"…we must have improved methods of teaching, better textbooks and more good treatises on advanced subjects "[1]

As a college lecturer teaching Mathematics to students in Primary Education, Secondary Education and a Natural Science ASc. Programme, I face a dilemma.  My education students are told about and encouraged to use learner-centred teaching methods. I expect my student teachers to guide their students in discovering mathematics by way of hands-on activities, cooperative learning, group assignments and class discussion.  Having set this standard for my students, are they allowed to expect the same of me?  

To create a learner-centred environment I will have to find activities which provide opportunities for exploring abstract concepts while imparting sufficient knowledge of the subject.  I will be constrained by time and the need to cover a syllabus heavy in content.  The challenge is to achieve the following objectives in a balanced way.

! Provide students with sufficient knowledge to competently teach the subject, pursue further studies or apply mathematics in the workplace.

!  Provide experiences which allow students to explore, make connections between concepts and related topics, as well as develop an appreciation for the relevance of mathematics.

Almost 100 years ago Fiske [1] addressed members of the American Mathematical Society on the topic Mathematical Progress in America.  He called for improved methods of teaching, better resources for advanced subjects and closed with this advice:

“…increase and improve the opportunities offered those interested in mathematics to meet one another for the purpose of exchanging their views upon mathematical topics.”

Recently Artigue [2] presented a European perspective in her article The Teaching and Learning of Mathematics at the University Level.  She indicates why the challenge is yet to be met after so many years.

 “…although the benchmarks acquired by research allow one to understand better the difficulties that students have, as well as the dysfunctions of our teaching, they more rarely give us inexpensive means of action to make teaching better in an immediate and sensible way.”

Discouraged by the lack of progress, it may seem wise to simply prepare a lecture.  However, before giving up efforts to provide better teaching it may be worthwhile to look at what others are doing and consider what more may be done.

Artigue discusses research, done in a number of countries to measure the ability of university students to determine whether integral calculus may be applied in solving a problem of modelling.  The results of this research present a bleak picture.  Fortunately, she continues, with optimism to describe research which involves asking students to calculate the gravitational force between a linear bar and a point mass.  Invariably, it has been found that working together as a class, the students propose a solution and then some will have doubts.  Because of their doubts the students seek ways to refine the process and solution, eventually discovering for themselves the need for the definite integral.  

Weirstrass shocked the mathematical world in 1872 when he provided an example of a continuous function which has no derivative.  Undergraduate textbooks such as [3] state this result, cite an example, and then skip over the details.  Surprisingly, Tall [4] describes an activity which guides students toward the creation of a function continuous everywhere but differentiable nowhere.  The hands-on nature of the activity is emphasized and a case is made for the utility of carefully drawn pictures in providing geometric insights, making formal results seem intuitive.  The article ends with a statement on the value of intuition.

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