Skip to content
Archive with last of tag-string Q31

Vn12401

Vn124.1 Analogical Guidance 2/23/78

This evening at dinner, my family enjoyed a good time at the expense
of Scurry, our Scotch terrier. Earlier in the day, Miriam had played
tug with Scurry, the object of their contention a squeaking toy mouse
she had given the dog. Scurry wrested the toy from Miriam and sat chew-
ing it out of reach. Miriam then rolled Scurry’s small ball across the
floor, and Scurry, the mouse grasped firmly in her teeth, bounded off in
pursuit. She caught the ball and appeared trying to pick it up but
failed because the mouse was still in her teeth and she was definitely
not willing to loosen her grip. After relating this story to Robby,
Miriam rose from the table to demonstrate Scurry’s bind by duplicating
the situation. Scurry would not cooperate; she kept the mouse and
refused to chase the ball. Robby wandered off and Miriam, a little
dejected, draped herself over the back of the couch.

But there, her interest rekindled, for she saw on my toyshelf a
puzzle she has been working at unsuccessfully for a week or more. This
is the Pythagorean puzzle (cf. Vignette 77, Geometric Puzzles) which I
have realized in both 5 and 7 piece forms. This is their form, each
assembled as a single large square:

Miriam failed to assemble the 7-piece puzzle despite her repeated attempts
this past week. It is a vanguard problem for her. When she dumped out
the pieces on the couch, I asked her to bring it to the table and pushed
aside the dishes. Miriam complained, “I can’t do it.” When I asked why
not rhetorically and advised her it was just like the 5-piece puzzle,
she responded, “But I can’t do that either.” Miriam has done the 5-piece
puzzle. Her statement may mean she can not do it at will, without trial
and error, that she does not comprehend the puzzle. Miriam brought both
puzzles to the table but had trouble locating the center square which,
when found behind the couch, I kept.

Miriam began with two congruent triangles, thus:

Vn 124-2 Yukky DIamonds

She declared the first a “yukky diamond” and tried, with no confidence,
to suggest the rectangle was a square. I gave her a hint: you have to
use all the pieces to make a square. Miriam then articulated a salient
bit of known knowledge: “This side has to be on the outside.” When
queried, she pointed to the hypotenuse of one of the congruent triangles.
She tried, in order, these intermediate configurations:

Vn 124-3 point to point

As Miriam attempted lining up the corners of the 3rd triangle, she
pushed away the second from the first, saw the configuration below, and
held out her hand to me.

Vn 124-4 pattern of insight

I gave her the center square, and she completed the puzzle. “Now do the
7-piece,” I challenged her.

Miriam laid the two complete triangles of the 7-piece puzzle on top
of the 5-piece assembly, arranged as in V, and noted, “I’m using the
same patterns.” She added the corner-cut-off congruent triangle; first
she put it in backwards, then as below. Her outstretched hand requested
the missing corner.

Vn 124-5 analogy by superposition

I gave her the missing corner and center square. Miriam tried the
smaller, similar triangle abutting the center square, then moved two
vertices to the periphery. She first tried the final piece backwards,
then completed the puzzle correctly.

Vn 124-6 7 piece completed

Miriam was pleased with herself. I removed the 7-piece puzzle and
asked, “Can you make a black-colored square?”

When she had done so, I asked if she could then make a gold one.
Miriam asked, “Do I have to use the square?” I said she should try to
without it.

Vn 124-7 golden square

Arrangement XIII showed Miriam she needed the little square which I
returned to her. The 7-piece puzzle followed. Miriam was stuck for a
while. One hint crystallized her completion of the 7-piece puzzle:
“Find a shape the same as this black part.”

Miriam located the end-cut triangle and fitted the smaller triangle
to it. She proceeded to build a column of these pieces, then added the
rectangle of 2 triangles in the appropriate orientation.

Vn 124-8 3 rectangles

Voila!

Relevance
Several themes seem to arise clearly here. Some relate to the
precipitating situation: stumbling into problems accidentally when
the materials are at hand; the existence of this unsolved puzzle as
an item on an internal agenda to be worked at till mastered.

Miriam retained a very specific piece of knowledge as a key element
of the solution: “the long edge goes on the outside.” She was able to
use generally formulated advice when it had specific and obvious appli-
cation (e.g. use all the pieces) as well as very specific direction
(e.g. find a piece with the same shape as the black area). Trial and
error plays a large role.

Finally, for this difficult 7-piece puzzle, the combination of a
few hints and an analogous, simple version about which 1 key point was
known operated as guidance as effectively as do the pictures on the
surface of a picture puzzle.

Vn12801

Vn128.1 Robby’s Topological Game 4/2/78

About a month ago, Robby was shown a paper-cutting game by a
classmate’s parent. The procedure to follow was this:

1. Cut two paper strips of equal length (8″ will do)
2. Draw a line down the middle of each (using lined paper makes
this unnecessary)
3. Bend each strip of paper into a circle and tape the juncture
4. Join the circles perpendicularly and tape the juncture
5. Cut around the mid-line of each circle.

When two strips of equal length are so connected and cut, the surprising
result is that, though having passed through a circular phase, the strip
halves end up taped together as a square.

Squaring two circles

Robby enjoyed this game when shown it. Yesterday, I removed
a paper form he had made in the past (an 8 x 11 sheet divided into 11
strips 8″ long) from my clipboard and gave it to him. When I inter-
rupted his reading to give him this sheet of paper, Robby recalled the
game and quietly took it up on his own. He was very happy when the
procedure produced a square and showed it to Gretchen and me. We neither
paid much attention.

Going on to three circles, Robby cut two of the three along
their mid-lines. He judged (in error) that he had finished by finding
a square with a bar (a double strip) across the center. It lay flat.
Still no one paid attention. Robby went on to four circles, and he
cut all the mid-lines. What he got was a confusion of floppy paper.
I advised him to try to get it lying flat. Robby again borrowed my
clipboard, clipping and taping the product to it. He was delighted
when he succeeded in flattening the strip-figure and subsequently
taped it to a large piece of cardboard. The resulting shape is this:

But why stop at 4? Robby went on to connect and cut 5 circles. Here
he met another surprise. When cut, the 5 circles separated into
identical, non-planar shapes. Robby likewise taped these to another
piece of cardboard. When he made a cutting of 6 circles, controlling
the floppy strip-figures became a big problem. Robby succeeded at
taping it to the box from which he had been cutting cardboard backing
pieces, but in doing so went over an edge. He decided the problem
was getting too complicated to be fun and quit.

This morning I told him I had been thinking about his paper
cutting game and asked Robby to find the figure made from three circles.
When he returned, I asked him if he had cut all three circles. Robby
thought so, but when I pointed out the middle bar in his square was
double thick, he agreed he had only cut two. Robby saw immediately
that his square would divide into two rectangles. He cut the center
strip. “The 5’s made 2 too. Hey! I’ve got a new theory: the odd-
numbered circles make 2 and the evens all stay together.” I agreed
that this was an interesting speculation and that I could believe it
might be true, but that I couldn’t see immediately why it should be.

Relevance
I see this incident as one exceptionally valuable for
characterizing how significant learning occurs very naturally in a
mildly supportive milieu. First note that the initial exposure to the
“phenomenon” was quite memorable and puzzling. (How can you make a
square from two circles?) Robby clearly marked this phenomenon in his
mind as one which he would explore later. This pending explorarion was
invoked by the accident of his seeing a piece of paper approximately
meeting the material requirements for use in the game. The circum-
stance was one of no pressure. (He had been reading all of Gretchen’s
collection of Oz books and was probably a little bored.) He had no
outside direction or motivation at all. Once Robby succeeded at
making a square, he continued executing the procedure with stepwise
complications all focussed on one variable — the number of circles.
(He might have chosen to make the strips of different lengths — a
possibility he mentioned.) With the 3 circles, Robby stopped prema-
turely because he had produced a result (a square with a bar) only a
little different from the next simpler case (2 circles make a square).
With 4 circles, the outcomes of cutting were apparently sufficiently
confusing that completion could not be judged from the product but
depended on verifying that individual steps of the procedure were
completed. With the figure of 4 circles he was excited and delighted
to have succeeded in imposing some sort of order on the tangle — and
that the final product showed a family resemblance to the earlier
products. Finally, Robby was quick to jump to conclusions (his new
theory) in explaining why some figures were connected and others were
not.

Post Script — 4/3/78

After writing the preceding, I spoke to Robby again of his
game and his theory, inquiring whether or not he could prove it correct.
His method of choice was to test the case of 7 circles (which, as he
later found, splits into two planar figures of overlapping near-squares).
I tried to introduce the idea of a proof in place of another case study,
suggesting he take all possible cuttings of 3 connected circles and
figure out which one cuts the strip-figure in half. He said he had cut
the center first one time and at another had cut from one end.

Robby then drew the two pictures below on my chalk board:

Vn128-2 intermediate state squaring circle

He argued that it is always the last cut that severs the strip-figure
in two, representing the situation as at the above left. By cutting
along the dotted line, one joins the two small circles (here he made
motions of pulling apart the strip pieces) into the one large one.
Note well that this argument is merely a restatement of how he
appreciates the deformation, but it contented him.

Vn12901

Vn129.1 Robby Computes a Tax 4/5/78

Robby caught on fire again today. He approached me inquiring,
“How much is half of 423?” Miriam responded to his question from the
other room, “2 hundred and 11 and a half.” I told her to stop butting
in and asked Robby how much was half of 400, then half of 22, then half
of 1. He came to his own conclusion of 2 hundred and 11 and a half.

But why this concern with the specific question? $423. was
the price of a swing set in a catalog the children had been perusing.
They had agreed to go halves on buying this much-desired super-toy.
I opposed their doing so and raised as an objection along the way the
observation that they hadn’t included the amount of tax they would
have to pay.

“Is there a tax on toys?” was the incredulous question. “If
food is taxed,” I responded, “should you not expect toys to be taxed
also?” When he asked how much it was, I explained to Robby that he
could think of the tax as a nickel for every dollar of the purchase
price. Here we got into complicated computations.

Robby tried to figure out how much money is 4 hundred nickels.
His confusion was great, even including such faux pas as “there are 200
nickels in a dollar.” Correcting to 20 to the dollar, he went on to
observe that $100. of the purchase price converted to $5. of tax. Here
he was stymied but began to add $5. and another. I complicated his
computation by suggesting he use the multiplication results he had
learned at school. He looked blankly at me. “How much is 4 times 5?”
I asked, and received an answer: “20.” “How much is 4 times 5 dollars?”
No answer was forthcoming. He came to $20. eventually (I believe by
adding). Robby then computed the tax for 20 dollars more (of the
original $423.), and with Gretchen’s reminder, added another 15¢ for
the last 3 dollars.

This incident required a surprising amount of time, as much
as 5 minutes, to develop.

Relevance
This was a very exciting incident for Robby — his first
computation of a sales tax. He brought the idea of “a tax” under
control as a comprehensible percentage, thus eliminating that
mysteriousness which has troubled his world of money since
at least last summer (cf. Vignette 54).

Vn13101

Vn131.1 Miriam’s 7th Birthday 4/8 & 9/78

4/8/ Miriam began planning her birthday party several weeks ago.
On the 3 x 5 cards of Addendum 131 – 1, she listed the friends to bo
invited, the candy, and her selection of party games. The children were
all from her class at school. The games are all familiar, the first
being a party standard, the second played at Meg’s party, and the third
one of Miriam’s favorites from gym. (She also spoke of playing Red Rover
outside and was much concerned that Brian and Miceal should be on opposing
teams.) Miriam thought of getting cards for invitations, but did not.
Thus at the last minute we had to make our own. Miriam liked the idea
of preparing invitations at Logo, so we made a special trip there and
used the letter-writing procedures and her pretty flower to create her
unique invitations (cf. Addendum 131 – 2). Yesterday morning was dedi-
cated to preparing the house. We pushed the furniture out of the living
are of the loft to make a big play area, nonetheless praying for sun
shine so that we would not have 12 active kids confined in our small
apartment on a rainy afternoon.

A week of allergy-driven fitful sleep left Miriam physically
depressed but cheerful on her party day. She donned the party dress made
by her great-grandmother and played in the courtyard waiting for guests.
As they arrived, Robby helped first by carrying in presents and then by
playing soccer with the boys. At the one point where all the guests had
arrived and were inside, I spoke above the pandemonium to announce that
we would have an ice cream cake about 3 o’clock but that otherwise they
should enjoy themselves in whatever way they chose. The children gathered
about while presents were being opened. . . and then began a problem. An
early-opened gift was a set of face paints, which appealed to everyone, and
some children went off to the bathroom and decorated themselves. Somehow
two girls ended up fighting in the hallway, pulling hair and crying. At
this pass, Robby led the boys off to the tree fort and either Dara or
Lizzie suggested playing on the space trolley out back. I joined that
group of girls for fear they might get too close to our land lord’s
horses. Miriam and two friends stayed inside with Gretchen. From the
space trolley landing, the girls could see the boys across the lawn and
made an assault on the tree fort. That short-lived battle was ended by
my recalling the children for the party meal.

The children sat in a circle on the floor, and Miriam asked if
Peggy could join them. Everyone sang ‘Happy Birthday’, had his soda and
goodies, and after a quick clean up played in the courtyard until parents
started arriving.

Miriam was disgruntled, mainly because her face-paint crayons
had been used against her will and some got broken. She was also disap-
pointed that no one played the games she had picked out. (We discussed
later whether I should have directed them to, and Miriam opined that I
was right in not taking over). Miriam cheered up a little when I gave
her my present, a small string art design in the shape of a heart (which
she had requested) with a large letter ‘M’ in the middle, and when she
chose the evening’s dinner (pizza).

4/9/ This birthday began on a cheerful note. Since I had neglected
to give Miriam her weekly allowance on Saturday, the normal day, and
because it is calculated as a dime for each year of her age, on this day her
allowance was 70¢, where yesterday it would have been 60¢. This joke,
heightened by my feigned aggravation, delighted Miriam.

After a good night’s sleep, Miriam was considerably more chipper
than yesterday and eagerly accepted my suggestion that we should go riding
the trolley cars of Boston. She, Robby, and I took the Riverside line to
the terminus. The conductor, finding the kids and I were out for a ride,
would take no money, so we enjoyed a free ride both ways as we headed
down into the city. At Park Street we took the red line to Quincy,
stayed on board when the train reversed directions, and emerged at
Harvard Square. After a late lunch at Brigham’s we returned home by
the red line and the Commonwealth Avenue trolley for a quiet afternoon
and a small party this evening for the 5 of our family.

Addendum 131-1

Party Planning

Vn 131-1 Party Planning

Addendum 131-2

Party Invitations Made at Logo

Vn 131-2 Party Invitations