Auf The Volokh Conspiracy (einer von Jura-Professoren betriebenen Webseite) gibt es seit Dienstag eine Meinungsumfrage “Ist 0 gerade oder ungerade?”. In 2 Tagen haben sich mehr als 3000 Leser beteiligt.
Nach aktuellem Stand (1.10., 23:55) sind von 3138 Lesern
– immerhin 56 der Meinung, 0 sei ungerade,
– 123 meinen, 0 sei gerade und ungerade
– 1309 (!) meinen, es sei keines von beiden
– und 1650, also gut die Hälfte, sagen richtig, 0 wäre gerade.
Ich möchte das eigentlich nicht weiter kommentieren.
Bemerkenswert ist aber vielleicht, daß diese Frage in der Didaktik ausgiebig diskutiert wird. Jedenfalls gibt es einen wirklich sehr ausführlichen Wikipedia-Artikel zum Thema Evenness of Zero.
Das Inhaltsverzeichnis des Wikipedia-Artikels:
1 In education
1.2 Students’ knowledge
1.3 Group discussions
1.4 Teachers’ knowledge
2 Numerical cognition
3.1 Motivating modern definitions
4 Mathematical contexts
4.1 Not being odd
4.2 Even-odd alternation
4.4 The empty set
4.5 Degrees of evenness
5 Everyday contexts
8 Further reading
und zu jedem dieser Abschnitte gibt es ziemlich lange Texte.
Zum Beispiel zu “Group Discussions”:
Often curious students will directly ask if zero is even; the Israel National Mathematics Curriculum reminds first grade teachers that zero is even, but advises that it is unnecessary to mention this unless the class brings it up. One study observed a class of 15 second grade students:
There was little disagreement on the idea of zero being an even number. The students convinced the few who were not sure with two arguments. The first argument was that numbers go in a pattern …odd, even, odd, even, odd, even… and since two is even and one is odd then the number before one, that is not a fraction, would be zero. So zero would need to be even. The second argument was that if a person has zero things and they put them into two equal groups then there would be zero in each group. The two groups would have the same amount, zero.
In another class of 22 third graders, Deborah Ball asked her students to reflect on “a particularly long and confusing discussion on even and odd numbers”. One student commented that hearing other ideas had helped her understanding, and she now believed for the first time that zero was even. At the same time, another student had originally thought zero to be even but “got sort of mixed up” and wasn’t sure whom to agree with. Ball finds it significant that the latter student expressed a desire to listen further to the discussion: in this sense, both students have learned something valuable about their own learning process.
Later on, during a discussion on fractions, Ball asked the class whether or not voting would be a good way to prove what is true in mathematics. One of the students returned to her experience of the discussion on zero:
Betsy: I have an example of why voting doesn’t work because when we were talking about zero, if it was an odd or even. A whole lot of people said that it was an odd but then afterwards we figured out that it was even and voting didn’t help us know if it was odd or even because the answer was opposite than what people had voted. Teacher: So how did we change our minds then if the voting doesn’t work? Betsy: Because the people found out patterns and the number line and they figured out that no, zero must not be a odd because when it goes up there it goes odd, even, odd, even, odd, even and so when you had an odd number like one and then you have zero, zero must be even because that’s the way it is.
oder zu “Teachers Knowledge”:
It is important for teachers of mathematics to understand that zero is even and other basic facts. Researchers of mathematics education at the University of Michigan used the true-or-false prompt “0 is an even number”, among many similar questions, in a 2000-2004 study of 700 primary teachers in the United States. For them the question exemplifies “common knowledge … that any well-educated adult should have”, and it is “ideologically neutral” in that the answer does not vary between traditional and reform mathematics. Overall performance in the study significantly predicted improvements in students’ standardized test scores after taking the teachers’ classes.
Unfortunately, many teachers harbor misconceptions about zero, although it is hard to quantify how many. The Michigan study did not publish data for individual questions. One report comes from Betty Lichtenberg, who wrote an article titled “Zero is an even number” in the journal The Arithmetic Teacher in 1972. Lichtenberg, an associate professor of mathematics education at the University of South Florida, draws on her experience with a course she and her colleagues taught on methods for teaching arithmetic. She reports that several sections of prospective elementary school teachers were given a true-or-false test including the item “Zero is an even number.” They found it to be a “tricky question”, and about two thirds answered “False”.
Data is also scarce for teachers’ attitudes on students’ attitudes. The National Council of Teachers of Mathematics’s Principles and Standards for School Mathematics records a first grader’s argument that zero is an even number: “If zero were odd, then 0 and 1 would be two odd numbers in a row. Even and odd numbers alternate. So 0 must be even…” In a survey of 10 college students preparing to teach mathematics, none of them thought that the argument sufficed as a mathematical proof. When they were told that it had been written by a first grader, most agreed that it was acceptable reasoning for that age level.
Einen Beweis, daß 0 gerade ist, liefert dieses Bild:
“A mirror partitions an empty collection of pencils; half (0) are real and half (0) are reflections.”