Gifted Students || MATHEMATICAL RESEARCH IN THE HONORS CLASSROOM

MATHEMATICAL RESEARCH IN THE HONORS CLASSROOMAuthor(s): HENRY BORENSONSource: The Mathematics Teacher, Vol. 76, No. 4, Gifted Students (April 1983), pp. 238-244Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27963458 .

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MATHEMATICAL RESEARCH IN THE HONORS CLASSROOM

By HENRY BORENSON Council Rock School District

Bucks County, PA 18954

In the typical classroom, the learning potential of

the student is decided before he/she steps into the

classroom. The teacher knows exactly what will be

covered and time is not allowed for original consider

ations.

?Paul Feldman, M82H Student

The M82H course at Stuyvesant High School is a college-level course in abstract

algebra and number theory. Although many of the students in this course are seniors who are concurrently taking ad vanced placement calculus, some juniors, sophomores, and an occasional freshman are also in the course. These students have exhibited an exceptional interest and abil

ity in mathematics and have taken the nec

essary prerequisite courses. Whereas some of the students in the

M82H class had engaged in various re

search projects and had entered mathemat ics and science fairs, many high-caliber stu dents had not pursued any mathematical

investigations on their own. To familiarize these students with the mathematical re search process, I presented the class with the opportunity to engage cooperatively in research in the classroom. This "research unit" took place early in the semester and lasted about two to three weeks. This arti cle describes the unit in the hope that other teachers might be encouraged to pursue a similar activity in their own classes with

gifted and nongifted students.

Objectives of the research unit

Through discussion with the students on

A special thanks to student Gregg Patruno for his

very precise and simplified notes on all the research

work reported herein.

Whereas a research unit in an honors class is appropriate, regular classes can benefit from some of the same strategies.

the benefits derived from the research unit, the following list of objectives was com

piled:

Allow students to experience the slow and deliberate process of developing and

formulating mathematics

Enhance students' awareness of the frus trations inherent in doing research

Enhance students' awareness of the

necessity for a clear, concise, and accu rate statement of propositions, conjec tures, and results

Promote independent thought and crea

tivity Encourage cooperation and the sharing of ideas

Afford students the opportunity to devel

op their own mathematical results and to see these results named after them

Inspire students to pursue research on their own

Selection of a problem

The question used to launch the research

process should ideally be one to which nei ther the teacher nor any of the students has an immediate answer. A good source for such questions is the "Elementary and Ad vanced Problems" section in each issue of the American Mathematical Monthly. Other sources include the Annual High School

Mathematics Examination and the Math ematical Olympiad problems that appear in the Mathematics Teacher. The teacher should have two or three such problems on hand in the event that the class makes very little progress in the solution of a problem and an alternative problem is required.

Classroom procedures

Although the initial question is posed by the teacher, the students make the major

238 Mathematics Teacher



mathematical contributions. Working indi

vidually or in groups, at home or in school, the students attempt to tackle the question in any way that they can. When a student has a conjecture about the question under

consideration, it is presented on the black board. Then, with the help of the class, the student tries to write the hypothesis.

The class next tries to search for excep tions to the conjecture. If some are found, the proposition is modified accordingly. If none is found, an attempt is made at its

proof. Often, a final proof is developed as a result of the contributions of a number of students over a period of several days. The students themselves are the final arbiters as to when a proof is acceptable. This re search process continues as students gener ate fresh conjectures requiring further in

vestigation.

A proof is attempted if no exceptions to the proposition can be found.

Role of the teacher

The teacher plays two principle roles in this process?namely, that of risk-taker and of facilitator. By venturing with the students into an unknown mathematical area, thereby giving up the role of dis

penser of knowledge, the teacher assumes the risk-taker role. The teacher has no

guarantee that any mathematical results will emanate from the research process. The students could be disillusioned by the entire experience, but more likely, what

they will learn about the research process will be rewarding in and of itself.

The teacher assumes a facilitator role in

seeking clarification and simplification of students' observations and in encouraging their formulation of propositions and

proofs. The teacher also facilitates by managing classroom behavior: grouping students so that they can share ideas, ac

cepting all students' comments for con

sideration, encouraging students to go to

the board to present their ideas, and pro moting a climate in which ridicule does not exist.

Application to the average classroom

Many of the techniques mentioned in the

preceding sections can be used to advan

tage in the average classroom in the pro cess known as guided discovery. After all, every time a class does not know some

thing, the teacher has an opportunity to

help the class investigate the situation and arrive at the correct "discovery." When a student makes an incorrect as

sertion, rather than correcting him or her, the teacher might challenge the class. "John proposes that

(a + b)n = an + bn.

How can we investigate whether or not this is true?" Usually a student will respond

with, "Try it out and see." Another student will suggest the substitution of numbers into the statement. The teacher can then elicit particular values of a, fc, and for consideration. One set of values can be as

signed to the first two rows, another set to the next two rows, and so on. The con clusion that, in general,

(a + b)n a + bn,n> 1,

should be the class's and not the teacher's. If the students have not already done so on their own, the teacher can raise the

question of what conditions, if any, im

posed on a, ?, or will make

(a + b)n = an + bn.

In this manner, the students will be partici pants in the process of developing math ematics.

A research unit

A three-week research unit began with the presentation of the definition of a ring and the consideration of various examples. The students' attention was then directed to example 6 in Introduction to Modern AU

gebra (McCoy 1964). There, a set is pre sented with addition and multiplication de fined by the following tables:

April 1983 239



abed

abed

bade

c d a b deba

abed

a a a a

abed

a a a a

abed

The students were then asked to verify McCoy's claim that under these operations

is a noncommutative ring. The students

easily verified that is closed under + and *, that + is commutative, that a is an

additive identity, that every element has an

additive inverse, and that multiplication is not commutative.

The teacher assumes the risk-taker role.

Three properties remained to be verified to demonstrate that is a ring, namely, associativity of addition, associativity of

multiplication, and the distributive law.

Certainly associativity of either operation could be determined through 43, or 64, direct verifications of all possible triplets of K. This process, however, would certainly be tedious. The students were then in formed that the previous year the M82H class had considered the question: "On the addition table for K, what is the minimum number of triplets that need to be verified for associativity to conclude that + is as

sociative on KT Their results, reducing the initial sixty-four cases to eight cases, were

then made available to these students to be read overnight. (See Borenson 1978.)

The following day, a student observed that the table for contained duplicates in the same row, whereas the addition table did not. This student then proposed the fol

lowing conjecture: "In the addition table for a ring, no element may appear more

or

than once in any row or column." The stu dent then offered the following proof :

Proof. If b + bx = and b + b2 = x, then b + bx

= b + b2. So

(-fc) + (6 + fc1) = (-fc) + (fc + fc2)

(-b + b) + bx = {~b + b) + b2 0 + bx

= 0 + b2

b, =

b2.

At that point, another student, Joel

Hirschhorn, thought he had a characteriza tion of the addition table given for and that it was this characterization that made

associative. Hence, Joel, with the support of Gregg Patruno, proposed what came to be referred as

Hirschhorn's Theorem. "Let be a set with 4 elements on which is defined a com mutative binary operation. Let each element of be its own inverse. Let the

operation be such that no row of its table contains the same element twice. Then the

operation is associative on K.

The proof of this theorem was derived

through the contributions of various mem bers of the class. The students reasoned that the triplets of containing the ident

ity element a are all trivial cases. All the other triplets are of one of the following types: (l)(p, q, r), (2Xp, q, q\ (3)(p, p, q\

where p, q, and r are distinct, nonzero ele

ments; and (4)(p, q, p), where and q are not necessarily distinct.

Case 1. + (q + r) = (p + q) + r,

+ p = r + r, a = a.

(Note: Why must q + r = and + q = r?)

Case 2. + {q + q) = (p + q) + q, + a = r + q,

y p

= p.

Case 3. + ( + q) = ( + ) - <?,

+ r = a 4- q, y

q =

q.

Case 4. + (q + ) = + ( + q) = (p + q) + p.




(Remember, the operation is assumed to be commutative on K.)

The class had, in effect, generalized the

original problem and then presented a

proof to cover the more general case.

Consequently, any binary operation satis

fying Hirschhorn's conditions on a set of four elements must be associative !

The students were next presented with the following problem proposed by F.

David Hammer in the November 1978 issue of the American Mathematical Month

ly. Advanced Problem #6238 (p. 770) states:

To see if a binary operation on a set with elements is associative, one might think it necessary to verify directly n3 instances of the associative law. Often, however, for instance if the operation is commutative and has an identity, considerably fewer cases need be verified. Is there a set of elements and an operation on them for which all n3 verifications are necessary?

By the following Monday morning, Chris Slawinsky had answered this

question in the negative.

Slawinsky's Theorem. For every set on

which is defined a binary operation, there is at least one triplet whose associativity

follows from that of the others.

Proof For an arbitrary e , either (1) + = or (2) + = y, y . If (1),

then ( , , ) is associative, since

4- ( + ) = ( + ) + and

+ = + .

If condition (2) holds, we can show that the associativity of A(x, x, x), B(x, x, y), and C(y, x, x) implies the associativity of

(x, y9 x).

Lemma, + y = y + .

Proof Since + ( + ) = ( 4- ) +

(by A), we get + y = y + x (definition of

y). Now,

(x + y) + = (y + x) -h x by l?mma = y + (x -f x) by C(y, , )

associativity = ( + ) + y by definition of y = + ( + y) by B(x, x, y)

associativity = + (y + x) by lemma.

Hence, Chris demonstrated that at most 3 ? 1 triplets need be verified for associ

ativity to determine that a binary operation on a set of elements is associative. But could this number be reduced still further? Leonid Fridman, another student, studied this question for a few days and proposed the following theorem:

Fridman's Theorem. Given any set with elements and with a binary operation on

it, at most 3 ? triplets need be verified

to determine that the operation is associ ative.

For the proof of this theorem, see the M82H solution to Advanced Problem

#6238 in the May 1980 issue of the Ameri can Mathematical Monthly.

All student responses are considered.

Further considerations of Hirschhorn's theorem

Next, the class chose to further pursue Hirschhorn's theorem. Paul Feldman indi cated that he had come up with the follow

ing theorem in his attempt to generalize Hirschhorn's theorem:

Feldman's Theorem. No associative binary operation on sets of 5, 6, or 7 elements

satisfy the conditions of Hirschhorn's the orem.

Proof Assume associativity. Let b, c, and d be distinct, n?nidentity elements such that d = be (possible since be b, be c, and be a). Then

bd = b(bc) = (bb)c = ac = c,

or

bd = c.

Also,

cd = (bd)d = b(dd) = ba = b, or

cd = b.

Hence, if we consider a set = {a, fe, c, d,

April 1983 241



e} of five elements with an associative oper ation satisfying Hirschhorn's condition, we

get the following entries :

abode

bade

c d a b deba

e a

Since no element may appear in the same row more than once, it becomes clear that be = f where f $ K. Likewise, ce = g and de = , where g and h are distinct and dif ferent from any of the other elements.

Hence, if an associative binary operation satisfies Hirschhorn's conditions on a set

containing more than four elements, that set must contain at least eight elements !

Joel Wein demonstrated that Hirsch horn's conditions could not hold on any set

containing an odd number of elements.

The teacher helps the class investigate.

Wein's Theorem. For any set containing an odd number of elements, > 1, no oper

ation exists satisfying the conditions of Hirschhorn's theorem.

Proof Suppose that such a binary oper ation exists on a set =

{au a2, ..., a?}, where is odd. Furthermore, assume for the moment that each element ak of the set

appears exactly times in the table. Let af be an element other than the identity ele ment. Then at appears above the main di

a??nal, say, k times. By commutativity, it also appears k times below the main diag onal. Since each element is its own inverse, a{ does not appear ori the main diagonal itself. Hence, ax appears a total of 2k times in the table. This means that 2k = n, which contradicts the assumption that is odd.

Joel Wein then proceeded to show that each element ak e does indeed appear exactly times in the table. Suppose an

element ak e appears more than times in the table. Since there are only rows in the table, it follows by the Pigeon-Hole principle that at least one row must con tain an element more than once. Now, if ak appears fewer than times in the table, there would be fewer than n2 entries in the table. Consequently, each element ak e

appears exactly times in the table, and Wein's theorem is proved.

Extending the conditions of Hirschhorn s

theorem to a set containing 2n elements

Lisa Randall conjectured that for any set

containing 2n elements, e I+9 there exists an associative binary operation satisfying th? conditions of Hirschhorn's theorem. Lisa noted that such an operation defined oil a set of two elements could be extended to a set of four elements in a systematic

manner. Thus, if each A and entry of the table

A

A A

were replaced, respectively, by A = . \b a

and = A, with a? c and b-> d, the re

sulting table 1 would irideed define the desired binary operation on a set of four elements. (This operation satisfies the con ditions of Hirschhorn's theorem. The fact that it is associative follows from Hirsch horn's theorem itself.) Furthermore, it ap peared as if this "cloning" process could be extended to a set of eight elements by re

placing each A and entry in the preced ing 2x2 table, respectively, by

A =

TABLE l

abed

a b

b a

c d d c

c d d c a b

b a




and = A, with a?> e, fe?>/, c-^g, and d?>h. See table 2. That the operation so

defined is associative on the set of eight elements was noted by Paul Feldman. He demonstrated that this operation was

equivalent to multiplication modulo 24 on

the set of elements relatively prime to 24,

according to the mapping: a? 1, b? 17, c-> 5, d-> 11, e-+ 23J-+ 7, g-* 19,

? 13.

TABLE 2

Even though the class seemed to think that Lisa's extension process was a viable

one, they sensed the need for a greater for malization of the idea. Brian Greene ac

complished this task.

Randall's Theorem. On a set containing 2n

elements, e I+, there exists an associ ative binary operation satisfying the con ditions of Hirschhorn 's theorem.

Green's proof. If = 1, the operation de fined by 1 a b

a b

b a

is associative and satisfies Hirschhorn's conditions. Now, assume Hirschhorn's con ditions hold for an associative binary oper ation defined on a set containing 2k ele

ments, {al5 a2,a2}, k e 7+. Now con struct a set S =

{au a2,..., a2 ; a'l9 a'2,...,

a2) containing 2*+1 elements. Extend the

binary operation to the larger set as fol lows:

x' * y'

= * y

x' * y = ( * y)' * y' = ( * y)'

a

a b

b a

a

a b a' b'

b a b' a'

a' b' a b

b' a' b a

Illustration of Greene's extension process for k = l

The binary operation so extended meets

Hirschhorn's conditions on the new set S. If it can be shown that this operation is as

sociative on S, then the inductive step will have been made. Call the elements of the

original set normals and the additional ele ments primes. All triplets of 5 are then of one of the following types, where JV = nor

mal and = prime: (1)(P, , P), (2)(P, JV, ), (3)(P, P, J\f), (4XP, JV, JV), (5)(JV, JV, JV),

(6)(JV, P, P), (7)(JV, P, JV), (8XJV, JV, P). We now prove that any triplet of each type is associative. Consider the following:

(1) ( , , ): (x' + y') + z' l x' + (/ + z') (x + y) + z' = x' + Cv + *)

((x + y) + z)' = (x + (y + z))'

(2) (P, JV, ): (x' + y) + z' = x' + (y + z') (x + y)' + z' = x' + (y + z)'

(x + y) + = + (y + )

(3) ( , , JV): ( ' + y') + = ' + (/ + )

( + y) + = ' + (y + )'

( + y) 4- = 4- (y 4- )

(4-8) can be proved in a similar fashion. By the principle of induction, Randall's theo rem holds for all e I + . Q.E.D.

When to end the research unit

Although the students had already ac

complished a great deal during this three week period, I posed one additional

question:

Does there exist an operation on a set of elements such that no triplet is asso

ciative? If so, is there one for which all n3 verifications are necessary to con

clude the nonexistence of any associative

triplets?

April 1983 243



Zachary Franco submitted a solution to the first part of the question.

Franco's Theorem. For all ^ 2, there exists a binary operation on a set with elements such that no triplet is associative.

Proof. Define our binary operation on

S = {au a2,..., an} by the following rule:

at * Qj =

ai+l9aH+1 =

av

Now assume that

?, + (a, + a*) = (a? + <ij) + <*k, ?i + ?j+l =??+1 +?k>

and

ai+i = a +2? Contradiction.

Hence, under this rule, no triplet is associ ative.

The second part of this question was not

pursued by this M 82H class. The students had been involved in an intensive research

experience for three weeks. By this time, everyone, including the teacher, was exhausted as well as exhilarated by this

very demanding, creative work. It was now time to move on to the regular course

work. The class, therefore, proceeded with the basic theorems concerning rings.

REFERENCES

Borenson, Henry. "Promoting Discovery in Algebra." Mathematics Teacher 71 (December 1978):751-52.

-. "Promoting Mathematical Creativity in the Classroom." Educational Forum (May 1081):471 76.

Borenson, Henry, and M82H Class. "Verifying Associativity, Solution to Advanced Problem

#6238." American Mathematical Monthly 87

(May 1980):409-10. McCoy, Neal H. Introduction to Modern Algebra.

Boston: Al?yn & Bacon, 1964. Torrance, E. Paul. Education and the Creative Poten

tial. Minneapolis: University of Minnesota Press, 1963.

OF NEW HAMPSHIRE ^/\^

offers a stimulating, Informal atmosphere for learning while earning a

RriS-T. degree In MATHEMATICS

The Master of Science for Teachers degree is designed especially for teachers of s^ondary school mathematics. The Program is carefully

geometry, aigebra arti analysis.

1963Sutoroer ^ from June 27 to August 19.

The University is located in a smalt New England town with easy access to the beaches of New Hampshire, the White Mountains and

Boston,

Write: Richard H? B?lomenos, Chairman, Department of Mathematics,

Kingsbury Hall, Durham, N.H. 03824.

Mathematics Teacher



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Gifted Students || MATHEMATICAL RESEARCH IN THE HONORS CLASSROOM