Posts Tagged ‘Shape and space’

After Study Block 1, taught session 2, we were given a reading to look through and digest.

We were given Chapter 5 (pp. 79-104) from “Children’s mathematics 4-15: learning from errors and misconceptions” by Julie Ryan, Julian Williams. (McGraw-Hill International, 2007)

The mistakes children make in mathematics are usually not just ‘mistakes’ – they are often intelligent generalizations from previous learning. Following several decades of academic study of such mistakes, the phrase ‘errors and misconceptions’ has recently entered the vocabulary of mathematics teacher education and has become prominent in the curriculum for initial teacher education.

The popular view of children’s errors and misconceptions is that they should be corrected as soon as possible. The authors contest this, perceiving them as potential windows into children’s mathematics. Errors may diagnose significant ways of thinking and stages in learning that highlight important opportunities for new learning.

This book uses extensive, original data from the authors’ own research on children’s performance, errors and misconceptions across the mathematics curriculum. It progressively develops concepts for teachers to use in organizing their understanding and knowledge of children’s mathematics, offers practical guidance for classroom teaching and concludes with theoretical accounts of learning and teaching.

Children’s Mathematics 4-15 is a groundbreaking book, which transforms research on diagnostic errors into knowledge for teaching, teacher education and research on teaching. It is essential reading for teachers, students on undergraduate teacher training courses and graduate and PGCE mathematics teacher trainees, as well as teacher educators and researchers.

Anyway, I found an online version of the text via Google Books, which is embedded below. However, pages 82, 83, 87, 88, 94, 95, 101 and 102 are not included in the preview due to copyright reasons. Although the content that is there gives a reasonable account of the subject and includes many examples of good practice. Clearly, the lack of the full chapter really doesn’t help!

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.

Tomorrow, I work with Year 3, completing some of the activities that were so successful with Year 6. I plan to introduce the session with a secret construction, along similar lines to the one made at the start of last Wednesday’s lesson with Year 6 – a simple house like structure using only a couple of colours. Hopefully this will be at a reasonable level for them to work securely.

After this, I’m moving outside with them to work in circles. I will be having 13 children, more than I would like – so I will have a group of 6 and a group of 7. Hopefully these groups won’t be too small for a successful string challenge – I can always combine them into a group of 10 with a few onlookers and rotate the children as and when so everyone gets an opportunity to be in the circle. I am hoping for dry weather – standing up and lowering the string was harder than I thought for the 6s and so if they were sat on the floor, it would help a lot.

My main concern is that I haven’t worked with children of this age for a long while now…

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!

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.

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