Posts Tagged ‘Math’

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.

Angle with highlighted vertex
Image via Wikipedia

The main reading for this study block (linked below) is a tricky and detailed account of current teaching relating to angles. It’s main findings are that not enough is done to develop the concept of ‘turn’ with children – that an angle can be defined at the amount of turn from one position to another, and that, if it is taught, the main focus is on right angled turns.

Beebots and roamers could be used in school to investigate the idea of turn, but why not use the school grounds? Create obstacle courses in the playground for children to be directed around – making sure that the turns aren’t always at right angles.

Mitchelmore and White state that children need to experience a wider range of angle concepts. They believe that teacher move too quickly on to the abstract idea of an angle – as shown here.

Their research looked at whether children could represent the movement of objects such as a door, or wheel in terms of diagrams and still understand what was happening – whether they could move from the physical to the abstract in one move.

For instance, the angle shown here could represent the movement of the blades of a pair of scissors, or the opening of a handheld fan, things that children could see happening, and represent in a diagram like this. However, children referred to these movements as ‘opening’, not ‘turning’.

Children were asked to represent the angles using bendy straws to demonstrate the movement and the associated angles. If children could see the angle of movement, and explain what was happening, the researchers were happy.

The researchers also looked at children’s ideas of slope, with the idea that this is an area that is overlooked in schools. I would consider this to be the case simply because of the difficulty of representing it as well as not always being able to see the angle that a hill slopes at, for instance.

…most students had some global concept of slope but that many did not quantify it by relating the sloping line to a fixed reference line. Unlike the wheel and door, where the second line may be suggested by the initial position and there is a global movement which can be copied, there is in fact very little to help a naïve student interpret a slope in terms of a standard angle.

We conjecture that many students have a global conception of slope as a single line and do not conceive it in terms of angles. Had the physical model of the hill consisted simply of a sloping plane without any supporting edges, it is likely that far fewer students would have indicated a standard angle interpretation.

Mitchelmore and White discuss how children find it easier to see the turns, angles and slopes when both elements are easily visible (the scissors, fan, etc.) and this is likely to be because it fits more readily to the idea of an angle as drawn above – something they are likely to encounter in class. They go on to say that, “the fact that the standard angle was used more frequently for the door and hill (where one arm must be constructed) than the wheel (where both arms must be constructed) supports the view that the crucial factor accounting for the rate of use of standard angles is the physical presence of the angle arms… 88% of the students used standard angle modelling when both lines were visible, 55% when only one was visible, and 36% when no line was visible.”

There are three main findings to this piece of research.

  1. That angle work can be related to the everyday concepts of corner, slope and turn.
  2. The fewer arms that are present in a particular angle context, the more that has to be constructed to bring it into relation to other angle contexts and, therefore, the more difficult it is to recognise the standard angle. It is only in exceptional cases that the relevant line has to be discovered. In most cases, it has to be invented through conscious mental activity.
  3. That many children form a standard angle concept early, but that this concept is likely to be limited to situations where both arms of the angle are visible. If the concept is to develop into a general abstract angle concept, children will need more help than is presently given to identify angles in slope, turn and other contexts where one or both arms of the angle are not visible. The slope and turn domains are particularly important for the secondary mathematics curriculum, the former because of the frequency of angles of inclination in trigonometrical applications and the latter because it provides a valuable aid in teaching angle measurement.

Again, the more hands on practice children have at experiencing  the different elements of angle, the stronger their knowledge is likely to be.

Having just finished my work with a bunch of Year 3 – who are far fussier than I remember – I feel quite pleased.

They were able to build and complete the secret construction quicker and more accurately than I first hoped, they used a good range of language – next to, on top of, it looks like a house (!) – and generally they worked well together.

I used magnetic polydron to build the shape for the secret construction again. In fact, I decided I would use the same shape entirely for the job, with the same colours too! It was clear that they had done this activity before. They knew what was expected of them – although that didn’t stop me from telling them – and how to go about getting to a satisfactory end. Certainly the secret construction is an activity that works lower down school than I am used to and I can see it being of value to them from a vocabulary, communication and shape point of view. The only problem with magnetic polydron is that currently we only have squares and equilateral triangles, which limits the number of shapes we can make. I have included some of their creativity with the equipment from the very end of the session when I let them have a little play…

During the string activity, it was clear that they had limited knowledge of shape names. They struggled to predict what shapes would be made – although a couple did correctly identify a hexagon. When a star shape was created with a heptagon inside, I already knew they wouldn’t have a clue what it was, so focussed on the outside shapes instead.

Now, because of troubles when lowering the string to the floor, we ended up with some unusual patterns around the edge. We had trapeziums and triangles… the trapezium intruiged me. So I asked them what they thought it was called. Instantly, we had the name ‘square‘, which lead me to ask why they considered it a square. The response I got showed their knowledge, but also immediately reminded them they were wrong. They said, “It has four sides… but they’re not the same size, so it can’t be a square.” Intelligent thinking! This lead another child to say, “Well it must be a rectangle then!” Prompting another to say, “But it doesn’t have four right angles.”

While I wasn’t expecting this at all, it showed that a simple thing can generate such a wonderful discussion. To me it doesn’t matter that they didn’t know what a trapezium was, it was valuable enough for me to go back to their teacher and tell him that those children knew the properties, roughly anyway, of a rectangle and a square. And thinking about it, that’s all they should know. After all, it’s the fourth week of their first half term in Key Stage 2 – their knowledge of shape hasn’t been touched since the back end of Year 2 anyway, and that would consist of looking at the names of basic shapes.

If it taught me one thing, it’s that I haven’t been around Year 3 enough lately, that I’ve become used to the language and abilities of Years 5 and 6 so much that I’ve desensitised myself from children further down the school.

I certainly need to make time to work with them more throughout this course.

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.

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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