London Maths ATM & MA Conference Part 3: Jo Morgan

Here is part 3 of the London maths conference and this was the talk I was most looking forward to; Jo Morgan's talk 'Angles in Depth' exploring the misconceptions and progression of teaching angles.If you don't know, Jo blogs at Resourceaholic. 

As a newly qualified teacher, really thinking about the pedagogy is so important to me and something I struggle with sometimes. Jo said the idea of exploring topics in depth came about from sharing a class with a non-maths specialist. She asked him to cover the topic of angles and he took two lessons to cover what Jo would have spent 6-8 lessons on. During my PGCE year, I felt pressure to move quickly through topics but felt sometimes I missed the opportunity to really check the students understanding. 

When thinking about planning a topic Jo's key tips of things to think about are:

• topic progression
• primary curriculum
• assessment (prior knowledge assessment & end of topic assessment)
• misconceptions
• resources
• stretch and challenge

A lot of the above I have considered but the primary curriculum was not something I really thought about. It was amazing to see some of the content that is covered in KS2. 

Here is a SATs question that tests students knowledge of opposite angles, angles in a quadrilateral and the notation of a right angle. I think I underestimate what is covered in KS2 so being aware of what they have seen is important. Equally important, however, is to not underestimate the forgetting that goes on over the summer between year 6 and year 7. A reason I was so surprised by what is covered is that a lot of my students don't come across like they have seen these things before. 

By the end of KS2 some of the things that have been covered in relation to angles are as follows:

• Rotations (quarter, half, 3 quarter turns)
• Obtuse and Acute angles
• classify shapes
• angles in a quadrilateral add up to 360 degrees
• angles on a straight line add up to 180 degrees
• Opposite angles are equal 

Jo then highlighted some common (and not so common) misconceptions that students have when dealing with angles.

1) The misunderstanding of what is meant by angles on a straight line add up to 180 degrees.

Students incorrectly believe that angle b is equal to 150 degrees as there are two angles and they are on a straightt line, therefore, believe they add up to 180 degrees. Jo talked about the importance of language in definitions and encouraged teachers to use definitions such as:

1) adjacent angles on a straight line add up to 180 degrees

2) angles at a point on one side of a straight line add up to 180 degrees

In conjunction with a really clear definition, showing examples and non-examples is really important so students know what fits the definition and what doesn't. I can see how that would really help reduce misconceptions. 

There were a few heated comments about the how much emphasis should be on the fac that an angle is a measure of turn. I will admit it is not something I'd ever really thought about and whilst I do think it is important, it is so easy to get lost in that core meaning. I do feel a little sad that some educators couldn't phrase their questions or beliefs in a more considered manner when someone kindly gave up their time to share some amazing ideas. 

Jo talked about introducing different notations early on in year 7 which I believe is really important. Students must be familiar with the different notations use to describe angles and triangles and straight lines and I think these help with other topics later on, like constructing triangles and vectors. 

Now I initially had doubts at how Jo felt teaching angles could take so long (with no parallel lines either!!) so she helped us by breaking down what should be covered.

Straight Lines

• standard horizontal straight line with one angle labelled and the other missing
• vertical straight line with one angle labelled and the other missing
• Straight line with angles hidden within other shapes
• 3 angles on a straight line
• a straight line with angles on both sides
• multi-step straight line problems 

Angles in a triangle

• measuring angles in a triangle
• demonstration that angles in a triangle add up 180 degrees
• basic two angles given, find the missing angles
• angles in a triangle combined with angles on a straight line questions
• substitution of algebra ( x=30, y = 40, z = ?)
• multiple triangles in one question
• triangles in a rectangle - using interior angles of a quadrilateral add up to 360 degrees
• equilateral and isosceles triangles rules to use 
• Algebra - single letters
• Algebra - single letters (in isosceles triangles)
• Algebra - letters and numbers (need to rearrange and solve)
• writing reasons for each step of a problem
• Angle proofs in triangles
• Problem-solving questions
• GCSE questions that are accessible for year 7

There was so much wonderful information to take in and I can't possibly do justice was Jo was saying and she has made her slides available here. I really hope I have a chance to hear Jo speak again and it has really inspired me to do some similar planning for some other topics as it has given me so much to think about. 

London Maths ATM & MA Conference Part 2 : Kate Gladstone Smith

Here is part 2 to the London Maths ATM/MA Conference. Kate Gladstone Smith, and her colleague, both from Langdon Park School talked about their experimental year of setting half of their year 10 cohort and leaving the other half in mixed attainment groups (both higher and foundation were fixed). Below are some box plots of the attainment and progress of the two halves of the year. 

 Whilst it is a small-scale study, it is really interesting to see that the students generally seem to be doing better in mixed ability groups. I have always been fascinated by the idea of not setting students as I do query the idea that students are equally weak in all areas of maths. Some kids may just not get on with algebra at all but with geometry, they can just see it. My middle year 9 set did transformations must quicker than I could because a lot of them could just see it.  I also feel that you are putting a ceiling on those students in the lower ability groups and whether you call the groups (or tables in primary school) 1, 2, 3 or names like alpha, beta, gamma, the students know where they are in the ranking of ability.

Whilst the idea of not setting really appeals to me, I will be honest and say I have absolutely no idea how I could do it higher up the school. I struggle with my year 7s who are very widely spread (and I work in a partially selective school), where in their end of year tests, the marks ranged from 18 - 89 out of 90. It was because of his I was so fascinated by the idea of mixed attainment groups at KS4 which I haven't heard of before.

Kate then spoke a little bit about the how. The maths department has designed and created a lot of their own resources that fit with their curriculum and the needs of their students. The students all have their own booklets and the teachers plan collaboratively to help work out how best to approach mixed attainment teaching.

The one thing that really hit me was what went on before any teaching occurred. Kate talked about how carefully constructed the classes were, to ensure the dynamic worked. At Langdon Park School they believe strongly that group work and collaboration is the key to success when it comes to teaching mixed attainment and this is why group dynamic is crucial. So, whilst ensuring there is a mixed spread of ability, it is important to have a balance of quiet students and loud characters, really hard workers and those that need more encouragement to get started. I've never thought about taking into consideration the personalities of the students and that does seem like something that should be really important to consider. Students are people first and foremost, not a level.

We were then given a task to do as a table (in groups of four). We all had to draw a net and construct a tetrahedron with sides of length 7cm (with the hint to remember to draw tabs). Another reminder was given after a few minutes that a tetrahedron was made up of 4 equilateral triangles. We were given coloured paper, scissors, glue, compasses, rulers, and pencils to complete the task. We were then told to assemble our tetrahedra into shape below (on the right).

Once we had done that, our next task was to create a net for the 3D shape that would fit in the void and construct it. This was quite a challenge and stimulated some really great discussions. As I said before, visualising isn't really my thing so I found this really challenging. Luckily I had some bright sparks on my table to help me out!

Ways that this could further be extended:

• Calculate the surface area of the tetrahedron/shape of void
• Calculate the volume of the tetrahedron/shape of void
• How many other polyhedra are there?
• How can you prove what you've found?

This was so fascinating. By working in groups, it really allows the students to be exposed to some incredible thinking and reasoning. If I had been given this task to do, I don't think I would have got there but having people explain it to me in different ways was amazing to see how students that just couldn't see it could eventually get to the stage where they are engaged in the activity.

Other tips Kate shared about mixed ability groupings

• Planning together with your colleagues and collaborating (this is a bit thing for Kate!)
• Planning open-ended tasks such as the one above (to really allow development and stretch but with scaffolding in place (suggests wildmaths for some ideas)
• Get the students to work in groups (and teach them how to work in groups - it is a skill!!)
• Buy some chalk pens and use them to write on tables and windows (a baby wipe is all you need to remove them apparently) as a way to get your pupils up and about and excited about group work.

This was such a wonderful talk with so much food for thought. Thank you so much, Kate, for sharing with us what you are trying out at your school and will definitely look at incorporating more open-ended tasks and group work with my classes. 

London Maths ATM & MA Conference: Mark Horley

On Saturday 8th July, I attended my first ever maths teaching conference. It had been a spur of the moment decision after Jo Morgan (@mathsjem) mentioned it on her blog Resourceaholic. I am a huge fan of Jo's blog and when I found out she was going to be speaking, I knew I had to go. 

The first speaker was Mark Horley (@mhorley), who I have followed on twitter for a while. The focus of his talk was around supporting all students in mixed attainment classes, especially in KS3. At the school I work out, students are not set in year 7 so this was really interesting to me. 

Mark talked mastery and really spending time with the pupils on ensuring they have truly grasped the concepts before rushing on. He used the examples of fractions, speaking about how he may do a whole lesson, just on adding fractions with the same denominator. He showed us a worksheet, which used scaffolding to assist students with adding fractions, but as the questions went on, the scaffolding was slowly removed, in a subtle way that meant students with low confidence didn't feel like they went from total support to no support.

Some examples of the questions were as follows: 

This is definitely something I will explore with my new year 7 groups, giving this to some of my weaker students and the same questions, with less scaffolding, to my stronger students.

Mark then went on to talk about algebra and the importance of introducing in early in year 7. One thing that really interested me was the idea of discussing the distributive and associative laws in relation to algebra as well as number. This is something I would probably have done but unintentionally but it will be something I will be more explicit about when teaching algebra as I haven't had much of chance to teach algebra before. We had some interesting discussions surrounding these questions below and others we may use when looking at what information we can get from an equation.

He then showed us a lovely true or false activity designed to highlight any misconceptions about place value. Mark also showed us a lovely way of getting a whole class check discretely by getting students to make a tick (one hand) or cross (two hands) at their chest to show whether they thought the statement was true or false. No whiteboard hassle and no (or at least) less copying. 

Here are some of the questions:

1) when you multiply by 10, you add a zero
2) 3.4 x 100 = 3.400
3) 4.56 / 100 = 0.456
4) 340 / 10 = 34.0
5) 0.83 x 1000 = 83
6) Dividing by 100 is the same as dividing by 10 twice
7) To multiply by 100, you move the decimal place 2 places to the right

These are some wonderful questions to stimulate discussion and check deep understanding. 

One thing I have used before but not really thought too deeply about is those maths problems in spider diagrams (no idea if there is an official name) where you have a number in the centre and operations to perform around it, like the image below.

These allow students to pick where they begin, which can help encourage students who dislike the structure and control of a worksheet by giving them some choice. Some students may under or overestimate the starting point they should take so some prompting of "You might like to start with this one..." if they are really struggling or don't like the freedom that comes with choice. Students that finish early could then be extended to add their own arrows which is a nice easy way to stretch those.

Another nice quick idea Mark mentioned was for filling in the boxes (which is a good follow on from primary) where you use a circle for operations and a box for numbers so it is clear if you have an operation and a number to fill in, which is needed.

We then went on the look at an activity called quarter the cross which I had seen before on Mark's website. Students would then need to shade in a quarter of the cross. What I had not thought about was getting the students to write a number sentence showing that what they have shaded is a quarter. Mark then showed us a complicated way of shading the cross (there are over 50 ways!) and we then had to try and write the number sentence to show it was a quarter. This activity could really be stretched but still accessible to those who struggle. 

This post has now got too long because there was so much great information to take away so I will be another post on the other speakers at the conference. Thanks to Mark for such a wonderful and informative talk. I also sat on a table with Mark for the rest of the sessions and he was such a lovely guy!