Math/Science Curriculum Alignment WorkshopAlignment Workshop

(May 22, 2009)

NSF/BMCC/John Jay

Calculus Courses: What Should We be doing?

Curricula and skills to support course work i F i S iin Forensic Science

Doug SalaneMathematics & Comp ter Science DeptMathematics & Computer Science Dept.

May 22, 2009

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

B k d ( d d t h l Background (courses, pedagogy and technology, expectations for students)

Dialogue between math and science faculty members

Math for Forensic ScientistsMath for Forensic Scientistsat John Jay

Precalculus (MAT141)

C l l I (MAT 241) Calculus I (MAT 241)

Calculus II (MAT242) Calculus II (MAT242)

Probability & Statistics (MAT 301)y ( )

If needed, College Algebra (MAT 104 or MAT 105)

The Salane Evaluation:(This is thoroughly inadequate (This is thoroughly inadequate preparation for a career in Science.)

No course in differential equations.

No course in applied linear algebra.pp g

No course in applied statistics (301 should be followed by a calculus based applied course )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 programScience program.

Do (and this may be the last chance you’ll ever ( y yhave) work to get the college to establish appropriate college-wide general education requirements for science and technical majorsscience and technical majors.

Calculus Reform Pedagogy Cooperative – students work in groups

Exploratory study by students – formal definitions 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

D fi iti d th f ll d b 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 coursesg g

Train students to use tools so they solve real world blproblems

Stress understanding concepts rather than symbolic 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 mathcommunities. (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 inis 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 importantsolution…numerical techniques are more important now than analytical techniques.”

A physicist’s view

“What we want is for students to bring a basic gunderstanding of fundamental concepts of calculus into their physics courses…right now they are good at taking a derivative mechanically but have littleat 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 numericalstudents more and more are looking at numerical and graphical representations and less and less at symbolic representations …I tell my students that the

t j it f thi d t h l b ivast majority of things do not have algebraic representations …their calculus training is too loopsided in emphasizing symbol moving …we p p g y gexpect 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 deeperIt encourages them to interact at a much deeper level of involvement…it opens doorways to them.”

The new calculus student Understands visual representations

C f t bl ith t li ti Comfortable with computer applications

Understands concepts of precalculus and calculus Understands concepts of precalculus and calculus (functions, limits, continuity, derivative, integral)

Able to explore models and simulations

Calculus reform curricula –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 By the way: What is Calculus Reform?

Calculus: Catalyzing a National Community for Reform : Awards 1987-1995 by William E. Haver (Editor) National Science FoundationHaver (Editor), National Science Foundation

Total awarded: $44 million by NSF Total awarded: $44 million by NSF

Harvard Calculus Group Text: Single and Harvard Calculus Group Text: Single and Multivariate Calculus (Huges-Hallet, et al, 1995,1998,2001)

The Reform Movement (2008)S ( id l id d b di i l 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 st dents to e perience n merical and geometricstudents to experience numerical and geometric representations of concepts.

H H ll t t t (3 d diti ) h b i d t 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: AACETeaching. 18 (2), pp. 95-103. Charlottesville, VA: AACE.

BYU Outcomes StudyGPA and Calculus Preparation

Reform-TLC(Mathematica)

Reform-Harvard

Traditional(Mathematica) Harvard

Physics II 3.32 3.09 2.99

Physics I 2 97 2 93 2 91Physics I 2.97 2.93 2.91

Analysis 3.16 2.44 2.72

Adv Eng 2 93 2 86 2 96Adv Eng 2.93 2.86 2.96

Statistics 3.33 2.99 3.02

Calculus and Science & Engineering 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 Create positive peer learning communities through early engagement in structured, collaborative learning activities.

Mathematics like all fields has the 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 Tools abacus, calculator, sophisticated problem solving environments such as Maple & MatLab

History of Calculus y

Precise (ε-δ) Definition of Limit ( b li i l & hi l (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?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 -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 y y g p y y (level and quality of instruction. )

T l S i E l Taylor Series Example

Concepts/Skills/Tools Applications 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

E ti l G th/D ( di ti d ) 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 students learn in preCalculus and Calculus courses?

Principles

Methods

Practices

Tools