Yes, you will get t^2 = 36, which gives you 6 hrs. (Either 6 hrs forward or backward depending on pos or negative root).

-- BarryGarelick - 30 Oct 2005

Looks like I copied the problem wrong. I used 5 pm for the second villager arrival time for some reason. The answer checks for this time, so since I did the problem slightly differently, I'll go through the solution to givr a slightly alternate way.

I'll use Barry's variables, along with D as the total distance from the first village to the noon spot and 1-D as the distance from the second village to the meeting spot.

I got the following 4 equations:

f = (1-D)/5 f = D/t s = D/9 s = (1-D)/t

4 equations and 4 unknowns means it's solvable.

let's solve for D first:

s = D/9 = (1-D)/t D/9 = (1-D)/t 9D + tD = 9 D(9 + t) = 9 D = 9/(9+t)

Now let's solve for t:

f = (1-D)/5 = D/t t(1-D) = 5D t - tD - 5D = 0

Let's substitute what we found for D to get down to one variable and one equation:

t - t(9/(9+t)) - 5(9/(9+t)) = 0 t(9+t) - 9t - 45 = 0 9t + t^2 - 9t - 45 = 0 t^2 = 45 t = 6.71 or approx 6 hrs and 43 mins Subtracting this from noon gives us 5:17 am

We can also solve for D, f, s by plugging this value back into the other equations:

D = 0.573 d's (an arbitrary unit of measurement) f = 0.0854 d/hr s = 0.0636 d/hr

I'll have to resolve with a time of 4 pm to see if I get the same answer as Barry and Jim milgram, but it looks in the right ballpark with the hour difference.

-- KDeRosa - 30 Oct 2005

Let’s discuss what happens if we use -3/2 = f/s

Here is my solution to the Russian villagers problem.

K De Rosa gave us 5:17 AM. This is a different answer, but not necessarily the right one. In fuzzy math philosophy there are many right answers to the same problem, so I will wait to see how K De Rosa explains how he arrived at 5:17 AM.

I arrived at 6 AM which Jim Milgram tells me is correct. Or rather, the answer he got.

Let t = the time in hours it took from sunrise until noon. (Which is when they both met each other)

Let s = rate of speed of the slow walker Let f = rate of speed of the fast walker

To solve the problem there are two relationships to work with. 1.

The distance covered by the slow walker in t hours = the distance the fast walker completes her remaining journey or: st = 4f

A similar relationship is expressed in terms of the fast walker: ft = 9s

2. The total distance each person walks is equal. We can express this as:

f(4+t) = s(9+t)

Now, let’s go back to the first relationship:

Solving each equation for t, we get t = 4f/s, and t = 9s/f.

Since both expressions equal t, we get 4f/s = 9s/f, or 4f^2 = 9s^2, leading to f^2/s^2 = 9/4.

Taking positive square roots, f/s = 3/2. This leads to the equation f = (3/2)s (I’ll talk later about the possibility that we can also take negative square roots for f or s, to obtain -3/2).

We go now to the second relationship and substitute f = (3/2)s into f(4+t) = s(9+t).

Doing this yields: (3/2)s(4+t) = s(9+t)

Solve for t. t = 6 hrs.

To obtain time of sunrise, subtract 6 hrs from 12 noon to obtain 6 AM.

Now, for the honors students, let’s discuss what happens if we use -3/2 = f/s

If you substitute f = -(3/2)s into the second equation, you obtain -6 hrs.

For Star Trek fans, this could mean that if there were a black hole between the villages and time went backwards, then the sun rise was at 12 noon – (-6 hr) = 12 + 6 = 6 PM.

For Lewis Carroll fans, this could mean that if the villagers used clocks that had a normal face, but the clock ran backwards (i.e, counterclockwise), their clocks would have shown the sun rising at 6 PM.

The mathematics gives us -6 hrs, and really doesn’t give a hoot what interpretation you choose to give it.

-- BarryGarelick - 30 Oct 2005

Answer is 5:17 am

-- KDeRosa - 28 Oct 2005

So how do you solve the villager problem?

-- KtmGuest - 28 Oct 2005

-- KtmGuest - 27 Oct 2005

i care.
that russian villagers problem is great;
i'll be sharing it with some math buddies soon.

milgram's my doctor-grandfather, btw:
his ph.d student jim davis directed my dissertation.
("james f. davis" in the published literature.)

-- VlorbikDotCom - 07 Oct 2005

"They teach like they used to in the U.S."

I had read somewhere, Liping Ma maybe, that China (I think) actually took western math textbooks (since they were the best anyone had at the time) and used them as the basis of their own math curriculum. Then, they figured out the deficiencies and improved upon them. They didn't try and reinvent the wheel (cough, NCTM, cough), they just stood on the shoulders of others.

Great stuff, Barry.

-- KDeRosa - 06 Oct 2005

Alice’s retort

Trespassers in Wonderland

In the 2004 National Assessment of Education Progress (NAEP) exam only 20 percent of fourth graders correctly calculated the answer to 314 x 12. Eighth graders’ performance was also disturbing: a question asked for the length of a line segment above a ruler, with one end at the 2 cm mark and the other at the 7 cm mark. Only 58 percent of eight graders got it right; and it was multiple choice. On the international front, anyone following how U.S. fourth and eighth graders fare in international tests in math (called TIMMS) have by now noticed that U.S. has come in about 14th or 15th, and that Asian countries top the list (Singapore is number one).

To put the issue of math education in context, one has to understand the prevailing attitude toward math education in this country. Two years ago, at a packed conference on math education. Jim Milgram, a math professor from Stanford, gave a talk in which he presented the following story problem which, he noted, seventh grade students in Russia are expected to solve: “Two people left their villages at sunrise and walked, each to the other’s village at constant speed. They met at noon and the first arrived in the others’ village at 4:00 PM while the second arrive at 9:00 PM. What time was sunrise?”

At this, a man sitting behind me articulated the following reaction typical of those who believe that U.S. students do not perform well in math because they are not taught how to apply it to the problems that actually occur in real life: “Who cares?”

Sentence first—verdict afterward

In May, 2005, the National Council of Mathematics Teachers (NCTM) in a statement which appeared in the Washington Post echoed the exact same “Who cares?” sentiments as the disgruntled man at the conference:

“For generations, mathematics was taught as an isolated topic with its own categories of word problems. It didn’t work. Adults groan when they hear ‘If a train leaves Boston at 2 o’clock traveling at 80 mph, and at the same time a train leaves New York...’ Whatever problems and contexts are used, they need to engage students and be relevant to today’s demanding and rapidly changing world.”

NCTM is a large organization based in Reston, Virginia which exerts considerable influence over how math is taught in this country. In 1989, NCTM published a set of curriculum and evaluation standards for math, and revised them in 2000. Some states have relied on these standards in framing their own. Such standards de-emphasize learning basic skills, is light on content and heavy on context-based learning otherwise known as “real life math”. Cathy Seeley, current president of NCTM is critical of math texts and programs that tell students "here's the rule, now do the problem" and says there is too much “teacher instruction” in the U.S. NCTM’s topsy-turvy approach to teaching math is more like “Here’s the problem, you figure out the rules needed to solve it”—an approach alarmingly similar to the Queen’s declaration at Alice’s trial in Alice in Wonderland: “Sentence first–verdict afterward.” Some real life problems

Here’s an example of a real life problem which can be found on NCTM’s very own web site in the section called “Illuminations”:

"Suppose you have saved $63. You find a used video game system that you would like to buy. The seller is asking $180. You earn $10 a week doing odd jobs. How long will it take you to earn enough money to buy the game?”

While this type of problem has been around for years, NCTM’s suggestions for how to “explore” the problem in class is what’s different. They explain that adults typically subtract 63 from 180 and divide by 10. While this would be a preferred approach for students to have mastered by the 5th or 6th grade–the grade level for this activity–NCTM describes with particular pride a student entering 63 into the calculator (no apology offered for calculators being used here), then adding his first week’s allowance, then the second, third, and so forth until the display showed that he had at least $180 (12 weeks).

NCTM explains: "Allowing students the freedom to use strategies that are intuitively obvious to them helps them to feel more comfortable in the problem-solving process. At some stage it also helps them appreciate the efficiency of standard algorithms." NCTM does not discuss when this stage will occur. One would hope that it occurs quickly so that the calculator-aided counting-on-fingers method can be supplanted with the more efficient method that students in Japan and Singapore have mastered by the third grade.

A Word from NCTM

Recently, when asked why U.S. students suffer from an inability to perform complex reasoning and mathematical assignments compared to students overseas, Cathy Seeley responded: “We’re not doing as much problem-solving of that type as we need to be.” In another instance, she said “We can definitely learn lessons from Singapore, Japan and China. But we have to look beyond their textbooks to determine what these lessons are.”

Even a faithful NCTM adherent would not fail to notice that in Singapore’s textbooks, problems require multi-step solutions that are considerably more complex than what we expect US students to solve at that grade level. From a sixth grade Singapore textbook: 3/5 of Mary's flowers were roses and the rest were orchids. After giving away ½ of the roses and 1/4 of the orchids, she had 54 flowers left. How many flowers did she have at first?

Looking beyond the textbook as Cathy suggests allows NCTM to throw the baby out with the bath water, and to reject problems that are good by saying “It’s not the text, it’s the teaching.” In fact, in Japan, Singapore and Russia, they do teach math differently. They teach it correctly. They teach content. They teach skills and facts as a foundation upon which understanding will be built. They teach like they used to in the U.S. Alice’s retort

In Alice in Wonderland, Alice tells the royal family “Who cares for you? You’re nothing but a pack of cards!” In real life, however, boards of education, school districts and state departments of education are bowing to a pack of cards that has made math education almost content free. Over and over, as parents, teachers, and world-class mathematicians protest how math is being taught, and tell school boards and administrators the type of content students should be mastering, they are viewed as trespassers in Wonderland. Story problems are met with groans, proclaimed not to be real life, and dismissed with a mighty “Who cares?”

“Who cares is not the point,” Jim Milgram says. “Let me give you an example of a problem that people had better care about since it will affect their very lives. Design a robot arm to select and lift items off an assembly line and place them on a second line correctly positioned for a second robot to work on them. There is no chance in hell that someone can do this if they can't do the Russian problem about the two villagers.”

Until “real life math” is recognized for the pack of cards it is, the influence of NCTM and their followers will continue, as will the unmistakable and irreversible harm to our children, many of whom do not know how to multiply two-digit numbers without a calculator, nor how to use a ruler.

-- BarryGarelick - 06 Oct 2005

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