Thursday 14 May 2015

A Maths Game with a Psychological Twist

Today I'm going to tell you about one of my favourite ways to kill a spare fifteen minutes in a maths lesson. It's great because it requires absolutely no preparation, no prior knowledge, and no differentiation. All you need is a class set of mini whiteboards and pens.

Tell students that you're going to ask them to write down a number. It must be a whole number (no decimals, fractions or surds) and it must be zero or greater. They have to write this secretly and not show anyone their number. Tell them you are going to count down from 5. At 2, everyone's pens need to be down. At 1, everyone needs to show their boards. The winner, and this is the important bit, is the person who has the smallest number that no one else has.

This should be simple enough to understand, and if a student doesn't quite get it, they can at least write down a number and participate until it clicks.

When students hold up their boards, first look for zeros. If there's just one, the zero wins. If there's more than one, tell all the zeros to put their boards down and then look for ones. Repeat until you have found a number that only one person has written. They are the winner. This doesn't take very long at all, even with 32 students like my year 7 class.

I have found that this game intrigues students immediately. They have to try to predict what everyone else is thinking. I have found that I can usually predict who in the class will write zero. Zero rarely wins, and I've never seen it win in the first round. I've found that the winning numbers vary massively from class to class. The results are also very different when I've played this with adults. I'm sure there's a lot that can be explored here in terms of psychology (and economics, actually) but I'll leave that for you to think about.

I like to try to encourage strategic thinking and annoying my students at the same time by saying things like "oh, so three won that time. That means you are probably thinking about choosing three this time. But if other people think that too, you won't win, so you'd better choose four instead. But now that I've said that, you can't choose four, it's too obvious, so..." The students usually interrupt me at this point and beg me to shut up because I'm ruining their strategy.

The beauty of this game is it can be done with year 7s, year 13s, and even the Maths faculty as a pre-meeting warm-up. I have never met a class (or group of adults even) that doesn't enjoy this game.

A more mathematical version of this game is to have the same rules but this time the winner is the student whose number is the closest to the mean of all of the numbers. This is too difficult to do as a whole class, so I do this in table groups. The four students each choose a number, they calculate the mean, then the winner is whoever's closest. They keep a tally of how many times each person wins so they can declare a winner at the end of the fifteen minutes.

What's really nice about this variation is that students will be calculating means with much more enthusiasm and motivation than if they were meaningless numbers on a worksheet. Also, by playing this game and trying out different strategies, students begin to appreciate the nature of the mean. There will always be a student who will write down a million, thinking it will skew the mean towards them. However, if the other three numbers are low they still won't win.

You can also play this game with the median instead of the mean. It doesn't quite work with mode! Actually, maybe one of my amazing readers could come up with a way of making a mode variation. Comment below!

Try one of these games out when you have ten minutes to spare. Let me know how it goes by commenting below.

Emma x x x

Thursday 7 May 2015

The Mathematics of Voting

I'm going to do something completely out of character and write a TOPICAL post! Yes, I may live underneath a proverbial rock (what, there was an earthquake recently?) but even I am aware that there is a General Election today in the UK. In fact, I was the first person at my polling station to vote this morning!

I've mentioned in a previous post that I love Lewis Carroll. He combines two of my favourite things: books and maths (Lewis Carroll is the pen name of mathematician Charles Dodgson. Please keep up!)

Dodgson became involved in college elections in the early 1870s at Oxford university where he was a professor. He became interested in the theory of voting, of the accuracy and fairness of different voting systems.

First Past The Post

Dodgson was not a fan of this voting system. He claimed "the extraordinary injustice of this Method may be very easily demonstrated". He then gives an example to show how stupid it is:
Suppose there are 11 electors and 4 candidates a, b, c and d. Each elector ranks the four candidates in order of preference. The 11 columns here show their choices:

 a a a b b b b c c c d c c c a a a a a a a a d d d c c c c d d d c b b b d d d d b b b b

It's easy to see that a is considered best by three of the electors and second best by the rest. But in actual fact, it is b who ends up winning, even though he/she was considered the worst by seven voters.

I don't think Dodgson looked at "Alternative Vote", although he did write about lots of other systems.

The Method of Elimination

In this method, each voter chooses their favourite, and then the one who gets the fewest votes is eliminated, and the process is repeated (a bit like Big Brother? The TV show, not the Orwellian thing). This method at first seems pretty flawless. However, consider the following situation:

 b b b c c c d d d a a a a a a a a a a a b c d c d b b b c c b d d c d c d d d b b c c b

Notice that a is everybody's first or second choice, and hence appears to be the best candidate. However, he/she will be eliminated first. c will be elected instead.

The Method of Marks

In this method, each voter is given a specified number of marks that they can divide between the candidates. Then the candidate who gets the most marks wins. Dodgson said that this method would be perfect as long as the voters divided their marks fairly: giving most to their favourite but some to the candidates that they wouldn't mind electing. But Dodgson commented that "since we are not sufficiently unselfish and would assign all our votes to our favourite candidate, the method is liable in practice to conicide with that of the simple majority [first past the post] which has already been shown to be unsound".

I hope you voted today! Let's get rid of the current education-ruining idiots!

Emma x x x

All quotes are from Robin Wilson's "Lewis Carroll in Numberland", a book I highly recommend.

Tuesday 28 April 2015

Are Our Students Thinking Too Much?

"It is a profoundly erroneous truism that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of operations we can perform without thinking about them."
-Alfred North Whitehead, An Introduction to Mathematics

How many times in your life as a maths teacher have you said to your students, "think about what you're doing!"?

And how many times in your life as a maths teacher have you thought to yourself, "how do you still need to use your fingers to know 4 times 5 is 20?"

I know that I have these two conflicting belief systems. I believe that knowing times table facts, number bonds, the values of sine, cosine and tangent for common angles, the quadratic formula, and probably loads more things are very very useful. I would go so far as to say important. Maybe even so far as to say, your exam success depends on it.

But at the same time, I believe that rather than relying on memory, we should have a solid understanding of maths and think about what we are doing and why at each stage of a process. This is why I am conflicted.

A good example is long division. Students in UK Primary schools are generally taught chunking as their primary method of long division. On the one hand, I'm in favour of this. Chunking is my preferred mental method, and I have even started doing it as my main written method. However, students (especially those who do not know their times tables very well) are slow with chunking. It also takes up a lot of space. In secondary schools, I think most teachers teach the traditional bus stop method of long division. Students find this difficult at first but after much practice they can use it efficiently and accurately. But when they are used to this method, they do it without thinking about it. Is this a good thing or a bad thing?

Does it make sense to "save your brain" for the more challenging bits and rely on your subconscious to do the easy bits? Or should we be mindful at all times?

Should we memorise the quadratic formula or derive it by completing the square every time we want to use it? Should we memorise that cos(30) is root three over two, or draw a 30/60/90 triangle every time? Does it free up space in our memory if we don't memorise these? Do we free up space in our working memory or unburdon our central executive if we do memorise them? Which is more beneficial?

I think that in the UK, we place much much less emphasis on arithmetic fluency compared to other countries. I'm frequently told by friends and colleagues who went to school in Asia or Europe that you simply did not move on to the next grade or year group until you had mastered addition, subtraction, multiplication and division. And mastered means you can do these things efficiently and accurately. Over here, we move on to algebra before students know their number bonds. We teach fractions and percentages before they know their times tables. We say, it's OK if you're no good at arithmetic, you might be really good at shape and space. Is it the lack of arithmetic fluency that is holding UK students back? When it comes to maths, are our students thinking too much?

[Is it just me, or is that last sentence totally Carrie Bradshaw-esque?]

The GCSE for maths now has a much larger emphasis on functional problem solving. Teachers say this will be a challenge because our students are bad at problem solving. I think they're wrong. I think our students are brilliant at problem solving. But they do badly on these questions because the arithmetic involved in these problems poses a much bigger problem to our students than, say, Korean students. If you know how to do arithmetic automatically, without thinking, you only have the problem to think about. If you also have to think long and hard about the arithmetic, it interrupts your thinking process and tackling the problem becomes much harder. It also means you won't as easily spot neat tricks or patterns that simplify the problem.

For example, there is a Foundation GCSE question with a rectangular patio 3m by 3.6m. Henry has 32 square paving slabs with side length 60cm. Does he have enough slabs to cover his patio? If you are good at arithmetic you may spot that 60cm goes into 3m and 3.6m nicely. If you don't know your six times table, you won't notice this. That means you may choose a longer, more complicated method, like finding the area of the patio and the area of the slab and doing a division. This was a non-calculator paper so now students will be having to calculate 3 x 3.6 and 0.6 x 0.6, then use long division. It is much, much simpler to see that you need 5 rows of 6 slabs, which makes 30.

So although I'm a big fan of exploring, discovering, reasoning, and understanding, I'm starting to think that a bit of rote learning of efficient, accurate methods, and drilling of times tables (and their related quotients) would actually improve students' ability to solve problems a lot more than simply getting students to practise problem solving.

What do you think?

Emma x x x

Wednesday 22 April 2015

Show Me Your Knowledge - An Alternative to Tests

I had become fed up of marking my year 10s' fortnightly assessments. They are aiming for a grade C, and following the Higher syllabus, which means the assessments are quite challenging for them. Marking takes a lot longer when the student has got almost everything wrong.

One day I decided to do something different. I offered my class the choice. They could either do the assessment as normal, or they could Show Me Their Knowledge (Patent Pending). I explained what I meant by this. I wanted them to pour out the contents of their brains on that particular topic (it was averages) onto an A3 sheet of paper. I told them they needed to make up their own examples to demonstrate all of the skills. I gave them a list of skills so they wouldn't forget anything: mean, mode and median from a list, mmm from a frequency table, mmm from a grouped frequency table.

What was quite interesting was all of the boys chose to take the normal test, whereas all of the girls chose to Show Me Their Knowledge. And the girls loved it, they were begging me to do this every assessment. This gender difference is worth exploring in more detail, so I will save that for a separate post.

The pages that were created were impressive. Most girls did an example for each of the three sections, and heavily annotated the solution with explanations. For example, "You find the midpoint here because you don't know the actual number" then an arrow from this led to, "To get the midpoint add them together and half it or just work it out by looking at it".

Some girls put up their hands at various points and said they didn't know how to explain a certain bit. I told them that means they maybe didn't understand that bit. I told them to leave a bubble or box where they wanted to write the explanation, so that they could see there was a gap in their understanding. I told them I would help them fill that gap in the follow-up lesson.

When it came to marking, I read through the explanations and tweaked some of the wording if it wasn't quite right. I did this very neatly because girls don't like you defacing their work (male teachers please take note of this). If I thought some bits needed further explanations, I drew bubbles or boxes (copying their style) but left them blank, or started the sentence for them. For example, I might have written, "If there are two numbers in the middle..." or "The question will usually tell you to r____ your answer...".

When I gave these back, they filled in the bits they could by themselves, then I talked to them on each table to explain any bits they didn't understand. They also talked to each other to fill in the gaps. By the end, they all had a poster explaining the whole chapter. That is now a very useful resource for their revision. The best looking one I could have copied and had laminated for future classes to use.

I will continue to use this method of assessment, although I will still do some normal tests too, as at the end of the day, they will have to sit a traditional exam, and they need to be prepared for this. I can't help but think the fact that girls prefer this method suggests that maybe they are disadvantaged by our exam system and that girls could do much better if we changed this. But again, that's a post for another day.

Emma x x x

Sunday 19 April 2015

Run a Reverse Pilot

It's the start of a brand-new term, and lots of us teachers will be returning to work with more energy and a renewed sense of purpose. It is when we are in this frame of mind that we are eager to read about new teaching techniques and "tweaks" and we are thinking about adding a few new things to our lessons.

Well, here's a new idea for you. Instead of starting to do something new this half term, do the opposite. Stop doing something old.

You will have all heard of the idea of a "pilot", where you trial something on a small scale to see if it will add value and if it does, you run with it.

Daniel Shapero, a director at LinkedIn, coined the term "reverse pilot" to refer to the exact opposite: removing an activity or initiative to see whether there will be any negative impact.

Is there anything you are currently doing that is taking up a lot of your time but you suspect is not actually adding any value? For example, maybe you have developed an elaborate points system with prizes and a "star of the week" award. This might have been effective at the start of the year, but maybe it is no longer doing anything. Try getting rid of it for a trial period, and see if it has a negative impact. If it doesn't, you have just saved yourself the time it used to take you.

Does your faculty have a weekly lunchtime meeting (sorry, not "meeting", calling it that would make the unions unhappy, so let's call it a "gathering") that takes up 15 people-hours and accomplishes very little? Try scrapping it, and see if anything bad happens.

Do you make sure to write "next steps" or similar after marking your students' books? Stop writing these, save yourself 3 minutes per book (an hour and a half per class per fortnight) and see if your students' progress starts to decelerate.

Are you currently running a weekly after-school revision class for your exam class? Have you done this every year since you can remember? Don't do it this year. See if the results are worse than normal. If they're not, then you've just saved yourself an hour a week for the rest of your life. You are welcome.

Of course, it could be that everything you do at the moment does make a positive impact on learning. But we shouldn't just assume that they do. To do so would be wasting precious resources. I flippantly said in the above paragraph that you could stop doing that revision class to save yourself time. Well think about redirecting that time towards something else - better quality feedback, or one-to-one tuition. These might have a bigger overall impact. That lunchtime meeting (sorry, gathering) could be spent producing the world's best corridor display. Or it could be spent, I don't know, eating?!

There are several cognitive explanations for why we are probably wasting a lot of time on activities that might be adding little or no value. These are loss-aversion, the sunk-cost fallacy, and the status-quo bias.

It has been shown that human beings are naturally loss-averse. We think that losing something will have a bigger negative impact on us than gaining something of the same value will have a positive impact. So it seems to you that stopping sending out your weekly teaching and learning newsletter (which takes three hours to write) would have a big negative effect, but gaining three hours of extra time per week would have a smaller positive effect, not enough to balance it out. When you write it down like this, it might seem that the choice is obvious, but most of us don't write it out like this, and hence we don't question whether carrying on doing the things we're doing is the best use of resources.

You spent hours setting up your praise and reward spreadsheet, with built-in mail merge facility and automatic colour coding. So spending an hour a week filling it in for all of your classes is definitely worth it, right? After all, you spent so long on building it, it would be a waste not to use it, right? This is the sunk-cost fallacy in action. The time you spent making the spreadsheet is just that, spent. You cannot unspend it. Therefore whether you use it or not makes absolutely no difference. Do you still have a dress in your wardrobe you've never worn, but you can't get rid of because it cost a bomb? Whether you keep it or not, you have already wasted the money. Let it go. Spent forty minutes waiting for a bus so there's no way you're paying for a taxi now? That forty minutes has been lost either way, do what you want. The only time the sunk-cost fallacy is your friend is with your gym membership. "I've paid all that money, I really have to use it!" this is a fallacy, but a healthy one. Don't fight it.

The "status quo bias" is where we have a tendency to keep doing something simply because we have always done it. Why do we write the learning objective on the board at the start of the lesson? Because we've always done it (and in my case, because my own teachers did it). Why do we begin each lesson with a starter activity? You can probably think of lots of examples.

I challenge you this half term to stop doing one thing. I suggest at first you don't replace it with anything else, just stop doing it. Feel the relief of having one fewer thing to do. If you observe no impact, then keep not doing it. If you observe a negative impact, instead of going straight back to doing the same as before, tackle the problem from scratch. You might find there is a better solution.

To read more about the cognitive biases I have mentioned in this post, I recommend The Art of Thinking Clearly by Rolf Dobelli.

To read more about the idea of doing less to achieve better results, I recommend Essentialism: the Disciplined Pursuit of Less by Greg Mckeown.

Emma

x x x

And by the way the weekly lunchtime meeting referred to in this post is a work of fiction and bears no resemblance to any real weekly lunchtime meeting, living or dead. Any resemblance to the weekly lunchtime meeting my own faculty has on a Thursday is purely coincidental.

Thursday 16 April 2015

The Chaos Game

Draw an equilateral triangle on a bit of paper. Draw it nice and big. Now pick a corner to start at. Next, get a die (a D-6) and assign two numbers to each corner. For example, the top corner can be 1 and 2, the left corner can be 3 and 4, and the right corner can be 5 and 6. Write these numbers near the corners on the outside of your triangle so you remember them.

Roll your die. The number you get tells you which corner you are heading towards. Find the midpoint of your current location and the corner you're heading towards. Use a ruler for this and try to be as accurate as you can. Mark this new point with a good-sized dot, that is your new location.

Repeat.

After about fifteen minutes your triangle should have lots of lovely dots, and at this stage you might even see a pattern emerging.

A pattern! (I hear you cry) How can there be a pattern, when I am moving randomly! Surely the dots will be scattered in a random manner, looking like the freckles on a pasty Irish face. But there is indeed a pattern. A very nice one in fact. A very familiar one, actually.

I will now insert some line breaks to stop you seeing the pictures below. Don't scroll until you're ready.

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I didn't rub out my construction lines because I've been taught well :-)

Can you tell what it is yet?

Holy fractal, Batman! That there is Sierpinski's triangle! Or as my students would say: Illuminati confirmed.

OK, it doesn't look exactly like Sierpinski but it's definitely getting there. I need to do another thousand or so iterations.

I am a mathematician, which basically means I have turned being lazy into a career. So at this point I started thinking, why am I drawing this s*** when I could be running a simulation instead?

Here is the spreadsheet I used to simulate the Chaos game. It was kind of fun to set up, so I suggest you try it yourself before reading my formulae. You can probably find a much more elegant way to do it, but I'm a mathematician, dammit, not a programmer!

Teachers: this is a cool way to kill an hour with a group of students who know how to use a ruler and divide stuff by two. You could do it as I have suggested, with a triangle drawn on blank paper, but you could instead do it with a triangle drawn on a coordinate grid, with the corners having the coordinates I used in my spreadsheet. This way, students can practise finding the mid-points of two points (a skill that is needed for Higher tier GCSE and for AS Level). However, the numbers get nasty pretty quickly (as there is a root 3 involved). Even if you just use this as an exercise in measuring with a ruler, I think students can get a lot out of it. The pattern is so cool and unexpected, your students may even have that "wow" awe and wonder moment.

Have fun!

Emma x x x

Wednesday 15 April 2015

Cheryl's Birthday

Imagine my delight when I found out last night that loads of people on the internet were talking about a maths puzzle! Some people are even saying #Cheryl'sBirthday was the new #TheDress. Not being a Twitter user (is it OK that I just don't get it?) I arrived a bit late to this party but I hope you don't mind.

So, this puzzle comes from a Singapore Maths competition which I think is probably comparable to the UKMT Intermediate Maths Challenge follow-up round (the Pink Kangaroo) which means it is supposed to be challenging. Can I just take this opportunity to brag and say I got the answer right in about three minutes? Please, if you haven't already, pause and work out the answer for yourself.

There are lots of explanations of the solution out there on the internet, but I'm a maths teacher and I can't help myself, I love explaining stuff!

Statement one:

Albert knows the month. This means he can't know when the birthday is, as there are no months with just one possible date.

However, Albert has deduced that Bernard cannot know either. The only way Bernard would be able to know is if the number he was told was 19th or 18th because they are the only dates with one possible month. So for Albert to know that Bernard does not know, these two options must not be possible. Albert only knows the month, so for him to know these are not possible, the months of these must not be correct. Therefore it is not May or June.

Statement two:

Now that Albert has said that, Bernard has deduced that it is not May or June, just like we have. This information is enough for him to know the correct date. This means it can't be the 14th because there are two months with the 14th. So it must be 16th July, 15th August, or 17th August. .Bernard knows which one of these it is because he knows the number. We do not know.

Statement 3:

Albert has deduced the same as us, and narrowed it down to those three dates. But Albert knows the month, and by knowing this, he knows the answer. So it must be July, as if it was August he still wouldn't know.

Therefore the answer is 16th July.

What a wonderful question!

This question is similar to those questions where nobody knows anything but by saying "I don't know" enough times everyone suddenly knows everything. Do you know the kind of question I'm talking about? My favourite is probably the one with the island full of brown eyed and blue eyed people: The Blue Eyes Logic Puzzle. This is so difficult to wrap your head round it but once you do, you feel like you have understood the secrets of the universe and your brain suddenly enters this state of ultimate clarity. Unfortunately this only lasts for a few minutes and then you stop understanding it again. I have read about this puzzle so many times now I can hold onto this state for a whole evening. I always wake up ignorant again.

Maybe I will do a full post on the Island puzzle one day when I'm feeling brave and I have a large supply of stimulants to hand.

Let me leave you with a link to my favourite place to find logic puzzles. These are great for teaching the Logic chapter in D2. They are also nice and short so you can do one at the beginning of every maths faculty meeting just to get everyone's brains warmed up.

And also, a maths joke: Three logicians walk into a bar. The barman asks “does everyone want a drink?” The first logician says, “I don’t know”. The second logician says, “I don’t know”. The third logician says, “Yes”.

Ha ha ha ha ha ha!

Emma x x x

Tuesday 24 March 2015

Motivating Students Without Rewards

"The reward of a good thing well done is to have done it" -Ralph Waldo Emerson

"The reward for solving a maths problem is to have solved it" -Me

I have been reading a book about habit formation: Better Than Before by Gretchen Rubin (author of one of my favourite books ever, The Happiness Project). It has nothing to do with teaching, but one part of it talks about rewards. Specifically, about rewarding yourself for doing something in order to reinforce the behaviour and help with habit formation. e.g. every time you go for a run, reward yourself by buying a magazine. This might sound like a good strategy, it is certainly a popular one. I myself have done it many times: I reward myself for exercising by letting myself take the bus to work the next day. I reward myself for losing weight by eating fatty foods. I reward myself for not shopping by buying new shoes. I think maybe I'm doing something wrong.

But even if the rewards you give yourself are sensible (for example, every time you remember to make your bed in the morning for a whole week in a row, you get a manicure), apparently this would not help reinforce the habit, but actually obstruct habit formation. This got me thinking about using rewards as a teacher. Do all twenty questions and I'll give you a positive point. Do this difficult question and I'll tell your house head how good you are. Do extra homework and I'll send a postcard home. Are these rewards effective at encouraging good maths work, or are they obstructing the habit formation?

Receiving a reward for doing something tells you that the activity is not worth doing for its own sake, and hence you start to associate the activity with inconvenience, boredom or suffering. One study by Lepper, Greene and Nisbett (1973) found that children who got a reward for colouring in a picture, later on didn't spend as much time on this activity as children who didn't expect a reward. The children began to think, "it's not worth doing if I'm not going to get rewarded", even though colouring in is an activity most children love. In addition, the drawings produced by the group who were rewarded were of worse quality.

By offering students a reward if they get all ten questions done, we're telling them that doing the questions is not a pleasant task, and that you wouldn't want to do it if there was no reward. Wouldn't it be better if students felt like the reward for doing all ten questions was the satisfaction of having all ten answers?

I have spoken before on this blog about intrinsic and extrinsic motivation. Extrinsic motivation is doing something to get an external reward or avoid an external punishment. Intrinsic motivation is doing something for its own sake. If you are intrinsically motivated to do something, you are more likely to keep doing it, and find it satisfying.

Thomas Malone and Mark Lepper (1987) identified seven sources of intrinsic motivation:
Challenge: we enjoy pursuing a goal that is difficult but not impossible.
Curiosity: we enjoy learning new things.
Control: we like feeling like we've mastered something.
Fantasy: we like using our imagination to make an activity more fun.
Cooperation: we enjoy working with others.
Competition: we feel good when we think we are doing better than others around us.
Recognition: we like it when others recognise our achievements.

Instead of thinking about rewards for doing good maths work, think about how you can motivate your students by providing opportunities for the above to take place. Set a difficult puzzle (that doesn't seem impossible), pique their curiosity, give them something they can do well so they experience the feeling of mastery. Invent a fantasy situation, have them work in pairs or groups, introduce an element of competition. And if you feel like you must give them a reward, don't give them a house point, just let them know you have recognised their achievement.

What are your thoughts on using rewards to motivate students?

Emma x x x

Alfie Kohn - Punished by Rewards
Daniel Pink - Drive

Thursday 19 March 2015

Why is 0! (zero factorial) equal to 1?

This post was originally written at the end of 2013.

Today I had a very typical Further Maths A Level lesson. Someone asked a very simple question, I started to answer it, and ten minutes later we were talking about how many imaginary sheep there were in the classroom.

Like I said, a typical lesson.

The question that I was asked was, "Why is zero factorial one?". This was asked by a female student I will refer to as H (to protect her identity- she probably doesn't want to be associated with this nerdy conversation). I'm not entirely sure why H asked me this, as she was supposed to be working on the Secant Method (otherwise known as the most painful mathematical process of all time).

But anyway, she asked me this, and my immediate answer was that very useful mathematical phrase: "by convention".

She responded with, "What do you mean?" to which another student, who I will call J, replied, "To make everyone happy", which pretty much sums it up. Zero factorial was defined as one to make everyone happy. What a lovely answer!

But I couldn't just leave it there, could I? Oh no, my geek sense was tingling. I tried to get my head back to where it should be (doing the register) but I just couldn't. Before I knew it a board pen had somehow leapt into my hand and I was on my feet.

Let's take a look at the factorial function.
3! = 3 x 2 x 1 = 6
2! = 2 x 1 = 2
1! = 1
0! =

Notice the deliberately blank space next to 0! =. Because that's what the answer is. A blank space.
"Three factorial is three times two times one".
"Two factorial is two times one".
"One factorial is one".
"Zero factorial is ...[silence]".

So the question is, what number is "...[silence]"? By the way, when you say "...[silence]" you should accompany this with a hand movement kind of like "ta-da!" but less dramatic. I might post a video up here later so you can see what I mean.

Sorry, I was saying, what number is "...[silence]" *hand movement* ?

Well in my opinion, it's one. To me, it's obviously one. It's not zero. Zero has too much meaning. Zero is a very definite nothing. Zero is a dangerous number - it can ruin all kinds of calculations. I think the "blank" number is one.

Here's a reason why:

What's 3x - 2x?
What's the coefficient of x?
But where's the 1?

The blank space in front of the x means one.

Another example:

Say you had some algebraic fractions to simplify by cancelling common factors. Look at the first two examples. Using similar logic, surely the answer to c) is a blank space? But we know the answer is 1.

It is easy to see why 0! has to be 1 when we look at combinatorics. 5C0 ("5 choose 0") means how many ways are there of choosing zero items from a choice of 5. The answer to this is one. Why? Well if you have to pick zero items, how many ways are there to do this? Well the only way to do it is to not do it, which is one way, so the answer is one. The formula for the nCr (choose) function involves factorials, and the only way for nC0 to equal one is if 0! = 1. So 0! has to be 1, or the formula won't work.

In other words, it's 1 to keep everyone happy. I should have just listened to J.

And in case you're wondering, I never did get round to doing my register.

PS No I will not explain the imaginary sheep thing. You had to be there.

Thursday 12 March 2015

How Do You Round a Negative to the Nearest Whole Number?

How should you round -1.5 to the nearest whole number?

Almost anyone you ask this to will reply without thinking: -2, because 5 rounds up.

Spot the mistake!

Rounding -1.5 to -2 is not, in fact, rounding up, it is rounding down, because -2 < -1.5.

Of course, that doesn't mean rounding to -2 is necessarily wrong, but it does disobey the general rule that "5 rounds up". But this rule of thumb that we maths teachers use, have we actually thought it through?

For now, let's just consider positive numbers, and the reason we round things that have .5 up. Numbers whose decimal bit starts with .5 and then has loads of numbers after it, e.g. 3.532423765 would obviously round up, as they are more than half way between the two whole numbers. By deciding that 3.5 would also round up, it means if you are scanning a massive set of data, you only have to look at the first number after the decimal to know whether it will round up or down.

However, surely if we always round things ending in 5 up, we are creating an imbalance somewhere? This might seem minor, but if you consider all of the millions and billions of transactions that take place in, for example, bureaux de change, where currency is changed, this will add up to a lot of money that someone will be unfairly losing (or gaining).

I myself run into this problem every month when my husband and I sit down to pay off our shared credit card. Our Google Sheet halves the cost of all of our shared purchases and totals up how much we each have to pay. And when we share a purchase that is an odd number of pence, we run into a little problem. Google dutifully rounds our individual costs up, but then we would overpay our credit card by a penny for every such transaction. My husband, having the amazing qualities of both a mathematician and a computer scientist, fixed this so that one of the values rounds up and one rounds down. And my husband, also having the tight-fisted qualities of a Scotsman, fixed it so his costs always round down, and mine round up.

So always rounding .5 up (officially known as "Round Half Up") can be a bit of a problem. The whole ends up being less than the sum of the rounded parts. Maths teachers also know that this is incredibly annoying when it comes to pie charts and stratified sampling. You know exactly what I'm talking about.

There are some ways of fixing the unfairness of Round Half Up . A lot of these methods are actually used without you even being aware of them. I bet you didn't even know that the method you usually use has a name. I'd bet even more that you aren't aware their are eight types of commonly used rounding methods.

The Eight Main Rounding Methods

Round Half Up

When it's a 5 you round up. So 4.65 rounds to 4.7 and -2.5 would round to -2. This is known as "asymmetric rounding" because it is positively biased - that is, we round up slightly more often that we round down.

Round Half Down

When it's a 5 you round down. So 4.65 rounds to 4.6 and -2.5 rounds to -3. This is hardly ever used. This is also known (confusingly) as "asymmetric rounding".

Round Half Away From Zero

When it's a 5 you round away from zero. So 4.65 rounds to 4.7, and -2.5 rounds to -3. This is probably what most normal people probably assumes happens. This method is symmetric because half the time 5 rounds up and half the time 5 rounds down. However, this is only fair if positive and negative numbers are equally likely. There are some situations that deal only with positive numbers, and then the method would still be biased.

Round Half To Even

When it's a 5 you round towards an even number. So 3.5 rounds up to 4 but 6.5 rounds down to 6. -2.5 rounds to -2, -3.5 rounds to -4. This method of rounding should be unbiased because even and odd numbers are equally likely, right? But zero is even, so aren't there sort of more even numbers than odd? That's a debate for another post. This method of rounding is probably the most commonly used, as it is the default method used in IEEE 754 computing functions and operators.

Round Half To Odd

This should be obvious, having read the previous paragraph. This method, however, is almost never used.

Stochastic Rounding

When it's a 5, flip a coin, and use that to decide if it rounds up or down.This should be unbiased, as it really would be a 50/50 chance. However, if you let your students use this method in their maths homework, you would have thirty students with completely different sets of answers. Whilst the unbiasedness of this method appeals to me, the fact that you would get different answers every time would just be annoying. Some of my students (many of my year 11s) actually do apply this method of rounding, but without a coin. It's otherwise known as guessing. They have a 50% chance of being right, which is good enough for me.

Round Half Alternatingly

The first time you have a 5, you round up. The second time, round it down. So if you had this list of numbers: 3.5, 6.5, 2.5, -1.5, you would round these to: 4, 6, 3, -2. This will be free of bias as long as you have an even number of data that end in a 5.

So there you have it. Eight different rounding methods, six of which are commonly used (although some are more common than others). And many people (including many maths teachers) have absolutely no idea our money, our personal data, and the data we are presented with in newspapers, have been subjected to these methods. We could be missing out on half pennies all over the place!

Another method worth mentioning is Supermarket Rounding, which is where if something is half price, they always round the price up. So 99p becomes 50p when half price. Interestingly, when the same supermarket advertises 50% off, 99p still becomes 50p, even though the 50% that is taken off should be the bit that is rounded. Hey, these half pennies add up you know!

So back to my original question, how do you round -1.5 to the nearest whole number? The answer is either:
Round Half Up: -1
Round Half Down: -2
Round Half Away From Zero: -2
Round Half To Even: -2
Round Half To Odd: -1
Stochastic Rounding: *flips coin* -1
Round Half Alternatingly: -1
Supermarket Rounding: N/A

Simple.

Emma x x x

Saturday 7 February 2015

You Know You're a Maths Teacher When...

You never give a direct answer to a question, but instead reply with another question, so the asker can figure it out for themselves. I have a lot of trouble with this one to be honest. My husband has complained on several occasions. For example, the other day he asked me where the bin bags are. Even though I knew where they were and could have just told him, I automatically responded with, "Well, when did we buy them?" He didn't really appreciate that I was giving him the opportunity to work out the answer for himself.

You find a protractor in your wallet (this happened to a colleague of mine a couple of weeks ago).

You feel like scrawling across Google Calculator "You must show your working out!"

You sit in whole-staff meetings and when the presenter mentions the word "average" you immediately turn to your colleague and snidely point out "They didn't say which average, so that statement is meaningless".

All of your shopping lists, to do lists, Christmas card lists, etc are written on squared paper. In green pen.

You can touch-type on a Casio calculator.

You carry a mini screwdriver in your handbag for emergency compass-tightening. (Please tell me I'm not the only one?)

You start to type "body" in a text message and "BODMAS" comes up as a spelling suggestion.(Seriously, who was I texting about BODMAS?)

Can you think of any more?

Emma x x x

What Does the O Stand for in BODMAS?

I recently sat down to plan a lesson for year 7 students about the order of mathematical operations. Here in the UK, I believe this is most commonly known as"BODMAS". The B stands for brackets, D for division, M for multiplication, A for addition and S for subtraction.

But what does the O stand for?

I've heard several different answers to the above question, none of them satisfactory. One of my colleagues told me he taught it as "Orders". This random website I found agrees. But what the heck are orders? According to the aforementioned random website, they're "numbers involving powers or square roots". I have never heard this definition before, and after consulting the oracle (Wikipedia) I found no mention of indices or powers on the page for Order (mathematics). So why on earth would we teach students the word orders when we never call them that in lessons? We in the UK usually refer to these as "indices" although I believe the Americans prefer "exponents" (but I'll get to them later).

Another colleague told me he teaches that the O stands for "of" as in "powers of", and I'm ashamed to admit this was what I was taught in school. I think this one is faintly ridiculous. Firstly, O cannot stand for "Powers of" because "Powers of" clearly begins with a P not an O. Kids may be getting dumber every generation, but I have a feeling they will notice this. Also, why does the word "powers" even need an "of"? Can we not just call them powers? It reminds me a bit of learning French when we were always taught to write the following preposition after certain words like "decider de" or "je pense que" to help you form sentences. This was actually excellent advice for learning French, but this does not dilute my point.

A third colleague (it is amazing how many of them are willing to contribute to my inane Monday-morning conversations) said that he teaches that the O stands for "Other" as in, any other operations not mentioned. This is quite nice actually, because it includes not just powers and roots but also sines, logs, factorials, etc. Very handy.

I then went into my year 7 lesson and asked them what they thought the O stood for. Interestingly, the most common response was one I had not heard yet: "operations". This is perhaps the one that annoys me the most. BODMAS is the tool we use to remember in which order we should do operations. If O stands for "operations", then we are basically saying, do the bit in the brackets first, then do the operations. Oh wait, what order do I do the operations in? Use BODMAS. So I do the brackets and then the operations. But what order do I do those operations in? etc etc. Thank you Primary school teachers. Thanks a bunch. You have just created an infinite loop in my head. You have given my eleven year-old students an acronym to learn that is actually a recursive formula. After infinite iterations they will still not have found the value of 3 + 2 x 5.

So I bet you're dying to know what I taught them in the end, right? Well I told them about the conversations I'd had in the maths office. I also told them about the American version: PEMDAS. Seriously. That's what they call it. The MDAS is obvious enough. The P is for "parentheses" which my students had never heard of but which is quite useful to know I suppose, and the E is for "exponents" as I mentioned above. My year 7s were not happy that our friends across the Atlantic do their multiplication before their Division though. "Surely they'll get different answers from us and then spaceships won't work!!" they cried. (I must have told them about the metric/imperial satellite mix up in a previous lesson). This led to a nice discussion about how those two operations are interchangeable and you would still get the same answer (or would you? I have just thought of a topic for a future post).

Anyway, in the end, I taught them the O stands for Indices. That's right, I'm on Team BIDMAS. All you BIDMAS haters out there can hate hate hate but if we refer to powers as "indices" the rest of the time why not in this? And if you have a problem with me not including trig functions or logarithms or whatever in my acronym well you shouldn't because by the time you're learning that sort of stuff you shouldn't need a mnemonic to help you remember which order to do stuff in anyway!

Over to you: what did you learn at school, and, if you're a teacher, what do you teach now?

Emma x x x