School mathematics education is largely based around
arithmetic and those kinds of calculus and algebra which
typically involve equations which are solveable, sets of
simultaneous equations which together are solveable and so
on. Given that most people don't go on to study mathematics
at university
level and thus don't have a need for this type of
eductaion as a basis for further mathematics, my idea is to see
at least some emphasis placed at school level on more
real-world mathematics - i.e. Those problems where the
problem itself is poorly defined and has multiple unknowns. In
real life these problems often cannot wait until they are
better defined but need answering straight away. Some
examples follow, expressed as the answers to example
questions. Points are given for intelligent assumptions and
getting a plausible 'order-of-magnitude' answer. [Note: I'm NOT
saying that the calculations below are right! I just thought of
them - Rather they express the kind of flexibility,
assumption-making and estimation that I think should be
encouraged in children].

===== Example 1
=====

Q: If the Great Pyramid at Cheops was reduced
to rubble, how long would it take for trucks to clear the
site?

Well, I don't know how big the thing is, but it's
pretty big - assume 100m tall. Six pyramid shapes would fit -
one to a side - in a 200m high cube, so the volume of the
pyramid is then 8,000,000m^3/6 or roughly 1,300,000m^3 of
rubble, or 1,000,000m^3 if we assume some of the pyramid is
empty. So far, so good. Now we'll assume a truck can carry
10m^3 of rubble and that 10 trucks an hour can load up and
take rubble away from the site, so that's 80 truckloads a day
or 400 a week or about 20,000 a year. So 200,000m^3 of rubble
can be moved a year, so it'll take 5 years.

=====
Example 2 =====

Q: How many trees per year are used
to provide the Japanese with disposable
chopsticks?

These disposable chopsticks are about 8"
long and between 1/4" and 1/8" thick. We'll assume that the
cross-sectional area of 10 pairs is one square inch, with
machining lossses. Let's say that our tree is a foot across and
has 100 square inches cross-section. Let's say it's 50 feet high
and (throw in some more machining losses as I think we've
been a bit tight so far) we can get one chopstick length per
foot. So we can get 50*100*10 = 50,000 pairs of chopsticks from
one tree. Then, Japan is about the size of the UK, so we'll
assume it's got 50m people. Say 40m off them use disposable
chopsticks and they use 2 pairs a day for 300 days a year -
that's 24,000,000,000 pairs or 4,800,000
trees.

========= ============

(waits for
someone to post links to pages conmtaining the actual
answers to both these problems!)

========= ============

More examples - added 5th Dec 2005

What are the chances that the breath you last took contains molecules of air from Caesar's dying ("Et tu Brute?") breath? (Surprising answer, this one. You'll also need to remember Avagadro's constant from school - roughly 6x10^23)

How long would it take the population of the world to empty all the oceans of the world, with teaspoons? (assume they've got somewhere to put all the water)

Ph 103c: Order of Magnitude Physicshttp://dope.caltech.edu/ph103c/info.html Baked. I took the course myself. Online class notes, problem sets, solutions. [egnor, Oct 28 2001, last modified Oct 21 2004]

(?) Google Links to Fermi Problemshttp://www.google.c...ch?q=fermi+problems Other questions of the sort posed above. [Guncrazy]

Pi-faced in Indiana
http://www.inwit.co...s/indianapilaw.html Dr. Edwin Goodwin came up with the Evil plan [Guncrazy, Oct 28 2001, last modified Oct 21 2004]

Pi-faced in Indianahttp://www.inwit.co...s/indianapilaw.html Dr. Edwin Goodwin came up with the Evil plan [thumbwax, Oct 28 2001, last modified Oct 05 2004]

Slide ruleshttp://www.hpmuseum.org/sliderul.htm Thought about running breaks, but trying to field a good link for calculus would be a job. [reensure, Oct 28 2001]

Google (in Latin)http://www.google.com/intl/la/ [hippo, Feb 22 2002, last modified Oct 21 2004]

Hmm. I've been agonising over this all day, for some reason. I think this is baked, because (at least in my experience,) kids are taught to construct their own mathematical formulae, and this is called problem solving. The last step in this process is simply plugging in the numbers.

You've all had questions like, "a pyramid is built from Q bricks, X x Y at the bottom rising P high. How many bricks are at the Nth level?" The last thing they do is give you the numbers to plug in - this makes it easier to mark, so long as you can show the working out...

If anything this is sort of WIBNI. Sure, I would have liked my teachers' to devote more of they're time to speling and grammar. It's all about priorities, I guess.

[sdm] - (not rising to your ("they're", "speling") bait) - interestingly, I believe one of the most useful things I studied at school was Latin - helped my English enormously. I agree, kids are taught to construct their own formulae, but generally to solve problems which have only one 'unknown' - part of my point is to give experience in making intelligent assumptions in the face of missing facts. [PeterSealy] That wouldn't be you shouting "Baked" with no supporting evidence would it? [egnor] Order of Magnitude Physics - good stuff, but why can't this be done at High School? [UB] But my 'Pyramid' answer would still have got good marks because I stated my assumptions and ended up with an answer within an order of magnitude of your answer. I deliberately didn't look on the web for facts to help me with the answers - I wanted to see how far assumptions would get me.

hippo: I heartily endorse your idea, but my son was explicitly asked to do estimation on some of his math problems in 5th grade. Mind you, it was not on problems as detailed as those you provide.

There's an absolutely great bit about a physicist--Oppenheimer?--at the Los Alamos project whose forte was this kind of estimation in realms mere mortals cannot conceive. He once witnessed a particular nuclear test, saw the fireball rise, looked at his watch, did a few calculations on an envelope, and announced the approximate kiloton yield. I wish I was better at that stuff. Er, not blowing things up, I mean off-the-cuff thinking.

I'm never one to avoid jumping on a bandwagon so Baked! cp's doing it in physics classes at school and so was I, 20 years ago. You really should have paid more attention in class, hippo.

I did two years of Latin at school, and I agree with [hippo] that it helps with English. I didn't take Greek but my brother did, and I picked up some useful stuff from that as well. I also remember doing something like this, but that was 30 years ago.

Even though it could be argued that this is baked I'm going to croissantify it becuse it's not baked enough in my opinion. We did one such problem in physics, which was to work out how fast helicopter blades need to spin to get it off the ground. The thought processes that it provoked are the same thought processes which I have often had to call upon in the real world. It was often referred to in university as 'engineering judgement'.

Croissant (many of 'em, but I ain't gonna be precise)

This is one of my complaints about people-in-general (PIG). Too many PIG have absolutely no sense of number at all, as some of the anecdotes here attest.

I think that one of the worst things done in education in the US is the use of calculators in grade school.

Yes, I said "calculators in grade school."

Kids dont' have to "memorize" multiplication tables (I never memorized them, I learned to understand how multiplication works). They don't have to "work it out long hand." They just punch it in and get an answer.

I'm with [stupop] in saying that it's not baked enough (reference my idea on Evolutionary Education). The key is to raise a generation of children who are willing to think about the problem. Accuracy will come with experience.

In the same breath, I have to disagree with [quarterbaker]. I was on the trailing edge of the last generation to have to go without calculators. In my mind, the important skill is knowing how to solve the problem, not the tools you use to do it.

Baked........sort of. This is taught in the first three grades, and it drives the brighter students absolutely bonkers because there are no single correct answers. It needs to be continued into the higher grades, with suitable adjustments. Incidentally, estimating is one of the things I miss about slide rules: you need to come up with the order of magnitude and can't stop thinking simply because you've got a number.

Without calculators, kids just crunch through the same tedious formulas by hand, wearing out their wrists with long division, and coming up with a number that's even more bogus, because not only did they get the operations wrong, they also forgot to carry a '1' in step 7.

With calculators, at least you have the freedom to run through things several different ways to get more of a "feel" for what's going on.

I'd rather see kids using *computers* -- spreadsheets with graphing packages and the like -- because then you really can see what's happenning, and it's not just a bunch of mind-numbing digits that have lost all meaning.

[egnor] I partially disagree. Mechanically slogging through
a long division is no fun for a child and may not teach
them anything. However, mechanically slogging through
the same thing with a calculator doesn't teach them
anything either. What's better is teaching the child good
mental arithmetic skills - and I speak from experience
here. 12 yr old kids can be accurate and quick at mental
arithmetic and it gives you such a command of numbers
which never quite goes away (even now, when I hardly do
any arithmetic). [UB] Thanks - It's nice to know
someone else did Latin at school (I bet you didn't do
Sanskrit though).

If you're going to blame a slide in education standards on the introduction of calculators, then maybe you need to go back and do some more work on formal logic.

[egnor] The school that I go to encourages the use of graphics calculators, which has both good and bad points. They are easily programmable and can display nice graphs like Excel and Gnuplot can. On the other hand, it's too easy to just rely on a pre-made calculator programme to solve a problem, mistype something, and get a dodgy answer. Also, because the calculator can do simple problems without thinking, it discourages some people from thinking about harder questions which (presumably) require more traditional algebra, etc. Either way, people should certainly be able to tell when an answer is _way_ off rather than just trusting a calculator (or their arithmetic), which is what this idea was about.

Hey, wait a minute. "Latin... is fundamental to understanding not only English, but all of the Latin-based languages... German, Spanish, Italian..." <spluttering in apoplexy> Spanish, Italian and French _are_ in the Latin group, but English is part of the Germanic language group; it's no more Latin-based than Lithuanian or Serbo-Croat. Sure, they're all Indo-European (like Sanskrit) but even languages within the same group can have fundamentally different grammatical systems. English abandoned much of its Latin-style grammatical trappings - all that inflexional bollocks - in favour of a more analytic grammar, in the transition from Old English to Middle English, in order to achieve a much greater flexibility. In the process, it's now moved so far from its roots that you might as well compare it to an agglutinative language like Sumerian, Magyar or Turkish as to Latin. OK, that's an exaggeration, but honestly, every time I hear a phrase like "split infinitive", or the hoary old chestnut about how Latin helps teach people to "understand English" (i.e. instruct them in grammar), I blow a gasket. Drilling meaningless Latin-based rules into kiddies' soft little skulls has done as much harm as good in developing our understanding of how the English language works. We could teach them _valid_ rules based on the _actual_ Anglo-Saxon, get them reading Beowulf in the original - or The Wanderer (that's a great poem!) - but nooooooo: indoctrinate them with Latin and turn them into pedants for life, hitting on everyone with their *correct* use of language codswallop, as if it's not just another privileging class-marker like RP, used to pick out the grammar school-<gah! gah! clutches heart, falls over>

On the other hand, I agree entirely with all other cantankerous curmudgeons in bemoaning the demise of mental arithmetic in schooling. Calculators? Infernal contraptions! It wasn't like that when I was a paper boy. If you can't calculate the change from a pound for a Sunday Mail and a Sunday Post (circa 1985), then ye should be bloody ashamed of yerself, etc.. Calculators, indeed!

Except for long division. I was always shit at long division.

Guy Fox, I have no problem with doing mental arithmetic (most of the time), but generally I find it far more convenient to use a calculator than to work out sums on paper. And I _was_ taught long divison, but have never had to use it in practice.

[cp] - nothing wrong with a calculator as long as (in my opinion) you know what it's doing, and you know roughly what answer it's going to come up with. [Guy Fox] - I wouldn't say that Latin helped me with English grammar - more with the meanings of words as so many English words have Latin roots.

Fair 'nuff, hippo. (In retrospect, I think I was obviously suffering from a nicotine deficiency this morning. I have chocolate now and all crankiness is back under control. *ahem*)

What hippo said, cp. I was thinking more of basic arithmetic than the sort of stuff that requires paper. I wouldn't even bother to attempt, say, (37 * 23) if I should ever need to, and, as I say, long division was always a bit of a bugger for me.

My daughter (10 years old, in California) has math estimation exercises on a regular basis. The most recent involved problems such as this --

Estimate the answer:

12/7 + 7/8

If I gave the proper interpretation, one should add the numerators and report the sum (19) over either the 7 or 8, instead of changing the problem to 96/56 + 49/56 = 145/56 or approximately 2.59. 19/7 gives 2.71 and 19/8 gives 2.37.

But what I started to say is that we must not confuse grammar with etymology. Between Latin and German, we can determine a large proportion of English words (wait -- and Greek too, ok, ok "among" the three) we get many etymologies, but using German or Latin grammar we would not gain much insight into that of English. Zum beispiel, German sentences often have the verb all the way in the end of the sentence gestuck. Not to mention the Latin tendency of "Gaul all of it into three parts divided is." Unless you're Yoda, that's not very useful seems to me.

daruma: if you want to follow the Greek or German model, you could always teach your daughter to use a reverse Polish calculator (the HP 12C, for example, if they still make them). That sum would then be, um, 12 [enter] 7 ÷ 7 [enter] 8 ÷ +.

Personally, I think there needs to be classes in counting change - sinec about half the country seems to work as register clerks, and manyof them are confused by the concept of exact change. Even worse, is when the bil comes to say, 2.57, and you dig out $.07 so you can get two quarters back (for the laundry), and they fall completely apart - it seems ther are little pictures on the register keys, a hamburger, an order of fries, etc., and it's quite possible these people cannot even read.

It's gotten better lately, test score always go up when the economy is doing well, but watch out in about three years, when the recession kids start graduating, it'll be a mess again, invest in Krylon and Whackenhut, is my advice.

I can always estimate my grocery or Wal-Mart bill, within about a dollar - just adding things, and rounding very roughly; a talent I developed after inadvertantly buying the store on a couple of occasions - "awww, it's just another buck two ninty -five...".

Gawd, what a long set of annotations; there must be something to all of this! Change the educational system by changing what people remember they studied? Damn good idea! There would go all of the folks that are stuck in the past and cannot gear up to the here and now.

Until more stragglers catch up, [Rods Tiger] I remember well the dens of burger hell. Actually recall writing my orders on a chalkboard over my register so's the prep crew could glance up and make the food. I usually had an 'estimate' of what the order would be and was happy to provide it to the customer (and was unhappy if I was off by more than the tax involved). I also usually had the change in my hand before the bill was tendered. Seems to me that I don't recall being short or over in my drawer or staying in late to recount.

Is the use of estimation only possible in a developing brain? Perhaps only in the sphere of social learning? It may be that the more concrete approach to behavior, that involving characterization, role modeling, and proof involves more rational learned mechanisms than is involved by more creative thinking. What my statement implies is that a youngster may be put at a disadvantage by over-relying on estimation techniques that lend to experimental processes beyond the social inquiry or preliminary outline of an idea.

Teaching kids to 'go at it' when seeking solutions to the questions they face, rather than to plot out a statement of the difficulty based on some predictable outcomes and then to solve for the variables involved in choosing a best outcome, only teaches kids to trust in the power of experimentation to achieve an end. Such a ideology will empower kids to choose for themselves, but are you choosing the school for your kids based on the degree of empowerment you think your kid will gain from attending? If you are, would your opinion change if you know that the school will 'estimate', that is to say 'project or forecast' the grades as if they were offered rather than earned?

Indiana, Bubba - see link. I refused to use a calculator in my school days - they had been approved for use in 1971-72 School year in my School. Throughout the course of my High School years, I had highest scores in County on all aptitude tests - SATs included as I came to find out in 1980 - coincidence? I think not. I am quite sure there were many students who were too reliant upon calculators and therefore would/could/should have outscored me had they not used calculators on a daily basis. I went to a party recently where there were only 2 of 30 people present who could total up a running figure for Gift Certificates - The rest thought we were geniuses. The 2 of us quietly determined later that we weren't geniuses, they were idiots. The Idea is for Estimation skills - and I approve of that.

I'm in Canada, but here i think all the disposable chopsticks are made out of bamboo. Bamboo is a renewable resource. (one species of bamboo is the fastest growing plant in the world, as far as i remember) Further, here the chopsticks go in the garbage, and ultimately a landfill. This allows us to deposit the carbon absorbed from the atmosphere back where it belongs - in the ground! (ie. where the carbon in your car exhaust came from)

Steve DeGroof: Just over a minute - 1m 20s, I'd say.

It'd take 13/14*20 minutes when driving at 70 mph, or 14/14*20 minutes at 65 mph. Thus, you'd save 20/14 minutes. 20/14 is slightly larger that 20/15 (1m20s).

one of my profs in university would knock off half of your exam marks if you forgot to write the units along with your answer (or if you got them wrong). he also proved to us that 32.2 = 1. he didn't do much estimating.

Twenty minutes is a third of an hour, and since 70 mph is slightly more than one mile a minute the math is easy, so guess a third of the difference of the rates 65 and 70 or 1.66 minutes, alternately, one minute 40 seconds.

Is this the kind of estimation that tells you to leave home a half hour early?

[mihali] Too true. Also, we used to get picked apart worse for not checking our answer.

mihali: I am pleased to say that my son figured out right away what was wrong when his teacher "proved" that 2 = 1 (it's the old divide-by-zero trick, the same one your professor used). And it's absolutely correct to deduct points if you miss off the units (just look at what happened to the Mars lander - some idiot left the units off and they lost the whole damn lander).

I can do the 'estimate value of supermarket trolley contents' thing , no problem. (Usually it's because they sent me a '£2 off £20-worth' voucher, and I'm trying to spend no more than £20.01).

The difficulty is not the arithmetic. The difficulty is avoiding making eye contact with someone I know, and having to mouth "Hello" and then finding that the momentary lapse in concentration lost me all the numbers, and I have to stand there in mid-aisle muttering into my trolley while working it all out again before I can move on.

I think I need a card I can hang around my neck saying "Estimation In Progress - Talk Later" or maybe a paper bag over my head so nobody recognizes me.

[angel] PostScript. Like PDF, only older and (arguably) better. You'll need a programme like GhostScript (which is free) to read a .ps file. If you're desperate, I'll track down the link for you.

I'm with you, waugs. I can't tell the difference between 2 miles and 10 miles. Y'know what, though? I used to not be able to tell the difference between 10 feet and 100 feet, but steady contemplation of massive, incomprehensible distances (like the space between planets) and conscious, deliberate effort to meticulously measure things and then just step back and get a sense of its 'bigness', knowing now exactly how big it is, then internalize the correlations of bigness and numbers has given me a sort of sense of space now, so I imagine that any sort of estimation could be learned.

At any rate, I'm all for the idea. I estimate that I would be much smarter had I been exposed to this sort of thinking in HS.

In my head, quickly
20x65=1300
20x70=1400
100 seconds difference=1:40 faster
disclaimer: Strange, but true - it is easier to multiply then subtract in the 'wrong' direction to come up with many answers, rather than decimalize or further break down fractions. Thinking on feet requires K.I.S.S.

A new question for you all: Can you estimate how many atoms are in an apple? I don't want anyone to start using what they know about atoms (e.g. just multiplying your estimated apple mass by Avegadro's number), what I really want to know is whether anyone can use ordinary everyday experience to get an appreciation of the scale of an atom.

It's just occured to me how I might answer this question, but I'm going to wait a bit before posting my answer.

//what I really want to know is whether anyone can use ordinary everyday experience to get an appreciation of the scale of an atom.//

Since it's rather difficult to infer the existence of atoms from everyday experience, let alone their mass or size, I am intrigued to find hippo's possible method.

My best would involve continuously halving the apple, but at some point you're going to have to ask that friend with the electron microscope for a hand.

(imagining cutting it with a knife/razor blade, I made 18 cuts before I had such a small lump of goo that I felt I couldn't cut it in two. There are therefore st least 2^18=262144 atoms in an apple. (oops, didn't include the stalk).)

My back-of-the-envelope method is based on the assumptions that when you drop a drop of oil on some water it spreads out into a layer 1 molecule thick, and that an apple has the same density of atoms as oil. Let's say our 3mm oil drop (volume about 25mm^3) spreads out into a circle with a diameter of 200mm (area, roughly 40000mm^2). That means the thickness of the layer is 6.25E-4, which represents the thickness of 1 molecule. If the molecule is 100 atoms thick, then the inter-atom spacing is 6.25E-6, or to put it another way, you could put 160000 of them in a line 1mm long or 4.096E15 in a 1mm^3 cube. If our apple has a volume equivalent to a 50mm cube (125000mm^3) then it contains 5.12E20 atoms. Probably completely wrong but hopefully you can see what I'm trying to do.

Nice. Lots of argument could ensue about the size and shape of the molecules involved but, who cares, it's a neat solution. I'd forgotten about the oil drop experiment.

This is my favorite approximation ever:
http://www.baltimoremd.com/humor/santaengineer.html
The first time I read it, I was unable to stop laughing. Believe me, after 5 minutes, it really hurts your belly... :)

I like the idea.
It's not just pure math which helps in life.. It's the corrctness to the expecting value which helps. To begin with.. the art of expectation is required.
The numbers that you fill at every stage tells you your visual abilitiy to identify things.
for example: How many time it will take for all the spectators to leave the stadium after match is over..?. It means considering lots of factors such as size of the stadium, the estimated flow rate, how far is the parking, how much time it takes for cars to come out of the gate..

The data which can only be obtained by experience, can be guess little bit..The skills obtained can definitely provide usefulness in different circumstance of life

it may be useful but it is a complete perversion of the mathematical arts, which is intended to be a creature born of unknowns that bare perfections... and how is calculating the time it would take a momument to be reduced to rubble be practical to 'most of us'... Big thumbs down

That all works... until you start changing the size of the cake on us.

My results pretty much matched UB's.
Area of sheet cake (9x11) ~ 100 sq in.
Volume of frosting tub ~ 18 cu in (say a little under 20).

Thickness of frosting on sheet < 20/100 ~ Just under one fifth of an inch.
Icing surface of cylinder cake: 16pi + 16pi + 40pi ~ 50 + 50 + 125 = 225 sq in.

1 tub does about 100 sq in, so I'll need another tub and a quarter. I'm obviously a little more stingy than Unabubba with my frosting. [later: or just estimated 1.5^2 badly.]

A couple of extra estimation problems added to the idea text (above). I've worked through estimated answers to both of these but I thought it would be more fun to leave them as questions.

As well as Avagadro's constant, don't you also need to know that that much gas occupies 22.4 litres (0.0224 m^3) at STP? Assuming that most of the atmosphere is about 2km thick, and that the CO2 content of the Divine Julius' dying breath of about 0.8l litres hasn't been locked away in some plant (or indeed that the O2 content hasn't been liquified and burnt in low Earth orbit), I estimate about five thousand. I remember a similar question about distributing all the molecules in a drop of water evenly through all the Earth's bodies of water, and then taking a tumbler at random, and estimating how many of the molecules from the drop the glass would contain. When I was at school, estimation was taught - partly to ensure you got in the right ball-park when using slide-rules (started at primary school) or log tables. Later my estimation skills were formally tested as part of the selection process for graduate recruitment into BT.

Should also be combined with sensory estimation training.

A fair number of people can estimate the weight of an object in their hands because packages of food are labeled, and a select few have bothered to pay attention to that fact and to calibrate their senses.

Or they lift weights at the gym.

Not so many could estimate what a foot-pound of torque feels like, how fast a bicyclist is going, or what a decibel increase in a sound sounds like. Experiences like these could help build a more vivid picture of the trigonometric and polynomial functions so important in everyday newtonian physics.

The likelihood of this working is about as high as the odds of
a 1-inch needle, randomly dropped onto a square grid with a
spacing of 2 inches, not lying across any line. [+]

I have not worn a wrist watch since 2002. I find that this has led to a nearly unbelievable improvement in my ability to estimate the time of day, quite often within 15 minutes of the real time many hours after last having seen a clock.

The human body can really pick up some interesting skills that you may hardly be aware you're learning at the time.