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Re: [Phys-l] Student engagement--GAIN

John Clement wrote:
Of course your puzzle is not the same because they start at the same time.
But my reply is do you know why the puzzle was presented in the activity?
It has a pedagogical goal.
*** One can easily suppose that subverting the usual student means to formulating an algebraic solution was intended in all good faith, in order to facilitate inspection of geometric or graphical means. I am inexorably reminded of the Arabian methods of solving equations of the first and second degree, known since the seventh century.
[ al-jebr w'al-muqabalah. “The Manipulation of Equations” , al-Khowarizmi (c 825) ]

Long centuries later, when the art had diffused to Italy, and developments soon led to cubic and quartic methods, I am amused to consider this concealment of solutions as comparable to the Italian Mathematicians' concealment of methods, related here in delightful prose...
*/from the/**/ Encyclopaedia Britannica,/** 9th Edition (1875)
and 10th Edition (1902)**, as** conve**yed** at** http://www.1902encyclopedia.com/*
“It was then (14^th C.) the practice among the cultivators of algebra, when they advanced a step, to conceal carefully from their contemporaries, and to challenge them to resolve arithmetical questions, so framed as to require for their solution a knowledge of their own new-found rules. In this spirit did Ferreus make a secret of his discovery: he communicated it, however, to a favorite scholar, a Venetian named Florido. About the year 1535 this person, having taken up his residence at Venice, challenged* Tartalea of Brescia*, a man of great ingenuity, to a trial of skill in the resolution of problems by algebra. Florido framed his questions so as to require for their solution a knowledge of the rule which he had learned from his preceptor Ferreus; but Tartalea had, five years before this time, advanced further than Ferreus, and was more than a match for Florido. He therefore accepted the challenge, and a day was appointed when each was to propose to the other thirty questions. Before the time came, Tartalea had resumed the study of cubic equations, and had discovered the solution of two cases in addition to two which he knew before. Florido's questions were such as could be resolved by the single rule of Ferreus; while, on the contrary, those of Tartalea could only be resolved by one or other of three rules, which he himself had found, but which could not be resolved by the remaining rule, which was also that know to Florido. The issue of the contest is easily anticipated; Tartalea resolved all his adversary's questions in two hours, without receiving one answer from him in return.

The celebrated* Cardan* was a contemporary of Tartalea. This remarkable person was a professor of mathematics at Milan, and a physician. He had studied algebra with great assiduity, and had nearly finished the printing of a book on arithmetic, algebra, and geometry; but being desirous of enriching his work with the discoveries of Tartalea, which at that period must have been the object of considerable attention among literary men in Italy, he endeavored to draw from him a disclosure of his rules. Tartalea resisted for a time Cardan's entreaties. At last, overcome by his importunity, and his offer to swear on the holy Evangelists, and by the honor of a gentleman, never to publish them, and on his promising on the faith of a Christian to commit them to cipher, so that even after his death they would not be intelligible to any one, he ventured with much hesitation to reveal to him his practical rules, which were expressed by some very bad Italian verses, themselves in no small degree enigmatical. He reserved, however, the demonstrations. Cardan was not long in discovering the reason of the rules, and he even greatly improved them, so as to make them in a manner his own. From the imperfect essays of Tartalea he deduced an ingenious and systematic method of resolving all cubic equations whatsoever; but with a remarkable disregard for the principles of honor, and the oath he had taken, he published in 1545 Tartalea's discoveries, combined with his own, as a supplement to treatise on arithmetic and algebra, which he had published six years before. This work is remarkable for being the second printed book on algebra known to have existed.”
Incidentally in the book this is the first time
they have been asked to do a kinematic problem, and it introduces motion
maps or what they call a strobe diagram.
*** I hold the view that the Royal path to continued scholastic success is by means of measured steps of modestly successful solutions, rather than by casting around after failure of the desired approach....
I suspect that most students couldn't do the alternate puzzle which was
presented by Brian.

John M. Clement
Houston, TX

Though the Arabians did not command the use of algebraic symbols yet, their 7th century methods did not differ greatly from this:

Let x be the distance from John's starting point to the point at which he meets Sandra.
Let t be the time elapsed to this meeting.
Say distance / time represents average speed.
Then...
x/t = 1.5 m/s and
(33-x)/t = 2.5 m/s But t is common to these two equations,
so x = 12.375 m and therefore t = 8.25 seconds.
This point is (18 - 12.375) meters before the lamp post, or 5.625 m.

Surely more students could hope to solve this equation?

Brian W
This is a pleasant puzzle, dressed up for obfuscation.
Say it were presented in this manner:

John sets out to meet Sandra, who is 33 meters away.
He steps towards her at 1.5 meters/sec. She steps out
towards him at the same time, at 2.5 meters/sec.
There is a lamp post at 18 meters from John's starting
point. This post is initially 15 meters from Sandra
of course.
Where do they meet, in relation to the lamp post?
I fancy more kids could answer this algebraically.

The dressed up puzzle reminds me of the puzzle
starring an insect flying to and from between two trains:
and how far it travels before being squashed when
the locos collide. This one too, has a need
to be 'unfolded' for ready solution.

Brian W
p.s. This is not to minimize the conceptual visualization
value of the graphical approach over the algebraic method
for the puzzle as given.

John Clement wrote:
/snip/
Here is a specific problem that students in general either find impossible to do, or impossible in a short time to do by algebra. /snip/
http://srri.umass.edu/files/mop_samples/Act016.pdf /snip/
John M. Clement Houston, TX

Actually there are some problems which are very difficult using equations, and are much easier using either graphs or motion maps.