Limits underlie everything in calculus and analysis. To see this, simply look at some old textbooks that use infinitesimals (I do not mean the infinitesimals from non-standard analysis that require hyperreals) or spend some time programming functions for numerical integration or calculus operations for some CAS in general (or look at the content of a real analysis textbook). You can’t actually understand anything much in calculus without understanding limits. Yet your standard textbook introduces them in chapter two, reintroduces them somewhat when defining Riemann integrals (and then again with improper integrals), and of course with sequences and series. But apart from that 2nd chapter introduction, limits are never covered in any kind of comprehensive, detailed way; rather, they are mostly introduced to move on to differentiation and then used when needed to introduce various other components of single or multivariable calculus.
Is there a good reason for this? Is there a way to teach calculus at least somewhat via limits as the foundation they are (as in e.g., classic textbooks such as Courant’s)? Is there a good reason not to?
Indeed: it's hard. In general, students are used to being handed rules and treated like bad calculators: apply rules/algorithms to problems set-up in particular ways such that the mechanically applied procedures will yield correct answers without any real need for a conceptual understanding of the procedures, the problems, or the topics.
The problem is that since around the beginning of the 20th century, the calculus has been far less important a tool than previously, while linear algebra practically IS applied mathematics (yes, probability and some other widely used areas of mathematics make little use of abstract or linear algebra, but from discrete mathematics to multivariate statistics to quantum mechanics, the sciences are FILLED with topics from or related to linear algebra).
Linear algebra is hardly the only conceptually challenging course that your would-be engineer, mathematician, statistician, physicist, and so forth will encounter. It's simply often the first encounter a student has in her or his experiences in mathematics where all the rote application of rules and computational skills are basically useless: the computations are elementary and the rules require understanding conceptually challenging notions almost from the beginning (such as first thinking about operations in spaces that cannot be visualized and become easier through our experiences only if these are via mathematics, as are perceptual experiences fail rather completely here).
So, we make students learn a conceptually difficult topic and often require calculus to be a prerequisite even though most linear algebra textbooks use no or almost no calculus because there isn't really a way to make linear algebra "easy" the way we do with calculus (i.e., make calculus basically pre-calculus computations with some new terms thrown in that won't be understood because the foundation for the entire enterprise is skipped over with minimal treatment).
In other words, when we can't skimp on the conceptual challenges, we force students to take them on. Moreover, we see an increasing number of transition courses and textbooks (A Transition to Advanced Mathematics: A Survey Course by Johnston & McAllister, Mathematical thinking and writing: A transition to abstract mathematics by Maddox, A Transition to Advanced Mathematics by Smith, Eggen, & Andre, etc.). Interestingly, not only do these courses seem to be more and more necessary, but those who design them or have introduced them have encountered with frustration the fact that students' mathematical literacy is so poor they struggle with the TRANSITION to advanced mathematics courses, let alone the actual courses (see e.g., Epp, S. S. (1986). The logic of teaching calculus. In R. G. Douglas (Ed.) Towards a Lean and Lively Calculus. MAA.).
So, we make things "easier" such that students who merely take single variable calculus in a pre-college or college course can pass it, learning virtually nothing that can be applied anywhere and perhaps instead of learning e.g., logic, statistics, probability, and other much more useful courses both from the perspective of overall critical & abstract thinking and applications, while those who go on to take multivariable calculus, differential equations, and more advanced mathematical courses will end up continuing not to really understand many of the topics covered whilst simultaneously taking courses that require the very conceptual challenges we are shying away from by teaching how to solve calculus problems using pre-calculus.
For "most scientists and engineers", the rule-based approach is by definition NOT intuitive as intuition relates to concepts, not procedures. It is easier, and therefore easier for researchers to waste several semesters learning a bunch of rules so that hopefully they will grasp enough of the computational tools they will require (most of which aren't in the calculus courses taught, as most of the material will have to be re-learned anyway, from vector-analysis to integration theory). It's also a waste of time. Most of what is needed from calculus requires linear algebra, and there are textbooks of varying degrees of difficulty that teach multivariate calculus and linear algebra together (the two I know best are Hubbard & Hubbard's excellent text, and a similar but simpler text by Shifrin). Basically, the "easy " route is a waste of time.
Of course, I may be wrong (I've been told it happens). Which is why I am looking for arguments that we actually get something from the "easy" approach other than students who either take calculus and get nothing from it because they end their mathematical education their, or continue on to re-learn most of what they were already "taught" through calculus courses that do not prepare them for more advanced mathematics.
This is a good question.
From present calculus and analysis, we face a truth that infinite relating “numbers” really exist in mathematics and limit idea and limit theory is a special quantified technique and theory of treating infinite relating numerical substances in mathematics. We have to teach limits because it is really a useful tool to quantify infinite relating numerical substances in mathematics.
Many people just take limit theory as a tool and never care how it works. For a tool-user, this is ok and never worries us if we remember the “operation manual” and follow other people’s footprints.
If one asks why limit theory work that way, it would be in big trouble. Let’s see following example.
There are two “reasonable” limit operations in convergent- divergent proof of Harmonious Series:
1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．． （1）
=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．． （２）
>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．． （3）
=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity （4）
（1） During the whole process in dealing with infinite substances (infinitesimals) in limit calculations, no one dare to say “let them be zero or get the limit”. So, the infinitesimals in the calculating operations would never be too small to be out of the calculations and the calculations dealing with infinitesimals would be carried our forever. This situation has been existing in mathematics since antiquity------ those items of Un--->0 never be 0 all the time and Harmonious Series is divergent, so we can produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonious Series and change an infinitely decreasing Harmonious Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity. Here, with limit theory and technique, we see a strict proven modern version of ancient Zeno’s Paradox.
（2） During the process in dealing with infinite substances (infinitesimals) in limit calculations, someone suddenly cries “let them be zero or get the limit”. So, all in a sudden the infinitesimals in the calculations become too small to stay inside the calculations, they should disappear from (be out of) any limit calculation formulas immediately. This situation has been existing in mathematics since antiquity-------those items of Un--->0 must be 0 from some time and Harmonious Series is not divergent, so we cannot produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or…from infinite Un--->0 items in Harmonious Series. But if it is convergent, another paradox appears.
I have published a few articles with my colleagues on student learning, including an article on College Algebra. This publication was used by lobbyist to pass legislation in Florida by which Algebra II became a mandated course in high school just a few years ago. The roots have to be strengthened before students come to college.
https://www.researchgate.net/publication/265535647_A_Study_of_College_Readiness_for_College_Algebra?ev=prf_pub
Maybe this is somewhat associated with what your question is about, Andrew. Many people don't want to teach more math.
Article A Study of College Readiness for College Algebra
Dear Mr. Raid Amin, how do you think of above “strict proven modern version of ancient Zeno’s Paradox by the limits”?
Regards, Geng
Dear Raid Amin:
Thank your for the paper! It actually touches on another issue I have (I spent several years working for the largest test prep company in the US, teaching & tutoring students to prepare for the SATs & ACTs, as well as the GREs and both SAT & GRE subject tests, and worked for this and another company tutoring high school students in various subjects, including mathematics). Personally, I agree with about the limited value of pre-college scholastic aptitude tests as reliable predictors of college performance.
However, one of the reasons the mathematical component of SATs (not so much the ACTs) differ from mathematics grades is because of a far more fundamental problem. In my experience, despite the fact that several years before the "standard" age a student begins her or his college career it becomes possible for students to be introduced to mathematical concepts (rather, the foundations for mathematical/quantitative REASOING vs. COMPUTATION), generally the extent to which the conceptual bases underlying mathematics are taught remains extremely limited. More importantly, a great deal of the topics covered hinder students in multiple ways.
For example, the emphasis on computation vs. abstract reasoning in pre-college and early undergraduate education tends to dissuade students who are not particularly strong when it comes to the mechanical procedures that students rely on in their rote application of memorized rules to solve problems of questionable value.
Put more simply, pre-college mathematics tends to be oriented in such a way as to "prepare" students for college mathematics. However, as most students will take few if any college mathematics courses (perhaps because they have been persuaded by years of learning what appear to them to be pointless, difficult, and irrelevant math topics), why do we teach subjects such as algebra II, which are fundamentally oriented around achieving success in calculus by
1) Supplying the pre-calculus skills that will be necessary to pass elementary and intermediate calculus courses because these don't teach calculus as much as they introduce calculus notions/terms that the students then use pre-calculus to solve
2) Continuing to provide algorithms to solve carefully prepared problems which do not help students to either appreciate mathematics or understand what it typically involves, because they will do this in their earliest undergraduate mathematics courses.
I have before used the example of systems of equations, matrix operations, and other linear algebra topics introduced in algebra II and pre-calculus courses as examples of utterly useless but challenging topics for high-school children. The idea that there is anything to be gained by teaching pre-college students matrix multiplication, determinants, etc., is absurd (if for no other reason that, to make anything useful out of such topics, students will have to course in linear/matrix algebra anyway, and the same topics will be covered from scratch).
Most of the mathematics that non-scientists may use and that are certainly important in terms of general mathematical competency are minimized or ignored altogether in favor of e.g., manipulating fractions with square roots so as to prepare for the evaluation of such algebraic expressions in terms of limits; of course, the decision to contrive limit problems that need such algebraic manipulations was arbitrary in the first place, and is pointless in general (it allows students to solve limit problems without understanding limits, but is not in any sense of any mathematical import because there is no general problem whatsoever with fractions that contain square roots).
Set theory, logic, statistics, probability, and combinatorics all involve the foundations of mathematics: algebra, proofs, logic, abstract reasoning, etc. Most students who pass pre-college mathematics learn few if any such topics, and understand less. Instead they are taught mostly seemingly worthless rules, procedures, formulae, etc., so that they can then take another math class or two in college where these can be applied to NOT learn the subject taken (i.e., NOT learn calculus because it is “easy” for students who have spent so much time as calculators to do more computations than to actually teach them calculus).
However, the pre-college curriculum is far less likely to change than the approach to undergraduate mathematics. In your opinion, are there ways to teach the underlying concepts of mathematical foundations in pre-college classes like algebra II but also include plenty of topics readily applied or at least easily understood in terms of application (even if the application is just abstract/quantitative reasoning)? If you could change aspects of pre-college mathematics, what might you change and why (and if not, why not?)?
Thanks, and thanks again for the paper!
-Andrew
Let us consider a physician trained in treating diseases. If his/her training is just to ask symptoms and making decisions on those symptoms to prescribe a medication, then that is not a perfect training we want, as symptoms may be more profound effects of the actual disease which might elude the actual cause of the ailment. To make the real diagnosis and analysis of the ailment, the training of the physician is paramountly fundamental not only in symptom analysis but in searching and understanding the actual causes of the ailment which offers a lasting and true medication for the patient.
This analogy works in learning mathematics from deep and invisible building behaviors of mathematical objects. These understanding enable one to learn and know the root causes of the actual behaviors of these objects, as to why and how such and such properties are valid. Knowing the valid properties of objects and using them whenever need arises, is not a purposeful act to know why and how but merely to use.
Mathematicians are trained to do mathematics, i.e, to know the building blocks and behaviors of mathematical objects, to analyse, extrapolate the horizons/domains so that a better and generalized behaviors are obtained or to search a higher order behavior. Therefore we have to be not only artists and consumers of the outside and the visible but also of the innermost and the invisible.
Limits are innermost and fundamental quantum process of converging values in mathematics such as continuity, differentiation, integration, measurability of sets, etc,. Our study of mathematical process should therefore include to know these processes to create real mathematicians not only users of mathematics. At the same time it is also equally valid to use established results of mathematical truth in sciences as it is not required to be trained to know why but to use when.
Dear Andrew,
At least in Greece we teach limits! The real question however is not if we teach limits, but how we introduce them. My suggestion is to use a kind of "intuitive" modern infinitesimals, to introduce notions like derivative and integral, and then step by step introduce limits and prove the equivalence of the two.
Dear Andrew, the way you want limits to be taught (limits as the foundation of derivatives) was taught in the USSR. And then a lot of practice with the limits! Thus many great mathematicians came out of this teaching methodology! If you like my answer, please use a little green arrow.
Stancl and Stancl wrote calculus with limits as its foundation. The foundation of derivatives is limits.
In Italy we teach limits both in highschool and in university when introducing analysis, and we treat them quite carefully.
Limits are one of the most useful tool for calculus for analisys of functions, convergence of series, function series, power series etc.
I sense the frustration, Andrew.
The function collapses, at 5. That is its value after integration. That is the function's LIMIT of holding our attention. Little attention is paid to the infinitesimal boundary layers approaching 'greatest number' upper or lower bounds. There is a universe in these limit bounds, but they are considered insignificant or as commonplace as the constant of Avagadro's number. Accept that thats the way it works because we don't really care how it really work in our macro world.
In present traditional finite—infinite theory system, people have been creating many new “understandings” on “infinite”, “potential infinite” and “actual infinite” since Zeno’s time 2500 years ago. But it is difficult to solve those infinite related problems produced by the fundamental defects disclosed by the infinite related paradoxes since Zeno’s time, because within the present traditional finite—infinite theory system, “the infinite related problems” are strongly interlocked together with the foundation. So, though trying very hard willing to solve “some infinite related problems” with some new “understandings” within the present traditional finite—infinite theory system, but people finally discovered that nothing can be done because “everything is perfect” in present traditional finite—infinite theory system.
I know few people agree with me, but this is true.
Limit is an important subject for understanding calculus (especially continuity .. etc). I think that students should learn how to use limit as a tool in a rigorous way (the epsilon delta arguments! ) and in fact this is the way we teach our (mathematics major) students in the first year. Unless this important tool is mastered the students will not be able to understand higher level analysis courses.
Dear Ahmed，I agree with you that limit theory and skill is an important tool in present infinite related area in mathematics. But I am very sorry to say that we sometimes have to stuff our students some other mysterious things.
Just see following divergent proof of Harmonic Series, very elementary and important, which can be found in many current higher mathematical books written in all kinds of languages:
1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．． （1）
=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．． （２）
>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．． （3）
=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity （4）
We teach our students that we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonic Series by “brackets-placing rule" ?!
The more we try to explain, the more doubts be aroused and we feel more helpless.
I think it is miserable.
Regards, Geng
I think that understanding of limits and ability to calculate them is a necessary part of training for mathematicians in universities. I do not know why it is not done in some of them today. While my first graduate education was more in physics than in mathematics, we used good textbooks by such authors like V.Smirnov (see http://www.amazon.com/Elementary-calculus-course-higher-Mathematics/dp/B0007K2EZ2 ) and G.Fichtenholz (https://en.wikipedia.org/wiki/Grigorii_Fichtenholz). Fichtenholz's books about analysis are widely used in Eastern European and Chinese universities due to its detailed and well-ordered presentation of material about mathematical analysis. Due to unknown reasons, these books do not have the same fame in universities in other areas of the world.
I heard that mathematical education becomes worse worldwide, first at the level of schools and even in some universities. When many students cannot do arithmetic without calculator, it might be better to give them some practical tools how to differentiate or integrate elementary functions rather then to study limits.
Limits are essential in many proofs, and pure mathematicians should be able to use them to prove theorems. It might happen that there is decreasing benefits from calculation of complex limits that were often a part of calculus in good math schools of he 20th century. Differentiation ability is more important for analytical work of non-mathematicians. For example, optimization in economics requires an ability to differentiate analytically. Now students are often turned to computer when they see a simple differential equation or integral that can be solved analytically.
With modern limit theory and skill, can we really change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant?
This is a “strict proven modern version of ancient Zeno’s Paradox by the limits”. The more we try to explain the more doubts be aroused and we feel more helpless.
So, why don't we teach limits in calculus?!
Regards, Geng
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