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Catch up on the latest M-casts

Feb 24, 2012: Michael Bulmer, The Island: Letting Students Experiment and Collect Data, recording

Feb 10, 2012: Rebekah Isaak and Andrew Zieffler, Identifying Modeling Misconceptions recording and the Google document listing misconceptions.

Jan 29, 2012: Isabel Darcy, A Calculus-for-Biology Web Site recording and the BioCalc wiki page itself.

Nov 18, 2011: Rebekah Isaac and Andrew Zieffler, Developing a Concept Inventory for Modeling recording and slides

Nov 4, 2011: Danny Kaplan Calculus Operators in R recording and slides

Oct 21, 2011: Lisa Dierker An inquiry-based introductory statistics course recording and slides

Sept 30, 2011: Randall Pruim, A mosaic Sampler: A Package for R recording and slides Comments and Suggestions for the mosaic Package

Sept 23, 2011: Rachael Miller Neilen, Computational/Modeling Project recording and slides

Sept 16, 2011: Terry Speed. Simulation and Theory: recording and slides and link to IMS Bulletin article

July 15, 2011: WeBWorK Automated Homework recording and slides

July 29: New Program in Biomathematical Sciences recording and slides

Upcoming M-casts

January 27, 2012 A Calculus-for-Biology Web Page

Presenter: Isabel Darcy

Abstract: I will describe the creation of a webpage which will contain teaching material for calculus for biology where educators can contribute teaching modules, twitter questions, and other items of interest.

February 10, 2012 "Identifying Modeling Misconceptions"

Panelists: Rebekah Isaak, Andrew Zieffler, Daniel Kaplan

Time: 1pm CST (2pm EST, 11am Pacific)

Abstract: An on-line meeting to discuss student misconceptions about modeling.

December 2, 2011 Coupled oscillators: Joggers, fireflies, and finger coordination

Tanya Leise (Amherst College)

Originally broadcast: noon Eastern (11:00am Central, 9:00am Pacific) video and slides

Abstract: I'll describe a simple oscillator model that can describe the position of a jogger going around a circular track, a firefly blinking on and off, or the motion of a finger waving back and forth. We can add a second oscillator to model two joggers, two fireflies, or two fingers trying to coordinate their motion. I'll demonstrate how to use some basic analysis involving derivatives to analyze the various scenarios, emphasizing the role of derivative as rate of change.

to be scheduled Roundup of resampling methods: simple technology for introductory statistics

Nicholas Horton (Smith College)

Time: tbd

Abstract: Resampling based inference (bootstrapping and permutation testing) plays an increasingly important role in introductory statistics courses. But instructors and students have been hampered by the lack of easy to use interfaces to these approaches. In this M-cast, I will discuss the advantages and disadvantages of several implementations, then facilitate a discussion about the joys and challenges of teaching these concepts.

Digital Ink

Eric Marland and Erick Hofacker (tentative)

Time and Date: To be arranged.

M-CASTs Topics to be Scheduled

Modeling Trebuchets


What's an M-CAST?

M-Casts are 20-minute seminars broadcast over the Internet on the 2nd, 4th, and 5th Friday of each month. They are part of Project MOSAIC, an NSF-sponsored project to improve undergraduate STEM education by better integrating Modeling, Statistics, Computation, and Calculus. M-Casts are designed to provide a quick and easy way for educators to share ideas, get reactions from others, and form collaborations.

M-Casts are recorded and posted on the Internet soon after the event.

This document is part of an interactive MOSAIC Wiki, which provides a forum for sharing additional ideas and materials, for discussion, and for reports of experiences using the ideas presented in the M-Cast. Access to most of the Wiki requires a login account, available to educators by request. Contact Danny Kaplan for an account or more information.

If you have a MOSAIC login, click on the title of any M-CAST to access the Wiki pages for materials, discussion, etc.

Tuning in to an M-CAST: Instructions

M-CASTS are broadcast using the ReadyTalk computer conferencing system:

  1. Direct your browser to the ReadyTalk web server and enter access code 2923887. This provides the video component of the M-CAST.
  2. For audio, telephone 866-740-1260 and at the voice prompt, enter the same access code: 2923887.

The audio is two-way: a conference call that befits the seminar style of the event. To avoid excessive background noise, please use the mute feature of your phone until you want to speak.

Recordings of the M-CAST are posted soon after the event.

Contact for Questions

For questions about the M-CAST schedule, please contact the "convener" listed in the schedule.

Do you have an M-Cast to propose or a topic to request? Enter an abstract or a description of your request here .

For more detailed information on tuning in to M-Casts, see the instructions for participating in an M-Cast.

Fall 2011 Schedule

September 16, 2011 Simulation and theory: necessity is truly the mother of invention

Terry Speed (University of California at Berkeley, CA, and Walter & Eliza Hall Institute of Medical Research, Melbourne, Australia)

Time: 4:00pm Eastern (3:00pm Central, 1:00pm Pacific, 6:00am Eastern Australia Standard Time on 9/17)

Abstract: For most of my career, simulation was something researchers did when they couldn't do the math. Things started to change for me around 1984, when I learned about the bootstrap and the Gibbs sampler. More importantly, around then I learned (or tried) to view computers as tools for solving problems not solvable in any other way. It now seems to me that we are heading into an era when all statistical analysis can be done by simulation (though theory, and lots of it, will be needed as well). In this m-cast, I will review these transitions, ways that not paying attention to them has hurt me, and observations and speculations for the future. Link to ISM Bulletin Article

September 23, 2011 A Computational/Modeling Project using DEs and Data

Rachel Miller Neilen (Duquesne)

Originally presented: 1pm EDT recording and slides

Abstract: We will discuss a simple computational project that one can assign in an undergraduate calculus, differential equations or modeling course to introduce the basic concepts in mathematical modeling. We illustrate these concepts using a biological example in which an ordinary differential equation is used to describe fish growth. During exposure to low dissolved oxygen (DO), the instantaneous growth rate of the fish decreases. Students are asked to use data from published lab experiments to estimate parameters in the model, answer biological questions regarding various exposure scenarios using the model, and present their results in a lab report. Analyses and simulations may be carried out in Matlab, R, or Excel. Link to the project write up and the link to the Stierhoff, Targett, and Miller article from which the article is drawn.

September 30, 2011 Teaching statistics using the mosaic package in R and RStudio

Randall Pruim (Calvin College)

Originally presented 2:30pm Eastern recording and slides Comments and Suggestions for the mosaic Package

Abstract: The Project MOSAIC team has been working on an R package to simplify teaching statistics with R and RStudio. The mosaic package for R is now available on CRAN, so it's time to put it on public display.

This M-cast will demonstrate a few of the features implemented in this package, including

  • simplified statistical summary functions that accept formulas and generate cross tables
  • augmented statistical inference functions that provide easy syntax for common situations
  • augmented plotting functions
  • a simplified syntax for randomization methods using do()
  • examples taking advantage of the manipulate features of RStudio.

We will also invite discussion about other things that could be added in the future.

October 21, 2011 An inquiry-based introductory statistics course

Lisa Dierker (Wesleyan College)

recording and slides

Originally broadcast: 2:00pm Eastern (1:00pm Central, 11:00am Pacific)

Abstract: Statistical analysis plays a significant role across the sciences and is arguably the most salient point of intersection between diverse disciplines given that scientists constantly communicate information on varied topics through the common language of statistics. Despite its central importance however, the teaching of statistics is limited by numerous challenges that are not easily overcome with traditional pedagogical approaches. In this presentation, I will describe individual and synergistic components of a collaborative, project-based first course in data analysis and applied statistics that can serve large numbers of students and cross both divisional (natural and social sciences) and departmental (biology, chemistry, neuroscience, computer science, astronomy, earth and environmental science, social science) boundaries. The role of computation is key to facilitate analysis of these real-world multivariate datasets.

The goal is to increase the number of students exposed to modern statistical methods; to inspire students to pursue advanced course work and opportunities in statistics; and to increase interest in statistics among women and students from underrepresented groups.

November 4, 2011 Calculus Operators in the mosaic Package

Daniel Kaplan (Macalester College)

recording and slides

Originally broadcast 2:00pm Eastern (1:00pm Central, 11:00am Pacific)

Abstract: Last year we introduced R operators for differentiation and integration. These have been extended to provide support for an easier notation, multiple variables, and symbolic differentiation (to support high-order differentiation). An important issue is notation: how to make the link between the traditional algebraic notation students learn in high school and the notation used in computational work. We'll discuss the trade-offs between two approaches --- the close-to-traditional notation used, e.g., in Mathematica, and the R-function notation used in our first round of R calculus operators. The mosaic package uses a new, intermediate notation, e.g. sin(2*pi*x)*exp(k*t) ~ x & t. We'll report on the student experience with the system, which we will have been using at Macalester for three months.

November 18, 2011 Developing a Concept Inventory for Modeling

Rebekah Isaak and Andrew Zieffler, University of Minnesota

recording and slides

Originally broadcast: 1:30pm Eastern (12:30 Central, 10:30 Pacific)

Abstract: To teach modeling, you need to know what modeling is. To assess how well students are learning modeling, you need to be able to break down the overall framework into components around which assessment items can be designed. The main purpose of our M-cast is present the framework that we have been designing along with a process for developing assessment items. We're hoping to solicit feedback from the audience about the comprehensiveness of this framework, the ways that we've split it between statistical and mathematical concepts, and the process for developing the concept inventory instrument, starting with open-ended questions to identify student misconceptions and eventually translating this to a multiple choice format. In the end, we hope to have an instrument that can guide the instructor in covering modeling in a comprehensive way.

Spring 2011 Schedule

Feb 11 and March 4, 2011 Teaching about Units in Calculus

Time: 2pm EST, 1pm CST, 11am PST

Dan Flath and Danny Kaplan

Most students are introduced to "units" in elementary school: liters and pints and meters and miles. That's often as far as it goes. We think this is a mistake, and that units should be an important component of modeling and calculus courses. In this two-part M-CAST, we will show how we introduce the concept of dimension to our students in Applied Calculus and then build on it as we teach modeling and calculus. Far from being peripheral to calculus, units and dimensions illuminate and make concrete the fundamental operations of calculus.

  • Part I --- presented on Feb 11: Video recording and slides
    • Basics about units and dimensions
    • Arithmetic and units
    • Connecting units to calculus
  • Part II --- presented on March 4: Video recoding and slides
    • Unit conversions
    • Simple dimensional analysis
    • Units in economics, biology, etc.
    • Description of Homework Exercises

March 18, 2011, The Notebook Interface to Sage Mathematics Software

Time: 12:30 EDT (11:30 CDT, 9:30 PDT)

Karl-Dieter Crisman, Gordon College Video recording and the Sage notebook.

This introduction to Sage will be about the big picture: the range of functionality, what the notebook can do for collaboration and students, etc. I'll demonstrate some basic commands, show a few servers, and how we have used Sage pedagogically. I'll also show how to use R in the notebook and how to publish worksheets.

convener: Randall Pruim

March 25, 2011 Toward a MOSAIC R Package for calculus and statistics

Video of M-CAST

Time: 1:00pm EDT (12:00 CST, 10:00 PST)

Randall Pruim

A team of Project MOSAIC collaborators, led by Randall Pruim, are developing an R package with data and utilities for Calculus and Statistics. Come see a demonstration of some of the features currently implemented and join a discussion of potential features to add. Some of the topics on our agenda:

  • Functional derivatives and anti-derivatives
  • Simplifying permutation and resampling methods
  • Additional (easy) graphical tools
  • Data sets for teaching statistical methods
  • Interactive graphics for teaching (using RStudio's manipulate())

For more details about our plans, see the mosaic R packages wiki.

The mosaic package is already available at R-forge or from R using the command

install.packages("mosaic", repos="")

We anticipate submission to CRAN later this spring.

Interesting in keeping the conversation going? Join the mailing list by subscribing subscribing here.

April 8, 2011 Teaching Modeling with Mosquito-Borne Disease Epidemics

video recording of presentation and slides

Time: Originally presented 1pm EDT, noon CDT, 10am PDT

Jeff Knisley, Eastern Tennessee State University

The Symbiosis project is a Howard Hughes Medical Institute funded curriculum that teaches calculus and elementary statistics to biology and mathematics students though an emphasis on modeling within biological contexts. In particular, there are many biological questions that can be explored via the modeling and statistical analysis of simulations. In this M-Cast, we will introduce the idea of simulation-based modeling and statistics as a means of educating first year students in mathematics, statistics, and biology. Our focus will be epidemiological with a focus on Mosquito-borne epidemics.

April 15, 2011 Dimensional Analysis and Atomic Bombs

Originally broadcast at 2pm EDT (1pm CDT, 11am PDT)

recording and slides

The Buckingham Pi Theorem is a tool from dimensional analysis that can be used to deduce how natural laws depend on parameters even when the governing equations are not known. Many calculus students find this to be a surprising and powerful way to apply their knowledge of units and dimensions. I'll discuss the basics of the Buckingham Pi Theorem, show one or two example problems, and describe a project that I assign in my Applied Calculus courses. In this project, students re-create the historic work of physicist G.I. Taylor, who figured out the energy of the United States' first atom bomb test explosion -- a top secret piece of information -- using pictures of the explosion that were published in the press.

Chad Topaz

April 22, 2011, Modeling in Sage: Love, War, and Zombies

Originally presented 1pm EDT (12 CDT, 10 PDT) recording and slides

David Joyner, US Naval Academy

Systems of differential equations can be used to mathematically model the weather, electrical networks, spread of infectious diseases, conventional battles, populations of competing species, and, yes, zombie attacks.

This talk looks at some of these models from the computational perspective using the mathematical software Sage.

April 29, 2011, A panel discussion on modeling-based calculus

Originally presented 2pm EDT recording and slides

A discussion involving several faculty from several types of colleges and universities who actively use modeling to teach calculus and/or calculus to teach modeling. We'll talk about models that we use to motivate or inform calculus topics, ways in which the operations of calculus are important for the modeling process (it's not just that calculus is used to solve models --- it's an important language for framing models), and modeling topics that are linked to calculus concepts.


  • Mariah Birgen, Wartburg College
  • Laurie Heyer, Davidson College
  • Joe Mahaffy, San Diego State University

Moderator: Daniel Kaplan, Macalester College

May 6, 2011 Reading and Writing about Derivatives

Time: 2pm EDT (1pm CDT, 11am PDT) video and the document from the M-CAST

Sommer Gentry, US Naval Academy

Phrases like "open-ended problems" or "word problems" are often about modeling: translating from an English-language understanding of the situation into a mathematical formulation. I'll describe some of the exercises I use in my calculus courses to get students to interpret newspaper reports in terms of the underlying mathematical concepts and vice versa, translating mathematical concepts into English-language descriptions that convey the mathematical ideas in a faithful way.

An example of such an exercise and the Instructor's notes

Moderator: Daniel Kaplan, Macalester College

May 13, 2011 Using active learning to teach math methods

recording and slides

Originally presented: 1pm EDT (noon CDT, 10:00am PDT)

Gary Felder (Smith College)

Many physics and engineering curricula gather many math topics together in one "math methods" course. Students in these courses are asked to rapidly learn a wide variety of challenging math subjects with little motivation or context and then to recall them when they are needed for a later course. The results, as anyone who has taught these later courses knows, are not always impressive. One tool I use to address this problem is "exercises" that have the students work through a series of problems before they have learned the material in a given unit. Some of these exercises are designed to show the students why certain physical problems require particular math techniques. For example I have them set up the differential equation for a nucleus in a crystal and show that it can't be solved analytically, and then I give them an approximate equation for the force on the nucleus, have them show that it works as a good approximation to the true force for small displacements, and have them solve the resulting simple harmonic oscillator equation. This example motivates the unit on Taylor series, where I tell them that they will learn how to derive such approximations. Other exercises are designed to have them work out key math concepts on their own, such as solving an ODE using separation of variables. My expectation, which I plan to test in an upcoming NSF-funded research study, is that the use of these exercises will increase student learning during the course and retention in later courses.

Two supporting documents from the M-CAST: an exercise on vibrations in crystals and an exercise on separation of variables and

July 15, 2011 Using Webwork for activities and homework assignments

Joe Mahaffy, San Diego State University

Originally broadcast 11am PDT (2:00 EDT, 1:00 CDT) Video/audio recording and slides

Webwork is a system support by the Mathematical Association of America that allows instructors to post on-line activities and homeworks for their students that can be automatically graded, and parameters can be set to allow retries. There is a very large collection of existing problems, and instructors can write new problems in a language that closely resembles Latex with some PERL inserts. It's easy to learn by drawing on the already published problems. The webinar will introduce some examples of Webwork problems, and show how to get started writing your own. Indications are that using Webwork substantially increases students' engagement and participation in their mathematics courses.

July 29, 2011 New academic program in Biomathematical Sciences

Originally broadcast 1:00pm EDT (noon CDT, 10:00am PDT) Video/audio recording and slides

Christophe Golé (Smith College)

Abstract: We started a "Concentration in the Biomathematical Sciences" [1] this year at Smith College. Concentrations are neither majors nor majors, but seek to integrate practical experience for undergraduate students with the curriculum. I will talk about some of the curricular issues arising with the concentration, especially as they pertain to the agenda of the MOSAIC Project. I will also talk about some of the administrative growth pains. Finally I will also touch on the 4 College Biomath Consortium that the concentration is helping us move into existence.

Fall 2010 Schedule

August 13: Daniel Kaplan, Finite-Differences as Derivatives

Originally broadcast at 1:00 CDT (2:00 Eastern, 11:00 Pacific)

Video/audio recording and slides

Abstract: Calculus students encounter differentiation as a set of rules for symbolic manipulation, x^2 => 2 x or sine => cosine and so on. I'll show a computer operator for differentiation that can be written from scratch in about 30 seconds in languages such as Matlab, Mathematica, R, etc. It implements a finite-difference version of the derivative, taking a function as an input and returning a function as an output. What's the point? Why use finite differences rather than the real derivative?

  • The computer notation emphasizes that differentiation is an operation on a function and produces another "derived" function.
  • The link between derivatives and slopes is explicit, rather than implicit as in the symbolic transformation rules.
  • Students can investigate the consequences of NOT taking a limit, which leads naturally to the idea of examining how good an approximation is.
  • Students can explore higher-order derivatives from the beginning, without having to master symbolic manipulation first.

Convener: Nick Horton

Additional Resources

August 27: Eric Marland, Animating Bifurcations

Time: Noon CDT (1:00 Eastern, 10:00 Pacific)

Video/audio recording, slides, and the student video shown briefly in the M-CAST.

Abstract: A project for an introductory course in differential equations using Maple. Students learn how to create animations, then investigate bifurcations by creating animations of the phase plane. Not only do the students become more adept with the software, but they begin to understand, by actually "seeing", the difference between quantitative and qualitative changes to the solution to a differential equation.

I will explain the project, share several options for coding, and share some of the students' work. One group even set their animation to music!

Convener: Danny Kaplan

September 10: Vittorio Addona, Helping Students Understand Regression Coefficients: An Example of Modeling Body Fat Percentage

Video recording and Slides

Originally Broadcast: 11:30 CDT (12:30 Eastern, 9:30 Pacific)

Abstract: Once students have understood univariate regression models, the move to multivariate models can be rather straightforward, if the explanatory variables are uncorrelated. Quite commonly, however, it is important to interpret coefficients in a multivariate model as partial changes in the response variable, holding the other explanatory variables fixed. This language is hard for most students to understand without a specific context. We discuss a data set (freely available on StatLib) which has successfully served as a tool for aiding students comprehension of multiple regression coefficients. The data deals with measurements of male body fat percentages, and we seek to build models for this response variable using more easily measured quantities. At Macalester College, introductory students learn about multiple regression in their first statistics course. Many other institutions do not broach this material until a second course. This talk will be helpful regardless of when students first encounter multiple regression.

Convener: Danny Kaplan

September 24: Nicholas Horton, Being Warren Buffett: a classroom and computer simulation of financial risk

Time: 11:30 CDT (12:30 Eastern, 9:30 Pacific)

Video/audio recording and slides


Students have a hard time making the connection between variance and risk. To convey the connection, Foster and Stine (Being Warren Buffett: A Classroom Simulation of Risk and Wealth when Investing in the Stock Market [2] The American Statistician, 2006, 60:53-60) developed a classroom simulation. In the simulation, groups of students roll three colored dice that determine the success of three "investments". The simulated investments behave quite differently. The value of one remains almost constant, another drifts slowly upward, and the third climbs to extremes or plummets. As the simulation proceeds, some groups have great success with this last investment--they become the "Warren Buffetts" of the class. For most groups, however, this last investment leads to ruin because of variance in its returns. The marked difference in outcomes shows students how hard it is to separate luck from skill. The simulation also demonstrates how portfolios (weighted combinations of investments) reduce the variance. In the simulation, a mixture of two poor investments is surprisingly good.

In this M-CAST, the activity and a computer simulation will be briefly demonstrated, followed by a discussion of goals, context, background materials, class handouts, and references.

Convener: Danny Kaplan

October 8: Randall Pruim, Golfballs in the Yard: An Introduction to Hypothesis Tests

Video/audio recording and slides

Abstract: One challenge in any introductory statistics course is helping our students understand the logic of hypothesis testing. In this M-Cast I'll demonstrate one of my favorite examples for doing this. The data are a sample of golfballs. The hypothesis is that the number on the ball is equally likely to be a 1, 2 ,3 or 4. Using a function written in R, I allow students to design their own test statistic and then produce a graphical display of the sampling distribution and calculate empirical p-values.

Convener: Danny Kaplan

October 22: Eric Cytrynbaum, Science One - a multidisciplinary first year Science program at UBC

Time: 1pm Eastern (12 noon Central, 10am Pacific)

Video recording and Slides

Abstract: In this webinar, I will introduce a novel first year Science program at UBC. The program admits a select group of ~75 students and consists of a single multidisciplinary course that covers all of their first year Science requirements (Mathematics, Physics, Chemistry and Biology). I will give an overview of how we organize the course with an emphasis on the role of Modeling, Statistics, Computation and Calculus in the program.

November 12: Eric Marland, Using Difference Equations for Introducing Derivatives

Time: 1pm Eastern (12 noon Central, 10am Pacific)


Abstract: Over the past decade many calculus texts in the life sciences have introduced discrete modeling in the early portion of the course. Why hasn't this caught on in the standard calculus course? Should it? What are the advantages or disadvantages. In this M-Cast, I want to discuss these issues and provide some ideas on how it might be accomplished successfully.

Convener: Nick Horton

December 3: Dan Flath, Using the Wiki to Share Curricular Ideas

Link to the recording

Before the conversation, we'll have a tour, showing how the kind of email conversations and in-the-hall conversations we always have can be easily and usefully posted, showing how to assemble example banks (as for contour diagrams), and how to do day-to-day postings based on experience in class. The wiki becomes more useful when everyone joins in - don't be shy! This is not intellectual judo. No one is going to criticize. We all want help in designing and teaching our classes and are willing to pay for it by offering our experience in return. Community can be a pleasure.

December 10: Danny Kaplan The antiD Operator

Time: 12:30 Central (1:30 Eastern, 10:30 Pacific)

video and slides.

One of the most beautiful notations in mathematics is the sweeping S-curve of the integral sign. But carrying out the operation itself is challenging. There are substantially different procedures for similar-looking functions, the results are often elaborate and complicated, and many basic functions don't have any integral that can be written with elementary functions.

Integration is an important technique in many branches of science and technology. To make it accessible to students, particularly when introducing it in a first calculus course, it would be nice to be able side-step the algebraic difficulties of integration, presenting it as a unified technique that can be explored and applied.

I'll present a way to do this, a computer operator that I call antiD. It carries out integrals quantitatively, returning an anti-derivative function. It involves hardly any algebra. Since the output of antiD is a function, you can differentiate it, you can integrate it, you can evaluate it. It works with just about any function you like, not just the standard forms of traditional calculus, but splines and other piecewise-defined functions, sigmoidals, the gaussian, ... whatever.

For the Future

Pending M-CASTS not yet scheduled.

TBD, Vases and calculus (TBD)

This project uses the fundamental concepts of calculus in a practical way. Students fill up a vase 1/4 cup at a time and record the height of the liquid at each step. They are then asked, from this data, to reconstruct the shape of the vase. Their analysis relates the different functions obtained with concepts of calculus, using ideas and tools developed in the Mathematica tutorial MathematicaIntro. As students go through the project, they become aware of ways that derivatives and integrals relate to the activity. Students are asked to formalize these relationships at the end of the project.

Jeff Knisley, ETSU

The Quantitative Modeling Track of the Math Major

Michael Bulmer

The Island: A detailed simulation of life, love, death, and disease (among other things) on an Island of population 20,000.

Karl-Dieter Crisman

The Interact facility in SAGE and its uses in introductory calculus.

A Conversation: Engaging Students in Calculus

A discussion. How do we engage students in the first week of a calculus course. Bring your ideas and let's share them. Some of the ideas:

  • Billiards as a canonical calculus problem.
  • Developing the idea of divide and conquer.
  • Simple models to start students off.