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So first I gave my class the quiz problem: Compute $$\lim_{h\rightarrow 0} \frac{\frac{1}{3+h} - \frac{1}{3}}{h}.$$ Upon finding that they could not do that (no real surprize) I asked them to compute $\frac{1}{3.01}-\frac{1}{3}$ in hopes that they would recognize the kernel of the former problem in the latter, and in hopes of indicating that it is perfectly reasonable for an entering college student to be able to add fractions.

A disturbingly large number of students could not perform the latter arithmetical calculation even though i had made comments about how to add fractions within class.

I imagine my experience is not unique. And I think that the current forum may have a sufficiently large readership to deliver an informed opinion about whether or not calculator use is inhibiting algebraic ability. If it is not the calculator, then what is the cause?

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closed as not constructive by JSE, Suvrit, quid, Felipe Voloch, Noah Snyder Mar 1 '12 at 4:50

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Calculator use does not inhibit algebraic ability. Failure to be taught algebraic abilities does.... – Mariano Suárez-Alvarez Feb 11 '10 at 20:46
I think the main cause is just like Mariano said; it's that there's not much of an emphasis in many courses on calculation these days. I blame it on the "it's better to know concepts" trend; as if concepts could be somehow separated from getting your hands dirty... – Jason Polak Feb 11 '10 at 21:05
Here's my take on this sort of question. I think professional mathematicians can easily slip into "grumpy old man" syndrome, usually of the form "kids these days seem much less bright than kids x years ago" when x years ago is when the speaker was a kid. I think this is just a myth. When I was a kid I was better at maths than most people around me, but that's no surprise because it's me that became the professional mathematician. I think my view of "what kids aged 19 should know" is indelibly tainted by what I knew at 19. I think you just have to step back and realise "you're not the norm". – Kevin Buzzard Feb 11 '10 at 23:49
I hope that this question does not become (1) too discussiony, or (2) devoid of any actual data. Alas, I fear that questions like this are bound to become (1,2). – Theo Johnson-Freyd Feb 11 '10 at 23:59
I agree with Kevin and Theo above. Next year I will be teaching elementary and middle school teachers how to teach math so this question interests me, as a Math Overflow question it is inappropriate. Cause and effect in education is a very subtle topic. – Jason Dyer Feb 12 '10 at 1:09

13 Answers 13

I did not want to answer this question at first, but now that we end up with teacher bashing let me put in my 2 cents. Facts first: I teach kids between 13 and 19 (grade 7 to 13), and I do know some maths; so do my colleagues, although they know somewhat less than I do, of course.

I have no problems explaining why we add or multiply fractions the way we do, or why the product rule holds. I think it is idiotic to assume that problems with simple arithmetic magically disappear when we start explaining the mathematics behind it, and in this respect there is no difference between the addition of fractions or the product rule: your explanations reach only 5 % of the class, the rest will patiently wait for the recipe.

In fact I think the problems start when instead of teaching children how to add an substract, we try to make them understand why the algorithms work. I don't think I cared a lot about the fine points of the decimal system when I was 11, and I don't think that today's kids do. Problems with calculators set in when the kids are 13 upwards; simple arithmetic is not supposed to be trained because the calculators can do it. By the time they graduate from high school (gymnasium in Germany), my students can form the derivative of $e^{\sin x}$ without problems, but many of them have problems if they have to manipulate $\frac1x - \frac1{x-1}$.

The main problem I am having with our current approach (over here) is that, at least in my neck of the woods, being able to use the calculator (TI 83) is compulsory for graduating. So yes, the problem is not the existence of the calculator, it is the teaching; but: the curriculum is designed in such a way that basic arithmetic plays only a minor role, and the reason why it is designed that way is -- the calculator.

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"I think the problems start when instead of teaching children how to add an substract, we try to make them understand why the algorithms work" - This is madness! I simply cannot express how fundamentally we disagree. – Steven Gubkin Feb 12 '10 at 13:24
I am not sure whether what you call "people" are the children I'm trying to teach. I can't name a single person who would be interested in why long multiplication works before he has figured out how it works. As for myself, I didn't understand what a class group is until I had computed several dozen of them. Now I might be abstraction impaired, but so are the kids I know. They don't understand Dedekind cuts but happily multiply the square roots of 2 and 3. – Franz Lemmermeyer Feb 12 '10 at 13:39
Steven, while I have written about what you speak (… ), you have a very generalized idea of what students like. The are an enormous number of strategies developed to reckon with the problem, but one can never presume teachers just aren't trying. As David Cox wrote recently, "I do my best to make my students think, but they still try to become good little algorithm followers." ( ) – Jason Dyer Feb 12 '10 at 14:17
If I could teach students one at a time instead of 30, I too could do incredible things. Oh, and I don't like your jumping to conclusions about what I do and what I don't do in class. – Franz Lemmermeyer Feb 12 '10 at 16:27
I totally agree with Franz! (and being realistic, as an analogy: vastly more people want to learn how to drive, than to the entire body of science, math, and engineering, that makes the driving possible). Franz is being very realistic---and full respect to him for it. – Suvrit Mar 1 '12 at 4:12

(Turning what was originally a comment into an answer, for I think it is the answer.)

Calculator use does not inhibit algebraic ability: failure to be taught algebraic abilities does....

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I'm not sure you can be taught algebraic manipulation/ability. I believe that this is a question of fluency, like in a language. You only become fluent by practising it. – José Figueroa-O'Farrill Feb 11 '10 at 21:21
So - duh - you teach algebraic manipulation by giving your students plenty of practice at it. – TonyK Feb 11 '10 at 21:28
OK -- maybe it's semantics, but I don't call that teaching. For instance, I learnt how to integrate by doing many integrals. I don't remember being "taught" to integrate, although I do remember discussing integration tricks with my cohort. – José Figueroa-O'Farrill Feb 11 '10 at 23:12
I think that practice is important, but it has to be practice which includes actual thought. If you are having people mindlessly apply algorithms you will just deaden them. People are not computers. If you are not thinking about distributing when you are practicing the multiplication algorithm, you are wasting your time. If you don't understand the place value system well enough to understand what is really going on in the algorithm for addition, and you are not using that understanding to guide you calculations, then it will not be surprising when you think math is a BORING FORMAL GAME. – Steven Gubkin Feb 12 '10 at 12:42

Do I ever?! (to your first question)

I do not know the cause. Of course, I'm in the UK and therefore our mileages will certainly vary. I suspect, though, that in the end the problem is simply lack of practice.

It is alas not uncommon for our students (at least in exam conditions) to be unable to successfully finish a problem, not because they do not understand how to go about solving it, but because once they've done the hard stuff, they are bogged down by what ought to be simple arithmetic, trigonometry,... I have talked at length about this problem with colleagues at my university and several explanations were proffered:

  1. Students do not necessarily learn any maths in School: they learn to pass exams. (I think that "maths" can be replaced by pretty much anything else, and you'll still get a true sentence.)
  2. Exams are much more modular now in the UK. It used to be that students were examined precisely twice during the four years of "high school": once at the end of the first two years (so-called "ordinary level") and once at the end of the last two years (so-called "advanced levels"). (Strictly speaking this is in England and Wales. In Scotland the system has always been different.) Now one can get examined on less material which seems to favour "cramming".
  3. Students do not spend nearly enough time solving the sort of routine problems which hone their calculational skills.

I am not sure whether one can blame this on the use of calculators.

At any rate, I agree that it is a disturbing trend and one we would like to reverse. If anyone has had any success at this, please share your thoughts!!!

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Well, being bogged down by simple arithmetic, trigonometry and friends is part of the problem: they are also bogged down by their inability to properly read and understand the problem descriptions, and to write down an understandable answer. – Mariano Suárez-Alvarez Feb 11 '10 at 21:21
Yes, this is true as well. There's a lack of understanding of basic mathematical language: students are often unsure of what a question is asking, but this they learn quicker. And I also agree that they often have difficulty writing a coherent answer with narrative and not just symbols scattered on the page. To their credit, they get better by their 4th year. – José Figueroa-O'Farrill Feb 11 '10 at 21:25

For what it's worth, there is a fairly specific villain to blame for this problem in the school district where I attended high school. In this district - which is not the district I grew up in - there is an awful math curriculum called CORE which is taught from first grade on and which emphasizes students "discovering" concepts on their own, etc. in place of teaching them basic skills. My understanding is that this is a holdover from reaction to (?) the "New Math" movement, and as far as I can tell, what it produces are college students who cannot divide 42 by 7 without a calculator. (I experienced this while tutoring an otherwise very bright friend of mine in calculus, and while the calculator plays a pernicious role in this story I don't think it's the culprit.)

So at least where I come from, the problem - at least as it seems to me - is that the curriculum has changed for the worse. I don't know how serious an issue this is in other parts of America or in other countries, though.

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I was subject to New Math, and I do not recall having had to discover anything... – Mariano Suárez-Alvarez Feb 11 '10 at 22:05
I think the problem here is that, to do that kind of thing well, you have to actually know math at a pretty intimate level to be able to properly guide students through it. This works very well when, e.g., I tutor people to supplement what they're learning in classes, but is a terrible failure when someone who doesn't understand how everything is related to everything else (in other words, all primary and secondary school math teachers) tries it. – jeremy Feb 11 '10 at 22:28
Many of my courses were "new math." I learned to manipulate rational functions before learning to add fractions, but once I learned the latter (which I perceived as more difficult because I had not developed the facility with factorizations), I was able to see in retrospect, they were the same idea. – Scott Carter Feb 12 '10 at 0:49
My apologies; yes, I think I was a little confused. The point is that I think there was some sort of ideological impetus for this program to exist, and that it was a bad idea. – Qiaochu Yuan Feb 12 '10 at 7:07
This "discovering math on your own"-thingie is the latest fad here in Germany; I still have to find a difference between students whom I let "discover" the product formula or the power series expansion of the exponential function on their own and those who were taught the "traditional" way: they don't differ in performance, and not in attitude towards mathematics. And whereas this method may work (if you have enough time on your hand, that is) for 17 year olds, it does not work for 9 year old kids if "working" means being able to work out 42 : 7 without a calculator. – Franz Lemmermeyer Feb 12 '10 at 19:28

[original answer by Chris Leary; tidied slightly by YC]

I am in sympathy with Kevin Buzzard's opinion that we mathematicians can become "grumpy old men." For several years (I was perhaps very naive), I labored under the belief that my students had a secondary school math background similar to mine. I have abandoned that belief. I have been at the same college now for over 25 years. I have noticed a decline in the preparation, but mostly in attitude, among our recent students. I wish I could say why this is the case, but I can't really.

As far as technology is concerned, I remember an article published in some journal on technology in math education. The article appeared during the height of the calculus reform movement in the US and was based on the authors' experiences at Oklahoma State. One of their conclusions was that, in the hands of talented students, calculators et al enhance students performance, but for less talented students, and I still remember the phrase, technology "adds one more layer of obfuscation" between the student and the material.

I believe there is something fundamentally wrong with how the US mathematics educational system functions in primary and secondary school. I don't think technology itself is the main culprit. How the technology is used is crucial.

A bigger problem is teacher preparation. My college has a school of education and the struggles of the elementary education people with mathematics are legendary. They actively resist learning anything about the math they will be teaching and only want to learn algorithms for solving problems. Even prospective secondary school teachers are not immune. A former student of mine in abstract algebra was incensed at having to learn about factoring polynomials, claiming that she was going to be a teacher, already knew how to factor, and didn't see any value in learning about polynomial rings. Unfortunately, she displayed an amazing inability to factor quadratics on an exam. So student attitudes are sometimes working against us as well.

What's wrong, and how to fix it, are not simple questions. I think there is a complex mixture here. Technology is a convenient target (and the crticism is not wholly unjustified). However, educational philosophy and policy, and societal factors, probably play a significant role as well. I'll stop here, because the more I think about these issues, the more discouraged I become.

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Students are not being taught that mathematics is something you can think about: it is being taught as a mindless system of rules. I would guess that less than 5% of people in the world can really explain why (3/5)(2/7) = 6/35. There is a very clear picture which explains why, but no one ever gets taught this picture. Try asking any of your students why this is true: the answer you will get is "because that's the rule".

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It only gets worse after a typical AP calculus class. How many students actually understand the product, quotient, or chain rules? l'Hopital's rule? – Qiaochu Yuan Feb 11 '10 at 21:56
Only those who are going to go on to be mathematicians. – David Mandell Freeman Feb 11 '10 at 22:21
Exactly! And the problem here seems to be that the people teaching just don't know, because they often don't know any math at all! Surely, most of my grade school teachers did not know math at a level above the error-ridden textbooks, and did not know the context that "definitions" come about in, nor where the rules come from, or how they are related to them. I know I relied on calculators a lot in school, but once I started learning why things worked, doing calculations myself was trivial, and calculators could actually be helpful to discovering new things by myself. – jeremy Feb 11 '10 at 22:26
There is some hope. My kids (grades 1 and 3) are in a school district that uses this thing called "Everyday Math" (somewhat of a misnomer). I was warned by a chemist friend of mine that it's a terrible system, but I think I have come to a different conclusion. I think it helps develops the exact skills we thought were disappearing. Unfortunately, we're hitting rock bottom with the next few in-coming classes of college freshman. But I think (I hope) it gets better from there. – Ian Durham Feb 12 '10 at 2:47

I feel both human-ability and technological-assitance should go hand-in-hand. We have to give equal importance to making students use a calculator and also learning how to do it by hand. I also feel we should encourage students to use softwares like Mathematica and Matlab. Otherwise, what advantage does a future mathematician have over old-timers!

With this background, I feel there should be a clear emphasis on the interpretation of the results a student obtains on performing a calculation.

            'The purpose of computing is insight, not numbers.' -Hamming.

For example, we can use the series

$\frac{1}{1+x} = 1-x+x^{2}+\cdots,$ for $|x|<1$ to demonstrate the fact that if $|x|<<1$ (|x| is far far less than one) then $\frac{1}{1+x} \approx 1-x$ and show the results in a calculator.

Say, $(1.001)^{-1}$ can be easily seen without the use of calculator as approximately equal $1-0.001=0.999.$ Division problem can be turned into a simple subtraction problem. After showing algebraic manipulation, we can show the calculator result and ask students to interpret the precision and give a good explanation.

We could also enhance Mathlete competitions and make students learn to calculate mentally faster than a calculator, for which we need calculators!

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At least in some locations in the US, the problem is summed up in this video. Students in some places aren't being TAUGHT arithmetic anymore. At all. Add to that the general lack of computational repetition that's been trending upwards these days, and I think that this sort of thing explains a large fraction of the problem.

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Her problem is that she doesn't understand the stick method or the lattice method for computing products. Since she doesn't understand, she critiques them. Using four crossed sticks is pretty amazing, and A SKILLED elementary school teach could train students to learn that (a+b)(c+d)=ac+bc+ad+bd by using sticks. A SKILLED teacher could go from multiplying to represent area to sticks. Too often elementary school teachers hate math and have no deep understanding. – Scott Carter Feb 11 '10 at 23:15
But as we know, methods that are excellent in the hands of a skilled teacher can be disastrous in the hands of an incompetent or average one, and we have a majority of the latter (re:math, especially) in elementary and secondary education. I also found the books' lack of faith disturbing. – Charles Siegel Feb 12 '10 at 1:47
I really liked the "focus" multiplication algorithm - I think a student would have a better chance of understanding what was going on that way. If you ask the average child why they "carry", or why they line up numbers the way they do in the algorithm, they will have no idea. What is the point of having people remember these algorithms if there is no understanding of how they work? To steal Lockhart's words: What is the point of having a whole generation walking around with "Minus b plus or minus root b squared minus four a c divided by 2 a" floating around in their head? – Steven Gubkin Feb 12 '10 at 14:14

I've not made my mind up about this, and I've had the privilege of teaching little, and teaching students with elite educations who did not have these problems.

From my own recollection, being drilled in arithmetic made me bored and rebellious in class. Polynomials got my attention, and transformed the way I did mental arithmetic.

The thing about constructivist pedagogy is that I can see how it might be successful, and produce enthusiastic, resourceful students, but it seems to place huge demands on teachers to get that result. We can't fill any country's classrooms with hundreds of thousands of Seymour Papperts. So choice of curriculum has to be pragmatic, fitting the needs of the students you have with the abilities of the teachers you have. And throwing out today's practice because of the dream of a brighter tomorrow sounds unlikely to work out well.

But I don't know enough to feel I can pass judgement. No doubt when my daughters reach school age, I will feel differently.

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Am I the only person here who thinks that new math had some appeal? – Harry Gindi Feb 12 '10 at 7:32
What is new math for those living outside USA? – Andrea Ferretti Feb 12 '10 at 8:56
"New Math" was an attempt in the 1960s US to catch up to the allegedly-superior Soviet mathematics education. The idea was to children based on the way that mathematicians organized it -- for example, you would teach students basic set theory, logic, algebra, and so on, so that when they learned concrete arithmetic and geometrical skills they would be able to see how these tools fit together. With a mathematics teacher who really understood mathematics, this was often wonderful. Unfortunately, many didn't, and led to catastrophic algebraic muddle plus a failure to learn basic computations. – Neel Krishnaswami Feb 12 '10 at 10:52
Basically everything concerning education that happens in the US reaches Germany with a time lag of about 10 years. We got our "New Math" in the early seventies, with the same (predictable) results. – Franz Lemmermeyer Feb 12 '10 at 22:10
New Math is amusingly presented in a Tom Lehrer song: – Elizabeth S. Q. Goodman Feb 29 '12 at 6:57

I think the use of calculators at an early level is a great thing. For one thing, calculators give kids a sense that math actually works, a solid thing that can be checked and thus grasped without guidance. Being only 25 myself, I don't know how that felt for disinterested students back in the days of slide rules.

Most practical curricula only teach how to repeat math, that is, to follow well-known recipes in order to find answers people need, and maybe we are seeing disheartening effects of lack of practice doing algebra and arithmetic. I think most people only use the math they've really practiced and feel comfortable with. But if you think about the benefits of playing with math when it comes to pattern recognition and developing logical ideas, then perhaps the more time spent on calculators and other toys, the better.

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Not a fan of Asimov's The Feeling of Power, then? ;) – Yemon Choi Feb 29 '12 at 9:11
(looks up reference) That's a rather extreme idea. To be clear, what I'm really in favor of are interactive calculational tools and/or games. For example, a derivative-taking program that allows you to graph a function, pick a point by clicking, construct any secant line by clicking another point, and calculating its slope. I'm talking about tools that present mathematical concepts in terms of actually doing the math, rather than formulas. Formulas are a time-saving way of representing mathematics; computer-assisted graphics are too, and less hard to interpret. – Elizabeth S. Q. Goodman Feb 29 '12 at 19:23
I've had previous colleagues say good things about judicious use of GeoGebra and the like. (Though I've not tried such things myself.) – Yemon Choi Feb 29 '12 at 23:52

The OP asks for comments from university-level professors on whether (a) they have seen a decreasing trend in arithmetic skills among their students over time, and whether (b) such a trend might be attributable to the use of calculators. 1973 was roughly the last year in which one could teach freshman calculus to a group of students who had not been exposed to electronic calculators. Anyone who was teaching freshman calc in 1973 is at least ~64 years old, so at most we will have a very thin cohort of teachers who can comment on how their own students in 2012 compare to their past students who used slide rules.

It's also very risky to use anecdotal or subjective evidence to measure such trends. The best objective evidence that I'm aware of is in a book called Academically Adrift, Arum and Roksa, 2011. The authors document that certain downward trends have indeed occurred over the last 50 years. Two such trends are a marked decrease in the time students spend studying and a decrease in the amount of improvement in critical thinking and writing skills that occurs while students are in college. These trends persist even when one controls for such factors as the greater percentage of the population that now attends college.

I have been teaching physics at a community college in California since 1996. In my experience the main difference between students who have taken a calculus course and those who haven't is an increased probability that they will be fluent in basic arithmetic and algebra (e.g., being able to solve a=b/c for the variable c). This may indicate that they can't pass calculus with a C without these skills, or it may be an example of self-selection.

I find that very few students who have passed calculus can do any of the following without extensive coaching and remediation: Differentiate or integrate any function that is expressed in terms of variables other than x and y. Differentiate or integrate an expression containing symbolic constants. State the geometrical interpretations of the integral and derivative. Find the value of $x$ that maximizes $-x^2+x-2$. State under what circumstances $\Delta y/\Delta x$ is a valid measure of a rate of change, and under what circumstances $dy/dx$ is needed instead. Determine whether a car's odometer performs differentiation, or integration.

In other words, if we label the courses in a college-level math sequence with successive integers, what I find is that students who have passed course $n$ can only be counted on to display some level of competence in the material covered in course $n-3$ or $n-4$.

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I would like to know what kind of "prerequisite exams" you give/have given to your students. Such a simple test of competence before embarking on a study of calculus not only could serve as a measure of what to study, it could serve as a handy cheat sheet of what algebraic manipulations to perform for various problems and how to check ones work. Gerhard "Ask Me About System Design" Paseman, 2012.02.29 – Gerhard Paseman Mar 1 '12 at 0:32
@Gerhard Paseman: My only relevant experience is that ca. 1997, we wrote a diagnostic exam for the students in our algebra-based physics class. The test was mainly about area, volume, ratios, and proportionalities. E.g., if $y$ is proportional to $x^2$, and $x$ goes up by a factor of 3. By what factor does $y$ change? Most couldn't even decode the statement of the question. Many also were unable to distinguish area and volume, e.g., to tell which was more relevant when deciding how much paint was needed to paint a car. We didn't find much correlation between the exam and success in the course. – Ben Crowell Mar 1 '12 at 0:52

I have the same issue with business students in my class so I guess the problem is more widespread than just math undergrads. In order to combat the issue, I re-designed my course so that repetition is the key theme. In other words, the same concept/formula is emphasized via in-class examples that are solved by me, via out-of-class graded assignments, via in-class ungraded assignments and sample exams. Students are allowed to work with each other on assignments and sample exams but the exams are individual exams.

My hope is that repeated use of the same formula/concept in different contexts and allowing them to talk to each other for assignments/sample exams will help them internalize the ability to answer questions involving basic algebra/arithmetic on the exam as well. I am not sure to what extent my answer generalizes to math or if it gives you any ideas for your own course. Their performance on the first exam (scheduled for next week) will probably tell me if my approach is working or not.

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Pure Mathematics is some kind of art. You cannot teach art: You may teach history of art, You may teach what works was created by some art-enabled people. So in Math is the same. If You want to educate new mathematicians You should follow with guidelines which are used in Art Academias. They are good: they accept creativity, fresh look etc. whilst still they teach about history of art, some movements, achievements of the future etc. Look: real artist is also great craftsman in his area!

In Math there is also practical aspect: applied math. Here You have to learn more crafts than art, but sometimes art goes from the air.

The only solution to problem You mentioned is to learn math in a way which is seen as interesting and important for Young people. They have ability to learn much more complicated things than calculus or even abstract algebra. If You try to play one of this newest computer games You will see that it is sometimes more difficult than proof of something maybe even non obvious. But it is different: it creates emotions. Try to create emotions learning math and You will be welcomed by them, and they will be glad to learn math! That is the way our society works: we like to entertain ourself!

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This issue is a huge component in what I have observed of kids' disengagement from mathematics, not only k-12 but college and grad students in math: first, as this answer notes, people are quite able to do complicated things when_they_care. Second, presenting math as at best puzzles, at worst drills, often quite disconnected from the rest of life, is not nearly as compelling as anything else. I fear there is really no solution, since some of that seeming disconnectedness is the very power of mathematics... A difficult subject to teach. – paul garrett Feb 29 '12 at 17:52

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