There is a lot of advice out there for teachers. As an NQT, every experienced teacher you meet will probably start their first conversation with you by imparting some "words of wisdom". Often this begins with, "I don't know what they've taught you on your PGCE, but in my experience [insert gratuitous advice here]".
I have been given a lot of bad advice is my short time as a teacher. Some of it was from my PGCE course, some from books, and some from the soapbox in the corner of the staff room.
So, here is my list of the top ten stupidest teaching tips (in no particular order):
"Don't smile until Christmas"
This is probably the most oft-quoted teaching tip. But does anyone actually do this? I think it's terrible advice!
I love maths. I want my students to love maths. Can you share your enthusiasm for a subject without smiling? If I explain a topic in maths completely straight-faced, doesn't that imply to my students that I don't find that topic interesting/exciting/inspiring? And they'll be thinking, even Miss finds this boring and maths is her favourite subject!
Also, the first term of teaching a new class is when you're trying to make relationships. Is it possible to make a relationship with someone without smiling?
Of course you want to lay down the law and show them who's boss, but without coming across as cold, boring, and unrelatable.
"As your students leave the room, stand by the door and tell each of them you love them".
You probably think I've made this one up. But no, I read this in Rocket up Your Class! which I actually thought was a great book until I read this particular titbit.
However, being the diligent student that I am, after reading this advice, I decided to try it out. I don't think I can really call it a success. The first student out the room was definitely uncomfortable, the last person out the room only got a very lacklustre "Uvyoo" (you try saying "I love you" thirty times in thirty seconds!) The students didn't even find it funny, they just thought I was weird.
The main problem I have with this is that to be quite honest, I don't "love" my students. They're perfectly nice people (some of them) and all, but my love is reserved for immediate family, pets past and present, and chocolate-covered popcorn. Sorry kids.
"Positives and negatives should be given out in the ratio of 4:1"
I've already written about my scepticism of this. I know teachers who are highly effective and have excellent relationships with students who very rarely use praise, and never use rewards.
I hate over-praising. It just devalues the whole currency. I'd rather be praised meaningfully once in my whole seven years with a teacher, than shallowly every lesson.
"Never take work home with you"
If it were possible to not mentally take work home with you, I might have thought this was good advice.
In terms of physical work: marking, lesson planning, reports, various admin (far too much of it, considering teachers aren't technically supposed to have to do admin)... the way I see it there are two choices.
Go home lugging your books on your back (or in the boot, if, unlike me, you have a car), get home at 4:30, make yourself a cup of tea by filling the kettle from the kitchen sink, not from the communal toilet's sink (the closest water source to the maths office), using milk that's not perpetually "on the turn", in your favourite mug, not whatever vessel you can find that hasn't been appropriated by the faculty slob. You can immediately change into your jammies, put on your guilty-pleasure music (a certain adorable quintuple springs to mind), and mark/plan/do stuff whilst stuffing your face with roasted chickpeas (too smelly to eat in the office).
Or...
Stay at school, do your work, get home at 7pm, and smugly say to your spouse/kids/tamagotchi: "I don't take work home with me. Check out my work-life balance!"
"Don't do more than one of the same type of question"
This advice was actually given to me by an OfSTED inspector. Despite my new-found respect for HMIs, after they showed excellent judgement in awarding me the coveted title of "outstanding teacher", I still feel I have to disagree with this particular tip.
I talked at length about this in this post. I won't bother repeating myself.
"Never touch a student"
Way too many teachers are terrified of being called the P word. A gentle hand on the arm isn't going to cause any trouble! A pat on the back isn't going to put you in jail! (Obviously if the student in question has ASD or is a CP case then exercise caution).
On January results day just gone, one of my year 12s asked me for a hug. I am ashamed to say I almost said no. But then I thought, screw it, this girl was so nervous this morning and now she's overcome with relief, of course I should give her a hug! She shouldn't have had to have asked, I should have offered her a hug! It's a one-off ting, it's not a big deal. Stop being scared.
"Show students the insides of your wrists to show them you trust them"
This wee gem was imparted on me by someone at university when I was doing my PGCE. Not only is this advice bizarre, it is also incredibly hard to do. I challenge you all to attempt this in one of your lessons this week. Unless you have a curiously-shaped mole on your wrist you can somehow incorporate into your starter activity, I don't think you'll manage it.
"Keep your hands above waist height to demonstrate power"
Another PGCE one. I should point out these were external speakers, not my actual teachers, who were all brilliant.
My question is this: where can you put your hands above your waist? On your shoulders? Under your armpits, in the manner of a gorilla? Stroking your chin with your left and scratching your head with your right? Or arms spread wide, as if welcoming in your students' ideas and basking in their insights. Hmm.
Oh dear, a quick google has revealed to me that this advice is not only given to teachers but to business people who have to make important presentations and speeches. Listen up guys, here's my public-speaking advice: if you need to resort to body-language tricks to get people to agree with you, the content of your speech is probably rubbish.
"Do vocal exercises in the shower every morning"
We had quite a few sessions on looking after your voice during my PGCE. Teachers talk a lot compared to people in other jobs, and voice strain is definitely something that could be an issue. I know that the first day back after a holiday my voice always hurts by the end of the day.
But do any teachers ever actually do vocal exercises? Every morning? In the shower? Don't get me wrong, I enjoy my sessions with the loofah-microphone as much as the next person, but certainly not at 5:45am, and anyway, my performances are limited to actual songs, not odd bird-like squawks and exaggerated vowel sounds. I can't believe I attended those lectures.
"Get a gun"
No, I've not gone crazy. Read this article. I'm not even going to pass comment. Except perhaps to say: oh dear, America.
Got any stupid teaching tips for me? Post them in the comments!
Emma x x x
Thursday, 25 April 2013
Saturday, 20 April 2013
Factorising: a Divisive Topic
See what I did there?
We had a faculty meeting last week, which can only mean one thing: a raging argument about the best way to teach something. We're an opinionated bunch.
The topic in question was quadratic equations. More specifically, factorising, although we touched on solving them in general.
What method would you use to factorise this?
6x2 + 26x + 20 = 0
The mathematician answer, of course, is I wouldn't, I would just use the quadratic formula.
The more observant amongst you will have divided by two first and then found the answer quite easily.
For the purpose of this exercise, dividing by two is not allowed, and neither is taking two out as a factor in the beginning.
This is the method that was put up on the board for us to discuss:
6x2 + 26x + 20 = 0
20 * 6 = 120
Two factors of 120 which add to make 26 are 20 and 6.
6x2 + 20x + 6x + 20 = 0
2x (3x + 10) + 2(3x + 10) = 0
(2x + 2)(3x + 10) = 0
I would estimate that about a third of our faculty looked at this and immediately said, yes, that's how I teach it. Another third said, I know that method but I don't really teach it, and the last third said, I've never understood that method, I think it's stupid.
OK, that last group was probably smaller than the other two, but it felt bigger because I was in it.
I hate this method! It took me ages to get my head round it. I finally got round to proving it to myself so I feel happier, but there is no way I would teach this to a class, because first I'd have to prove it to them to show them where it comes from. I don't think they'd be able to use this method without understanding how it works.
But apparently, I am wrong. There are a lot of students who are successful using this method, and many teachers swear by it. The alternative, trial and error kind of method that I use is obviously quite annoying (hence why in those cases where the coefficient of the x squared term is not 1 or a prime, I would never bother factorising) and it probably puts some students off.
So here's the big question: should we teach students a method they do not understand if it makes getting the answer faster?
Primary schools have moved away from teaching un-understandable methods like column subtraction, bus-stop division, long multiplication in columns, etc towards using methods that students can understand how they work, like chunking, partitioning, open numberlines, the grid method, etc. Surely we should be doing the same higher up the key stages?
I would love to know your opinions on this. If you're a maths teacher, which method do you teach? If you're not, what method were you taught when you were in school?
Emma x x x
Appendix: Explanation of above method
ax2 + bx + c
= (kx + l)(mx + n)
=kmx2 + knx + lmx + ln
=kmx2 + (kn + lm)x + ln
So we are looking for k, l, m and n such that:
km = a
kn + lm = b
ln = c
Aim to split b up into kn + lm
ac = kmln = kn * lm
So we can factorise ac such that the two factors sum to make b.
Working forwards:
ax2 + bx + c
= kmx2 + knx + lmx + ln
=kx(mx + n) + l(mx + n)
= (kx + l)(mx + n)
We had a faculty meeting last week, which can only mean one thing: a raging argument about the best way to teach something. We're an opinionated bunch.
The topic in question was quadratic equations. More specifically, factorising, although we touched on solving them in general.
What method would you use to factorise this?
6x2 + 26x + 20 = 0
The mathematician answer, of course, is I wouldn't, I would just use the quadratic formula.
The more observant amongst you will have divided by two first and then found the answer quite easily.
For the purpose of this exercise, dividing by two is not allowed, and neither is taking two out as a factor in the beginning.
This is the method that was put up on the board for us to discuss:
6x2 + 26x + 20 = 0
20 * 6 = 120
Two factors of 120 which add to make 26 are 20 and 6.
6x2 + 20x + 6x + 20 = 0
2x (3x + 10) + 2(3x + 10) = 0
(2x + 2)(3x + 10) = 0
I would estimate that about a third of our faculty looked at this and immediately said, yes, that's how I teach it. Another third said, I know that method but I don't really teach it, and the last third said, I've never understood that method, I think it's stupid.
OK, that last group was probably smaller than the other two, but it felt bigger because I was in it.
I hate this method! It took me ages to get my head round it. I finally got round to proving it to myself so I feel happier, but there is no way I would teach this to a class, because first I'd have to prove it to them to show them where it comes from. I don't think they'd be able to use this method without understanding how it works.
But apparently, I am wrong. There are a lot of students who are successful using this method, and many teachers swear by it. The alternative, trial and error kind of method that I use is obviously quite annoying (hence why in those cases where the coefficient of the x squared term is not 1 or a prime, I would never bother factorising) and it probably puts some students off.
So here's the big question: should we teach students a method they do not understand if it makes getting the answer faster?
Primary schools have moved away from teaching un-understandable methods like column subtraction, bus-stop division, long multiplication in columns, etc towards using methods that students can understand how they work, like chunking, partitioning, open numberlines, the grid method, etc. Surely we should be doing the same higher up the key stages?
I would love to know your opinions on this. If you're a maths teacher, which method do you teach? If you're not, what method were you taught when you were in school?
Emma x x x
Appendix: Explanation of above method
ax2 + bx + c
= (kx + l)(mx + n)
=kmx2 + knx + lmx + ln
=kmx2 + (kn + lm)x + ln
So we are looking for k, l, m and n such that:
km = a
kn + lm = b
ln = c
Aim to split b up into kn + lm
ac = kmln = kn * lm
So we can factorise ac such that the two factors sum to make b.
Working forwards:
ax2 + bx + c
= kmx2 + knx + lmx + ln
=kx(mx + n) + l(mx + n)
= (kx + l)(mx + n)
Tuesday, 2 April 2013
Minus versus Negative: Some Mathematical Grammar
Ooh, today you're getting a discussion of maths and grammar, aren't you lucky?
We had a "moderation day" last week. It's an INSET day where teachers moderate their coursework. As you can probably imagine, the maths department was incredibly swamped that day. NOT!
Maths doesn't have coursework, so theoretically, we didn't have anything to do. In practice, however, we had absolutely loads to do, because we are still teachers, and a teacher's work is never done. That sentence has far too many commas. Should I really be writing a post about grammar? You can always put bad grammar down to style, can't you? My style is to use too many commas. Like, this.
Anyway, the maths department decided to take a long lunch on this moderation day, and went to a popular pizza restaurant armed with multiple two-for-one codes. The joke "how many maths teachers does it take to split a restaurant bill?" comes to mind.
We spent most of the meal maths debating. This often happens to us. Luckily the restaurant was almost empty, or it could have been quite embarrassing. We were scrawling equations on napkins using board markers (the only pen we ever have on us) by the end.
The subject of the debate? Should the word "minus" only ever be used as a verb?
Read the following sentence out loud: The weather today is -2 degrees. How did you say it? Did you say "minus 2 degrees"? Or did you say "negative 2"? I would guess that if you are from the UK you probably said minus. I know that's what I say. It's definitely what the weather people say on TV.
Now read this out loud: x - 5 = 7. Did you say "minus" again? You might have said "take-away", possibly "subtract", or even "less". I would say minus, probably because this is what all of my maths teachers used to say to me.
Third test: what rule were you told about why -5 x -2 = 10 not -10? Say this rule out loud. Did you just say something like "a minus times a minus makes a plus" or "two minuses makes a plus"?
Can you see a slight issue? We're using "minus" as an adjective, meaning negative, and we're also using it as a verb* meaning subtract. And thirdly, we're using it as a noun when we say "a minus" meaning a number less than zero.
We're all quite comfortable with this word having several meanings. But what about students? When they first learn about "directed numbers" (as they're known), does this odd quirk of English confuse them?
I can see why some might think this. I understand that the language of mathematics should be used very carefully. I've always been very interested in grammar, which was why I did A level French (which, incidentally, everyone thought was weird: most of my teachers assumed I would be studying maths and the three sciences). At uni I took some modules that were about logic, which is pretty much just another word for language. I've taught enough EAL students to know that you need to choose your words carefully. But to be honest... I'm not completely convinced.
In a "number sentence" (a wonderful expression, thank you primary school teachers), I will always pronounce a dash (or hyphen, or em dash, or en dash) as "minus". Let me tell you why: that little symbol represents two things at once; it's the operation of taking away, and it's also to indicate something that is being negated (notice that both of these things are actions: I'm not saying it represents a negative number, I'm saying it represents something that is being negated). You absolutely need this symbol to represent both at once, because you want to be able to swap between the two meanings depending on how you feel.
Take this example:
3 - 4 (x - 7) = 10
If you wanted to solve the above equation, there are a few ways you could do it. Before reading ahead, please solve it.
How did you treat the minus before the 4? Did you see it as indicating something you're taking away from 3? Or did you see it as attached to the 4, making it a negative 4?
Did you do this:
3 - [ 4x - 28] = 10 (expanding the bracket with 4 as the multiplier)
31 - 4x = 10
etc
or this:
3 [- 4x -- 28] = 10 (expanding the bracket with -4 as the multiplier)
3 - 4 x + 28 = 10
etc
Or something different?
Can you see that if I, as a teacher, had indicated in some way that the minus before the 4 was a negative symbol, the first method wouldn't really make sense? And if I had indicated it was a take away, the second method doesn't really make sense, because students aren't taught that subtraction follows the distributive law.
The duplicity of the minus is one of those mathematical things that makes sense when you are mathematically fluent. Just like in English, how we have words that look the same and sound the same but mean two different things. As a fluent speaker of English, I don't even notice these. Look, I just used one! Notice! I didn't have to think: wait, is this the verb to notice, or a kind of sign stuck on a wall? I just used the word. And guess what, when I learnt French, my professeurs didn't just remove all homophones from the syllabus so that, as a learner, I wouldn't get confused, they left them in, so that I could aim to become fluent. Why should maths teachers do this? Don't we want our students to become fluent in maths?
Yes, we should be careful with our language in maths lessons. We should make sure when we say "line" we don't mean "line segment". But we cannot protect our students from the difficult to understand bits. We need to expose them to these things.
What do you think?
Emma x x x
*Technically it is a preposition rather than a verb. But these days we use it as a verb, saying things like "minusing" and "you minus the five from both sides". I know technically these uses are wrong, but it's what we say. Just like how we say "timesing" and "timesed" because we use the word "times" as a synonym for multiply now.
We had a "moderation day" last week. It's an INSET day where teachers moderate their coursework. As you can probably imagine, the maths department was incredibly swamped that day. NOT!
Maths doesn't have coursework, so theoretically, we didn't have anything to do. In practice, however, we had absolutely loads to do, because we are still teachers, and a teacher's work is never done. That sentence has far too many commas. Should I really be writing a post about grammar? You can always put bad grammar down to style, can't you? My style is to use too many commas. Like, this.
Anyway, the maths department decided to take a long lunch on this moderation day, and went to a popular pizza restaurant armed with multiple two-for-one codes. The joke "how many maths teachers does it take to split a restaurant bill?" comes to mind.
We spent most of the meal maths debating. This often happens to us. Luckily the restaurant was almost empty, or it could have been quite embarrassing. We were scrawling equations on napkins using board markers (the only pen we ever have on us) by the end.
The subject of the debate? Should the word "minus" only ever be used as a verb?
Read the following sentence out loud: The weather today is -2 degrees. How did you say it? Did you say "minus 2 degrees"? Or did you say "negative 2"? I would guess that if you are from the UK you probably said minus. I know that's what I say. It's definitely what the weather people say on TV.
Now read this out loud: x - 5 = 7. Did you say "minus" again? You might have said "take-away", possibly "subtract", or even "less". I would say minus, probably because this is what all of my maths teachers used to say to me.
Third test: what rule were you told about why -5 x -2 = 10 not -10? Say this rule out loud. Did you just say something like "a minus times a minus makes a plus" or "two minuses makes a plus"?
Can you see a slight issue? We're using "minus" as an adjective, meaning negative, and we're also using it as a verb* meaning subtract. And thirdly, we're using it as a noun when we say "a minus" meaning a number less than zero.
We're all quite comfortable with this word having several meanings. But what about students? When they first learn about "directed numbers" (as they're known), does this odd quirk of English confuse them?
I can see why some might think this. I understand that the language of mathematics should be used very carefully. I've always been very interested in grammar, which was why I did A level French (which, incidentally, everyone thought was weird: most of my teachers assumed I would be studying maths and the three sciences). At uni I took some modules that were about logic, which is pretty much just another word for language. I've taught enough EAL students to know that you need to choose your words carefully. But to be honest... I'm not completely convinced.
In a "number sentence" (a wonderful expression, thank you primary school teachers), I will always pronounce a dash (or hyphen, or em dash, or en dash) as "minus". Let me tell you why: that little symbol represents two things at once; it's the operation of taking away, and it's also to indicate something that is being negated (notice that both of these things are actions: I'm not saying it represents a negative number, I'm saying it represents something that is being negated). You absolutely need this symbol to represent both at once, because you want to be able to swap between the two meanings depending on how you feel.
Take this example:
3 - 4 (x - 7) = 10
If you wanted to solve the above equation, there are a few ways you could do it. Before reading ahead, please solve it.
How did you treat the minus before the 4? Did you see it as indicating something you're taking away from 3? Or did you see it as attached to the 4, making it a negative 4?
Did you do this:
3 - [ 4x - 28] = 10 (expanding the bracket with 4 as the multiplier)
31 - 4x = 10
etc
or this:
3 [- 4x -- 28] = 10 (expanding the bracket with -4 as the multiplier)
3 - 4 x + 28 = 10
etc
Or something different?
Can you see that if I, as a teacher, had indicated in some way that the minus before the 4 was a negative symbol, the first method wouldn't really make sense? And if I had indicated it was a take away, the second method doesn't really make sense, because students aren't taught that subtraction follows the distributive law.
The duplicity of the minus is one of those mathematical things that makes sense when you are mathematically fluent. Just like in English, how we have words that look the same and sound the same but mean two different things. As a fluent speaker of English, I don't even notice these. Look, I just used one! Notice! I didn't have to think: wait, is this the verb to notice, or a kind of sign stuck on a wall? I just used the word. And guess what, when I learnt French, my professeurs didn't just remove all homophones from the syllabus so that, as a learner, I wouldn't get confused, they left them in, so that I could aim to become fluent. Why should maths teachers do this? Don't we want our students to become fluent in maths?
Yes, we should be careful with our language in maths lessons. We should make sure when we say "line" we don't mean "line segment". But we cannot protect our students from the difficult to understand bits. We need to expose them to these things.
What do you think?
Emma x x x
*Technically it is a preposition rather than a verb. But these days we use it as a verb, saying things like "minusing" and "you minus the five from both sides". I know technically these uses are wrong, but it's what we say. Just like how we say "timesing" and "timesed" because we use the word "times" as a synonym for multiply now.
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