Mathematics Conglomerate is Nimble, Responsive, Maintains a "Just in Time" Inventory

Over the last 10 years, mathematics faculty at Milton have collaborated on an in-house suite of “products” that take “what’s best for the consumer” seriously. Focusing on the math experiences of their consumers—Upper and Middle School students—mathematics department faculty teach, in large measure, with materials they have developed themselves, in lieu of standard texts.

The department’s years long commitment to shaping their own teaching materials reflects their love of math and the fun they have with mathematical ideas, as well as their genuine understanding of adolescents, those who are mathematically gifted as well as those who may not be enthusiastic about math (initially). Teaching this way is different, challenging, and ultimately much more rewarding: it requires constant dialogue among colleagues and unremitting attention to what happens among students in each class, each day. The faculty’s decision to compose and collect course materials grew from practical matters, and ultimately harnessed faculty creativity and teaching skill. Each year, math faculty members found themselves conferring and ordering new texts for each course, only to be disappointed once they began using the books. “We were supplementing so much,” John Banderob comments, “that the cost of the books seemed wasteful. We were looking to strike a better balance between providing skills-based problems, and asking students to use skills in new applications.”

The introduction into the classroom of the graphing calculator (in the early 1990s) changed the teaching of math so significantly that this, too, pushed faculty toward the decision. “With the calculator you are able to treat ideas and concepts inductively,” explains Keith Hilles-Pilant. You can do a hundred experiments quickly: setting a hypothesis and proving a hypothesis. The advent of new software in geometry, Geometric Supposer and its sequel, Geometer’s Sketchpad, also shifted the classroom dynamic toward discovery. “Students ‘uncover’ a theorem through manipulating a figure,” Hal Pratt explains, “rather than doing exercises that apply a textbook theorem to a static figure on a page.” Finally, the department was moving toward “modeling” within the curriculum: “We wanted a manipulable application for every concept we introduced,” John recalls.

“Using the materials we develop, we’re able to determine how we spend time in each course, and how we approach the material,” says Jackie Bonenfant, department chair. “We spend less time on the repetitive practice of skills, in the abstract, and more on presenting a stream of situations, asking students to determine what they need to know to solve the problem. We help them develop mathematical ideas and skills by working on them in a context—a more intriguing, less routine treatment of math for students.”

Members of the department agree that this “discovery and extension” method of studying math is much closer to what mathematicians do in a research environment. Faculty ask students to understand a concept and then see where else it may apply. “In pre-calculus, for example, together we take a look at a special situation, establish a set of criteria, learn a lot, and then zoom out to test where else those criteria might apply,” says Keith. “They might apply to circular motion, for instance, or a field of objects that work in a similar way.”

Writing your own teaching materials takes time and work, and it fosters a collegial environment that members of the department who have come from other schools experience as rare and intellectually invigorating. “You understand,” says Jackie, “that to do the best work with students, you need to trust and depend upon your department colleagues.”

“As a department, and as a group of individuals, we have had to think and talk about what we are teaching, why we are teaching it, and how best to teach it; it’s an essential and ongoing conversation,” says Terri HerrNeckar. All those who teach sections of a given course meet once each week; teachers of several courses have many meetings. They discuss how classes have gone and roadblocks that have appeared; they agree upon common homework assignments and who will write an upcoming quiz. The discussions include: “What way would you use to solve this problem?” or “I want to introduce this concept. Do you have an effective problem to do that?”

The outcome of teacher collaboration and attention to the craft of teaching is a curriculum that is responsive, efficient, customized, open-ended. “I teach two classes that each have a single section of students,” Erica Banderob says. “I write something up after each class. It’s not the same as last year; it fits exactly. When I see a need, I respond with the right thing, tomorrow!” Rather than following the preordained sequence in a textbook, “having a data base of our own materials gives us the confidence to change the flow, based on the students,” Terri notes.

When he arrived at Milton almost 20 years ago, Keith Hilles-Pilant designed Math 7 for students who have already taken at least two years of advanced placement-level calculus. “The content is completely different every year,” Keith says, “based on the interest of the students. I poll the incoming students and together we establish the syllabus for the coming year. This year, the students were interested in studying mathematical physics, so we are doing that. Along with the students, I generate the materials for the course. The result is a book, or perhaps a research journal is a more appropriate name for it. My hope is that the students will not only learn, but also discover that they can and want to do their own research.”

Not surprisingly, students respond well to math that is designed just for them. Each student keeps a notebook, compiling daily the teaching materials from faculty members along with their own work. “When you give students a new section of the ‘text’ each day, every day is important. The students know the pages are the work of the faculty in the department and they sort of share in our pride of ownership,” says Steve Feldman. “When a student asks a teacher why he or she is studying this or that, a faculty member can easily answer the question.”

Students are responsible for building their notebooks; therefore, they are in charge of the key resource for the course. (The faculty keep a close eye on the Class IV notebooks, as they emerge.) In this environment, note-taking increases in importance: students can’t easily look up in a book the information they miss, although they might find helpful information on the Internet. Students learn to use other resources, too, like each other.

Students are much more aware of their homework; they have a sense of ownership for it, and it becomes part of their overall math notebooks. “Having the same homework as others in different sections of the course is helpful, too,” Heather Sugrue notes. “They’re likely to work on it together, in the library, the student center, or the dorm. The conversations stretch across sections.”

“We have the technology now to produce much more professional-looking materials,” John says, and reworking the materials is constant and demands time. The transition from textbooks to faculty developed materials is the most difficult part; the up ramp is steep, and it takes two to three years to come up with a good critical mass of material. After that, teaching with these materials is easier, mainly because it’s not boring, it’s more effective and much more efficient.

New technological developments that are affecting the teaching environment and sparking creativity today are digital projectors and SMART Boards. “The next big question,” Jackie says, “is whether or not we’ll use a computer algebra system (CAS). Students can use calculators with a CAS on standardized tests and AP tests, so that’s the next issue we will need to address.”

Providing the best possible math experience for students has been the math department’s abiding goal, as they worked closely together to shape pertinent, challenging material year after year. “This style of teaching,” says Heather, “is what makes me want to stay at Milton. Not many schools are doing this: I feel I can contribute to students’ success on a daily basis.”

Cathleen Everett

 

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