This topic appeared on the AP-Calculus thread and I don't want to belabor it there. But it is very indicative of what happens in education and I am certain it is not new. But I have decided that it is a phenomena that deserves some attention and that phenomena is basically this:
"What if we just out of the blue teach math this way (in this case, teach calculus backwards)."
The reasons why this never works are simple.
1. These subjects, including calculus have been taught in various fashions since their discovery. The order and pattern in which they are taught did not just happen overnight and were not dictated by some cult. They evolved and arrived at their current form via a lot of thought and natural selection over much time.
2. This "natural order" is almost never questioned by those that grasp the math. It is either questioned by those that do not or in reference to those that do not. In either case, the idea that you could randomly choose an order to these topics, especially one that is counter the order that reasonably and naturally evolved over 100's of years, without something reasonably on par with all that, is ludicrous. You can't just say, hey what the heck, let's teach it backwards.
3. These alternatives never work because obviously they didn't work. It may be new to those that are floundering or to those that have been tasked with students that are floundering, but they are certainly not new in the annals of time. Millions of people have taught calculus.
Now, I know that Lou is going to belch out a reference to a mathematician or maybe even Socrates that thought about the order that calculus is taught, and ergo since such an esteemed individual simply thunk about it, it must be true. My counter to that is that since it went nowhere it is highly probable that it is not true. Time tells the truth.
Back to the order of teaching calculus. I proposed that the conceptual reason for the current (evolved) order is that the integral or anti-derivative depends on having the notion of a derivative to begin with. But furthermore, the mapping of functions to their derivatives is far more solid than functions to their anti-derivatives.
Maybe an actual mathematician (i.e. Dave) would be kind enough to define this "function closure" I am trying to describe. But nonetheless, as an analogy, consider multiplication of integers (basically the way it is learned). This progresses easily because it is closed on that set. You can multiply anything by anything. Now you move on to division and immediately you are hit by a problem. Division is not so neat on the set of integers and so you have to further develop the numbers, which has to be done simultaneously with your development of division obviously.
And besides this closure issue, there is the issue of hamming order. Now, I have repeatedly pointed out that conceptual knowledge does not follow a simple hamming order like these standards we see written in crayon. It is not A then B then C. It is a body of knowledge that is continually cross connected and that grows in sophistication and requires intelligence. But at a very high view there is an order. You don't learn calculus prior to algebra (and I am not talking about TI calculus). Likewise, even within these subjects there is a general order in which this sophistication grows. And this is true about calculus and the body of mathematical reasoning gained with limits and derivatives and then how to undo derivatives. Now I am sure someone will say "well what about limits and then integrals and then how to undo integrals" and I guess my answer back would be "differential equations".
Now, there were a couple posts that did not necessarily say "Integration" but instead said "Definite Integrals". One post was clearly talking about the idea that starting a discussion of "area" rather than "slope" was easier. I can see some merit in that, but in its obvious effect of introducing the notion of "limits", not integration per se, well certainly not the notion of the anti-derivative, or I guess I should say the anti-thing. And then of course, after a few sentences it's Riemann, Riemann, Riemann, and Riemann.
Dave posted a couple historical references to books that did approach the subject integration-first. I would be interested in what the reasoning was, since I obviously do not have his historical perspective on this. The Apostle book looks very rigorous and I would even appreciate the insight behind that one.
A lot of this discussion is moot because most of the posts were prefixed or suffixed by notes like "My students didn't know algebra". And you know, what can I really say about teaching kids calculus that didn't get algebra, other than "whatever". I guess eighty six dollars from a failing kid spends just like eighty six dollars from a successful kid.
Other than my interest in the calculus curriculum itself, as I said in the beginning, my interest is also towards this phenomena of "what the heck, let's do it backwards". In remedial land which seems to have completely lost its mind, this is not an issue. But when this stuff leaches into non-remedial classes and there is no "honors" safe harbor it is actually very sad.
Robert Hansen posted:
> Now, I know that Lou is going to belch out a
> reference to a mathematician or maybe even Socrates
> that thought about the order that calculus is taught,
> and ergo since such an esteemed individual simply
> thunk about it, it must be true. My counter to that
> is that since it went nowhere it is highly probable
> that it is not true. Time tells the truth.
Apparently you aren't familiar with Tom Apostol's calculus book, which went quite a lot of places. Perhaps you should do a little research. Apostol is not the only one who has considered teaching integration first.
Oh, and you would do well to listen to Lou - he has had a lot of excellent things to say over the years I have been following AP Calc forum.
I just read through the discussion of this topic on the ap-calculus list. To state that Robert Hansen is misrepresenting reality about what was said there is perhaps the only civil thing I can say about his post. I guess Courant and Apostol were a couple of math-hating "reformists" who didn't know much mathematics (surely not compared to Robert Hansen).
It's one thing to have a viewpoint that opposes teaching calculus or any other mathematical subject in an order different from the one used in one's own education. It's quite another to pretend that such an alternative must be the result of idiots who don't know math trying to work with ill-prepared and remedial students. But when the discussion on the AP list is so readily accessible to everyone, what is the point of so blatantly distorting what was said over there, as well as the history that was presented by several contributors?
This sort of nonsense appears to be nothing more than one person's ego-tripping efforts to represent himself as some sort of expert and all those with whom he disagrees as abject idiots. The reality of what has been posted, easily examined on the ap calculus list, does not support Mr. Hansen's tale, however. I, for one, will judge his future posts here with continued skepticism, if I bother to read them at all.
Someone actually asked me about the "remedial" reference and I replied. And this conclusion is not rocket science buddy. Why else would you alter the sequence? Because the kids were doing so well that you wanted to challenge them more?
"I am not sure where you came up with the idea of "remedial students"
At the bottom of that article they talk about students from diverse learning experiences with "faulty knowledge or perceptions" of math. In addition, the posts here are followed with "the students are not good in algebra" etc.
I just did a survey of first year calculus offered at 27 engineering schools.
If you score a 4 on the AB calculus exam (sometimes a 3) you will start out with Integral Calculus. So there is an example of kids starting college with integration with a happy ending.
But I did not get that from that article nor from the posts here. I do not think that the original poster's reason for thinking about teaching calculus backwards was because the kids were advanced or because they were "getting it". I think it was as an alternative as a means to help struggling students.
Unless of course we just have all this free time to think up random curriculum arrangements because all of the kid learn calculus so well that we are bored.:)
And obviously, in my opinion, I don't think this is the answer. I think the answer is before they get to calculus. Seriously, calculus is a biggy. It is kind of odd to have remedial discussions about calculus.
I think I was pretty clear in my post that I was asking for more insight on Apostle's approach. If you have any, please offer it. Thanks.
Yes, one poster on the other list mentioned remedial students. That's one (1). I've already requested you address the many other reasons given, the highly regarded texts and mathematicians who have used the approach you so sneeringly dismiss as being for remedial kids. That's not rocket science either. You just misrepresent the bulk of the replies on another list to make your audience here think you know what you're talking about. And clearly, you do not.
Robert posted on 10/02/09:
> I think I was pretty clear in my post that I was
> asking for more insight on Apostle's approach. If you
> have any, please offer it. Thanks.
Based on this statement in your previous post:
> These alternatives never work because obviously they
> didn't work. It may be new to those that are floundering
> or to those that have been tasked with students that are
> floundering, but they are certainly not new in the
> annals of time. Millions of people have taught calculus.
I would disagree that you made it clear you were interested on more insight. It seemed to me that anyone who offered such insight would be dismissed as someone who clearly doesn't know their subject well.
His name is Tom Apostol (not Apostle - you would have to go to the Bible for those...). I would look him up. In fact, it is has been suggested on the AP Calc forum, as well as many other places that his Calculus book is one of the "must-have" references for any calc teacher worth their salt.
Going back a little further, but no less influential, is Courant as Michael has suggested.
Robert Hansen Posted: Oct 1, 2009 7:20 PM
> my interest is also towards this phenomena of "what
>the heck, let's do it backwards". In remedial land
>which seems to have completely lost its mind, this is
>not an issue. But when this stuff leaches into non-
>remedial classes and there is no "honors" safe harbor
>it is actually very sad.
Someone else, like Bishop or Renfro could probably give you a better answer, but I can briefly say that I do not think you have this quite right.
For centuries, the body of mathematics that gave rise to the differential calculus and the body of mathematics that gave rise to the integral calculus were different fields of investigation, in much the way algebra and geometry were separate fields. And, just as algebra and geometry were unified by Rene DesCartes ("I corrected the faults of one with the other") and became analytic geometry, the integral and the differential calculi were unified under The Fundamental Theorem.
As you know, the point of The Fundamental Theorem is that differentiation and integration are two sides of the same coin. Same coin. It cannot matter which you teach first, and the pedagogical order we are used to is an unimportant accident of history.
To put it another way, although I have not investigated this question, I will bet you a cup of coffee that the conventional approach is just as unsuccessful as Apostol's approach.
In fact, I will say this in favor of Apostol. These days, lots of students come to college with all sorts of strange ideas about the calculus (and I quote, "What's up with this limit shit?"), having studied what passes for the calculus in high school (teaching the calculus in high school is one of those truly bad ideas, IMHO). Apostol has the virtue of shaking the students out of their intellectual ruts. It is that bracing slap in the face, calculated to produce an intellectual awakening.
As a rule, students resent being intellectually awakened. This is why I think the issue is irrelevant. It certainly does not matter mathematically, and I am confident it does not matter pedagogically.
Keep The Change
"To put it another way, although I have not investigated this question, I will bet you a cup of coffee that the conventional approach is just as unsuccessful as Apostol's approach."
If you read my post I specifically said in the context of successful students. I actually agree with you in what I think is your context. Or I guess I will just say what I think and let you judge if this is not similar to what you are saying...
The reasons students fail to get the math are many. There is ability, home life and culture just to name a few. The importance of the order in which topics are covered is relatively miniscule to just those other factors alone. Assuming we aren't doing something really crazy like starting with calculus and working our way to addition.
However, when we are talking about classes where the kids do get it then the order of topics is no longer "relatively" miniscule and there is always going to be a reasoned and natural process of ordering and that has already occurred. Could you teach calculus backwards to strong students? Yes. But for the various reasons I posted, starting with differentiation has apparently won the show.
It also depends on what you mean to "teach calculus". To me, it means that the students have a solid analytical understanding of the operation of differentiation and integration. I can best explain this via what I found in the college survey...
Almost all colleges had three calculus strands.
1. Liberal Arts version - A slower pace and not as deep (still no calculators mostly). This is for a student that will NOT have to tackle differential equations. This is still a fair treatment of the subject but this is the end of math as far as the student is concerned.
2. Engineering version - This is what we generally think of as traditional calculus. This has provide a pretty study basis for multi variable calculus and differential equations. This is for students that will actually use the math. Yet still, this is the end of math as far as engineering students (in general) are concerned.
3. Honors version - Well, first off this is obviously for mathematicians and it introduces more analysis so that they will be prepared for it when math begins to be all analysis. It is also for engineering subjects like physics where a deeper dive will be handy later for current models. And technically, there is no "end to math" in this strand.
I point out all these strands because they ALL must be grounded on the analytical understanding of the two main operations of calculus. And I think the natural order of delivery that attains this understanding most efficiently is differentiation first.
I will concede that it is debatable exactly how you teach the liberal arts version.
Regarding Apostol's calculus (thanks Jim/Michael, I think I got the spelling right now), from my first glance at it, it appears to be quite rigorous. And it is still in use, such as at Univ of Penn in the Honors Calculus class...
As you will notice, the teacher reordered the sequence to be differentiation first.
I don't know why Apostol did the order he did. Maybe he wanted to stand out. Hopefully someone can add to this. I do know that the circumstances and what he may have been thinking probably have very little bearing on what the original poster's issues were.
I was not debating whether or not bright students could handle calculus backwards. I will say that the forward treatment is probably much more efficient.
I was debating whether this type of conversation is even neccessary with students that are struggling. Is it even valid to say "Apostol did it" in that context?
This is like the ADP Exam results. Because the 8th grade students (a fraction of a fraction of a percent of all the students) did well on the exam, the conclusion is "the secret to algebra success is taking it in the 8th grade". For whatever reason, the obvious reality that these were very bright kids and that is why they were taking algebra in the 8th grade entirely evaded the committee analyzing the test results. How did that happen? How are these educators being so brainwashed that they now miss the obvious. This is why they keep running smack into walls. Someone told them walls don't exist.
That is what I am addressing.
"In modern calculus courses the treatment of differentiation and the construction of tangent lines usually precedes the treatment of integration and the calculation of areas under curves. This is a reversal of the historical sequence of discovery; as we have seen in the preceding chapters, the calculation of curvilinear areas dates back to ancient times. However, apart from the simple constructions of tangent lines to conic sections (with the static Greek view of a tangent line touching the curve in only one point), and the isolated example of Archimedes' construction of the tangent to his spiral, tangent lines were not studied until the middle decades of the seventeenth century.
Then beginning about 1635, a number of different methods for the construction of tangent lines to general curves was rapidly discovered and investigated. It was the combination of these new tangent methods with area problems and techniques, during the last third of the seventeenth century, that produced the calculus as a new unified method of mathematical analysis."
This passage above is the Introduction section of Chapter 5 of the book by Charles Henry Edwards, Jr., *The Historical Development of the Calculus*. Although David Parrot in his article "Integration First?" (<http://www.sci.usq.edu.au/staff/spunde/delta99/Papers/parrott.pdf>) quotes part of this introduction, I thought it might be useful to include the whole introduction. Some might find this article by Parrot interesting; here below is an excerpt from the article:
Why is Differentiation taught first?
In considering this question, one is naturally led to look at the historical development of the calculus. However, to quote Edwards, from his book, The Historical Development of the Calculus : 'In modern calculus courses the treatment of differentiation and the construction of tangent lines usually precedes the treatment of integration and the calculation of areas under curves. This is a reversal of the historical sequence of discovery; . . . the calculation of curvilinear areas dates back to ancient times.' Certainly the work of the ancient Greeks, and Archimedes in particular, laid the foundation for the modern treatment of the definite integral. Archimedes' work on the estimation of the area of a circle by using inscribed and circumscribed polygons and his determination of the area in a parabola surely were the first important and very significant steps towards the ideas now used in the integral calculus. Archimedes and other Greek mathematicians also derived other are!
as and volumes of geometrical figures and it was their works which formed the starting point for the 'giants' who preceded Newton and Leibniz in their development of the calculus. As far as the derivative is concerned, Archimedes did give an isolated example concerning tangents to curves in his work on the spiral. For most of Greek mathematics however the tangent was a 'static' geometric construction.
If there is no historical justification for teaching differentiation first, why do we do it? I offer some suggestions although I doubt this can be attributed to simply one or two particular reasons. The development of a rigorous basis for the calculus in the last century strongly influenced our approach to mathematics teaching in this century. The usual discussion of the derivative lends itself to such rigorous methods. General formulae for derivatives can be derived from first principles with much less complicated algebra than that needed to develop a similar result in integration. For example, compare the derivation of the derivative of x^2 from first principles to the calculation of the area under y = x^2 using Riemann sums. Further, the derivatives of all the elementary functions can be calculated from a few special cases by a 'mechanical' application of general rules; whereas the determination of corresponding indefinite integrals (if possible) is much more complicated !
and is often referred to as an 'art'. Another possible reason is the importance of applications from Physics, and more recently Biology, where rates of change are certainly more intuitive when expressed in terms of a derivative compared to an integral.
Should we teach Integration First?
In giving his reasons for teaching integration first, Rodin suggested that the quickest way for students to tackle an interesting mathematical problem is to compute the area under a curve such as a parabola. One of the most basic problems in mathematics is that of finding areas. Students are naturally led from calculation of areas of rectangles and triangles to an attempt to find the area of a circle, and then later to areas bounded by other curves they meet, such as the parabola. This natural progression mirrors the historical development of the calculus where the Greeks, and Archimedes in particular, developed methods for approximating areas. While merely following the historical development of a subject is not sufficient reason for determining how to present topics in a subject, surely the history does indicate how ideas developed and which problems may be the most natural to consider first.
As I have suggested above the treatment of the definite integral first will allow students to appreciate both fundamental problems considered in the calculus, before they see the connections between the two methods. They should therefore gain a better understanding and appreciation of this relationship as expressed in the Fundamental Theorem. The availability of computers and other electronic technology enables students to actually compute Riemann (or upper and lower) sums when first calculating the definite integral. This thereby removes, or at least diminishes, one of the stumbling blocks which is encountered when first discussing the definite integral; namely the laborious calculations necessary to find even simple areas. Further the existence of such technology reduces the need for analytic solutions to many problems and applications, and therefore decreases the importance of the indefinite integral.
There are some new approaches to the teaching of calculus that introduce differentiation and integration almost together. In the text, Calculus in Context , the authors first consider mathematical modelling and differential equations, thereby including ideas relating to both the derivative and integral. However when they do consider these mathematical principles they also discuss differentiation first. The other approach is that of Hughes-Hallett  in which the definite integral is introduced following an introductory chapter on the derivative, but before the techniques for computing the derivative are considered. This approach has been trialled in our alternative first year subject Mathematics 1M in the past two years with reasonable success. Since the students in this subject have also been exposed to the current teaching practice in schools the same comments made above concerning Mathematics 1 apply equally here.
Should we teach Integration first? There is of course no right answer to this question. It appears that current teaching has consistently introduced the derivative first. It is also clear that many students do learn, understand and appreciate the definite integral and the Fundamental Theorem under the current approach of introducing the derivative first. However it is time for us to at least look at the alternative to ensure that we do adopt the best approach, especially in light of the availability of the new technology.
Apostol's calculus is a great book for advanced students; my guess is
that nearly everyone in U Penn Honors Calculus already had AP
Calculus AB and the majority, BC, so this is not their first look at
calculus. AP Calculus has its problems, too much emphasis on use of
calculators, *far* more schools still enthralled with Harvard
Consortium Calculus that died (after an honest trial at top-notch
universities as well as mediocre ones), ill-prepared teachers, etc.,
but use of such an advanced introduction to calculus is not one of
its problems. Lack of theoretical competence is a problem; not an
overemphasis on it.
Historically, integration (or some form thereof) preceded
differentiation by a couple of millennia; Archimedean Method of
Exhaustion (think of the historical proof of the theorem that
guarantees a number called pi; i.e., that the ratio of the
circumference of a circle to its diameter is constant). That said,
the idea is irrelevant. Hansen is correct and it is not, as MPG
asserts, because *he* had differentiation followed by integration but
because that is the (almost) universal approach; i.e., because we all
did. Changing from the status quo without substantial evidence that a
change needs to be made is ed industry silliness, not pedagogical soundness.
I do lots of advisement of incoming students, the majority of whom
come from our feeder community colleges but many from other colleges
and universities, and it is taken for granted (thank goodness!) that
Calc I and Calc II, etc., mean something. In fact, once established
whether semesters or quarters, it almost means everything. Having a
website such as this one:
would simply not be possible but it is a great boon to community
college students and university advisers such as myself. Specify a
college (e.g., East Los Angeles College, one of our primary feeders)
and a second one (e.g., California State University LA) and a
department (e.g., mathematics) and a great deal of helpful
information is at your fingertips (well, monitor screen but...):
The situation is more difficult for colleges that have not been
"articulated" but the basic idea remains. Offbeat topic structure (as
with offbeat unit titles in reform math curricula) makes everyone's
work harder. If it were clearly superior to not follow the tried and
true, it would be worth verifying that fact in several well-studied
situations (elementary school, middle school, high school, college,
grad school, take your pick). If the evidence were conclusive that a
change is justified, spread the word widely with the evidence and
schools clearly identified and reform the world but, without that
proper preparation, stick with the status quo. Please.
At 10:46 AM 10/2/2009, Robert Hansen wrote:
>Regarding Apostol's calculus (thanks Jim/Michael, I think I got the
>spelling right now), from my first glance at it, it appears to be
>quite rigorous. And it is still in use, such as at Univ of Penn in
>the Honors Calculus class...
>As you will notice, the teacher reordered the sequence to be
>I don't know why Apostol did the order he did. Maybe he wanted to
>stand out. Hopefully someone can add to this. I do know that the
>circumstances and what he may have been thinking probably have very
>little bearing on what the original poster's issues were.
>I was not debating whether or not bright students could handle
>calculus backwards. I will say that the forward treatment is
>probably much more efficient.
>I was debating whether this type of conversation is even neccessary
>with students that are struggling. Is it even valid to say "Apostol
>did it" in that context?
>This is like the ADP Exam results. Because the 8th grade students (a
>fraction of a fraction of a percent of all the students) did well on
>the exam, the conclusion is "the secret to algebra success is taking
>it in the 8th grade". For whatever reason, the obvious reality that
>these were very bright kids and that is why they were taking algebra
>in the 8th grade entirely evaded the committee analyzing the test
>results. How did that happen? How are these educators being so
>brainwashed that they now miss the obvious. This is why they keep
>running smack into walls. Someone told them walls don't exist.
>That is what I am addressing.
>__________ Information from ESET NOD32 Antivirus, version of virus
>signature database 4477 (20091002) __________
>The message was checked by ESET NOD32 Antivirus.
Actually Wayne, I did what I should have done in the first place and contacted Dr. Apostol directly. His reasons (and they are also mentioned in the preface) are that this order is historically and pedagogically accurate. And he obviously disagrees with my notion of the anti-directive.:)
I wrote more about this on the ap-calculus thread...
But in a nutshell, if the historical development is accurate than that is a positive for the pedagogical order. We discover these things in increasing order of sophistication and thus rediscover (learn) them in basically the same increasing order of sophistication.
But I have a conflict to resolve because integration appears to require much more sophistication than differentiation, especially when you reach differential equations.
1. The discovery or development of "calculus" actually occured in that small window when it all came together and we might never know the actual "order". Not way back with notions by Archimedes that were never actually resolved.
2. My idea of sophistication regarding integration is faulty.
I'll have to think about it.
Regarding his book, it is quite rigorus and yes, he did do a rigorus mathematical development of the integral all on its own. But like Dave Renfro's reference to an analytical development of the Logarithm, these types of treatments presuppose an almost perfect understanding of the math. And that is a bit different than what I am stating regarding the development of sophisticated reasoning. I am speaking with regards to a student's first experience with calculus.
In any event, I will have to look into it more. I don't expect a rush of integration-first books because if it was that significant, I think we would have seen it by now.
So the history of mathematics according to Robert Hansen is not terrifically accurate. This is non-surprising. What is also non-surprising is the shallowness of his analysis of the whole issue that he managed to turn from a simple inquiry by someone on ap-calculus into one of his predictable (and predictably wrong) pronouncements of his absolutist philosophy. And now he's buried himself in error and trying to dig his way out with a teaspoon.
On the surface, the fate of the Dvorak keyboard might suggest a similarity to this issue of calculus topic order. But with very little additional thought, the analogy falls apart.
The Dvorak keyboard failed not because people thought it was better or worse than the QWERTRY format. It failed because you cannot simply reteach everyone on how to type. The qualifications of the keyboard had nothing to do with it. It was a simple matter of physics. To actually reteach everyone how to type was insurmountable, in any stretch of the imagination. Now, if Dvorak had invented something that replaced typing itself, this would have been a different story.
Mathematics curriculum can change on a dime. As you yourself have pointed out, districts can replace their entire collection of math with crap in one calendar year. Each batch of students is brand new and there are no insurmountable barriers to changing math curriculum. At this point in this post, I think we can firmly discard Dvorak or Dvorak like analogies.
Regarding the order of sophistication in discovery and in teaching, I think everyone is pretty much on the same page. I think the quibbling here is when and how calculus was discovered. I have yet to see much said on my comparison of sophistication between integration and differentiation, which was actually the key issue.
Regarding publishers, they obviously have a lot of pull. I don't like that. But it does not seem to affect the assortment of curriculums out there. You cannot deny that they get ample days in the sun. What kills them is that they keep going up against much stronger curriculums and they simply cannot compare. I guess it would be difficult for them to marketed to "slower" students for obvious reasons. But that is certainly how they are written and how they end up looking.
And back to the natural order of calculus that has evolved over a few hundred years. It has had many incremental e changes over that time. It was never static and dictated. And as you and I have both shown how easy it is to entirely change a curriculum in a very short frame, I would suggest that is a pretty strong argument that it is the way it is because it works that way and there is probably little or no gain in changing it, regardless of the reasoned arguments concerning sophistication of topics.
In reply to this post by Michael Paul Goldenberg
On Oct 4, 2009, at 11:08 AM, Michael Paul Goldenberg wrote:
Oh, it's accurate enough. Things just happened in the wrong order by a historical accident.
Regarding this sophistication argument...
I could mathematically fully develop division before multiplication. But from a learning sense, would that order seem natural? Hardly. When two operations are that intertwined maybe we can't actually unwind the history of their development, like arithmetic versus algebra. Yet, the natural order of multiplication and division, in a learning sense, is still obvious. And to me, so far, so is the order of differentiation and integration, from a learning sense I mean.
I think the best answer so far is that integration and differentiation are mathematically intertwined, like multiplication and division, and it is virtually impossible to unravel the history of "notions" of either pair of these operations. But at some point when things all come together into a unified realization, there certainly does appear to be a natural (hamming like) order to their sophistication.
Quoting Robert Hansen <[hidden email]>:
"What is the point of these historical arguments, however? Even if Mr. Hansen has game film of the history of human intellectual development (lots of hidden cameras with timers placed all over the globe as needed?) proving that multiplication came first, what will he have proved about the order in which we "should" teach arithmetic to kids?"
Well, first off, the game film thing was funny.
The original purpose of the discussion was the phenomena of "Hey everybody, let's just try it this way" and why I thought that these were bold statements considering the time that these curriculums have had to mature and evolve. And the fact that the current curriculums seem to have attained a decent "natural" order.
And then that led to me having to defend what I meant by "natural order". And I am not complaining because I gained a lot more insight into the matter.
Regarding your question of why mathematics is so hard for so many, I don't have an answer. It sucks that it worked out that way. You see me jabbering about those ADP test results? It sucks! I think they were so freaking bad it might have just killed 8 years of work. It looks like we are back to seeking out hidden honors algebra classes with old tried and true textbooks along with the usual tried and true smart white (lol, and asian now) affluent kids, and smart poor and/or kids of color need not apply. They created a special math for them over there.
You seriously misunderstand my agenda.
Quoting Robert Hansen <[hidden email]>:
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