Posts Tagged ‘Child’

Key points:

  • The article discusses how algebraic equations can be shown as visual sentences. The example of x+y=4, where x>y is used. The Reception aged children are given two rules to colour them in: “they have to colour in four snails, and the number of brown-coloured snails must be more than the number of yellow-coloured ones.”
  • The author states how remarkable it is that the children of this age can complete this algebraic idea and that staff argue it should make it easier for the children to manipulate equations later in life.
  • The approach to teaching also requires children to discuss their ideas in groups, challenging each other’s answers, explaining their reasoning and arguing with the teacher who deliberately makes mistakes to generate such discussion.
  • It seems that both children and teachers are capable of exceeding perceived expectations through innovative thinking. Clearly, this is just one example and it’s hard to draw conclusions but it would be interesting to see where those children of 2006 are now in Year 5.

Main Reference:

Original Article:

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.

On Thursday I followed up Wednesday’s work outside by taking it a step further.

My Year 6 pupils were asked to create mini versions of our large circle using wool and paper plates. I had pre-cut the plates to have a range of evenly distributed and more random slots around the edges. Pupils were then asked to secure the string by sliding it into the slot once, looping it at the back, then sliding it down the same slot.

They were invited to invent their own rules to create a pattern (such as missing every other slot, or missing two slots to go down the third). Finally, they were requested to photograph the resulting patterns, a selection of which, representative of all ability groups, are shown below.

As you can see the potential for discussion about shapes is huge. There are a range of polygons, angles, regular and irregular shapes all visible. Also, with ones that haven’t quite worked, we can look at the reasons why. Interestingly, the plates with evenly distributed slots were far easier for them to use then the irregular patterned ones.

Yes, I could have followed the outdoor work up on paper with pre-drawn circles and dots around the edge as suggested, but I thought that this related to what was done outside, was a little more fun and – important in this world of changing curricula – helped to develop hand-eye coordination skills.

Many boxes were ticked here and all sorts of interesting conversations were had with the children about their predictions for the shapes they would produce and about the ones they created.

Next week, I work with Year 3 children along very similar lines. I haven’t worked with Year 3 for about three years now and so I’m not too sure what to expect as the main differences in outcome – language will clearly be different, especially with it being so early in the school year. As for what else will be different, I’m waiting to find out!

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