The teaching of mathematics requires critically examining the ways in which students understand basic concepts and build upon these to formulate new ideas. The New Haven Public Schools’ Mathematics Department has initiated a collaborative endeavor called “vertical teaming” to bring together math teachers across grades 5-12 in clusters roughly consisting of pathway schools for students. Vertical TeamTM is a trademarked term for a concept developed by The College Board to bring together teachers from different grade levels in a discipline to work cooperatively in developing and implementing a vertically aligned program aimed at helping students acquire the academic skills necessary for success in the Advanced Placement Program.3 On the one hand, it may appear to be a clever marketing strategy to assure future participants in AP courses and (paying) customers for The College Board’s AP exams--and it is. On the other hand, who among us does not want to have our schools “build rigorous curricula; promote access to AP for all students; introduce skills, concepts, and assessment methods to prepare students for success when they take AP; and strengthen curriculum and increase academic challenge for all students”?4
Some New Haven teachers have begun talking to each other about their goals and frustrations in teaching math with the aim of developing a consistent set of expectations for student accomplishment at each grade level. Knowing what is expected in the grades-before and grades-following the one(s) you are teaching will hopefully lead to better opportunities for students to progress in mastery of skills and in complexity of problem solving. Numeracy tasks introduced in elementary classes need reinforcement right through grade 12. By working together, vertical teams hope to stop the blaming of prior teachers for failing to adequately teach fundamentals (i.e., fractions, decimals, percent, graphing, vocabulary, technology) and recognize our shared responsibility for helping all students attain their greatest success along a continuum of learning.
So, in a nutshell, the idea is to “improve academic performance for all students in earlier grades by introducing skills and concepts needed for success in AP and other challenging courses.” 5
Vertical teaming is a “win-win” situation. Isn’t this what all schools have been striving to do all along with curricular standards, curriculum outlines and grade level goals in each discipline? Yet, how often have teachers from all grade levels in a discipline been encouraged and enabled (given the time and yes, even the money) to meet together regularly over a long period of time to really look at and talk about what they and their students are doing? In our seminar, Professor Howe floated the idea that perhaps schools should be “teacher-centered” instead of “student-centered”. After all, who knows best what your next year’s incoming class can and cannot do in arithmetic, algebra and problem solving? Teacher-driven reform lifts up the hope of real staying power in education, because teachers know that reform takes constant tending and checking, adjusting and praising. This seminar has been one form of vertical teaming--we listened to each otherr’s struggles with our teaching at different levels, tried to solve problems together and began to form ideas of what, when and how to unfold mathematical understandings with our students.
A book that should be read in developing a critical foundation for the teaching of word problems is Magdalene Lampert’s
Teaching Problems and the Problems of Teaching
(Yale University Press, 2001). In this book-length reflection on the teaching of fifth grade math by a university professor, what stands out is her patient questioning and listening processes while working with students along with her deep reflections afterward in preparation for the next day’s class. Lampert had the extraordinary opportunity of doing her research on teaching and thinking by instructing an elementary class one hour daily for a year, along with her college teaching responsibilities. Her book is itself a strong argument for a “vertical” view of mathematical learning. She brings to her fifth grade class both an unclouded focus on the importance of the work each individual is doing and clear sense of how that work is connected to the lifelong development of math’s “powerful ideas”. Of course, as teachers in demanding urban schools, my seminar colleagues and I marveled at the depth of reflection and the sheer time Lampert could afford to try to grasp the intricacies of each fifth grader’s struggle toward understanding.