# Four Ways to Use Your Learners as Equipment in Maths Lessons

I cannot take the credit for these ideas; I got them from an excellent session run by John Suffolk at the Association of Teachers of Mathematics conference. John explained that it is a good idea to use your students as equipment because then you don’t have to carry the equipment around from one classroom to another. Another advantage of this kind of teaching method is that taking an active part in the lesson, rather than just listening passively, helps students to learn.

1. Learning about multiplication tables, factors, and prime numbers. Ask ten students to stand up. Can they get into a solid rectangle shape? (5 x 2) Can they do it a different way? (A long rectangle 10 by 1). Now ask two more learners to stand up. Now how many different rectangles can they make? Now ask a thirteenth person to join in. What happens now? Why is this?

2. Sum of the angles in a polygon. Three students form a triangle. A fourth student moves around the inside of the triangle, being gently  guided by the three corners to make sure they turn through all the angles. When the student returns to where they started, they will be facing the other way because they will have turned through 180 degrees. Now try with a rectangle. What happens now? What about other polygons?

3. Algebraic Graphs. This requires a big space. A playground would be ideal. Mark out the x and y axes, either with chalk or with cards. To start with, each student stands on the x axis and they are told their x value according to where they are standing (x = -2, x = -1, x = 0, x = 1 etc.) and they all hold onto a long piece of string. You can give the more able students the negative numbers for differentiation. Then you give the students an equation such as y = 2x – 1. Each student has to move to the correct position. They should note what shape they make – a straight line for a linear equation and a curve for a quadratic. You can even solve simultaneous equations in this way – use two groups of students and two pieces of string. The point where the two strings cross over is the solution to the equations.

4. Loci. This is best done outside where there is plenty of room, but can be adapted for the classroom. Ask the students to stand approximately two metres from you. Ask them to note what shape they are making. (They should be roughly standing in a circle.) Now if you have two trees nearby, ask them to stand so that the trees are equidistant from them. You could also use two chairs, two students, or any other objects. What shape are you making now? The students should be in a straight line. This activity can also be used to show the loci of all points a certain distance from a straight line (just draw a line on the floor), the bisector of an angle (stand so that two walls are the same distance from you). Perhaps you can think of others.

# Seven Reasons Why I Hate Graded Observations

This week is the dreaded observations week in the English and Maths department. Everyone hates observation week. Here’s why.

1. It’s so stressful! We find out on Monday morning that we will be observed at some point during the following week. This means we are in a state of anxiety for at least a week and possibly two weeks. I teach about twenty lessons a week and I could be observed in any of them. My anxiety levels rise just before the start of each lesson and then I relax again when it becomes clear that it won’t be this lesson.

2. It literally gives me nightmares; a few nights ago I dreamt I was in a huge open plan classroom with teachers and students all over the room, tucked away in corners and behind curtains, and I didn’t know who I was supposed to be teaching or what I was supposed to be teaching them.

3. It’s so subjective. We don’t know who is going to observe us until they turn up at the classroom door. It could be a subject leader or head of department from any department in the college. Although in theory they are all using the same criteria, there is bound to be subjectivity. Will they even know what an outstanding maths lesson looks like?

4. The outcome affects our pay. If I don’t get a grade 1 or 2, I won’t go up a point on the pay scale this year. That seems unfair when I might be observed with a lovely, compliant, enthusiastic class or it might be a group of unruly reluctant learners. (Thankfully I taught my big group of boisterous rugby boys today and I wasn’t observed!)

5. It wastes so much paper. Normally my lesson plans are kept digitally, but in observation week they must be printed out to give to the observer along with a scheme of work, class profiles and other paperwork.

6. It wastes so much time. Teachers spend hours writing lesson plans in minute detail, remembering to indicate where they have embedded English, maths, Equality & Diversity and now also British Values.

7. It does nothing to improve learning or teaching. Even a bad teacher can pull it out of the bag for one week and then go back to teaching badly the following week. And good teachers teach well whether or not they are observed once a year. There are better ways of improving teaching than stressing teachers out for a week.

So wish me luck. It is Tuesday evening now and I haven’t been observed yet but it will happen at some point over the next three days.

# Four Christmas Maths Activities

Here are some Christmas maths ideas for all of you who are maths teachers or who just like doing maths.

1. This picture shows two Christmas tree shapes. The first shape has eleven corners. How many corners does the second shape have? How many corners would the 11th shape have? Can you write an expression for the number of corners on the nth shape? Would there be a shape with 100 corners?
2. Santa is buying clothes for his elves. He has trousers, jackets and hats in red, blue and green. He wants all the elves to be different. How many combinations of the three items of clothing can he make with the three colours? (They can have two or three items the same colour.) What about if he had some yellow clothes too?
3. There are approximately two billion children in the world. If Santa has to give each child a present in 24 hours, how many presents does he need to deliver each second?
4. My friend collects toy cars. They come in cube shaped boxes with length y cm. I have bought her 32 of them for Christmas and I want to wrap them. What will the surface area be, in terms of y, if I wrap them individually? I could wrap them all together in a cuboid shape to save wrapping paper. How many different ways can I put 32 cubes together to make a cuboid? Which way would give me the smallest surface area? What would the surface area be?

# One Hundred and Twenty Reports to Write

I have committed myself to writing a blog post a day for the whole of November for National Blog Posting Month. Today is my 29th blog of NaBloPoMo but I really have other things to do that need my time. I have to write a report for each of my 120 students.

I am  a maths teacher. I teach GCSE maths in a Further Education college to students who failed their GCSE maths when they were at school. I love mathematics and I love my students but I sometimes find my job disheartening because my students do not love mathematics, no matter what I do to share the love with them. I spend hours looking for new ways to teach maths that are different  from the ways they were taught at school, because the ways they were taught at school didn’t work for them. I enthuse about the beauty of mathematics. I show them awesome visualisations like this one.  I tie them to the table leg and make them walk round the table to teach them about loci.

But they moan and say that this won’t be on the exam paper. They just want to do past papers.

What they really want to do is go out on their horses or their tractors, or go and play rugby or football. They went to college to study agriculture, equine or sport, not maths. They already feel like failures because they failed their maths. Now they have to sit in a classroom and do maths while their friends who passed their GCSEs are out on the football pitch. They hate my subject and they are not too keen on me either since I am the one making them study it.

And now I have to write a report with a positive comment for each of my students. It’s going to take up a lot of my blogging time!

Here is an excellent article explaining why my job is so hard.

# Six Things that Maths Teachers in Shanghai Do

Teaching for Mastery

I was privileged today to observe a maths teacher from Shanghai teach a maths lesson to a year 7 class in Cheltenham. This was organised by the GLOW Maths Hub for teachers from Gloucestershire, Oxfordshire and Worcestershire. After the lesson we talked about some of the differences and similarities between maths teaching in Shanghai and the UK.

1. The answer is only the beginning. In Shanghai, maths teachers put more emphasis on how a student arrived at the answer, rather than the answer itself. Students are encouraged to discuss and explain their reasoning and their methods, thus deepening their understanding and clarifying their ideas.
2. Variation in examples and representations are used – pictures, numbers, diagrams, and symbols.
3. Mistakes are encouraged. “Mistakes are good for us,” says the teacher, “because we can notice something and avoid doing it again.”
4. Homework is set every day to practise and consolidate learning. Because this is done every day at home, it means that all the time in the lesson is used to introduce, discuss and explore the ideas rather than for mechanical repetition.
5. Teachers introduce “wrinkles and confusion” into their lessons – deliberate mistakes which address common misconceptions and explore the students’ understanding.
6. Key ideas and rules are repeated and recited by the whole class. “Keep the denominators the same and add the numerators,” they chant.

1. Be positive. If you don’t like maths or think you are bad at it maths, don’t pass these negative ideas on to your kids. Show enthusiasm when your child brings home maths homework. Tell them you love it when they have maths homework!
2. Don’t put too much emphasis on speed. Sometimes children get the idea that if they are not fast at maths they are not good at maths. They think the people in class who are the first to put up their hands with the answers are the “maths people” and the others who are slower, are not. This is not true. Some of the best mathematicians are not fast at maths. This is because maths is about depth, not speed. It’s important to have a deep understanding of the concepts, not just to be able to do things quickly.
3. Maths is not about memorising. Don’t worry if your child finds it hard to memorise facts. Many children decide they are “rubbish at maths” at the point in school when they are asked to learn their multiplication tables. Memorising may not come naturally to them, and they think that this means they are not good at maths.  Although it is useful to be able to recall multiplication facts quickly, being good at memorising is not the same as being good at maths. Encourage your child to look for patterns in numbers that can help them to work out their tables easily.
4. Maths is not about calculations. It’s much wider and deeper than that. Encourage investigating and problem solving. Help them to see maths investigations in everyday life. If you go out for ice-cream, count how many different flavours there are. How many combinations would that make, if you can have 2 scoops? What about if you can have 3 scoops? What difference does it make if you allow the scoops to be of the same flavour, or if they have to be different? Can you see any patterns in the results of your investigations? Maths is about connections, creativity and communication, so talk to your child about mathematical ideas.
5. Encourage mistakes. Don’t be afraid of making mistakes! Recent brain science shows that when you make a mistake, your brain grows, because it means you are thinking deeply about something you find difficult. When you get a question right, on the other hand, your brain doesn’t grow.
6. Encourage a growth mindset. Some people believe that either you are a maths person or you are not. These people have a fixed mindset. people with a fixed mindset don’t do as well at school as students with a growth mindset, who believe that anyone can do well in maths. If your child has a fixed mindset, they won’t want to ask questions and make mistakes in maths lessons because they will want to show the teacher that they are good at maths. As a result, they are less likely to make progress. Children who have a growth mindset ask questions and are not afraid to make mistakes, so they learn and do well. This is particularly important if you have daughters because girls are more likely than boys to have a fixed mindset and to believe that they are not “maths people”. Let your children know that anyone can achieve in maths.
7. Believe in your children and encourage them to believe in themselves. Brain science shows that when you believe in yourself, your brain works differently and you are more likely to achieve.

For more information about maths, mindset and brain science, visit youcubed, the website of Jo Boaler, professor of mathematics learning at Stanford University.

# Three Maths Problems to Tease Your Brain

After work yesterday afternoon I went to a Teach Meet – an event organised by the local Maths Hub for maths teachers to meet and chat and exchange ideas. This meet was on the theme of problem solving – a topic which is gaining importance in the curriculum. People who know me know that I love a good Professional Development event – a chance to meet other maths teachers, talk about maths and eat cake. Yesterday afternoon’s event did not disappoint – and neither did the cake. I also love a good maths problem, so here are three problems from the event.

1. There are five single digit numbers. Their mean, median, mode and range are all the same. What are the five numbers?
2. ABCDE x4 = EDCBA (where ABCDE is a 5 digit number) what digits do the letters represent?
3. The 24 Game. Using all four numbers 4, 6, 6 and 8, but using each number only once, there are over 60 different ways of getting the answer 24 by adding, subtracting, multiplying and dividing. How many can you and your friends find? You can try it with different numbers – some websites will randomly generate four numbers for you, time you, give you a score, and cool stuff like that.

Here are 3 websites where you can find lots more challenging maths problems. These websites are great for parents, teachers, home educators, or for maths geeks who just enjoy spending their spare time doing this stuff.