Different points of view lead to different views. Talking from the point of view of an applied mathematician – this will lead to a particular biased applied mathematician’s point of view, a utilitarian point of view, as well as an environmental scientist’s point of view.
He speaks as a customer. His students need certain skills, knowledge, etc. He is the one who needs the results of mathematical education at technical schools and university activities.
A lot of abstraction leads to unrealistic ideas: Who cares about the trigonometric functions applied to a 30 degree angle?
Fenando Pessoa: The preacher of his own truths…
Do students share our enthusiasm with what is taught in the classroom?
In fact, do WE share any enthusiasm? Often we teach a given subject begrudgingly!
Students can read this dissatisfaction! Why should they be enthusiastic?
Will this ever change?
We have so many excuses: Papers to finish, committee meetings to go to.
So do the students.
So do the bureaucrats – they don’t want to mess with what seems to have been working for the last 150 years. Why should you change the teaching of calculus when it seems to be working!
What’s the deal here for traditional mathematics?
Maths classes are about: Existence and uniqueness of results
Mathematical and scientific truths: Correct answers (for odd-numbered exercises only at the end of the text-book). This does not exist in real life!
However, beliefs do change in science.
None of this “seeing mathematics in everything”, but more importantly needing it!
- Bus prices: When bus prices go up, this is because politicians have done calculations. However, they may be corrupt.
- Where do we place a new bridge? Again, corruption may come into the decisions. They don’t call mathematicians.
- Where to build a new airport? Variational problem in linear algebra.
- How to control sewage/oil spills?
What do these have in common? No unique, true, exact answers!
But isn’t this what we teach linear algebra, calculus, analytic geometry?
We teach truths and exactness for topics which don’t necessarily have a true, exact answer in the real world.
Using mathematics in the classroom (as well as in life) is really a trapeze act with no safety net.
However, there is something else for the classrooms:
- The students’ own knowledge and problems:
- A student might not to know anything about the first derivative in a mathematical language, but intuitively anyone who can cross the road does know this.
- If you don’t know any differential geometry, but intuitively you know how to kick a ball along a specific trajectory.
- What are the problems on campus, in the community, or general social difficulties and ethnomathematical issues (bricklayers use ethnomathematics)?
- We can give them projects related to the above.
Do we obtain students’ co-responsibilities in learning processes?
When we talk to the students, they figure as indirect objects, presumed to be in a situation when we talk about “I’m not going to be able to teach this topic this year”.
Do they know why they have to learn limits? Do we know? Of course limits have real world applications, as have Fourier series, ODE techniques, integration rules, cubic splines, etc.
The set of all exactly solvable mathematical problems is vanishingly small compared with all mathematical problems – why do we bother teaching them these rules which furnish them with solutions only to a tiny subset of all problems?
Why should students care about that if we don’t?
Is what we teach useful? and how so?
How can we fit real world problems into the calculus syllabus? How do we use mathematical modeling in such a course?
All mathematics is applied mathematics: Even if it’s applied to itself (ie. pure mathematics)
A really provocative statement: If the teacher does not know what it is, forget it! In other words, calculus professors must learn, as well! This starts with listening! Calculus teachers don’t listen too well!
Is this possible? We have to change the culture in teacher’s schooling. We have to communicate with other areas. We have to realize that mathematics itself changes in time!
Why are technological tools so fundamental? IQ versus patience. Machines have zero IQ and infinite patience. We have IQ and limited patience.
Do our students learn from us how to listen, create, criticize etc?
In both pure and applied mathematics trial and error do exist! Generally we don’t let our students do that.
When is a mathematical model any good?
- When will we move from Riemannian to Lebesgue integrals? Derivatives in a distributional sense?
- When will concepts be more important than results?
- When will we recognize that the absolute truth exists, but are unattainable?
So, how about a system where there is no safety nets for teachers!
- No unique answers: Multiple possibilities for answering – this is like the real world
- No correct answers: Use numerical approximations – they are not going to face easy problems
- No precise elegance
- No safe sequential themes: Use mixed up mathematical subjects – but really so!
- No transmission of truths: Relative truths depend on hypothesis
- No universal truths whatsoever
What can teachers rely upon?
- Teachers’ histories and formations
- Students’ histories
- Community problems
We must also rely on insatiable ambition! Deep down inside, we want to change history for the better!