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.
Wednesday, 20 March 2013
Why Are You Just a Teacher (part ii)
Remember back in May when I wrote this post? Give it a quick re-read if you want. I'll wait. Ho hum.
Ready? OK, well I had a similar sort of conversation a few weeks ago. To be honest, I have this conversation fairly often. Every single one of my classes has asked me at some point, why are you just a teacher? I'm used to it now, but something one of my students said a few weeks ago really hit home, and I'm still thinking about it now.
I was talking with one of my A level groups, and we must have been discussing universities or something. The question that is the bane of my life (see title, I can't face typing it again) came up, and as usual I deflected it with some verbal hand-waving. But then a student, in the most sombre voice you've ever heard an eighteen year-old use, asked me, "Do you feel like you've achieved your dream?"
That's one of the problems with my school: we're so big on promoting following your dreams and believing you can achieve them. The students are so full of the optimism of youth. It's quite sickening really.
I just sort of stood there with my mouth trying to form words that my brain wouldn't supply. I needed to say yes. I needed to for them, because they need to believe that their teacher loves teaching them, and I needed to for me, because the opposite would mean admitting to myself that I have failed.
But I said no. I gave the word an extra two syllables, as if stretching it over three would distribute the weight of its meaning. It came out like this, "...Nuh-oh?-oh..." That's right, the question mark was in the middle of the word. I didn't even know that was possible. Try it.
I think at that point I distracted them with the estimated monetary value algorithm. But I kept thinking about it.
I think a teacher was what I was meant to be. But maybe I'm not. Yeah, I love maths, and I love sharing my passion with others, and I love speaking in front of an audience, and I love teenagers, but is teaching the only job that fits with all that? And what about my other passions: dance, fashion, reading, writing blogs?
Well there's probably not a job that encompasses all of those things. But this notion of a "job" is so old fashioned anyway. These days, you don't have to have a "job". In this day and age, there are people out there making money from just about everything. At the weekend I read a couple of books on this subject: Be a Free Range Human and Screw Work Let's Play. These books are all about escaping the 9-5 and making money from something you enjoy. I would not quit teaching at this stage, I'm far too attached to having a roof over my head (call me materialistic), but there are some great ideas for side projects that can be done alongside your regular job. The idea is that when these become profitable you quit your day job.
I became a little bit inspired. What could I do to achieve my dream whilst still teaching full time?
One of the suggestions was blogging. As much as I enjoy writing this blog, I have to accept that it probably won't ever make any money. I know some people do make money from blogs, with advertising and sponsorships etc, but if they say that sex sells, then a blog about teaching maths is going home at the end of the day with a full briefcase and an empty wallet.
Do you feel like you've achieved your dream?
Emma x x x
Ready? OK, well I had a similar sort of conversation a few weeks ago. To be honest, I have this conversation fairly often. Every single one of my classes has asked me at some point, why are you just a teacher? I'm used to it now, but something one of my students said a few weeks ago really hit home, and I'm still thinking about it now.
I was talking with one of my A level groups, and we must have been discussing universities or something. The question that is the bane of my life (see title, I can't face typing it again) came up, and as usual I deflected it with some verbal hand-waving. But then a student, in the most sombre voice you've ever heard an eighteen year-old use, asked me, "Do you feel like you've achieved your dream?"
That's one of the problems with my school: we're so big on promoting following your dreams and believing you can achieve them. The students are so full of the optimism of youth. It's quite sickening really.
I just sort of stood there with my mouth trying to form words that my brain wouldn't supply. I needed to say yes. I needed to for them, because they need to believe that their teacher loves teaching them, and I needed to for me, because the opposite would mean admitting to myself that I have failed.
But I said no. I gave the word an extra two syllables, as if stretching it over three would distribute the weight of its meaning. It came out like this, "...Nuh-oh?-oh..." That's right, the question mark was in the middle of the word. I didn't even know that was possible. Try it.
I think at that point I distracted them with the estimated monetary value algorithm. But I kept thinking about it.
I think a teacher was what I was meant to be. But maybe I'm not. Yeah, I love maths, and I love sharing my passion with others, and I love speaking in front of an audience, and I love teenagers, but is teaching the only job that fits with all that? And what about my other passions: dance, fashion, reading, writing blogs?
Well there's probably not a job that encompasses all of those things. But this notion of a "job" is so old fashioned anyway. These days, you don't have to have a "job". In this day and age, there are people out there making money from just about everything. At the weekend I read a couple of books on this subject: Be a Free Range Human and Screw Work Let's Play. These books are all about escaping the 9-5 and making money from something you enjoy. I would not quit teaching at this stage, I'm far too attached to having a roof over my head (call me materialistic), but there are some great ideas for side projects that can be done alongside your regular job. The idea is that when these become profitable you quit your day job.
I became a little bit inspired. What could I do to achieve my dream whilst still teaching full time?
One of the suggestions was blogging. As much as I enjoy writing this blog, I have to accept that it probably won't ever make any money. I know some people do make money from blogs, with advertising and sponsorships etc, but if they say that sex sells, then a blog about teaching maths is going home at the end of the day with a full briefcase and an empty wallet.
Do you feel like you've achieved your dream?
Emma x x x
Thursday, 14 March 2013
Official Outstanding Lesson: Permutations and Combinations
In this post I am going to describe to you the lesson that I taught this week which was graded as Outstanding by an OfSTED inspector.
Obviously I can't guarantee that if you do this lesson you will get a 1, because a lot of it is in the delivery, and in how your students respond. There was a sort of magic spark in my lesson, which I think was there because of the good relationship I have with the group.
My lesson was structured as a series of problems relating to permutations and combinations. I didn't tell my students what perms and combs were until right near the end of the lesson. Not one of my students could have told you the learning objectives, until the very end. But who cares about learning objectives when you're busy doing PROPER MATHS? FYI, the only resources you need for this lesson are a deck of cards and, optionally, some mini whiteboards and pens.
1) The Dramatic Intro
Here is a rough transcript of what I said at the start of the lesson:
"All of you in here are extremely lucky. I am going to show you something no one has ever seen before. *dramatic pause*. This is a once in a lifetime experience, literally. In the whole history of the human race, no one has ever seen this before, and there's a very good chance that no one ever will. I'm not just doing this because we have a special visitor today (nervous glance at inspector) but because I love you all very much [I tend to babble when I get nervous]. Are you ready? *Another dramatic pause*"
Then I took out a deck of cards and shuffled them. Then I fanned them out and showed them all of the cards. Then I just looked at my students, paused, then said, "Amazing, yeah!" In my most excited voice.
There was kind of a confused awkward silence, followed by a chorus of "no"s. I feigned disbelief. Then I explained to them that that particular order of cards has never been seen before by anybody, living or dead, and probably won't be seen again for an extremely long time.
I then explained the maths. I said, let's assume that playing cards have been around for about 500 years, because that's roughly what Wikipedia suggests. Let's assume the population of the world has been a constant 6 billion for the past 500 years. And let's assume that everybody shuffles one deck of 52 cards every second. How many orders of cards would that make? I let them work it out. I gave them mini whiteboards to do their working out on, because pupils are much more likely to start attacking a problem if they're not worried about making mistakes. With MWBs they can erase all evidence of stupidity later. They worked out the answer (I leave that to you as an exercise!) and then I got them to calculate the number of possible orderings (permutations) of 52 cards. Both tables of pupils worked out very quickly that it would involve factorials. Again I will leave the answer to you as an exercise.
[*edited to add* I've just remembered a terrible joke I made. This is pretty cringey, but I'll include it anyway. A student asked the sensible question of what to do about leap years. I said to ignore leap years, and pretend that everyone gets the day off from shuffling cards once every four years. Then I said, "oh no, but that contradicts the song!" So they're all like, "What song?" and I said, "You know, Everyday I'm Shuffling!". Pretty good considering I thought of it on the spot! ]
We had a discussion about the hugeness of the result. The students were pretty amazed, I think. The inspector was too!
2) The iPhone Problem
I moved swiftly on to my next problem. Here is roughly how I introduced the problem:
"You're sitting with your friend, and their iPhone is on the table. They leave it there and get up to go to the toilet. You want access to their phone, because you want to frape them, or whatever [is frape an appropriate word to use during an observed lesson?] but they have a passcode on it. It's four digits, and you happen to know each digit is different, so there are no repeated numbers. The question is, do you have time to try every possible passcode before your friend comes back from the toilet?
The thing I like about this question is that students have to make certain assumptions in order to answer it. Like how long the friend will spend going to the toilet (actually that ends up being kind of irrelevant once you've worked out the answer), and how long it takes you to type in the four numbers. My two tables ended up with different lengths of time that it would take to try all the possibilities, because they made different assumptions.
You can have a little discussion then about phone security and whether a four-digit passcode is enough. That will take care of your PMSSC. I didn't really go into it that much though.
3) The Baby Chris Problem
"Chris' mum gave Chris five wooden alphabet blocks to play with. She gave him the letters A, H, M, S and T. She left the room to make a cup of tea, and when she came back, baby Chris had spelt out the word MATHS. So Chris' mum is going around saying he's a genius. Do you think her claim is valid?"
The students calculated the number of possibilities and decide whether the probability is low enough to claim he's a genius. The class decided it was just a coincidence.
So then I gave them part two:
"This time Chris' mum gives him all 26 alphabet blocks to play with. She leaves the room, and when she comes back, he has taken those same five blocks and spelt out MATHS. Now do you think her claim is valid?"
This is trickier to calculate, and the first thing many of my students wrote down was 26!. I had to question them a bit to steer them in the right direction. The probability is about 1 in 8 million, so we concluded that he probably is a genius.
At that point my inspector left, but I'll tell you the rest of the lesson anyway.
4) An exam style question
Each of the digits 3, 4, 5, 6 and 7 are written on a separate piece of card. Four of the cards are picked at random and then laid down to make a four digit number.
a) How many even numbers are possible?
b) How many odd numbers smaller than 5000 are possible?
I then formalised the idea of permutations slightly by showing them the notation. I didn't give them the formula though, because I find the formula is quite unhelpful. I think it's easier to just think about the question and work it out. Someone said, "Oh! That's what that's for! I always wondered that!" Meaning the key on the calculator.
I only spent a couple of minutes on that, and moved swiftly onto the next problem.
5) The lottery
My students have all memorised by now that the odds of winning the lottery are 1 in 14 million. I suppose I must go on about it a lot! But none of them has ever thought to work out where that number comes from. I think they were quite excited to actually be able to work it out. The key thing I wanted them to notice is that it is a different type of question because in the lottery, the order of the numbers doesn't matter, so it's not the same as finding permutations.
They worked out that they needed to divide by the number of arrangements of 6 numbers. Then it took them a little to work out what that was. The most common guess was 36.
I took their workings out and demonstrated that what they'd done was divide 49P6 by 6!. I said that finding the number of combinations will always be like that. Then I said, "If only there were a key on your calculator which could work it out for you!" At which point they all realised that what they'd just done was work out nCr. They'd used nCr in binomial expansions in core maths but hadn't realised (or forgotten) the connection with combinatorics. They were very relieved that there was a shortcut! But I think it is so important to show them where the formula comes from. I proved the formula using the nPr formula that we'd already derived.
6) The Cricket Team Problem
This was just a question I took from the textbook. A cricket team consists of 6 batsmen, 4 bowlers, and 1 wicket-keeper.
So there you go, my outstanding lesson on perms and combs. I really enjoyed teaching it, and I think my students enjoyed it too.
If you decide to use this lesson, please let me know in the comments below how it went for you.
Outstanding Emma x x x
Obviously I can't guarantee that if you do this lesson you will get a 1, because a lot of it is in the delivery, and in how your students respond. There was a sort of magic spark in my lesson, which I think was there because of the good relationship I have with the group.
My lesson was structured as a series of problems relating to permutations and combinations. I didn't tell my students what perms and combs were until right near the end of the lesson. Not one of my students could have told you the learning objectives, until the very end. But who cares about learning objectives when you're busy doing PROPER MATHS? FYI, the only resources you need for this lesson are a deck of cards and, optionally, some mini whiteboards and pens.
1) The Dramatic Intro
Here is a rough transcript of what I said at the start of the lesson:
"All of you in here are extremely lucky. I am going to show you something no one has ever seen before. *dramatic pause*. This is a once in a lifetime experience, literally. In the whole history of the human race, no one has ever seen this before, and there's a very good chance that no one ever will. I'm not just doing this because we have a special visitor today (nervous glance at inspector) but because I love you all very much [I tend to babble when I get nervous]. Are you ready? *Another dramatic pause*"
Then I took out a deck of cards and shuffled them. Then I fanned them out and showed them all of the cards. Then I just looked at my students, paused, then said, "Amazing, yeah!" In my most excited voice.
There was kind of a confused awkward silence, followed by a chorus of "no"s. I feigned disbelief. Then I explained to them that that particular order of cards has never been seen before by anybody, living or dead, and probably won't be seen again for an extremely long time.
I then explained the maths. I said, let's assume that playing cards have been around for about 500 years, because that's roughly what Wikipedia suggests. Let's assume the population of the world has been a constant 6 billion for the past 500 years. And let's assume that everybody shuffles one deck of 52 cards every second. How many orders of cards would that make? I let them work it out. I gave them mini whiteboards to do their working out on, because pupils are much more likely to start attacking a problem if they're not worried about making mistakes. With MWBs they can erase all evidence of stupidity later. They worked out the answer (I leave that to you as an exercise!) and then I got them to calculate the number of possible orderings (permutations) of 52 cards. Both tables of pupils worked out very quickly that it would involve factorials. Again I will leave the answer to you as an exercise.
[*edited to add* I've just remembered a terrible joke I made. This is pretty cringey, but I'll include it anyway. A student asked the sensible question of what to do about leap years. I said to ignore leap years, and pretend that everyone gets the day off from shuffling cards once every four years. Then I said, "oh no, but that contradicts the song!" So they're all like, "What song?" and I said, "You know, Everyday I'm Shuffling!". Pretty good considering I thought of it on the spot! ]
We had a discussion about the hugeness of the result. The students were pretty amazed, I think. The inspector was too!
2) The iPhone Problem
I moved swiftly on to my next problem. Here is roughly how I introduced the problem:
"You're sitting with your friend, and their iPhone is on the table. They leave it there and get up to go to the toilet. You want access to their phone, because you want to frape them, or whatever [is frape an appropriate word to use during an observed lesson?] but they have a passcode on it. It's four digits, and you happen to know each digit is different, so there are no repeated numbers. The question is, do you have time to try every possible passcode before your friend comes back from the toilet?
The thing I like about this question is that students have to make certain assumptions in order to answer it. Like how long the friend will spend going to the toilet (actually that ends up being kind of irrelevant once you've worked out the answer), and how long it takes you to type in the four numbers. My two tables ended up with different lengths of time that it would take to try all the possibilities, because they made different assumptions.
You can have a little discussion then about phone security and whether a four-digit passcode is enough. That will take care of your PMSSC. I didn't really go into it that much though.
3) The Baby Chris Problem
"Chris' mum gave Chris five wooden alphabet blocks to play with. She gave him the letters A, H, M, S and T. She left the room to make a cup of tea, and when she came back, baby Chris had spelt out the word MATHS. So Chris' mum is going around saying he's a genius. Do you think her claim is valid?"
The students calculated the number of possibilities and decide whether the probability is low enough to claim he's a genius. The class decided it was just a coincidence.
So then I gave them part two:
"This time Chris' mum gives him all 26 alphabet blocks to play with. She leaves the room, and when she comes back, he has taken those same five blocks and spelt out MATHS. Now do you think her claim is valid?"
This is trickier to calculate, and the first thing many of my students wrote down was 26!. I had to question them a bit to steer them in the right direction. The probability is about 1 in 8 million, so we concluded that he probably is a genius.
At that point my inspector left, but I'll tell you the rest of the lesson anyway.
4) An exam style question
Each of the digits 3, 4, 5, 6 and 7 are written on a separate piece of card. Four of the cards are picked at random and then laid down to make a four digit number.
a) How many even numbers are possible?
b) How many odd numbers smaller than 5000 are possible?
I had to help them quite a bit with this question, but it's a nice one. FYI, the answers are 48 and 30 respectively.
I then formalised the idea of permutations slightly by showing them the notation. I didn't give them the formula though, because I find the formula is quite unhelpful. I think it's easier to just think about the question and work it out. Someone said, "Oh! That's what that's for! I always wondered that!" Meaning the key on the calculator.
I only spent a couple of minutes on that, and moved swiftly onto the next problem.
5) The lottery
My students have all memorised by now that the odds of winning the lottery are 1 in 14 million. I suppose I must go on about it a lot! But none of them has ever thought to work out where that number comes from. I think they were quite excited to actually be able to work it out. The key thing I wanted them to notice is that it is a different type of question because in the lottery, the order of the numbers doesn't matter, so it's not the same as finding permutations.
They worked out that they needed to divide by the number of arrangements of 6 numbers. Then it took them a little to work out what that was. The most common guess was 36.
I took their workings out and demonstrated that what they'd done was divide 49P6 by 6!. I said that finding the number of combinations will always be like that. Then I said, "If only there were a key on your calculator which could work it out for you!" At which point they all realised that what they'd just done was work out nCr. They'd used nCr in binomial expansions in core maths but hadn't realised (or forgotten) the connection with combinatorics. They were very relieved that there was a shortcut! But I think it is so important to show them where the formula comes from. I proved the formula using the nPr formula that we'd already derived.
6) The Cricket Team Problem
This was just a question I took from the textbook. A cricket team consists of 6 batsmen, 4 bowlers, and 1 wicket-keeper.
These need to be chosen from a
group of 18 cricketers comprising 9 batsmen, 7 bowlers and 2 wicket-keepers.
How many different ways are there
of choosing the team?
This question is actually really easy using nCr, but it was useful because they could see how to use nCr rather than working it out fully themselves. This was a good question to end on, because they found it quick and easy, which felt like sort of a reward for having derived the formulae earlier.
Then I set the homework. I didn't really feel like I needed a plenary, because we'd been plenarying all the way through.
So there you go, my outstanding lesson on perms and combs. I really enjoyed teaching it, and I think my students enjoyed it too.
If you decide to use this lesson, please let me know in the comments below how it went for you.
Outstanding Emma x x x
Winning OfSTED
So, this week we had a little visit from the inspection fairies. It's funny: we were sort of expecting them last week (just because it was exactly a year since our preliminary inspection), and when they didn't come, we sort of decided they weren't going to come until after Easter. I took my eye off the ball a bit. When the phone call came, it definitely caught me off guard.
Strangely enough, I felt like staff morale actually went up. I think we were all more excited than anything else. The maths department was definitely brought closer. We had all spent some time over the past few months jointly planning series of lessons, and so it had this sense of it being a team effort.
When the results of our individual lesson observations came back, our spirits got higher and higher. We just kept knocking them out of the park! Historically, our department hasn't really done that well in formal lesson observations, but we were killing it this week!
I was observed last out of all the maths teachers, which wasn't very nice, especially as most had been done the previous day! I will blog about my lesson separately, later. But the headline is this: my lesson was Outstanding!
I think on the whole, the staff knew the final result before it was given. But it still felt so good to hear our principal say those words.
Here are my tips for winning OfSTED:
-Carbohydrates. Bring in some muffins for your colleagues. Chances are, people will have been too nervous to eat breakfast, so they will need something to keep them going. And the great thing about muffins is that they're dual-purpose: comfort food for if you do badly, celebratory food for if you do well.
-Club together. Show each other your lesson plans, and offer some tweaks. I think one of the things that made my lesson so good (well, outstanding) was one of the tiny tweaks that my head of department made less than an hour before my lesson.
-Share your excitement, nerves, and success with your students: I sort of briefed my students before their lesson. I went into their previous lesson and said (slightly manically, I'll admit), "ARE YOU EXCITED!?" which I think got them in the right kind of mood. As soon as I found out my result, the first thing I wanted to do was find my students and tell them.
-Come and work in my department. I have such amazing colleagues. They really made this horrific experience not just bearable, but enjoyable. There are just some people who really know how to lift the mood of a group. And luckily for me, my department is full of them!
This whole year, and this term in particular, have been pretty stressful. The pressure was on us for such a long period, that it was really starting to wear some of us down. But it feels like none of that matters now because it all paid off and we (hopefully) won't be having another "O Day" for six years! Now that's a good feeling!
Happy Pi day everyone!
Emma x x x
Saturday, 9 March 2013
When Multiple Wrongs Make a Right: Parrondo's Paradox
If you have a maths degree, I recommend that instead of reading this post, you read this page instead. This is a very interesting article but not very accessible to non-maths grads. I have tried to make my version more simple and easier to understand.
Imagine you are at a pretty dodgy-looking carnival stand and there is a game you can play in the hope of winning some money. The odds of winning are less than half, which means if you play for a little while, you will probably end up broke.
Opposite this stand is another one, just as dodgy-looking, and with the same premise. Again, the odds of winning are less than half, so again, you will end up broke if you keep playing.
Common sense, and your maths teacher, would tell you not to play either of these two games and to go and buy some candy floss instead. But then a mysterious man called Parrondo appears and tells you to play both games. He says that by playing them both in a particular sequence it is likely to earn you a tidy profit.
How can this be possible? Two bad things surely can't make a good thing, can they? These two games are not tied to each other, you can play them independently, so surely one can't affect the other's odds?
Let's take a look.
Let's keep game A nice and simple. Let's say it's a coin toss game, where if it's heads you win a pound, if it's tails you lose a pound. But because it has to be a game you're more likely to lose, let's make it a biased coin. We'll say the probability of winning is, oh, I don't know... 0.495 (I'm pretending to have chosen that randomly, I'm guessing you can tell I didn't!) and hence the probability of losing is 0.505. If you kept playing this game for a while, you will gradually lose your money. You are likely to lose a pound for every 100 games you play. Anyone can see that this is a bad game.
Game B will have to be slightly more complicated, but bear with me. Game B involves tossing a coin as well. But there are two different coins, and the one that you use is determined by the amount of money you have at the time. If your balance is a multiple of three, you flip the bad coin, otherwise you flip the good coin. Both coins are biased, but differently.
The bad coin has a a 0.095 chance of winning £1, a 0.905 chance of losing £1.
The good coin has a 0.745 chance of winning £1, a 0.255 chance of losing £1.
This game is also bad, as the bad coin is soooooo bad. Even though you will only have to use the bad coin a third of the time, the chance of losing is so great that it overrides any wins you get from the good coin. Unfortunately, my knowledge of Markov chains and such like is not good enough to compute the expected win/loss, but running a simulation shows roughly the same results as with game A. Although it's not obvious, take my word for it, this is a bad game.
I can sense that some of you are already putting two and two together here: does playing game A somehow make you less likely to have a balance that is a multiple of three, and hence less likely to use the bad coin on game B? Obviously you can't only do game B when you have a certain balance, as the carnies would see what you are up to and throw you out. The idea is to plan the sequence of games before starting so you don't look suspicious. The best sequence to choose happens to be ABBABBABBABB... Although there are others you could use to win. But how does this work?
I simulated this game 100 times with 102 goes each time. The average (mean) winnings at the end of 102 goes was £9. The highest was £42, and the lowest was -£12. Out of the 102 simulations, only 10 ended in a loss. I think this is enough evidence to suggest that this combination of games is worth playing.
My simulation spreadsheet can be downloaded here.
Let's think about how this works.
Say your balance is a multiple of 3, and you are about to play game A. If you lose, you're more likely to win next time, because you'll be flipping the good coin, and also the one after.
Now say your balance is 1 more than a multiple of 3, and you play game A. If you lose, you will probably lose the next one, but win the one after.
If your balance was 1 less than a multiple of 3, and you play game A, If you lose, you will probably win the next one, and also the one after.
So, without putting probabilities on it, it just sort of looks right that you'd end up winning.
So there we have it, two wrongs can make a right!
Now I'm off to alternately buy lottery tickets and put 10p coins in one of those slidey things.
Emma x x x
Imagine you are at a pretty dodgy-looking carnival stand and there is a game you can play in the hope of winning some money. The odds of winning are less than half, which means if you play for a little while, you will probably end up broke.
Opposite this stand is another one, just as dodgy-looking, and with the same premise. Again, the odds of winning are less than half, so again, you will end up broke if you keep playing.
Common sense, and your maths teacher, would tell you not to play either of these two games and to go and buy some candy floss instead. But then a mysterious man called Parrondo appears and tells you to play both games. He says that by playing them both in a particular sequence it is likely to earn you a tidy profit.
How can this be possible? Two bad things surely can't make a good thing, can they? These two games are not tied to each other, you can play them independently, so surely one can't affect the other's odds?
Let's take a look.
Let's keep game A nice and simple. Let's say it's a coin toss game, where if it's heads you win a pound, if it's tails you lose a pound. But because it has to be a game you're more likely to lose, let's make it a biased coin. We'll say the probability of winning is, oh, I don't know... 0.495 (I'm pretending to have chosen that randomly, I'm guessing you can tell I didn't!) and hence the probability of losing is 0.505. If you kept playing this game for a while, you will gradually lose your money. You are likely to lose a pound for every 100 games you play. Anyone can see that this is a bad game.
Game B will have to be slightly more complicated, but bear with me. Game B involves tossing a coin as well. But there are two different coins, and the one that you use is determined by the amount of money you have at the time. If your balance is a multiple of three, you flip the bad coin, otherwise you flip the good coin. Both coins are biased, but differently.
The bad coin has a a 0.095 chance of winning £1, a 0.905 chance of losing £1.
The good coin has a 0.745 chance of winning £1, a 0.255 chance of losing £1.
This game is also bad, as the bad coin is soooooo bad. Even though you will only have to use the bad coin a third of the time, the chance of losing is so great that it overrides any wins you get from the good coin. Unfortunately, my knowledge of Markov chains and such like is not good enough to compute the expected win/loss, but running a simulation shows roughly the same results as with game A. Although it's not obvious, take my word for it, this is a bad game.
I can sense that some of you are already putting two and two together here: does playing game A somehow make you less likely to have a balance that is a multiple of three, and hence less likely to use the bad coin on game B? Obviously you can't only do game B when you have a certain balance, as the carnies would see what you are up to and throw you out. The idea is to plan the sequence of games before starting so you don't look suspicious. The best sequence to choose happens to be ABBABBABBABB... Although there are others you could use to win. But how does this work?
I simulated this game 100 times with 102 goes each time. The average (mean) winnings at the end of 102 goes was £9. The highest was £42, and the lowest was -£12. Out of the 102 simulations, only 10 ended in a loss. I think this is enough evidence to suggest that this combination of games is worth playing.
My simulation spreadsheet can be downloaded here.
Let's think about how this works.
Say your balance is a multiple of 3, and you are about to play game A. If you lose, you're more likely to win next time, because you'll be flipping the good coin, and also the one after.
Now say your balance is 1 more than a multiple of 3, and you play game A. If you lose, you will probably lose the next one, but win the one after.
If your balance was 1 less than a multiple of 3, and you play game A, If you lose, you will probably win the next one, and also the one after.
So, without putting probabilities on it, it just sort of looks right that you'd end up winning.
So there we have it, two wrongs can make a right!
Now I'm off to alternately buy lottery tickets and put 10p coins in one of those slidey things.
Emma x x x
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