
Economist's View: "The Mathematics Generation Gap" - ColinWright
http://economistsview.typepad.com/economistsview/2011/05/the-mathematics-generation-gap.html?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+EconomistsView+%28Economist%27s+View+%28EconomistsView%29%29&utm_content=Google+Reader
======
tel
The points brought up here, especially by Thoma but less so by the quoted
passage by Woolley, are very valid in my experience. Often times the shortest
way surpass a difficult abstract challenge is to find a practical instance of
it and then slough through the mechanics. You abuse the natural pattern
searching of the human brain very productively that way.

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.

~~~
run4yourlives
The argument though is that it is simply impossible to use a calculator
effectively without first understanding the basics.

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.

~~~
ColinWright

        > The argument though is that it is simply impossible to
        > use a calculator effectively without first understanding
        > the basics.
    

I don't think that's the case. I think the author is perfectly happy that
people know how to use the calculator. I think the point is that without
working through the basics underneath, mind-numbingly repeatedly, you don't
gain any real insights about what's going on.

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.

~~~
kenjackson
_I think the point is that without working through the basics underneath,
mind-numbingly repeatedly, you don't gain any real insights about what's going
on._

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.

~~~
jules
This is even true in college, where for example a large part of a physics
education is spent computing integrals and solving differential equations in
special cases, using approximation methods by hand. It would be more
appropriate for an intuition of the physics to use symbolic and numerical
methods on a computer a lot more. You realize that doing arithmetic by hand is
a largely pointless exercise when you get a calculator, and we should
similarly recognize that doing integrals and differential equations by hand is
largely futile when you have things like maple and scipy.

~~~
kwantam
I can tell you right now I'd never hire someone who wasn't able to compute the
result of a linear differential equation without using a computer, because
it's something that every person I hire needs to do very frequently while
designing circuits. The difference in quality of design between one who can
analyze a circuit on paper or in the head and one who relies on simulation to
discover basic properties of the circuit is massive. Iterated simulation is
not a tenable approach to the design of any sufficiently complex circuit.

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.

~~~
jules
It is definitely useful to be able to do linear differential equations by
hand. It's not useful to keep doing these things by hand. Just like it's
useful to know the algorithm for multiplying two numbers, and it's not useful
to keep doing multiplication by hand.

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.

------
torstein
"Conrad Wolfram says the part of math we teach -- calculation by hand -- isn't
just tedious, it's mostly irrelevant to real mathematics and the real world.
He presents his radical idea: teaching kids math through computer
programming."

[http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_m...](http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html)

~~~
hugh3
Y'know, maybe what we should really do is to completely decouple "mathematics"
from "arithmetic". Arithmetic is useful but boring and you mostly learn it
from age 5 to 12. Mathematics is harder, more interesting, and either useful
or useless depending on what you eventually do with your life (no doubt 60% of
people could go through life without understanding any mathematics at all --
all they need is arithmetic).

------
noblethrasher
Tangentially, up until grade 6 or 7 (in the late 80's), my math books also
included BASIC code related to the concepts --- usually at the end of the
chapter. It's how I became curious about programming and more so about math
(though in my case none of the teachers ever did anything with the material, I
either just typed the code into my family's TI-99A or tried to mentally work
out what the program was doing).

~~~
fferen
That's amazing, I wish any of my math books had that; we would probably see a
lot more young programmers these days. I can seriously say that the only
reason I even got into programming was discovering TI-Basic in 7th grade
(although the habits it taught me had to be slowly unlearned as I started
using a real language ;)

~~~
T-hawk
Well, the entry barriers to programming were actually lower in the 1980s.
Pretty much every home computer came with a flavor of BASIC built right in to
ROM or at the worst bundled with it on a disk. Anybody with a TI-99/4A, C64,
various Apples, Atari 400/800/ST, or many others could jump right in to that
kind of programming.

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.

~~~
knshaum
Every desktop of laptop OS I know of comes with at least one JavaScript
interpreter installed.

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.

------
droz
Back in the 90s there was an episode of the Outer Limits that touched on this
subject. Effectively there was a nexus of information that everyone connected
to through their minds. Anytime they wanted to know something, they just
looked it up by thought and presto, instant expert. One guy couldn't connect,
so he had to learn things the old fashioned way by research and practice.
Naturally, the nexus goes down and people can barely function. He goes from
being the village idiot to the village savior.

~~~
mkr-hn
There's an Asimov story similar to this:
<http://en.wikipedia.org/wiki/Profession_%28short_story%29>

~~~
GregBuchholz
Also, "The Feeling of Power".

[http://www.themathlab.com/writings/short%20stories/feeling.h...](http://www.themathlab.com/writings/short%20stories/feeling.htm)

------
akikuchi
The crux of the issue as I see it is an extension of what mathematics
educators call "number sense." (<http://en.wikipedia.org/wiki/Number_sense>)

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.

------
DanielBMarkham
Seems to me there was an article a few months back on how the use of GPS in
cars was handicapping drivers -- damaging their ability to form and manipulate
abstract mental maps.

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.

~~~
run4yourlives
In my mind, math is a language like any other. Whereas you wouldn't be
considered fluent in English until you have memorized a significant number of
words and phrases, the same holds for mathematics.

Al that rote memorization that everyone hates is the same as understanding a
basic sentence.

~~~
derrida
What rote memorization?

------
44Aman
I think the problem with Economics in this case (I'm an undergraduate) is that
it has become overly mathematical, often for its own sake.

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.

~~~
billswift
I like the way Sowell did it (recounted in his _A Personal Odyssey_ ); when
teaching economics to engineers he concentrated on a "literary" conceptual
teaching style, since he already knew they were competent in the needed math.

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.

------
JabavuAdams
I remember doing "Mad Minutes" in Grade 4 (1986). We'd have to do 30
multiplication problems (up to 12 * 12) in 60 seconds. It was a game to see
how many we could complete.

I think I might need to revive these for my technical-college math students.

~~~
marshray
About that Mad Minute.... Sure it may be great for for kids that are already
good at it. The fact that you thought it was a "game" says a lot.

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.

~~~
run4yourlives
_it just raises their anxiety level, even to the point of near-panic._

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.

~~~
marshray
You seem to think I'm against working hard on multiplication tables - I'm not.

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.

~~~
JabavuAdams
Yes, this is a concern. You wouldn't expect someone to compete athletically
without training first.

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".

~~~
marshray
_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._

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.

------
vixen99
From the original article 'Students today might be taught by a teacher who is
himself unable to work out 37+16 without help' I take it that this is extreme
exaggeration to make the point. Such a person would clearly not be fit to
teach any subject let alone one requiring numeracy skills at its core.

~~~
mkr-hn
I think most people who seem unable are just unwilling. I can do that in my
head, but it's a lot of effort without practice. Someone less patient might
put "37" and "16" in their mind, start adding, and interpret the block they
hit when trying to carry the 1 as "can't do it."

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.

------
rmc
Conservative generation gap nonsense. The prevelance of Google Maps means
people can't plan directions and hence can't read some graphs?

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.

------
run4yourlives
The big issue here, since most reading this are probably of the generation in
question, is found in the first comment:

 _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.

~~~
silvestrov
I think we should aim for "quantitative intuition" by teaching how to do rough
calculations instead of spending time teaching precise calculations.

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.

~~~
run4yourlives
_I think we should aim for "quantitative intuition" by teaching how to do
rough calculations instead of spending time teaching precise calculations._

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.

~~~
jerf
"Math is precise."

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.

~~~
run4yourlives
I don't seen how quantitative intuition and precision are somehow opposed to
each other. 4*5 is 20. Not 21. Not 18. 20.

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.

~~~
lurker14
When is the last time you had a use for exactly 1/12 oranges?

"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...](https://secure.wikimedia.org/wikipedia/en/wiki/Significant_figures)

~~~
run4yourlives
I think you misunderstood my example. My point was that knowing that there is
5 1/12 oranges (as opposed to 5) is the important precision, and it works
logically to a child's mind.

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.

------
chopsueyar
Don't blame this on affordable technology.

It is the curriculum.

~~~
arethuza
I agree - calculators have been generally available for a _long_ time, well
over 30 years.

~~~
tokenadult
_calculators have been generally available for a long time, well over 30
years_

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

<http://xkcd.com/768/>

~~~
chopsueyar
In my elementary school, we had math superstars. These were math teaser
problems.

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...](http://cgi.ebay.com/Vintage-Radio-Shack-EC-490-Scientific-Calculator-
Case-/290571906416?pt=Calculators&hash=item43a76f6d70)

Math Superstars link: [http://it.pinellas.k12.fl.us/schools/curlew-
es/studentconnec...](http://it.pinellas.k12.fl.us/schools/curlew-
es/studentconnect/mathsuperstars/mathss.html)

------
nradov
I see the lack of mental arithmetic ability all the time when scuba diving.
There are many experienced divers who are totally dependent on wrist computers
or pre-calculated tables, and can't do simple things like figuring out
decompression schedules or gas consumption rates in their heads. If you have a
good sense of numbers then it's really easy to do these things in your head
after you memorize a few simple rules. Yet so many divers think it's some kind
of black magic, or even that it's somehow "dangerous" to make your own
calculations.

~~~
wazoox
It's a good idea to keep a table and a wrist calculator when scuba diving
anyway, in case you're subject to nitrogen narcosis, which makes you really
stupid.

~~~
nradov
Nonsense. It's a good idea to not do stupid things like breathing narcotic
mixes at significant depths.

~~~
wazoox
Did you actually dive at sea? You may feel the narcosis at varying depths,
depending upon the temperature, your fatigue and some other parameters. You
may usually dive at 50m breathing ordinary air without problem, and some other
day get completely stoned at 40m. This happens sometimes, particularly when
you dive for the first time after the winter break for instance.

~~~
nradov
I was diving at sea last weekend. I don't dive air at 40m. Your other comments
about narcosis are sheer nonsense, completely off base. Don't breathe narcotic
mixes.

~~~
wazoox
> _I was diving at sea last weekend. I don't dive air at 40m._

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?

~~~
nradov
I don't care what's "allowed" or not. Where I dive there are no scuba police
to tell you how deep you can go or what gasses to breathe. There is no
evidence that most divers breathe air down to 50m. What you subjectively
"feel" about narcosis has little relevance to what's actually happening in
your body or how impaired you are. As for your comment about feeling "stoned"
the first time diving after a break you seem to be implying that divers can
somehow adapt to narcosis over time, which of course is completely wrong.
[http://www.us.elsevierhealth.com/product.jsp?sid=EHS_US_BS-S...](http://www.us.elsevierhealth.com/product.jsp?sid=EHS_US_BS-
SPE-954&isbn=9780702025716&lid=EHS_US_BS-DIS-1&iid=null)

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.

~~~
wazoox
> _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.

------
tristanperry
It is a difficult issue. I do a mathematics and computer science degree hence
there's obviously no real lack of mental arithmetic maths skills for this
degree (well, at least the basics are consistent across the maths students).

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.

------
aufreak3
Skills become outdated over time. The skill of mental calculations, I think,
is not really that valuable on the average nor is it unlearnable later on if
you sit at a cashier's desk at some point ... but the skill of doing
_quantitative_ calculations mentally greatly helps me make connections that I
suspect I won't make otherwise. I think this is because I'm morphing pictures
in my head instead of crunching numbers.

Next step - lets allow all kids to use google search during their tests.

------
tokenadult
There has been a profound change in mathematics education in the years
indicated, and the author of the submitted article is on to something. One of
my favorite authors on mathematics, Professor John Stillwell, writes, in the
preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998):

"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...](http://www.aft.org/newspubs/periodicals/ae/fall1999/index.cfm)

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.

~~~
kenjackson
I side much more Conrad Wolfram on this than Stillwell [1]. Computation simply
is a lot less important now as we have tools that do them with far greater
speed and accuracy. And I'm from the generation that did tons of computation,
but I honestly never had a great grasp of it. I just knew there were steps to
be taken (leave an extra space to right as you do long multiplication) -- and
I was one of the top math students in the district.

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-...](http://blog.wolfram.com/2010/11/23/conrad-wolframs-ted-talk-stop-
teaching-calculating-start-teaching-math/)

~~~
ntkachov
As a student in a fairly good university(Umass), Math tends to be taught
exactly how you describe it. "Heres the FT. This is where you use it. This is
how you do it". Most students will be able to do the FT, and most students can
actually find the correct answer. However, because teachers and books tend to
formulate very artificial situations where the a specific tool students often
are used to looking for very specific patterns inside problems to decide what
tool they will use for the job. When a problem falls outside the usual
patterns, students will have a hard time identifying the tool they should use.

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.

------
wmobit
I saw nothing about mathematics here; this was all about arithmetic.

------
fleitz
The best way to teach mathematics is to put it in a useful context. Want to
teach trig? Have students build a set of stairs. When students need it to
solve a problem they will figure it out and intuitively understand it. The
primary problem with mathematics teaching and education in general is that
students know the knowledge but have no idea how to apply 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.

------
reso
"I plan to remain hard-headed about this until I am convinced that abandoning
the rote sorts of exercises done in, say, a linear algebra class (which can
also be done on a calculator) does not hinder our ability to form intuition
about how to do proofs, etc"

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.

------
peterbotond
what i gained from learnign math using trig and log tables, pencil and paper
to calculate fractions, multiplying them, more numbers, more calculations, =>
feel for a solution. yes, it is subjective, by looking at a problem there are
no magic that pops out, just a feel for what would be a correct answer.
objectively, the experience how to simplify is a great gain as well. ... then
i studied logics. today i can calcualte areas, circumferences in my head,
faster then using my phones calculator. the precesion is acceptable for my
daily use.

------
chopsueyar
The worst part is, how old is all this stuff, and how was it able to be passed
along for generations until now?

------
swarzkeiser
I read years ago that taking notes about everything diminished our capacity to
memorize things. Now I can't remember where I left my notepad ;) (But
seriously, exercizing our natural gift of thought is essential for good brain
shape.)

------
rick_bc
Just count in binary, everything will be easier.

110 * 111 = 101010

Very easy, don't you think? :)

------
tybris
This might be true for any other field than math, but the basic understanding
of math you need today is no different from what Pythagoras and Euclid needed.

~~~
run4yourlives
You're arguing against basic mathematical education on the basis of not being
needed to a country with the highest personal debt levels the world has ever
seen.

~~~
whyenot
To me, US personal debt seems like more of a self control issue than a problem
of innumeracy.

~~~
run4yourlives
Self control however is much better maintained with a proper understanding of
the problem.

This is why payday loan sharks target poor areas: lack of education.

