Softcover ISBN:  9781470469894 
Product Code:  CLRM/71 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9781470475215 
Product Code:  CLRM/71.E 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
Softcover ISBN:  9781470469894 
eBook: ISBN:  9781470475215 
Product Code:  CLRM/71.B 
List Price:  $130.00 $97.50 
MAA Member Price:  $97.50 $73.13 
AMS Member Price:  $97.50 $73.13 
Softcover ISBN:  9781470469894 
Product Code:  CLRM/71 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9781470475215 
Product Code:  CLRM/71.E 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
Softcover ISBN:  9781470469894 
eBook ISBN:  9781470475215 
Product Code:  CLRM/71.B 
List Price:  $130.00 $97.50 
MAA Member Price:  $97.50 $73.13 
AMS Member Price:  $97.50 $73.13 

Book DetailsClassroom Resource MaterialsVolume: 71; 2023; 441 ppMSC: Primary 01; 26; 30; 54
“It appears to me that if one wants to make progress in mathematics one should study the masters and not the pupils.”
—Niels Henrik Abel
Recent pedagogical research has supported Abel's claim of the effectiveness of reading the masters. Students exposed to historically based pedagogy see mathematics not as a monolithic assemblage of facts but as a collection of mental processes and an evolving cultural construct built to solve actual problems. Exposure to the immediacy of the original investigations can inspire an inquiry mindset in students and lead to an appreciation of mathematics as a living intellectual activity.
TRIUMPHS (TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources) is an NSFfunded initiative to design materials that effectively harness the power of reading primary historical documents in undergraduate mathematics instruction. Teaching and Learning with Primary Source Projects is a collection of 24 classroom modules (PSPs) produced by TRIUMPHS that incorporate the reading of primary source excerpts to teach core mathematical topics. The selected excerpts are intertwined with thoughtfully designed student tasks that prompt students to actively engage with and explore the source material. Rigorously classroom tested and scrupulously edited to comply with the standards developed by the TRIUMPHS project, each of the PSPs in this volume can be inserted directly into a course in real analysis, complex variables, or topology and used to replace a standard textbook treatment of core course content. The volume also contains a comprehensive historical overview of the sociocultural and mathematical contexts within which the three subjects developed, along with extensive implementation guidance. Students and faculty alike are afforded a deeper classroom experience as they heed Abel's advice by studying today's mathematics through the words of the masters who brought that mathematics to life.
ReadershipInstructors interested in incorporating deep historical context into their teaching as a means to strengthen students' engagement with the mathematical content and methodology learned in courses on real analysis, topology, and complex variables.

Table of Contents

Introduction

Teaching and Learning with Primary Source Projects

PSP Summaries: The Collection at a Glance

Historical Overview

Real Analysis

Why Be So Critical? NineteenthCentury Mathematics and the Origins of Analysis

Investigations into Bolzano’s Bounded Set Theorem

Stitching Dedekind Cuts to Construct the Real Numbers

Investigations into d’Alembert’s Definition of Limit

Bolzano on Continuity and the Intermediate Value Theorem

Understanding Compactness: Early Work, Uniform Continuity to the Heine–Borel Theorem

An Introduction to a Rigorous Definition of Derivative

Rigorous Debates over Debatable Rigor: Monster Functions in Introductory Analysis

The Mean Value Theorem

Euler’s Rediscovery of $e$

Abel and Cauchy on a Rigorous Approach to Infinite Series

The Definite Integrals of Cauchy and Riemann

Henri Lebesgue and the Development of the Integral Concept

Topology

The Cantor Set before Cantor

Topology from Analysis

Nearness without Distance

Connectedness: Its Evolution and Applications

Connecting Connectedness

From Sets to Metric Spaces to Topological Spaces

The Closure Operation as the Foundation of Topology

A Compact Introduction to a Generalized Extreme Value Theorem

Complex Variables

The Logarithm of $1$

Riemann’s Development of the Cauchy–Riemann Equations

Gauss and Cauchy on Complex Integration


Additional Material

Reviews

Primary sources provide motivation in the words of the original discoverers of new mathematics, draw attention to subtleties, encourage reflection on today's paradigms, and enhance students' ability to participate equally, regardless of their background.
These beautifully written primary source projects that adopt an “inquiry” approach are rich in features lacking in modern textbooks. Prompted by the study of historical sources, students will grapple with uncertainties, ask questions, interpret, conjecture, and compare multiple perspectives, resulting in a unique and vivid guided learning experience.
David Pengelley, Oregon State University


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“It appears to me that if one wants to make progress in mathematics one should study the masters and not the pupils.”
—Niels Henrik Abel
Recent pedagogical research has supported Abel's claim of the effectiveness of reading the masters. Students exposed to historically based pedagogy see mathematics not as a monolithic assemblage of facts but as a collection of mental processes and an evolving cultural construct built to solve actual problems. Exposure to the immediacy of the original investigations can inspire an inquiry mindset in students and lead to an appreciation of mathematics as a living intellectual activity.
TRIUMPHS (TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources) is an NSFfunded initiative to design materials that effectively harness the power of reading primary historical documents in undergraduate mathematics instruction. Teaching and Learning with Primary Source Projects is a collection of 24 classroom modules (PSPs) produced by TRIUMPHS that incorporate the reading of primary source excerpts to teach core mathematical topics. The selected excerpts are intertwined with thoughtfully designed student tasks that prompt students to actively engage with and explore the source material. Rigorously classroom tested and scrupulously edited to comply with the standards developed by the TRIUMPHS project, each of the PSPs in this volume can be inserted directly into a course in real analysis, complex variables, or topology and used to replace a standard textbook treatment of core course content. The volume also contains a comprehensive historical overview of the sociocultural and mathematical contexts within which the three subjects developed, along with extensive implementation guidance. Students and faculty alike are afforded a deeper classroom experience as they heed Abel's advice by studying today's mathematics through the words of the masters who brought that mathematics to life.
Instructors interested in incorporating deep historical context into their teaching as a means to strengthen students' engagement with the mathematical content and methodology learned in courses on real analysis, topology, and complex variables.

Introduction

Teaching and Learning with Primary Source Projects

PSP Summaries: The Collection at a Glance

Historical Overview

Real Analysis

Why Be So Critical? NineteenthCentury Mathematics and the Origins of Analysis

Investigations into Bolzano’s Bounded Set Theorem

Stitching Dedekind Cuts to Construct the Real Numbers

Investigations into d’Alembert’s Definition of Limit

Bolzano on Continuity and the Intermediate Value Theorem

Understanding Compactness: Early Work, Uniform Continuity to the Heine–Borel Theorem

An Introduction to a Rigorous Definition of Derivative

Rigorous Debates over Debatable Rigor: Monster Functions in Introductory Analysis

The Mean Value Theorem

Euler’s Rediscovery of $e$

Abel and Cauchy on a Rigorous Approach to Infinite Series

The Definite Integrals of Cauchy and Riemann

Henri Lebesgue and the Development of the Integral Concept

Topology

The Cantor Set before Cantor

Topology from Analysis

Nearness without Distance

Connectedness: Its Evolution and Applications

Connecting Connectedness

From Sets to Metric Spaces to Topological Spaces

The Closure Operation as the Foundation of Topology

A Compact Introduction to a Generalized Extreme Value Theorem

Complex Variables

The Logarithm of $1$

Riemann’s Development of the Cauchy–Riemann Equations

Gauss and Cauchy on Complex Integration

Primary sources provide motivation in the words of the original discoverers of new mathematics, draw attention to subtleties, encourage reflection on today's paradigms, and enhance students' ability to participate equally, regardless of their background.
These beautifully written primary source projects that adopt an “inquiry” approach are rich in features lacking in modern textbooks. Prompted by the study of historical sources, students will grapple with uncertainties, ask questions, interpret, conjecture, and compare multiple perspectives, resulting in a unique and vivid guided learning experience.
David Pengelley, Oregon State University