Unfortunately, the scenario that Strogatz cited manifests arrogant mathematicians' grand cop out for many generations.
"I know something that you don't know ... and I'll tell you what it is ... even though you cannot understand it until you have encountered it several times ... if you survive the math curriculum for that long. .[I can say that, because that's what happened with me ... and surely, you can fare no better.] So just play along, and I will grade you on how well you do that ... even if you don't have a clue. And don't worry, "math anxiety" is normal."
Pedagogical conservatism at its best/worst? It presumes that development of functional personal mathematical intelligence necessarily follows "the path through which I came ... which is about how our textbook is written ... which really can't be commonsensible to you, until you later have grown a great deal."
It is a pervasive theme of pedagogical irresponsibility. "Mr. Johnson" refused to accept the responsibility for making delta-epsilonics fully commonsensible to his students ... which might have meant that he did not know enough mathematics to be able to do so. There is no stronger case for "indirect instruction." It is a safeguard against telling non-sensible mathematics to students.
It is easy to make the concept of "continuity of number valued functions" fully commonsensible to anyone who is well grounded in pre-calculus. Calculus textbook authors do not do so, because they do not endeavor to learn HOW to do so. Instead, they arrogantly tell what they think they know ... and perpetuate teachers' beliefs that students who "don't get it" simply are lacking in "aptitude" ... yes, lacking in aptitude for swallowing "math" that makes no sense!
Until mathematics educators accept the challenge of making core-curricular (presently, K-calculus) fully commonsensible to all functional students, their personal mathematical health will continue to be a worldwide disaster. But that requires also accepting the challenge of *learning how* to do that. The only viable route for that kind of progress is clinical case-study research.
Fortunately, as Kahn has clearly demonstrated, all that is needed in today's world is for someone to make the delta-epsilon criterion for function-continuity fully commonsensible via "the (cyberspace) cloud" ... and for all concerned to know that the revelation is there, and how to find it.
Yes, some readers on this list will ask (a ) what has case-research disclosed about that concept, and (b ) why is a (MACS) Mathematics-As-Common-Sense^TM disclosure not already out there. The answer to (b ) is "full plate" ... with the MACS Project's higher priorities recently being given to lower-school mathematics. For (a ), this is not the medium for a full report. But the essence is that: (1) students first must grasp that the line of rational numbers is dense ... but also having a density of "irrational" holes, and (2) also perceive how the "zoom in" squeeze on a function, at each point within or outside its domain, converges to a (sometimes empty, sometimes single-point, sometimes otherwise) "vertical" interval. With that kind of conceptual under-standing, the delta-epsilon criterion can result as a common-sensible *theorem.* But for purposes of conceptual understanding, it is a LOUSY "definition."
Bottom line: the major reason for "aging" to be necessary for personal mathematical comprehension of curricular mathematics is that educators have failed to make it commonsensible to students on their first go. So we wastefully must re-teach and re-teach, ad nauseam ... because educators are too lazy to learn.
From: Robert Hansen Sent: Friday, November 09, 2012 1:59 PM To: firstname.lastname@example.org Subject: Re: How teaching factors rather than multiplicand & multiplier confuses kids!
On Nov 9, 2012, at 2:26 PM, Joe Niederberger <email@example.com> wrote:
That's why I like "meta" better than "symbolic". There are meta levels upon meta levels, not just two levels. The distinctions are always relevant no matter how far one gets. "Meta" also captures better the idea of studying "expressions with variables" as objects in their own right, rather than simply viewing them as ways of expressing things *about* integers or rationals or what-have-you.
I certainly agree there. That's my big deal, development.
There is also an aging aspect involved, which I cannot state better than does this quote from the book "The Calculus of Friendship"...
"When we were learning about the rigorous definition of continuity - a very fundamental, difficult concept in calculus - Mr. Johnson told us something I?d never heard a teacher say before. It was ominous. He said he was going to present some ideas we wouldn?t understand, but we had to go through them anyway. He was referring to the epsilon-delta definition of continuity: A function f is continuous at a point x if, for every epsilon > 0, there exists a delta > 0 such that if [x ? y] < delta, then | f(x) ? f(y) | < epsilon. He said we?d need to see this four or five times in our education and that we?d understand it a little better each time, but there has to be a first time, so let?s start."
Strogatz, Steven (2011-03-07). The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math (Kindle Locations 136-141). Princeton University Press. Kindle Edition.