# Things some teachers get wrong

Discussion in 'Mathematics' started by Subjectknowledge, Nov 30, 2011.

1. Does anyone else have their favourite examples of things some teachers get wrong? Here are some that I have seen:
1. Getting the definitions of segment and sector the wrong way round (seen more than once)
2. Getting students to face the front, then turn 90 degree to the left, then another 90 degrees to the left, then another, then another, and saying that this demonstrates rotational symmentry of order 4.
3. Misapplication of BIDMAS/BODMAS to argue that addition takes precedence over subtraction, e.g. demonstrating to the class that 10 - 3 + 4 should be interpreted as 10 - (3 + 4) = 3, rather than working left to right as in 10 - 3 + 4 = 7 + 4 = 11.
4. Telling students that 0 is not a number because 'it is the absence of number'
5. Telling students that 1 is not a prime number because a prime number can be divided by itself and 1, but 1 can only be divided by itself.
6. Telling students that 1 is a prime number because it can be divided by itself and 1.
Some are a bit more subtle:
7. Telling students triangle 'has' 180 degrees.
8. Talking about 'the function f(x)' rather than the function f.
9. Telling pupils that arcsin is the inverse of the sine function, rather than saying it is the inverse of the sine function with domain restricted to [-pi/2 , +pi/2]. If not convinced, try graphing y = arcsin(sin(x)) and comparing it with y = x.
10. Telling pupils that 'no-one knows what the biggest prime number is' rather than 'there is no biggest prime number' or that 1 - .9999...(recurring) is 'point 0 recurring with a one after it'
11. Telling students that an event is 'something that could happen' rather than a set of possible outcomes or a subset of the sample space, or that P(A|B) is the probability that A occurs given that B has already occurred
12. Having no answer to the question 'But what exactly is a random variable?'
It strikes me that a collection of mistakes and confusions might be a useful resource for professional development.

2. Does anyone else have their favourite examples of things some teachers get wrong? Here are some that I have seen:
1. Getting the definitions of segment and sector the wrong way round (seen more than once)
2. Getting students to face the front, then turn 90 degree to the left, then another 90 degrees to the left, then another, then another, and saying that this demonstrates rotational symmentry of order 4.
3. Misapplication of BIDMAS/BODMAS to argue that addition takes precedence over subtraction, e.g. demonstrating to the class that 10 - 3 + 4 should be interpreted as 10 - (3 + 4) = 3, rather than working left to right as in 10 - 3 + 4 = 7 + 4 = 11.
4. Telling students that 0 is not a number because 'it is the absence of number'
5. Telling students that 1 is not a prime number because a prime number can be divided by itself and 1, but 1 can only be divided by itself.
6. Telling students that 1 is a prime number because it can be divided by itself and 1.
Some are a bit more subtle:
7. Telling students triangle 'has' 180 degrees.
8. Talking about 'the function f(x)' rather than the function f.
9. Telling pupils that arcsin is the inverse of the sine function, rather than saying it is the inverse of the sine function with domain restricted to [-pi/2 , +pi/2]. If not convinced, try graphing y = arcsin(sin(x)) and comparing it with y = x.
10. Telling pupils that 'no-one knows what the biggest prime number is' rather than 'there is no biggest prime number' or that 1 - .9999...(recurring) is 'point 0 recurring with a one after it'
11. Telling students that an event is 'something that could happen' rather than a set of possible outcomes or a subset of the sample space, or that P(A|B) is the probability that A occurs given that B has already occurred
12. Having no answer to the question 'But what exactly is a random variable?'
It strikes me that a collection of mistakes and confusions might be a useful resource for professional development.

3. I'm not sure how critisising collegues in this way is helpful. The only purpose I can see in your post is to inflate your ego, I shouldn't be surprised with a user name like yours though.

Attitudes like this serve only to undermine others in the profession. Have you ever considerd that " ... an event is .... a set of possible outcomes or a subset of the sample space." might confuse 9 set 3?
Also
Getting the definitions of segment and sector the wrong way round (seen more than once). Although I know the difference, I can understand where the confusion lies if people talk about orange segments!

4. Didn't Cantor say that there were many infinities? Or am I thinking of something else?

5. Indeed there is an infinity of infinities.

6. I tend to think of infinity as a notion rather than a number, although different levels of 'infinity' are categorised within the area of transfinite numbers rather differently - it isn't something that I know enough about although I do know someone who did her PhD thesis on it. I'll have to get in touch with her and ask about it.

7. Post 1...do you [OP] ever make a point of looking something up in a book?
I find it interesting that no one has flamed you for item 3 on your list.
as for item 4 ... is not the statement "0 is a number just like any other" also untrue?
Is it not, therefor, acceptable to warn students that 0 is not like other numbers?

Do mathematics teachers have to perfect then?
I find this sort of thing rather tedious.
Do you think it is ok to say "I'm not sure, but I will check"?
I
find some students waste far too much time trying to catch the teacher
out instead of concentrating on learning the things that are pointed out
to them.
I am reminded of the Horizon Documentary about Fermats
last theorem when Wiles made his proof. Shimura spoke of his late
colleague Taniyama [of Taniyama-Shimura conjecture latterly theorem]
"being wrong..but in a good way".
The problem with the excruciating
exactness of some of the definitions in post one is that they are far
beyond the conceptual abilities of younger children.
I see nothing
wrong in telling bright children in year 6 that there are "proper"
factors and roots which are positive and exploring the 'exciting' world
of number types other than the positive integers.
[quote
user="Subjectknowledge"]It strikes me that a collection of mistakes and
confusions might be a useful resource for professional
development.[/quote]
It strikes me that there might be a lot more
students [and teachers] interested in mathematics if we stopped assuming
that everybody is immediately ready to work at the level of the
intellectual giants of the past.
I have seen many colleagues do a
sterling job of teaching students without having a degree in
mathematics but knowing when to point them in my direction.
I can see professional development could be done in a positive way, however the original post does not strike me as ultra positive.
My
favourite quotation on this topic is from a linear algebra book, which
has a short bio of Gauss, I must paraphrase "generally considered to be
last person to know everything that there was to know about mathematics
at the time." [apologies to Andrew Wiles].

8. Just a general musing, is subject knowledge important? Yes we need a certain amount, but i would argue subject comfort is more important.

I think there is a problem where students see their teacher as a maths God, spitting out answers and knowing everything (i think some teachers love to show off). I think this makes it seem that teachers or people who are good at maths just spring out of the ground. I tend to admit the bits that get me confused or that i struggle to remember, show you're human. If a question throws you ADMIT it! why has it? does it not sit with how you view things? why not? be honest with your students. Why can you and your students find out something together?

Not doubting subject knowledge is important, but on the things i'd look for in a good teacher i'm not sure it'd get into my top 5.

9. Wrong sport! 10. Well it's never too late to change.

11. Not likely! Thanks for pointing out the lack of avatar though. I'll get on that...

12. You look better parading around in all that unnecessary garb than you did when you were naked anyway.

13. ??

14. naked = avatarless

15. Ah, I'm with you!

16. I think sharing misconceptions like this is helpful. We all have them.
Re 'misconception' number 5, the number 1 is NOT a prime.
The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since "in exactly one way" would be false because any n = n x 1. In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states "Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable." As more simply noted by Derbyshire (2004, p. 33), "2 pays its way [as a prime] on balance; 1 doesn't."
(Thanks to http://mathworld.wolfram.com/PrimeNumber.html for the above)

17. Well, it seems pretty clear that it means that the sum of the angles is 180 degrees. What else could it mean?
Would it really confuse anyone if you didn't say that every time?
Mathematical rigour is important, but so is balancing this rigour with clarity and accessibility. There is so much more to teaching maths to pupils than this and I am sure I make far more "mistakes" in dealing with pupils on a personal, human level than I would if was a little bit sloppy with my definitions now and then.

18. so is 1 prime or not?

19. No it isn't prime.

20. The mathematical dictionary I have on my desk at school (http://www.amazon.co.uk/Oxford-Mathematics-Study-Dictionary-Tapson/dp/0199151180/ref=sr_1_2?s=books&ie=UTF8&qid=1322934730&sr=1-2) defines a prime number as having two factors; 1 and itself. Therefore, 1 is not a prime number because it only has one factor.....