PGM 400 – Module Assessment v2

PGM 400 – Model Assessment: Portfolio and Commentary -2013-2014 Word Count:4249

Portfolio

During SBT1, there have been instances which have left me thinking ‘could I have done this differently?’ or ‘Just because it has always been done this way, is this the right way?’ As I probed deeper into these instances, I realized this is what Tripp meant by incidents which could be termed critical if I deemed to see them as critical.

My two incidents are in direct contrast of each other in regards to children’s subject knowledge, in incident One my dilemma was that the children did not know about roundings when I had been told they had covered this in a previous lesson. In incident Two the children were determined to show me their knowledge of measures, so much so, that I was left with ‘how are we going to get everything done in 55 minutes?’

My focus to my analysis of my incidents will be on the four approaches as outlined by Tripp and how I can apply these to improve my teaching practice. These four approaches are: Thinking strategies, the why challenge, dilemma identification and personal theory analysis.

I have listed two critical incidents based on my Primary With Route (PWR) which is Mathematics.

There are two sets in Mathematics in year 5, one Red group who are the higher ability class (this class is taken by the year 5 leader) and two Green groups who are the lower ability class. I am with one of the Green class.

On 16/11/13, I took six children (AP, JT, AM, AN, AR) out of the class, to an open area used by the teaching assistants, to teach them about roundings. The children were well known to me as I had worked with them for several weeks and we had built up good relationship of mutual trust. They found mathematics quite hard and were on a table which needed extra help and support. Usually the class teacher sat with this group.This lesson was observed by my tutor.

I started the lesson well in control: all the resources were in place, each child had a white board and pen (which worked). I stood at the front, by a larger board. I wrote five numbers in pounds and pence which I wanted the children to round either up or down (£12.63 £8.31 £71.78 £2.20 £6.50).

The lesson plan which I had prepared and had been approved by my mentor, showed that the children had previous knowledge of money worded problems. They knew about change or money leftover. ‘Surely roundings would be a walk in the park?’ (I thought).

I asked them what they would round £12.63 to. JT said £23, AB said £51. I asked where they were getting these numbers from? No response. I asked if they knew what £12.63 meant? ‘Yes’ they all said ‘12 pounds and 63 pence’. So they understood the money amount. ‘So, what about roundings, what do I

Student Number:U1302686Group:P8

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mean by roundings?’ AR said ‘Miss, is it something to do with the number 5?’ ‘Yes’, I replied ‘this number is key to roundings’. Can you tell me what the rule is?’. No response. I think my group needed some help. I asked ‘Could you round £12?’ ‘Easy’ shouts JT ‘it’s £10’. So could I deduce from this that the decimal point was causing the problem? I spent 5 minutes trying to work out what this group knew and how far we could progress. Roundings was meant to be a warm up exercise with the main lesson being on solving word problems involving money. It was ten minutes into the lesson and we were still on question one of the starter.‘What do I do?’ I thought, ‘Do I use my lesson plan or do I start again slowly going through the basics?’

On 04/12/13, I took a whole class lesson in mathematics with the Green set on worded problems involving measures. The class knew there were three types of measures (length, weight and capacity) as they had been looking at measures in the previous two lessons.

I wanted to keep the introduction to 10 minutes and no more.In my mind I knew the sequence I wanted my lesson to follow, I would:Explain to the class that they will be estimating and measuring the length of different objects today. Show the class the object.Ask them to estimate the length of the object in their groups.Ask them to feedback answers to whole class.Model how to record this information in their exercise book.Ask each group to measure the length and compare the result to their estimation.Which group was closest?Does it matter if our estimation is wrong?Get them to solve at least 5 problems in their books.This was a very engaged class, they were very keen to show me how much they knew about both estimation and measures. Giving all the children a chance to show their skills proved to be quite a challenge for me, bearing in mind I wanted them to start their independent work.

Thinking strategies as stated by Tripp ‘..offer a process that helps frame the kind of questions that will begin to produce a deeper reading’ (2012, pp.44). By this he means we always look at what happened in an incident but if we turn this on its head we could consider what did not happen and ‘seeing what did not happen often reveals the significance of what happened’ Tripp calls this type of thinking strategy ‘non-events’. Tripp identifies five type of thinking strategies which are open for thought.

Tripp (2012) summaries de Bono’s (1987) programme as he finds this the most useful of the thinking strategies which is the one I ,too, will follow.


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As a mathematician I know only too well that to balance any equation, the left side and the right side need to have equal weight. If this is taken into consideration when reflecting on an incident, then ‘every situation has both good (plus) and bad (minus) points about it’ (2012, p.45) makes sense and seems fair. I found this part of the extract very true ‘…for it is a matter of viewpoint whether a particular point is classified as good rather than bad, or vice versa’ (2012, pp.45) as I too have always felt our values and beliefs shape our thinking. In my second incident where the children were keen to estimate the board using their hand, I wanted each one of them to come to the front and have a go but I knew this was neither practical nor productive to their learning. The good part was the keenness shown by the class, the bad point was I could not allow all of them to show their skills as time management is a vital part of good teaching practice as stated by the teaching standards.

The non-events which Tripp mentions are the alternatives. So in my first incident where none of the children could round £12.63 up or down accurately was the non-event. It did not happen even though the previous lessons indicated the children should have had this knowledge. I have been informed both by UEL and my mentor that in every lesson, the children should learn something new. What new thing did they learn in the roundings incident? Roundings. This was a surprise.

In a classroom there is the class teacher (me), the mentor, the tutor, the teacher assistants and the children. All parties have their own opinion which may be similar or differ greatly. Comparing each others views helps to get a clear picture of what was good practice and what needs improving. In incident 2, the mentor appreciated my using my hand span to estimate the board, the teaching assistants did not offer an opinion even when asked and the children showed a lot of enthusiasm. TB came up to me and said ‘Miss, I love your lessons, will you be teaching us Maths tomorrow?’ I replied ‘Yes, I rather think I will be’. I have never seen anyone jump so high in the air, arms pumped up shouting ‘yeah’. He made me feel 10 feet high!

Teaching requires subject knowledge and class room management but it does not stop there according to Tripp ‘Because teaching is a social practice, we must examine our attitudes, values and judgements and work on those too’ (2012, p,45). In theme 8 of her book, Barbara Bassot asks us to look at our ethics and values and how this impacts on our professional practice. I agree with Corey, Schneider Corey and Callanan (2007) that it is not possible to be a completely independent thinker and leave behind our values in our professional life. What is important here is to determine what our beliefs are and see how we can constructively apply this to enhance our professional development. I believe that all children have talents, it is a matter of bringing it out into the open. Just because it is hard to find, does not mean it does not exist. We just have to try harder. In incident One, AR began her sentence with ‘Miss, I don’t know, I have no idea about roundings, I can’t do it.’ This is a repeated pattern I have observed in her, she seems to lack in confidence.


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The challenge for me is, I know she is capable of a lot more than she believes, how to improve my practice so she can benefit?

This is the opposite of what we actually do in our teaching. ‘Whether to do something or not is the basic dilemma of all our teaching decisions’ (2012, p.46). In both my incidents, I felt ‘if I had done the opposite would that have produced a better result?’ I had already argued this out in my head with the conclusion being, I had to stick to my lesson plan. This was my professional duty and took precedence. Moreover, this is what I had agreed with my mentor.

When we analyse our incidents we need to make sure we have thought of everything. Tripp mentions that our analysis of an incident should continue beyond when we have understood some part of it as to stop at this point is too soon. I wonder if it is possible for me to do this. In both my incidents, my lesson plan was very thorough but the time management became an issue for me. So although I understood this, according to Tripp, I should continue my reflection further as this will help me develop as a reflective teacher.

To ask ourself why something happened and keep asking until we reach some sort of conclusion. Repeatedly asking this question can lead to different results as mentioned by Tripp ‘But answers to Why? Do not necessarily follow one path’ (2012, p.47).In the roundings incident, JT rounded £12.63 to £23. Following the Why questioning went as follows:

Why £23?Because there is a 2 and 3 in the questionWhy not £16?NoWhy not?It does not make senseWhy does it not make sense?We can’t change the order of the original numbersBut we did change the order didn’t we when we rearranged the numbers to give £23 as the answer?I wasn’t thinking ….

This went on for a while. I know this is not exactly sticking to just asking Why but in essence I felt it did satisfy the Why criteria.

In dilemma identification, Tripp very neatly states ‘..the great stress of teaching comes both from the number of decisions teachers have to make every minute and from the nature of the decisions they have to take.’(2012,


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p.49). Both my incidents caused me quite a dilemma. As I mentioned earlier, should I have abandoned my lesson plan in incident 1 and concentrated on making sure the children understood roundings? In incident 2, should I have allowed the children to come to the front of the class to model estimation of measures?

Personal theory analysis by Tripp highlights. ‘…values, as one set of beliefs, influence our judgement’ (2012, pp.51). To my way of thinking, this is ingrained to the ‘parts and qualities’ thinking strategy, in which I have discussed the importance of my beliefs to my professional development, as just as I can’t separate my arm and leg from my body, likewise I cannot separate my attitudes and values from my professional judgement. A study by Rigby (1984) relating to authority showed significant differences between countries with some ‘..favouring .. authority more strongly.’ I too feel that I must respect and support the authority figure unless it goes against my personal beliefs.

Commentary

Haylock (2010, p. 167) emphasises that rounding should be taught in terms of real life context. He feels that rules can easily be forgotten but if children are given real life examples to consider then they will remember the topic better.He gives two contrasting examples. If we need to buy wallpapers we tend to round up as we want all the wall to be covered but if we had to catch the 9:05 train we would round down as we would not want to miss the train so we would try to reach the station before 9:05.

Further in the book, he also mentions what he calls ‘learning and teaching point’ He reminds the reader of the area to concentrate on in each topic.So when we consider the rule of rounding up when the next digit is 5 or more, Haylock’s learning and teaching point is ‘emphasize the idea of recording measurements ‘to the nearest something’ when doing practical tasks.’

Some examples of this are: at the petrol station , I buy 38.2 litres of diesel, this is recorded to the nearest tenth of a litre. My weight on the bathroom scale reads 11 stone 3 pounds, I am recording my weight to the nearest pound. If my niece runs the 100 metres in 9.96 seconds, I can conclude that the measurement is recorded to the nearest one-hundredth of a second.

During observed lessons, I noticed the class teacher frequently used real life examples which children responded to very well. In a lesson on averages, she asked each child to work out their height in metres and centimetres. She then asked them to calculate the average height in the class. The answer came to 1.4732 metres, this could be recorded to the nearest hundredth which would be 1.47m or 1m 47cm.


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My question to the children ‘What would you round £12.63 to?’ does not seem detailed enough. The alternative thinking strategy here would be to consider real life exampes as emphasized by Haylock in my modelling and to get the children to come up with examples as well. I should have set the scene so this could be understood by the children. How were they to know if I was looking for a whole number as the answer or a tenth or even a hundredth?If whole number, the answer is £13.If tenth, £12.6.If hundredth, £12.63. I obviously had not considered all the implications.

Cockburn (2005, p.4) makes an important point that it is not necessarily bad for students to make mistakes because if they were getting everything right then perhaps they are not being challenged. Similarly, mistakes can benefit children claims Melis (2004). From the omissions thinking strategy and the measurement lesson, perhaps the lesson was too easy for them. Had I thought of their levels, their pre-requisite knowledge in enough details? Is it possible for 30 students to all be so engaged at the same time?Applying the reversal thinking strategy, instead of being so worried that they were making errors in roundings, I should have questioned them more deeply. To improve my practice, I must concentrate on questioning the class more and use appropriate differentiation.

As problem solving skills take centre stage in the new national curriculum, Melis’ abstract mentions that through mistakes children can learn life skills such as problem solving, communicating and reasoning.As a teacher, I need to go through the steps with the children and see where the misconception occurs and how they can be corrected.

I noticed when the children used rulers to measure the length of paper some of them were not starting from zero. I stopped the lesson and asked one of the children to model this for the class. We discussed the importance of measuring from the correct starting point. Cotton (2010, p.162) mentions another common misconception ‘..the mass of an object is directly linked to the volume.’ He had several different size and shape boxes with items inside them, he asked the children to place them in order of mass. Then he asked them to lift each box. He had put a heavy item in the smallest box. He found the children were very surprised that the smallest box would be the heaviest.

Does the teaching of language help in solving the rounding problem?An article by Mercer (2006) considered a more general version of this question. He found that teachers who were able to encourage the children to think logically by providing ‘..guidance and practice in how to use language for reasoning’ were able to solve maths problems.By applying other point of view thinking strategy to teaching a topic, I need to be able to come up with practical examples using good language skills which encourages the children to think of ways of finding answers. It is very important that I can communicate at the right level for the class I am teaching. One of the areas I need to improve on as highlighted by my mentor has been that my lessons are more like lectures.


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In her study on math thinking Wakefield (1998) mentions children’s prior knowledge is essential in helping them to learn new concepts. ‘Children cannot see, hear, or remember that which they cannot understand. If the mental structures are not in place to support what is seen or heard, there will be no mental connection, and consequently, it will not be remembered.’ Also, Sengul (2013) observed that primary school children have low number sense so they may struggle with maths as there will be too many rules for them to remember.

With measures, the children may have helped their mother to bake cakes or measured their heights standing against the kitchen wall with a pencil and measuring tape. This could account for their excellent understanding of length, weight and capacity. I should be providing an environment that encourages children to explore and make their own connections. Wakefield (1998) feels it is fine to make mistakes because once we have corrected this mistake, we won’t make it again but rather this inspires autonomy and confidence in the child.

Applying thinking strategy of parts and qualities, I need to consider the children’s knowledge as a collection of parts. I need to know their prior knowledge, how do they like to learn (visual, auditory or kinaesthetic) and set up activities to guide them and let them discover the result for themselves.

Raynor (2010, p.65) mentions a common error occurs when numbers are rounded too early in a calculation. This became very obvious during our PWR week, when we observed the class teacher going through a lesson on how fair trade is better for the workers even though the percentages stay the same. The children had to fill in the ‘amount recieived’ part of the tables below.From Old Ford Weekly Mathematics Plan, the teacher started the lesson by telling the children to round up £1.60 to £2.00 and round down £2.20 to £2.00. This resulted in both sets of answers being the same and defeated the purpose of the lesson which was to compare the amounts received by organisations participating in non fair trade and fair trade as shown below.

Non fair trade: Lindt Chocolate: £1.60 rounded up to £2.00Fair trade: Green and Blacks Chocolate: £2.20 rounded down to £2.00

Percent received Amount receivedSupermarkets 22% £0.44Chocolate companies 40% £0.80UK government 17% £0.34Non cocoa companies 10% £0.20Farmers 3% £0.06Plantation owner 8% £0.16


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Non fair trade: Lindt Chocolate: £1.60


Fair trade: Green and Blacks Chocolate: £2.20


From the other point of view thinking strategy, the teacher would have benefited from discussing with colleagues on how best to tackle this question. Likewise for myself, once I have planned the lesson, I need to show my plan to my mentor and professional tutor, discuss it in detail and make acceptable adjustments which will benefit my practice.

Jones lesson plan shows how well roundings can be taught. I made many omissions in my lesson, I had not defined roundings or modelled any examples and the instructions were not clear (I had not specified the answer to the nearest ‘something’). There was no link to practical examples nor were there any illustrations. By contrast, the measurement lesson contained all these criterias and was very well received. Below is an extract of Jones lesson plan.

1. Introduce the lesson target to students: Today, we will be introducing the rules of rounding. Define rounding for the students. Rounding means reducing the digits in a number while trying to keep its value similar. The result is less accurate, but easier to use. Example: 73 rounded to the nearest ten is 70, because 73 is closer to 70 than to 80. Discuss why rounding and estimation is so important. Later in the year, we will get into situations that don’t actually follow these rules, but they are important to learn in the meantime!

2. Draw a simple hill on the blackboard. Write the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 so that the one and 10 are at the


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bottom of the hill, on opposite sides, and the five ends up at the very top of the hill. This hill will be used to illustrate the two tens that the students are choosing between when they are rounding.

3. Tell students that today we will focus on two-digit numbers.

4. With a number like 29, this is easy. We can easily see that 29 is very close to 30. But with numbers like 24, 25, and 26, it gets a little more difficult. That’s where our mental hill comes in.

5. Ask students to pretend that they are in a car. If they drive it up to the 4 (as in 24), and stop, where is the car most likely to head? (Back down to where they started.) So when you have a number like 24, and you are asked to round it to the nearest 10, the nearest 10 is backwards, which gets you right back to 20.

I could have created other possibilities for myself by the use of rounding rhymes which Smith (2005, Maths Cats) put on the internet:

1 through 4 stay on the floor 5 through 9 climb the vine

Underline the digitLook next doorIf it’s 5 or higherAdd one moreIf it’s 4 or lowerJust ignore

1 2 3 4 or less let it rest5 or more raise the score


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There are several articles (Rote learning equals maths confusion by ‘Helen Ward and ‘Rote learning: easy as ABC but is it as useful? Channel 4 news) which consider rote learning to be detrimental to students learning. A study published in TES states ‘lessons should focus on logic, not just arithmetic’ Applying my personal theory analysis, I would argue that how can reasoning be applied if the basic knowledge is not there?As a subject mathematics becomes harder as children move to secondary school, so when the schools minister, Nick Gibb, championed the importance of rote learning by saying ‘learning the times table by heart and being able to do long division helps children do well in algebra in their teens’ he was spot on. This was further emphasised by the education secretary Mr Grove who insisted ‘Only when facts and concepts are committed securely to the working memory – so that it is no effort to recall them and no effort is required to work things out from first principles – do we really have a secure hold on knowledge.’

In summary, I can improve my teaching practice considerably if I can through careful discussion and lesson planning do the following: model practical examples and give the children plenty of opportunities to come up with their own examples, allow them to make mistakes, let them explore and make their own connections, use age appropriate language in my lessons, know the children and their learning styles as well as their prior knowledge of a topic. I also need to question them more and apply appropriate differentiation.For me, rote learning is an essential part of student learning. It is very encouraging to see its comeback in the national curriculum. Reasoning and questioning play an equally important part once the basics have been grasped.

The final word has to go to Ward (2012) ‘The real need is to concentrate on how a curriculum manifests itself in a learning experience in classrooms… Teachers’ enthusiasm for maths, subject knowledge and pedagogy (are) by far the most important (elements) in that.’


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Bibliography

Cockburn, A.D. (2005) Teaching Mathematics with Insight. London & New York: RoutledgeFalmer Taylor and Francis Group.

Cotton, T. (2010) Understanding and Teaching Primary Mathematics. Edinburgh: Pearson Education Limited.

De Bono, E. (1987) The CoRT Thinking Course, London: Pergamon.

Haylock, D. (2010) Mathematics Explained for primary teachers. 4th edition. London: SAGE Publications Ltd.

Jones, A. Lesson Plan:Rounding.Available at: http://mathlessons.about.com/od/thirdgradelessons/a/Lesson-Plan-Rounding.htm (Accessed: 28 April 2014).

Melis, E. (2004) ‘Erroneous examples as a source of learning in mathematics’, German Research Center for Artificial Intelligence, Abstract.

Mercer, N. and Sams, C. (2006) ‘Teaching children how to use language to solve maths problems’, Language and education, Vol.20, No.6.

Rayner, D. (2010) GCSE Mathematics Revision and Practice. 5th edition. Oxford: Oxford University Press.

Rigby, K. (1984) ‘The attitudes of English and Australian College Students towards Institutional authority’ The Journal of Social Psychology 122, 41-48.

Sengul, S. (2013) ‘Identification of Number Sense Strategies used by Pre-service Elementary Teachers’, Abstract.

Tripp, D. (2012) Critical Incidents in Teaching: Developing Professional Judgement London: Routledge.

Wakefield, A.P. (1998) ‘Support Math Thinking’, Education Digest, Vol. 63 Issue 5, p59. 6p.

Ward, H. (2012) Rote learning equals maths confusion’ TES Magazine.


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PGM 400 – Module Assessment v2

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