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Mathematics as the Science of Patterns (Hardcover)
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Mathematics as the Science of Patterns (Hardcover)
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The value of patterns to the teaching and learning of mathematics
is well understood, both in terms of research and application. When
we involve or appeal to pattern in teaching mathematics, it is
usually because we are trying to help students to extract greater
meaning, or enjoyment, or both, from the experience of learning
environments within which they are occupied, and perhaps also to
facilitate remembering. As a general skill it is thought that the
ability to discern a pattern is a precursor to the ability to
generalize and abstract, a skill essential in the early years of
learning and beyond. Research indicates that the larger problem in
teaching mathematics does not lie primarily with students; rather
it is with the teachers themselves. In order to make changes for
students there first needs to be a process of change for teachers.
Understanding the place of patterns in learning mathematics is a
predicate to understanding how to teach mathematics and how to use
pedagogical reasoning necessary in teaching mathematics.
Importantly, the lack of distinction created by the pedagogical use
of patterns is not immediately problematic to the student or the
teacher. The deep -seated cognitive patterns that both teachers and
students bring to the classroom require change. Chapter 1 opens the
book with a focus on mathematics as the science of patterns and the
importance of patterns in mathematical problem solving, providing
the reader with an introduction. The authors of Chapter 2 revisit
the work of Po lya and the development and implementation of
problem solving in mathematics. In Chapter 3, the authors present
an argument for core pedagogical content knowledge in mathematics
teacher preparation. The authors of Chapter 4 focus on preservice
teachers' patterns of conception as related to understanding number
and operation. In Chapter 5 the authors examine the role of visual
representation in exploring proportional reasoning, denoting the
importance of helping learners make their thinking visible. The
authors of Chapter 6 examine patterns and relationships, and the
importance of each in assisting students' learning and development
in mathematical understanding. The authors of Chapter 7 examine the
use of worked examples as a scalable practice, with emphasis on the
importance of worked examples in teaching fraction magnitude and
computation is discussed. In Chapter 8, the authors expand on the
zone of proximal development to investigate the potential of
Zankov's Lesson in terms of students analyzing numerical
equalities. The authors of Chapter 9 focus on high leverage
mathematical practices in elementary pre-service teacher
preparation, drawing into specific relief the APEX cycle to develop
deep thinking. In Chapter 10, the author focuses on number talks
and the engagement of students in mathematical reasoning, which
provides opportunities for students to be sensemakers of
mathematics. Chapter 11 presents an epilogue, focusing on the
importance of recognizing the special nature of mathematics
knowledge for teaching.
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