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by groups of two to four students. Each member of the group hasa specific assignment. Each member may have a different job suchas recording and keeping records, collecting data, making measure-ments, or operating instruments.There are no limits to the possibilities for mathemetics outside

your classroom. If you run out of ideas, ask your students to plantheir own lesson. You may be surprised by the ingenuity and complexityof the lessons they propose.

How Do You Read "~ x"?Charles Vanden Eynden

Illinois State UniversityNormal, Illinois 61761

"Minus jc" is what my teachers taught me. My university students,on the other hand, say "negative jc," to, I claim, their disadvantage.My main objection to "negative jc" is that while a black cow is

black and a hot stove is hot, negative jc may not be negative. Whenan adjective, such as "negative," is applied to an object, we havea right to expect that object to live up to its description. "Negativex" is false advertising.Of course the inconsistencies of the English language are too

well known for me to base my case on grammatical principles alone.It has been my repeated experience that "negative jc" causes realconfusion. The word "invariably" should be interpreted literally inthe account that follows.

I define absolute value to my class. I remark that |x| is nevernegative. Later we calculate that |x| = -x if x = -3. Invariablya student asks, "How can the absolute value of x be negative xwhen you said the absolute value is never negative?" Sometimesthe line "| x\ = - x if x < 0" in the definition provokes the objectionright away.My diatribes to colleagues on this subject have uncovered several

defenses of "negative" over "minus."

(1) If one reads "- x" as "minus x" then "minus" carries twodifferent meanings, since the same word is used to vocalize "a -b."

True enough. Dictionaries would be skinny books if every wordwere allowed only one meaning. "Negative" has the undisputedmeaning "less than zero"; saying "negative jc" gives it another. Thepoint is that in the latter case the multiple usage causes confusion.

How Do You Read "- x"? 135

(2) You say "negative x" should be negative. Doesn’t the same gofor "minus x"?No, because "minus" is not an adjective. One does not say (I

hope) that a number is "minus." What part of speech "minus" isI am not prepared to say; an "operator" might be a good name.I liken it to "ex," as in "ex-president." To call a person "ex" aloneis meaningless.

(3) Admittedly "negative x" is troublesome. But you cannot objectto "negative 3."

Yes I can. "Negative" is still an adjective. Saying "negative 3"leaves the impression that there are two 3’s, one positive and onenegative. If a dead horse is a horse that is dead, why isn’t negative3 a 3 that is negative? The objection doesn’t apply to the operator"minus." No one thinks the ex-president is commander in chief.Using argument (3) seemed especially poor tactics to me since I

assumed the surrender of "negative x" could only mean the victoryof "minus x," leaving "negative 3" indefensible. I was wrong. Indeedmy colleagues in mathematics education have provided me with UICSMmaterials explaining the concept of "oppositing." Therein a distinctionis made between -3 (negative 3), which is a negative integer, and-3 (opposite of 3), which is also a negative integer, the same onein fact, but is considered as formed by "oppositing" 3. (I mighthave written "positive 3," for emphasis, but it would have givenme a painful pain.) The way is clearly prepared for "the oppositeofx."Thus three separate ways of reading the symbol "-"are provided,

and "7 - - -8" may be read either "7 minus the opposite of negative8" or "7 minus the opposite of the opposite of 8." The symbol"+" in the expression "+ (3 - 5)" is said to denote "saming,"but "plus 3 minus 5" is the suggested vocalization! (Italics mine.)The explanation for the teacher claims "+" also has three meanings.Evidently there is 3 and there is positive 3 and there is plus 3.That the above distinctions are made to school children would

astonish many a working mathematician, pursuing a successful careerentirely unaware of them. They are not, after all, mathematicaldistinctions. For example, -3 is the same number no matter howit is read; the distinctions referred to are purely semantic.A similar hullabaloo is often made about the difference between

a number and a numeral. The difference is real enough. Just as realis the difference between Napoleon and "Napoleon." The first wasa French general, the latter his name. One hopes, however, that nottoo much time is spent in history classes explaining the difference.History is about people, not words, just as mathematics is not aboutnumerals. Grammar should be left to English classes.

136 School Science and Mathematics

I recognize that the introduction of saming and of oppositing, andthe three plus signs, and the three minus signs, all represent a sincereattempt to make elementary mathematics logical and precise. Thegoal is admirable, but the methods used to achieve it are, I feel,unnecessarily complicated, inconsistent (if "plus 3 minus 5" is allowedwhy not "minus 3 minus 5"?), contrary to English usage, and confusingto the student.

I would like to present a counterproposal of my own for explainingand naming the symbols "+" and "-." Let "- 3" ("minus 3")be defined as an abbreviation for "0 - 3," just as "America" isan abbreviation for "The United States of America." This ties inclosely with the way negative numbers are commonly introduced.In the same way "+3" ("plus 3") is defined to be an abbreviationfor "0 + 3." The same definitions handle - x and + x. Under thissystem the symbols "+" and "--" would carry only one meaningeach and would always be read "plus" and "minus"; likewise"positive" and "negative" would have a single application.The idea of oppositing was introduced to me for the first time

only a few weeks ago. I had asked my calculus class to work aproblem at their seats. A student claimed to have the answer andI prepared to write it on the blackboard. He gave (correctly) hisanswer as "the opposite of the quantity b minus a times the absolutevalue of c." Never having heard of the opposite of a real numberbefore, I didn’t know what to write. My point is that the only purposeto the names we give the symbols of mathematics is communication.My student had evidently learned his lessons well, but at that particulartime he was not communicating with me. Perhaps sometimes I don’tcommunicate to my students because my words mean different thingsto them than they do to me. What is needed is a nomenclature thatis simple, logical, consistent internally and with common usage,universal, and stable.

TRICKY FISH

You can’t teach an old dog new tricks, but you may be able to teacha young fish a few things. Scientists at the University of Wisconsin-Madisonhave successfully taught migratory fish�like salmon and trout�to homein on particular rivers and streams by imprinting them to synthetic odors.

It is known that the home stream odor is an important guide to fish headingback to their home streams to spawn. Also, it is believed that each streamhas characteristic odors that the fish learns�or is imprinted to�early inlife and remembers on into its adult life.

It should be possible, scientists state, to raise fish in a hatchery, andimprint them artificially to some chemical odor. Then, after releasing theminto the lake or ocean, they can be attracted back to central locations simplyby adding the imprinting chemical to the water.The results have been very encouraging. If they continue to be as good,

the work will have great implications for fisheries’ management.