Posts Tagged ‘Geometry’

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:

Key points:

  • “Early algebraic thinking in a primary context is not about introducing formal algebraic concepts into the classroom but involves reconsidering how we think about arithmetic.”
  • “arithmetic thinking focuses on product (a focus on arithmetic as a computational tool) and algebraic thinking focuses on process (a focus on the structure of arithmetic) (Malara & Navarra, 2003)”
  • “The aim was to assist 5 year-olds to come to an understanding of the structure of equations, and in particular the use of the equal sign.”
  • “When asked to find the unknown for 7 + 8 = ? + 9, many students express this as 7 + 8 = 15 + 9 = 24.”
  • “Three add 2 is different from 4 add 2. There is a different number on each side. Three add 2 is not equal to 4 add 2. It is different from 4 add 2″. Arithmetic thinking is required when computing the value of the two expressions. Algebraic thinking is required when placing the appropriate language between the expressions. Equals is only applicable if the two expressions are equivalent, that is, have the same value.”
  • “It is interesting that both Brianna and Ethan spontaneously introduced symbols as short hand for the object: p for pears and b for bananas or an iconic picture of each. In the secondary context this is commonly referred to ‘fruit salad’ algebra where the letter stands for an object instead of variable, and is thinking that we want to discourage. For example, a common misconception in the secondary context is that 3a + 3b stands for 3 apples and 3 bananas instead of a and b standing for any number. In the early years it is important to make the distinction between how we verbally describe number problems and how we represent these problems with symbols. While we say, “Three cars and five trucks,” the convention is to represent this as 3 and 5. Number sentences are made up of numbers.”

References:

  • Malara, N., Navarra, G. (2003) ‘ArAl Project: Arithmetic pathways towards favouring pre-algebraic thinking.’ Bologna, Italy: Pitagora Editrice.
  • Warren, E., Mollinson, A., Oestrich, K. (2009) ‘Equivalence and Equations in Early Years Classrooms.’ Australian Primary Mathematics Classroom, Vol. 14, no. 1, pp. 10-12

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!

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.

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