Posts Tagged ‘Mathematics’

Below are the general responses to the questions posed – I recap the questions and the views are generalised notes from talking to a range of teachers from nursery, through Key Stage 1 and 2. I provide the detailed breakdown of two colleagues from Years 3 and 4.

Throughout these you can see that fractions is a huge, varied and tricky concept to think about, teach and learn. I have found that, throughout my teaching career, children have always found fractions hard, although there are a core of children who grasp it quickly, these are the exception rather than the rule.

It is clear that a hands-on, physical approach is needed at the beginning of fractions work – indeed practical maths was a key talking point of all the teachers throughout the Primary phase. But also, I feel that language is a huge barrier to children’s learning. The language of fractions is often misused throughout life and if they don’t have a solid understanding in the first place, their will only serve to cloud the issue further. Another difficult aspect is the link with division – children who don’t know their multiplication and division facts can’t begin to develop their ideas of fractions

So, how can we bring all these parts together to make up one cohesive whole?!

I think it’s a case of reviewing the way it is taught throughout the Primary Phase. Children have to encounter teachers who are confident in approaching fractions and the subject needs to be taught consistently, removing areas of conflict and making sure that each part of their learning isn’t conflicting with another area. If it is to be taught alone, then it needs to be done until that child grasps that particular stage of learning. My belief however, is that is must feature in each part of mathematics – there can always be a question relating to fractions in whatever is taught. This may help to break some of the barriers to learning that exist – I currently feel that children are very negative towards them.

One key to learning is children’s difficulty with fraction language. Maybe teachers are trying to make too many jumps at the same time, moving too quickly. Maybe this is down to pressures from the curriculum. It is often better to avoid comparisons. For instance, when focussing on halves, it would be better to focus on and describe objects or models that are either halves or not halves, rather than giving objects other labels (much in the same way that children find it easier to learn that bricks are heavy and feathers are not heavy rather than comparing them as heavy and light. Floating and sinking is another example, it is easier to get children to understand things that float and things that don’t float BEFORE investigating things that sink – it’s too confusing).

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I tried this with my current Year 6 group of 24 children to disappointing results. The children break the stick of 18 Multilink cubes, describe what they have done and put something on the sheet of paper that will show what their actions to the rest of the group.

They were sat in two circles of 12 each with a large sheet of flip chart paper in the middle, three pens to record their work and a stick of 18 multilink cubes. I chose 18 for its number of factors: 1, 2, 3, 6, 9 and 18. This gives many possibilities for different sums being created. 12 is also a good number (with factors of 1, 2, 3, 4, 6 and 12), as demonstrated in the taught session. Of course outcomes could be expressed as aspects of addition, subtraction, multiplication, division, fractions, proportions, ratio and percentages (there may be others, arrays could be used for example too).

I wanted them to keep discussion to a minimum and they worked silently for the most part. I had to keep reminding them to be mathematical as the activity progressed, but that was all I said. I didn’t say that anything they had written was right or wrong, remaining neutral and fairly detached throughout. I also made a point of not mentioning any potential things they could record and stated that if the stick had been around the circle once to carry on – especially after seeing the responses… this was in the hope that they might actually use some mathematical ways of recording!

One group in particular, took this as an opportunity to create a silly theme. Their recordings were written sentences of what they had done, such as, “I dropped it on the ground and 9 fell off. 9/18 = half ½” or “I dropped it and 14 came off” and even “I headbutted it and 13 came off.” Needless to say, I wasn’t particularly happy with that group as their work showed little thought or care for the maths they were doing nor did they relate this task to the fractions work we had done previously.

The other group’s responses were more considered. Although the first four responses were variations of the sum 18-5=13: “-13 = 5 left” and “5=13left” being two of their recordings. The rest of their responses try to take the form of ratio statements, which is a great piece of thinking from them – although they don’t quite get it right as they write “6:18 [which is followed by] Took off 6 cubes which makes 12 cubes” or “8:18”.

In all, this tells me that my class need a little more guidance and structure when it comes to tasks. Although I had a feeling that this may not be a useful activity with these children, having taught them for a year and a half now and knowing how they can be, I didn’t expect the outcome to be this far removed from my expectations. I suppose it shows that not all good activities work with all classes.

A reflection through an axis followed by a reflection across a second axis parallel to the first one results in a total motion which is a translation.

Image via Wikipedia

The discussion at our last MaST meeting about parallel axes of reflections lead me to think about this a lot. And it wasn’t until I sat down with some paper and drew things that I could see it clearly.

At the last meeting, we looked into exploring reflection using a skipping rope and children mirroring each others movements. We looked at reflecting points, with people moving freely and easily with this concept.

We moved on to creating a triangle with three people holding hands. The reflection of this object was made simple by each of the adults matching themselves to someone on the imaging side, each movement was then copied by the imagers so that each point remained the same distance from the axis of reflection.

The major sticking point was when parallel axes of reflection were introduced. We couldn’t agree on how things should be. I actually think, on reflection (pun fully intended) that we got it right straight away and all the discussion only served to confuse things even more.

In the diagram below, the purple line is Reflection Axis 1, the black line is Reflection Axis 2. The lines are parallel, that is lines that do not intersect or meet.

Also, the object is always on the far left, image 1 is middle and image 2 is on the right.

The red lines show the distance between the object and image 1. The blue lines show the distance between image 1 and image 2.

The second diagram shows what has happened after the object has been moved further to the right, away from the first axis of reflection.

As the changes in the coloured lines show, as the object is moved further from reflection axis 1, object 1 moves further away – the length of the red line has increased. As the 1st image moves closer to the 2nd axis of reflection, the 2nd image moves closer to the 2nd reflection line – the length of the blue line has decreased.

Furthermore, the movement of the 2nd image related to the object is the same effect as translation.

So, parallel lines of reflection are fairly straightforward.

I’m not entirely happy with the way yesterday went. For a start, we didn’t manage to fit in everything I had originally expected, which stops me from trying out all three activities with the small group as planned. Also, the group had grown by two to thirteen, making it a little on the large side to do much meaningful investigative work with the string.

Positives:

  • They all enjoyed a slightly different way of working.
  • They all appeared to be engaged throughout both activities.
  • They all gave a range of input into discussions – the outdoor environment, while far from perfect at my school – encouraged a freer feel.
  • It flew by. The fifty minutes scheduled for a Wednesday numeracy lesson honestly only felt like fifteen minutes.

Negatives or interesting outcomes:

  • My highest of high flyers really struggled with the secret construction – more on that later.
  • A feeling that more should have got done – did everyone make progress in that lesson? It’s hard to tell. Maybe my pacing was off.
  • The larger group number made it difficult to get the most from the outdoor activity.

My fabulous mathematician, the sort of child anyone would want in their lesson as a human calculator, confirmed my long help suspicions – that his mathematical talent lays mainly with number and most other aspects of the subject are weaker for him.

For instance, in the opening task, the secret construction, he failed to notice that the colours his partner was using were the backs of the magnetic pieces, therefore all black. I’ve recreated the shape they had to make and the outcome he instructed.

He used all the correct pieces, just back to front. Also, when discussing the shapes made with string, he was adamant that that a turn between two sides would be around 70° when it was an obtuse angle – something we had been discussing only he day before.

A mixed one this so far…

I am developing these ideas further today with Year 6 and have a session planned to work with some Year 3 children next week along similar lines.

Tomorrow I am trying out two of the three activities we were shown during the first meeting with 11 of my Year 6s – the rest are having a cycle training course, so I have a reduced number. For once this is a helpful thing!

I plan on starting with a secret construction task – starting with a simple house shape made of two colours. This will be the first time they have tried anything like this. I almost feel I’m not going to be pushing them far enough this time around. However, the cycle course is over two days so I will repeat this activity both days, amending the difficulty as needed.

I’ll be listening carefully for the language children will be using – my worry is that the simple shape won’t get much vocabulary out of them. Although clearly an entry level objectivie is needed to begin with – it’s the first time for me too!

To follow up, I’ll be taking them out into our playground where there is a clock face. I’ll be trying out the string shape making activity – passing a ball of string around the people in the circle following a given rule (pass the ball to every 2nd person, for instance) and seeing what internal shape is made when the string is placed on the ground and kept tight. As the clockface has equally spaced dots around the edge, the shapes produced should be regular – my teaching assistant will be the 12th man. However, I only expect to introduce the idea in this way. I plan on moving on to a less evenly spaced, less circular space to investigate irregular patterns. Then, in class, I’ll be following this up with coloured thread on some paper plates with notches cut into the edges. The pupils will be creating their own rule and seeing what shapes are made when moving the string around the plate. For recording purposes, I’ll encourage them to use the school’s digital cameras to take photos of their shapes, which will then make an ideal display.

On Thursday, I’ll be carrying out the other activity – the multilink plan views – with the same Year 6s. This will be more of a challenging thing for them. I’m not sure that visualisation is much of a strong point for them yet. Time will tell!

I have further plans to repeat the string activity with some Year 3 children next week.

I will most certainly update you on what happens.

The NCETM has a series of tools for analysing how confident you feel about various areas of mathematics. Part of the MaST programme requires me to complete each area over time. I also have to complete the sections for a range of Key Stages – 1, 2 and 3 – to demonstrate a broad knowledge of the subject.

Here are my results for the Understanding Shape/Geometry sections. (1 is not confident and 4 is very confident)

Key Stage 1Understanding Shape

  1. How confident are you that you understand the relationship between angle as a measure of turn?
  2. How confident are you that you can give relevant examples to illustrate the meaning of reflection?
  3. How confident are you that you can give relevant examples to illustrate the meaning of line or reflection symmetry?
  4. How confident are you that you know common side, angle and symmetry properties of polygons?
  5. How confident are you that you know common side, angle and symmetry properties of triangles?
  6. How confident are you that you know common side, angle and symmetry properties of squares and rectangles?

I answered 4 for each of these, giving me an outcome of very confident. I chose 4 for each of the answers as, reading through the examples given, I use the techniques described and go deeper too, being a Key Stage 2 teacher.

Key Stage 2Understanding Shape

  1. How confident are you that you understand through practical activity and the use of ICT the meaning of congruence?
  2. How confident are you that you understand through practical activity and the use of ICT the meaning of translation?
  3. How confident are you that you understand through practical activity and the use of ICT the meaning of reflection?
  4. How confident are you that you understand through practical activity and the use of ICT the meaning of rotation?
  5. How confident are you that you understand through practical activity and the use of ICT the meaning of Reflective (or line) symmetry?
  6. How confident are you that you understand through practical activity and the use of ICT the meaning of rotational symmetry?
  7. How confident are you that you can establish through practical activities the side, angle and symmetry properties of regular polygons?
  8. How confident are you that you can establish through practical activities the side, angle and symmetry properties of equilateral, isosceles, scalene and right-angled triangles?
  9. How confident are you that you can establish through practical activities the side, angle and symmetry properties of squares, oblongs, parallelograms, rhombuses, kites and trapeziums?
  10. How confident are you that you can establish through practical activities the nets of common 3D solids?
  11. How confident are you that you understand the terms right angle, acute angle, obtuse angle, reflex angle?
  12. How confident are you that you can show that the sum of the angles in a triangle is 180° in two different ways?

Again, I answered 4 for each of these questions, giving me an outcome of very confident. I chose 4 because of the ways I have used to teach shape over the years in Years 5 & 6. I use pull up nets to show how the 5 Platonic solids are made, regularly discuss the properties of shapes – especially the range of triangles – with my class. One minor concern was the use of ICT in the first 6 questions, but I consider my SMART Notebook slides to be using ICT and I rarely teach a maths lesson without one.

Key Stage 3Geometry

  1. How confident are you that you are aware of a range of visualisation activities to help pupils to appreciate properties and transformations of shapes? (3)
  2. How confident are you that you understand through practical activity and the use of ICT the meaning of translation? (4)
  3. How confident are you that you understand through practical activity and the use of ICT the meaning of reflection? (4)
  4. How confident are you that you understand through practical activity and the use of ICT the meaning of rotation? (4)
  5. How confident are you that you understand through practical activity and the use of ICT the meaning of enlargement? (4)
  6. How confident are you that you know the meanings of alternate angles, corresponding angles, supplementary angles, complementary angles? (3)
  7. How confident are you that you can prove that the exterior angle of a triangle is equal to the sum of the two interior opposite angles, the sum of the angles in a triangle is 180° and the sum of the exterior angles of any polygon is 360°? (4)
  8. How confident are you that you can prove that the opposite angles of a parallelogram are equal? (3)
  9. How confident are you that you know the conditions for congruent triangles and can prove that the base angles of an isosceles triangle are equal? (3)
  10. How confident are you that you know how to establish through geometrical reasoning the side, angle and diagonal properties of quadrilaterals? (3)
  11. How confident are you that you know how to execute and prove the standard straight−edge and compass constructions? (3)
  12. How confident are you that you know how to describe simple loci? (2)
  13. How confident are you that you know how to explain and prove some circle theorems? (3)
  14. How confident are you that you understand Pythagoras’ theorem and its application to solving mathematical problems? (4)
  15. How confident are you that you can explain the conditions for similar triangles? (4)

This held some trepidation for me, as I haven’t ever really considered the Key Stage 3 curriculum before now for geometry while teaching. My answers are in brackets above and a mainly a mix of 3s and 4s with one 2. This gave me and outcome of confident. The 2 is for the question about simple loci – a choice made because I can’t remember having done any locus work in years! (The locus of a point is its path when it moves according to given rules or conditions. The plural is loci.) I think, having read the examples on the NCETM site, that I could certainly do the work for myself, but would probably struggle to teach it.

Where I chose 3, it is often because I felt I fully understood most of the content but there were areas where I may not have been able to give examples. In question 6, for instance, I would be fine with alternate angles, corresponding angles and complementary angles but may confuse supplementary angles.

Clearly from this, I need to develop my knowledge of some of the Key Stage 3 geometry material.

Here we go then…

I attended the first meeting of many over the next two years as I begin my professional journey to a richer, greener, hopefully much improved, teaching field. I have many ideas about what I would like to achieve from this course – specific, actual targets aren’t something I’ve thought about. Yet.

I know where I want my career to go eventually in that I’d like to be someone who is a creative teacher of maths, and other subjects. I want to be known by the people I work with as someone they can mine for ideas or suggestions. I have plenty of ambition, plenty of drive, I’m willing to try anything at least once if I think it will help my pupils get a richer educational experience – although I’m slightly skeptical before I try something without knowing it’s worked elsewhere. This reason is precisely why I’m a follower of many teaching professionals on Twitter, why I spend time reading all kinds of educational web sites and forums.

Anyway, on with the course. The idea behind the Mathematics Specialist Teacher role is to become a “Mathematics Champion”. Someone who is, according to the course handbook:

…a confident and competent mathematician who can inspire children and teachers and be truly regarded as a Champion of Mathematics in the schools in which they work.

The initial development of this role was outlined in the 2008 Williams Report (Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools – WMR Final Report).  Recommendation 3 of which states:

There should be at least one Mathematics Specialist in each primary school, in post within 10 years, with deep mathematical subject and pedagogical knowledge, making appropriate arrangements for small and rural schools. Implementation should commence in 2009 and be targeted initially to maximise impact on standards and to narrow attainment gaps.

Now, this is likely to go out of the window in the future as the admittedly much needed money saving cuts are put in place by the current government. Indeed, funding is in the air for future cohorts. I am part of the second such cohort in Kirklees, a member of a group of 40 individuals – but Calderdale will only have one cohort, consisting of just 10 people as their second cohort funding was not approved.

The three main aims of the programme are to develop:

  1. a deep understanding of the subject
  2. an understanding of pedagogy
  3. an ability to support the mathematical and pedagogical understanding of colleagues in school

These aims are to be covered through taught sessions, school based tasks, readings, directed tasks and a learning journal – this is my attempt at the latter part.

Welcome.

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