"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
Tuesday, 24 March 2015
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.
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
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!"
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
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
Wednesday, 15 October 2014
Another New Way to Teach Dividing Fractions
Remember my post from June 2013 about a new way to teach dividing fractions? Well the other day I came up with another new method!
You might be wondering why I need a new method anyway, when the normal method (flip it and times it) works so well. Here's why: because that method is not intuitive. Well, it is if you understand reciprocals properly, and that the multiplicative inverse of a is 1/a. We don't normally go into the axioms of fields though, when we teach year 8 fractions.
Here's the way my method works. Say you want to do 3/4 divided by 2/3.
The way I usually think of integer division in my head is to make it into a multiplication. So 20 divided by 4 becomes 4 times something is 20. And then I think of what the something is. I think this is the way many students think about division.
So applying that to my question:
But 2 times something makes 3 and 4 times something makes 4 is quite difficult. So what we'll do is find an equivalent fraction for 3/4 so that 2 goes into the numerator and 3 goes into the denominator.
Then we just have to work out 2 times something is 18 and 3 times something is 24. Easy!
I thought of this method because there was a question in the year 9 MEP textbook that my students came across that was something like 3/4 x = 5/7 and you had to solve for x. But not having done algebra recently, my year 9s didn't think to divide both sides by 3/4. This led to them trying to find the answer by the method above. Interestingly, this year 9 class is the same class (then in year 7) that provided inspiration for the previous blog post on this topic!
What do you think, a waste of time, or a nice way in?
Emma x x x
Saturday, 11 October 2014
Challenging Gifted and Talented Students Accidentally
This post is about challenging our most able young mathematicians.
I always thought I was good at giving students challenging learning activities. But on Friday I learnt something surprising. Students can be challenged a lot further than I ever thought.
I decided to use the UKMT's Team Maths Challenge resources from last year's competition to run an internal team maths competition with a class of high-ability year 8 students. The Team Maths Challenge is designed to be done by a team of two year 8s and two year 9s, and it takes place in March each year. Running it with just year 8s meant the students would have to work harder as they wouldn't have the older students to help them. Also it is only October, so they have got 5 months less experience than they should have when doing this competition, so that makes it even more challenging still.
I decided to only do the group challenge round and the crossnumber round, as I only had two hours. They started with the group challenge. They responded very enthusiastically, and there was much dialogue between the groups of four. The jottings they were doing were vast and full of impressive maths. When I went over to see what individuals were doing, I had a few comments saying it was hard, but most were too absorbed to even talk to me. One student, about two-thirds of the way through, announced that he had finally got the answer. The other teacher who was with me asked him, "What, you've spent all that time just answering one question?" (as there are ten questions altogether). The student gave the best reply I could ask for. He said, "Yes, but it was worth it".
They got all the way through the forty minutes without giving up, and when I announced there was only one minute to go, the room reverberated with pencil scratchings and the discussions became noticeably higher-pitched. I collected up the answer sheets. It was then that I noticed.
You see, there is a twist to this story. I mentioned above that the challenge would be difficult for the year 8s, for various reasons. But what I hadn't taken into account was that I had accidentally photocopied the wrong materials for them. Yes, I had in fact given them the Senior Team Maths Challenge. Yes, the one designed for year 12 and 13 A-Level students. Yes, the one that even students with targets of A and A* in A-Level maths find difficult. I gave them that.
So these intelligent little 12 year-olds sat for forty minutes tackling problems designed for those who have learnt a lot more mathematical techniques than they. Take a look at the problems I gave them here. They were able to access the questions, give them a good go, and even answer some of them correctly! Yes, many of the groups got one or two answers correct! And what is wonderful is that they weren't put off by the fact that there were several questions that they had no idea how to attempt. They just took the questions they felt they could make a start on, and ran with them. Like the boy who spent 25 minutes on one question - which was worth it. He never even checked with me afterwards to see if he had got it right. The satisfaction came from reaching a conclusion.
We may feel that we challenge our students. But this suggests that students can be challenged more than we give them credit for. What's great about the questions I gave them, was that they don't necessarily rely on knowledge of mathematical techniques (although I think some involved Pythagoras or Trig) so they can be accessed by all ages.
May I suggest you try this with your year 8s and see if you get a similar reaction? It would be interesting to see if they respond similarly or if they give up. The group that I did this with have been taught by a very skilled teacher whose strength is in getting students to be resilient and independent, so this may be why this worked.
Oh, and if you've never entered your students into the UKMT Team Maths Challenge, make sure you do this year!
Emma x x x
I always thought I was good at giving students challenging learning activities. But on Friday I learnt something surprising. Students can be challenged a lot further than I ever thought.
I decided to use the UKMT's Team Maths Challenge resources from last year's competition to run an internal team maths competition with a class of high-ability year 8 students. The Team Maths Challenge is designed to be done by a team of two year 8s and two year 9s, and it takes place in March each year. Running it with just year 8s meant the students would have to work harder as they wouldn't have the older students to help them. Also it is only October, so they have got 5 months less experience than they should have when doing this competition, so that makes it even more challenging still.
I decided to only do the group challenge round and the crossnumber round, as I only had two hours. They started with the group challenge. They responded very enthusiastically, and there was much dialogue between the groups of four. The jottings they were doing were vast and full of impressive maths. When I went over to see what individuals were doing, I had a few comments saying it was hard, but most were too absorbed to even talk to me. One student, about two-thirds of the way through, announced that he had finally got the answer. The other teacher who was with me asked him, "What, you've spent all that time just answering one question?" (as there are ten questions altogether). The student gave the best reply I could ask for. He said, "Yes, but it was worth it".
They got all the way through the forty minutes without giving up, and when I announced there was only one minute to go, the room reverberated with pencil scratchings and the discussions became noticeably higher-pitched. I collected up the answer sheets. It was then that I noticed.
You see, there is a twist to this story. I mentioned above that the challenge would be difficult for the year 8s, for various reasons. But what I hadn't taken into account was that I had accidentally photocopied the wrong materials for them. Yes, I had in fact given them the Senior Team Maths Challenge. Yes, the one designed for year 12 and 13 A-Level students. Yes, the one that even students with targets of A and A* in A-Level maths find difficult. I gave them that.
So these intelligent little 12 year-olds sat for forty minutes tackling problems designed for those who have learnt a lot more mathematical techniques than they. Take a look at the problems I gave them here. They were able to access the questions, give them a good go, and even answer some of them correctly! Yes, many of the groups got one or two answers correct! And what is wonderful is that they weren't put off by the fact that there were several questions that they had no idea how to attempt. They just took the questions they felt they could make a start on, and ran with them. Like the boy who spent 25 minutes on one question - which was worth it. He never even checked with me afterwards to see if he had got it right. The satisfaction came from reaching a conclusion.
We may feel that we challenge our students. But this suggests that students can be challenged more than we give them credit for. What's great about the questions I gave them, was that they don't necessarily rely on knowledge of mathematical techniques (although I think some involved Pythagoras or Trig) so they can be accessed by all ages.
May I suggest you try this with your year 8s and see if you get a similar reaction? It would be interesting to see if they respond similarly or if they give up. The group that I did this with have been taught by a very skilled teacher whose strength is in getting students to be resilient and independent, so this may be why this worked.
Oh, and if you've never entered your students into the UKMT Team Maths Challenge, make sure you do this year!
Emma x x x
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