College Mathematics in the 21st Century

Google and other online graphers

2011-12-22T10:11:00.000-08:00

You can enter an expression like sin(x),cos(x) into the Google search box, and it will give you a graph that you can zoom in on and trace! Pretty simple - nothing to download! Can ditch the graphing calculator for just plain graphing.

You can also check out the Desmos online calculator, which has many features. I found out about it recently, from Maria Andersen's blog.

Math Jax

2011-11-20T17:15:00.000-08:00

MathJax is a web-based JavaScript application that you can use to include LaTeX in your web pages, like the following.
\[ \cos^2x+\sin^2x=1 \]
\[ \int x \, dx = \frac{x^2}{2}+C \]
No GIF's to export, no complex, additional markups to learn etc. It just works!
If you want to use it in Blogger, click here. Planning to use it to create more math pages with interactive features, using Wolfram Alpha widgets and GeoGebra.

Quick quizzes for fun

2011-06-17T17:26:00.000-07:00

A fun quiz to put on your web site can be created easily at xat.com. This one's for a mini review for calculus class I'm teaching in the summer.

Get Your Own Quiz!, More Quizzes

Some thoughts on math videos online

2011-04-24T14:13:00.000-07:00

By now, most folks in the math teaching world have heard of Khan Academy. Other free video sites for math include Math TV and Brightstorm. I do refer students to these sites for review and reinforcement. In YouTube, these types of instructional math videos get a large number of views, and so they obviously address a need out there.

However, I am not convinced that these types of videos actually illustrate concepts very well. There are now a wide variety of technologies that can illustrate math concepts and can be interactive. GeoGebra, graphing calculators, Microsoft Math, OCW applets in math (from MIT), and lots of applets spread throughout the Internet do a great job of illustrating ideas that are hard to do with just chalk and talk (or their video counterparts). But a simple survey of YouTube videos describing their capabilities showed that they attracted far fewer viewers.

Is it just that students are used to wanting something reinforced the same way they've seen it in their lectures? Do they just want to get their (online) homework done? Are these types of resources used often enough in classrooms to illustrate ideas?

I don't think any of these technologies takes away from basic skills. I think they make conceptual understanding easier for many students - if they are guided along those lines. Not everyone can make abstract connections solely on the basis of processing symbolic notation. If one wishes to progress to higher level math or use math in other subjects, it is necessary to be able to think about math in addition to the skills, whether at the beginning algebra level or at the calculus level.And many of these technologies that have been around for a while now can make that task a little simpler.

Wolfram|Alpha widgets for an online course

2011-04-09T05:44:00.000-07:00

I've blogged about using Wolfram|Alpha (W|A) widgets for general use. Since I teach an online intro stats course every year, I thought it be handy to embed the widgets right in the topics section within the course management system. W|A widgets can fetch some data distributions - it's a lot easier to insert these types things at point of use rather than to send students to web sites all over the Internet. Also, I do not have to type extra instructions so that students can obtain the proper data. Here is the one I plan to use for some health statistics. ( I noticed that the units on the plot for height and weight seem to be inconsistent with the text - puzzled by that one, since W|A is usually pretty accurate.)

Online LaTeX Editors

2011-02-26T07:53:00.000-08:00

Although my primary word processor is MS Word 2007 for class materials, I use LaTeX for assignments in my upper level courses. Many of the assignments I create are not too lengthy and so I started searching for a quick browser-based online LaTeX compiler. Here are three I am experimenting with. The first one is totally free. The other two offer a free limited version.

latexlab.org : works with your Google account sign-in, but it's not affiliated with Google. Advantage is that your tex files are in the Google docs folder. I found the previewer to be somewhat clunky and it seemed to run the slowest of the three I found. But it is free and has no restrictions.
scribtex : worked very quickly on my sample latex file and rendered a nice PDF file in a separate tab. The interface is very minimalist, though. The free account has restrictions, but will be fine for my use of creating assignments
verbosus: Has a nicer interface than scribtex, with the tex file and pdf file side by side, but a lot of the window's real estate was taken up with file and project windows. Maybe there is a way to customize it, but I didn't check it out. The free account has restrictions as well.

I think I'll work with latexlabs and scribtex for now. But I thought verbosus was worth checking out.

Elearning on a shoestring

2010-09-15T06:55:00.000-07:00

Once again, I'm teaching an online statistics class this Fall. While the online materials that come with the text are really superior, I still needed a platform for my virtual lectures every week. I have used VYEW in the past, but that has a maximum capacity of 20 students for the free version.

I an now using Wiziq, which allows many more students, even for the free version. The interface is much improved compared to what it was in the past. Since many of my students wanted the recordings of the lecture, I upgraded to their premium version (just $50/ year) to get the ability to store the recordings. And students don't have to register to view the lecture.

I have also used Jing in the past, but upgraded that one too (for a whopping $15/ year). Now, I can publish my videos directly to YouTube, which is a big plus for me.

Colleges without a large budget for online learning and/or just a few online courses can use tools like WizIq or VYEW as opposed to enterprise level software such as Elluminate or WebEx.

Wolfram Alpha Widgets

2010-07-30T07:25:00.000-07:00

You can now customize your Wolfram Alpha (W|A) queries using a W|A widget right on your web site. It's currently in the Beta stage. You'll see a widget,which I cooked up in a few minutes, below this post. I was able to control the output to just what I wanted. I did not need all sorts of complex number representations and other stuff that comes with the standard W|A query for a simple derivative. Also, with the widget, you don't leave the web site you're on. I plan to put some of these on my class web pages.

Here's another one.

Writing Math in Word 2007

2010-07-22T06:33:00.000-07:00

In the past, I have not required my students to typeset math in their homework. They had the option of using Equation Editor or doing it by hand within their word processed documents. I have also encouraged them to use Google Docs, which has a TeX-like interface for math equations.

I recently started using the MS Word 2007 equation editor, which is quite different from the MathType editor they had before. For one thing, you type everything in-line, not in a separate window. You can also use a TeX like syntax to build up your equations, although the point and click feature is what is on the menu. The syntax is not documented within Word 2007, though, and I suspect most people will simply use point and click. You can find more about the Unicode syntax for building up equations here. It's still a much better equation editor, and next time my students have an assignment using the word processor, I will require that they use some type of equation editor.

"No Excuses" Software for Math

2010-05-30T03:29:00.000-07:00

Just finished teaching a short course on software for use in math classes.It's a course required for math majors, and the objective is to familiarize students with the various types of software to do math.

Way back when, like ten years ago, math software was the domain of licensed software housed in college computer labs. The usual stuff happened - the computers got updated, but the software license did not get renewed. And excuses for not using the software abounded - you could only use the software in the labs, students couldn't afford the software for their home, etc. etc.

Fast forward to 2010. Students bring in their own laptop and WiFi is freely available. I built my software course around free or easily available software such as Wolfram|Alpha, Geogebra, and Excel. Of the three, Geogebra was quite a hit. Students could download the software for free on their own laptops or use the web based applet. And they were up and running without mastering any rarefied syntax.

Wolfram|Alpha was a close second. There was some frustration that the "natural language" interpretation of W|A was not so natural! Nevertheless, the students did find W|A to be a good tool for doing math.

Then there is the lowly spreadsheet. Found basically on every desktop, students were quite surprised you could do actual math in Excel. They used it to model data, forecast trends and examine limits and difference quotients.

I put all the course material on our campus Blackboard. I plan to transfer it to my Moodle account so that the course material is accessible to everyone.

Integrating math technology easily - no licenses, no fees, no excuses...

Proofs and Wolfram Alpha

2010-03-26T11:12:00.000-07:00

In my Introduction to Proofs course, I discussed the proof the following: The cube of an integer is of the form 9k, 9k+1, or 9k+8, for some integer k. The problem is from the text I use, How to Think Like a Mathematician: A Companion to Undergraduate Mathematics

The big idea here is to to note that any integer can be written as 9q+r, q some integer, r=0,1,2,..,8. Then simply find (9q+r)^3 and examine the form of the expression of the nine different cases for r. The algebra is a bit tedious - and so Wolfram Alpha comes to the rescue in the form of the command

expand((9q+r)^3) for r from 0 to 8

The output is here:

Note that the proof itself consists of the big idea of writing an integer as a multiple of 9 with a remainder. W|A simply did the grunge work for us of expanding the polynomials and substituting the values of r. In a course such as this, W|A can be a great timesaver in doing these types of calculations and students see the value of such software as efficient helpers in solving larger problems.

Teaching teachers to teach

2010-03-15T10:41:00.000-07:00

An article in last Sunday's New York Times discusses the complex issue of teacher preparation. The writer of the article includes a lengthy discussion of math teaching in particular. I found it to be quite interesting - the work of math educators doesn't usually make it to mainstream media. Hopefully the Times will continue its coverage of key education issues such as this one.

Getting students to be quantitatively literate

2010-03-05T10:46:00.000-08:00

One of the main aims in teaching a Math for Liberal Arts course is to get students to have a better appreciation of general mathematics. And what better way to do that than to have them read math articles or blogs aimed at a general audience? Sounds like a good plan, but I needed to make it a graded assignment so it would get done.

In my online course I am now making up questions related to articles in Steven Strogatz's NY Times blog. Steven Strogatz is a mathematics professor at Cornell University. He makes basic math so clear and brings in so many connections that anyone who reads it will get something out of it, no matter what their education level is. Using his post on algebra, I asked my students the following as part of a longer homework assignment:

Suppose the length of a hallway is y when measured in yards, and f when measured in feet. Write an equation that relates y to f.

I gave them the wrong answer of y=3f and asked them to figure out why it's incorrect. Not exactly rocket science, but having them read something about math other than what's in a textbook is a big thing for many of my students. When we get to the probability and stats part later in the semester, I plan to do something similar with the abcnews.com articles by John Allen Paulos, math professor at Temple University. These articles are not as direct as Strogatz's, but do aim at probabilistic understanding for a general audience.

Online math courses and academic integrity

2010-02-24T07:10:00.000-08:00

For the past three years, I have been teaching an online math course every semester. To ensure that the students taking the class are really the ones who signed up, I have always used in class midterm and final exams. I check their ID and my grading system system reflects a heavy weighting toward the in-class tests. I encourage students to use whatever resources are available, including Wolfram Alpha, to study and do their online homework. They know that the in class exams are closed book and so the outside resources mainly serve as a study guide. All information about my online classes are publicized well before the class starts, and so the students know what is expected.

I do wish there was a better way to monitor academic integrity other than in-class tests. A presentation by Judy Baker, Dean at Foothill College, CA, discusses many aspects of online learning and academic integrity.

Academic Integrity in Online Courses

View more presentations from Judy Baker.

I tend to agree with her that most students are honest. But the anonymity of an online course can certainly tempt students to compromise their sense of ethics, especially those who are math-phobic. High tech security measures are too cost-prohibitive to use in an academic online setting. As online programs and courses are expanding in colleges and universities, there certainly needs to be a discussion on how to check identities of the students in the classes, and to ensure that they are the ones doing the work. One suggestion is to require students to make a presentation using a webcam. Would a one-on-one short oral exam using something like vyew.com and a webcam work for math courses? It is worth thinking about. But I think that an online course with no face to face interaction, either virtual or real, is opening itself to all sorts of compromises.

Counterexamples in Calculus

2010-02-06T07:24:00.000-08:00

One way I motivate critical thinking in my Intro to Proofs class is by using counterexamples. The book, Using Counter-examples in Calculus by Mason and Klymchuk, provides an accessible set of ideas to think about. Producing counterexamples is an important step to thinking about proofs in general, especially for students who are used to computations. What I really liked about the book were the graphs that accompanied many of the counterexamples. Here is one statement to think about:

True or false: The tangent to a curve at a point is the line which touches the curve at that point but does not cross it there.

The book is actually intended for first year calculus students and could work in that context if the course emphasizes concepts along with the standard skill set.

And the counterexample for the statement quoted above is x cubed at the origin.

Are some online math homework systems frozen in time?

2010-01-22T09:09:00.000-08:00

I'm teaching an online math course this semester on liberal arts math. I taught it online three years ago, and since then, a lot of innovative technologies have sprung up on the web. However, when I opened up the learning management system (LMS) associated with the textbook, I didn't find any incorporation of any of the new technologies that could streamline my teaching. Rather than rant about its limitations, I have simply chosen to add on the new technologies on my own.

The main reasons for using the LMS from the textbook publisher are the ebook and algorithmic homework. But online learning ought to be more than just text and worksheets on the web! So here are a few of the free tools I'm using on top of the usual stuff you get with the LMS that comes with a textbook.

vyew.com This is a free online meeting tool where I schedule my "virtual lectures" with audio and Powerpoint slides. Students can interact via a chat column on the left of the screen. The students need not register, and the presenter registers for free. I used it during Fall 2009 and it worked quite well. By the way, check out Maria Andersen's tips for effective webinars .
Google docs Aside from the automated homework, I also have students hand in exercises that they have to write up. In the past, I had them email them to me or drop it in a "digital dropbox" within the LMS. Either way, I had to do the open-edit-save cycle for each document. With google docs, the student creates a file on the web and simply sends me the link as a collaborator. I just click, edit and save in the same location and I'm done! No attachments to email back and forth. This is the first semester I'm implementing this. Our university's email system is through Google, and so all students already have access to Google docs. Even otherwise, they can create one for free.
Wolfram Alpha This computational engine not only solves equations, but also has access to many data sets. I am planning to use it for students to make specific queries and write up the implications of what they find.

Hopefully, the publisher issued LMS's will evolve to include tools such as these. My virtual lectures especially have tended to create a lot of interaction among the students and myself. It can really help reduce a feeling of isolation in an online course.And Google docs and Wolfram Alpha are technologies our 21'st century students should become familiar with anyway.

The paradox of higher math standards in high school

2010-01-06T07:34:00.000-08:00

Those of us who regularly deal with entering college freshmen are all too familiar with their inadequate math preparation. But in fact, high school mathematics has been ramped up quite a bit in terms of content. What happened? An article in the American Physical Society discusses this paradox. The author of the article, Dr. Joseph Ganem, is a professor of physics at Loyola University, Maryland. He is also a parent of children in high school. He noticed that over the years, the math homework his children were assigned required his frequent intervention. Since this is a parent with a Ph.D. in physics, he had no trouble helping them. But what about the many other students who do not have such an educated parent or access to tutoring services? Sounds like many students are just muddling through their jr. high and high school math without retaining much of anything. He points to two major flaws in the high school curriculum:

1. Confusing difficulty with rigor. It appears to me that the creators of the grade school math curricula believe that “rigor” means pushing students to do ever more difficult problems at a younger age. It’s like teaching difficult concerti to novice musicians before they master the basics of their instruments. Rigor–defined by the dictionary in the context of mathematics as a “scrupulous or inflexible accuracy”–is best obtained by learning age-appropriate concepts and techniques. Attempting difficult problems without the proper foundation is actually an impediment to developing rigor. (emphasis mine)
[....]

2. Mistaking process for understanding. Just because a student can perform a technique that solves a difficult problem doesn’t mean that he or she understands the problem. ...

Pushing students too early to do algebra is not really the answer to our perennial problem of students being noncompetitive in the global mathematical landscape. We ought to invest in better teacher prep programs in colleges and provide better incentives for pursuing a career in K-12 teaching. And the math that is taught should be focused and connected, not just a drill based collection of disconnected topics, nor the latest reform program designed by math educators from the education college.

Efficient Math Commands for Wolfram Alpha

2009-12-17T08:27:00.000-08:00

Wolfram Alpha is a free, online, computational engine. It provides some of the power of a computational algebra system (CAS) such as Maple or Mathematica without having to learn the proper syntax. However, it takes a little bit of practice to get WA to give exactly what you want.

Here, I'll give a list of queries that are commonly used in math classes:

Solving an equation in one variable: solve 3x^2-1=0
Solving an equation for one variable in terms of another: solve 3x+2y=-8 for y
Factor an expression: factor (4x^2-16y^2)
Simplify an expression: simplify (x+3(x+2)^2-2(x-y))
Plot a function: plot x^3-x^2
Plotting a function on a given interval: plot sin(x), x=-pi..pi
Take the derivative of a function: derivative of (x^3+sin(x))
Higher derivatives: second derivative of (x^3+sin(x))
Higher derivatives, symbolic: d^2/dx^2(x^3+sin(x))
Integration: integrate (x^3+sin (x))
Definite Integrals: integrate (x^3+sin (x)), x=-2..3

There are many, many more cool things you can do with Wolfram Alpha and I have only scratched the surface here. But the commands above should take care of most of the solving equations and graphing chapters in algebra, and the differentiation and integration material in calculus. I've told my students about it so that they can check their homework.

One thing to note is that WA will often give both graphical and algebraic representations for solutions of equations. It really helps students to connect those concepts. However, it is not going to help a student set up and solve a word problem or help them to interpret a solution or a graph. If you're interested in teaching or learning both math concepts and skills, then WA is an extremely useful tool.

Free Multimedia Algebra Review

2009-12-02T08:43:00.000-08:00

There are seemingly an infinite number of free math resources on the Internet. However, very few provide the comprehensive content that are usually found in textbooks. One of these rare sites is Hippocampus. It is part of the Monterey Institute for Technology and Education and provides free multimedia ebooks for a variety of high school subjects and introductory college level courses. Of particular note are the algebra and calculus ebooks that they have on hippocampus.org. I found their online material to be engaging, and have directed my students to the site for review purposes. They have also added a new "mini-site" just for algebra review.

Instructors looking for really good online videos an a wide range of math topics can also check out Brightstorm and MathTV.

(Not affiliated with any of these folks - just passing along some good study aids.)

Research meets teaching

2009-11-24T19:25:00.000-08:00

When I was asked to give a talk on some aspect of the history of mathematics a couple of years ago at Suffolk County Community College, I tried to bridge my research and teaching interests by presenting on the history of numerical algorithms. My Ph.D. was in numerical analysis and this was an opportunity to use my research background to present how using technology in math requires a great deal of conceptual understanding. You can view the presentation below. (It's really not as boring as the title suggests!)

Numerical Algorithms

View more presentations from Reva Narasimhan.

Teaching Reading in a Math class?

2009-11-19T07:06:00.000-08:00

An often asked question from my students is "but this is a math class - and you want me to read and write?" When teaching my upper division Intro to Proofs class, I find a certain discomfort among students in extracting information from a math text. Most students are used to skimming over some examples and finding one that matches the homework problem. I don't really count that as "reading" - just a sort of search and replace operation. And so when we have to prove something - uh oh - the search and replace strategy no longer works.

I thought finding a readable text would be a solution. Well, a readable text is only good if it's read! So now I am finding myself teaching higher level reading skills and critical thinking skills. This is way tougher than teaching math. I've been looking at material from my college library on how to teach this type of reading. Here are some ideas from this literature I've adapted for college level math:

This is actually something I haven't seen in many intro to proofs books: Have students read and interpret lower level math material such as theorems from precalculus; if they don't understand how to read and interpret those, how can they understand a theorem in abstract algebra or real analysis?
Start the intro to proofs course with topics in discrete math and nonstandard problem solving to jump start their thinking skills. These problems are not easily amenable to the "search and replace" approach to math.
This is an old idea - an online reading quiz before class using Blackboard or some other LMS. You can also use Google forms very quickly for this.
Too late for this semester - but for the start of next semester I'm going to have the students do a "mind map" to help sketch out their proofs. There are several available on the web and Maria Andersen has information about how she uses mindmaps in her blog

If critical thinking and critical reading skills in mathematics were taught in K-12 and in the computational courses in college, I may not have this problem at such a late stage in an undergraduate math student's career. Or, at least, it would not be so severe.

Explorations with Geogebra

2009-10-27T09:29:00.000-07:00

I just started using GeoGebra, the open source dynamic geometry software, to create an exploration activity for my Intro to Proofs class. The activity itself was an extension of a discussion in class about the Fundamental Theorem of Algebra. It can also be used in a precalc class which stresses concepts.

(Click here to see in a larger window.)

Your browser does not support iframes.

I was surprised by how easy it was to create - no need to learn Java and the user interface for GeoGebra was very intuitive. I plan to do more with this software, given the speed with which I can make some very interesting acivities.

Not so Elementary Mathematics

2009-10-07T07:24:00.000-07:00

A thought provoking article on mathematics for elementary school teachers appeared in the recent issue of the American Educator. It was written by Dr. Hung-Hsi Wu, Professor Emeritus of Mathematics at UC- Berkeley. He writes that the basic operations of addition, subtraction, multiplication and division involve more conceptual processes than most people realize, and suggested that there should be separate fourth and fifth grade teachers in math.

Of note in his article is the very basic notion that mathematical thinking involves breaking up a complex task into several easier components. To add 15+16, it is conceptually easier to break up the numbers into 10's and 1's. Even if the rote algorithm is taught at some point, teachers definitely need to understand the ideas behind place value that are inherent in the standard algorithms for arithmetic. I mentioned to my students, in my graduate level math course for high school teachers, that 79*85 can be rewritten as 79(80+5), which leads to the distributive property used in algebra.

Many had never thought it about that way. They quickly pointed out that students are less likely to make errors when calculating 79(80+5) as opposed to the standard way of multiplying. Both methods require the same number of arithmetic operations. The partial products method simply requires a little more space. At any rate, it does require some organization of thought for the student, which is another aspect of mathematical thinking.

I should note that the same issue of American Educator also has an excellent article on the "science wars" and sheds some light why one needs both content and reasoning in science. That is, scientific reasoning cannot exist without content. Likewise, understanding mathematical concepts cannot exist without sound mathematical content.

GeoGebra

2009-10-03T15:50:00.000-07:00

My latest tool for interactive math is GeoGebra. Just started to explore it. A great resource outlining many possibilities is this wiki page by Dr. Linda Fahlberg-Stojanovska.

I'm looking forward to using GeoGebra in my math for elementary teachers course as well as my graduate course for high school math teachers.

Connecting concepts with Wolfram Alpha

2009-09-13T07:49:00.000-07:00

I posted earlier about using Wolfram Alpha to create worksheets with conceptual understanding. One of the most central ideas in college algebra and precalculus is the connection between zero, x-intercepts and factors of a polynomial. However, it is an idea that many students have trouble grasping. So I made a short worksheet using Wolfram Alpha that engages students in studying this connection more closely.

Wolfram Alpha

2009-07-17T04:01:00.000-07:00

There has already been a lot of buzz about the online computational engine called Wolfram Alpha (WA). You can check out this article in the Wall Street Journal, for instance.

My main interest in WA is to figure out how to integrate it into the classes that I teach, which range from developmental mathematics to graduate level numerical analysis.
In a summer 09 class I taught in Intermediate Algebra, I told all my students about WA and encouraged them to check their homework answers using it. Here's a worksheet on lines that I wrote using WA.

The answers are already given by WA - I'd like the students to figure out how they were arrived at. Also, WA automatically ties in the algebraic and graphical perspectives. This makes connecting concepts much easier. I plan to do more worksheets of this sort in the future for material in precalculus and calculus and will be posting them here.

Math in Art - kolams of South India

2009-04-23T06:26:00.001-07:00

Math and art is always a fascinating topic in a math for liberal arts class. To broaden the students' perspectives, one could include kolam designs. These are designs that women draw in front of their houses in southern India, especially in the state of Tamil Nadu. In northern India, a similar type of threshold design is called rangoli.

Kolams are made with dots and loops around the dots The classic kolams are done only in white rice powder. However, the colorful rangolis from North India have influenced many kolam designs as well. Click here for a lot of interesting material on kolams, including connections to computer algorithms and knot theory.

(Kolam Image Source - http://pudukkottai.org/archieve/pongal-2003/index.html)

Math in the Digital Age Presentation

2009-04-19T11:06:00.000-07:00

For those seeking to include modern applications in their math courses, here's my presentation that includes some examples.

Stats on dev math and college graduation

2009-04-15T11:16:00.001-07:00

I've been trying to find out for a while now about longitudinal studies on the graduation rates of students in developmental programs in college. The National Center of Education Statistics (NCES) has published a study which provides the data on that issue as well as many others. In fact you can create your own tables with their Quickstats feature. The study I looked at was the following.

Beginning Postsecondary Students Longitudinal Study (BPS), which follows first-time students beginning their postsecondary education, typically over a period of 6 years. The 1995-96 cohort was followed through 2001.

The data only talks about any remedial course, not math specifically. Nevertheless, one can see from the table that 65% of students who took any developmental course in 1995-1996 did not obtain any degree by 2001. Out of students not taking any developmental courses, 45% did not obtain any degree in that same time period.
So I do wonder if developmental courses in college are helpful only to some limited extent.

Embedding Documents

2009-03-30T16:52:00.000-07:00

By way of Kevin Jarret's blog on ed-tech, I'm trying our embedit.in, a web site that allows you to embed many types of documents in your web site. Here's a presentation I recently gave at IMACC 09, a conference of community college educators in Illinois.

Math using Google Earth

2009-03-20T09:07:00.000-07:00

From time to time, I teach a freshman level course on liberal arts math. In the section on similarity of figures, there are a few exercises about reading maps and estimating distances. My students saw this as a pointless exercise in the age of GPS and Mapquest. And my comments about being stuck with only a paper map in the boonies usually go nowhere.

Enter Google Earth and its "ruler" tool. I can have them look at a picture of the Pentagon, and with one side measured with the ruler tool, ask them to find its area by subdividing the pentagon into triangles. (Using technology always has the advantage of cranking up the conceptual level of a problem.) There are some interesting math lessons using Google Earth at www.realworldmath.org.

Some articles on faculty development

2009-03-17T19:05:00.001-07:00

The March 2009 issue of PRIMUS, a journal for undergrad math education, focuses on faculty development. The articles are an interesting read for those who want to learn more about student centered math classrooms and varying types of assessments.

Also, the January 2009 issue of PRIMUS has three articles on the use of wikis in math classes. The one on the use of wikis in a senior capstone course was authored by me and the other two articles are on the use of wikis in a general math course and in a real analysis course.

The formula that ate your 401(k)

2009-03-04T09:04:00.000-08:00

There have been quite a few articles recently discussing the mathematical models of the valuation of mortgage backed securities, and the house of cards that fell as a result. One appeared on WIRED and another in The New York Times .

So what's the take-away for college math education? Most of our math students are not in math intensive majors. However, majors in business and social sciences must have a very good understanding of the mathematics that they do use - typically, a subset of elementary statistics and elementary algebra. It is not clear to me that students are achieving a deeper understanding of these elementary concepts. They should at least understand that mathematical models are limited and there is much more to applying math than simply substituting variables into a range of formulas.

Bits and Bytes : Math of Image Restoration

2009-03-03T11:59:00.000-08:00

I've been looking for a simple introduction to image processing that would appeal to students with modest math backgrounds. Most image processing materials are written for electrical engineers and are way above the level of a typical non-math major.

So I was really glad to see this paper from the Electronic Proceedings of the Eighteenth Annual International Conference on Technology in Collegiate Mathematics, Orlando, Florida, March 16-19, 2006.

I'm planning to use it in my math for liberal arts course next time I teach it.

Bits and Bytes: Math Behind the Megapixel Myth

2009-03-02T17:58:00.000-08:00

Just about everyone seems enthralled by all things digital - iPods, digital cameras, and so on. But hardly anyone stops to think about the math that's behind the digital craze. About a year ago, I started introducing examples using references to digital items that students are familiar with. They perked up - "hey - this is something I can relate to..."

One of the really interesting examples I use is the "Myth of the Megapixels". Ask anyone about digital cameras and they're likely to tell you that more the megapixels, the better the camera. Well, this is not necessarily true. David Pogue, of The New York Times, posted an article about this. Here's a small excerpt:

Let me tease you first with this question: How much bigger can I print a 10-megapixel photo than a 5-megapixel photo?

Most people answer, “twice as big” or even “four times as big.”

People assume that the length and width of the picture will be doubled.
He shows the math in his article. The gist of the calculation is this: the megapixels refer to the number of pixels in the area of the picture. So even if the area (number of pixels) was doubled, the length and the width of the picture are not doubled. For if the length and the width of the picture were doubled, the area of the picture would be four times as much, not twice. Here are Pogue's calculations:

A 5-megapixel photo might measure 1944 x 2592 pixels. When printed at, say, 180 dots per inch, that’s about 11 by 14 inches.
A 10-megapixel photo (2736 x 3648 pixels), meanwhile, yields a 180-dpi print that’s about 15 by 20 inches—under three inches more on each margin!

The reaction from my students, you ask? "I'll remember that next time I go shopping."
I don't get that reaction when I teach a topic like completing the square!