Exploration and Inquiry in an Introductory Course for Mathematics Majors

Helmut KnaustThe University of Texas at El Paso

hknaust@utep.edu

January 12, 2015

The Problem

• Math majors take 4-5 courses geared towards science and engineering majors, not math majors.

• They get the wrong idea that Mathematics is all about computations, and “set in stone”.

• These courses are often mostly lecture-based and not very interactive.

The Problem (cont’d)

• The Calculus sequence may not be the ideal setting to recruit math majors.

• Community college students often arrive on our campus “behind schedule”: Before completing Calculus II there is really no other math course for them to take.

The Course

• "Introduction to Higher Mathematics"

– Sophomore level– Only co-requisite: Calculus I– Students work in pairs on 6-

7 “laboratories” on a variety of mathematical topics (two weeks each)

The Laboratories

test conjecture

refine conjecture conduct experiment

devise experiment

formulate conjecture

The Two-week Laboratories (cont’d)

• Intriguing open-ended problems

• Short exposition by instructor, maybe 15 minutes

• Ultimate goal: Students come up with mathematical conjectures and try to prove them

• Students write a substantial laboratory report • 10-20 pages• Students can resubmit for a new grade once per lab

• Instructor provides guidance and feedback all along

• What happens to this sequence as n gets large?

• How does the answer depend on a, b and xo?

Sample Laboratory: 1. Iteration of Linear Functions

xn = a xn-1 +b ; initial value xo

-1.5 -1 -0.5 0.5 1 1.5a

-1.5

-1

-0.5

0.5

1

1.5

b

-1.5 -1 -0.5 0.5 1 1.5a

-1.5

-1

-0.5

0.5

1

1.5

b

Convergence only if x0=b/(1-a)

Convergence only if b=0

Convergence always

Convergence never

The Laboratories (the textbook offers 16 choices)

1. Iteration of Linear Functions

2. The Euclidean Algorithm

3. Parametric Curve Representation

4. Sequences and Series

5. Iteration of Quadratic Functions

6. The p-adic Numbers

Sample Laboratory: 2. The Euclidean Algorithm

1. Exposition: Explanation how the EA works.

2. Students read about why the EA works.

3. Key Questions for Student Investigations:

1. How fast does the EA work? Any clue what to expect?2. How often are two “random integers” relatively prime?3. How does the EA work with pairs of integers from the

Fibonacci sequence?

Sample Laboratory: 3. Parametric Curve Representation

Study of parametric curves of the form

x(t)= sin(p t) + cos(q t)y(t)= sin(r t) + cos (s t),

where p,q,r,s are positive integers

Key Question: What parameter choices lead to what symmetries of the parametric curve?

For their investigations, students use Mathematica notebooks that my colleague Art Duval and I wrote.

A hint I gave last time:

• What students like:

– Everybody has an “Eureka” moment a few times during the semester.

– They can work at their own pace.

• What students do not like:

– Lots of writing (each student group writes 100 pages or so during the semester)

• What I like:

– Seeing students make progress during the semester: they get noticeably better at exploring, conjecturing, and writing.

– Once in a while students ask me a question, and I have to explore and conjecture…

• What I do not like:

– Lots of reading to grade papers: • 6 projects • 10 student pairs• 2 submissions per project

= about 2,000 pages

Helmut Knausthknaust@utep.edu

The Textbook:Laboratories in Mathematical Experimentation: A Bridge to Higher Mathematics.Spinger-Verlag, 1997.ISBN-10: 0387949224