
The mathematics generation gap - ColinWright
http://worthwhile.typepad.com/worthwhile_canadian_initi/2011/05/bridging-the-mathematics-generation-gap.html
======
mathattack
This article has an economics spin to it, which is interesting as only
insiders seem to understand how much mathematics is required in that field.
Greg Mankiw lays out a similar defense of mathematics in what we might assume
to be a more logic or intuition driven field.
([http://gregmankiw.blogspot.com/2006/09/why-aspiring-
economis...](http://gregmankiw.blogspot.com/2006/09/why-aspiring-economists-
need-math.html)) His advice for aspiring Phd students in economics suggests as
much mathematics as many undergrad programs require
([http://gregmankiw.blogspot.com/2006/05/which-math-
courses.ht...](http://gregmankiw.blogspot.com/2006/05/which-math-
courses.html)).

What's this mean about calculators and learning the basics? There are concepts
that you can't mentally understand if your mind can't map the basics of math.
I see it in real world examples (people can't make quantitative business
judgements on the fly) as well as abstract reasoning. One of the critiques of
Google is similar - by reducing what we need to remember, we reduce our
ability to process certain ideas on the fly.

Like many things, it's ultimately a trade-off. Are the things we mentally miss
from utilizing calculators and Google more or less important than what we gain
in added productivity? (Is it worth the health hit of driving 5 miles to work
rather than walking? Can you mitigate it by spending half the saved time in
the gym?)

------
zeteo
People who can do basic arithmetic on paper have an advantage with back-of-
the-envelope calculations and ballpark estimates, and will end up using such
simple and effective tools more often. The mental cost of pulling out a
calculator and concentrating to enter all digits in sequence is just too high
sometimes.

There is also an often overlooked educational advantage in that pupils get
early exposure to working with simple algorithms. Becoming used to the idea of
following a sequence of unchanging steps, for varying inputs, comes in handy
for understanding more complicated algorithms later on (such as those involved
in algebra, calculus, and computer programming).

~~~
CWuestefeld
Having an understanding of the meaning of a mathematical concept does not
require continuous, ongoing, manual, exercise of its mathematical calculation.

For example, I'm willing to bet that virtually all of you reading this fully
grok the concept of "square root". Your understanding of this is probably
strong enough that you can wield it in day-to-day usage, successfully
employing it to build back-of-the-envelope estimate, etc. Yet how many of you
actually know how to calculate a square root by hand?

I've been shown how to do it, but have never done it myself but for once or
twice. I also know how to estimate it via via calculus and interpolation. But
I don't know how to do it now, and I've not even done it enough times to have
gotten an understanding of the workings of "square root" by way of repeated
manual application of the mathematics.

So, how to explain the fact that I (and presumably you) are so comfortable in
employing the concept?

~~~
zeteo
Knowing how to do a square root efficiently by hand is probably a bit
overkill, although it can still be useful sometimes. I was rather arguing
about addition, division etc., which come up much more often in back-of-the-
envelope calculations; and which also provide a much easier and earlier
introduction to the kind of algorithmic thinking that is required in algebra,
programming etc.

------
lmkg
_Calculus_. Not arithmetic, calculus.

Economics, even the basic stuff when taught at the college level, involves a
couple of integrals. If you have a good, solid, intuitive grasp of what an
integral is and what it represents, all this stuff is obvious. Obvious to the
point that when the prof says "this is an integral" that will explain the
concept better than the English definition. On the other hand, if integrals to
you are voodoo witchcraft that puts numbers on a physics exam, then when you
reach the same point in the lecture you'll be dealing with something extra you
have to learn, rather than something that helps you learn. You'll be worse at
economics because you can't switch back and forth between the concepts and the
models. If someone gives you an equation, you won't be able to interpret it,
and if someone defines a new concept, you won't be able to calculate it.

~~~
Homunculiheaded
I missed Calc in hs/undergrad but need it much more later on (and felt pretty
innumerate without it), I remember picking up the classic "Calculus Made
Easy". I was immediately stunned by it's simple description of integral:

"Now any fool can see that if x is considered as made up of a lot of little
bits, each of which is called dx, if you add them all up together you get the
sum of all the dx's (which is the same as the whole of x). The word 'integral'
simply means 'the whole'"

After reading this and more I asked a few people who I know knew calculus how
they would explain an integral (figuring that they would give a similar
definition and would delight in finding a book which explained it so clearly)
and was shocked to discover how many could calculate it without really
understanding what it was or at least being able to describe what it was in
clear terms

~~~
sliverstorm
I think of everything in the form of signals, so I define the integral as "the
area under the curve".

------
jriddycuz
I vehemently disagree. The gap we should be worried about is the reason gap.

While I enjoy math tremendously, part of my deep dissatisfaction with
economics as a field is its _incredible over-reliance on math_ as a tool for
analysis. I'm speaking as someone who dreamed of being an economist all
through high school and early college. Classically, economics had very little
to do with (numeric) math, and much to do with reasoning about how people
behave.

However, as modern statistics (and math as whole) began to develop rapidly in
the early 20th century, and as logical positivism became a dominant philosophy
(<http://en.wikipedia.org/wiki/Logical_positivism>), economists took note and
begin applying these tools liberally to their field. They started collecting
and compiling tons of data on anything they could measure. Data is compelling:
numbers give a sense of precision and clarity that mere reasoning does not.
But this appeal is also what makes numbers dangerous. Though rigorous
empirical testing of hypotheses in science is clearly one of the greatest
advancements of the last 200 years, it has often been misapplied to other
fields where the same controls are hard to apply. And experiments without
controls can produce essentially meaningless data. Economic data is
particularly complex, and there is still much debate as to how to calculate
even very basic oft-quoted economic figures like inflation, unemployment, and
GDP.

Though there was significant debate about the usefulness of these new tools,
they became enshrined by the two dominant mainstream schools of the 20th
century: Keynesianism and Neo-Classicalism. This bastardization of the field
has made economics into a cargo cult science, where researchers regularly base
their knowledge on data that is only slightly more controlled and scientific
than corporate accounting.

This is not a trifling academic concern. So much of our lives is affected by
what economists do and say. The bigger concern I have for young economics
students is that their lack of mental math skills will make them more inclined
toward the kind of overly precise large number manipulation that computers and
calculators make so easy. I hope, for all of our sakes, that these less
mathematically-inclined students will instead be wary of the numbers and
critically apply reasoning to the models and assumptions they have been
taught.

~~~
thaumaturgy
It sounds like you're calling for the application of reasoning without
measurement. How would that work?

~~~
jriddycuz
What I'm calling for is for economists to drop the pretense and misconception
that using empirical methods for studying macroeconomics makes it scientific.
And I'm suggesting that being good at math is not useful unless you're using
useful data. In the study of logic, arguments can be considered _valid_ if
they are formally correct, but still _unsound_ if their premises are false.
Much in the same way, one can perform any number of valid mathematical
transformations on data but still be left with unsound conclusions if those
data were gathered incorrectly.

I am not saying that empiricism is inherently flawed, or that we should stop
collecting economic data. And I I do not intend to advocate any particular
school of economic thought here. All I'm advocating is that students be taught
how to think critically about what they are being taught. So much of a modern
economics education consists of looking at the changes in figures over time
that very little is spent focused on a more general kind of reasoning.

The kind of reasoning I'm calling for is not easy to define. This is one of
the tremendous advantages numbers have over argument in most minds. This kind
of reasoning takes into account the notion that most of the information we
obtain is not perfect or complete, and that many of our determinations are
really judgment calls on what is more likely to be true. If empiricism is
reasoning with your eyes, this is reasoning with your nose. It is a trained
skill that allows you to recognize dubious premises and unspoken assumptions.
When refined, it allows you to distill the essence of arguments down to a set
of axioms that you can use to build a coherent model of the situation at hand.
It is this _theoretical_ side that allows you to understand how to construct
experiments that test hypotheses, or whether that is even possible in each
case.

To demonstrate the importance of gaining an understanding of the theory and
rules behind something before testing it, I offer a parable:

 _The commissioner of the NFL once decided that teams were punting too much
and he hired an econometrician (economic statistician) to study the situation
and provide a solution to this problem. The econometrician applied his skills
to the task at hand, aggregating data from several seasons to find
correlations. He noted that there is an incredibly strong correlation between
forth downs and punting, and he recommended that the commissioner ban fourth
downs. In the next season, offenses were only given three downs. To the
econometrician's surprise and the commissioner's chagrin, teams actually
punted more frequently, as the fewer number of downs dramatically limited
offensive opportunities._

The econometrician's misunderstanding was based on something rather obvious
(if you understand American Football): a failure to separate correlation and
causation due to an ignorance of the rules of the game. And compared to a
global economy, football is a very simple game, with very simple rules.
Applying reasoning to the example is very straightforward, but applying the
same thing to a world of dynamic human behavior is much more subtle. Which is
why students ought to be trained to question assumptions and sense where logic
and math have separated themselves from the reality they are supposed to help
us describe.

People will disagree about when things correlate to reality, and about what
things make sense in parables. But almost anyone can learn to recognize when a
number seems too specific, just like most decent coders learn to recognize
"code smell." Just the other day, someone told me confidently that 65% of
communication is non-verbal. Now, while I almost agree intuitively, I
immediately asked where they heard that, and how someone could have arrived at
that figure, which seemed oddly specific for something (communication) that I
don't think is frequently quantitized. Every student of a soft science needs
to have this skill strongly developed, or they will begin to take these kinds
of things at face value.

------
usedtolurk
"students can now solve problems that were previously too time-consuming to
attempt, and can focus on underlying concepts."

The question should be whether the more time-consuming problems are helping to
understand the underlying concepts.

In my experience this is not always the case. My high school maths teacher
made us do extensive graph plotting by hand and it was slow and tedious.

At university, my physics course used software to visualise vector fields. We
could tweak the inputs and see WHAT it did immediately. We could examine much
more complicated scenarios and it seemed we were gaining an intuitive feel for
the subject, but we didn't really learn HOW the inputs lead to the outputs so
we couldn't apply the knowledge to other scenarios (even simpler ones) without
the software.

2 decades on, and I wouldn't know where to start if I had to tackle even a
trivial vector field but I can still picture reasonably complex graphs in my
head just from looking at a formula which I find surprisingly useful in
everyday life.

Perhaps mine was just not a very good course (in an otherwise excellent
program) but care should be taken to ensure the technology really does aid the
understanding of the underlying concepts and not distract from them.

------
Sukotto
Reminds me of the classic story of Richard Feynman vs the Abacus:
<http://www.ee.ryerson.ca/~elf/abacus/feynman.html>

I never really memorized basic math tables in school and it annoyed me at a
low level for years. So a few weeks ago I added addition (0+0 ~ 50+50) and
multiplication (0x0 ~ 12x12) tables to my flashcard app to try and remedy that
lacking. It's been depressingly and frustratingly difficult so far.

~~~
3pt14159
Out of all the comments this one made me the most depressed. 11x12 is
something you need to think about?

At my high school we had these expensive graphing calculators that were
supposed to last for four years. I of course lost mine in early grade 10. Best
thing by far that happened to me.

I was always that the top of my class for mathematics, so I was usually pretty
bored with the homework, but all of a sudden I thought of a new challenge: Do
all the homework without any calculator whatsoever. Yes, even things like 3^4
+ 5^(1/4). At first of course it was painful, but at least I was doing
_something_. I had a much better understanding of actual graphing than anyone
else as well, because I would boil the functions down to their key points and
play with them until I saw what they did at infinity.

I ended up getting my calculator back in late grade 11 (funny story, I had to
sub in for a math teacher at the last minute because there was a wave of
sickness at the school, so I was teaching people one year younger than me) and
I saw my trusty calculator on the desk in front of me (I had engraved my name
with a knife). I picked up and said "hey thanks, I've been looking for this."

Since then I've always been able to do math in my head and it is unbelievably
useful. When people say "oh so that is log(800), what is that" and I respond
with "roughly 2.9"* they are flabbergasted, but I know how the curves actually
work, I haven't outsourced that knowledge to a machine.

Programming is great and we should have graphing calculators in some part of
the math curriculum, but students should be able to do this type of math by
hand/in their head.

*I checked this after answering it in my head and was mildly amused. Here was my thought process: log(1000) = 3, log(100) = 2 a log graph has a consistently degrading slope, but one that will still take you to infinity, so the last 200 units in the 100-1000 range will be worth less than the first 200 units. And since halfway there is roughly log(300) and we assume a kinda-semi linear path form there (rounding down because 800 is higher on the scale) it comes out to roughly 2.9.

~~~
sliverstorm
_Out of all the comments this one made me the most depressed. 11x12 is
something you need to think about?_

I don't find that especially depressing. If you have to sit down with a
pencil, yes, you need to play catch-up a bit, but is it so terrible not to
have it memorized? I personally always get by with things like "12x12 = 144,
so 144 - 12" or "12x10 = 120, so 120 + 12". I just don't have a use for
memorizing the complete table, so while I knew it 2 decades ago, I have lost
pieces of it.

~~~
rdouble
To multiply 11 by 12 you just "split" the 12 and put a zero between the
digits: 1_0_2. Then you add the digits in the 12: 1+2 = 3. Then you add the
result of the 2nd operation to the zero in the first split: 1_0+3_2 = 1_3_2 =
132.

This technique works for any 2 digit number multiplied by 11.

What IS especially depressing is that the slowest and least efficient
arithmetic techniques keep being passed on for generations. Mainly because
it's easier to teach the dumb way and teachers are too lazy to learn how to
teach anything new. The result of this is that most students end up hating
arithmetic and calculation, and subsequently any further math, because they
are never taught the fast and fun way to do things.

If you ever have a job interview at a hedge fund or trading firm, you will not
get past the phone screen if you can't do this sort of arithmetic in your
head.

~~~
sliverstorm
Well, to each his own. I always _hated_ the kind of math where you have to
remember a basketful of little tricks, like your method of computing products
of 11. I have always much preferred "the-method-I-invented-on-the-spot", my
approximate method using nearby known numbers being one of them.

The most exciting math tests for me were always the ones where I couldn't
remember the 'trick' for half the problems, and would re-invent the solution.
It didn't always go well, but the GREAT SCOTT!!! moments were some of the best
in all of college.

~~~
rdouble
Approximate estimation is what you need to multiply other two digit numbers
quickly. It just happens that there's a trick for multiples of 11. There's
actually probably less than 20 other "tricks" that help with rapid
calculations.

~~~
AlexandrB
I don't understand why this trick is necessary. For 11x<number> you can just
do 10x<number> \+ <number> which is easy until you get to 3 digit numbers, but
even then continues to work.

~~~
rdouble
Speed. The "trick" breaks down the problem so there's always a simpler
summation to perform. In half of the possible products, one only has to add
the sum of the digits to zero. For the rest, just slot the 2nd digit of the
sum where the zero goes, and add 1 to the first digit. Most people can do
either of those two simple additions faster than they can do sums like 210+21
or 390+39. If one is already quick with the latter kind of addition, then the
trick is unnecessary.

------
rmc
Is there any evidence that being unable to do mental arithmetic has any
correlation with understanding complicated mathematics?

All to often with articles like this I can't help but think of grouchy old
people with congnitive dissonance who are now conviced that (a) young people
are stupid and (b) things were better in their day and (c) since they had to
learn mental arithmetic the young one should learn it too.

~~~
tokenadult
_Is there any evidence that being unable to do mental arithmetic has any
correlation with understanding complicated mathematics?_

That's a very good question. This question has not been studied as rigorously
as it should have been. Here are some suggestive observations. Professor W.
Stephen Wilson surveyed many colleagues (other research mathematicians) once
to ask if they thought advanced mathematics could be learned without a basic
understanding of arithmetic. The responses he received

<http://www.math.jhu.edu/~wsw/ED/list>

included comments such as "I am shocked that there is any issue here" and
"That it is even slightly in doubt is strong evidence of very distorted
curriculum decisions" and "One of my favorite attacks is that we are _helping_
the students by insisting that they do things by hand because otherwise they
can waste a lot of time when the calculator would fail them." One especially
thoughtful comment, by a mathematician who has long thought deeply about
teaching mathematics, was "It might be argued that we do not really require
students to fiercely add, subtract, multiply and divide in our university
courses - which is true. But we do require an automatic understanding of these
operations and why they work because WE BUILD FROM THERE."

The longer story about understanding arithmetic--REALLY understanding it--and
how that relates to learning beyond arithmetic can be found in the book
Knowing and Teaching Elementary Mathematics by Liping Ma

[http://www.amazon.com/Knowing-Teaching-Elementary-
Mathematic...](http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-
Understanding/dp/0805829091)

(well reviewed by two mathematicians who study mathematics education)

<http://www.ams.org/notices/199908/rev-howe.pdf>

<http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf>

A classic article on the subject is "Basic skills versus conceptual
understanding: A bogus dichotomy in mathematics education"

<http://www.aft.org/pdfs/americaneducator/fall1999/wu.pdf>

A recent effort to embody strong conceptual understanding of basic skills into
a mathematics textbook is Prealgebra by Richard Rusczyk, David Patrick, and
Ravi Boppana,

[https://www.artofproblemsolving.com/Store/viewitem.php?item=...](https://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra)

which points to what young students should be able to do WITHOUT a calculator
if they really understand mathematics well.

One example I know, related to me by an economics professor, is teaching a
lecture on economics in which the professor (the one who told me this story)
said, as part of a calculation, "20 percent," and then was interrupted by a
student who asked, "You just said '20 percent,' but you wrote '.2' on the
blackboard. Why did you do that?"

~~~
jerf
What it really all boils down to from my point of view is that the question is
"Can we really teach mathematics ( _qua_ mathematics, as opposed to raw
computation) without at some point creating a firm intellectual
foundation?"... and there are some people arguing _yes_ , which just boggles
the mind.

Yes, you may never in life have to subtract 1/4 from 3/4, but even so, in the
grand scheme of mathematics manipulating fractions is the _easy_ part, and
still useful if only as a really good place to practice and pick up useful
manipulations on relatively concrete items, before you move on to more
abstract versions of the same operations, which _are_ used everywhere in
mathematics.

(I'd also point out I phrased it as "at some point", not "at the beginning";
in general we can't solve the chicken-and-egg problem and just start
Kindergartners on pure set and number theory. But at _some_ point you need a
decent foundation laid down or you will never build a strong structure.)

~~~
rprospero
The argument isn't that we can teach mathematics without a firm intellectual
foundation. The argument is that arithmetic is not that foundation.

No one would argue that you should know the turn by turn direction to drive
from the White House to the Liberty Bell and use that to generalize how to
read a map. We don't force kids to memorize Harry Potter and then later expect
them to generalize that our into literacy. History courses don't start with
Supreme Court cases and wait until students generalize out the Constitution.

By starting with the abstraction and moving toward the concrete, we give the
information context and allow the students a good mental framework to build
upon as they learn the details. Starting with the concrete and moving towards
the abstract is needlessly confusing and robs the students of the firm
intellectual foundation that they need to understand the subject.

~~~
jerf
I'm not arguing arithmetic is that foundation; I'm arguing its an unremovable
_part_ of the foundation. There's more to it than that, but for all the well-
documented foibles of true mathematicians when it comes to simple arithmetic,
how much math can someone _really_ be learning when they are staring blankly
at 3 times 5?

There's a lot more to programming than creating and calling functions, but I'm
yet to see an interviewee struggle with how functions work in their putative
"best" language _but_ they're otherwise brilliant. You can't be progressing
very far if you're burning _that_ much mental effort on the very, very basics
of the task at hand.

By starting with abstraction and moving towards the concrete, you're
completely fighting everything about how we experience life and learn. Nothing
else works that way. The only abstractions you can teach that way are series
of words that students can mouth without understanding, but have no true
comprehension of.... gee, that sounds a bit familiar.

~~~
rprospero
I'll completely agree with you that a student who can't multiply three by five
doesn't understand mathematics. However, I'd argue that I'd rather have a
student spend ten hours figuring out that 3*5 = 5+5+5 =
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 = 15 than to have a trained parrot that simply
squawks the answer. The first student was slower, but it's the parrot that's
mouthing a series of words without understanding.

As for the rest, I disagree that I'm fighting that standard learning pattern.
In nearly every subject, we start with a generality and then move into
specifics. We learn about the Roman Empire before we learn about Cicero. We
learn about teeth before we learn about incisors. We learn about protons
before we learn about quarks. We learn about function calls before we learn
the standard library. We learn about China before we learn about Beijing.
Almost all learning is performed through iteratively refining a central
abstraction with increasing detail.

------
pnathan
It's well-acknowledged that those who don't understand the fundamentals of a
subject are hampered.

Whether it's _computer use_ , _programming_ , or _mathematics_.

Of course, the prof would do well to become fully familiar with a TI-89 or a
HP-50 graphing calculator. It's ignorant to bash what you don't understand.

Since having a sense of math is required for being a mature citizen in modern
life - think about all the numbers we are flooded with regarding public policy
- it is no surprise that as numeracy goes down (helped by our good friend
Calculator), mass voting wisdom goes down with it.

Speaking as a US citizen.

~~~
rmc
The fundamentals of mathematics are logical deduction, reason and abstraction.
Not begin able to do 134 * 23.

~~~
shadowfox
Unless you are a savant of sorts, being able to do 134 * 23 in your head
requires all three.

For example, you note that 134 * 23 is really 134 * (20 + 3), which is just
2680 + (100 + 34) * 3 which leads to 2680 + 300 + 102 = 3082. And that is just
one way of doing it.

I do agree that arithmetic ability isn't all that great. But there is more
logic and reasoning involved than you would think at first sight.

------
ams6110
Funny I feel the same way about IDEs as the author does about calculators.
They have their place, but they can also be a crutch that contributes to
mental weakness. I know devs who can't write code without Visual Studio
prompting them every step of the way.

~~~
euroclydon
You should at least explain why that is bad.

~~~
ams6110
There is a contention in the subject article that "deep understanding of
mathematical concepts is related to basic number sense" and that e.g. students
who must use a calculator to compute 3 X 5 = 15 are lacking that.

I was implying my feeling that there is something similar going on in
programming... if you are helpless without an IDE, then you are lacking some
"basic sense" about your craft. Though I can cite no studies to back that up.

~~~
arethuza
Someone who can't develop outside of Visual Studio probably is missing some
"basic sense" - but will that really do them any harm?

~~~
mrj
Absolutely. They'll never be able to experiment with a platform that Visual
Studio doesn't support. Or with the many fine languages that don't have a
fancy IDE.

They will artificially limit their career and opportunities based on a tool.

~~~
georgieporgie
_They will artificially limit their career_

What languages provide more jobs opportunities (i.e. career options) than
C/C++[1], C#, and Java?

[1] no, I'm not saying C and C++ are the same thing, but I haven't seen a C++
IDE which isn't also a C IDE.

~~~
mrj
Using only Windows is very limiting.

~~~
georgieporgie
Eclipse is cross-platform. As is Mono, come to think of it.

------
hermannj314
I haven't memorized Taylor series for all known functions, so I occasionally
have to reach for a calculator. I guess I am mentally weak and the sine of a
broken education system.

Oh wait? We've arbitrarily choosen multiplication tables and fractions as "the
thing" to memorize because that's what was done before calculators were
ubiquitous. Thank God. Now I can go back to being smug and imposing my value
system on people I've never met about when calculator use is appropriate. :-)

~~~
jarek
> the sine of a broken education system.

:D

------
scott_s
_Graphing the production function F(x)=ln(x) by entering the function into a
graphics calculator and copying down the result just seems like cheating._

Well, yeah. But math courses that require a graphing calculator don't ask,
"write down the graph for f(x) = ln(x)".

The problem here is two-way. This professor does not understand what the
graphing calculators are capable of (which he admits), nor does he understand
what the previous math courses were like that relied on them.

~~~
Fliko
I've been in a few math courses that have not required a graphing calculator,
we were required to memorize basic functions to help us solve problems more
quickly, and to understand more of the intuition behind the math, and what the
graph is actually telling us.

~~~
scott_s
My intro to calculus class forced us to memorize the standard values for sin
and cos for the quarter-values of pi for the same reason. For those quizzes,
we could not use a calculator, of course. But for the rest of the class we
used a calculator.

------
alphaBetaGamma
I may be old school, but I believe that technology can only very mildly help
with mathematical understanding. Math is build of bricks, one on top of the
others. And to understand advanced concepts, you need a deep and solid
understanding of the lower layers. I think you only get this understanding by
thinking hard, looking at examples and building your own.

Calculators are fine if you are dealing with numbers, but if you work in a
quantitative field you often don’t deal with numbers directly: you deal with
expressions involving variables. You manipulate the expressions, and at the
very end you plug in numbers. A calculator is of no help, and I hope you did
not use one when you learned how to deal with numbers. You better be able to
manipulate fractions, and know the distributive & associative laws, and know
when to complete the square, and the exp(ln(x)) trick, etc… What, you say I
could use Mathematica? Of course, and I do -- when I know exactly what to
compute. But generally I do not: I have these relations and I try to make
sense of them. Moreover, when the final result is nice, concise and elegant,
it means that I do not fully understand the problem. Examining the computation
will help me understand what is happening: what part of the equations cancel
with what other part, and do I understand why? Good luck doing that with
Mathematica.

Another example: continuity. Nothing is simpler: you plot a few graphs, and
the functions are discontinuous where there are jumps. Who need this
epsilon/delta gibberish?

\- Functions that are discontinuous everywhere? Ok, I can wrap my head around
that. Still no need for epsilon deltas.

\- A function that is discontinuous on the rationales and continuous on the
irrationals [1]? Good luck understanding that with your graphic calculator.

\- And the topological definition of continuity [2]? This is a beautiful
definitions, that unifies all the definitions of continuity you have seen for
all these functions of THIS space into THAT space. Well, thinking and well
chosen examples are going to help you understand the definition, not
technology.

[1] <http://en.wikipedia.org/wiki/Thomae%27s_function>

[2]
[http://en.wikipedia.org/wiki/Continuous_function#Continuous_...](http://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces)

~~~
Dn_Ab
I must sincerely and strongly disagree with you. Technology can help elucidate
mathematics beyond plug and chug. Teaching is one of the best ways of
learning. a rough paraphrase of something I read someone say is they write a
new book every time they wish to learn something new.

Writing a computer program which embodies a mathematical concept is teaching
the most retarded entity that is capable of more than just arithmetic.
Certainly anything which is non-constructive falls outside of this, but in
laying a motivation and providing a foundation that reduces the amount of
problems you need to do by say an order of magnitude? Technology is unmatched.
Although the requirement on constructive* maths seems restricted you would be
surprised that both your [1] and [2] level of abstraction can be tackled with
such tools. <http://www.cs.bham.ac.uk/~mhe/papers/entcs87.pdf>,
<http://haskellformaths.blogspot.com/>,
[http://blog.sigfpe.com/2006/08/algebraic-topology-in-
haskell...](http://blog.sigfpe.com/2006/08/algebraic-topology-in-haskell.html)

You do it this way, vary enough examples and try to anticipate results, you
will develop a number sense that is required to be comfortable with maths. It
worked for me.

* I am skeptical in the reality of arbitrarily real numbers because I am skeptical in the reality of hypercomputation.

~~~
slowpoke

      Writing a computer program which embodies a mathematical
      concept is teaching the most retarded entity that is
      capable of more than just arithmetic.
    

I have to agree with this. Solving a problem - this isn't even limited to just
math - with a computer program often means to _generalize_ it. Generalization
requires understanding. Thus, if you manage to generalize something, you have
understood it.

------
orangecat
Economists could do students a huge favor by switching supply and demand
graphs so that the independent variable is on the X axis, like every other
graph in the world.

------
alan-crowe
> And because I've heard these calculators are programmable,...

I'm happy to suggest that a student get himself a programmable calculator and
use it like this:
[http://www.reddit.com/r/learnmath/comments/jl9gz/evaluate_th...](http://www.reddit.com/r/learnmath/comments/jl9gz/evaluate_the_limit_without_lhopital/c2d2u26)

Notice though that I'm assuming that the student is completely on top of place
value and arithmetic. I assume that they can look at 0.17157, 0.16713,
0.16671, 0.16667, and recognise 1/6 emerging from the murk.

------
meric
Forget about "Recent research", look in the history books!

"However, this switching from counting rods to abacus to gain speed in
calculation was at a high cost, causing the stagnation and decline of Chinese
mathematics....In Ming dynasty, mathematicians were fascinated with perfecting
algorithms for abacus, many mathematical works devoted to abacus mathematics
appeared in this period, at the expense of new ideas creation."

<http://en.wikipedia.org/wiki/Chinese_mathematics>

Weee, its repeating!

------
TheCapn
I'm going to speak from the role of recent graduate in Software Engineering, a
strong focus on math here...

1) Visualization of mathematical concepts is a skill that many students simply
lack. When someone asks me to crunch out 1.5 * 3.67 mentally I don't see much
of an issue because its a simple concept for me to break it down and go 3/2 *
11/3 and work from there. Many people don't make the connection and the root
of that issue exists beyond the whole calculator discussion in my honest
opinion.

2) The first reply on that page is a very important one, students with a
strong mathematical foundation are found in the natural sciences. It is rare
to find students practicing strong math skills outside of these faculties. Its
the same argument we saw with Engineers lacking english language skills, the
students who excel in that field are drawn that way so its only natural to
have some deficiencies.

3) Profs from a different generation gap are part of the problem in the idea
that they're slow to adopt a technology-centric system. My honest opinion is
that formula sheets should be allowed in all exams with the criteria that: i)
they're hand-written ii) they're of limited length (1 standard 8.5x11 sheet of
paper, single sided) and iii) no photo-copies This forces students to look at
the information they've been given and make rational decisions for what
formula are important, which ones can be memorized, and enforces studying. My
electrostatics course forced memorization on us and I spent more time shoving
formulas into my noggin instead of applying concepts to solve actual problems.
As soon as the final was complete those formuli leaked from my brain onto the
floor and are now forgotten. I honestly wish I was allowed more time to solve
problems then to attempt memorization.

Basically what I mean to say when I state that "profs are slow to adapt" is
that we live in a technology centered world now. Information such as formula
or constants are accessable almost instantly from google and the likes so
instead of using valuable time to have students memorize these things time
would be spent better if they allowed "cheat-sheets" and instead required a
higher amount of work or knowledge per semester. I'd gladly trade the time
spent memorizing different formula for magnetic flux in favor of another two
or three chapters from the book giving more than the "ideal scenario" cases we
were given.

------
CesareBorgia
Economics demands a strong background in math and any undergrad program worth
its salt will require at least multivariate calculus (if not analysis), linear
algebra and a fairly robust course in statistics. Beyond this, a proper
understanding of metric spaces, fixed point theorems and rigorous stats are
all useful and almost essential tools. All of these courses should be proof
based. I've said it before and I'll say it again: If it's not proof based,
it's not mathematics. It's computation (and I don't mean computer science).

Unfortunately, even programs at many "top" universities sometimes skip these
requirements. There are very few undergrad economics degrees worth the paper
that they are written on.

As to the theory that arithmetic is any determinant of success in real math, I
have only anecdotal evidence that this is false. Having gone to an undergrad
university with a top 5 math department, I can assure you that people who are
some of the most significant contributors to their fields sometimes struggle
at the chalk board when it comes to "basic math." I think it is not clear to
most people that arithmetic is not math. It is not the "base" of math, and
skill in this area does not provide a "strong foundation" for math. Math is
the study of quantity, structure, space and change (to borrow a leaf from
wikipedia) among other things. It is the most fundamental of the sciences in a
metaphysical sense (it may be argued that philosophy is the real root, but I
would say that at their most fundamental, there is significant convergence
between math and philosophy). Arithmetic is not related in any form to this.

------
megaframe
I used to work part time as IT support for a public school system back in
College(07), and it shocked me to see every single student in a 6th grade
class pull out TI-89s or better! to handle logs and exponentials. I'm not
talking about solving them to get the decimal number, no, this was to do
things like solve for x in: e^3 * e^4 = e^x Yes in the real world we all rely
on calculators to some extent but if you can't piece together basic math
you'll struggle to solve unfamiliar problems.

------
TeMPOraL
I think that learning to do mental computations vs. relying on technology is a
trade-off. As hermannj314 pointed out [1], there's an issue whether memorizing
multiplication tables/trigonometric tables is a matter of fundamentals, or
it's just how we used to live before calculators to be able to compute
anything. Well, I think it's both in some sense.

Human brain is good at some things (pattern matching, quick look-up, aka.
intuition), and terribly bad at others (computation, explicitly running
algorithms). I do believe that we should augment our mental skill whenever
possible - we're already thinking using our environment [2], so unless one
rejects pen and paper, or even looking at things around him/her, one is using
his environment to increase his mental powers. So why stop at handwriting? Why
not use slide rules, calculators and Google?

There's the other side of the trade-off though. Learning tables with numbers,
or multiplication tricks, or doing lots of exercises with plotting functions
by hand is the way we leverage the power of our brains. Our minds seem to be
good at caching stuff and quickly looking things up, so the more such mental
tools we incorporate, the more power we can get of our thought process.

In a very simplifying way:

Our cognitive powers = internal 'skills' in thinking * external tools we have.

The best strategy is to properly invest on both sides of the multiplication.
Boosting only one side is suboptimal.

[1] - <http://news.ycombinator.com/item?id=2924937>

[2] - <http://consc.net/papers/extended.html>

------
fleitz
Mathematics is a system of learning and reasoning, arithmetic is the process
of actually computing an answer. While it might be nice to be able to compute
compound interest in your head no one is seriously proposing that anyone do
multivariate linear regressions with a pen and paper, we simply have much
better tools for _computing_ the answer.

In a modern society with large amounts of non-human computational power it's
far more economical for humans to focus more on math and less on arithmetic.

------
kiba
I am baffled by the idea that today students can't do basic math problem
without resorting to calculators. Big tedious problems, I can understand.

I am a 20 years old.

~~~
Someone
Just curious: where does 'basic math' end for you? 13 * 7? 43 * 76? 123 * 42?
sin(PI/3)? 12534 + 3 * 26327?

Which ones do you need paper for? Which ones would you use paper for if it
were readily available?

~~~
klbarry
I would need paper for all but the first, and could not solve the 4th without
a calculator. Not sure what the norm is, here's my data point.

~~~
TheCapn
More of a curious question but are you saying you're _incapable_ of solving
the last one without paper or just that its really not worth the time. Any of
those example questions I could do without paper (I'm 23 for a reference) but
I certainly wouldn't attempt the last one without it unless someone was
challenging me since it'd take more time than what its worth.

~~~
Someone
With the problem in front of me, I would not think doing the last one on paper
would speed up things. On paper, it would feel like

    
    
        Load X
        Triple
        Store Temp
        Load Temp
        Load Y
        Add
    

I would optimize that store/load away. If, on the other hand, someone told me
the problem, I probably would not remember Y by the time I needed it (seven
plus or minus two is about two, apparently)

------
igrekel
"Calculators make fractions obsolete."

This is just nonsense. the digits after the dot are just fractions in powers
of 10 and they are much less useful if you actually need to do something in
real life with your result, like cutting pie to use a classic in fractions.

