How Maths Helps a Democracy

As another election passes, we have been subjected to a lot of arguments about money. One party will promise to spend £800 million on something, another party will respond saying their figures don't add up and it will actually cost £2.5 billion, and the arguments go on and on. It seems like everyday there is a new spending promise followed by a row over it.

But do voters actually grasp all the figures correctly? One thing we also hear about is employers moaning about a lack of numeracy skills among employees, so I would expect many voters aren't able to properly compare figures. And if you can't understand the costs of the promises, then you are left to trust the newspapers to accurately report on them, and given the British media's lack of impartiality, I don't think that is a good idea. So can voters be properly taking part in a democracy if they don't have the numeracy skills to evaluate a party's promises?

I've been talking to students this week about the costs and many have no idea at all of what a million, billion, or trillion really is, other than they are "big numbers". I asked some average year 9 students and some lower ability year 11s (grade E/D) to write in figures 1 million, 1 billion, and 1 trillion. About a third got 1 million correct, and only about 10% got 1 billion and 1 trillion right.

Many students who got 1 million wrong wrote it as 10000. When I put a comma in it for them, they were able to see that 10,000 was ten thousand and realised they had made a mistake. Unfortunately the same wasn't true when we moved on to billions and trillions.

The incorrect answers often had 1 million as 10000, and they then added an extra 0 for 1 billion, and another 0 for 1 trillion. If you think about it, this makes perfect sense to them. The "names" for numbers they learn start with tens, then hundreds, then thousands. And each time we add an extra 0. So the next "name" is million, so why not just add an extra 0 for that? They don't seem to think of say "10s of millions" as another column name in the place value chart. They view them as a quantity of millions, and their place value understanding doesn't extend to numbers above a certain size.

Place value problems seemed to be behind much of the msitakes. For the students who could write 1 million out, I asked what 20 million looked like. The answer was correct: 20,000,000. But then when asked what 2 billion looked like they followed through their incorrect answer for 1 billion, and wrote 20000000 (20,000,000 but they weren't putting commas or gaps in). They then decided that 2 billion was the same as 20 million. Looking at what they thought 1 billion was and what 10 million was (the same thing for them), they decided that that meant that a billion was ten times more than a million.

So when the government says that they will cut £9 billion from the welfare bill, and then there is also talk of CEOs earning over £10 million, that cut to the welfare budget might not seem like such a big deal. Voters can't make informed choices of who to vote for if they can't understand the maths involved. Maybe by the time people are adults everyone is better at understanding these figures, but that's certainly not what is implied by the reams of stats put out by bodies like National Numeracy.

What can be done about this? Well for a start, when students enter secondary school, and we maths teachers look into their place value understanding, we should also try to extend it. Help them develop fluency with millions, billions, and trillions. We could also regularly look at newspaper headlines about the nation's finances (there are plenty these days) and discuss them, work out what the numbers mean and how much they are worth. This would all help to develop student's number sense which is a huge issue with many low achieving students. We can also have a good answer the the question "when will I need maths?" - "Everyday if you're to participate in our democracy."


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Seating Plans

New school year, new seating plans to be made. The first lesson always ends up with several students not particularly happy with where they're sat. After gently explaining that it's tough luck, I like to do a bit of problem solving.

I give the students this problem to work on. They have to make a seating plan for a small group of fifteen students sitting at three tables of five. The students have all given their preferences for who they want at their table, in order. This is real data from a few years ago which always interests them.

The activity is a great one for a new class: it gets them working together on a non-trivial problem that there is no correct answer for. Some groups will inevitably tell me they're finished quickly, but I just tell them to try other arrangements and see which is best. The scoring system is a great motivator to keep them working at improving their answer.

At the end we always have different seating plans from all the different groups, and the scores are generally quite similar. I then point out that it was only for fifteen students at three tables. Imagine what it'd be like for thirty students!

If you are teaching decision maths at A-Level then it's a good way to introduce(or extend) the concept of a maximal matching, and if you're teaching bottom set year seven maths, they'll still get engaged by it, so give it a go!

Resource used: Make a seating plan.


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Random Questioning

Every school year I try to make small changes to improve my teaching. I've always thought that if I try to change too much it could all go wrong and it'd be impossible to evaluate the changes properly, but by doing one or two small changes I can integrate them properly into my teaching and see if it was worth it.

This year I thought I'd look at questioning. I've always done no hands up teaching and when a student doesn't know the answer I'd try some scaffolding to help them get there, but this year I wanted to go further.

I started by getting some ice lolly sticks and put the names of the students on them. That'd give me truly random targets for my questions. This is really important and I think everyone should give it a go. One comment I overheard from a student to his friend at break one day was: "The trouble with Mr F is that he makes you answer a question even if you don't have your hand up. I can't even think about something else cos he might pick me." Perfect, I thought.

But that wasn't enough. Soon some students took the view that if I picked on them they could just say "I don't know." No way could I let that happen. So anyone who said that got repeated follow up questions to elicit an answer from them. The first girl to try this in my year 10 class decided to answer the question "Is 10 prime?" with "No idea". So I said "What do you know about a number if it ends in 0?" "Nothing" she said. "Well what number do you know goes into it?" "No idea" came the response. She was clearly determined not to answer my question. This went on for a further four or five questions before she relented and told me that 10 wasn't prime because it had four factors.

After that I had no further problems with students in that class refusing to answer a question. In my other classes there'd always be at least one student who'd try to get away with refusing to answer but by keeping questioning them they soon realised that they'd have to give me an answer so they may as well have a go first time.

I overheard one difficult year ten boy saying "Mr F just won't take no for an answer with his questions, he just keeps pestering you with more questions if you say 'no', and you just have to answer properly to get him to move on". He saw this as a bad thing, but I was delighted.

One thing I've heard a teacher say is that they use differentiated lolly pop sticks - green for the bright ones, orange for the middle students, and red for the weak ones in a class. I understand why someone would do that, but if combined with effective scaffolding and follow up questions there is no need to do it. Every student should be able to answer every question with appropriate questioning, and doing truly random questioning is a great example of "teaching to the top".

So, I'd recommend this approach to questioning to anyone looking to improve their practice. It's easy to start, and once you and the students get used to it you'll witness much more engagement from the students.


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The Problems with Anchoring Expectations When Marking

Sometimes I wonder if it is possible to ever get my head round effective marking and feedback. I suspect that if I were to discuss it all with a psychologist they'd just tell me to give up trying to be outstanding at this and settle for "the best I can do".

I was marking some GCSE stats coursework recently when my thoughts on "knowing my students" and using data when marking were challenged. I first marked a boy from a top set (we'll call him Tom), probably the most intelligent student in the year. He had spent all year amazing me with his contributions to lessons, and was predicted an A*. His coursework was very poor for him, just scraping a B by one mark. But knowing him and his past performance I went through his coursework trying to see if there were more marks in there that I'd missed, but failed to find any.

A few pieces of work later I marked one for a girl from my middle set(we'll call her Jane). She'd struggled all year, and had been recently getting a C. At the end of marking her work I added up the marks and she had comfortably achieved an A. Finding this very surprising i went back through it to see if I'd made a mistake, but I hadn't: it was definitely A grade work.

At the time I didn't think much more of this, but a couple of days later I was reading the book Thinking Fast and Slow by Daniel Kahneman (all teachers should really read this, it's brilliant; all maths teachers must read this for all the interesting problems in it). In the book he describes the concept of anchoring: a type of cognitive bias where we rely on information we already know when making judgments or decisions. The example he uses in the book is house prices: if the asking price on a house is £400,000 then you will have that in your mind when making a judgment on its value, regardless of whether or not it is actually worth £400,000; when deciding on its value you'll be choosing a price related to £400,000 even if it is a complete dump.

Now we come across many anchors in education: target grades, predicted grades, past performance, etc. One I repeatedly came across last autumn was after the GCSE results came out and the grade boundary for a C in the AQA maths paper had gone from 45 to 54. I heard plenty of people talking about how much the grade boundary had changed and how much harder it had been to get a C, using it as a reason for certain students not achieving their grade. Looking at the paper however, it was an easier paper so of course the grade boundary had gone up. But as it had been around 45 for the previous few years, people were anchoring their view of what a C grade should be to that 45 marks rather than the actual difficulty of the test. I'm not saying the grade boundary change hadn't made it harder to get a C, just not 20% harder.

Having read the book I realised that I had been doing the exact same thing with the two students above, and did it really make sense? Why should I go looking for extra marks in one student's coursework and look to deduct marks from another's? The only student who scraped a B who I then double-checked was Tom, despite there being several students with the same mark: the others had a target grade of a B or C and so it was no surprise when they got that.

So how do we get around this? Ideally we could all just be aware of our tendency to anchor our expectations. But that's unlikely to work as it is our cognitive instinct to do it. The only real solution is to try to mark anonymously. Obviously this presents real difficulties: over time we get to recognise different student's handwriting or style, and they need to put their name on the work somewhere. But even if it doesn't work perfectly, I thought I'd give it a try.

I've been doing some RAG123 marking recently: at the end of the lesson I just ask the students to leave their books open on the page they've stopped on in a pile on their tables. That way I don't see the front of the book when I am marking it and so don't know who it belongs to until I've done the marking (it also helps that it's a new class to me). This approach certainly involves having high expectations of all and "marking to the top". Some students feel they have been marked too harshly but in general it has made students try to meet those expectations and I haven't unconsciously expected less of them. Now I just need to find all the other ways I'm not marking in an outstanding way; I'm sure there are many of them.


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WWW/EBI Feedback

Lots of schools use the What Went Well, Even Better If model of feedback but does all the evidence point to it being the most effective model? I was reading a great article on feedback that got me thinking.

When I get observed and then get feedback afterwards it always starts with some comments about the good parts, followed up by the grade, and then some parts that might be improved. I normally get a 1, sometimes a 2, and I'm usually pretty confident about how the lesson went during the feedback session. But I still spend the entire feedback session just waiting to hear the negative comments which is what I inevitably think of the "how to improve" section as being.

So if that's how I feel about it, why wouldn't students feel the same way when given the WWW/EBI feedback? Would they hear the positive comments or would they just not really hear them because they're waiting to be told what they do wrong?

It would make sense to me to start separating the two parts of the feedback to ensure students properly hear what they've done well. Perhaps the WWW at the beginning of a lesson and EBI later on, or else just at what seems like appropriate times based on lesson content. We teachers put so much time into writing feedback we ought to make sure we get maximum effect from it.


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Using Whiteboards around the Room

One of the best things about visiting other schools is when you see one simple idea that could really change your own lessons for the better. Last week I taught a one-off lesson in another school and the classroom had eight whiteboards, approximately A2 size, mounted on the walls around the room:

Picture showing whiteboard

I decided to use them in the lesson straightaway and it was brilliant. The students got up out of their seats and worked in threes at the boards solving a problem about tessellations. The atmosphere was brilliant, there was a real buzz, and the communication between students was great to hear. I've seen research(which I now can't find of course) suggesting that getting students to stand up and work on their feet changes the way they work, raising their energy levels, and improving their learning in the process. From just this one session it's hard to conclude too much myself, but talking to other teachers who use them everyday, they seem like a brilliant idea.

I immediately began coming up with ideas of how I'd use them if I had them in my room: (these apply to lots of subjects, not just maths)

1: Problem Solving Methods

Put a different problem or question on each board, then students start working on it in pairs/groups. After five minutes have them move to a different board; they now have to see what the other group did on their problem and then try to solve it in a different way. Not only would they be understanding other students' work but they'd also have to develop their flexibility in finding different ways to solve a problem.

2: More Problem Solving

Again, put a different problem or question on each board. Students have to put their first step in solving the problem on the board. They'd then all move to another board and put the next step they'd do in solving the problem. In maths it'd be a good idea in solving an equation for example.

3: Peer Feedback

Students solve a problem on the whiteboard and then another group evaluates their answer and gives feedback comments on it, what was good about it, how it could be improved etc.

4: Find the Mistake

Problems are put on all the boards and the first group has to solve it, but purposely make a mistake. Then all groups move to another board and try to spot the mistake.

5: Find as many...

On each board, put a question of the form "Find as many ____ as you can". It could be countries, synonyms for said, or ways of making 48 with 3 numbers. Students do as many as they can on their board and then they all go on to a different board and try to add more answers to what the previous group had found.

6: Standard Ideas

In addition to the above ideas, students could do mind maps, stick loop cards on to the board when doing a loop card activity, or just generally do some group work on the board.

All of these ideas could of course just be done using big pieces of paper on desks and passing them around groups, but using the whiteboards with the students standing round them generates some great energy and engagement that sitting and working at a desk just doesn't achieve. I hope to get some of these in my room soon!


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Peer Tutoring

Five years ago I was thinking about the different ways I learnt best during my education. Aside from the obvious ones, I thought about how at both school and university students in higher years would go through a concept with me, explaining it in a different way to teachers, and how often I would gain from this experience. So I decided to start trying it myself at school.

It's a concept that I'm hearing more and more about recently: peer tutoring. There is a lovely graph:

Graph showing Peer Tutoring is Cheap and Effective

showing what a good idea it is: effective and cheap. I've been running a successful peer tutoring scheme for a few years now and thought it was time to share what I've learnt.

Which Students to Pick?

Choosing the correct students to pick is probably the single most important decision. I've done it with year 11 students as both tutors and tutees, but other schools have used students in different years. I don't think the age of the students matters so much as the relationship between them.

The tutors need to be confident in their subject knowledge, but not arrogant. They need to be good communicators as well as patient, understanding, and good listeners. Really, they need to have most of the qualities of good teachers!

The tutees shouldn't be too far removed in achievement from the tutors. I found that putting together top set and bottom set students (A* and F/G grades) wasn't very successful. The language used to explain things by the tutors could be inaccessible to the tutees, and the tutors would tend to assume more knowledge than the tutees had. The tutees would often just nod and agree without actually taking much in. The best mix seemed to be more-able students with students that were expected to get a C or D.

What Topics; Skills or Problem Solving

This might not be true for many subjects, but maths can generally be broken up into skills and problem solving. Think solving an equation versus forming an equation from a word problem and then solving it. It was definitely important to focus on the skills side of things. That shouldn't be too much of a surprise - from what I've read, problem solving is hard to learn without a teacher to help guide students through. It also meant that the tutors could rely on having a nice closed topic to go through and didn't need to worry about getting confused solving a hard problem themselves.

What Topics for Each Student

This is probably the second most important point: to get the most out of the sessions it is really important to pick topics on a student by student basis, making sure then are topics the tutee is weak in. They must also be specific topics. In maths, just going through "algebra" was a waste of time, but going through factorising quadratics is easy for a student to learn and practice in a half hour session.

It also doesn't make much sense to try to cover a topic that the tutee has absolutely no idea about: they must have at least heard it from a teacher before and just need to consolidate it, or brush up on their skills in that area.

In order to get the topics to cover, I went through the tutee's most recent tests to pick the topics, and this proved very fruitful: the tutees understood they were weak in that area and appreciated the personalised help.

How a Session Should Go

I ran the sessions for half an hour during a lunchtime. The students could bring their lunch if they liked and I provided biscuits(this was very popular). It helped to have the same pairings of tutor and tutee each time, although this wasn't always possible. Some students would tell me if they didn't think their pair was working and I could change it around.

I would give a "lesson plan" to the tutor, just explaining what they'd be talking about that session, some key points to talk about, and some sample questions to go through. They would talk about this for fifteen minutes, using a mini-whiteboard to help them.

Then for the last fifteen minutes the tutee would work through some exam questions with the tutor helping them where needed but generally letting them get on with it.

Problems with the Sessions

The biggest problem I've had is attendance, particularly from the tutees. Some of them wouldn't turn up because they couldn't be bothered, some because they forgot, and some because they felt embarrassed.

Dealing with the forgetful ones was easy: they were reminded every week that they had a session that day. The embarrassed ones I'd have a chat with and really emphasize that the sessions weren't because we thought they weren't good at maths, but because we thought that they were good at maths but just needed a bit of extra help to make them realise their potential. I'd also then pair them with the friendliest tutor I could find and it always seemed to work. The ones who couldn't be bothered were definitely the hardest to deal with and there were three different ways to approach it: (1) phone home and get their parents on-side so they could be encouraged to attend; (2) give rewards for regular attendance (our year 11s get £5 off the prom for every 10 revision sessions they attend, so we started counting the tutoring as revision sessions); (3) just physically go and collect the student from their lesson before lunch and deliver them to the tutoring room. If none of these worked, then that was the student's loss really.

Getting a list of topics for each student that they needed to work on was difficult. Asking the students themselves didn't yield much other than "any shape" or the like, and asking teachers also didn't get a specific enough set of topics. It is worth seeing the tutees actual class work or tests to get a good idea.

Is it Worth it?

The idea that peer tutoring is cheap ignores just how much work it is to start it up and organise it, not to mention a teacher giving up their lunch once a week to sit with students. It is hard work, but if done right then it is definitely worth it. When everything is going as planned, looking at the students working together in a room and feeling the buzz is brilliant.

I've spent a lot of time looking at data to see if it has an impact and it definitely does. I recently ran six sessions last term prior to the year eleven mocks and students had improved in approximately 80% of the topics they had covered. The more a student commits to it and the better their attendance, the more they get out of it. Talking to the tutees, the ones who bought into it right from the beginning all said they felt it had a positive impact on their learning and was worthwhile.

So... take the time to do it properly and you will reap the rewards. There will no doubt be problems along the way, but once it starts running smoothly it is a great and different way to help students in their learning.


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Feedback - Using Model Answers

Using model answers can be a valuable part of feedback on a piece of work. They show the students what an excellent piece of work should look like and it can be easy to see how their work is different to that. If doing a quiz or homework I like to use one of the student's work as the model answer; I find the students respond well to seeing another student's work and it seems more achievable than a teacher's model answer.

Using model answers are another nice, easy way to do some positive marking.

Below are some ways of using model answers that have been successful for me in the past:

1

With a homework or quiz, photocopy the model answers and hand them out to students in pairs. Ask the students to discuss together what is good about the piece of work and what they can learn from it. Here is an example of a model answer I have used for this:

Improving work image

Next discuss it as a whole class. This is a great opportunity for some high quality questionning: what is the same about the answers to questions 3 and 4? Why have they answered it in that order? What is different about questions 5 and 6 that makes the answers different?

Finally, collect the model answers and give the students their own work to improve. You should have only ticked the correct parts of their work, and leave them to improve their work from there.

2

Give students the model answers and their own unmarked answers. With those ask them to find out what they did correctly and incorrectly. Now give them some new but similar questions and have students answer them using the model answers and their own, now corrected homework to help them.

3

Make copies of the model answers, but make sure you haven't written any comments on it. Ask students individually or ideally in pairs to write comments explaining what the student has done well on the model answers, the type of comments that a teacher might write such as "inverse operations applied in the correct order" or "good use of descriptive language".

4

Take the model answers and remove some parts, e.g. tipp-ex one of the steps in solving an equation. Then ask students to write in what they think is missing. This removes some of the pressure to find a correct answer to a whole question as it is mostly there.

I would suggest that all of these should be followed up by students doing a simliar piece of work in silence to give them the opportunity to show they have learnt from activities. Using student's work as model answers has proved both effective and popular with students and is definitely something I'd recommend!


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Positive Marking

What is the point in me marking homework and keeping a record of it? There are a few answers I've heard:

  1. to record student attainment over time;
  2. because students like to get a grade on their homework;
  3. so I can give them a written target on how to improve.

Looking at (1) I don't believe the grade students get on their homework is anything other than an indicator of how hard they have worked. Some of the cleverest students I teach get some of the worst results on their homeworks purely because they are quite lazy when it comes to doing homework. When I was at school the grade I got on a homework was an indicator of how good I was at doing it either at break time or in the previous lesson. Looking at the correlation between homework grades and exam grades, it is a weak positive correlation. Proper in class assessments of various kinds are a far better way to truly assess what grade a student is working at.

When it comes to (2), student liking to get a grade, I have heard this said many times but I'm not convinced. There are definitely some students who do like to be given a grade but equally there are others who hate to be given a grade and use it as an excuse not to put their full effort in: if you don't try very hard you can always use that an excuse for getting a poor grade. Grades can be a good incentive for students, but can make them ignore what they did wrong and how to improve.

As for targets, it is something I have done over the years in various different ways, but never been particularly happy with. When I first started teaching I was told how I needed to write a written target such as "To move to a level six you must learn to solve equations with the letter on both sides." That was a lot of writing on the 150 or more homeworks I'd mark every week. So a few years ago I started using codes: I'd write "P" or "G" and put the key to the codes on the board. The student would then write something like "P: learn to find the gradient of perpendicular lines." Then three years ago I extended that to putting follow up questions on the board for each code. This seemed great, but it really only told the students what topics they weren't good at, not how to improve.

Recently I've been doing things differently, doing some positive marking. This involved just ticking what the student has done correctly, and then giving them a hint on where to go next if they've made a mistake. So in solving an equation a student might correctly perform the first couple of steps and then make a mistake. So I tick the first two steps, and then give a simple comment such as "now factorise" or "remember to do things in the correct order: SAMDIB".

We'd then spend twenty minutes or so doing some really high quality DIRT, letting the students improve their work. As I'd go round I'd then award them more marks when they improved their work, culmanating in a new grade at the end. Below are some examples, with my comment written in red, and the student's new improved work highlighted in yellow.

Improving work image

Improving work image

Improving work image

Improving work image

The results have been excellent, with a really good atmosphere developing in the classroom when I say we're going to improve our homeworks. Students are all engaged in trying to improve their work and they really want to improve their grades now. Speaking to a few of them about it, they said it seemed fairer because they got some help but still had to do most of it themselves. Having given a few mini quizzes in the weeks after I'd done this, the students had certainly learnt from their mistakes.

I will still use targets sometimes, particularly for year 11 students needing to know where to revise, but they'll be based more on quizzes and tests. For homework marking I'll focus much more on this positive marking: it's the first time I've felt genuinely satisfied that all my marking was going to benefit the student's attainment, and would definitely recommend it.

I will definitely blog more in 2014! To find out about updates,

Practical Proportionality

A year or so ago I read a fascinating blog called What it feels like for a sperm. The main subject would have been a bit too much for a year 11 maths lesson (just imagine all the comments from the boys!), but the blog gave a formula for something called Reynold's number:

The formula is very simple and immediately struck me as being a perfect way to talk about proportionality, something which students can often find difficult to understand. Reynold's number is all about fluid dynamics and used in things like airplane design but I'm not sure I completely understand the science behind it so I just simplified everything and did a very practical lesson with my year 11s. The result was a fun lesson that really engaged the students.

I started by explaining we would be looking pushing a sphere through a medium, either water, yoghurt, jelly, or peanut butter. I asked them to think about what different physical properties of sphere or medium would effect how easy or hard it would be to move the sphere.

Luckily they quite quickly came up with: size of sphere, mass of the sphere, and the viscosity of the medium. This gave us a good place to talk about and define density: was it the mass of the sphere, or was it about two spheres of equal size but different mass?

Then the experiments started! I had brought in the yoghurt, peanut butter, jelly, and water, along with different balls: a marble, some balls from a Hungry Hippos game, a large bouncy ball, and a very light weight ball off the same size.

Some volunteers came up and we first of all had them try to push the marble through the different mediums, coming to the conclusion that as the viscosity goes up, it gets harder to push the ball, so viscosity was inversely proportional to the ease of pushign the ball. We then tried to push the differnt balls through the peanut butter, coming to the conclusion that as the sphere got bigger, it became easier to push the ball, a directly proportional relationship.

The students were getting really engaged in the experiments (partly cos of all the mess). We next tried pushing the two spheres of the same size but different densities through the peanut butter, deciding that as the density went up, it became easier to push the ball, another directly proportional relationship. The experiments gave lots of places to ask great questions about proportionality, and to make predictions based on what we'd found so far. It was also an excellent place to look at graphs of directly or inversely proportional relationships.

Knowing that we weren't being entirely faithful to Reynold's number, I decided to define something a bit different: Johanson's number. We decided that a high Johanson's number would mean it was easier to push the sphere, and a low Johanson's number would mean it was hard to push the sphere. We put together all the results of the experiments, to come up with a formula: Johanson = (density x radius of sphere) ÷ viscosity.

We could then look at how changing some of those quantities could change Johanson's number, and discuss whether any of the variables needed to be squared or otherwise changed.

The students loved the lesson. Everyone stayed engaged throughout, and are still talking about it a week later. Some went home and looked up this Johanson's number on the internet but were disappointed to find out it didn't really exist. It took a lot of cleaning up (yoghurt and peanut butter is not a good mix), but was well worth it. Linked below are a basic lesson plan and the PowerPoint file that went with it.

Documents: Lesson Plan   PowerPoint file.

Practical Maths for Top Sets

26 September 2013

 

In my teaching I always try to be creative, and use more practical tasks for the students to do mathematics with.

But when teaching Key Stage 4 top sets, with all the algebra and number skills that are there for them to learn, it is too easy to stop doing practical lessons. The students work hard, and are keen to learn, so they don't really complain, but it's not really fair on them.

So I have been trying to do more creative lessons with them recently, and one that was particularly successful was Chocolate Volume.

The students were given different chocolate products - KitKat, Malteser, and Rolo, and asked to work out what percentage chocolate they were. Before we began we went through what they'd do for a made-up chocolate - the Chocywoc, and then I gave them the task sheet and some chocolate.

They all estimated that the chocolate content would be between five and twenty percent, and were all very interested to find out that actually they were more like forty or fifty percent.

There was a lot of interesting maths involved, beyond just working out the volumes of compound 3D shapes, which the chocolates provided a good visual way of thinking about. They had to estimate lengths, and then decide what effect that would have on their answer. They had to decide what shapes to approximate their chocolate to: the Maltesers were mostly spherical, but there was a lot of of variation; for the Kit Kats, should they approximate it to a cuboid or some type of frustrum. We would get a variety of different answers and had to decide which ones to ignore; we also talked about sampling and quality assurance

Once they'd done it for a Kit Kat or Malteser, I gave them a Rolo to try and see how fast they could do it. In only a couple of minutes, they came up with an answer. They were really engaged the whole lesson and the learning and progress was clear for all to see.

I did exactly the same activity with a set 3 (out of 7), and although they took longer to answer the questions, they were also very successful.

The students enjoyed it so much that I'll be getting some of them to do it at Prospective Parent's Evening next week!

I'd highly recommend trying this activity out, the resources are linked to below.

  • The chocywoc - to look at finding the chocolate volume as a class. Chocywoc.docx
  • Task Sheet - give out to students with the chocolate of their choice. Task Sheet
  • Help sheet for Malteser (for students who need it). Malteser Help
  • Help sheet for Kit Kat (for students who need it). Kit Kat Help

Philosophy for Children in Maths

12 July 2013

 

A few years ago I attended a CPD session on "Philosophy for Children", an approach that puts developing the student's own thinking skills at the heart of the lesson. It sounded wonderful, and exactly the type of activity I'd love to do, and I finally got round to doing it recently.

At the centre of a P4C session is a "stimulus". This can be a picture, story, object, or anything else that will make the students start thinking with enquiry in mind. The stimulus kicks off questionning and we then just go where the student's own thinking takes us.

Before beginning, some basic rules need to be applied:

  • Be respectful of others opinions
  • Listen to the speaker
  • Give reasons for any opinions you state
  • Comment on the point, not the speaker

If possible the students should sit in a big circle so everyone can see who is speaking and they feel they are speaking to the whole class, rather than just a few students or the teacher.

Once we went over the rules we got started and I gave them the stimulus.

One of the reasons I took so long to do a P4C lesson was that I lacked a mathematical stimulus. A few years ago I saw a scatter graph that showed that in American states, as the religious attendance rate rises, so does the murder rate. It was a moderate positive correlation with a Pearson's value of 0.57.

I knew it could be a controversial topic, and the opinions expressed could easily offend some students, but I spent time talking to the class about this and the need to be respectful and to think through everything we say before saying it. The results were excellent: they were all polite and some students even stopped themselves when they realised they were about to say something dodgy!

Once students are given the stimulus they are given a few minutes to think about it, and then in small groups them come up with some enquiry questions (questions for thinking, not factual questions). Each group chose one that they wanted to pursue and then we read them out as a class and picked one to discuss.

The type of questions they thought of were:

  1. is the murdering just about their religion?
  2. are religion and murder really related?
  3. is the data reliable?

As a teacher I really had to take a step back and let the students own opinions dictate how the lesson went. The results were really good and I got excellent feedback from the students. Some people who rarely said things in lessons were really getting involved, and as the session went on the student's thinking really developed: they were putting forward more and more sophisticated ideas.

Something that really impressed me was how much thought they put into it and how well they listened to each other. Someone would say something, there would be ten seconds or so of silence, and then some responses would come from other students. They really respected each other even if they didn't agree.

It is quite hard to come up with ideas for P4C lessons for maths, but this was definitely a good place to start. I've been reading "A Mathematician's Apology" by G.H.Hardy and will be doing another session of P4C using an idea from his book about what is more real, maths or science.

The resources I've used are links below. Feel free to use them and good luck, it really is worth it!

Resources used: Scatter Graphs Religion v Murder, What is more real: Maths or Science?

Using Computers in Maths

13 June 2013

 

Computers are the future, and we should be using them in our teaching of maths. Everyone can agree on that, but how is it done?

There are numerous approaches to using computers in maths, from specific software packages to programming languages. But they all require both teachers and students to be proficient in their use and this takes time. For years I have been trying out different approaches, from using graphics calculators to dynamic geometry programmes to using the Python programming language. I have a background in computer science so I find it easy to use these, understand how they work, and know what to do when something goes wrong.

But whenever I have tried to share this with other teachers it hasn't worked. Teachers without much background in computers really struggle particularly when something isn't working as expected. When there is an error on the graphics calculator, if you don't have much experience with it, then it can really ruin your lesson when you can't fix that error.

So I've been thinking a lot over the last year about how we can use computers in the classroom in a more teacher friendly way. Then it hit me - Microsoft Excel. All teachers are familiar with it - just think how much data entry/analysis we end up doing on it! Students are also familiar with it from their ICT lessons and know the basics of it. The more I thought about it the more it made sense. Whilst Excel can be quite boring to look at, it is powerful and allows students to work on a variety of different problems.

It also struck me that it was perfect for many of the problems students work on. I was doing an investigation with my year eight class which involved arranging the numbers one to nine in a three by three square, multiplying all adjacent pairs, then adding the answers together. We were looking for the highest possible total. The idea was that they would try out different arrangements, look for patterns such as "the total gets higher when I put big numbers in these positions", and then move towards an answer. Unfortunately, by the time they did all the calculations they only had time to do three different squares and so couldn't really notice many patterns or come to conclusions. The point of the lesson was not to do lots of calculations, but to do some investigating, so really the lesson had been a failure. But if we had done it on Excel, getting the spreadsheet to do the calculations, we would have accomplished much more.

The next day I booked us into a computer room, quickly explained one thing about Excel - to write a formula you start with an = - and asked them to find the smallest total they could. The lesson was fantastic: they quickly got to grips with the task, tried out different arrangements, noticed patterns that they were then able to articulate, and most students were able to find the minimum possible total.

Since then we have been back to the computer room many times to work on problems, and I have also taken other classes, particularly my year tens to work on statistics.

The resources I have created and used so far have been very popular with students and have resulted in some great learning. Some of the advantages of using computers rather than working by hand have been:

  • Independent work has flourished: students have been able to work by themselves and have had to choose their own approaches to problems, thinking everything through.
  • Investigation skills have developed well: students were able to look for patterns in problems as the calculation were done by the computer, allowing them to solve problems much quicker.
  • Deep questioning: going round the class room while they were working, I was able to ask some really probing questions, such as "What would a list of numbers with a negative skew and high standard deviation look like?" Students were even asking themselves these questions, and instead of spending ages doing the calculations, once they'd set up the spreadsheet to do it they could just look at the patterns.
  • Engagement: students would really look forward to going on to the computers and would be keen to get to work on problems.
  • Experimentation: students could come up with their own "what would happen if.." questions and then try to answer them

One of the concerns I've heard voiced about using computers regularly in maths is that they're not allowed them in exams so we shouldn't use them too much in lessons. I had some concerns about this myself, but was greatly reassured by an experiment I did.

I took two fairly similar classes, and did an investigation into mobile phones, comparing different deals, with each. One group did it by hand, and the other wrote a spreadsheet to calculate the costs of different deals depending on how many minutes and texts a customer used each month. I then set them a homework, on paper, to compare two different mobile phone deals for a particular customer.

The class that had done the initial investigation on the computers produced some fantastic homework. It was organised, clear to follow, and correct! The class that had done the initial investigation on paper strugggled with the homework, approaching the problem in a very unorganised way. It may not have been a randomly controlled double-blind study, but it was enough to convince me that students who regularly use computers in the classroom won't suffer in exams as a result, and would probably do better.

Going back to my original point about having a technology that other teachers can use: I have shared these resources with other teachers and done some team-teaching using them. It has been far more successful than when I have used other technologies with other teachers.

Using Excel is easy, user-friendly, and allows for some great problem solving to be done. I think there are also plenty of opportunities for cross-curricular learning by working on science or business problems that involve maths.

Go to the Computer Maths section to check out some resources I have created so far.