Posts Tagged ‘Study Block 1’

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.

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.

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