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

Further Reading:
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?
Answer: x.
What's the coefficient of x?
Answer: 1.
But where's the 1?
Answer: you don't need it.

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


Emma x x x