Transcript here - Michael A. Karls Home Page

IMPLEMENTING AN
INTRODUCTORY MATHEMATICAL
SOFTWARE COURSE
Michael A. Karls
Department of Mathematical Sciences
Ball State University
ABSTRACT
We will look at the challenges of designing and
implementing an introductory mathematical
software course;
 Provide an overview of topics covered and
resources used for each type of software;
 Give examples of homework problems, exam
questions, and project topics chosen for the
course;
 Address issues that have arisen.

2
INTRODUCTION
 In
fall 2007, I piloted a course, entitled
MATHS 159 Introduction to Mathematical
Software at Ball State University.
 The course was designed to introduce
mathematical science majors to
“mathematical software” – in particular
Excel, Mathematica, and MATLAB.
 MATHS
159 was a direct result of work
done with colleagues Irene Livshits and
Richard Stankewitz as part of an
Enhanced Provost Initiative Grant.
3
DEPARTMENTAL RATIONALE (PORTION)
 Currently,
most of our math majors have
difficulty with the technology they encounter
in many of their classes.
 One reason may be that most students in our
programs don't have a solid foundation in
both the mechanics of the software and the use
of the software as a tool for doing
mathematics. …
 MATHS 159 … taken early in a student’s
program, will provide a systematic
introduction to mathematical software
packages that are used in many of our upper
level courses.
4
DEPARTMENT OF MATHEMATICAL
SCIENCES OVERVIEW

Ball State University

Public institution
 Located in Muncie,
Indiana
 20,000 students


Program Enrollment (fall
2008)



Majors and minors:
Actuarial Science
Mathematical Sciences
(pure and applied
mathematics options)
 Mathematics Teaching,
(middle and secondary
school options)
 Mathematical Economics
(joint with Economics)
 Minor in Mathematics


240 undergraduate majors


97 Actuarial Science majors
112 Mathematics Teaching
majors
31 Mathematical Sciences
majors
Common core for all
majors
Calculus (three semesters)
Linear algebra
 Discrete mathematics
 Senior capstone course
 MATHS 159 (starting in
fall 2008).


5
SOFTWARE USE IN DEPARTMENTAL
COURSES




Mathematical software use in
courses varies by course, major,
and instructor.
Calculus sequence –
approximately three
Mathematica labs per course.
Linear Algebra – Mathematica
or MATLAB component.
Mathematics Teaching majors
take a junior-level course on
teaching with
software/technology that
includes graphing calculators,
Excel, Logo, and Geometer’s
Sketchpad.



Actuarial Science majors use
Excel extensively in their
courses
Mathematical Science majors
use MATLAB in their
numerical analysis courses.
Other courses that routinely
integrate mathematical
software include




Most actuarial science and
statistics courses (Excel,
Minitab, SAS, R)
Mathematical Models
(Mathematica, Excel,
MATLAB).
Differential Equations
(Mathematica)
Operations Research
(Mathematica, Excel, MPL,
LINGO).
6
INITIAL CHALLENGES
For MATHS 159, at first, all I had for a course
guide/template was a master syllabus and a list
of technology topics.
 From this, I had to create a set of course
materials at a level appropriate to freshman who
would be concurrently enrolled in first-semester
calculus.

7
EXCEL RESOURCES

Introductory Resources




Ball State mini-course
Greg, H., Excel 2003 for Dummies.
Walkenbach, J., Excel 2007 Bible.
Online Resources





Microsoft Office Online. Microsoft
Excel – math and trigonometry
functions.
http://office.microsoft.com/enus/excel/CH100645361033.aspx
Microsoft Office Online. Microsoft
Excel – list of worksheet functions.
http://office.microsoft.com/enus/excel/HP100791861033.aspx
James, B.W., Excel Tutorial.
http://people.usd.edu/~bwjames/tut/ex
cel/
French, T. Excel IF functions - Using
IF functions in Excel Spreadsheets.
About.com.
http://spreadsheets.about.com/od/exce
lfunctions/a/if_funct_hub.htm
Solver Add-On Tutorial


Finance and Simple Linear
Programming



Hillier, F. S. & Lieberman, G. J.,
Introduction to Operations Research
(8th ed.).
Lial, M. L., Miller, C. D., &
Greenwell, R. N., Finite
Mathematics and Calculus with
Applications (4th).
Mooney, D. & Swift, R., A First
Course in Mathematical Modeling.
Population Growth and
Optimization


Olinick, M., An Introduction to
Mathematical Models in the Social
and Life Sciences.
Stewart, J., Calculus (early
transcendentals). (5th).
8
MATHEMATICA RESOURCES

Introductory Resources



Built-in Help Files
Burkhardt, W., First
Steps in Mathematica.
Help References

Mathematica Tutorial and
Lab Resources
Karls, M., Mathematica
Tutorial.
http://www.bsu.edu/web/mk
arls/MathematicaTutorial0
8.nb.
 Emert, J. & Nelson, R.,
Calculus and Mathematica.

Wolfram, S.,
Mathematica: A System
for Doing Mathematics
by Computer (2cd)
 Calculus Resources
 Wolfram Research.
 Hollis, S., CalcLabs with
http://www.wolfram.com/
Mathematica.
 Wolfram Research.
 Stewart, J., Calculus (early
Wolfram Screencast and
transcendentals). (5th).
Video Gallery.
http://www.wolfram.com/
broadcast/.

9
MATLAB RESOURCES

Introductory Resources




Calculus with MATLAB



Built-in Help Files
University of Utah. MATLAB
Tutorial.
http://www.math.utah.edu/lab/
ms/matlab/matlab.html/.
Sigmon, K., MATLAB Primer
(3rd)
Jenson, G., Using MATLAB in
Calculus.
Most of the mathematics for
this section was calculus based,
at the level of differentiation,
Riemann sums, and
integration.
We also touched on
programming and graphing,
using examples from the
previous sections.

Linear Algebra Resources
(MATLAB is matrix-based)



Anton, H. & Rorres, C.,
Elementary Linear Algebra (7th)
Lewand, R.E. Cryptological
Mathematics.
To help tie the other software to
MATLAB, we looked at how to
solve linear systems in Excel
and Mathematica.

Systems of linear equations and
matrices Excel files. Supplement
to Finite Mathematics, (7th ed.),
by Margaret Lial et al.
http://wps.aw.com/aw_lial_finite
math_7/0,1769,12520-,00.html
10
INTRODUCTORY SURVEY

Given on the first day,
20-30 min allowed.
 Problems to be done
by hand.
 All questions appear
in the first Excel
lecture!

1.
For the integers n =
1, 2, 3, … 99, 100,
find each of the
following:
a.
b.
c.
d.
n^2
n^3
4n^3 + n^2 -1
1 + 2 + … + n (Note:
For n = 1, the sum is
1. For n = 2, the sum
is 1 + 2.)
11
INTRODUCTORY SURVEY
2.
The data in the
following table are the
land area in Australia
colonized by the
American marine toad
(Bufo marinis).
a.
b.
c.
Plot this data with
Year on the x-axis and
Area on the y-axis.
Find a function that
can be used to predict
this growth.
Estimate when
Australia will be
overrun by the toads
(Australia’s area is
7,619,000 km2).
Year
Area(km^2)
1939
32800
1944
55800
1949
73600
1954
138000
1959
202000
1964
257000
1969
301000
1974
584000
12
INTRODUCTORY SURVEY
3.
Sort the following
list of birthdays of
famous
mathematicians
according to their
a.
b.
c.
Birth Year
Name
Birth Year followed
by Name
Birth Year Name
Birth Year Name
1584
Vernier
1840
Henrici
1646
Flamsteed
1852
Le, P.
1736
Bring
1900
Aiken
1739
Klügel
1878
Bernstein, F.
1939
Baker
1909
Black
1852
Frattini
1920
Pillai
1888
Courant
1946
Margulis
1924
Cohn
1880
Bernstein, S.
1942
Hawking
1796
Bienaymé
1814
Wantzel
1801
Cournot
1867
Bôcher
1819
1883
Keynes
1883
Schouten
1816
Wolf
1910
Turán
1905
Dubreil-Jacotin
1910
Koopmans
1906
Feller
1911
Kakutani
1922
Marchenko
1939
Kingman
1818
Joachimsthal
13
TOPICS OVERVIEW

Topics in Excel









IN
Definitions and Terminology
Finance Applications – Simple
and Compound Interest
Finance Applications –
Annuities and Amortization
Logical Functions and
Conditional Formatting
Mathematical Functions
Recurrence Relations
Population Models
The Excel Solver
Topics in Mathematica



Basic Syntax and Basic
Commands
Entering Text, Formatting, and
Using the Help File
Formatting and Working with
Tables
ORDER PRESENTED

Topics in Mathematica (cont.)








Calculus Commands
Parametric Curves and
Surfaces; Animation
Importing and Exporting Data
Optimization
Numerical Computation
Simple Programming
Systems of Equations
Topics in MATLAB





Elementary Matrix Theory
Numerical Differentiation and
Integration in MATLAB;
Function M-files
Working with Data in MATLAB
The Symbolic Toolbox
Programming in MATLAB
14
SAMPLE EXCEL HOMEWORK QUESTION

Using the data from the Toads Example,




Create a table in Excel with a column for years and a column for area.
Make a scatter plot that includes a labeled best-fit line.
Find the area predicted by the Excel best-fit line, error, and relative error.
Compare the best-fit line model values and resulting errors to the graphs.





Repeat with y = 13568.1x -2.63405*10^7, found via Mathematica’s “Fit”
command.
Which linear model is better – Excel’s or Mathematica’s? Why?
Use the “better” best-fit line to help estimate graphically when Australia
will be overrun by toads (Australia total area is 7,619,000 km^2).
I have gotten a lot of mileage out of this set of data – related
questions and examples include:




THEY DON’T AGREE – DUE TO ROUNDING IN THE TRENDLINE FORMULA!
Calculating the best-fit line slope and y-intercept several different ways
(formulas, built-in functions, Excel’s Solver);
Treating toad growth as a recurrence relation;
Fitting the data to exponential, logistic, or polynomial models (in all three
types of software).
Source: Blanchard, P. (1994). Teaching differential equations with a
dynamical systems viewpoint. College Journal of Mathematics, 25
(5), 385-393.
15
SAMPLE MATHEMATICA LAB QUESTION






For the function f(x) = x^3+x-4,
Define the function f[x],
Choose an x-coordinate x = a,
Define a function m[h] that
computes the slope of the secant
line between (a,f(a)) and
(a+h,f(a+h))
Reproduce the given graph that
includes a tangent line at (1,-2) and
secant lines that will converge to
the tangent line as the slider
controlling h is moved.
The curves are made with a Plot
command, the dots are made with a
ListPlot command, and the two
plots are combined with a Show
command.



h
0.632
f x
6
4
2
x
0
2
4
6
1
0
1
2
3
f[x_]:=x^3+x-4
a=1;
m[h_]:=(f[a+h]-f[a])/h.
16
SAMPLE MATLAB EXAM QUESTION





Let g(x) = exp[-(1/2) x^2].
Find the actual derivative of this function, g’(x).
Find the numerical derivative of g(x) on the interval
[-1, 1].
Use a Riemann sum to numerically estimate the
definite integral of g(x) on the interval [-1,1].
Extra Credit: Find the integral of g(x) on the interval
[-k, k] symbolically.


Then find the value of this integral if we replace k with
infinity, two ways – numerically with increasing k-values
and symbolically with a limit.
The idea for this problem comes from similar
examples in Jenson’s Using MATLAB in Calculus.
17
FINAL PROJECT



Part of a student’s course
grade is based on a final
group project which includes
a class presentation.
Each project applies ideas
from calculus to real-world
problems or situations and
can be investigated with the
software tools studied in the
course.
Source: Straffin, P. (Editor).
Applications of calculus:
Volume 3. MAA Notes (29).
Washington, DC:
Mathematical Association of
America Press.
Project
1
2
3
4
Calculus
Area of
Ideas Used Application
Minimizing Derivatives to
quadratic
find local
Fitting Lines polynomials extrema,
Statistics
to Data finds the
logarithm and
regression line exponential
to a data set functions
Minimizing Derivatives of
Trigonometric trigonometric
Somewhere
functions to functions,
Optics,
within the
explain
finding
Meteorology
Rainbow
properties of a maxima and
rainbow
minima
Average
Integrals of value of a
Measuring polynomials function,
Political
Voting give a
integration of
Science
Power measure of
polynomials,
voting power mathematical
induction
Differentiatio
Trigonometric
n&
How to Tune integrals
Signal
integration of
a Radio explain tuning
Analysis
sines and
a radio
cosines
Title
Topic
18
COURSE LOGISTICS




The course meets 200
minutes per week, with
at least one 100 minute
(double) period.
The double period works
well for in-class exams.
PowerPoint or other
appropriate electronic
formats as well as
chalkboard are used for
interactive lectures.
Lectures and files for
examples are posted on
line before each class
period.


Three in-class exams,
one for each type of
software – software use
is required.
Students may use
5” by 8” index card (one
side)
 Help files
 No other course material


Electronic files for exam
solutions are emailed to
me (and a course
graduate assistant for
back up).
19
COURSE ISSUES


Since the course is different than
“traditional courses”, I have had to
work out bugs as they arise – most
of which can be classified as one of
the following:
Technological issues related to
using computers in general, such as





Not being able to log into a network
to get to the class web page,
Not knowing where a file is saved,
Trouble with saving a file in a
certain format,
Not knowing how to email a file.
Logistical issues related to running
the course, such as


Software issues related directly to
specific course software, such as


Getting errors while trying to work
out an example in class or on
homework problem.
Mathematical issues related to the
course content, especially questions
on labs and exams, such as




Being able to interpret a question,
Knowing how to start a question,
Figuring out how the question
relates to the course lectures,
Using the software to answer the
question.
Finding ways to effectively collect,
grade, and return student lab work
and exams.
20
COURSE SOLUTIONS

Technological




Many of these issues occur only once
and are unpredictable, but can
usually be “fixed” immediately.
Having a graduate assistant in the
course to help answer student
questions has worked well.


Logistical



Several attempts were made,
including submitting work to
Blackboard (which doesn’t recognize
Mathematica files), turning in hard
copies of work to be graded by hand,
and saving exam work to laptops.
Overall, email has turned out the be
the best way for students to turn in
their work.
Homework is graded electronically
and returned via printed summary
sheet.
Software

Most problems with the software
boil down to entering incorrect
syntax and are “easily” remedied
once the syntax error is found.
As each software section (and the
semester) progresses, these errors
occur less frequently.
Mathematical


Ignoring syntax issues,
mathematical issues tend to arise
from lab or exam questions that
students do not understand or know
how to interpret.
Since I am making up or getting the
questions from various sources, I
revise, rewrite, or throw out
questions as needed and provide
liberal hints in lab, office hours, and
via email.
21
COURSE SUCCESS

Overall, I feel that the course has been a success,
based on







Feedback from students (in person and on evaluations),
Comments from colleagues who have taught the course,
Student performance on exams,
Student pass rate (77% earned a grade of C or higher),
Departmental support for the course,
Student confidence in using the software by the end of the
course, especially Mathematica.
Informal input from colleagues who teach courses that use
the software:


More students in our introductory statistics course report
familiarity with Excel,
Our students in numerical analysis are more confident and
fluent with MATLAB than in the past.
22
STUDENT PERCEPTIONS

Fall 2007 students felt
that


the course was valuable
to them and wished it
would have been offered
earlier in their college
career.
Mostly upper class
majors, 12 students.

Fall 2008 students
indicated that



they didn’t see the value
of the course to their
major.
Mostly first-semester
freshman, twice as
many students.
Similar sentiment for
the versions taught in
Fall 2009 – the course
was a mix of upper and
lower level students.
23
AN UNEXPECTED ISSUE
MATHS 159 is required of all freshman majors.
 Sections are capped at 25 students.
 Only two sections offered per semester.
 This leads to a potential “bottleneck” for students
entering our programs.
 For this reason, since fall 2010, the course is now
taken in a students’ sophomore year after
completing one semester each of calculus and
discrete mathematics.
 In addition, since Mathematics Teaching majors
already have a technology course, only Actuarial
Science and Mathematical Science majors are
required to take the course.

24
THE FUTURE!
As the course progresses, in whatever form it
takes, many more examples, homework
problems, exam questions, and final projects will
have to be created.
 I look forward to the challenge of continuing to
find ways to help increase students’ awareness of
the usefulness of software for their mathematical
careers!

25
REFERENCES

For more detail and a list of all references used
for the course, please see:
Implementing an Introductory Mathematical
Software Course. The Electronic Journal of
Mathematics and Technology. Vol. 4, No. 3, 232-250,
October 2010.
 https://php.radford.edu/~ejmt/

26
EXTRA SLIDES – COURSE TOPICS DETAIL
27
TOPICS IN EXCEL – DETAIL




Finance Applications – Simple and Compound Interest: Simple
Interest, Compound Interest, Present Value, What-If Analysis
(Scenario Manager and Goal Seek)
Finance Applications – Annuities and Amortization: Sequence
(Fibonacci, geometric); Series (partial sum, closed form); Annuities as
partial sums; Annuities (ordinary, annuities due); Future Value of an
Ordinary Annuity; Sinking Fund; Using Excel’s built-in functions to
perform these calculations; Present Value of an Annuity; Mortgage;
periodic payments; Amortization Schedule
Logical Functions and Conditional Formatting: Logical
Functions (TRUE, FALSE, NOT, AND, OR, IF, IFERROR); Nested
IF; LOOKUP; Conditional Formatting
Mathematical Functions: Common Mathematical Functions
(SQRT, ABS, EXP, LN, LOG10, POWER, ROUND, SIN, COS, TAN,
CSC, SEC, COT, PI, RADIANS); Best-Fit Lines (SUMPRODUCT,
SLOPE, INTERCEPT); Excel’s Trendline Feature (number of digits,
exponential); Rates of Change (average, instantaneous, tangent line,
derivative); Engineering Functions (BIN2DEC, HEX2DEC,
CONVERTCOMPLEX, IMPRODUCT); Excel Add-Ins
28
TOPICS IN EXCEL – DETAIL



Recurrence Relations: Recurrence Relation (factorial
function, Fibonacci sequence, compound interest); Initial
Conditions and Order; Closed-form solutions (compound
interest, Fibonacci sequence); Exponential Function;
Exponential Growth and Decay; Damped or Undamped
Oscillation; Fixed Points and Stability
Population Models: Finding a recurrence relation from a
closed-form model (exponential growth for toad population
data); The Logistic Model; Using Excel to study growth rates
with different carrying capacities and a fixed growth rate;
Fixed points in the logistic model; Two or more populations
(predator-prey, host-parasite, competitive hunter, arms race);
Predator-Prey (foxes and rabbits) in Excel – recurrence
relations; Revised Predator-Prey Model which includes
logistic growth
The Excel Solver: Maximizing an area enclosed by a fence;
Minimizing least squares error in an approximating function
(SUMXMY2); Maximizing office storage space (simple linear
programming, naming a range of cells); Excel’s Solver Add-In
29
TOPICS IN MATHEMATICA – DETAIL




Basic Syntax and Basic Commands: Basic Syntax;
Constants and Commands; Grouping Symbols (mathematical
grouping, function syntax, list syntax); Types of Equals Signs
(=, ==, :=); Calculator Commands (trigonometric functions, N);
Symbol Manipulation (Clear, Expand, Factor, Solve, Apart,
Together); Graphing Commands (Plot, ListPlot); Defining
Functions ([ ], Simplify, /;)
Entering Text, Formatting, and Using the Help File:
Entering Text and Formatting (changing text, palettes,
shortcuts); Using the Help Browser (Plot3D example);
Formatting and Working with Tables: Table Command
(TableForm, GridBox, FrameBox); Extracting or Working with
Elements in Tables ([[ ]], Length); Tables of Graphs
Calculus Commands: Finding a Limit (Limit);
Differentiating a Function (D, prime notation, higher-order
derivatives); Integrating a Function (Integrate, NIntegrate);
Evaluating Sums and Series (Sum)
30
TOPICS IN MATHEMATICA – DETAIL




Parametric Curves and Surfaces; Animation:
Parameterized Curves and Surfaces (ParametricPlot,
ParametricPlot3D); Real-Time 3D Graphics (rotating,
zooming); Animation (ListAnimate, Animate, Manipulate)
Importing and Exporting Data: Import and Export
(Head, $Path); Filetypes (CSV, DAT, XLS, etc.); Put and
Get (FilePrint, PutAppend, Needs, Flatten);
Optimization: FindMinimum and FindMaximum (local);
Minimize and Maximize (global); Constraints; NMinimize
and Nmaximize; Replacement Operator (/.)
Numerical Computation: FindRoot (Accuracy,
WorkingPrecision, Chop); NSolve (solving polynomial
equation or system polynomial of equations); NLimit vs.
Limit; Nintegrate (Timing, MaxRecursion)
31
TOPICS IN MATHEMATICA – DETAIL

Simple Programming: Definition of Computer Program;
Logical Operators (And, Or, Not, Xor, Nor, Nand); If
(piecewise defined function, absolute value function
example using If vs. /;); Differentiating each version –
which works better as a plot; Differentiating each version –
which “works”; Which (piecewise defined function – more
than two “pieces”, differentiating yields a function defined
via Which); Switch; Nest (recurrence relations); Print
(Hello World example, new line (\n)); Do (sum of squares,
including a Print statement, adding an If statement,
Return[ ]); For (local vs. global variables, Module,
Continue[ ], Break[ ]); While (Coin Toss example,
RandomInteger, using And to ensure While loop stops);
Newton’s Method Example (by “hand”, simple Do loop, Do
loop with intermediate outputs printed via Print, For loop,
maximum 50 steps, tolerance checked via If)
32
TOPICS IN MATLAB – DETAIL


Systems of Equations: System of two equations in two unknowns;
Review of substitution and elimination; Solving a System of
Equations with Technology (calculator, Mathematica, Excel); Using
Excel’s Solver to solve a system of equations (2 x 2 case); Using
Mathematica’s Solve command to solve a system of equations (2 x 2
case); General 2 x 2 system – what choices of coefficients are allowed?;
Solving general 3 x 3 case (by hand, calculator, Mathematica, Excel,
Other, when a solution is guaranteed, Mathematica solution)
Elementary Matrix Theory: Matrix definition and examples (row,
column, vector); Matrix Notation ([A]ij = aij, square matrix);
Arithmetic with Matrices (addition and subtraction, matrix
multiplication, scalar multiplication); Identities and Inverses; Review
of Real Number Properties (additive identity, multiplicative identity,
additive inverse, multiplicative inverse); Matrix Properties (zero
matrix, identity matrix, additive inverse); Multiplicative Inverse (2 x
2 case, setting up algebraic system to solve to find inverse); When is
matrix A invertible? (determinant of a matrix, how to check if a
matrix is invertible via determinant.); Linear Systems of Equations
Revisited (2 x 2 linear system in matrix form, solve via multiplication
by multiplicative inverses.)
33
TOPICS IN MATLAB – DETAIL


Introduction to MATLAB: MATLAB Basics (Desktop,
help, lookfor, doc, type, up – arrow, clc, Command History
window); Entering Matrices; The Colon Operator (pulling
out parts of a matrix, concatenating matrices); Matrix
Operations (+, -, *, ^, ’, .’, inv, \, / ); Array Operations with
matrices A and B, scalar k (k*A, A.*B, A./B, A.^B); Special
Matrices and Matrix Functions (eye, zeros, ones, det, clear,
clear all); Linear Systems of Equations with MATLAB (2 x
2); Making a Table of Values for a Function; Creating an
M-File; MATLAB’s Path and Running an M-File; The Plot
Command (plot, title, xlabel, ylabel, grid, legend); Multiple
plots and piecewise-defined functions
Numerical Differentiation and Integration;
Function M-files: Numerical Differentiation (linspace,
diff, graphical comparison via subplot); Numerical
Integration (Riemann sums via sum); Function M-Files
34
TOPICS IN MATLAB – DETAIL



Working with Data: Useful built-in commands for
data manipulation (fzero, sum, min, max, length,
mean, median, std, sort); Finding minimum of a
function on an interval (numerically, discretely, via
fzero); Represent a polynomial as a vector of
coefficients (polyval, polyfit); Fit a polynomial to a set
of data via least squares; Importing data from text file
into MATLAB via Import Wizard; Fit data with
polynomial; Compare to Excel’s Trendline or
Mathematica’s Fit command
The Symbolic Toolbox: MATLAB’s built-in
symbolic commands (syms, diff, int, simplify, pretty,
subs, double, ezplot); MATLAB’s function calculator
(funtool);
Programming: Plotting Toad Data; Making a Loop;
Working with Symbolic Functions; Newton’s Method;
Competitive Hunters Model
35