Teaching and Learning Mathematics Online

James P. Howard, II, John F. Beyers, James P. Howard, II, John F. Beyers

• Publisher: Chapman and Hall/CRC
• Year: 2020
• eBook ISBN: 9781351245562
• Edition: 1
• Topic: Matematica
• Subtopic: Matematica applicata

Online education has become a major component of higher education worldwide. In mathematics and statistics courses, there exists a number of challenges that are unique to the teaching and learning of mathematics and statistics in an online environment. These challenges are deeply connected to already existing difficulties related to math anxiety, conceptual understanding of mathematical ideas, communicating mathematically, and the appropriate use of technology.

Teaching and Learning Mathematics Online bridges these issues by presenting meaningful and practical solutions for teaching mathematics and statistics online. It focuses on the problems observed by mathematics instructors currently working in the field who strive to hone their craft and share best practices with our professional community. The book provides a set of standard practices, improving the quality of online teaching and the learning of mathematics. Instructors will benefit from learning new techniques and approaches to delivering content.

Features

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• Based on the experiences of working educators in the field

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• Assimilates the latest technology developments for interactive distance education

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• Focuses on mathematical education for developing early mathematics courses

PART 1

Course Design

CHAPTER 1

Teaching Cross-Listed Mathematics Courses Online

Laurie Battle, Atish J. Mitra, and H. Smith Risser

CONTENTS

1.1 Introduction

1.2 The Courses

1.2.1 Groups and Geometry

1.2.2 Advanced Linear Algebra

1.2.3 Mathematical Modeling

1.3 Adapting the Courses to Online Delivery

1.3.1 Groups and Geometry

1.3.2 Advanced Linear Algebra

1.3.3 Mathematical Modeling

1.4 Technical Issues

1.4.1 Creating the Lectures

1.4.2 Online Interactions with Students

1.4.2.1 Asynchronous Tools

1.4.2.2 Synchronous Tools: Office Hours

1.4.3 Assessment

1.4.4 Ensuring Rigor

1.5 Conclusions

Bibliography

1.1 Introduction

This purpose of this chapter is to provide a detailed description of three cross-listed (combined upper-level undergraduate and graduate) classes that were taught online in the summer of 2017 at Montana Tech. All three courses were part of a funded project designed to prepare current high school teachers for teaching Dual Credit Mathematics Classes. Current state-level policies require all Dual Credit teachers within the state of Montana to have a Masters degree and 9 credits of graduate-level math. The rural nature and size of Montana are barriers for current teachers wishing to take face-to-face graduate courses. As a result the funded project was designed to support teachers taking online graduate-level mathematics. Mathematical Modeling and Groups and Geometry were previously taught face-to-face to upper-level undergraduates. Advanced Linear Algebra was designed specifically for the project. However, some topics (vector spaces and numerical methods for matrix algebra) were previously taught by the instructor in other face-to-face courses within the undergraduate curriculum.

As mentioned above, all three classes had both undergraduate and graduate students enrolled. The students in the undergraduate section were typically current undergraduate students majoring or minoring in mathematics. The students in the graduate section were typically adults working in K-12 or higher education. The cross-listed nature of the courses, along with the online delivery of the materials, provided unique challenges. Each of the courses had to accommodate different expectations in learning objectives and different approaches in evaluation. For example in Advanced Linear Algebra, most of the graduate students needed substantial review of the prerequisite material in order to succeed. Similarly in Groups and Geometry some of the students did not have a previous background in group theory. This required the instructor to introduce the concept of a group in the special case of groups of transformations. In this chapter, we discuss how we dealt with challenges like assessment, technology use, ensuring rigor, and providing support to students at a distance. This paper would be relevant to any instructor wishing to teach graduate courses or dual-enrollment courses online, especially those including graduate students with nontraditional preparation.

1.2 The Courses

1.2.1 Groups and Geometry

The face-to-face version of Groups and Geometry was not cross-listed. It was intended for advanced undergraduates, typically seniors, who had already completed a course in abstract algebra. Groups and Geometry introduced students to the unity of abstract mathematics by demonstrating interconnections between geometry and algebra. Additionally, it guided the students into developing and communicating mathematical proofs in both oral and written forms.

Groups and Geometry was developed as a natural successor to a year-long sequence of abstract algebra. The instructor assumed that the student had at least a rudimentary background and understanding of algebraic structures, and guided the student through an exploration of various geometries and their algebraic connections. Linear Algebra was the formal prerequisite of the course. The course met twice a week for a total of three hours.

The face-to-face section started with a gentle review of the algebra of complex numbers and its connections to plane Euclidean geometry. This topic took about three weeks. By the end of this first topic the students would prove results in Euclidean geometry, starting from simpler propositions such as the concurrence of medians/angle bisectors of a Euclidean triangle, to more esoteric aspects such as the fact that the adjacent trisectors of the angles of an arbitrary triangle form an equilateral triangle. At this stage the proofs largely required only a level of comfort with manipulating algebraic expressions involving complex numbers, and understanding their geometric significance.

Once students had a good understanding of the application of complex numbers to Euclidean geometry, the concept of isometries of the complex plane was introduced with the aim of a complete classification of the isometries of the Euclidean plane. This part of the course lasted about seven to eight weeks and was the heart of the course. Here the level of the proofs became more conceptual than the algebraic manipulations in the first part of the course; students were able to see that non-trivial geometric structures can be reduced to study of algebraic structures. Here the group structure of the set of Euclidean isometries was introduced, and several non-trivial results, such as a study of the subgroups of the Euclidean isometry group, could be reached at this stage.

In the final part of the course the students saw how an algebraic structure such as a finitely generated group could be viewed as a metric space. This part lasted about two to three weeks and was a gentle introduction to geometric group theory.

The grade for the face-to-face version of the course was based on homework assignments and exams. The homework assignments typically were extensions of concepts discussed in class.

1.2.2 Advanced Linear Algebra

Advanced Linear Algebra introduced students to theoretical concepts from algebra using matrices. The first part of the course focused on building a theoretical understanding of algebra and vector spaces. Then the course showcased applications of linear algebra to other disciplines (e.g. the singular value decomposition applied to regression). Finally, students learned how numerical methods are applied to problems described within the course. The instructor had previously taught some of the topics in face-to-face Linear Algebra and Numerical Computing courses.

The face-to-face Linear Algebra class had a prerequisite of Calculus 2 and included elementary row operations, vector spaces, and eigenvalues. The class was taught in a lecture format and met three hours per week. Students were expected to complete homework assignments from the textbook. These homework assignments included both computational work and short proofs. Students were also assessed through face-to-face examinations. The questions on the examinations were similar to those completed in the homework assignments.

The Numerical Computing class had a prerequisite of either Linear Algebra or Differential Equations. The curriculum included a unit on numerical linear algebra. Both direct and iterative methods for solving systems of equations were covered. The class was taught in a lecture format and met three hours per week. The homework assignments required students to use MATLAB^® [1] to solve systems of linear equations. Homework assignments were completed in groups. Each group analyzed the errors and floating point operations for different types of matrices (e.g. ill-conditioned, banded). The homework assignments required students to compare results for the different methods. Students were also assessed using face-to-face examinations which required students to solve questions by hand and to analyze errors produced by numerical algorithms.

1.2.3 Mathematical Modeling

The Mathematical Modeling course taught students how to devise and analyze models in the form of difference equations as well as differential equations, including systems of both types of equations. Applications were taken from a variety of fields with an emphasis on population modeling. Students learned methods for finding analytical and numerical solutions, as well as geometric representations, including phase lines and phase planes. Spreadsheet software was the primary technology that students used to analyze difference equations numerically. Matrix calculators were used to assist with eigenvalue and eigenvector analysis of differential equations. The instructor had previously taught this as a face-to-face class, but the content was modified for the online version. The face-to-face class focused on differential equations only. Methods of analysis included finding equilibrium solutions, stability analysis, and finding both analytical and numerical solutions. MATLAB was the primary technology used. The content for the online course included difference equations in addition to a less in-depth selection of topics in differential equations. The primary technology used for the online course was spreadsheet software. The face-to-face version required Differential Equations as a pre-requisite.

The face-to-face version met three times per week for 50 minutes. This time was used primarily for lectures and exams. The types of assessment included homework, three quizzes, three exams, three group projects, and a final exam. The homework exercises came from the textbook and were collected weekly. Students worked in groups of two to three on the projects, and each group selected a project from a list of several options. Some of these projects came from the textbook, and some were written by the instructor. Each group prepared a written report and gave a presentation to the class. Students were asked to use MATLAB on homework and group projects, but they did not have access to MATLAB for quizzes and exams. As a result, the quizzes and exams tended to emphasize theory while the other assignments ex...

Table of contents

• Cover
• Half-Title
• Title
• Copyright
• Dedication
• Contents
• Preface
• Acknowledgments
• Editors
• Contributors
• PART 1 Course Design
• PART 2 Student Interaction
• PART 3 Using Technology
• PART 4 Teacher Education
• PART 5 Commentary
• INDEX