## 15 February 2016

### Common Errors Made by Maths Teachers

I've recently started working with a couple of trainee maths teachers. It's already proving to be a useful experience for me - I'm sure it will have a positive impact on my teaching. It's fascinating to sit at the back of the classroom alongside my students.

There's been a couple of moments lately where I've seen a trainee teacher tell a class something that's not quite right, or that I've felt has needed further clarification. It's interesting to see how a teacher's subject knowledge - particularly their knowledge of common misconceptions - develops quickly over the first few years of their career.

I've started to pull together a list of some of the mistakes or oversights that I've seen both inexperienced and experienced maths teachers make. I'd be interested to hear what you've seen too - please let me know so I can create a comprehensive list that will be of use to trainees and NQTs.

Square root
The square root symbol denotes only the positive root. So if you write √9 = ±3 on the board, you've made a mistake.
 Source: Wikipedia
If you're solving the equation x2 = 9 then of course there are two solutions - both positive and negative three work here. You're not wrong about that. But make sure you use the radical sign correctly.

Angles
What angle fact is depicted below? Some of my students have been incorrectly taught to say 'angles in a circle sum to 360o'.  It should be 'angles around a point sum to 360o' (as per GCSE mark schemes) or 'angles in a full turn/revolution sum to 360o'.

Whilst we're on the subject of angles, I've only recently found out that saying 'angles in a triangle are supplementary' is wrong. Supplementary means that two (and only two) angles sum to 180o.

Prisms
I don't recommend asking maths teachers on Twitter whether a cylinder is a prism! I did that once - the argument went on for days and I was none the wiser as a result.
Since this Twitter conversation, I avoid defining a cylinder as a prism in case I'm wrong, but I do tell my students that we find the volume of a cylinder in the same way we find the volume of a prism.

Interestingly, the GCSE formula sheet uses a controversial picture of a prism.
Frustum
It's a frustum, not a frustrum. This word is often misspelt.
The order of operations
Try this question: 3 - 3 x 6 + 2.

If you don't get an answer of -13 then you need to look again at the order of operations. The unhelpful acronym BIDMAS (which incorrectly implies that you should add before you subtract) confuses some teachers as much as it confuses students.

Vocabulary
I recently observed a lesson on angles in polygons with a Year 8 class. The student teacher asked the name of a seven sided shape and a boy put up his hand and said it was a heptagon. The teacher said he was correct, then moved on. She didn't notice that another boy was very bothered by this. He angrily protested to the students around him. I watched from the back. This boy had been taught that a seven sided shape is called a septagon - which is also correct - and he thought that the teacher had made a mistake. At first he was angry, but then he doubted himself. His confidence took a knock. Later in the lesson, when the opportunity arose, I reassured him that both terms are acceptable. He was relieved. He felt vindicated. It was a minor oversight by the teacher but it had quite an impact on this student.

I probably do things like this too sometimes. We say things that, rightly or wrongly, differ from what students have been taught previously. This causes confusion and mistrust.

I'd be interested to know what mistakes and oversights you've seen teachers make, so please comment below or tweet me so I can pull together a comprehensive list. We're always talking about the importance of students learning from each other's mistakes, so let's do the same.

1. I have noticed a lot of new teachers use incorrect vocabulary regarding expressions and equations... you can evaluate or simplify an expression and you can solve an equation but you can never 'solve' an expression.

1. Yes, same here, noticed a lot of using incorrect vocabulary expressions and equations

2. Don't forget rounding! https://cavmaths.wordpress.com/2014/02/10/the-rounding-pandemic/

Too many get that wrong.

I came across a GCSE question where they'd spelled fustum 'frustrum'!

1. "I came across a GCSE question where they'd spelled fustum 'frustrum'!"

Ah ... Hmm.

2. I've seen a website where they'd spelled frustum 'fustum'.

Oh the irony, glass houses, all of that.

3. Great post. I've definitely made some of these mistakes in the past. Most of the points made on this post could be discussed by the students and make for some interesting homeworks. In my early days (as a teacher, not in life) I was asked by a year 7 is zero was a negative or positive number. I didn't want to give an an answer (because I didn't know) so I set it as a homework and some students did some serious research. It was great.
We cannot be expected to know everything, and get everything right especially as Maths teachers these days do not have Maths degrees.... But we should be prepared to read up and always want to improve our own knowledge. Your blog gives some of us that opportunity so thank you.

1. There is a good reason why zero is even. The "congruence modulo n" relation (where n is a positive integer) is defined as follows:

The integer x is congruent modulo n to the integer y if and only if n divides the difference x - y. In this case, we write x ~ y mod n. Effectively, x and y have the same remainder upon division by n.

This is an equivalence relation on the set Z of integers, so it partitions it into n mutually disjoint equivalence classes:
class(0) - all integers congruent to 0 modulo n
class(1) - all integers congruent to 1 modulo n
...
...
class(n-1) - all integers congruent to n-1 modulo n.

Now, when n = 2, there are just two classes: class(0) and class(1). What, in fact, are class(0) and class(1)?

Well, class(0) is the set of integers x for which x - 0 or x is a multiple of 2 i.e. the set of even integers.

And class(1) is the set of integers x for which x - 1 is a multiple of 2 i.e. x is of the form 2k + 1 for some k; the set of odd integers.

As classes are disjoint, 0 can only be in ONE of the above two sets. Since 0 - 1 = -1 is not a multiple of 2, it follows that 0 is in class(0) and is an even integer.

Hope this helps.

2. There may well be a good reason for zero to be even but that's not what the post above was talking about.

3. @1violalass You are absolutely right, although I am quite sure that when I wrote the above it was in response to the issue of whether 0 was even or odd, not whether it was positive or negative. It is possible, however, that I may have misread the comment or perhaps confused it with a different comment (on whether 0 is even or odd), which appeared somewhere else. :)

Anyhow, I apologise for the confusion.

For completeness, I would say that 0 is neither negative nor positive. Positives are greater than 0, negatives are less than 0 and (poor) 0 is in the middle of it all, an eternal sentinel!

4. If you look up prism on Wolfram Mathworld, it does specifically say that a prism is a kind of polyhedron. It also says that cylinder is the more general idea. So actually a prism is a kind of cylinder, and what we call a cylinder is actually a circular cylinder. They recommend using the term "general cylinder" to describe all shapes with constant cross-section.

5. I've seen tutors teach my students to find the volume of a frustum by taking the average of the diameter of the circle at the top and bottom, using this to work out the average area of the cross section and multiplying by the height. This does not work.

1. Well that's a worry! The 'proper' method is so intuitive, it's not necessary to make up an incorrect method!

6. The use of frustum rather than frustrum seems a fairly modern spelling formalisation. If you google frustrum you find quite a few books from the C19th.
I have seen 'converted' maths teachers who thought that 22/7 was not an approximation for pi but the 'actual' value.
The 'round' prism issue also applies to 'round based' pyramids!
'Septagon' is an americanism which should automatically be resisted!
When Paris had restricted traffic days based on odd or even number plate numbers, the police allowed those ending in zero to travel on either day!

1. Interesting!

I have no problem with the word septagon, though I know that some don't like it because it is a greek-latin hybrid.

2. When I teach it, I use heptagon, mostly because that is what is located in the textbooks. However, I can see how students can get confused (or better yet-easily remember) the septagon version. After all, if you think about our calendar, it would parallel very well:

december - 10th month; decagon - 10 sided
november - 9th month; nonagon - 9 sided
october - 8th month; octagon - 8 sided
september - 7th month; septagon - 6 sided

When I did a quick search, I found that both septagon and nonagon are Latin/Greek hybrids. Odd that one is "allowed" and the other isn't. Apparently the all Green name for a nine-sided figure should be enneagon. Maybe ease of spelling and pronounciation trump the use or non-use of hybrid words.

3. Are you sure September, October, November and December are the 7th, 8th, 9th & 10th months respectively? I always thought they were the 9th, 10th, 11th and 12th months....

4. Stephen,
That is how they got their name originally. Probably for your lifetime, they have been the 9th through 12th months.

Kind of reminds me of the Hitchhiker's Guide to the Galaxy trilogy--that has 5 books.

5. It is because the year used to start with March, not January. Hence these months were 7th, 8th etc.

6. No, I think the Romans added July and August when they changed the calendar. Named after their Caesars.

7. I taught my students both "nonagon" and "enneagon" :-)

7. I get frustrated when teachers tell students that something "cancels", which could mean anything from two numbers adding to equal zero, two factors dividing to equal one, squaring a square root, etc. That is confusing to students, who then think that anything can just "cancel" in algebra. Teachers need to use precise vocabulary and refer to the properties of algebra. Of course I didn't when I was a new teacher either.

8. Hi Jo,
Such an important post -thanks for sharing. When I was a student teacher, I was teaching the area of a circle after deriving that the circumference is Pi x D. We concluded that area is Pi x radius squared, however a curious child wanted to know what it will be in terms of the diameter. After deducing with the year 7's that it would be (Pi x diameter)/4 , my mentor shouted from the back to say that it was incorrect. You can imaging the chaos that ensued! I was humiliated and left second fussing myself, however I managed to get the class on side and in to practicing finding the area using the radius. After the lesson I confirmed again that the formula was indeed correct. I called my Uni and asked to be moved from the school or quite the program (there were other issues with my mentor). Luckily I was moved to my first placement and the rest is history!

1. Should be (Pi x diameter squared)/4

:)

9. I'm curious is it cm squared or square cm? I say squared units but revert to saying cm squared which I believe is wrong... I need a why to stop ;)

1. There's been some discussion on Twitter about this today... It should be square cm. I will include this on the final list!

2. I was taught that 4 cm squared = 16 square cm, i.e. "4 cm squared" means constructing an area with sides of 4 cm each.

3. I think both are okay. 1 cm squared is 1 lot of a unit of area measuring 1cm by 1 cm in the shape of a square i.e. (1cm)^2. "4 cm squrared" is 4 lots of a unit of area measuring 1cm by 1cm in the shape of a square. This could be seen more clearly if you pause after pronouncing the number i.e. 4 cm squared. I think people pronounce it square cm to avoid any potential confusion. The way I have described is consistent with the way we interpret algebra: would you say 4x^2 means 4x by 4x?

4. About the last bit: I would interpret 4x^2 as 4 . x . x

To get 4x . 4x, I would need to write (4x)^2

Otherwise, I agree with what you say about pronouncing 4 cm^2.

10. I had a student teacher who kept referring to a horizontal lines as a straight line. This is while he was instructing on linear functions. I had to keep telling him they were all straight

11. A common mistake is the overuse of degrees symbols. Such as in a geometry problem in which the x has a degrees sign, and we are asked to find the value of x. I have seen teachers penalise students for not writing the degrees symbol in their answer.

1 as a prime number is another common one.

12. using the word sum for any calculation not just when adding is required

13. Thanks all for the excellent suggestions. I will collate everything here and all the responses on Twitter and publish a complete list.

14. The wrong use of "=" is a common one which leads to problems as pupils move up the school.

15. One question about Bodmas. You can add or subtract and multiply or divide ... It is not supposed to matter which one you do first. Is that correct ? In which case you can get a different answer from -13. -17?

1. The convention (in UK secondary schools, I can't speak for other systems) is that if you have both addition and subtraction in the same calculation then you do them in the order they appear (from left to right). This is what a calculator does and I believe this is the only correct answer for the purpose of GCSE exams. Mathematicians say this convention is nonsense though, and in reality one should use brackets, spacing or context to understand what one is actually trying to calculate rather than follow a pre-defined order. For the purpose of teaching at school I would stick with the convention to be consistent with assessment requirements ie grouping first, then powers, then multiplication/division (working from left to right), then addition/subtraction (left to right). But I would use contextual examples (such as the example in my Shanghai post http://www.resourceaholic.com/2014/11/hublaunch.html) to help students make sense of things, and always encourage them to make lots of use of brackets.

2. 3 - 3x6 + 2 should give us the same answer as -3x6 + 2 + 3 or 3 + 2 - 3x6. The sign in front belongs to the number.

That's how I think of it in my head although I've yet to find a good way of explaining it to the kids that need it. Moving the numbers around like that has tended to confuse them.

3. How about saying that multiplication binds tighter than addition/subtraction? An alternative would be that multiplication takes precedence over addition/subtraction.

16. Much of this is pedantic nonsense. This is the stuff that turns students off maths.

1. Hello anonymous. I knew someone would say something like that. I quite like learning more about my subject, you're very welcome to ignore it all. Thanks for commenting!

2. I've been thinking about this comment... I agree that *some* of this stuff is a bit pedantic. But you can't deny that accuracy and precision of language is important in mathematics.

There are some suggestions that I may choose to ignore - though others would protest. For example I say the number -5 as 'minus five' even though I have been told that I am wrong (I should say 'negative five'). I've always called it minus five and I think it is perfectly acceptable, so I'm not going stop saying 'minus five'.

Although I would call some of this stuff pedantic, I find it all interesting and I certainly wouldn't call it nonsense. I can't see how it turns students off maths in the slightest.

3. I agree Jo, and I don't think it is pedantic to want correct and precise mathematics or vocabulary. I'd wager no one accuses English teachers of pedantry ilwhen they correct students who have used the wrong "their" or "your" or "too"

4. I agree totally with Jo and Stephen here.

5. I agree that some of these things are not of concern to students but this was clearly a post for teachers, not students.

17. Reading all of this is fantastic - we need to work together and support each others' learning and delivery, especially with the political upheaval and changes ahead, we have a hard enough job as it is, so agreeing the best or better approaches to ensure students understand their work, as well as developing consistency is surely a good thing.
Thanks everyone.....
My own bug bear is solving equations: the 'Ping and Flip' method is so frustrating, switching sides because a term appears on the left and now moves to the right???? This has major drawbacks, particularly when students get to upper school and are rearranging formulae, if they understand equations as balancing both sides, they can really see how the formulae change when rearranged, if they are asked to explain what they have done, they can, even if they have made an error to discuss in class, which provides great group discussion. The ping and flip method results in "I moved the 2x to the left because it's on the right", no real concept of the positive/negative symbols needed or operation when rearranged.....am I alone in feeling like this??

Great blog!!

18. Here is the problem with questions like 'is a cylinder a prism?' or 'is an equilateral triangle also an isoceles triangle?' It's that little word "is".

Without going into detail here -- those who wish should Google 'Korzybski' -- let me suggest a strategy that should help enormously when you are faced with "Is x a y?" questions that don't seem to be easily resolved: reword the question, along these lines: "Do we, and should we, USE the word 'y' to describe 'x'?" or "What is implied by using 'y' to describe/define 'x'?"

So ... it's not a question of whether a cylinder "is" a prism, but of whether we -- or the community of mathematicians -- have, for good reason, decided that the word "prism" should always be used to apply to a figure with a polyhedral cross-section -- leaving open the possibility that there is no consensus here.

This focuses the discussion on WHY we include or exclude various objects from our definitions, and emphasises that these are human decisions, not objective features of nature.

I believe that by frequently pointing out that language -- and the mathematical symbol system -- is not a simple one-to-one reflection of reality, but rather has involved human choices, sometimes arbitrary and even contradictory, we will help our pupils learn to think.

Why is the perimeter of a circle called a "circumference"? No good reason, it's just a legacy of the past. Why is pi defined in terms of the diameter, rather than the radius ('tau')? No good reason, just a legacy of the past. Why does a circle have 360 degrees instead of, say, 256 or 1000? Just a legacy of the past, probably having to do with the ancient astronomers' approximation of the number of days in the year and apparent horizontal movement of the sun's appearance on the horizon day by day, plus the fact that 360 has lots of integer factors.

Why is a meter the length it is and kilogram the mass it is? Look to the French Revolution and the desire to put measures on a rational basis -- a meter is one ten millionth of the distance from the equator to the poles, and a kilogram is the mass of a thousand cubic centimetres of water at a specific temperature.

All the result of human choice, and often irrational and contradictory. An "isoceles" triangle is a "same=sides" triangle in Greek, just as an "equilateral" triangle is also a "same-sides" triangle in Latin. Consistency would have us speak of trigons, tetragons, pentagons and hexagons -- and if we wanted to make it easier for our pupils to learn mathematics instead of having to wrestle with arcane irrational vocabulary, we would speak of "three-siders", "four-siders", "five-siders" and so on.

But we're stuck with this burden -- "he" has become "he or she" or possibly "they", but mathematics students will have to carry the load of a pointlessly-confusing vocabulary, left over from when those few children who did get an education learned Latin and Greek and already knew what a "gon" and a "hedra" were, and what "poly" meant, thus automatically knowing the difference between a polygon and a polyhedron. Suffer the little children!

Language can make a real difference: I teach my pupils never to say 'centimetres squared', because this doesn't evoke a mental image, but rather "square centimetres", which does. Likewise for cubic centimetres. And in physics, I teach them to always annunciate the expression for acceleration as "meters-per-second, every second" rather than the meaningless "meters per second squared".

1. I agree with most of what you say, but I would like to offer an ounce of help. You say that 'An "isosceles" triangle is a "same=sides" triangle in Greek'.

Actually, this isn't correct. The "sceles" in "isosceles" comes from the Ancient Greek word "σκέλη" (= "scele"), meaning legs. See https://en.wiktionary.org/wiki/%CF%83%CE%BA%CE%AD%CE%BB%CE%BF%CF%82

In this respect and because "iso" = "equal", an isosceles triangle is one that has equal legs; one is supposed to imagine the two equal sides that stem upwards from the (not necessarily equal) base and join up at the vertex
as a pair of legs.

Therefore, an isosceles triangle is not meant to be the same as an equilateral
triangle. For completeness, the Greek word for "equilateral" is "isopleuron" where "pleura" means "the side".

19. This has been a very informative article all round and I would be happy to see it augmented, enhanced and become a reference for misconceptions. Apart from the comments I made elsewhere in reply to various people, I have a few more points:

Square root
I normally write ±√a to indicate the solutions to x^2 = a for a >= 0. I feel that this serves to reinforce the fact that √a is the non-negative root.

The order of operations
I think that these two facts, namely:

1. brackets are done first, because they enclose calculations and "protect" them from the outside
2. multiplication takes precedence over addition

is all one needs to get by here. One should also remember that division and indices are variants of multiplication, and subtraction a variant of
addition. This should cover the BIDMAS rule.

20. Thanks for all the fascinating comments on this post - please keep them coming!