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.