At the same time, we should instill tolerance in the classroom to allow for and to encourage risk-taking. In an effective classroom, the student’s self-confidence will grow as will his/her self-esteem.

Our students should come to appreciate that mathematics provides a rich store of concepts that help us to understand relationships within the world and that it also provides a vehicle for learning how to think quantitatively, spatially and logically. Our students should come to realize that while mathematical abstractions are rooted in the concrete, mathematical thinking is distinguishable from concrete thinking. This discipline, by its very nature, is often a difficult one, but we hope that our students, in meeting the challenges that they face, will see mathematics as fulfilling and satisfying.

Because mathematics is more than a store of concepts, however, because it is a way of seeing and thinking, we expect that students will not merely be responsible for the acquisition of a well-defined collection of techniques. We expect, as well, that students will learn to express their ideas clearly, both in writing and orally. We encourage our students not only to generate their own solutions (and a variety of solutions, if possible) but to generate their own problems as well. We expect that our students will engage in open-ended projects, perhaps even ones for which the teacher hasn’t an answer. Problem-solving should be a process in which the students are actively engaged. Mathematics should also be seen as a human endeavor which is rich in history and replete with interesting characters. Mathematics was traditionally allied with other disciplines and it continues to offer opportunities for such alliances, and we encourage initiatives in developing interdisciplinary courses. Perhaps the most significant developments in mathematics in recent years are the advances in computer and calculator technologies; these have altered our teaching of mathematics and we should continue to explore opportunities to incorporate these technologies in order to support and enhance our teaching. Consistent with these several principles, we expect that evaluations of our students will include more than in-class tests and quizzes.

In order for teachers to continue to be effective in the classroom, they must continue to grow professionally. Taking courses and attending workshops and conferences provide opportunities for such growth. Less formally, discussions about curriculum and teaching methods serve, as well, to promote professional development when all points of view are accorded respect and thoughtful consideration. And if teachers are doing interesting, enjoyable math problems beyond the classroom, joining in the process of discovery, this too enhances a teacher’s effectiveness.

The primary goal of the department is to cause our students to master mathematical concepts and skills but there are other important skills that we also strive to help our students to develop. As our students mature, they must learn to accept greater responsibility for their own learning and to take the initiative if they need help. Students should learn to use resources other than their teachers, such as other students and books. We hope that our students will become resourceful thinkers who are productive (rather than merely reproductive) problem-solvers. We want our students, at the end of their time here, to be able to deal with unfamiliar situations. At the same time, our students should come to realize that what they have learned is not all there is to learn about this discipline or any subject within it and that the learning process has only just begun.

From the Classroom

Math Exposition Night

One evening in the spring, math and computer science students share their culminating coursework, or independent projects, in dynamic and hands-on ways. This year, student math enthusiasts showcased projects ranging from a robotics obstacle course to a stock market game; taught about sequences and series through knitting; analyzed probabilities of poker; and demonstrated a variety of computer programs involving various mathematical concepts.

A big value in studying math is that a student can face a challenging problem and not know how to do it at first; but they can start with what they know, build from there, and work toward the solution. The point isn’t that every student will learn to love math and become a mathematician, but to show students that they can face a difficult problem and figure it out.

LeeAnn Brash

Math Department Chair