Should I Encourage My Students Not to Fast During Exam Season?

You may or may not have noticed that Ramadan largely coincides with A2 exam season this year. All of the A2 maths exams this year are during Ramadan. As I have 18 Muslim students in my A Level maths class (and one Christian) this could have a big impact on my results. As a teacher (and form tutor), should I try to gently persuade my students that fasting might not be a good idea?

During Ramadan, Muslims do not eat or drink (even water) during daylight hours. This means staying up pretty late (in summer) to eat, and going hungry and thirsty most of the day. All exam advice I've ever read definitely says make sure you get plenty of sleep, eat a good nutritious breakfast, and keep your brain well-hydrated. Therefore surely fasting is going to lead to worse results?

Kids do not have to fast, technically, only adults, but most of the young people I know do choose to fast (or possibly their parents force them?) and are often very proud of the fact that they do. In case you are worried for their safety, I should point out that Muslims break their fast when they are ill (and make up the time afterwards) and girls do not fast when they are menstruating. Fasting is not about self-punishment or about inflicting suffering on yourself.

So should I try to convince my students not to fast? I do believe that fasting makes a difference to exam performance, and obviously I do want my students to do their best (because I care about them, not because I may get a good performance-management rating, for once). However, I don't want to encourage them not to fast. I can see that fasting is important to them, and helps them connect to God. Surely that is more important than exam results at the end of the day (or the end of life, in fact)?

As an Atheist, I don't believe that pleasing God is going to have any kind of positive effect in this life or the next, but I do believe that spirituality has an overall positive effect on a person's happiness, far more so than getting a good exam result. As a non-religious person, I'm almost jealous that these guys have something they have dedicated their life to, that they can practise in such a simple yet meaningful way. Who am I to take that away from them? As a white, non-Muslim person, who am I to advise Muslims on what parts of their religion (or even their culture) they should follow and which they shouldn't? And so what if non-Muslims have an unfair advantage in this year's exams? I'm sure Islam teaches that life doesn't always seem fair, but God will make sure you're all right in the end (Insha Allah, as my students would say).

So I'm not going to encourage my students not to fast, even if their results suffer.

I'm just going to hope that all of my female students happen to be on their periods for their C4 exam.

Emma x x x

Forget about Becoming a "Better Teacher"

Earlier this week I had a small epiphany. From reading The Teaching Gap, which I talked about a lot in my previous post, I discovered that in Japan, teachers are not judged in the way they are in the UK, as "outstanding", "good", "requires improvement" or "inadequate". In Japan, individual teachers are not judged at all. Instead, the lessons themselves are scrutinised routinely to make sure they are as effective as possible. Ofsted inspectors say they inspect lessons, not teachers, and your senior management team at school probably claim to do the same, but if your lesson is bad, the implication is that you are a bad teacher and you need to improve as a teacher. In Japan, as the lessons are planned by a group of teachers, and based on a previous group of teachers' lessons, the teacher who delivers the lesson is only held partly responsible for the effectiveness of the lesson, and if the lesson is bad, this is not seen as a reflection on the teacher.

Over the course of the five years or so I've been writing this blog, I've contemplated my development as a teacher and how I've become a better teacher over the years. Perhaps for my first five years as a teacher that was useful, but I've decided to move away from that kind of thinking. I will no longer attribute external circumstances to my internal failings (a very female way of thinking, according to Sheryl Sandberg), or even attribute successful learning to my skill in the classroom, and instead approach teaching from a more objective point of view. If a lesson does not work, I will look at how to change the lesson. I will not tell myself I'm a bad teacher. I will ask other teachers for lessons that worked well for them, and try them out. I will not tell myself, "well it worked for them because they're a better teacher than me", and will instead use a scientific approach: "can the results of their experiment be replicated with a different class?" 

I am also going to teach myself to respond to all setbacks (by which I mean ineffective lessons) in the manner of Boston Philharmonic's Ben Zander and simply say "how fascinating!"

Here's a before and after as an example:

Before: "the students learned nothing this lesson, I'm a completely useless teacher."

After: "how fascinating! The students learned nothing this lesson. I wonder which aspects of the lesson could be changed to improve this?"

I believe that this way of thinking would really help departments to improve together as a team. Some departments have a few "outstanding teachers" who are often asked to share good practice or be observed, and often one or two "inadequate teachers" who are given a support plan and are under a lot of scrutiny. Wouldn't it be better if we removed these labels and stopped thinking individually, and instead focused on improving all lessons by planning together, testing out lessons, and continually refining these lessons until learning improves throughout the school? How often do these "outstanding teachers" actually share with their department detailed lesson plans for others to use? It is more likely that they share general techniques (like assessment strategies, or three ways of differentiating a lesson) rather than specific learning episodes (like: this is how I explain carrying remainders in bus stop division). 

My department has started to move towards this way of working. We all split into pairs and worked on planning a series of three or so lessons on specific topics. These lessons have been saved in a central location so everyone can access and use them. What we really need to do now is make sure that every time a teacher uses one of these lessons, they document how effective the lesson was, what bits didn't quite work, and give suggestions for adaptations so that the next teacher can try a slightly refined version. If we repeat this process every time, by the end of the year we will have a good set of lessons, and after five years we'll have an amazing set of lessons. 

Does your department do something similar to this and how effective is it? Do you have any other ideas for implementing a more Japanese approach to professional development?

Emma x x x

Changing the UK's Teaching Culture

This week I have been reading another excellent book about mathematics teaching. This one is called The Teaching Gap by James W Stigler and James Hiebert and it's about a large-scale study that was carried out on year nine maths lessons in the United States, Germany, and Japan. The authors sought to find out why America was lagging behind in the international comparisons of school mathematics achievement, and why Japan was doing so well. All quotations in this post are from this book.

Let me quickly summarise what they found from studying dozens of filmed lessons from each country:

• American lessons usually involved a teacher showing the class an example, followed by the students copying the method used to solve very similar problems. There was a big emphasis on learning definitions and procedures. Lessons always started by going through the previous homework.
• German lessons involved very challenging mathematics, and usually involved the teacher going through a very difficult maths problem on the board with the class contributing. There was little time for practice in the lesson, and this was set as homework instead. Lessons always started by going through the previous homework.
• Japanese lessons started by recapping the main point of the previous lesson (they did not see any homework in Japanese lessons) and then the teacher would give a problem for the students to solve without any kind of indication of a method. The students would then work to solve it, sometimes working in groups. The teacher would then ask students to explain their methods and the teacher would discuss the merits of each method. 

I would say that in the UK we are probably somewhere between Germany and the US. Clearly the Japanese method is working best, as the PISA results show that mathematics achievement is best in Japan. What's more interesting to me, however, isn't the Japanese approach to teaching maths, it's the Japanese approach to improving teaching and learning.

Stigler and Hiebert talk about the fact that teaching is a cultural activity, much like a family dinner. Cultural scripts are learned by observing and participating in the activity. We know how to behave in a Sunday lunch situation because it is cultural. We don't study this, we just absorb the knowledge from being there. For some people of the older generation, using a computer is not a cultural activity, and they had to sit down with a manual and learn how to use it. For children in this day and age, using a computer is a cultural activity, because it is something they have just picked up from observation and by being around computers often. Teaching is a cultural activity and it is learned from spending thirteen years in a classroom as a student. We only spend one year training to be a teacher formally, and this pales in comparison to the thirteen years we spend informally learning how to teach.

Cultural activities never suddenly change, they evolve slowly over time. The UK cannot suddenly say, "Hey, Japan teaches better than us, let's now teach like Japan!" This would be too inconsistent with our current mental cultural "script" that we have about teaching. Teaching systems are made up of elements that interact with each other. If we change one significant element, we can throw the system off-balance. The system then rebalances itself by shifting things around. The result is that the change that was made is adjusted so that the system can function as it did before, so no real change has taken place. An example of this would be for your school to tell all of the maths teachers they must use a problem-solving approach in every lesson. This huge change throws the teacher's system off-balance. The teacher then subconsciously redresses the balance to their system by giving too much direct instruction to the students whilst they are problem-solving. Thus the huge change looks like it has been implemented when in fact it has been completely swallowed up by the system and nothing has changed.

"It has now been documented in several studies that teachers asked to change features of their teaching often modify the features to fit within their pre-existing system instead of changing the system itself". 

The students are part of the system too, and they also have a script in their minds about what their role in the classroom is. When teachers change their teaching system, often they fail to account for the fact that their students have not changed their learning system. The students' internal systems will over-compensate to deal with the teacher's new system, leading to normality.

"Trying to improve teaching by changing individual features usually makes little difference, positive or negative. But it can backfire and leave things worse than before. When one or two features are changed, and the system tries to run as before, it can operate in a disabled state". 

Teachers can sometimes be sceptical of new ideas and reforms. This is because the cultural beliefs about teaching are so fully integrated into a teacher's mindset that they are not questioned or even noticed. There are some things we would not even consider changing about maths teaching in this country, and there are some things we may like to change but we view them as unchangeable. For example, in the UK we put our students into ability sets, often from year 7. Many teachers I have spoken to about the subject have said it would be impossible (or at least very very difficult) to teach mixed ability classes in maths. And yet, we're pretty much the only country in the world who doesn't have mixed ability classes!

So is it impossible for us to change our teaching in any significant way? No, we just have to approach change very differently.

Stigler and Hiebert suggest we start by becoming more aware of the cultural scripts we are using. We need to find out: what are our fundamental beliefs about mathematics teaching? We can change these beliefs and we can change our scripts, but only if we are aware of what they are like in the first place.

Despite years of reform, new textbooks, excellent materials provided by the government and mathematics teaching groups (like the Standards Unit), maths teaching in the UK hasn't really changed that much, and neither has our standing in the international league tables. However, in Japan, teaching has changed greatly over the past fifty years, and so has their students' achievement levels. This is because Japan uses a very different model for improving teaching. The Japanese system leads to slow, steady, maintained improvement over a long period of time.

Japanese maths teachers take responsibility for improving lessons. And the key word there is lessons: they do not strive to improve themselves as teachers, they strive to improve individual lessons. Lessons which can be used by the other teachers in their school and even in their country. This continuous professional development is called Kounaikenshuu, and consists of teachers working together in small groups within their school to plan a specific lesson together, in great depth, try the lesson out with a class and observe how it goes evaluate the lesson afterwards, then revise the lesson and teach it again, to a different class. Notice that the focus is on improving a lesson, not on improving an individual teacher. When you develop a good teacher, their impact will only last the forty years they teach for and only affect their students, whereas developing great lessons can have impact for hundreds of years and benefit thousands or even millions. The lesson observations in Japan are focused completely on the structure of the lesson and how the learning takes place. No judgement would be given to the teacher themself.

Because Japan has a national curriculum, and because Japanese students are taught in mixed ability classes, these lessons that are created can be used by many different teachers year in, year out. So it makes sense to invest heavily in getting the lessons right. These lessons will be improved every year through these research groups, and the teachers involved write reports that are published in compilations of lesson studies for other teachers to read. These are kept within the school unless they are particularly ground-breaking, in which case they are sent to the authorities to publish nationally.

"Lesson study is a process of improvement that is expected to produce small, incremental improvements in teaching over a long period of time. It is emphatically not a reformlike process". 

What I like so much about Kounaikenshuu is that the teachers feel as if they are contributing to the teaching profession itself, not just themselves. Yeah, it's great when you receive an "outstanding" rating, but isn't it better to feel as though you have improved mathematics learning in your entire country, even just a little bit?

This idea of lesson study does exist in the UK and is happening in many schools. But let's make sure that this is not just another fad that will be absorbed into our systems and thus make no difference to our practice. My school introduced lesson study last year as the main part of our CPD. However, this year it has vanished, which is a shame. I think it was a great idea and I'm so glad we did it, but I didn't engage in it fully, and I didn't get much (if anything) out of it, and I believe that is because of the following reasons:

• The groups we met in were too large (eight to fifteen teachers I think). If we had met in threes or fours I think we could have had more focused discussions.
• We worked on our own independent projects, not in groups. We observed each other in order to help each other with our projects, but it would have been more useful if we had worked on exactly the same project, as they do in Japan.
• We were not in subject groups. In fact, we were encouraged to split the faculty up into different groups, the idea being we could then bring back to the department a variety of findings. But I think it would have been more effective if we were in small groups of teachers from the same department, and teaching very similar classes.
• We were not focusing on lessons, but on general strategies. Some of our areas of research were: increasing independence, encouraging a growth mindset, using flipped learning, and closing the gap between Pupil Premium and non-Pupil Premium students. I feel that it would have been much more effective if instead we had been focusing on: how to teach adding negative numbers, or how to introduce differentiation to year 12s. By focusing on specific lessons that get taught again and again and again by many different teachers, our findings would be immediately applicable and have a wide impact.

Of course, you don't need a group in order to conduct lesson studies, although it will not be as effective alone. I have decided to start my own lesson studies, on a much smaller and less impressive scale. I have decided to plan one lesson each week in great depth (not my usual back-of-an-envelope planning) and then evaluate the lesson afterwards. I'm keeping the lesson plans and post-lesson notes in a special notebook (#stationerynerd) and I'm going to keep these notes to refer to the next time I come to teach the same topics. The problem is, of course, that as a teacher of setted children, my lesson plans aren't as widely applicable as they would be in Japan. This year I am teaching year seven set eight. Next year I will most likely be teaching set four. The year after that will be set two. The classes will probably respond completely differently to the same lessons. But I'm going to try. My plan is to change my practice one lesson at a time, using the Japanese art of Kounaikenshuu.

The Elitist Nature of Mathematics

I have always received good grades in maths. This made me feel special. I remember that day in year six when I, along with just two other students from my large-ish Primary school, sat the level 6 Maths SATs paper in a special room. Why was it just us three? Because we were special. There was only a level 6 paper for maths, not for English or science. Why? Because maths is special. Because maths can reach levels of difficulty that English and science cannot.

I remember the day in year seven when in the very first week we had a maths test. Because maths is about showing how much you know. I remember the day, a week later, when we were put into sets for maths. Not any other subject, just maths. Why? Because in maths, ability is important. In maths, the clever students need to be separated from the bad students so that they can learn hard maths unhindered.

I remember the day in year nine when I sat my SATs, and again, English and Science only went up to level 7, but maths went up to level 8. Only the top class got to do that paper. It was a paper for special people, for the chosen ones who were good at maths.

I remember the day in year eleven when I chose my A-Level subjects. I remember being told, "you're good at maths, so you should do maths". Other people were told, "you enjoy English, you should pick English," or "you enjoy design, you should pick design". Because maths is something you do because you're good at it, not because you enjoy it.  

I remember the day in year thirteen when I chose the courses to apply for at university. I considered doing Psychology. I loved Psychology. But lots of people are good at Psychology. I was the best in my school at Maths. So I chose Maths. Not many people are good at Maths. I chose my university because it was the best non-Oxbridge, non-London university for Mathematics research. The cleverest mathematicians work there, so it must be the best place.

I found university maths very hard. I was not that good at it. So I stopped liking maths. Maths is not fun when you're falling behind. There was no help given at my university. The professors lectured but they did not teach. Because if you need help learning university maths, you're not good enough to be here. 


Look at my bolded statements. These are the beliefs that I developed as a result of my education. These beliefs are shared by many whether they consider themselves good at maths or not. If you ask someone what maths was like for them at school, they almost always answer by commenting on their performance, e.g. "I was always good/bad at maths". If you ask someone what, say, Geography, was like for them at school, they almost always comment on their enjoyment of it, e.g. "I found it boring/interesting". This is because maths is seen to be a subject that is all about ability and performance.

I remember the day, not too long ago, when I was looking at year eleven students who had chosen to do maths A-Level next year.

I said, "She's in set four [out of nine], is she deluded? She can't do maths A-Level!"

I said, "Students shouldn't be encouraged to pick maths unless they're in set one"

I said, "Maths isn't like Psychology, not just anyone can do it!"

I said, "They need to realise maths is really hard, it's not like other A-Levels"

I said, "I'm sure she'll try really hard but she just doesn't have what it takes"

I cannot  believe that I said these things just a week ago. Since reading Mathematical Mindsets by Jo Boaler, I have completely changed my beliefs about maths as a subject. No one is born good at maths. No one is born bad at maths. Maths is not about answering questions correctly. Maths is not about passing tests. Maths is about connections and communication. I almost want to petition to have maths re-named to that in my school. While I'm at it, maybe I should re-name my blog to "not just tests".

Here's what I think we should do to overcome this problem:
-Stop testing students in maths in the first week of year seven.
-Don't put students into sets in maths until you absolutely have to (do we ever have to? Finland doesn't).
-Stop praising students for getting answers right in maths lessons.
-Don't ever tell a student they are good at maths (I hope it goes without saying to never tell a student they're bad at maths).
-If your top set year eleven study Additional Maths or Further Maths GCSE, don't make it just for top set, make it an opt-in subject for any student who loves maths.

Let's eradicate this ridiculous notion that maths is different from every other subject. As maths teachers, it is up to us to end this.

Emma x x x

Is Homework Racist?

The other day I started reading Jo Boaler's new book Mathematical Mindsets and I think it is so far the book that that has had the biggest impact on my beliefs about teaching and learning in mathematics. When I have finished reading it I will give a full review and summary on this blog but for now I just want to talk about one of the things that really stood out to me as I was reading.

Homework is racist!

At the school where I teach, homework is given very high importance. We set a lot of it, especially in maths, where the policy is that homework is set after every lesson. We have a whole-school detention in place for students who do not complete homework. One quarter of every student's end of term report to parents is about homework quality and quantity. I tend to spend at least 25% of every lesson going through the previous day's homework. I used to say all of these things proudly. But now, knowing what I know now, I feel uneasy, and even slightly ashamed. And, well, racist.

So let me explain. Students from disadvantaged backgrounds and belonging to some ethnic minority groups are less likely to do their homework in the first place, are likely to spend less time on their homework, are less likely to have help on their homework from family members, are less likely to have the tools they need to do their homework at home (like a desk, a quiet place, stationery supplies), and are less likely to have the time to spend on homework, due to looking after younger family members, doing house work, or working a part-time job.

If I set homework and a student does not do it, that 25% of the lesson I spend going through the homework has little benefit to that student. If that student never does their homework, they are effectively only receiving 75% as much teaching as the other students (and that's assuming the homework itself was of no educational value). You might say it's the student's fault they didn't do their homework, or it was the student's choice. However, when you consider on a national scale that some ethnic groups are less likely to do their homework, we are actually indirectly discriminating against those ethnic groups.

In America, some students take an Algebra course in eighth grade (year 9) then if they pass it they can move on to Geometry in the first year of high school (year 10). However, whether they are put into Geometry or back into Algebra in high school is at the teacher's discretion. In 2012, the Noyce Foundation studied student placement in nine school districts in San Francisco and found that 60% of students who should have moved onto Geometry were actually placed into Algebra. This is damaging for those students because it means they would not be able to do AP Calculus in their final year, and hence may not get into college to study a STEM course. When the Noyce Foundation looked more closely at the data, they found that of the 53% of African American students who passed Algebra in eighth grade, only 18% were put into Geometry, and of the 50% of Latino/a students who passed Algebra in eighth, only 16% took Geometry in ninth. (For Asian students, 52% became 52%, for white students, 59% became 33%). Given these facts alone, you would conclude that the teachers who were deciding which classes to put students in were racist. The Silicon Valley Community Foundation hired lawyers who found that the schools were actually acting illegally. But the teachers did not see themselves as racist, nor did they realise they were discriminating. They had simply been choosing which students to advance by looking at test scores and homework completion. What they did not take into account was that the homework completion criteria disproportionately impacts minority students, and is hence still illegal discrimination.

Something similar may be happening at your school. Have you ever seen a student move down a set for not completing their homework or for having poor behaviour? As some ethnic groups are less likely to do their homework and more likely to have poor behaviour, this may be a case of illegal racial discrimination.

PISA, the organisation which compares mathematics achievement of students around the world, did a study in 2015 into homework and achievement. They had data from 13 million students, and found that homework perpetuates inequities in education.

 "Homework may then have the unintended consequence of widening the performance gap between students from different socio-economic backgrounds". (PISA, 2015)

Read the full paper here: Does Homework Perpetuate Inequities in Education?

Interestingly, Finland and Korea, two of the highest achieving countries in terms of maths, set the least amount of maths homework.

You might be tempted to say that instead of reducing the achievement gap by eliminating homework, we should instead focus on getting ethnic minority or Pupil Premium students to do their homework. After school homework clubs, often made compulsory for certain students, could work to address this. However, after reading around the topic, I am starting to doubt whether there is actually any point in maths homework at all.

Here are a list of studies that have started to convince me to ditch homework:
Baker and LeTendre (2005)  found no positive link between amount of maths homework and maths achievement.
Mikki (2006) found that the countries that set the most homework have the lowest achievement.
Kistantas, Cheema and Ware (2011) found that the more time students spent on homework, the lower their maths achievement across all ethnic groups.

Author Alfie Kohn has written extensively on the subject.

So setting maths homework might not have much of a positive effect on the students that do it, but it does have a negative effect on the students who don't do it. This is almost contradictory, but if it's true, it makes setting homework a completely ridiculous thing to do.

I am going to do some more research before throwing out homework altogether (not that my school would let me, anyway). In the meantime, I am going to change the type of homeworks I set so that they are more in line with my new beliefs about growth mindset and mathematics learning, which I will talk about more in future posts.

Emma x x x

A Maths Game with a Psychological Twist

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

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

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

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

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

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

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

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

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

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

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

Emma x x x

The Mathematics of Voting

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

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

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

First Past The Post

Dodgson was not a fan of this voting system. He claimed "the extraordinary injustice of this Method may be very easily demonstrated". He then gives an example to show how stupid it is:

Suppose there are 11 electors and 4 candidates a, b, c and d. Each elector ranks the four candidates in order of preference. The 11 columns here show their choices:

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

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

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

The Method of Elimination

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

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

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

The Method of Marks

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

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

 Emma x x x

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