Seeing and Remembering Mathematical Connections: Growth and Development Of Mathematical Knowledge

 

Mercedes McGowen, William Rainey Harper College

 

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.