An Especially Interesting EQUATIONS Match Between Sixth-and Seventh-Graders
by Layman E. Allen
The match that follows actually occurred between
two seventh-grade girls and a sixth-grade girl in the
spring of 1975. it was a three-player match played
over the telephone. The sixth grader was playing
from a rural school in Dexter, Michigan; one of the
seventh graders was playing from an inner-city
school in Detroit; the other seventh grader was
playing from suburban Ann Arbor.
There were four other matches occurring concurrently over the same telephone hook-up, and all five of the matches were being monitored by a member of the Instructional Gaming Project staff. That is how this particularly interesting match was ultimately made available for inclusion in this DIG Math (Diagnostic Instructional Gaming) program.
RESOURCES: + – - X / / ^ √ √ 1 2 2 3 3 4 6 9 9
In this match you will find the symbol ‘√’, which stands for the root operation, and the symbol ‘^’, which stands for exponentiation. For example, ‘A^B’ indicates “A to the power of B”, and ‘C√A’ indicates “the Cth root of A”.
There were four other matches occurring concurrently over the same telephone hook-up, and all five of the matches were being monitored by a member of the Instructional Gaming Project staff. That is how this particularly interesting match was ultimately made available for inclusion in this DIG Math (Diagnostic Instructional Gaming) program.
RESOURCES: + – - X / / ^ √ √ 1 2 2 3 3 4 6 9 9
In this match you will find the symbol ‘√’, which stands for the root operation, and the symbol ‘^’, which stands for exponentiation. For example, ‘A^B’ indicates “A to the power of B”, and ‘C√A’ indicates “the Cth root of A”.
Player 1 starts the match by setting a Goal of ’36′.
PLAYER 1′S ACTION: Goal = 36 GOAL: (36) RESOURCES: + – - X / / ^ √ √ 1 2 2 3 4 9 9 FORBIDDEN: PERMITTED: REQUIRED:
In analyzing the Goal set by Player 1 (P1), it was apparent to P2 that the three claims made about the situation being determined by setting 36 as the Goal are all true: (1) the mover has not made it too hard — the remaining Resources can be played so that a Solution can be built, (2) the mover has not made it too easy — the mover has avoided (if it was possible for her to do so) allowing the next mover to move just one more Resource and then build a Solution, and (3) the mover has not failed to correctly challenge the previous mover for making a false claim in moving. Since all three of these claims are true with respect to P1′s move, Pl has not made a false claim — P1 has not flubbed. We can only speculate about P2′s thoughts were when she
PLAYER 1′S ACTION: Goal = 36 GOAL: (36) RESOURCES: + – - X / / ^ √ √ 1 2 2 3 4 9 9 FORBIDDEN: PERMITTED: REQUIRED:
In analyzing the Goal set by Player 1 (P1), it was apparent to P2 that the three claims made about the situation being determined by setting 36 as the Goal are all true: (1) the mover has not made it too hard — the remaining Resources can be played so that a Solution can be built, (2) the mover has not made it too easy — the mover has avoided (if it was possible for her to do so) allowing the next mover to move just one more Resource and then build a Solution, and (3) the mover has not failed to correctly challenge the previous mover for making a false claim in moving. Since all three of these claims are true with respect to P1′s move, Pl has not made a false claim — P1 has not flubbed. We can only speculate about P2′s thoughts were when she
decided to move the ‘+’ to Forbidden.
PLAYER 2′S ACTION: F+ GOAL: 36 RESOURCES: – - X / / ^ √ √ 1 2 2 3 4 9 9 FORBIDDEN: (+) PERMITTED: REQUIRED:
Because there are so many Solutions that involve the ‘+’, P2 might well have been trying to simplify the total situation some by getting rid of a large number of the more simple Solutions. There were at least 16 Solutions snuffed by P2′s F+ move. Now many of them do you see in this situation?
The following 16 Solutions were snuffed by P2′s F+ move: (3+1)X9 (9+9)X(3-1) (9X(3+2))-9 (9-((4+1)-2))^2 (2+2)X9 (9+9)X(4-2)
PLAYER 2′S ACTION: F+ GOAL: 36 RESOURCES: – - X / / ^ √ √ 1 2 2 3 4 9 9 FORBIDDEN: (+) PERMITTED: REQUIRED:
Because there are so many Solutions that involve the ‘+’, P2 might well have been trying to simplify the total situation some by getting rid of a large number of the more simple Solutions. There were at least 16 Solutions snuffed by P2′s F+ move. Now many of them do you see in this situation?
The following 16 Solutions were snuffed by P2′s F+ move: (3+1)X9 (9+9)X(3-1) (9X(3+2))-9 (9-((4+1)-2))^2 (2+2)X9 (9+9)X(4-2)
(4+2)^2
(2^(9-3-1))+4
(9+9)X2
(9-3)X(9-(2+1))
(9-(2+1))^2
(3^(4-1))+9
(9X3)+9
(9-3)X(4+2)
((4+3)-1)^2
(9-2)^2)-(9+4)
More were snuffed. Do you see any of them?
In her analysis of P2′s move, it is apparent to P3 that the three claims made by P2 in making a move are all true; hence, that P2 has not flubbed. P3 knows that it is safe for her to move. She moves the ‘X’ to Forbidden.
PLAYER 3′S ACTION: FX GOAL: 36 RESOURCES: – - / / ^ √ √ 1 2 2 3 4 9 9 FORBIDDEN: + (X) PERMITTED: REQUIRED:
In her analysis of P2′s move, it is apparent to P3 that the three claims made by P2 in making a move are all true; hence, that P2 has not flubbed. P3 knows that it is safe for her to move. She moves the ‘X’ to Forbidden.
PLAYER 3′S ACTION: FX GOAL: 36 RESOURCES: – - / / ^ √ √ 1 2 2 3 4 9 9 FORBIDDEN: + (X) PERMITTED: REQUIRED:
Again, we can only speculate what was going
on in the player’s mind — but what P3 might
have been trying to do by such a move is to
precipitate an erroneous P-flub challenge by
P1 or P2. P3′s FX move snuffs at least seven
different Solutions that contain the ‘X’. How many
of them do you see? Here are six Solutions snuffed
by P3′s move:
9X4 9X(2^2) (3^2)X(9-4-1) 9X(9-4-1) (3X2)^2 (3^2)X4
With the ‘^’ still available in the resources, there are still some Solutions possible that these experienced sixth- and seventh-grade players of EQUATIONS readily recognize.
9X4 9X(2^2) (3^2)X(9-4-1) 9X(9-4-1) (3X2)^2 (3^2)X4
With the ‘^’ still available in the resources, there are still some Solutions possible that these experienced sixth- and seventh-grade players of EQUATIONS readily recognize.
PLAYER 1′S ACTION: F^
GOAL: 36
RESOURCES: – - / / √ √ 1 2 2 3 4 9 9
FORBIDDEN: + X (^)
PERMITTED:
REQUIRED:
After her analysis indicates that it is safe for her to move, P1 then pulls a surprise. She moves the ‘^’ to Forbidden. The effect of the move is to snuff at least the three following Solutions:
(9-3)^2 (9-(4-1))^2 (9-3)^(4-2)
The interesting question is whether there are any Solutions still possible after this F^ move. Do you see any? If you do not, then if you were in P2′s shoes, you should make a P-flub challenge. Maybe it will spur your curiosity (or efforts) to know that neither P2 nor P3 challenged this move of P1′s, although each of them had the option to do so by the game rules.
After her analysis indicates that it is safe for her to move, P1 then pulls a surprise. She moves the ‘^’ to Forbidden. The effect of the move is to snuff at least the three following Solutions:
(9-3)^2 (9-(4-1))^2 (9-3)^(4-2)
The interesting question is whether there are any Solutions still possible after this F^ move. Do you see any? If you do not, then if you were in P2′s shoes, you should make a P-flub challenge. Maybe it will spur your curiosity (or efforts) to know that neither P2 nor P3 challenged this move of P1′s, although each of them had the option to do so by the game rules.
PLAYER 2′S ACTION: F/
GOAL: 36
RESOURCES: – - / √ √ 1 2 2 3 4 9 9
FORBIDDEN: + X ^ (/)
PERMITTED:
REQUIRED:
In her analysis of P1′s move, P2 concluded that none of the three claims made by the move were false; therefore, there was no flub, and it is safe to move. She moves the one of the ‘/’s to Forbidden. This move has the effect of snuffing at least three Solutions involving the concept of division by fractions:
9/(1/4) 9/(1/(9-3-2)) 9/(2/(9-1))
But what is P2 trying to do in making the F/ move? If either of the other two players make a P-flub challenge, will they be correct? Such a challenge will throw the burden of proving that there still is a Solution possible on P2.
In her analysis of P1′s move, P2 concluded that none of the three claims made by the move were false; therefore, there was no flub, and it is safe to move. She moves the one of the ‘/’s to Forbidden. This move has the effect of snuffing at least three Solutions involving the concept of division by fractions:
9/(1/4) 9/(1/(9-3-2)) 9/(2/(9-1))
But what is P2 trying to do in making the F/ move? If either of the other two players make a P-flub challenge, will they be correct? Such a challenge will throw the burden of proving that there still is a Solution possible on P2.
Would she be able to sustain such a burden of
proof? Would you? A further prod — neither of
these other players (a sixth-grader and a seventh-
grader) challenged. They still had more Solutions.
PLAYER 3′S ACTION: F/ GOAL: 36 RESOURCES: – - √ √ 1 2 2 3 4 9 9 FORBIDDEN: + X ^ / (/) PERMITTED: REQUIRED:
After analyzing and seeing that it is safe to move in this situation, P3 moves the other ‘/’ to Forbidden. This move has the effect of snuffing at least three Solutions involving the concept of fractional root operations:
(1/2)√(9-3) (2/4)√(9-3) (4/(9-1))√(9-3)
PLAYER 3′S ACTION: F/ GOAL: 36 RESOURCES: – - √ √ 1 2 2 3 4 9 9 FORBIDDEN: + X ^ / (/) PERMITTED: REQUIRED:
After analyzing and seeing that it is safe to move in this situation, P3 moves the other ‘/’ to Forbidden. This move has the effect of snuffing at least three Solutions involving the concept of fractional root operations:
(1/2)√(9-3) (2/4)√(9-3) (4/(9-1))√(9-3)
This sixth-grader is claiming by such a move that
there still is at least one Solution possible after the
F/ move.
There may be some readers who are still wondering why the above three expressions are Solutions. Let’s consider the first one.
(1/2)√(9-3) = (1/2)√6 = 6^(2/1) = 6^2 = 36.
This result follows from the fact that for some numbers, the root operation and exponentiation are inverse operations (just as ‘+’ and ‘-’ are inverse operations, and as ‘X’ and ‘/’ are, too). One of the effects of this inverse relationship between them is that for any non-zero numbers A and B and any positive number C,
(A/B) √ C = C ^ (B/A).
There may be some readers who are still wondering why the above three expressions are Solutions. Let’s consider the first one.
(1/2)√(9-3) = (1/2)√6 = 6^(2/1) = 6^2 = 36.
This result follows from the fact that for some numbers, the root operation and exponentiation are inverse operations (just as ‘+’ and ‘-’ are inverse operations, and as ‘X’ and ‘/’ are, too). One of the effects of this inverse relationship between them is that for any non-zero numbers A and B and any positive number C,
(A/B) √ C = C ^ (B/A).
There were no Challenges from the two seventh-
graders after P3′s move. Instead the match just
continued with the players moving all the
remaining resources into Permitted except one of
the ‘-’s. At that point P2 declared a Force-Out, and
the burden of proving that there was a Solution
possible was cast upon all three players.
PLAYER 2′S ACTION: Declares Force-Out GOAL: 36 RESOURCES: - FORBIDDEN: + X ^ / / PERMITTED: 1 2 2 3 4 9 9 – √ √ REQUIRED:
Then, they each came up with a Solution. Do you see one?
They each had the same Solution, the following one:
((1-2)√2) √ (9-3).
PLAYER 2′S ACTION: Declares Force-Out GOAL: 36 RESOURCES: - FORBIDDEN: + X ^ / / PERMITTED: 1 2 2 3 4 9 9 – √ √ REQUIRED:
Then, they each came up with a Solution. Do you see one?
They each had the same Solution, the following one:
((1-2)√2) √ (9-3).
The concept in this Solution that was not
involved in any of the Solutions snuffed
previously is that of negative root operations.
((1-2)√2) √ (9-3) = ((-1)√2) √ 6 = (2^(1/(-1))) √ 6 = (2^(-1)) √ 6 = (1/(2^1)) √ 6 = (½) √ 6 = 6^(2/1) = 6^2 = 36.
Perhaps most interesting of all about this performance by this sixth-grader and pair of seventh-graders was the fact that they were never ever directly taught anything about negative root operations or fractional root operations by their teachers or any other adults.
They learned about these ideas by playing EQUATIONS with their classmates in classroom tournaments and by working through manual versions of kits like the ones presented here in
((1-2)√2) √ (9-3) = ((-1)√2) √ 6 = (2^(1/(-1))) √ 6 = (2^(-1)) √ 6 = (1/(2^1)) √ 6 = (½) √ 6 = 6^(2/1) = 6^2 = 36.
Perhaps most interesting of all about this performance by this sixth-grader and pair of seventh-graders was the fact that they were never ever directly taught anything about negative root operations or fractional root operations by their teachers or any other adults.
They learned about these ideas by playing EQUATIONS with their classmates in classroom tournaments and by working through manual versions of kits like the ones presented here in
the DIG Math program.
This sample match was included as part of a paper presented at a session of the 4th International Congress on Mathematical Education, Berkeley, California, USA, August 10-16, 1980, sponsored by the International Commission on Mathematical Instruction, the National Academy of Sciences, and the University of California.
The assembled group of more than sixty professional mathematics educators were challenged to match wits with the three young girls who had played this match by the statement of the following hypothesis at the beginning of the paper:
“The outrageous claim that I went to make to you before we start is that these motivated twelve-year old students (one of them a black youngster from what was one of the toughest inner-city schools in Detroit) each will have handled this problem better than most of you here today. Yes, you heard me
This sample match was included as part of a paper presented at a session of the 4th International Congress on Mathematical Education, Berkeley, California, USA, August 10-16, 1980, sponsored by the International Commission on Mathematical Instruction, the National Academy of Sciences, and the University of California.
The assembled group of more than sixty professional mathematics educators were challenged to match wits with the three young girls who had played this match by the statement of the following hypothesis at the beginning of the paper:
“The outrageous claim that I went to make to you before we start is that these motivated twelve-year old students (one of them a black youngster from what was one of the toughest inner-city schools in Detroit) each will have handled this problem better than most of you here today. Yes, you heard me
correctly. These motivated sixth- and seventh-
graders will do better on this math task than
most experts in mathematics education.”
A bold hypothesis, indeed! But one would not venture such a claim to such a group if there had not been a great deal of prior experience that suggested that it might be so. And what were the results of this little experiment? One, and only one, of the assembled professionals in mathematics education did as well as the three young girls in applying elementary mathematics concepts to the problems encountered in this match. Only one came up with a Solution after the last move had been made. Think about that for a minute or two.
The extraordinary result was that in comparing what the more than sixty assembled world experts in mathematics education had learned during their entire professional careers with what three young girls learned during a single school term about the relationships among some elementary arithmetic and algebraic operations, just a single
A bold hypothesis, indeed! But one would not venture such a claim to such a group if there had not been a great deal of prior experience that suggested that it might be so. And what were the results of this little experiment? One, and only one, of the assembled professionals in mathematics education did as well as the three young girls in applying elementary mathematics concepts to the problems encountered in this match. Only one came up with a Solution after the last move had been made. Think about that for a minute or two.
The extraordinary result was that in comparing what the more than sixty assembled world experts in mathematics education had learned during their entire professional careers with what three young girls learned during a single school term about the relationships among some elementary arithmetic and algebraic operations, just a single
one of the experts did as well as each of the three
young girls did in applying the concepts to a
problem that was generated in the course of
playing the EQUATIONS game.
The experience of the author of the paper had led him to write the following in the concluding passages of the paper (written before the paper was presented and before the experiment was conducted during that presentation):
“What does the experience of this experiment say about using EQUATIONS to learn mathematics? What does this say about the bold claim that, motivated by playing EQUATIONS, sixth- and seventh-graders would learn the mathematics of elementary operations better in some sense than most contemporary experts in mathematics have learned those operations? If this sample of such educators is representative, it would seem to say that the bold claim was grossly understated. It should have been, instead, that motivated by playing EQUATIONS some sixth- and seventh
The experience of the author of the paper had led him to write the following in the concluding passages of the paper (written before the paper was presented and before the experiment was conducted during that presentation):
“What does the experience of this experiment say about using EQUATIONS to learn mathematics? What does this say about the bold claim that, motivated by playing EQUATIONS, sixth- and seventh-graders would learn the mathematics of elementary operations better in some sense than most contemporary experts in mathematics have learned those operations? If this sample of such educators is representative, it would seem to say that the bold claim was grossly understated. It should have been, instead, that motivated by playing EQUATIONS some sixth- and seventh
graders would be inspired to learn the mathematics
of elementary operations better in some sense than
virtually all contemporary experts in mathematics
education!”
To the extent that this result mirrors the full reality about the effectiveness of EQUATIONS for mathematics education, it surely should motivate somebody to try to do something about getting such instructional tools used more widely.
To the extent that this result mirrors the full reality about the effectiveness of EQUATIONS for mathematics education, it surely should motivate somebody to try to do something about getting such instructional tools used more widely.
- Table Of Content