Efficient Math Commands for Wolfram Alpha

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

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

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?

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

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.)




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

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

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

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

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

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

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

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

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

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

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)

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

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

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!