Instruction with instructions: teaching mathematics in a structured environment - my teaching phillosophy 

Lourdes Juan

Pow-Wow, the Indian Boy loves all the animals in the Wood

I strive to make every course I teach a valuable learning experience for my students. Lecturing to students of differing experience and capabilities in the same classroom can be challenging. I believe that we are better prepared to meet this challenge if we understand well the human as well as technical sides of teaching and learning mathematics. So I have taken active steps in this direction by exploring and comparing different ideas and teaching techniques.

An essential component of my teaching is the conveyance of my enthusiasm for the discipline. I make lectures interesting by explaining the intuitive ideas behind the technicalities. My students seem to appreciate it--- the enrollment in my classes is usually high and many enroll because of recommendations of my former students.

In all the undergraduate and graduate courses that I teach, I encourage my students to be critical, independent and creative.

Regarding my undergraduate teaching, my philosophy of ``instruction with instructions'' has proven quite successful in my classes. It is my impression that many students do not do well in mathematics because they don't know how to study it. So I do my best to provide structure for them. I explain to them that learning mathematics is somehow like learning how to drive: you don't try to memorize the solution to every homework problem you have done as you would not try to memorize which traffic signs are posted at every intersection of the town. In mathematics you learn some key things and ideas (as for driving you learn how to operate a vehicle and traffic rules) the rest comes with practice.

Structure is especially important for lower level undergraduates. To them I give detailed instructions for writing their homework emphasizing that they must provide sketches of graphs when appropriate, clearly indicate the formulas that they use as well as carefully write all the important steps of their solutions. For the courses that is available I use the homework program Webwork but also have the students submit hard copies of the solutions. The hard copies are only graded for presentation while the correctness is checked through Webwork and quizzes. As a result students write better exam solutions than they did in the past when I only had them focus on the correctness of the homework problems. The students can immediately see the connection between doing the homework with care and doing well in the course.

I am also a vigorous proponent of technologies that extend or enhance the delivery of the traditional mathematics course. I use internet technologies to expand the classroom in many ways. Through my website I distribute a variety of course information and materials, including links to websites that relate to the content of the course. I do this in a way that is appealing to the students and will stimulate their learning.

I use Maple when I teach honors sections of Calculus and Differential Equations and in the graduate course on Grobner bases.

Working with graduate students through courses and individual advising is also a very interesting and rewarding aspect of teaching at a Ph.D. granting institution like Texas Tech. I make my graduate courses a form of research experience. Depending on the level of the class I may even not adhere to a particular text but rather encourage the students to go to the library and try to find relevant literature. I also encourage the use of computer algebra systems such as Maple and Mathematica as a research aid. Through interaction in graduate courses the students can learn about my research and see the viability of a possible advisor-student relationship.