My first year of math teaching was in 1995 at an innovative private high school in San Francisco named Lick-Wilmerding. Students worked in small groups to explore algebra and geometry, reporting their findings in documents written as if they were lab reports. Students would receive direct instruction when necessary, but frequently, questions were explored and answered by members of the groups themselves. I adapted hesitantly to this new structure, as I myself had been taught by teachers and professors using lecture combined with extensive homework practice. Over my subsequent years of teaching in the states I explored a range of methods tailored to the various cultures of the schools at which I taught. This included straight lecture and homework, a mixture of lecture and independent class work, and collaborative group work supported by a textbook. Over the last 8 years of teaching at La Jolla Country Day in San Diego and two international schools in Spain, the American School of Valencia and the American School of Madrid, I’ve developed Courage To Core which allows my ninth and tenth grade math students to explore mathematical concepts as if exploring scientific concepts in laboratories.
As educators we hope to develop students who are inquiring, drawing upon their own innate curiosity to explore the world around them and develop and delight in an understanding which persists beyond high school. We aspire to help them become communicators, who work readily and effectively with others to enrich their own understanding and are capable of verbalizing and writing about their ideas. We hope they become deep thinkers who can apply technical skills and conceptual understanding to complex new problems. We strive to help them take intellectual risks, to share novel ideas and pursue creative lines of thinking, and to respond to failure with resilience. My approach to helping students become resilient, self-directed, collaborative problem solvers is to give them missions which require them to work together. In the missions they investigate mathematical ideas in real-world contexts, recognize patterns, share conclusions and practice applying learned skills. As a teacher I work to facilitate, advise, challenge, motivate and instruct, depending on the needs of the group.
The Math Intuition
Research shows that not only do we have an innate sense of mathematical relationships, but that sense is rather complex. Research suggests that a child’s numeracy is likely logarithmic rather than linear, which means they compare the relative numbers of things by thinking exponentially. More broadly, a young child recognizes that a toy car is the “same thing” as a real car, and his ability to model things demonstrates more than just a sense of proportionality, but also a capacity to represent the real world symbolically. A graph is precisely such a symbolic representation if it is a graph of profits increasing over time, or the curvature of a rocket’s path, or the pace of a heart beat as a runner increases her speed. It is incumbent upon us to harness the natural capacity of students to understand the world through intuitive numerical relationships and models. For this reason CTC strives to connect numbers and functions to real world scenarios.
Collaborative Investigations with CTC
With Courage To Core, students work in small groups of three or four to explore mathematical concepts and procedures. Missions guide students with leading questions and a logical flow which directs their work to a learning objective. At the same time the investigations are replete with open-ended questions which require students to discuss and decide how best to represent the given situation. They work with each other at a pace determined by their group. Sometimes they will solicit help from another group, or from the teacher. The teacher walks around the class observing and providing hints and guiding questions as needed. The teacher's role is that of a consultant, a counselor, a tutor, a manager, a resource and a lecturer. The teacher is looking for determined collaboration and creative problem solving, and expecting students to make mistakes and then correct them. The student's ability to persist is more important on a daily basis than quick arrival at a correct solution. Correct solutions can be arrived at by many methods, and that they will arrive eventually through effort.
The Recommended Structure of the Class for CTC
Students work in small groups for about two weeks before the group members are shuffled. Every couple weeks, they complete a set number of missions and correct any errors in their with the answers. Students take frequent group tests in advance of individual tests. For group assessments, all students must record the same answers, but ultimately one student’s assessment is graded at random and everyone in the group receives that grade. Group tests allow students to grow comfortable with formative assessment and get the most out of it. Although assessment is often associated with high-stakes multiple choice testing, group assessments provide a great alternative. Research shows the value of frequent assessment in learning, which is consistent with the idea that students need opportunities to demonstrate their understanding verbally and in writing.
CTC is Fun!
Students often enjoy working in groups. We want to see students connecting socially and intellectually with each other in class. Research also shows that social skills developed during school are among the most important factors in future success, and CTC provides students an opportunity to be social and productive members of the school community. With CTC, students are given a space to engage their curiosity, work with others, make mistakes and correct them, and develop a deeper connection to mathematical ideas through real world scenarios. It's fun!