Ye olde math bookAuthor(s): Diane MasonSource: The Mathematics Teacher, Vol. 87, No. 3 (March 1994), pp. 216, 218Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27968805 .

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not know what that appraisal is. I would want my daughter to be so open and not be afraid to acknowledge her thoughts and beliefs on any subject, regardless of its sensitive nature.

The statement "Math class is tough* of itself tells us little of Barbie's assessment of mathe matics class. It is but a meaning less array of isolated symbols devoid of a culturally embedded context. To interpret Barbie's words (symbols) one must add the context. Here one could limit the context to one's own personal experience or open it up to broad er possibilities. The controversy appears to be centered on the word tough. I doubt if it would have caused a stir if Barbie had said, "Math class is tun," though the premise presented here would change little. The same symbols, presented as letters or numerals, can be given different meanings depending on their interpretation. Miriam Webster's Collegiate Dictionary, 2d ed. (Springfield, Mass.: Miriam

Webster) defines tough as "very difficult,""toilsome," orfslang] "fine," "excellent."

Even keeping to the first defin ition listed, Barbie's words "Math class is tough," are less bother some than "Mathematics is tough" or "I'm not good at mathe matics" or "I'm a girl, I'm not supposed to be good at mathe matics." Her statement, as is, could have myriad meanings. Consider the following:

The work in math class is too abstract.

It is not presented in a way understandable to me. I've never been challenged in math before now. Th? problems posed in math class really make me think.

Thinking is hard work. Math class is so intellectually engrossing that it exhausts me

physically.

Conversely, if Barbie had been programmed to say, "Math class is fun," or if her original state

ment had been interpreted as the slang version, "Math class is fine or excellent,* we continue to be faced with a decontextual sym bolic array needing interpreta tion. The new statement ("Math class is tun") would satisfy some. On deeper reflection, however, multiple meanings emerge:

Math class doesn't challenge me.

It's "fun" to be called on (or not) in class.

Intellectual stimulation is rewarding.

Manipulatives are fun to play with.

Understanding math concepts, through the use of manipula tives (or not), is mentally satis fying. Passing notes in math class is fun.

Working with my friends on a common problem that chal lenges us to collaborate toward a common solution produces a "Eureka'' experience.

Perhaps the deeper message in the whole affair is that it has forced us to think about underly ing issues, assumptions, and interpretations in our society, historically and today. The pro duction of such a talking doll has made gender concerns, as well as multiple interpretations, matters for public discussion, and for this we should thank Mattel.

Elizabeth Senger Louisiana State University Baton Rouge, LA 70803

Active learning Our class recently investigated the effect the size of the square cut from each corner of a 9-inch by-12-inch piece of paper would have on the volume of the open box formed by folding up the resulting tabs to act as sides. Working in groups of four, stu dents made the following conjec tures:

Group 1: The volume will be constant.

Group 2: The volume will decrease as the square gets larger.

Group 3: The volume will first increase, then decrease.

Each student was then given a different measurement, from one half inch to four inches, to use as

the size of the cutout. The stu dents measured, marked, cut, folded, and taped to form their boxes from the colored thin card board.

Using the four boxes they had made as visual models, the stu dents in each group were asked to reassess their conjecture. Each group then explained the changes they made in their conjecture and the reason for those changes. Using approximately uniform pieces of Styrofoam packing mate rial, the students filled their boxes to capacity and counted the con tents. The students entered these quantities as 'Volume'' next to their square-cutout size in a chart on the chalkboard. Once again each group revisited its conjecture and reported to the class.

The teacher then led a discus sion concerning other methods of detennining each box's volume. Students suggested that each box could be measured for length, width, and height and the results multiplied. These theoretical vol umes were computed, added to the chart on the chalkboard, and compared with the original "vol umes." After a discussion that concluded that length, width, and height did not have to be mea sured but could have been com puted from the known square size cut from the original 9-inch by-12-inch sheet, the groups then worked on the task of deteraiin ing the volume for a square cutout of size inches. The result of this discussion, V = x{12 - 2 )? (9 - 2x\ was then graphed using graphing calculators and comput er software. Groups then made a list of conclusions suggested by this graph. All groups shared their conclusions with the class.

The next day students wrote in their journals regarding this experiment. Some entries were these:

Stefan: "Do not assume things without data to back up a

hypothesis. My group came to a consensus and felt volume would not change. This conclusion was based only on the starting sur face area [being] the same. Once I saw the sizes of the various boxes I realized volume was

dependent on the size square that was cut out." Amanda: UI learned that I

should always be willing to look at a problem from more than one

perspective. Perhaps ifi would have been more willing to under stand my teammates' reasoning, I could have predicted more accu rately the outcome of our own experiment. Secondly, I learned the difference between indepen dent and dependent variables and their importance."

Song: I learned that when cut ting size changefs], [the] volume of [the] box also change[s], from decrease to increase to decrease."

Joi: "From the mathematical project we did in class I learned a few things. Our group's hypothe sis that all the boxes would have the same volume was proven wrong. All the boxes differed in volume. Also, I learned that from the data we collected we could come up with an equation that could be used to show volume and that could be graphed on our calculators."

Alan P. Hallee Nashua Senior High School Nashua, NH 03062

Ye olde math book I couldn't help but notice an

inconsistency in the September 1993 issue. Specifically, the advertisement on page 521 implies that old mathematics books are boring and only new textbooks can create excitement in the classroom with their appli cations. As one of a growing num ber of teachers who incorporate the history of mathematics into our classrooms, I didn't care for the advertisement the first time I saw it, and I care for it less now. Consider two articles that appeared in the same issue, "Humanize Your Classroom with the History of Mathematics" and "Back to the Present: Rumina tions on an Old Arithmetic Text." In the first article, James Bidwell clarifies that the old is not neces

sarily boring nor obsolete (p. 463). Frank Swetz points out in the other article the value that an old mathematics textbook

(Continued on page 218)

216 THE MATHEMATICS TEACHER

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(Continued from page 216)

might have in discussing how the subject of mathematics inter relates with the subject of sociol ogy, political science, economics, and ecology (p. 494). In addition both articles make apparent how application-oriented these old textbooks are. In a day when students continually question, "When am I ever going to have to use this?" wouldn't it be great if we all had such a resource as 'Te Olde Math Book" available to answer their question?

Diane Mason Northeast Missouri State

University Kirksville, MO 63501

Swetz responds: Or better yet, 'Te Newe Math Book"! My expe rience has shown that marketing books has very little relevance to how good they are or how useful they are. "Cosmetically appeal ing" has little to do with peda gogical quality.

What's this button for? I recently had an experience in

my mathematics class that reminded me of how important it is truly to listen to the questions

my students ask, to consider the merits of their questions, and continually to watch for "teach able moments." One of my stu dents asked me a question during a lesson I was teaching on frac tions: "What's this button for?" She was referring to the factorial button, [xT], on one of our TI SICO calculators. The student stated that she was curious because she had pushed the but ton several times and gotten a lot of "weird" numbers. My first inclination was to "tell" the stu dent what the button was for and how it generated the "weird" numbers she was seeing. Howev

er, instead I asked if anyone in the class could explain this "but ton" to the perplexed student. The ensuing silence was broken

only by the sounds of students

pushing the factorial button on their calculators to see what

would happen. No one had a clue what that button did or how it did it. Impressed by the level of interest piqued by this innocent question, I decided to drop the lesson on fractions and challenge the class to experiment with this button and to give me a written

explanation of how it worked. I encouraged them to be organized in their research, take good notes, and work together if they desired.

The results of this exploration were very encouraging. Students who didn't normally get excited about mathematics were excited about this pursuit. They got a chance to use their problem solving skills to unlock the secret of this important key on their cal culator; they worked together in groups; they developed, rejected, and modified strategies as they worked together; and they com municated their ideas and results in written form. All the students

made progress on the problem, and many gave very good solu tions. We brought closure to this experience the next day by talk ing over what had happened and discussing the merits of the vari ous strategies used and some of the pitfalls that led students away from viable solutions.

I found that I really enjoyed this experience and shudder when I think of how close I came to missing out on it by simply answering my student's question. The students also would have missed out on a great opportuni ty to flex their mathematical muscles and develop their own mathematical power. I would encourage all teachers to keep their ears and eyes open for these types of "teachable moments." They will rejuvenate both the student and the teacher and help to make the mathematics class room a place where mathematics is experienced and discovered rather than endured.

Eric L. Preibisius La Jolla High School La Jolla, CA 92037

Kudos for connections Hasn't every mathematics stu dent asked at one time or other, "Why do we have to learn this stuff?" NCTM's curriculum stan dards (1989) talk about making connections and using mathe matics in daily life, but where will these connections come

from? Sue Barnes supplies an answer (September 1993, 442): "Involve the Community." What an exciting discovery this must have been for the students to see how such diverse occupations use mathematics! I congratulate Barnes on this imaginative pro ject, and I hope that mathemat ics teachers everywhere will reread this article and imple ment its ideas.

Jerome E. Tuttle Mercantile & General

Reinsurance Company of America

Morristown, NJ 07962

Students aren't the problem Debbie Quaal (September 1993, 441) is one of the luckier casual ties of poor mathematics teach ing. She discovered what was not done for her early enough to avoid failure. I make a good liv ing tutoring our system's fail ures, pulling children back from the edge of academic disaster. Overwhelmingly they are bright kids badly taught. I hope that no one writes a response that tries to blame Quaal for the short change she got. Students aren't the problem, and blaming them can't be part of the solution.

James Vamador e Box 16472 San Diego, CA 92176

All solutions I have been retired from teaching for five years. A parent recently brought me a problem that her daughter had been assigned in a mathematics class for the gifted. Given the following:

ABC + DEF XYZ

in which the letters represent the numerals 1-9 with no repeats, find a solution.

Now that I have too much free time on my hands, I decided to find not only one solution but all the possible solutions. If the reader feels compelled to do the same, please work out the answers to the following: 1. How many solutions are possi

ble? 2. What are the smallest and

largest values of ABC and XYZ?

3. Other than the fact that the

XYZs always contain at least one even numeral, what do

they all have in common?

The author would like to hear from anyone who has an inter esting method of attack. For a copy of all the solutions, send the author a self-addressed stamped envelope.

Gordon Speer 3304 Woodlawn Rond Sterling, IL 61081-4144

Birthday problem At her thirty-first-birthday party, my sister-in-law remarked to me that her son would be thirteen years old next month. She won dered if this reversal of digits in their ages would occur again. I immediately used a birthday napkin and scratched out an interesting generalization that I have since used as a discovery exercise in my algebra classes.

Conclusion. (1) This phenome non occurs when the difference in the ages of the parent and child is a multiple of 9. (2) The pattern repeats itself every eleven years.

Justification, l? is the tens digit and y is the units digit in the age of the parent, then

(10x+y)-(l0y +x) = d, where d is the difference in age. Since

9(x-v) = d, d must be divisible by 9. Suppose that d - 18, then (x-y) = 2 and the condition is satisfied for ages of (20,2), (31,13), (42,24), (53,35), (64, 46), (75, 57), (86, 68), and (97, 79). Replacing by + 1 and y by y + 1 to get the next pair insures that the phenomenon occurs in intervals of eleven years.

James W. Roche LaSalle High School Wyndmoor, PA 19118

A mean geometry problem Recently we encountered an inter esting problem that uses the idea of a mean proportional applied to geometric shapes. The following problem appears in Mary Dolciani et al.'s 1970 high school textbook,

Modern Introductory Analysis (Boston: Houghton Mifflin Co., p. 87, prob. 33):

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218 THE MATHEMATICS TEACHER

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