Language in mathematics

I am aware that although I am a maths specialist I haven't actually written anything about maths yet. Here is an assignment I did very early on about some misconceptions children in mathematics. Sadly the figures don't want to appear. 

Introduction

Language, and our interpretation of it, is the key to understanding and being understood in mathematics. Definitions change from person to person and can take many forms depending on the situation and how the thinker views the world. Johnston and Mason, (2005, p. 20) remind us that “Meaning depends on context. When a learner says something that seems wrong to the adult, it may be that the learner is stressing something that the adult is ignoring and vice versa.”

How we use language could mean the success or failure of a lesson and it is important to remember that children don’t always see the world as we do. If, when children are learning to define shapes, they see only one example of that shape, that image then becomes that shape. This is called the “prototype phenomenon” (Hershkowitz, 1990, cited in Fujita). Any other orientation of the shape is something else. There is a good example of this phenomenon in the introduction of Johnston and Mason’s, (2005) book, Developing Thinking in Geometry, where a young girl is show a triangle like figure 1.1. She is asked to name the shape, but can’t. Eventually she points to the girl sitting opposite her and says “It’s a triangle for her.” This example clearly shows that children will link imagers to definitions, which as Mooney, C et al (2007) points out, can aid with the meaning, but it is not necessarily a good thing in the long term.

I will look at a series of phrases said by children studying geometry. Each statement is incorrect in some way, so I will attempt to identify what is incorrect and where the problem arose from, be it from skewed definitions or misunderstood concepts. By doing this it will help me to understand some of the misconceptions about geometry that children have and reinforce my own understanding.

Children’s Misconceptions

“This cube’s got spots on all six sides!” 

The child saying this has done well to identify that shape she is observing is a cube, a three dimensional shape made from six squares, and that there are indeed 6 sides. The inaccuracy here is one of terminology. A three dimensional shape has faces instead of sides. It is quite understandable why she will have referred to the cube as having six sides instead of six faces, as in everyday English when referring to three dimensional objects, such as buildings, we would say “around the side of the building” not “around the face of the building.” Even when being taught about shapes at home or in school or nursery, it is very likely that the faces of a cube will have been called sides. This problem links  back to the idea from Mooney, C et al (2007) that definition is learnt through experiencing various examples, and not-examples, which will refine a child’s definition of a word. If a cube is never referred to as having faces then the child will never learn otherwise. As the only objects to commonly have faces instead of sides are clocks and cliffs it is little wonder that many children, and teachers, will say a cube has six sides.

“Symmetry?  Yeh, I know what it means – it’s when it’s exactly the same on both sides.”

When I look at this statement I can understand what the child means and how they came to that conclusion because until now I have never thought of what symmetry actually is. I can see that the statement is wrong but I can’t say why because I don’t know. By drawing several shapes and their lines of symmetry I have come to the conclusion that a line of symmetry splits the shape in two. Either side of the line is exactly half and a reflection of the other. So a line of symmetry can also be called a reflective line of symmetry.

After reading about symmetry in Mathematics Explained by Derek Haylock (2010) I can now see where the child and I went wrong with our understanding. There are in fact two kinds of symmetry, reflective symmetry and rotational symmetry. Reflective symmetry is a phenomenon where a shape reflected on a particular mirror line matches exactly with the other side. Rotational symmetry is where the shape rotated about a point and will fit exactly with the original shape without rotating a full 360 degrees. I think the child is only referring to reflective symmetry, which may be the only type of symmetry known to him, and should have said “it’s an exact reflection/copy on two halves/ sides of a mirror line, of the shape.” That would be a more accurate description of symmetry. It is an easy definition to forget or not understand as it does appear that the shape is “exactly the same on both sides.”

“It’s not a parallelogram, it’s a square.”

The statement is incorrect simply because a square is a parallelogram and there are many possible reasons why the child has had this misconception. According to van Hiele’s model as mentioned by Fujita (no date) this child is most likely to be at “Level 1- Visualisation: Identifying shapes according to their concrete examples.” The child most likely has what Fischbein (1993, cited in Fujita) calls a figural concept, which is a collection of imagers and definitions of a parallelogram, which does not include a square, so anything that does not conform to this image is not a parallelogram. This image comes from the way in which we teach children the names and definitions of shapes. It is highly likely that every time a parallelogram was mentioned the teacher will have pointed to something like figure 1.2 and said “this is a parallelogram.” This may be a good method initially to teach what different shapes look like, as it does follow the definition, it has two pairs of parallel sides. But unless over kinds of parallelograms are shown to also fit that rule the definition is void as a parallelogram will always take the original figural concept.

“[Referring to a pentagon drawn in her book…] you only need to check the angles to know its regular.”

All the interior angles of a regular pentagon are 108º and as the exterior angles of any simple polygon add up to 360º then each of the exterior angles must be 72º. A regular or simple polygon has all the angles equal and all the sides. So just checking angles, although a good indicator, won’t definitely mean that it is a regular pentagon. It is possible to have a pentagon with all angles 108º that is not regular, that is why you also have to check the length of the sides as well. This misconception has likely arisen from misunderstanding when talking about pentagons. If an emphasis is placed on angles of a regular pentagon and very little mention of the side lengths then the angles will be the main focus of the child when identifying regular pentagons. It is easy to overcome this problem by ensuring that children are aware of what makes a regular polygon. All angles must be equal and all sides must be equal.

"They can't be parallel lines cos they're not the same length"

Misconceptions about parallel lines seem to be common place in the classroom, Johnston and Mason, (2005) talked about a similar example where a boy said the lines weren’t parallel because they were not straight (vertical or horizontal). It is likely that this child has formed a figural concept weighted towards imagers rather than definition. When he thinks of parallel lines he will probably recall lines of the same length or specific shapes like squares and rectangles that are made up of straight lines the same length. This may be due to having more emphasis on imagers and not definition when learning about parallel lines. The definition parallel lines states that they never meet no matter how far they extend, so if more emphasis had been placed on definition and varying examples then this misconception would not have happened.

English Essay

Ok as promised here is my essay, minus the reference list.
Please give any thoughts, criticisms (not many) or praise.

‘Explain, with examples, the contrasting approaches to the teaching of reading that you have experienced whilst on school placement and throughout the module. How do speaking and listening support the development of literacy skills? ’

Language is a major part of what makes us human and separates us from the rest of the animal kingdom. It gives us the power to imaging, lie, joke and plan. With language we are able to think, and express, ideas and concepts that aren’t real, and talk about objects and occurrences that aren’t there. Bancroft (1995, p. 48) says this is because language uses arbitrary symbols; “there is no necessary connection between the symbol, be it word or gesture, and the object or idea to which it refers”. This is an important aspect of language as it also gives us the ability to write. Writing is not natural; it is a distinctly human form of communication. As Stephen Fry (2011) said “Writing was not invented for the purposes of writing love poems, or novels, or prayers, but actually for the rather more mundane purpose of taxation and accountancy.” This is a rather dull birth for something that can produce such emotion and passion, but it can help us to explain why the use of written language can be so hard for children to learn. Writing was born from necessity, as a tool to be used, this means that it is not innate, but a skill to be learnt and developed. Anyone can make marks, call it writing and find some meaning in them, but being able to take marks on a page you had no part in the creation of and make sense of, or read them, can be extremely tricky.

By the time children leave school they should be able to read, write and do simple maths. These are the skills which successive governments have promoted over the years, and with good reason. These are basic life skills that in our society and culture it is nearly impossible to get by without. Many policies have been brought out on how best to teach these core topics, (insert references later), but recently the teaching of reading has been top of the list to get right. The Rose Review (2006) was commissioned by the government in 2005 to look into the teaching of synthetic phonics as best practise in schools. The recommendations made by the report have been the major cause of the government’s stance on the mandatory teaching of synthetic phonics in schools.

Major research programmes across the globe conducted by England’s Department for Education and Skills (2006), the American National Reading Panel (2000) and the Australian Government, Department of Education Science and Training (2005) shows that the use of systematic phonics programmes for the teaching of reading are more effective than non-systematic or non-phonological programmes. Graaff, S. D. et el (2009) define systematic phonics programmes as “those programs in which prespecified sets of phonics elements such as simple grapheme–phoneme correspondences and onset and rimes are taught sequentially.” This means that synthetic phonics falls into this bracket. The move to mandatory synthetic phonics was a controversial one and there have been many researchers, like Wyse and Goswami (2008), against the decision because of the lack of evidence supporting it.

It is true that none of the major reports concluded which programme of phonics study was the best but the teaching of synthetic phonics is certainly not causing any harm. Assessment data from the primary school I was recently placed at shows how much of difference phonics is making with the reading ability of pupils. There are pupils on every phase of the ‘Letters and Sounds’ programme in every year throughout the school. The year 5’s and 6’s did not have phonics in key stage one, so some are now having to take phonics lessons to catch up with younger higher ability readers.

Research evidence summarised by Oakhill (1995) also proves the positive relationship between phonics skills and reading ability but there is some discussion on which way the relationship goes. Does phonics support reading or does reading support phonics? Or is the relationship much more complex?

The school I was placed in takes great pride in its phonics provision for all of the key stages. In key stage one there are twenty minute phonics lessons every day and the children are taught by a separate teacher for each stage. There is a very good spread of children in all of the stages and many of them are at a level far below where they should be. This is because of the high amount of speech and language difficulties that many of the children have. The area surrounding the school is high in deprivation and many of the children come in not being able to speak coherently or never having seen a book. The importance of the link between spoken language and reading ability is stressed by Riley (1999) who talks about how recognition of words and rhymes in speech is essential to being able to pronounce individual phonemes and blend them to form a whole word. This is recognised by the DfES (2007) programme ‘Letters and Sounds’, which includes many speaking and listening activities. It can hardly be expect that a child who cannot say the words and phonemes correctly can read or write them as well.

The letters and sounds programme is primarily based on teaching the children to read through the identification of graphemes and their corresponding phonemes. This is a ‘bottom up method’, where the child has to decode the word by recognising the letters, any grapheme phoneme correspondence (GPC), then blending the phonemes to make the word and hopefully get meaning from it. A lot of what I witnessed was aimed at the children learning specific digraphs and trigraphs. Each session would focus on one particular phoneme or grapheme and engage the children with a variety of activities involving words which contained said sound and spelling pattern. There are many phonemes with more than one grapheme, and vice versa, so the session would either focus on one or all of the different patterns. This is not the only method taught by the ‘Letters and Sounds’ programme, there are also a number of tricky words that just need to be learnt.

Even though we teach synthetic phonics so that children can decode what they read, words like “was” and “said” cannot be broken down into easily recognisable phonemes. These words need to be learnt, so at the start of every phonics lesson the children read through a bank of ‘tricky words’ together. The repeating of these words every day is aiding them in whole word recognition techniques, described by both Oakhill (1995) and Riley (1999), as committing the word to the child’s lexicon. This stores the word as a logogen so that the word can be recognised on sight and meaning taken straight from the memory store. This is similar to, but not exactly, the top down approach to reading. In this method children are taught to take meaning from the text, pictures and expectations of the book. Although this is a good way teaching very early readers, young children find far easier to pick out individual words instead of letters and sounds, it does mean, as Oakhill (1995) states, that the child will need to remember around 50,000 words to read almost fluently. This does not mean we should not encourage prediction, as fluent reading involves both orthographic and whole word awareness, but there is a real need for children to decipher words phonologically.

To be able to teach what is relevant to the child’s learning it is necessary to have some way of assessing their reading ability. This can really only be achieved by listening to the child read in paired or guided reading. On placement I observed the teacher taking guided reading sessions with different ability groups, listening to each child read and prompting where necessary. Solity and Vousden (2009) say there is some controversy over the use of reading but schemes but the school uses a mixture of reading schemes and ‘real’ books to facilitate learning. The reading scheme is taken home or read with an adult in school where as the ‘real’ books are used to foster a love of reading in school. To assess the reading ability of pupils the school uses the “simple view of reading” proposed by the Rose Review (2006) to plot where pupils reading ability stands and make informed decisions on what to work on next.     

     

There are many different theories and ideas about how to teach reading. Years of research has been conducted by psychologists, education practitioners and governments on how the brain learns to read, which method supports the child best and what brings out the best results, but there are still many different schools of thought on what is right. It has generally been accepted that phonics provides the most effective route to learning to read, but which phonics programme is best, no one knows. Phonics has to be backed up with comprehension because there is no point being able to read what you cannot understand but there are still many methods of teaching this. The most effective way of teaching, and what I have witnessed in school, is to use a combination of different methods that fit best with your class. There is not one method that fits all so judging what is right for the class and the individual will bring out the best results.  

          

 

 

       

Assessing reading

Nearly all of the children enjoy books. Or perhaps they enjoy owning books. We went to the library so they could get a book that took their fancy and read it over a few weeks. They love getting their books out and holding them but I didn’t often see them looking inside. They would carry them around the class and take annoyance at anyone who touched a book that didn’t belong to them. Many of the children had books which were far beyond their reading ability and understanding some of them just liked the pictures or the colour of the cover.

The standard of reading varied greatly throughout the class. Phonics provision was very good but because of the many other factors affecting the children’s home lives many of them had difficulty with reading. Even still I was very impressed with the ability of most. The level of comprehension was good as well. I saw no cases where the child could read but not comprehend a sentence.

One of the boys I heard read had some speech difficulties but this did not show at all in his reading, accept in some pronunciation, as he was a very confident reader. He had no trouble with less common words and could easily answer my questions about the text. I heard him read from the reading scheme only which is designed to give the children exposure to the GPC’s and words in increasing difficulty and also make the stories interesting for the child. The books were filled with pictures to aid with comprehension, give clues to tricky words and probably to make it more interesting, but this child took no interest in the pictures at all. In fact it took quite a bit of prompting to get him to answer questions about the hidden information in the pictures. This is either because he’s not very observant or I think he just didn’t see the point in looking at the pictures, after all he was meant to read the words not look at the pictures.

I got the impression that most of the children didn’t read at home, except for home work on the reading scheme. They would most likely be placed in front of the television or computer and left there. They like to be read to, sometimes, and do enjoy stories of their choice but reading seems to be a thing reserved for school only.     

An assessment of a pupil’s writing ability

I have had ample opportunity to observe the year one’s writing, and it has been quite interesting. The two main points I want to look at are spelling and letter formation because examples of these represent the child’s level of development and how they view our language. I chose not to look too closely at handwriting as it’s based more upon fine motor skills than the knowledge of language.

A lot of the children in this year 1 class have speech and language difficulties, which makes it very hard to teach them how to spell. If they can’t say the word properly or sound it out then they can hardly be expected to get it right. There is a lot of very good phonics provision within the school so subsequently most children are good at sounding out words, but even though they can get the phonemes right on how they say the word, the actual way they pronounce it is wrong. Strong accents can be a cause for some very strange spellings, to give an example of this I actually saw a child write cus instead of because.

There is a mix up with letter sounds and letter names. I noticed a rather amusing example of this in a phonics lesson; the children were asked to write tight and all of them wrote tit instead. This is quite excusable as the letter i is pronounced eye. They have been taught the tri-graph igh but don’t apply it. The other common spelling mistake similar this is writing pla and mak instead of play and make. Again the mix up is with the a sound. Strangely was is often spelt correctly and I put this down to the fact it has a spelling rule that has been well remembered, although there are plenty examples of wus.

The children are well trained to get aids to help them sound out but not all know what to do with them once they have them. Good examples of use of are using white boards to practice writing tricky words before writing it in the book. Coupled with this is using the jolly phonics sheet to find the correct grapheme. At the same time I witnessed a child who took a white board and didn’t do anything with it as well as having a help sheet they didn’t look at. She then complained that she couldn’t do it. Both children were at the same level. An incident my partner spoke about was that of a child using the letter sheet to trace the letters before writing them down. This is great use of tools but sadly it had very little effect and the work was still illegible.   

There are some key letters that nearly all the children seem to have trouble with writing. The letter e is often written backwards. The letter y is written as a u for some children and sometimes they mix up w and u. What I have observed in one child is her tendency to represent words with a single letter. She is not in a low ability group but will very often write down just a letter for a word. Some children only write the first letter of a word but this child will write a whole string of irrelevant letters. I asked her to write found and she wrote y. MY partner noticed that she will sometimes just write hihihi. This same child also puts capital letters I strange places, even when copying what I have written she will use capital letter. One child who is on P levels will only write the letter J.

Nearly all of the children enjoy writing, writing their name is enjoyable. What does happen is that they will get carried away or rush their work and what I get is a piece of work that is just a string of letters. Sometimes these can be made out to words that make sense but other times they mean nothing. What the children see is another matter. They sometimes know what it says, when they don’t they can sound it out. 

Literacy observations

The following posts are two literacy observations I was asked to write up for English. You might find them interesting. 

I will also upload my English essay after the deadline tomorrow. 

I know I haven't written much recently but I will get round to it.