Intellectual Mathematics - Now There's a Grabber Title
Intellectual Mathematics is a website written and run by Viktor Blåsjö of Utrecht University in the Netherlands. He has a keen sense of what should be important in teaching undergraduate mathematics. He has a Blog with very interesting ideas in convincing prose. There is also a 107 page Free online text, Intuitive Infinitesmal Calculus containing a prominent 10 page chapter on Differential Equations which is introduced early in his calculus treatment. Moreover it is spiced, almost driven by applications.
There is a VERY interesting Manifesto on the Teaching of Mathematics. In his First Axiom he says, ". . . we must not introduce any topic for which we cannot first convince the students that they should want to pursue it. This is a standard very rarely met in mathematics. Everyone likes to tell themselves that they are giving motivations for what they teach, but very little of what passes for motivation stands up to critical scrutiny as a motivation in the sense of the learning ideal outlined above. In all such cases, therefore, the student has no reason to pursue the topic in question other than obedience to the dictatorial authority of the teacher. In my view we cannot fault a student who hates mathematics in such circumstances; if anything, I would sooner fault a student who did not." He pulls no punches and talks straight. We could all benefit from a good doeas of his thoughts.
In his 10 August 2016 Blog entry, A criterion for deciding if something is worth teaching, illustrated with examples from Calculus I he speaks on behalf of the early introduction of differntial equations and of the use of realistic situations to motivate students. He says, "Don’t teach things that don’t serve a purpose. Or to put it differently: Pick up a calculus textbook and open an arbitrary section. Look at the problems at the end of the section and ask yourself: Is there any reason to want to know the answer to these problems? Are the problems inherently interesting, and the substance of the section a means to answering them? Then this is a meaningful topic and it should be taught. Or are the problems artificially concocted for the sole purpose of testing you on the material just introduced? Then it’s a crap topic and should not be taught. Students have no reason to work on such problems except subservience to the instructor, and therefore their effect is to suppress independent thought."
On the intdroduction of differentil equations Blåsjö states, "Include differential equations. The usual battery of integration techniques are usually accompanied by “tan3(lnx)”-style problems, much like the limit sections attacked above. Should they therefore be committed to the flames also? Not at all. The situation could not be more different, although students (and perhaps not a few teachers) in a traditional calculus course wouldn’t know it. Unlike the nonsense real analysis material artificially shoehorned into Calculus I, integration techniques do serve a very credible purpose that is very easily made evident to students. It only takes one simple reshuffling of the order of the topics: teach differential equations as early as possible, as I do in my book. This simple recipe at once changes the entire nature of drill problems on integration techniques. Without differential equations the students will conclude, with good justification, that these problems are nothing but a cruel obstacle course with no purpose. But with differential equations the student cannot draw the same conclusion without denying the value of studying population dynamics, the motion of rockets and projectiles and planets, and a thousand other fascinating and useful things besides. Hence we must either teach differential equations in Calculus I, or accept widespread hatred of mathematics as a rational outcome of our own doing."