Seeing and Remembering Mathematical Connections: Growth and Development Of Mathematical Knowledge
ABSTRACT: How can we explicitly emphasize connections, and assist students to construct relationships between parts of mathematics that they see as different? We addressed this issue with a class of preservice elementary teachers. The principal agent for change was the focus on building connections between different representations of a problem, coupled with reflective writing assignments. The isomorphic problems involving binary choice we chose were unlikely to allow a solution by a remembered formula. By the end of the semester students expressed a different view of mathematics. Their language grew more precise, what they focused attention on initially changed, and mathematical marks became truly symbolic. Algebra and algebraic language became a means of expressing relationships between objects and processes a way of talking about and describing experiences in pattern and arrangement. In this session, we describe the assignments and report the effects of students'oral and written reflections on their understanding of mathematics and on their views of mathematics.
In our experience students need to be brought
face-to-face with their unexamined belief that mathematics is only about
applying formulas to find answers. We not only wanted to change students'
attitudes and beliefs about mathematics, we wanted to know how to enhance
students' flexibility of thinking and their ability to see and value
connections. Reports of recent brain research support earlier recommendations
that an essential component of learning and the establishment of long-term
memories is reflecting on what was observed, what was done, and what processes
were utilized. Incorporating reflective assignments, both oral and written,
into our courses, together with a changed focus of instruction resulted in
changed attitudes and deeper understanding about mathematics. We asked ourselves whether the signs and
squiggles we refer to as "mathematical symbols" mean something to our
students in a deep, satisfying, human sense or do they merely indicate actions
to be carried out, as when we say "sit!" to a dog? What is the
alternative to training (we refrain from writing "educate") students
who fail to see mathematics as having meaning, but who know how to push syntax
around?
For us,
evidence of deeper understanding of mathematics means that students demonstrate
they are thinking more flexibly and systematically, are able to articulate what
they have done, justify why a method was used, generalize their results,
establish connections between problems in different contexts and between
various mathematics topics, and recognize the role of definitions and proofs.
They make sense of the mathematics they are learning and build
connections-having realized that in order to make connections, they "have
to have something to connect.
How can we
explicitly emphasize connections, and assist students to construct
relationships between parts of mathematics that they see as different? We
addressed this issue with a class of preservice elementary teachers at Harper
College, Illinois during the Fall 2000 semester. They began, as many
pre-service elementary teachers do, expecting to apply formulas and get correct
answers in order to be "mathematical." By the end of the 16 week
course they expressed a different view of mathematics: one that most students
characterized as more relational. The emphasis in the first four weeks of the
course was on making connections between different combinatorial problems and
on multiple ways of interpreting answers. All deal with different aspects
and representations of a systematic counting problem related to binary choices.
Students work on and discuss the problems in groups. After completing the
sequence of problems, students are encouraged to explain individual insights to
the entire class. We ask them to write reflectively after each of the combinatorial
problem sessions, and to explain connections they have made. Re-writes of
reflective assignments are encouraged. Multiple opportunities for making
connections with their earlier work are provided during the semester in
questions students haven't seen previously on group and individual exams. Over
the course of the semester, their language grows more precise, what they focus
attention on initially changes, and mathematical marks become truly symbolic.
Algebra and algebraic language become a means of expressing relationships
between objects and processes-a way of talking about and describing experiences
in pattern and arrangement.
The
principal agent for change was the focus on building connections between
different representations of a problem. Problems as simple as enumerating all
towers of height 4 and 5, built with one or two colors, are sufficient to set
this process in motion. In this session, we describe a sequence of
isomorphic problems involving binary choice. The problems we chose were unlikely
to allow a solution by a remembered formula. In this paper, we describe the
assignments and report the effects of the oral and written reflections on
students' understanding of mathematics in their views of mathematics.