RE: [math-fun] Article on math education
I was recently asked by a senior manager of purchasing for a large school district - What is 3/8 + 3/8 ? I expect this manager deals in many millions of dollars annually. It makes me wonder how important math is for the non- mathematician. John McKay
John: I'm not so worried about his inability to do 3/8 + 3/8 -- after all, most of us can't do present value analysis in our heads, either. What would bother me would be if he couldn't get his calculator and/or spreadsheet to do these calculations. Henry ----- At 11:39 AM 12/13/2004, you wrote:
I was recently asked by a senior manager of purchasing for a large school district - What is 3/8 + 3/8 ?
I expect this manager deals in many millions of dollars annually.
It makes me wonder how important math is for the non- mathematician.
John McKay
In such matters, context is everything. I've been intrigued for many moons by the fact that each morning I hear the market analyst reel off the changes in bond rates in 32nds and then give the resulting rates in % to two dec places. Does his machine quote these things thus? R. On Mon, 13 Dec 2004, Henry Baker wrote:
John:
I'm not so worried about his inability to do 3/8 + 3/8 -- after all, most of us can't do present value analysis in our heads, either.
What would bother me would be if he couldn't get his calculator and/or spreadsheet to do these calculations.
Henry ----- At 11:39 AM 12/13/2004, you wrote:
I was recently asked by a senior manager of purchasing for a large school district - What is 3/8 + 3/8 ?
I expect this manager deals in many millions of dollars annually.
It makes me wonder how important math is for the non- mathematician.
John McKay
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Probably a table-lookup on a hand-held. Interest tables are some of the oldest known artefacts. I don't know offhand how old, though. Some businessmen would claim that the invention of interest ranks right behind the invention of money in the promulgation of commerce. I'd be interested in knowing approx when interest was invented. At 12:43 PM 12/13/2004, Richard Guy wrote:
In such matters, context is everything.
I've been intrigued for many moons by the fact that each morning I hear the market analyst reel off the changes in bond rates in 32nds and then give the resulting rates in % to two dec places. Does his machine quote these things thus?
R.
On Mon, 13 Dec 2004, Henry Baker wrote:
John:
I'm not so worried about his inability to do 3/8 + 3/8 -- after all, most of us can't do present value analysis in our heads, either.
What would bother me would be if he couldn't get his calculator and/or spreadsheet to do these calculations.
Henry ----- At 11:39 AM 12/13/2004, you wrote:
I was recently asked by a senior manager of purchasing for a large school district - What is 3/8 + 3/8 ?
I expect this manager deals in many millions of dollars annually.
It makes me wonder how important math is for the non- mathematician.
John McKay
Date: Mon, 13 Dec 2004 13:24:44 -0800 From: Henry Baker <hbaker1@pipeline.com>
Interest tables are some of the oldest known artefacts. I don't know offhand how old, though.
About 5000 years, according to the below.
Some businessmen would claim that the invention of interest ranks right behind the invention of money in the promulgation of commerce. I'd be interested in knowing approx when interest was invented.
Will Goetzmann, a finance professor at Yale, has been writing a book called _Financing Civilization_, about just such things. Chapter 1, "Borrowing in Babylon" is on the web: http://viking.som.yale.edu/will/finciv/chapter1.htm It's been on the web for years; dunno if he's ever gonna finish it. Some of the things Goetzman says in re interest: * Excavations at Uruk dating ca 3700-3500 BCE found hollow clay envelopes called bullae, containing tokens for animals, bread, jars, and so on. They were apparently contracts for the future payment of some amount of commodities. The exact interest rate isn't calculable, since their symbols for time are either not found or not understood. Encasing them in fired clay prevented tampering. * Around 3100 BCE they began to use cuneiform, and contracts from that day survive. Fired clay tablets not only survive for millennia, but also prevented tampering by the parties involved. Shortly after that, there were professional scribes and accountants. So call it 5000 years old for the oldest tables of interest. * Early words for "interest" are intriguing: - Sumerian "mash" means "calves" - Greek "tokos" means "offspring of cattle" - Latin "pecus" (root of pecuniary) means "flock" - Egyptian "ms" means "to give birth" ... so perhaps they thought of interest like the natural multiplication of flocks of animals. -- Steve Rowley <sgr@alum.mit.edu> http://alum.mit.edu/www/sgr/ ICQ: 52-377-390
*I'm reminded of this recent entry at http://learningcurves.blogspot.com, by a math professor who taught "Mathematics for Teachers" this semester: -*snip*- Question:* Which of the following four statements is *true*? Circle the true statement. * 1/13 < 0.13 * 1/13 > 0.13 * 1/13 = 0.13 * 1/13 can not be compared to 0.13 *Answers:* All four choices were popular, with each being selected by about a quarter of my students. *Source of Question:* This question was taken directly from the item sampler provided by my state's Department of Education for the State Comprehensive Assessment Test given to all 8th graders. -*snip*- Richard Guy wrote:
In such matters, context is everything.
I've been intrigued for many moons by the fact that each morning I hear the market analyst reel off the changes in bond rates in 32nds and then give the resulting rates in % to two dec places. Does his machine quote these things thus?
R.
On Mon, 13 Dec 2004, Henry Baker wrote:
John:
I'm not so worried about his inability to do 3/8 + 3/8 -- after all, most of us can't do present value analysis in our heads, either.
What would bother me would be if he couldn't get his calculator and/or spreadsheet to do these calculations.
Henry ----- At 11:39 AM 12/13/2004, you wrote:
I was recently asked by a senior manager of purchasing for a large school district - What is 3/8 + 3/8 ?
I expect this manager deals in many millions of dollars annually.
It makes me wonder how important math is for the non- mathematician.
John McKay
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_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
At the time I took math, all the statements below were true. Things may be different now. Steve Gray
Question:* Which of the following four statements is *true*? Circle the true statement.
* 1/13 < 0.13 * 1/13 > 0.13 * 1/13 = 0.13 * 1/13 can not be compared to 0.13
I wonder how those prospective teachers would have fared comparing 7^sqrt(8) vs. 8^sqrt(7) instead of 1/13 vs. 0.13 It seems this problem is deviously difficult. When it was posted to the newsgroup sci.math only two people offered correct proofs. I encourage fellow funsters to give it a try first before looking below to see my quick arithmetic proof [1]; it requires < 2 minutes of purely mental arithmetic (no pencil, paper or calculator needed!). In fact, only arithmetic of two-by-one digit integers is employed. SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW SPOILER BELOW Note 8*29^2 > 7*31^2 via x=30 in 8(x-1)^2-7(x+1)^2 = x^2-30x+1 so sqrt(8) > 31/29 sqrt(7) by taking sqrt of above. Thus to prove sqrt(8) sqrt(7) 7 > 8 it suffices to prove 31/29 sqrt(7) sqrt(7) 7 > 8 or 7^30/8^28 > 8/7 but (7^5/2^14)^6 > (42/41)^6 since 7^5 = 7(50-1)^2 > 7(2500-100) = 16800 and 2^14 = 2^4 2^10 < 2^4 1025 = 16400 > 1 + 6/41 since (1+x)^n > 1+nx [binomial theorem, x>0] > 1 + 1/7 since 7*6 > 41 Another proof [2] proceeds via calculus, namely the inequality (2n+1) All n: x f (a) > 0 => f(a+x) > f(a-x) Does anyone else know any other "interesting" proofs? Below are said sci.math threads. [1] Bill Dubuque, sci.math, 1996/06/08 http://google.com/groups?threadm=WGD.96Jun8060426@berne.ai.mit.edu [2] Don Davis, sci.math, 2000/11/24 http://google.com/groups?threadm=dtd-2411002306240001@ppp0c005.std.com --Bill Dubuque
I wrote about:
comparing 7^sqrt(8) vs. 8^sqrt(7) instead of 1/13 vs. 0.13
In case it was lost in the details it's worth pointing out that both comparisons are easily obtained by looking for a "simple" interpolant c between the two numbers a,b. Here "simple" means it is simple to show say a<c, c<b so a<b by transitivity. For 1/13 < 0.13 a simple interpolant is c = 1/10 = 0.10 For 8^sqrt(7) < 7^sqrt(8), I chose c = 7^(31/29 sqrt(7)), and then I further employed the following simple interpolants: (7^5/2^14)^6 > (16800/16400)^6 = (42/41)^6 > 1 + 6/41 > 1 + 1/7 Of course this is a very trivial example of breaking up a complex problem into "simpler" steps - a process that becomes almost subconscious to well-trained mathematician. That student math teachers lack such basic skills seems to indicate that there is a major gap in the teaching of mathematical problem solving. No doubt some of the blame lies on the widespread availability of computers and calculators. As Gauss once said "the purpose of calculation is insight". In the pre-calculator days one developed good insight and problem-solving skills because one was forced to in order to make hand/mental calculations tractable. But nowadays most students never develop such insight because they've been limping so long on computer crutches. They will be at a loss as soon as they are presented a problem that doesn't fit mindlessly into the prepackaged set of problems soluble by brute-force by their symbiotic partner. It should come as no surprise then that US math students keep performing poorly vs. other nations. They are the victims of the ubiquity of the technology and it's rampant abuse in the US educational system. Until this problem is rectified, calculator brain drain will continue to sap their minds. Btw, here's a precise link to the 1/13 vs 0.13 article http://learningcurves.blogspot.com/2004/12/from-final.html --Bill Dubuque
Am I allowed to say that sqrt is increasing more rapidly than log, so that sqrt 8 sqrt 7 ------ > ------ ln 8 ln 7 sqrt 8 sqrt 7 and hence 7 > 8 ? [ I've multiplied by ln 7 ln 8 and taken antilogs] R. On Wed, 15 Dec 2004, Bill Dubuque wrote:
I wonder how those prospective teachers would have fared
comparing 7^sqrt(8) vs. 8^sqrt(7) instead of 1/13 vs. 0.13
It seems this problem is deviously difficult. When it was posted to the newsgroup sci.math only two people offered correct proofs. I encourage fellow funsters to give it a try first before looking below to see my quick arithmetic proof [1]; it requires < 2 minutes of purely mental arithmetic (no pencil, paper or calculator needed!). In fact, only arithmetic of two-by-one digit integers is employed.
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Note 8*29^2 > 7*31^2 via x=30 in 8(x-1)^2-7(x+1)^2 = x^2-30x+1
so sqrt(8) > 31/29 sqrt(7) by taking sqrt of above.
Thus to prove
sqrt(8) sqrt(7) 7 > 8
it suffices to prove
31/29 sqrt(7) sqrt(7) 7 > 8
or 7^30/8^28 > 8/7 but (7^5/2^14)^6 > (42/41)^6 since 7^5 = 7(50-1)^2 > 7(2500-100) = 16800 and 2^14 = 2^4 2^10 < 2^4 1025 = 16400 > 1 + 6/41 since (1+x)^n > 1+nx [binomial theorem, x>0]
> 1 + 1/7 since 7*6 > 41
Another proof [2] proceeds via calculus, namely the inequality
(2n+1) All n: x f (a) > 0 => f(a+x) > f(a-x)
Does anyone else know any other "interesting" proofs?
Below are said sci.math threads.
[1] Bill Dubuque, sci.math, 1996/06/08 http://google.com/groups?threadm=WGD.96Jun8060426@berne.ai.mit.edu [2] Don Davis, sci.math, 2000/11/24 http://google.com/groups?threadm=dtd-2411002306240001@ppp0c005.std.com
--Bill Dubuque
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participants (7)
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Bill Dubuque -
Henry Baker -
MCKAY john -
Richard Guy -
Steve Gray -
Steve Rowley -
Thane Plambeck