The Element of Surprise: An Effective Classroom TechniqueAuthor(s): David R. JohnsonSource: The Mathematics Teacher, Vol. 100, 100 Years of Mathematics Teacher (2007), pp. 56-59Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27972375 .

Accessed: 24/04/2014 12:52

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 82.4.251.231 on Thu, 24 Apr 2014 12:52:47 PMAll use subject to JSTOR Terms and Conditions

Mathematics Teacher 66 (January 1973): 13-16

The Element of Surprise:

An Effective Classroom

Technique David R. Johnson Nicolet High School Milwaukee, Wisconsin

In his article, David Johnson uses both subject matter and pedagogy in developing an "element of surprise" and reminds us, in 1973, that

what we teach and how we teach are inextricably connected to the

quality of student learning. He also emphasizes that "telling" can not serve as the sole pedagogical technique for teaching mathematics since it often stifles the element of surprise and prevents students from thinking about and exploring mathematics for themselves.

Johnson's article provides many examples of topics that may be

surprising to many students?and teachers as well. Even more sur

prises await students today. The visual and exploratory features of

graphing calculators have expanded the topics that students might find surprising and the techniques used in facilitating this surprise. Let's take Johnson's advice and begin incorporating the element of

surprise.

To come upon suddenly and unexpect edly; to discover suddenly; to strike with a sudden feeling of wonder that arrests

the thoughts, as something unexpected or extraordinary. These are a few defi

nitions of the word surprise found in a standard

dictionary. Read these definitions again! Is the ele ment of surprise present in your classroom NOW? I believe the element of surprise belongs there! In fact, I feel that the element of surprise can be one of the

major teaching techniques used daily by mathemat ics educators. But wait a minute! What is "surpris ing" about arithmetic, algebra, or geometry? After

all, isn't mathematics dead, dull, and difficult? Let us focus on two specific areas in which the

element of surprise maybe effectively introduced into our classrooms.

SUBJECT MATTER Where is the element of surprise in our modern mathematics curriculum? Content surprises are

everywhere. But there is one problem. Most math ematics teachers are so familiar with the content

of the courses they are teaching that the real "sur

prises" are no longer surprises to them. Surprise is a motivation device that we as educators should not

overlook or underrate. A few examples in the junior high school and

high school mathematics program that can be used

56 MATHEMATICS TEACHER | Vol. 100 ? Special Issue ? 2007

This content downloaded from 82.4.251.231 on Thu, 24 Apr 2014 12:52:47 PMAll use subject to JSTOR Terms and Conditions

You hold the answers in your hand!

in conjunction with the technique of surprise are

listed below:

.9 = 1

2. The average rate of the total trip of a car going into the country at the rate of 45 miles per hour and returning at the rate of 55 miles per hour.

3. a2 = b2is not equivalent toa = b. 4. The algebra of matrices does not include prop

erties such as these:

a) The commutative property with respect to

multiplication b) ab = 0<H>a = 0vb = 0

c) There are at most two square roots for a

given element of the set 5. Pascal's triangle and its relation to the

following: a) Number of subsets of a set

b) Triangular numbers

c) Binomial expansion d) Pyramid [sic] numbers

e) Probability 6. The probability of at least two people in a room

of thirty people having the same birthday 7. The graph of the solution sets of such state

ments as these:

a) g = \x\ b) \g\ = \x\ c) g =

- 2

8. The practical applications of the Fibonacci

sequence

9. (-x) is not necessarily a negative number 10. "Union" distributes over "intersection" and

"intersection" distributes over "union."

11. The need for the completeness axiom in our

real number systems.

12. The sum of the infinite series

1 1 1 ! + _ + _ + _ + ...

2 4 8

13. The sum of the infinite series

1 1 1 1 + - + - + - + ???

2 3 4

14. The following statement is not true for all real numbers a, b, c: \ia<b then ac < be.

15. There exists an infinite set of real numbers that are smaller than their respective square roots.

16. \a + b\ < \a\ + \b\

This is just a limited sample list of the many surprises found in the content of the high school

mathematics program.

The items listed above all have potential to be real

surprises in the classroom! In other words, the specific

This tiny slide becomes a colorful 4X6 loot graph ot inequalities on your chalkboard. And you can write on it!

GET READY GET SET

GO METRIC METRIC CONVERSION TABLES Includes the most frequenily used conviions in an easy*ic-u?e format. Comprehensive with soparme tables for tenglh and speed, aria, volumi:, capacity, weight, temperature, and price. 53.95 METRICS MADE EASY is fully ?lustraMd concise guide lh.il coniai ni many useful conversion value; and chaits. With an explanation o! mettici in j ianguago that everyone cm understand. S 75

THE METRIC BOOK OF AMUSING THINGS \VTO DO, illustrated >n three ca?an, gives youngsters practical experience v.itn the metric

system. Wild games, c??les. and pro-eels lo learn about terms and quantities. S 1.95 METRIC CONVERTER is .in inexpensive alida device that wit! be of value to students, trnvclers. consum?is. Wilh equivalencies for temperature, weight, volume, length, are.!.

PUBLISHED BV SARRONS EDUCATIONAL SERIES, INC., WOOOBURV. NEW YORK

Clockwise from top left,

these advertisements

appeared in Mathematics

Teacher 63 (March 1970), 66 (January 1973), and 69

(November 1976).

content involved provides the opportunity for the teacher to use the technique of surprise. Too often, we, the classroom teachers, pass up this opportunity and

usually for the reason that we think we cannot afford the time. We employ the expedient teaching technique of lecturing and tell the students everything. In the

process we walk right over the many opportunities for

surprise. Lecturing, or as I call it, "telling," should not

be our major teaching technique. Dr. Robert F. Mager, in his book Developing Attitude toward Learning (Feron Publications, Palo Alto, California, 1968), expresses my belief well: "If telling were the same as teaching, we'd all be so smart we could hardly stand it."

Our teaching techniques should be built on more

than the qualities of expediency and efficiency (from the teachers' point of view). It is time that our teaching techniques be determined by their effectiveness in promoting real understanding, developing meaningful skills, and developing posi tive attitudes toward mathematics.

I believe the technique of surprise will do that! With your leadership, opportunities to make mini discoveries (reread the list of items above) should be provided in your classroom. In addition, the stu dents will receive a bonus: Discovered concepts will be more easily remembered and more meaningful to the learner in future applications.

THE CLASSROOM ROUTINE Developing the element of surprise in the classroom routine is not achieved by using one single aid or

technique but is based on many meaningful class room activities placed together in an appropriate arrangement. Consider the list below:

Vol. 100 ? Special Issue ? 2007 | MATHEMATICS TEACHER 57

This content downloaded from 82.4.251.231 on Thu, 24 Apr 2014 12:52:47 PMAll use subject to JSTOR Terms and Conditions

1970 An Editorial Panel, consisting of five people serving five-year terms, replaces the field editor as the leadership of MT.

1975 Thelma M. Sparks becomes the first female and first African American editorial panel member.

1977 The first September issue of the journal appears. Volume years now consist of nine issues per year, September-May.

Computer-assisted proof of the four-color theorem is published, although not in MT.

1978 Leroy C. Dalton becomes the first person named to serve as board liaison to the Editorial Panel.

1979 NCTM establishes the task force "Problems in the Mathematics Education of Girls and Young Women."

1. Vary the order of the basic projects to be included during the period from day to day.

2. Share some of the routine basic responsibili ties with appropriate stud ents.

3. Prepare activities that will make possible a

student-learning-centered rather than a teacher

dominated classroom. Learning by doing is still a most effective method of learning. It is our respon

sibility to encourage students to do a greater por tion of the "doing" in the classroom. We as teachers

must learn to become leaders and motivators rather than "dictators" and "the authorities."

4. Develop a technique of questioning that pro vides an opportunity for all members of the class to

respond at the same time to the same question. Try the following class activity: Request each of your students to use a sheet of scratch paper and instead of

responding to your questions by raising their hands and shouting out the answers, have them indicate their solution or answers on paper. This will make

possible a response and commitment from each stu dent. As they are writing their solutions to your ques tion, walk around the room; this provides an opportu nity for you to check the results and get an immediate feedback as to whether or not they understand the

concept under discussion. After a reasonable length of time given to write the responses, a student inter

change of ideas can follow. You will find that more

students will be involved in the discussion because

they have made a prior commitment that they are now

interested in verifying and evaluating. Examples appro priate in this type of activity are listed below:

a) Picture on a number line all real numbers that are smaller than their reciprocals.

b) The following statement is either true or false (if false, supply a counterexample): "The set of irrational numbers is not closed under the operation of multiplication."

c) Picture the set of real numbers for which the following statement is true: |x|

= -x.

d) Find a counterexample for the following statement: *4a + b = + Vfe.

e) Give two sets A and such that A c A e .

f) Write all the subsets of the set A if

A={1,0}. 5. Have activities that permit different-sized

groupings during the class period. For example, schedule an activity for groups of two or three stu

dents, followed by a total-class activity, followed by an individual activity.

6. Avoid long periods of time on any one type of

activity or class organization! This includes individual

learning activities. Working alone during the class

period can be just as boring, unexciting, and dull as

any other classroom grouping?and maybe more so. This is particularly true for the gifted or slow student, who unfortunately is confronted with this "uninspir ing" activity more than his fellow students. Long peri ods of working individually prohibit meaningful student-to-student and student-to-teacher interaction

and the opportunity for students to develop the tech

nique of sharing ideas. Individual study activity often

places great emphasis on reading; this cancels out

many opportunities for student discovery and inquiry learning. Discovery learning means more than locat

ing the answer in a textbook or on a worksheet. 7. Provide occasional opportunities for students

to move around the room during the period. 8. Incorporate meaningful contests occasionally

to check understanding and encourage teamwork. 9. Let students explain their own solutions to

problems and provide opportunities for them to share unique solutions. Students will surprise most instructors in their ability to communicate mean

ingfully with their classmates. 10. Do not announce to your students exactly

what activities will take place. That is, let even the next activity be a surprise to the members of the class.

11. Select or create meaningful assignments.

CONTRIBUTE TO THE 1MCTM BUILDING FUND

Mathematics Teacher 64 (May 1971)

58 MATHEMATICS TEACHER | Vol. 100 ? Special Issue ? 2007

This content downloaded from 82.4.251.231 on Thu, 24 Apr 2014 12:52:47 PMAll use subject to JSTOR Terms and Conditions

The problems found in most textbooks appeal to

few students. You should be the designer of the

curriculum for the course. Don't let your textbook

be your master! Decide the order of topics and the

major concepts to be presented. Following a math

ematics textbook page by page without any devia tion is in itself a boring activity for most students.

Supplementing the text or replacing the text with

your own effective material is a responsibility and

professional opportunity for the teacher. 12. Whenever possible, use well-designed

physical models to illustrate basic concepts, new

theorems, and definitions. A picture, as you well

remember, is worth a thousand words. A physical mathematical model is worth thousands of words.

Encourage students to help you build the appropri ate models in and out of class. This building activity will in itself clarify the concept under discussion.

13. Incorporate appropriate unanswered ques tions into each period to build up the element of

mystery and interest. Using open-ended questions that go unanswered until the following day is an

effective technique to introduce new concepts and to generate interest for the next day's class.

14. Provide the opportunity for meaningful guessing before solving the problem or presenting a

(the) solution. 15. Provide opportunities for students to find

counterexamples to statements made in class by fellow students or the instructor. Do not always state mathematical statements that are true. Intro

duce statements that are only true if the universe is limited. Let the students select the correct universe. In other words, it is appropriate for students to learn how to prove statements false as well as true. In deciding the truth value of statements, students undertake a meaningful classroom activity. The result is often a surprise.

16. Enthusiasm is contagious. Provide activities where students can let their enthusiasm show.

17. Vary the seating arrangement from class activ

ity to class activity. A room arrangement of six rows

with five desks per row contributes very little toward

creating an atmosphere for the interchange of student ideas. Try arranging the desks in concentric circles or semicircles. Place your instructor's desk in an

insignificant corner of the room. Let your students be the focal point of the room rather than your lectern.

Prepare the stage for student involvement.

Incorporating suggestions like those listed above will aid in bringing the element of surprise into

your classroom. Yes! The element of surprise will be present when the teacher is helping his students uncover mathematics rather than covering the sub

ject matter for his students. Consider the enthusi

asm, sheer excitement, and anticipation of a young

Minicalculators in Schools

NCTM Instructional Affairs Committee

h the decrease in cost ol

potrai contributo of ?.? ^M?^tl.?ta??v.

w.? ,0 retai."*

THE POCKET CALCULATOR AS A TEACHING AID

Encourage your students to use their minicalculators to discover patterns and verify math^n.^i -

POSITION STATEMENTS

USE OF MINICALCULATORS

The following position statement was approved by the Board of Directors in 1974.

With the decrease in the cost of the minicalculator, its accessibility to students at all levels is increasing rapidly. Mathematics teachers should recognize its potential contribution as a valuable instructional aid. In the classroom, the minicalculator should be used in imaginative ways to rein force learning and to motivate learners as they become proficient in mathe matics. In the January 1976 issue (pages 92-94), the Instructional Affairs Committee

offered a variety of problems to indicate how minicalculators could be used in the schools. This committee is now in the process of reviewing the status of minicalculators in the schools and will be making recommendations to the Board of Directors. The Board of Directors has adopted the following goal for the Council: "To

encourage the development and evaluation of curriculum and instructional materials incorporating the use of the hand-held calculator and other comput ing devices." A regularly updated bibliography, "Minicalculator Information Resources," is available free on request from the NCTM Headquarters.

COMPUTERS IN THE CLASSROOM

The fallowing statement was approved by the Board of Directors in Septem ber 1976.

Although computers have become an essential tool of our society, their diverse and sustained effects on all of us are frequently overlooked. The astounding computational power of the computer has altered priorities in the mathematics curriculum with respect to both content and instructional practices. Improvements in computer technology continue to make comput ers, minicomputers, and programmable calculators increasingly accessible to greater numbers of students at reasonable cost. An essential outcome of contemporary education is computer literacy.

Every student should have firsthand experiences with both the capabilities and the limitations of computers through contemporary applications. Al though the study of computers is intrinsically valuable, educators should also develop an awareness of the advantages of computers both in inter disciplinary problem solving and as an instructional aid. Educational deci

i makers, including classroom teachers, should seek to make computers readily available as an integral part of the educational program.

Headlines and official NCTM statement in (from top)

Mathematics Teacher 69 (January 1976), Mathematics

Teacher 69 (October 1976), and Mathematics Teacher 71

{May 1978)

lad who is opening a colorfully wrapped birthday gift from his grandfather. The element of surprise is indeed developing as the young man lifts the cover off the box. The excitement of anticipation and mystery is often as memorable and rewarding as the gift itself. Experiences like these are seldom

forgotten by the child involved. But compare this

activity to the assigning of this lad to the task of

wrapping up and mailing a box at the post office. It is a chore! There is no excitement, no anticipat ing, and seldom an opportunity for surprise. In a

sense, this characterizes the difference between the teacher who uncovers the subject as compared with the teacher who covers the textbook.

Isn't it time we all begin to consider seriously the art of uncovering our subject? Let us start by incorporating the element of surprise. <?

Vol. 100 ? Special Issue ? 2007 | MATHEMATICS TEACHER 59

This content downloaded from 82.4.251.231 on Thu, 24 Apr 2014 12:52:47 PMAll use subject to JSTOR Terms and Conditions