A Common Vision for the Undergraduate Math Program

in 2025: Our RoleFriday

2:00 – 2:50S112

Presenters today include:

Julie Phelps

Rob Kimball

Rob Farinelli

A Common Vision for the Undergraduate Math Program in 2025: Our Role

A Common Vision for the Undergraduate Math Program

in 2025: Our Role

Goals of this session:

Encourage you to take steps to modernize the undergraduate mathematics curriculum.

Share the COMMON VISION.

Why do you teach the content you

now teach?

A Common VisionDiscrete Time Difference Equations

Recurrence Relations

=

𝑁𝑡+ 1=𝑁𝑡 exp [𝑟 (1−𝑁 𝑡

𝐾)]

Biological SciencesComputer Science

Economics

A Common Vision

In 1202 the Italian Fibonacci published Liber Abaci which

contained the difference equation that models the

population growth of rabbits.

𝑥𝑛=𝑥𝑛−1+𝑥𝑛−2

A Common Vision for the Undergraduate Math Program

in 2025

Funded by the National Science Foundation.

PI: Karen Saxe

Co-PI: Linda Braddy

Web: www.maa.org/faculty-and-departments/common-vision

The Leadership TeamKaren, Linda, John Bailer, Rob Farinelli, Tara Holm, Vilma Mesa, Uri Treisman, Peter Turner

A Common Vision for the Undergraduate Math Program

in 2025

A Common Vision

The primary goal is to develop a SHARED VISION in the mathematical sciences community of the need to modernize the undergraduate mathematics program, especially the first two years.

It is time for collective action to coordinate existing and future efforts … to improve undergraduate education in the mathematical sciences.

A Common VisionThe crisis in mathematical sciences education is well documented in high-profile reports such as the U.S. government’s PCAST report on STEM education and the National Academies’ report on The Mathematical Sciences in 2025.

• Committee on the Undergraduate Program in Mathematics Curriculum Guide

• Modeling Across the Curriculum• Undergraduate Degree Programs in Applied Mathematics• Partner Discipline Recommendations for Introductory College Mathematics• Beyond Crossroads• Guidelines for Undergraduate Programs in Statistical Science• Guidelines for Assessment and Instruction in Statistics Education

A Common Vision“One of the most striking findings is that all seven guides emphasize this point, in particular:

The status quo is unacceptable.”

• Committee on the Undergraduate Program in Mathematics Curriculum Guide

• Modeling Across the Curriculum• Undergraduate Degree Programs in Applied Mathematics• Partner Discipline Recommendations for Introductory College Mathematics• Beyond Crossroads• Guidelines for Undergraduate Programs in Statistical Science• Guidelines for Assessment and Instruction in Statistics Education

A Common VisionCommon themes in all the guides:• Curriculum• More pathways• Modeling and computation• Connections to other disciplines• Communication• Transitions

• Course Structure• Pedagogy• Technology

• Workforce Preparation• Faculty Development and Support

A Common VisionPredictive Analytics

Predictive analytics is the practice of extracting information from existing data sets in order to determine patterns and predict future outcomes and trends.

A Common Vision

Predictive Models

Descriptive Models

Decision Models

Analytical Customer Relationship Management (CRM) is a frequent

commercial application of Predictive Analysis.

A Common Vision

Predictive Models

Descriptive Models

Decision Models

Predictive analytics can help analyze customers' spending, usage and other

behavior, leading to efficient cross sales, or selling additional products to

current customers

A Common VisionFORBES: $16.1 Billion Big Data Market: 2014

Predictions From IDC And IIA

• Growing 6 times faster than IT market• Employees work in teams and establish

best practices using operationalizing and management models

• Talent Shortage: over 100 programs at universities exist where analytics

and data sciences “are in focus”

A Common VisionGeospatial Analytics

See patterns and trends in a recognizable geographic context, so they're easier to understand and act upon.• Anticipate and prepare for possible changes due to

changing spatial conditions or location-based events.• Develop targeted solutions to business challenges that

may require different responses in different locations.

A Common Vision

Geospatial Analytics

Each red dot represents at least one student and is denoted by a POINT in the student layer. The student layer contains geographic as well as academic data about each of the 17,572 K-12 students.

Queries Into Big Data

- Attribute Table

ID School Grade Street # Street FRL AG EOG

135233 East 3 132 Adam 0 0 4

324133 West 11 4234 Booker 0 1 3

324112 West 9 324 Carlton 1 0 3

241553 East 4 144 David 1 1 2

155001 South 6 23 Eden 1 1 1

242113 South 12 42 Frank 0 0 5

Grade<9 AND GRADE>5 AND (FRL=1 OR AG=1)

Queries Into Big Data

- Visual Representations of Data

Median Income (ACS 2011)

VISUALLY

REPRESENTING

DATA

Density of K-12 Student Population

A Common VisionReport to the PresidentEngage to Excel: Producing one million additional college graduates with degrees in science, technology, engineering, and mathematics (February 2012)

Three imperatives underpin this report:• Improve the first two years of STEM education in

college• Provide all student with the tools to excel.• Diversify pathways to STEM degrees

www.whitehouse.gov

A Common VisionThe Mathematical Sciences in 2025

www.nap.edu

A Common VisionChapter 2 - VITALITY OF THE MATHEMATICAL SCIENCESThe Topology of Three-Dimensional SpacesUncertainty QuantificationThe Mathematical Sciences and Social NetworksThe Protein-Folding ProblemThe Fundamental LemmaPrimes in Arithmetic ProgressionHierarchical ModelingAlgorithms and ComplexityInverse Problems: Visibility and InvisibilityThe Interplay of Geometry and Theoretical PhysicsNew Frontiers in Statistical InferenceEconomics and Business: Mechanism DesignMathematical Sciences and MedicineCompressed Sensing

A Common Vision

Chapter 3 - CONNECTIONS BETWEEN THE MATHEMATICALSCIENCES AND OTHER FIELDS

Chapter 4 - IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES

A Common VisionResearch in the Mathematical Sciences is on a ROLL

The vitality of the U.S. mathematical sciences enterprise is excellent.

<Mark Green>

A Common VisionMarket Basket Analysis (Recommender Theory)

How Does Amazon Know What You Want?

A Common VisionMarket Basket Analysis (Recommender Theory)

How Does Amazon Know What You Want?

A Common VisionMulti-objective Optimization

Optimization is an important mathematical topic. But, let's be honest; many of the problems we assign in a calculus text are based on what someone else is doing; e.g., find the optimal amount of a product to produce.

Many real scenarios are based on what others are doing or producing. This slides into a "multiple optimization" problem where what I do influences what you have to optimize and what you do determines what I have to optimize. <Don Saari>

A Common VisionMulti-objective Optimization

Multi-objective optimization involves minimizing or maximizing multiple objective functions subject to a set of constraints. Example problems include analyzing design tradeoffs, selecting optimal product or process designs, or any other application where you need an optimal solution with tradeoffs between two or more conflicting objectives.

A Common VisionDesign DecisionsAspect RatioDihedral AngleVertical Tail AreaEngine ThrustSkiin Thickness# of EnginesFuselage SplicesSuspension PointsLocation of Mission ComputerAccess Door Location

F/A-18 Aircraft

ObjectivesSpeedRange

Payload CapabilityRadar Cross Section

Stall SpeedStowed VolumeAcquisition CostCost/Flight Hour

Engine Swap TimeAssembly Hours

MAX

MIN

Simulations – when the lab isn’t big enough

A Common VisionThe Mathematical Sciences are being used everywhere

Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and much more.

A Common VisionUseful mathematics in R & D

• Modeling and Simulation• Mathematical Formulation of Problems• Algorithm and Software Development• Problem-Solving• Statistical Analysis• Verifying Correctness• Analysis of Accuracy and Reliability• Explaining and modeling complex

phenomena with mathematics

A Common VisionTwo major drivers in the expanding role of the mathematical sciences:

• The ubiquity of computations simulations – they build on concepts and tools from the mathematical sciences

• Exponential increases in the amount of data available for many enterprises

The Internet, which both creates data and makes large quantities of data readily available, has magnified the impact of these drivers.

A Common VisionThe changing role of mathematics impacts what mathematics/statistics students need.

• Motivate math by how it is used…in other disciplines• Incorporate multiple modes of mathematical thinking• Provide new entry-points and new pathways• Partner with other disciplines to create a compelling

menu of lower-division courses• Diversify teaching methods, engage with online

education• Share in a community-wide effort to bring successful

experiments to scale

A Common VisionEvolutionary Game Theory

Evolutionary game theory originated as an application of the mathematical theory of games to biological contexts, arising from the realization that frequency dependent fitness introduces a strategic aspect to evolution.

Recently, however, evolutionary game theory has become of increased interest to economists, sociologists, and anthropologists--and social scientists in general--as well as philosophers.

A Common Vision

“The real value of de Mesquita’s” work is not only

in his predicting how a corporate event might unfold.

It is also in figuring out how to influence that event.

New York Times article about Bruce Bueno de Mesquita – one of the world’s most prominent applied game theorists.

What would it take to modernize / update the

undergraduate mathematics curriculum?

From the INGenIOuS Report:

We acknowledge that changing established practices can be difficult and painful. Changing cultures of departments, institutions, and organizations can be even harder. But there is reason for optimism.

WAKE UPChange is unquestionably coming to

lower- division undergraduate mathematics, and it is incumbent upon

the mathematical sciences community to ensure that it is at the center of these

changes and not at the periphery.

VISION is a community-wide effort to cooperatively spark improvement.

Our Role AMATYC

• Inform members about the VISION project through • Affiliates• Publications• Academic Committees

• The recommendations of the Vision report will influence the• Update of CROSSROADS• New AMATYC Strategic Plan

Our Role You• Read the publications mentioned in

this presentation• Keep up with the VISION Initiative• Rethink the curriculum• Rethink how you teach• Work with colleagues (in your

school and/or affiliate) to take steps to modernize the curriculum

Thanks

A thank you to Ben Braun, Mark Green, Ron Rosier, and Don Saari for helping with materials for this presentation.

A Common Vision

www.maa.org/faculty-and-departments/common-vision

TOPICS THAT DESERVE MORE ATTENTION

• Predictive Analytics– Data analysis– Modeling and Simulation– Algorithms

• Recursive Equations• Probabilistic and Statistical Thinking• Optimizing• Game Theory• Topology• Graph Theory

A Common VisionAlgebraic Topology

Aspects of algebraic topology (e.g., index theorem, fixed point theorems) are often used in economics (for instance, to prove the existence of equilibria) and they are beginning to be used in psychology. Indeed, part of Kenneth Arrow's Nobel in Economics was based on his using fixed point theorems to prove the existence of a price equilibrium. He used the techniques developed by John Nash to prove the existence of equilibria in game theory.

A Common VisionGraph Theory and Network Theory

Graph theory and network theory is a fast growing area in sociology, management sciences, and other social sciences. It is even being used to understand connections between crimes and trends in terrorism.

In fact, a recently developed procedure currently places about a million students each year into the appropriate math class: ALEKS. ALEKS was discovered and developed by a mathematical psychologist

A Common VisionFunctional Equations: F(g(x), x) = g(f(x))

Functional equations are very big in cognitive psychology. Experiments with human subjects define the functional relations; the mathematics of functional equations define the functions and resulting theory. A sizable portion of Duncan Luce's work depends on this approach.