Common Maths Misconceptions - let's make a list!

Discussion in 'Primary' started by Blueowl99, Jun 2, 2010.

1. bombaysapphireStar commenter

Sounds good to me! The confusion is which way round they go. The fact that the factor is smaller and goes into the multiple is clearly illustrated by this analogy.
I regularly put about 30 different numbers on the IWB. Each student has to pick a pair of numbers and say x is a factor of y or b is a multiple of a. I've had even students in top sets struggle to get the numbers the right way round.

2. sulas

It is clearly me!!
I know what factors and multiples are - i simply can't see how the analogy helps.
6 and 5 are both factors of 30 - however, 30 is also a factor of thiry - does the factor family 30 then walk into the multiple mansion 30? That sounds like a recipe for confusion if ever there was one.
Explain. How does that help children's comprehension??

3. bombaysapphireStar commenter

The case that 30 is a factor and multiple of itself doesn't cause any difficulties in my experience.
The fact that 5 is a factor of 30 but 30 is not a factor of 5 is the one that causes problems. Factor family walking into multiple mansion would help with that.
I have never used this but I can see how this would help the problems that I have seen and I am going to try it.

4. Blueowl99

Have just returned from a weekend away to 60 New Posts! Who knew you were all so het-up about maths misconceptions!
For my sins, will put the Word Doc of these ideas on resources - if I don't get shot down in flames that is by some of you particuarlly scary posters!
The ideas was essentially a quick staff meeting blurb as I think it's really important to offer Tas support for extending their maths knowledge in-house as some of our Tas have been there donkey's years and don't have the time / inclination to find out if they're passing on mis - information to the children and at least in a staff meeting, all staff are learning together, even if the teaching staff don't think they need it.
Shooting down in flames moment...what is discrete data ? - I teach bars with no spaces between, on bar charts in Y3/4 for information like - how many cms long their femur was or how many green smarties in a tube... what should my bar chart be like?

5. tafkamOccasional commenter

http://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/measureshirev1.shtml

6. Blueowl99

Thanks TAFKAM -
<h1>Discrete and continuous measures</h1>A discrete
variable can only take certain values.
A continuous
variable can take any value.
Example
The
number of people in a car can only be a whole number: 1, 2, 3, 4...
It is not possible to have 31/<sub>2</sub> or 21/<sub>4</sub>
people in the car.
So the number of people in a car is a discrete
variable.
The height of a person can take any value, such as
141.35cm, 127.371cm, and so on.

Do all lower key stage teachers teach this difference in data and the consequential impact on bar graphs? Why does it matter if the bars touch for continous data but not for discrete? Hoping that if I understand myself, I can teach it better!

7. florapost

i have seen textbooks show discrete values as line graphs, so i do
not fuss about bars touching
the dreaded borrowing - are you sure
they haven't been taught that at home - one child after another in y2
showing how they would subtract 46 from 72: 'you can't take 6 away from
2, so you make the 2 into a 12' 'why' 'because my mum says so'
rectangles
and oblongs - augh - but again, many a book .....i have been known to
tell children it is better to know you are right in your own head

2d shapes - that is a potential minefield - i teach gat kids and
we can waste hours merrily discussing the shape on the board having the
height of a molecule of chalk

but my real pet shape hate is 'a circle is a shape with one side'

oh - and 'one is a prime number'
probability - i'm
planning on looking all this up later this week for another thread, if
i'm feeling seriously geeky, but i have it in my head that 1/3 is a
probability, 1 in 3 a natural frequency
algebra: x - 2 = 5 - so
you 'move the 2 to the other side' - you may not teach algebra, but a lot of their parents do
on the multiplication issue
- is 6 x 8 not 8 sets of 6? if it is repeated addition, how do you
do/explain 6 x 8 1/2 ?
ditto division - if 60 divided by 10 is
repeated subtraction rather than 'how many sets of 10 can you make from a
set of 60' - what about 65 divided by 10 - i stand to be totally
corrected on this as it's only my musings and not something i actually
teach

i must stop now - i am sure i have a life out there somewhere

8. tromepic

Blueowl99 if the bars touch on discrete data such as number of people in a car you are implying that you can have 3 1/2 or 2 1/4 people in it which of course you cannot. As far as continuous data is concerned as you can have any value between 2 numbers the bars can join as the data is assumed to be rounded to the nearest value. I hope that helps.
As for Flora's comment about textbooks showing discrete data as line graphs, do you always believe what the textbook says or are you talking about bar line graphs rather than line graphs which quite rightly should be used only for continuous data e.g speed against time etc..

9. offbeatdave

In my Y6 class we don't borrow - we steal and spend the proceeds of our crime! Much more engaging, although morally dubious.
Sorry if this has all been said in this rather lively debate. After spending my youth being bought extra maths books by the school, doing a couple of maths A-Levels and a Maths degree, IMO, too much time is spent now on the reasons behind stuff which is why kids struggle to calculate quickly and efficiently. Surely the primary aim is to get kids to be able to do the maths. If they understand the reasons behind it great. If not, so what? I don't need to understand the workings of an engine to drive my car from home to work. The kids don't need to know how their pen works to write do they? Extreme egs I know. Methods for calculation work, have worked for years and will continue to do so. Use them! The child in my Y6 class operating at 2a/3c couldn't tell you why 138 +
147 = 285 but can do it using the classic formal method every time and
will go through life being able to do it.
Line graphs for discrete data is just plain wrong, but gaps really don't matter in bar charts. It still shows the same information whether there is a gap there or not. A 5 year old will still be able to tell you which is the most popular flavour of crisps in their class regardless of the bars touching or not. Being a mathematician myself, I get really wound up by others who are so precious about their conventions that really have no effect on anything.
Who cares if 6 x 4 is 6, 4-times or not. Learn the fact and that it can be reversed. If our Numeracy consultants spent less time worrying about this, they might have time to tell us something useful about the teaching of maths. Every chuffing training or cluster meeting it crops up. It really doesn't matter.
That all said, using sums for anything other than addition (and at a push, subtraction) winds me up!

10. rebeccagoad

Times IS 'lots of'.
When teaching multiplication, the first step is to illustrate it using arrays; and array for 4 x 6 is different to an array of 6 x 4.
4 x 6 is '4 lots of 6'; 6 x 4 is '6 lots of 4' and is a rotation of the original array.
Yes, multiplication is commutative and the children should be able to see the relationship between the two however, for depth in understanding, I feel that children should see multiplication in terms of 'lots of' because then it helps the to understand the inverse in terms of 'how many___ are in ___?'

This helps children to grasp the concept of multiplication in a more concrete way and I've found from personal experience that, particularly with lower abilit children, they are able to use and apply their understanding much better when seeing multiplication in terms of 'lots (or groups) of'.

11. hattonm2

You are absolutely right - a percentage or a decimal is an acceptable response to a probability question. It is true that fractions, decimals and percentages are equivalent, but none of the given examples are fractions, decimals or percentages! And 1:3 is the equivalent of a quarter, not a third!

12. hattonm2

Sorry - first post on these forums! That was a response to an earlier post (number 23 I think).
That will teach me not to use the preview button

13. hattonm2

I have used the idea that factors are the factories- they make other numbers

14. DamianHassan

I agree with offbeat dave - we can become too precious about this. In some ways, the original question has been hijacked, as
(1) we have ended up talking / arguing about conventions, and not misconceptions; and
(2) we are conflating misconceptions with simplifications.

The former is merely annoying. OK, so bar charts can work better with a gap between them, to emphasise the discrete nature of the data, and to avoid confusion with histograms. But many textbooks differentiate between a barchart for showing categorical data (such as numbers of car with a particular colour) and a histogram for showing data based on an interval scale (either continuous, or countable discrete such as numbers in a car). So the assertion that using a histogram to record car occupancy implies that there could be fractional occupancy - well, that is just nonsense! Other conventions: a diamond shape is called a rhombus. I would ensure my Y6 students know that, but they are not wrong in calling it a diamond (indeed, that terminology is used mathematically in other contexts - see Wolfram Mathworld for examples).

Simplifications can be useful when first introducing a topic. I would be delighted if Y1 students noticed that multiplying a digit by 10 could be achieved by adding a zero. If in Y2/3 they could extend that result to work out 67 X 10 = 670, I would continue to be pleased. The rule works - when multiplying an integer by 10, just add a zero. Of course, I would explain why the rule works. And yes, I would have to explain why the rule doesn't work when we start multiplying decimals in Y4/5, although I find referring to money when using decimals is a remarkably quick way of emphasising the point. Yes, there are still some students who will write 6.3 x 10 = 6.30 - but do you really think that these students would have been helped if in Y1/2 they had been told it was wrong to add a zero?!
(and for those of you still shaking your heads - I assume you are being consistent, and when you teach gravity in science you make sure that your students know that gravity is not a pulling force, but a result of the curvature of space-time ... of course you do)
Oh, and I do agree with a few points. I tell my pupils that I will quietly bury the first child who reads 3.14 as "three point fourteen". This is a dangerous misconception, and explains why some children think 3.3 < 3.14. I do make sure that my pupils see shapes in differing orientations. I emphasis that the words 'sum' and 'difference' have particular meanings in maths, and that they need to be thinking as mathematicians when listening to language in maths lessons. But I think we need to be careful in these debates to distinguish between ideas which will actively impede mathematical development, those which will work at the time but will need to be extended later, and those which are simply a matter of convention.

15. mariapaola

Well, I'm an Italian teacher and I understood the meaning of "move the decimal point" when I started teaching...
When I was a kid (and I was not a matematical genius!) my teacher explained me "move the decimal point" without explaining me that we did it because we divided by tenth, hundredth and so on...
For me, as a kid was something magic...
Now I understand why we did it... ;-)
MP&pound;

16. 5aran

" a cone has 2 faces - 1 curved face and 1 flat face"

definition of a face: a flat surface

Therefore a cone has 1 face, 1 curved surface and 1 edge

17. reedle

oops - misconception about misconceptions...1:2 is not the same thing as 1/2

1:2 means a split of 1/3 to 2/3 - my year 11 students often struggle with this.

18. DamianHassan

THis is why we must be careful when we are being pedantic; if you are allowed to define 'face' post hoc, then I can define edge as "the intersection of two planes" - a perfectly good definition for many mathematical discussions - and your cone has no edge. The point being that you must be careful about defining your terms in advance.

19. florapost

ah - that saved me mucking about on the internet
no of course i don't belive what textbooks tell me - i thought i'd established ubergeek status - what i was saying is, given such horrendous behaviour by textbook writers, failing to distinguish between joined and separated bar graphs seemed minor

20. bighsg

Then can I suggest you install "maths calc" to allow you to do that in Word or insert it as a Symbol - might not work in text only emails but using Rich Text or composing in word and copying it might.