Economist's View: "The Mathematics Generation Gap"
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Simultaneously, these arguments tend to be reactionary and ignore the benefits of calculators. A calculator, use effectively, allows for a far larger number of examples to be considered together than the brain is often able to achieve. This is was most apparent for me in pre-calculus classes where students should be gaining abstract intuition about the behaviors of functions. Here, graphing large numbers of functions varying their parameters allows one to quickly get a sense of a parametric family.
There's an implicit statement here that the "gap" is one such that the lower generation is worse off than their elders --- which is a pretty common human narrative, really. I think instead that this difference is less well-ordered than assumed. Technology is definitely capable of improving human cognition and learning by providing new capacity, and curricula need to explicitly study and take advantage of these capacities.
If you don't understand that 50*80 should give you something starting with 40, you don't even understand that getting 3745 as an answer on your calculator because you mistyped is horribly incorrect.
You can chalk it up to "kids these days" all you want, but if you do some basic math problems with a 60 year old the odds are they will simply leave you in the dust while you go looking for a calculator.
> The argument though is that it is simply impossible to
> use a calculator effectively without first understanding
> the basics.
It's certainly true for me. Many's the time, when I've had monster power under my fingertips, that I've brute-forced solutions and failed to gain any insight. Then I've worked a few small, single instances by hand, and realised that there was structure I hadn't seen before.
The pattern-matching, pattern-finding parts of the brain are phenomenal. Sometimes they are best exercised by getting machines to produce loads of examples, and loads of visualisations very quickly.
Sometimes they are better exercised by working tedious examples by hand.
Balance.
In my mind, if you don't know the concepts behind what you are doing, you don't know what you are doing. It's like copy and pasting code from the interwebs and then saying you are a python programmer.
But it isn't about balance, it's about using a tool to extend your abilities, not to replace them. If you are doing 36+19 on a calculator, I would posit that you simply aren't every good at math.
If you focus your vision on these sorts of example, the rise of calculators is reasonably disastrous-looking! My comment meant to destroy the artificial dichotomy that calculators are "good" or "bad". They are clearly capable of generating some appalling lazy mathematical habits, but they're also clearly able to extend your abilities, improve pedagogy, and garner insights impossible to consider unaided.
Science and mathematics have wholeheartedly embraced computers as a tool for analysis and exploration. Education cannot ignore that. Simultaneously, many abstract concepts that people must know to make use of computers are best learned by replicating the exact work the computer can save you later.
The benefits of trading off toward laborious computation are going to be especially pronounced while learning things, of course.
I don't think that's true. And I'll say something controversial. Programming is taught more effectively than math and its because you don't spend a lot of time mind-numbingly repeating the basics.
Half the people on HN recommend teaching with Python for just this reason.
And think about it, when you teach programming do you make your student walk through what every instruction does? When I learned recursion, I walked through fib(n) by hand -- once. That's the sum total of how many times I've done a full hand expansion of a recursion in my life. How many times have I run a full program on paper with the substitution model? Never. How many times have I iterated even a small loop by hand? Never. These are concepts that I understand as well as anything that I know, yet I've never "mind-numingly" worked through them by hand.
A lot of traditional math teachers want us to sit down with an instruction pointer, stacks, heaps, physical/virtual pages, laid out and have us repeatedly simulate program. That's really the mathematical equivalent of doing this rote computation. I say teach them the concepts and give them interesting and challenging problems that make use of the concepts. The concepts will stick better, they'll learn faster, and they'll be more engaged.
I went through this as a child, where you do the same problem over and over again. Some of the symbols change, but the operations are identical.
It reminds me of when I was in 1st or 2nd grade doing long subtraction. And I had informed the teacher that I wanted to create the problems for the class, so she let me (on the chalkboard). But I snuck in some problems where the result would be negative. The kids struggled with these problems, but not a single student in the class said anything about the structure of the problem. And why would they? They learned nothing about what they were actually doing. It was a sequence of subtractions and carrys and other magic.
the fact that when done badly it rarely yields insight does not necessarily mean it shouldn't be done.
I'm about to make a sweeping generalisation.
The problem is virtually no one in the world can teach math properly. Those who love teaching kids usually hate math and transfer their lack of skills and negative attitude. Those who really, really understand math are usually the type of people who would never teach 3, 4, 5 year olds.
People end up doing math despite the system, rather than because of it. They can usually point to one specific inspirational teacher. My specific inspirational teacher gave me lots of mind-numbing exercises and helped me find the insights.
Rote learning of mechanical processes will rarely, of itself, lead to anything other than a hatred of math and a lack of understanding about what math is. Repetition of mechanical processes, guided by someone who really understands stuff, and asks prompting questions, can lead to discoveries and genuine excitement and engagement - I've seen it happen.
How will we ever get really good math teachers?
There I have no answer.
Arithmetic algorithms still have value - for the insight they give on arithmetic and because following a complex algorithm is itself a skill with value.
But there is no need, as we do now, to insist on performing them so many times, or to do them very quickly in exams.
The main problem with this new approach, I feel, is that it makes learning mathematics harder. Building richer conceptual models which is necessary for both applications or theory is more interesting and meaningful, but also more difficult than following prescribed algorithms. It is harder to test in an exam, and where testable the problems are much harder.
This issue of algorithms vs conceptual understanding, is important at the undergrad level too. Eric Mazur has a nice video about this where he also talks about his way of testing conceptual understanding by asking very simple but illuminating questions - http://www.youtube.com/watch?v=WwslBPj8GgI
We're so "drill, drill, drill" focused that we lose sight of why we're doing the drills (Eric Mazur calls it "plug and chug"). It's possible that these drills will in rare instances create the likes of a Colin or an Andrew Wiles, but for virtually everyone else you have a group of students that can solve triple integrals, multiple 5 digit numbers in their head, factor matrices into any form desired -- yet not have a clue why.
Moreover, the nature of innovation in my corner of the mixed-signal circuit design world is such that said innovation rarely (if ever) comes as a result of a computer simulation. Much more likely, a person with a deep understanding of the fundamental underpinnings of his/her particular problem gains insight into its solution as a result of the same experience and intuition that leads to the aforementioned understanding.
I can have a computer calculate Fourier transforms for me all day, but it's vanishingly unlikely that any amount of such calculation will lead me to the kind of insight that sparked the invention of CDMA.
What lets people invent new circuits is their good intuition about circuits, not their ability to solve linear differential equations quickly by hand or to compute integrals by hand. When an expert is analyzing a circuit on paper he is thinking about "what happens if the input to this circuit is a sine wave with high frequency", he's not going to solve the differential equations by hand.
Rather than circuits look at how electromagnetism or quantum mechanics is taught in college. In my case it was integrals, integrals, integrals. Doing these by hand provided approximately zero intuition into the physics. We could have covered more ground if the instructor would just type these into maple, instead of doing them on the board or in the book by hand. Or how many times have I not had to compute eigenvalues of 2x2 or 3x3 matrices. How many times have we not applied crude approximations in class because doing it by hand was too difficult, when typing it into a compute would give you 100 digits of precision in a couple of milliseconds. One time one of my maths teachers how to compute tan(2) or something like that by hand. After half an hour of calculation he had 2 digits. Computing the integral of something to a crude approximation in an edge case strikes me as futile as computing tan(2) by hand.
I don't think there is much insight to be gained about programming in Python from repeatedly simulating the program at the level of tracing stacks, etc. - on that we agree. But I don't agree that doing so is the equivalent of learning how to add, subtract, multiply and divide fractions, or graph quadratics and cubics, or solve linear equations.
I believe the insights gained from doing quite a lot of arithmetic by hand are genuine.
Having said all that, if people aren't going to go on to careers or jobs in hard sciences then I believe the concept of qualitative computation is more valuable than doing any kind of mental arithmetic. But if you want to do proper math, or use calculus for real, or understand Fourier Transforms, or manipulate sheaves over higher dimensional topological spaces, then the abstract insights gained from arithmetic are, in my opinion, more than are generally suspected.
That's not to say that calculators are bad. Enormous computing power has let me get insights that I otherwise wouldn't have had. I'm just saying that not learning and internalising the underlying arithmetic basics is like not learning how to carve basic joints in carpentry.
To do so is to deliberately limit your skill set. Can't be good.
But what are the insights gained from doing them by hand? I agree that learning arithmetic is important. But we spend a lot of time (or at least did) on things like times tables. I remember looking at a page of 30 long addition problems as a kid. I knew that the hardest ones were at the end, so I'd start there. After about three of them I realized -- every one of these is virtually the same thing. But I have 27 more to do.
I was lucky. I actually had a teacher in 3rd grade who let me work at my own pace and actually got math books not in the school curriculum (and looking back, she probably did this out of pocket). But I imagine there are a lot of bright kids who just gave up on math and school out of sheer boredom.
But there are insights about differences of two squares, sums of cubes, divisibility tests, prime factorizations, smaller denominators imply larger numbers, greatest common divisors and lowest common multipliers, and many many more.
I find repeatedly that I show people small arithmetical tricks and they are intrigued and surprised. I then expand on the basic ideas and derive things like RSA and DHMW codes, or the fact that primes of the form 4k+1 are always the sum of two squares, or that for primes larger than 3, p^2-1 is divisible by 24.
And so on. People are often fascinated by these trinkets, and yet they are observations that for me arose from doing the arithmetic.
I don't deny that most math teaching is appalling, and that many bright kids give up out of sheer boredom, but without the basics they are equally ill-served. We need teachers who actually understand the math they are teaching, and not just regurgitating the curriculum they've been given.
That's why I spend around half my time going around talking about what math is really about, and how it can be interesting, useful, fun, and occasionally exciting.
Without a basic facility in arithmetic, so much of real math - as opposed to arithmetic - is denied. If every calculation you do requires that you reach for a calculator, or fire up a symbolic math package, you are slowed to a crawl.
It's like trying to programming without being able to type. The ideas can't flow when you are constantly held up by not having mastered an underlying skill.
And i suspect we are more in agreement than not, each colored by our own experiences. Mine were happy, full of discovery. Yours weren't. How can we make kids experience more discoveries if they won't actually play with the underlying basics?
Indeed. It was Knuth who said that mathematicians and computer scientists think in very different ways. With that I don't disagree.
How can we make kids experience more discoveries if they won't actually play with the underlying basics?
I'd argue that we should let them play with calculators and other tools. My discoveries came as I did programming. In 4th grade I wrote an arbitrary precision number package. I learned more about arithmetic doing that than all of the rote drills combined.
I have a feeling I'm never going to be able to convey what I'm trying to because you're not a mathematician, and I'm not a good enough writer. It's not just about becoming better at the arithmetic, it's about finding and creating structure, order, relationships and mappings.
But I'm going to stop now. It's clear that I'm just not expressing myself well enough to make the point, and I've spent far too long on it. I regret not being a good enough writer - and perhaps not a good enough mathematician - to explain in a way so as to make it clearer.
I almost love this analogy. There's certainly an aspect of "menial mathematics" which is like typing in that it directly translates into fluency of thought. There's also an aspect of discovery, though, that's missing from the metaphor.
There is a large difference between inferring that some equality holds based on abstract principles and actually performing the evaluation and directly tracing out why that equality holds (even very non-generally). I liken it to statistical modeling sometimes: models allow you to talk about and comprehend data on a high level, but only by directly plotting all of the data at high resolution can you let your brain's natural pattern seeking tendencies reach out for further insight.
That's a common theme there: granularity versus generality.
http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_m...
I'm of the opinion that you should always learn the thing at about two levels down in automation from where you normally use it. So, for programming, I'm all for assembler, compiler, and C programming skills, even though those might never be used in the real world. By the same principle, if you're learning navigation in an airplane, you should learn dead reckoning and a wet compass. If you're learning to driver and shoot a tank you should have pretty good concepts of how rifle combat works, etc.
In math and economics, however, I'm not sure what "2 levels down" means. Is math the rote memorization and repetition of stuff? Most definitely not. But does it depend on it? Maybe. Is it the application of pre-existing patterns in any fashion -- such as punching numbers into a calculator? I don't think so. I think it's much more about the ability to teach yourself to find and exploit patterns through trial and error. That's one of the reasons I've always thought so highly of High School Geometry classes -- when done well, they begin to teach how to think, not just what to think.
Food for thought.
Al that rote memorization that everyone hates is the same as understanding a basic sentence.
Major digression though.
http://www.themathlab.com/writings/short%20stories/feeling.h...
What's the equivalent nowadays? Windows doesn't really come with any programming environment. There's no standard for those math books to write towards. Yeah, there's plenty of free compilers for any language, but the funnel from even knowing they exist, to having a reason to get one, then finding it, and downloading and installing, gets pretty lengthy and technical for your average seventh-grader or whatever.
Heck, I would argue that TI calculator BASIC is the closest thing we have to a universally available development environment for secondary students today.
The real problem is the growth of phones and tablets as preferred client devices; while the JS interpreter is still there, it becomes a challenge to enter and run the code.
The trend has been away from this as programming has gotten more difficult and the general decline in resources for developing and updating school curricula.
It seems from many of the other comments is that people recognize that performing exact mental-arithmetic calculations is rarely necessary; however, the more intuitive understanding of how numbers relate in magnitude etc. is critically important. Estimation is something that I think many of us take for granted, but that has some significant mental pre-requisites.
Interestingly, a growing branch of mathematics education has been working to explore whether the traditional rote memorization is the most effective way of instilling this more hollistic understanding of numbers. If people were interested, I could ask some educator friends for more up-to-date links/citations on this topic.
A great article to read is here: http://www.jstor.org/stable/30042661 - much of UG level Economics can be taught and explained with the use of diagrams and graphs.
Overall, I think the generation gap argument is fairly sound - we aren't taught in the same way that our parents and their parents were, and for better or worse, this is how it is.
Presumably, though he didn't discuss this directly, he used more math with those who needed it to understand the mathematical underpinnings of the theories.
His idea was to present the coursework in a way that would make the students think about what they were doing. As he put it about the engineering students, if he presented it in the regular mathematical way, they could have just plugged the numbers into the formulas without necessarily understanding what he wanted them to learn.
I think I might need to revive these for my technical-college math students.
But for kids that haven't gotten the fluency with the arithmetic tables yet it just raises their anxiety level, even to the point of near-panic. Which of course shuts down exactly those parts of the brain that you need to be running well to be good at math. The anxiety association with math becomes the lesson learned and as a result students can end up absolutely hating math because they feel sure that "I'm just not good at it".
Of course, it could be a fun experiment to try on some college students.
If a child is this adverse to a simple competitive math challenge, (or any competitive challenge) the problem is much larger than hating math.
Not everyone takes to math, but not everyone takes to reading either. Both skills are essential to a certain level to function properly in our world. You wouldn't suggest that a child who finds reading difficult not be expected to read a novel, so I'm not sure why there is such aversion to multiplication tables.
Even if all the child does is mesmerize the patterns, they've still learned a useful construct that will serve them well for the rest of their lives.
What I'm saying is this particular test is notorious for turning borderline kids into outright failures. I spoke with an educational PhD specialist about it once.
There are issues here of personality, temperament, and biology. If you haven't experienced it yourself or seen it up close, it's very hard to appreciate but very easy to think that you do.
Think of a skill that takes a significant amount of gentle, enjoyable practice - until it clicks. Like whistling or riding a bicycle. We wouldn't expect kids to learn to whistle or bicycle by having daily contests to see who could whistle the most notes or bike the fastest in 3 minutes, would we? No, the kids who hadn't got the hang of it yet would simply be repeatedly crashing and burning in front of their peers.
Whistling and biking are relatively easy to pick up once you have the physical maturity, but the development that enables math kicks in over a range of several years in different people.
This is also the time the kids are developing likes and dislikes and academic self-image. At this age, it's better to learn to like math and to enjoy the experience of new concepts sinking in. Pressure drills which tend to exaggerate and reinforce differences can be downright harmful for elementary school kids.
Personally, I'm not familiar with these exact games, so perhaps I didn't grasp the full weight/level of competition they encouraged.
I do agree with you that learning anything requires a level of comfort prior to applying what you know. My assumption is that such a comfort would be achieved prior to using a competition as practice.
Food for thought, thanks.
I like the "gamification" aspect, and would prefer to frame this as self-competition.
The same "being beat down at math again" reaction can be turned around into "OMG I'm getting better at math" with some empathy and debugging. It's a wonderful thing to see.
I have students who can't do exponents and have trouble answering 3 x 5. We're trying to teach them symbolic and linear algebra. Dur. No wonder they can't keep up.
I can debug the exponents deficits (integer exponents) in about 3 hours with a small group. Perhaps another session for radicals and fractional exponents.
It's somewhat addictive, to see them realize "I don't have to suck at math".
It really is.
I have students who can't do exponents and have trouble answering 3 x 5. We're trying to teach them symbolic and linear algebra. Dur. No wonder they can't keep up.
I think the thing to keep in mind is symbolic math and 3 x 5 are almost completely different tasks.
We lump them together under "math" but I've met many teachers who are excessively attached to these false dependencies.
I'd love to bring back Euclid himself and sit him down in one of these third grade classrooms and see how he does on the Mad Minute. It's not like he knew how to do long division or anything. :-P
I can debug the exponents deficits (integer exponents) in about 3 hours with a small group. Perhaps another session for radicals and fractional exponents.
It's somewhat addictive, to see them realize "I don't have to suck at math".
Thank you for what you're doing.
It's hard to imagine not being able to do it if you're practiced, but it's as hard to do as anything else you never have to do when you never do it.
Ban television, radio, electric lights, the horseless carriage, automated looms, steam engines, the printing press, and writing while you're at it.
This sort of "the basics are really important" nonsense is sometimes heard from the old directed at the young because they can no longer argue against calculators themselves, so they make a proxy.
As a result they have no quantitative intuition, which means they have no idea if their arithmetic results (achieved by simply hitting buttons on a keypad) even make sense.
That's scary. It'll be quite difficult to teach my kids math if their own schools don't see it as a requirement.
There is a lot of "cheats" possible when calculating approximations that makes it much easier to learn/perform.
In real life there is often so many uncertainties in the source numbers that a 'precise' answer is not meaningful anyway. Many people think that all the digits displayed in the calculator are significant/meaningful.
If you are a painter that estimates an offer price to the customer, you don't need to be able to work out in your head that 10.5 * 21.5 is 225.75, the estimate 'approx 215-230' is almost always good enough.
This is like saying we should teach children how to swim by letting them splash around in the wading pool.
In real life there is often so many uncertainties in the source numbers that a 'precise' answer is not meaningful anyway.
Math is precise. That's the whole point of it. I don't want my banker, accountant, civil engineer, pilot, cartographer, or architect working on "good enough". I want them to be precise. Consistently.
Your brain is not. I've seen a lot more evidence of people bootstrapping from intuition up to mathematical precision than simply starting with mathematical precision. Taken to its logical conclusion (and I do mean that I believe this is the logical conclusion), this leads to trying to teach number and set theory to kindergarteners, berating them for failing to get it, throwing your hands up and declaring they just aren't suited for math. You have to start with quantitative intuition, the alternative is not to teach math at all. There isn't a "start them out on correct pedagogy immediately" choice. You can and should argue about what tradeoffs are best, but you will have to have tradeoffs.
That is logical, and it's vitally important in the understanding of all future concepts. You don't need to understand advanced calculus to grasp the concept that 2 3/4 oranges + 2 1/3 oranges is not 5 oranges altogther, but actually more, and that left over bit is in fact meaningful and relevant.
It's when you start to play with abstractions that mathematics becomes confusing, not when you are being precise.
"some bit more than 5" is plenty of precision for nearly any conceivable situation in which your example could appear.
https://secure.wikimedia.org/wikipedia/en/wiki/Significant_f...
It's quite a bit more complex to expect the child to discard the 1/12 and suggest there are 5 oranges. It's not that the 1/12 is useful for anything, it's the fact that it exists, and is accounted for.
No, this statement is too strong. There are many examples in mathematics where one cannot exactly calculate a result (e.g. because one doesn't know all inputs), but one can give an estimation and prove that it is good enough for a certain purpose.
Example: Chebyshev's inequality in probability theory: http://en.wikipedia.org/wiki/Chebyshev%27s_inequality
More importantly, I'd suggest that you need to know very basic math (simple addition, subtraction, multiplication and division) on the same level that you can type at your keyboard or drive a stick shift.
If you're spending brain effort on the basics, you've got less of it left for the complexities, making the entire exercise much more difficult.
It's a lot of work but it has gotten to the point that I feel like I'd be screwing my kids over to send them to school. Changes are in play that may make me comfortable with them sending their children to a 21st century school, but I can't stomach the thought of sending them to these 19th century monstrosities in their final days.
It is the curriculum.
Thirty years before 2011 is 1981. That's not true of graphing calculators generally being in the hands of high school students. By that year, many engineering students at university had calculators that could do calculus (usually with a "solve" algorithm) but the graphics plotting was just beginning for hand-held devices for university students. I was alive and studying math in the relevant years. Four-function or "scientific" calculators in K-12 schooling were just coming in during the 1970s.
After edit: while I disagree that calculators have been pervasive and encouraged in either elementary education or higher education for much longer than thirty years, I agree with the (grandparent?) comment that the curriculum in K-12 mathematics in the United States is lousy, and has been lousy through at least three different eras of curriculum fads, calculators or no calculators. The curriculum is indeed the key issue. Calculators are a useful tool, and today they belong in K-12 and in higher education. For a classic comment on how much calculator technology has progressed in the last fifteen years, see
One week, one of the problems actually required the use of a calculator (it was specified in the problem).
I remember my father taking me to RadioShack, where he purchased a calculator for me, only to solve this one particular problem.
This was in 1986. This one is from 1976...
http://cgi.ebay.com/Vintage-Radio-Shack-EC-490-Scientific-Ca...
Math Superstars link: http://it.pinellas.k12.fl.us/schools/curlew-es/studentconnec...
It was pretty badass.
For example, even the crappiest students in my class (in Physics) knew how to do a taylor expansion to approximate the sine of something, and they knew power series and all sorts of other stuff. You have to know how in order to simplify algebra in many cases.
If there is a generation gap, it's because professors aren't strict enough and unwavering on their decisions to not use calculators, or the courses aren't doing hard enough algebra and calculus to really merit not using a calculator. When you get into the really hard stuff, not even Maple can help you half the time.
Good for you. In France, you're allowed breathe air down to 60m. Most people commonly breathe air down to 40/50m, or did so recently enough.
> Your other comments about narcosis are sheer nonsense, completely off base. Don't breathe narcotic mixes.
My oh so numerous comments about narcosis? What are you talking about? Instead of giving condescending lessons, would mind explain what you mean?
I am being condescending because you are posting dangerous misinformation in a public forum. Divers who believe those lies and fairy tales tend to end up hurt. If you don't know what you're talking about then it's better not to post at all.
I didn't know that people came to HN to learn about diving. On the other hand, they go to wikipedia, where you should have a look.
Anywhoo, this is - IMO - something that the author's college should look at collectively. If there are some students and professors with massively different ideas of the required level of mental maths skills, perhaps the college should look at introducing a mandatory first year 'mental maths' crash course/module?
This would help to make things a little more consistent. If there's genuine confusion/disagreement between the students and Professors, this should - IMO - be addresses by a course-wide decision being made.
Regarding programmable calculators - they can be reset in about 2 seconds total (it's usually Menu -> Settings -> Memory -> Reset All).
In our University, the exam invigilators ensure that all programmable calculators are reset (with them watching them being reset, of course) before the start of the exam.
So I'm not sure why this (to me) fairly obvious idea seems to be overlooked in the article? As I say, it takes 2 seconds total.
An interesting article though; even though I think the author/prof is approaching things in a slightly muddled (for want of a better - non insulting- term!) way. The college should (IMO) decide on how they want to approach things, and then be consistent across all modules and all Professors.
Next step - lets allow all kids to use google search during their tests.
"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.
". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.
"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.
http://www.amazon.com/gp/product/0387982892/
I have a collection of analytic geometry and calculus books, accumulated as used books from various readers, that includes the books used by my late father in his higher education as a chemistry major during the Truman administration, followed by books from other previous owners reflecting "new math," "back to basics," and "reform" approaches to mathematics education. Plainly today's secondary and tertiary students of mathematics need to take advantage of current technology so that they can devote more time to THINKING about the mathematics they learn and less time to what even any mathematician would call "tedious calculation." But too few students have ever been guided to through the kind of insight-producing problems in which the tedious steps themselves and the false starts while struggling with the problem produce deep understanding. Stillwell gives examples of such problems in his books, and the minority of students who participate in math contexts or who voluntarily work the "challenge" problems not assigned in their textbooks may gain such insight, but most school textbook problems of all eras are mere exercises, and too few students do enough of those thoughtfully to have hope of learning mathematical concepts.
See "Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education," American Educator, Fall 1999, Vol. 23, No. 3, pp. 14-19, 50-52 for additional commentary on mathematics education,
http://www.aft.org/newspubs/periodicals/ae/fall1999/index.cf...
and see an earlier HN comment
http://news.ycombinator.com/item?id=2515796
for a FAQ on the distinction between problems and exercises in mathematics education.
In fact I'd argue that I never really understood much of any math until grad school. I was computational sophisticated, but lacked understanding.
And oddly, I seem to find quite the opposite problem from what the blog author describes. I find students who know 3x5. But struggle to understand when the Fourier Transform is appropriate. Sure, if they're looking at problem sets at the back of the chapter about Fourier Transforms then they'll start with it, but in the real world they lack the conceptual understanding of it. I've met students who can compute the SVD, they can tell you the text book definition, but don't actually intuitively know what it means. They don't know when it should be applied, or when it is applied, what it means.
[1] http://blog.wolfram.com/2010/11/23/conrad-wolframs-ted-talk-...
I see this all the time when I help someone with a programming assignment. They may understand the problem, they may understand each individual solution if you explain it to them. However, they usually do not know where they should start. They may understand what a hash table is, and they may even know how to implement one. However if a problem does not fall into one of the patterns they are used to for a "hash table problem" they may not realize right away that they could use such a solution.
Kids need to be taught how to break down a problem into its elements, and then realize which elements can be solved most effectively by what tool. This can also be learned through experience. Program enough and you will eventually start to break down problems yourself. But some kids don't seem to take initiative and work on themselves outside school. Which also is the reason why some kids from Umass CS are working at Facebook, Google, Microsoft, and others serve me coffee.
It is positive statements that require proof, not negative ones. If you believe that the introduction of calculators, google maps, etc., has negatively impacted number sense and human spacial reasoning, it is on you to prove it.
I didn't learn to program by sitting in a class having someone drone on, I learned it because I needed it to solve problems I had.
110 * 111 = 101010
Very easy, don't you think? :)
This is why payday loan sharks target poor areas: lack of education.