Transcript Math/Science Curriculum Alignment

Math/Science Curriculum
Alignment Workshop
(May 22, 2009)
NSF/BMCC/John Jay
Goals

Examine mathematics preparation for Forensic
Science students (concepts/skills)

Consider the role of new pedagogy and technology
in teaching mathematics (tools)

Suggest concepts/applications/problems in
mathematics courses that would benefit forensic
science students (problem solving abilities)
Method

Background (courses, pedagogy and technology,
expectations for students)

Dialogue between math and science faculty
members
Math for Forensic Scientists
at John Jay

Precalculus (MAT141)

Calculus I (MAT 241)

Calculus II (MAT242)

Probability & Statistics (MAT 301)

If needed, College Algebra (MAT 104 or MAT 105)
The Salane Evaluation:
(This is thoroughly inadequate
preparation for a career in Science.)

No course in differential equations.

No course in applied linear algebra.

No course in applied statistics (301 should be followed by a
calculus based applied course.)

No exposure to multivariate calculus.

Minimal or no exposure to algorithmic methods and computing.
The Salane remedy:

Don’t (unless you really want to become a service
department) add an extra year to the Forensic
Science program.

Do (and this may be the last chance you’ll ever
have) work to get the college to establish appropriate
college-wide general education requirements for
science and technical majors.
Calculus Reform Pedagogy

Cooperative – students work in groups

Exploratory study by students – formal definitions
and procedures evolve from practical problems

Multiple representations of concepts– symbolic,
geometric, numerical and verbal

Alternative assessments – student portfolio,
computer projects
Traditional Pedagogy

Instructor and lecture driven

Definition and theory followed by
applications/problems

Students work alone for the most part

Extensive drills (mainly symbolic problem solving)
used to reinforce learning
Characteristics of new Curricula

Employ modern tools to develop geometrical and
numerical understanding concepts

Emphasize practical problem solving to support work
in engineering courses

Train students to use tools so they solve real world
problems

Stress understanding concepts rather than symbolic
manipulation
Motivations for Calculus Reform

Increasing use of sophisticated problem solving tools
(MAPLE, MATLAB, Mathematica, AutoCAD, etc.)
and computer simulations in the workplace.

Demand from the science and engineering
communities. (Most Calculus students are not math
majors!)

Widespread agreement that students benefit most if
a calculus concept is understood numerically and
geometrically as well as symbolically.
A chemist’s view
 “I’m a bit embarrassed to admit that I haven’t
evaluated an integral in 40 years! That kind of thing
is not really what many of us need calculus for in
chemistry…it’s more important for students to
understand the meaning and application of
derivatives and integrals, set up a differential
equation and interpret the behavior of its
solution…numerical techniques are more important
now than analytical techniques.”
A physicist’s view
 “What we want is for students to bring a basic
understanding of fundamental concepts of calculus
into their physics courses…right now they are good
at taking a derivative mechanically, but have little
idea what the derivative tells them.”
Electrical engineer’s view

“Students must be able to interpret behavior of
solutions based on graphical output … engineering
students more and more are looking at numerical
and graphical representations and less and less at
symbolic representations …I tell my students that the
vast majority of things do not have algebraic
representations …their calculus training is too
loopsided in emphasizing symbol moving …we
expect them to be able to use computer packages.”
A biologist’s perspective
 “Computer simulations are an extraordinary tool for
involving students in a problem-solving environment.
It encourages them to interact at a much deeper
level of involvement…it opens doorways to them.”
The new calculus student

Understands visual representations

Comfortable with computer applications

Understands concepts of precalculus and calculus
(functions, limits, continuity, derivative, integral)

Able to explore models and simulations
Calculus reform curricula –
some reservations

What evidence is there that it works?

Do “cookbook” courses now become “clickbook” courses?

What topics are in and what topics are out?

Aren’t drills needed to build skills and understanding?

Today most students are deficient in algebra and trigonometry.
Does this just make things worse?
By the way: What is Calculus
Reform?
Calculus: Catalyzing a National Community
for Reform : Awards 1987-1995 by William E.
Haver (Editor), National Science Foundation

Total awarded: $44 million by NSF

Harvard Calculus Group Text: Single and
Multivariate Calculus (Huges-Hallet, et al,
1995,1998,2001)
The Reform Movement (2008)

Stewart (widely considered to be a traditional
approach) is used in over 80% of Calculus courses
nationwide.

Stewart and other Calculus texts now make
significant use of Computer Algebra systems to allow
students to experience numerical and geometric
representations of concepts.

Huges-Hallet text (3rd edition) has been revised to
include traditional topics and approaches. Pure
reform editions are out of print.
Statistical Findings

Armstrong, G. & Hendrix, L. (1999). Does Traditional or Reformed
Calculus Prepare Students Better for Subsequent Courses? A
Preliminary Study. Journal of Computers in Mathematics and Science
Teaching. 18 (2), pp. 95-103. Charlottesville, VA: AACE.
BYU Outcomes Study
GPA and Calculus Preparation
Reform-TLC
(Mathematica)
ReformHarvard
Traditional
Physics II
3.32
3.09
2.99
Physics I
2.97
2.93
2.91
Analysis
3.16
2.44
2.72
Adv Eng
2.93
2.86
2.96
Statistics
3.33
2.99
3.02
Calculus and Science & Engineering
curricula (Goals at Cornell)

Improve conceptual understanding and retention of math
content

Enhance ability to apply math to science and engineering
problems

Retain and nurture student interest in science and mathematics

Create positive peer learning communities through early
engagement in structured, collaborative learning activities.
Mathematics like all fields has the
following:

Concepts and principles -real number system,
function, limit, continuity, derivative, integral

Methods - of integration, of differentiation, algorithms

Practices - problem solving techniques

Tools - abacus, calculator, sophisticated problem
solving environments such as Maple & MatLab
History of Calculus
Precise (ε-δ) Definition of Limit
(symbolic, numerical & graphical
representations)

What radius is needed to manufacture a disk of area
1000 cm2?

If the machinist is allowed an error tolerance in the
area of plus or minus 5 cm2, how close to the ideal
radius in part must the radius be?

In terms of the ε-δ definition of
limx->af(x) = L, what is f, a, ε, δ and L. (Maple
Worksheet – Circular Disk).
Computer Algebra Systems Techniques to Enhance Learning

MAPLE and other computer algebra systems were
designed to simplify symbolic computations

With them students can begin to examine topics
symbolically, graphically and numerically (Enhance
level and quality of instruction. )

Taylor Series Example
Concepts/Skills/Tools Applications
for forensic Science

Physics: derivatives – s(t),v(t) and a(t); optimization – physical
principals, e.g. Snell’s Law in optics

Exponential Growth/Decay (radioactive decay)

Reaction Kinetics (simple differential models)

Biology (predator/prey models)

Statistics (normal distribution)

Economics: Compound Interest, resource allocation problems
So what should Forensic Science
students learn in preCalculus and
Calculus courses?

Principles

Methods

Practices

Tools