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Conclusion - an endless road ahead

Lessons I have learned so far:

I have really enjoyed my first few weeks on placement. I have found Schon’s in-action reflection is what I lean towards, and, rather than dwelling on them negatively, I am learning to embrace my mistakes and use the experience of them to build upon in my quest to become an excellent practitioner. 

Some of the learners I teach are quite challenging but I am glad that I did not assume the negative stance towards them that some of my colleagues have, instead I got to know them with a positive attitude and without preconceptions so that I now have a good working relationship with most of them and they know that I am interested in their progress and want them to succeed.

Lesson planning is an art form!  But practise makes perfect and I will be getting plenty in the coming weeks. I have struggled to explicitly apply learning theory, only to discover that I have been using them without realising so this is something I am going to continue to research and embrace, rather than be in fear of.  

I am not afraid of being in front of a class. In my research into behaviour management, and from watching teachers in action, I hope to build on my calm and authoritative manner to become a calm and consistent practitioner with an emotionally secure classroom where learners can feel safe and confident (Dix, 2007). 

Finally, an important lesson which I have learned: there is no end to learning, even the most seasoned practitioner needs to keep learning, so the practice of reflecting on my practice, evaluating and challenging my practice, values and beliefs and applying educational research as well as maintaining knowledge of my subject area that I am learning throughout this course I will continue to use throughout my career so that I am always up to date in my practice for the benefit of my learners. (PS1, PS2, PS8, PS7)

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Behaviour Management

I chose the Behaviour Management Professional Standard (PS11) in my first observation... was this a case of grasping the nettle?  Behaviour management is a hot topic, or perhaps a hot potato, nowadays and considered with some trepidation by those entering the profession. 

Having worked as an exam invigilator in recent years, I think I was probably somewhat less nervous than some of my fellow student teachers in this respect, and the insight gained from being the parent of two teenagers has probably helped too.

The class I taught for my observation are a variety of ages, mostly within the 16-19 age bracket but with a few adult learners too.  I think that I have built up a reasonably good rapport with them, we get along well but they know that I am not a pushover and that I have respect for them but reasonably high expectations of them too.  Generally they are respectful of each other as well.

In the lesson, they were attentive and generally engaged but I did need persist with one or two of the learners so that they did not wriggle out of giving an answer.  One learner arrived a little late and I got him down to the starter task as soon as he arrived.  He does have a habit of leaving his coat on so as he sat down and I delivered his starter worksheet, I gave him a quick reminder of the college policy by saying “Don’t forget to pop your coat off once you have warmed up” as it was pretty chilly outside.  Within a couple of minutes he had taken his coat off but left his beanie on.  So I used a version of Paul Dix’s drive-by method, by just pausing by his desk as I was circulating and saying “Hat off too please.”  He duly complied.  

This learner is in his mid-twenties and thus one of the eldest in the class, so I wanted to use a light and non-confrontational method, which didn’t draw the class’s attention to him but got the message across nonetheless.  I was pleased with how this technique worked as this learner is the one I have taught least out of the class and as he is rather quiet I am still getting the measure of him, I felt that this interaction demonstrated that I have gained his trust and respect but I need to work on being more robust with him when he is reluctant to answer questions in class. 

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Cutting the Mustard

My first formal lesson observation completed, I got my feedback.

First the good...

I chose PS3 Inspire, motivate and raise aspirations of learners through your enthusiasm and knowledge as my first Professional Standard and was awarded a 2 for this.  The starter dragged a bit and I acknowledge that I need to be careful about how much time they are allocated to attempt the starter, I have been rather too focused on them all completing it rather than keeping the momentum of the lesson and ensuring that the main thrust of the lesson is prioritised.  With this in mind, I now look to give them a fixed period to work on the questions and then go through them all, ensuring that, even if they didn’t complete every question they see them all modelled on the board to support their revision in this way.  I felt the class and I were reasonably well unified in our mission and that I managed to keep them engaged, and when one or two disengaged I was able to encourage them back in and regain their focus.  

I need to develop my strategies for employing with those learners lacking in confidence to ensure they are engaged and motivated in every lesson because, though I firmly believe in it, endless positivity alone will not cut the mustard!

In relation to my second Professional Standard for the lesson, PS11 Manage and promote positive learner behaviour, I was delighted with the feedback that my classroom presence was “calm and authoritative” and that I was graded 2 for this as well.  I have not struggled with behaviour management as much as I anticipated but I do think that life experience has helped me in this regard, both through my work roles, especially being an exam invigilator, and, of course, from being a parent.  I am a great believer in mutual respect and endeavour to model this to learners whilst being clear about my expectations.  If you are positive and keep your expectations high, then they are more likely to reach for them.  Low expectations will be readily fulfilled and not escalated to higher levels.  However, some learners are more challenging than others and I will need to maintain focus to ensure that my behaviour management continues to move in the right direction and that I improve my awareness of appropriate strategies.

The bad and the ugly?

The other 4 Professional Standards I chose (PS 13, PS14, PS17, PS18) I received grade 3s for, which, I am assured, is not really bad or ugly for your first lesson observation but will keep me from being complacent or thinking this is all easier than I thought!   

Lesson Planning is a subtle art and I think part of my difficulties were related to my lack of mastery on those banks of knowledge and understanding that underpin this art.  

I need to improve my understanding of learning theories and how to utilise these when structuring my lessons.  

I need to be more insightful and deploy Bloom’s Taxonomy effectively so that the Learning Objectives are more SMART and engage the learners’ interest and motivate them to achieve them in the lesson.  

I need to ensure that I thoroughly explore the specific needs of individual learners, looking past the obvious diagnosis and drilling down into their profile for greater understanding of how to ensure inclusion of them in every respect of their needs.

Though the pace of the lesson was considered to be good, at the same time I need to always be mindful of ensuring understanding and being rigorous in exploring any wrong answers and not sacrificing this in favour of a lesson timetable. 

I also need to explore expanding my repertoire of assessment techniques. Formative assessment within the class is essential due to the pace at which the exam specification has to be covered and the lack of connectivity for ICT based solutions is a frustrating limiting factor on options available.  I have been exploring how I might be able to employ Metacognition in classroom (Dix, 2010) as well as learning about how my mentor personalises targets so that those subjects an individual learner might be struggling with are not forgotten and re-assessed later after the learner has undertaken homework and independent study.

Overall my first observation was not the disaster I feared and it has given me confidence whilst highlighting to me where I need to focus to ensure that the underpinning of my practice improves.

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A Baptism of Fire?

 I think most PGCE students view the first observation with trepidation from the start of their course, the first test of them in the classroom and I was no different.  But the greatest part of this initial test, for me, turned out to be the paperwork.

Mathematics as a knowledge base is considered to be clear cut, well-articulated and unambiguous (Steele, 2005) and mathematicians whilst not limited to thinking in straight lines, are used to having an outcome or at least a defined destination, they know their subject and it is has a clear structure.  New destinations of knowledge will use the same well used principles and procedure, just in new ways and these new ways will have a proof, evidence of their validity, which will have been discussed and agreed - there is coherence, stability and accord within the mathematical community’s thinking for the subject in so far as it is studied up to the level of higher education and, largely, beyond.  

Pedagogy by contrast is more nebulous and less well defined.  The definite proof, the incontrovertible evidence that any knowledge is correct, is harder to obtain.  Knowledge from practitioner varies not only depending on their exposure to theories, and arguably this has, in the past, been limited once a teacher has qualified, and their own classroom experience and how it has related to those previously learned theories.  Steele (2005) found that teachers rely on their own practice-based experience as evidence for their preferred pedagogies’ effectiveness and their own opinion and values are utilised when they were discussing pedagogy with other practitioners.  

Two very different mindsets that have to be held simultaneously by teachers of mathematics; an exact science and an inexact one having to be interwoven to produce an effective practitioner.  I think that I started to be aware of this contrast when I was writing my first observed lesson plan.  I had a very defined amount of information which needed to be taught, with a clear methodology but I also needed to state what learning theories am I utilising and why, how am I ensuring that I have catered for all aspects of the learner cohort’s needs, have I ensured that I have embedded other important aspects of the learning community’s requirements, what are my motivations for all of the above planning?  All this seems so disconnected from how to plot a straight line graph and I struggled to do this. 

 So I have been learning more about learning theories such as those of Bruner, Bandura and Vygotsky, not to forget Descartes who was both mathematician and learning theorist back in the 17th Century.  I believe that to a certain extent I have been employing Vygotsky’s concept of MKO (more knowledgeable other) (Armitage, 2016) when mixing pairings in the classroom (eg giant graph paper activity in my observed lesson) (PS4). Slowly I am breaking down my own barriers and beginning to fully understand how learning theories do relate teaching maths

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Mastery - small steps in the right direction

The snowy weather threw a curve ball at the end of term, there was patchy attendance, both staff and learners, and the planned week of mock exams in the English & Maths Department was thrown into disarray. As a result I spent some sessions supporting the small handful of learners, who had made it into college, in the Resource Centre doing MyMaths revision tasks online.  This gave me a rare opportunity to promote the benefits of technology and support learners in its use (PS15) as the wifi in the college is very poor and cannot be relied upon in the classrooms.

The concept of Mastery in maths expects repeated practice to over learn and move the learner towards mastery of a topic and exercises on MyMaths certainly deliver this.  Exercises will first practice and consolidate their knowledge in one type of calculation and then move on to further exercises which test their ability to approach the topic from a different perspective.  A fundamental concept of mastery in mathematics is to be able to articulate the knowledge you have and use it in a different context for example.  This is particularly pertinent for when learners have to tackle the problem solving type questions that arise now in mathematics exams as it means that they will be better placed to spot the knowledge and skills that are required to answer a novel problem.

Sadly, my current learners are too old to have experienced mastery in their primary education and probably their entire secondary education as well.  The time constraints that are placed upon the GCSE re-sit programme are such that there is not sufficient time to employ the mastery technique with the learners, however I do feel there is scope for a limited introduction covering certain areas of the curriculum which form the underpinning of many other areas: a possibility for my planning in my post-course future.  Articulation of percentages and the use of algebra would be my first considerations in this regard.  In a short paper published by NCETM in 2014 it was cited that “Taking a mastery approach, differentiation occurs in the support and intervention provided to different pupils, not in the topics taught, particularly at earlier stages.” and this would be the case for those basic areas; all pupils need a robust understanding of them and the ability to articulate them in different scenarios in order to progress onto other topics so keeping a mixed ability group on the same topic in this way would be desirable.

I would argue that there is even more of a case for using mastery with the functional skills learners for whom there is less of a time constraint and a smaller curriculum.  For these learners who may never progress to GCSE, mastering key maths skills that they will use throughout their lives is an attractive target.  I would really like to take this idea further and will be exploring the idea with my mentor and colleagues in the coming weeks when planning lessons with the Functional Skills classes (PS4, PS6). 

One of the functional skills learners I worked with on MyMaths, really seemed to benefit from the practice and was pleased to complete an exercise which expressed the question in words rather than in the first steps which give the exact calculation to work out.  Currently working at Entry Level 3, she has found difficulty in understanding what is required of her when faced with this type of question where she has to work out which number you divide by and which you multiply by etc.  She was genuinely delighted to have got all 20 questions correct and I took the opportunity to reinforce this positive feeling and motivate her to continue by asking if I could take a shot of her marked work  for my assignment as evidence of her excellent progress (PS13).  She grinned from ear to ear and was happy for me to take the photo, which did not include her name for data protection purposes (PS12), after which she moved happily on to the next exercise.

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A moment of joy

My first lesson teaching sequences was a bit of a bumpy ride as mentioned in an earlier post, but it ended with a lovely moment. Part of the structure of maths lessons the college is to have a “mixed starter” and one of the questions in that lesson’s starter was algebra - expanding brackets. One of the less confident learners, let’s call her Bea, was having difficulty grasping the method that I modelled so I worked through it again, one to one. She said she found the other grid method confusing and just didn’t get it, so this time I decided to show her an alternative method, probably considered the old school method, and she seemed happier and able to replicate what I had modelled (PS1, PS4). Then, at the end of the lesson, she came over to me as the other learners were leaving and thanked me for explaining it to her the other way as she understood what to do now. 

At the time it gave me a lift; we had managed to progress her learning and she had been motivated to thank me… plus of course that she considered me approachable enough to ask for help in the first place (PS6).

Later I reflected on what this meant in the bigger picture. In Maths there can be more than one or two ways to work out the correct answer, so who chooses which one is taught? Who makes the decision as to which one is best when different learners think different ways? Is there always one best way or do we try and teach them all and the learners individually choose the one that makes sense to them, or does that just confuse matters for them?

Interestingly, Bea was educated until she was 16 in a different English speaking country and I have noticed, over the weeks, that her maths knowledge is patchy in places and eccentric in others. The grid method that I had used with the whole class, at the suggestion of my mentor, was based on the multiplication grid method for multiplying large numbers together. This is commonplace in the UK in KS2 & 3 and so, I suppose, is a comfortable, familiar methodology for those who have used it with number to then use with algebraic terms. But for someone unused to the methodology and already finding algebra a tricky topic, it was clearly a step too far.  Doubling back to an old school method proved to be the appropriate solution to overcome an individual barrier to learning (PS16).

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A Lesson Learned - Sequences

The week after my first planning effort I planned for the same two classes but this time I was covering the topic of sequences.  Sequences is one of those topics that comes very easily to some but not to others and is more to do with a person’s ability to see patterns rather than their mathematical aptitude.  Teaching someone how to articulate that pattern is less than simple and I used a revision guide to inspire how I demonstrated this to the learners.  My mentor had to come to my rescue as the explanation I had utilised lacked clarity for the learners and she demonstrated a simpler explanation.  The one I had used was correct and widely used by maths teachers but my mentor’s method was just easier to grasp – experience had taught her this!   I used the revised demonstration with the second class and there was not a moment of uncertainty that time (PS10) .

It is this accumulated knowledge and understanding of how learners may have previously been taught and the ebb and flow of trends within the teaching of the subject from primary age onwards that I need to feel more secure about so that I feel confident that I am deploying the appropriate technique for my learners so that I do not contradict their prior learning where that is secure. Whilst it is frustrating at present, time and experience are really the only way I can improve this, so I need to be aware of this, do my research thoroughly and seek advice from my mentor and other subject specialists, and ensure that other elements of my competency as a teacher grow quickly to support this (PS6, PS7, PS8, PS19).

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When I had to review an article on Roger’s Unconditional Positive Regard (Iberg 2001), I was extremely circumspect - this was about psychotherapy more than education, what was the point of considering this as a tool for effective teaching?  I should not have been so closed in my thinking, though I think some of the blame does lay with the author of the article I reviewed, it’s not the most accessible piece of scholarly writing! And now to reality... of course there is room for unconditional positive regard in the classroom!  One of the classes I have is a group of disaffected 14-16 year olds; the reasons for the learners being in this group are varied, but all are not in mainstream school and some are not there due to extreme behavioural issues that have led to multiple exclusions.  All the learners could be considered vulnerable in one way or another, be it due to mental health issues, specific educational needs, previous experience of being bullied or behavioural issues due to problems at home.  They, of course, viewed me, as a newcomer to the classroom, with a certain air of suspicion but I was smiley and positive in my demeanour from the outset and did not impose myself on them as I got a feel for how my mentor runs this class.  It was clear to see from my seat on the sidelines that many of these children were quite “jangly” constantly on the edge of an extreme reaction if things went awry... be that an explosion of temper or dissolving into tears and upset.  But my mentor is a great role model for how to keep things light and upbeat, her focus on improving attitude and positivity rather than some arbitrary maths learning objective.  Maintaining regular attendance and good behaviour are the primary objectives so that, gently, learning can happen over time whilst improving their attitude towards the subject so their levels of engagement in learning escalates (PS4, PS5, PS6, PS12).

She talks to them about anything and everything around the maths that she is teaching and progress is made.  Watching my mentor in action, I realised that I needed to emulate her by simply being kind and yet self assured in my manner. One of the learners with behavioural issues was quite aggressive in her attitude but when we were talking in the mid-lesson break, I dropped in an admiring comment about her make-up skills... her face was beautifully made up using the latest in-vogue contouring technique, no mean feat for a 14 year old!  The transformation in her expression was amazing, her suspicion scowling at me disappeared to be replaced by a beaming smile, “Really?” she said, “Oh yes I am in awe, I could never do that!” I replied, with my mentor echoing my sentiment too.  That totally unrelated compliment, without any hidden agenda, seemed to change her whole attitude towards the lesson and she worked for the second half of the lesson with effort and success.  So here it was, unconditional positive regard in action.

I have reflected on that one incident quite a bit and I feel that having an unconditional positive regard is particularly useful when working with learners who are feeling vulnerable and in the world of re-sits, many do. (PS2)  I wonder whether more will have a more positive attitude towards their maths, hiding inside, if only it can be unlocked... and sometimes the key is there ready to be turned, it just doesn’t say MATHS on it.

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