What Place Does Philosophy Have in Teaching Mathematics Preliminary report' - PowerPoint PPT Presentation

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What Place Does Philosophy Have in Teaching Mathematics Preliminary report'

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What is Philosophy of Mathematics? Ontology for Mathematics: 'Being' ... Roles for Philosophy in Teaching and Learning. For the Teacher/Mentor (T/M) ... – PowerPoint PPT presentation

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Title: What Place Does Philosophy Have in Teaching Mathematics Preliminary report'


1
What Place Does Philosophy Have in Teaching
Mathematics?Preliminary report.
  • Martin E Flashman
  • Department of Mathematics
  • Humboldt State University
  • Arcata, CA 95521
  • flashman_at_humboldt.edu

2
Abstract
  • In recent years discussion of the history of
    mathematics has grown in its treatment in
    mathematics courses from pre-calculus through
    advanced courses such as number theory, algebra,
    geometry and analysis.
  • The speaker will address the question of what
    role the philosophy of mathematics might take in
    these and similar undergraduate courses.

3
What is Mathematics? B. Russell
  • Mathematics may be defined as the subject in
    which we never know what we are talking about,
    nor whether what we are saying is true.
  • Bertrand Russell, Mysticism and Logic (1917) ch.
    4

4
What is Philosophy of Mathematics?
  • Ontology for Mathematics Being
  • Ontology studies the nature of the objects of
    mathematics. What we are talking about.
  • What is a number?
  • What is a point? line?
  • What is a set?
  • In what sense do these objects exist?

5
What is Philosophy of Mathematics?
  • Epistemology for Mathematics Knowing
  • Epistemology studies the acquisition of knowledge
    of the truth of a mathematical statement.
    whether what we are saying is true.
  • Does knowledge come from experience and evidence?
  • Does knowledge come from argument and proof?
  • Is knowledge relative or absolute?

6
Roles for Philosophy in Teaching and Learning
  • For the Teacher/Mentor (T/M)
  • Awareness of issues can alert the T/M to
    excessively authoritarian approaches.
  • Alternative philosophical views can allow the T/M
    to use and/or develop alternatives to traditional
    approaches.
  • Philosophical issues can illuminate the value of
    and need for developing a variety of mathematical
    tools for solving problems.

7
Roles for Philosophy in Teaching and Learning
  • For the Student/Learner (S/L)
  • Helps the S/L understand the context, goals, and
    objectives of the mathematics being studied.
  • Opens the S/L to considerations of the human
    values and assumptions made in developing and
    using mathematics.
  • Alerts the S/L to the use of authority and the
    value of different approaches to mathematics.

8
Exploring Example(s)
  • Following are two examples of topics that can be
    used to introduce and explore some philosophical
    issues in courses at a variety of levels.
  • Consider how these examples can be expanded or
    transformed to other aspects of the philosophy of
    mathematics.
  • Consider how these examples can be expanded or
    transformed to other mathematics topics and/or
    courses.

9
The Square Root of Two
  • Questions for Open Discussion
  • Ontological
  • Definition?
  • Does it exist?
  • What is the nature of this object?
  • Epistemological
  • How do we know it exists?
  • How do we know it is between 1 and 2
  • How do we know it is not a rational number?
  • Courses
  • Pre-calculus
  • Calculus
  • Transitional Proof Course
  • Number Theory
  • Algebra
  • Real Analysis
  • Numerical Analysis

10
The Square Root of -1 i
  • Questions for Open Discussion
  • Ontological
  • Definition?
  • Does it exist?
  • What is the nature of this object?
  • Epistemological
  • How do we know i exists?
  • How do we know i is not a real number?
  • How do we know that the complex numbers are
    algebraically closed?
  • Courses
  • Pre-calculus
  • Calculus
  • Transitional Proof Course
  • Number Theory
  • Linear Algebra
  • Algebra
  • Real Analysis
  • Complex Analysis
  • Technology/CAS

11
What to do? Two Suggestions
  • Organize workshops for college faculty
    introducing the issues of philosophy of
    mathematics and their relevance to the
    undergraduate curriculum
  • Organize conferences/communications for
    developing teaching materials and resources that
    involve college students at a variety of levels
    in the issues of philosophy of mathematics.
  • Funding? NSF, NEH, Corporate and Foundation

12
The EndThank you?
  • Questions?
  • Comments?
  • Discussion?
  • flashman_at_humboldt.edu
    http//www.humboldt.edu/mef2
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