Roser Gine and her New Mission High School colleagues have experienced the challenges and rewards of designing the mathematics curriculum for their recently founded, small, student-centered school. Based on a 2003 Fall Forum workshop led by Gine and New Mission teacher Stephen Cirasuolo, this article documents New Mission’s process of creating a challenging, meaningful, standards-based mathematics curriculum, offering compelling advice, curriculum examples and resources to educators and other new or redesigning schools. As well, Gin?© details the essential questions, mathematical elements, and habits of mind that form the foundation for New Mission’s evolving mathematics curriculum framework.
The Evolution of the Mathematics Curriculum
One of the major tasks teachers take on at New Mission High School (NMHS) is creating the curriculum. When I first arrived at the school in 1998, the mathematics curriculum did not yet exist, and the four math teachers had complete freedom to establish what we thought was important for our students to learn. We wrote an essential question, and taught integrated mathematics through classroom projects and activities that would help students develop tools they needed to respond to that question.
At the time, we only had two different grades, so we made the material accessible at two levels. But several problems were evident. First, we had not created a four-year plan, and when our school changed to accommodate four levels, the mathematics curriculum needed adjustment. Second, we had not established the graduation requirements for our students; this was crucial, as we wanted to build the curriculum by planning backwards, with clear goals and expectations. Third, in order for a curriculum to stand on its own, it must address what students need to learn; teachers needed to be informed of the national and state standards, as well as what other established mathematics curricula and educational research considered important. Additionally, we had to address problems specific to NMHS. These included very fast teacher turnover (especially in our mathematics staff), our students’ varying levels of preparation, our interest in integrating different disciplines, and the challenge of balancing the depth of the mathematics curriculum with the breadth of the state mandated test.
Along with other teachers, I took on the challenge of creating a mathematics curriculum framework for our school. The framework was organized around the following questions: “What do we want our students to know and understand when they graduate from New Mission High School? Why?” After consulting other teachers and mathematics professors, as well as reviewing existing mathematics initiatives, we wrote four essential questions (they continue to evolve) that would guide the mathematics teaching and would be addressed through the four-year student experience at the school.
At each grade level, students would work towards answering these questions through the development of outlined quantitative skills and habits of mind, and through exposure to relevant topics. In order to support this hierarchy, projects (which we call vehicles) and activities were designed for use in the classrooms. As student portfolios evolved into living documents exhibiting students’ progress and learning, I wrote portfolio strands that would reflect what I believed were the essential elements of our curriculum: a mathematical model, a mathematical proof, and a problem-solving strand. At this point, the framework was in progress, but still needed feedback and input from the teachers that would bring it to life.
Challenges still remained, including making a non-text based mathematics framework accessible to new teachers, getting feedback from other teachers, and making appropriate revisions, ensuring compatibility with national standards, providing academic support and test preparation for the MCAS (the Massachusetts Comprehensive Assessment System, our state test), integrating with other curricula, and putting the framework to use. The NMHS Science Coordinator and I were assigned as leaders in a new initiative supported by the Center for Collaborative Education in Boston, the Systemic Initiative for Math and Science Education (SIMSE), that would provide us with time and resources to realize our integration goals and to support our new teachers in working with the established framework; a mathematics coach was assigned to our school for the SIMSE work. As the Mathematics Curriculum Coordinator, I worked with the other three math teachers to make necessary changes, review national standards, and facilitate implementation. The school also gathered several teachers who volunteered to prepare students for the MCAS during after school and Saturday sessions.
The Three Mathematical Elements
The three elements I presented to the staff as essential for our students to develop are mathematical modeling, mathematical proof, and problem-solving. I questioned extensively why some students who would not pursue a scientific or mathematical field in higher education, or who would only encounter statistics in college in connection to the social sciences, or who maybe would not attend college, might need to develop mathematical thinking skills and processes. Additionally, I really wondered whether mathematics would be interesting to someone if they were taught skills and topics leading to the development of calculus, but never actually took a calculus course to understand their secondary mathematics education in its entirety. These questions along with the evolution presented above led me to identify the three elements outlined (they overlap in many ways) and to support the integration of mathematics with other disciplines and within itself (distinct mathematical fields are naturally connected).
The most important quality of the mathematics curriculum at NMHS is that it provides opportunities for the development of the thought processes not only needed to understand more complex and pure mathematics concepts, but also useful in exploring and making discoveries about other disciplines or other human endeavors. Learning how to create and assess a mathematical model allows a student to test her/his ideas, to determine shortcomings or limitations that arise from simplification, and to extract useful results (by making predictions, extrapolating from the model) from that same simplification. Through mathematical manipulation of an original model, the student may find a unique solution to a problem; she/he learns that reorganization, a new perspective, or a small change in a variable could yield an unexpected outcome, or even a contradiction. The creation of a math model in fact reflects the habits of mind embraced by Coalition Schools; perspective, supposition, and connection, three habits outlined at New Mission, are practiced when students engage in this work.
In addition to the mathematical modeling strand, we recognized the importance of deductive reasoning skills, again within the study of pure mathematics, and in connection to everyday life. A mathematical proof depends on deduction; an initial conjecture is made and tested, and a formal logical procedure developed and applied. When a student engages in this process, she/he makes a claim from observed patterns (or more typically, is given an idea to prove), and organizes information to determine the truth of the claim. He/she then ‘digs up’ stored units of mathematical knowledge, piecing them together to build a valid argument. Logical symbols may even be accessed in order to render communication of subtle relationships more effective. Once a claim is proved, it too becomes accessible mathematical knowledge. Much as it happens during the design of a mathematical model, a student’s habits of mind develop through the construction of a proof; evidence is used at each juncture as one premise builds on the truth of the previous one.
The third element, problem-solving, complements the other two strands in providing opportunities for the development of essential habits of mind. When students solve problems, they look for patterns as they explore what already exists and imagine extensions or alternatives. Patterns are tested through particular cases, and generalizations are made; through this process, students learn to use pictures, representational diagrams, and mathematical language to describe a situation. Additionally, they make connections from one problem to another, and often extend their ideas to more complex problems that arise in new situations. Habits continue to develop while skills sharpen and ideas emerge.
We continue to work on balancing the breadth and depth of the mathematics program; the NMHS curriculum is a work in progress, realized in different ways within the four grade levels. In our attempt to decide what we believe is important for our students to learn, we have designed a framework that guides the work within our particular school setting. It remains imperative that we find ways to measure the work our students are producing against our intent. The analysis of student data, including a close examination of student portfolios, will continue to steer the refinement of the framework and to help us close the gap between what teachers envision for their students and what students are really learning and understanding. Thus, the framework serves as a guide to spearhead our work and to mindfully return to as we engage our students in the learning process.
Roser M. Gin?© is a teacher, advisor, and mathematics curriculum coordinator at New Mission High School in Roxbury, MA. She is also a Leadership in Urban Schools doctoral student at the University of Massachusetts, Boston. She would like to acknowledge the support of Valerie Vasti, New Mission High School’s Curriculum Coordinator. Gin?© can be reached at firstname.lastname@example.org