Reflections on Teaching.

“It is important too, that rules are enforced firmly so that the pupils feel that there is a secure framework in which they know what kind of behaviour is allowed………..” (Backhouse et al, 1992, page 61).

At the commencement of the first lesson that I taught to the class, I waited at the doorway and pupils arrived and entered the class talking to each other. For safety reasons, pupils are allowed to enter the classroom on arrival. As pupils arrived, several of them called out, “Sir, are you teaching us today?” I raised my hand to acknowledge the questions but did not respond verbally:

“In the first meetings with pupils the status of the teacher is likely to be in question and it might be important to present oneself as one who initiates interactions rather than respond to the cues of the pupils.” (Robertson, 1981, page 32).

As the pupils were entering, I was ostentatiously looking at my watch. I still did not speak – just stood at the front of the class, acknowledging questions by raising my hand but not speaking. Gradually, the chatter in the class subsided and the class sat in silence looking at me. I said, “Do you realize that it took this class 6 minutes before it is quiet enough to call the register. I expect you to enter in silence and take your books out and begin the questions on the board. If I lose part of my next lesson waiting for the class to settle down, we will continue during break time.”

I felt this instruction complied with Tanners recommendation:

“Directions given to pupils should be concise and to the point” (Tanner, 1978, page 87).

During the remainder of the lesson, I referred to pupils by name, hopefully demonstrating “withitness” (Kounin) and that pupils would not be protected behind the anonymity of the class:

“The ability to name children in class will also suggest an alert awareness.” (Robertson, 1981, page 118).

I began each lesson by waiting for silence and then calling the class register while pupils worked on warm-up questions. I would then have a brief revision of the previous lesson before moving onto the concepts to be covered during the lesson. Expositions were generally structured around prepared resources:

 “If concrete materials or good visual aids are used, learners will generally find learning a concept less difficult than if an abstract approach is used.” (Backhouse et al, 1992, page 93)

and incorporated many examples:

“concepts of a higher order than the pupil already has cannot be communicated to him by a definition, but only by arranging for him to meet a suitable collection of examples.” (Skemp, 1978, page 32).

I would then issue an exercise for the class to perform. While the pupils were working, it gave me an opportunity to circulate amongst the class thus reinforcing CMC:

“Moving freely around the classroom will help convey that it is one’s own territory.” (Robertson, 1981, page 12)

and to assess the contributions made by pupils:

“Evaluation of the thinking and procedures employed by students usually is better done by careful observation and interview than by objective testing.” (Webb and Coxford, 1993, page 8)  

Furthermore, I planned lessons, whenever possible, to incorporate practical work, as I believed it would aid CMC:

“Control misbehaviour by keeping students actively engaged in classroom activities.” (Kounin quoted in Charles, 1998, page 34)

and it would provide a suitable mechanism to retain the interest of the pupils:

“Imaginative practical work can bring a particularly attractive flavour and enthusiasm to the mathematics classroom.” (Costello, 1991, page 54)

Whenever I circulated the class, I was able to assess who would benefit from being asked the questions later as I wanted to ensure that ALL pupils were willing to participate in the lesson.

A typical example of class interaction occurred whilst learning about shapes. I had organized the class into 6 groups of 4 pupils (2 absent) and each group had a pyramid, a cube, a cuboid and a prism to discuss. I issued the task for each group to “discover” at least 3 features about each of the shapes. I informed the class of the time allocated for this task, ensuring that it was a realistic target:

“When goals are realistic, it will discourage children from time-wasting or misbehaviour.” (Robertson, 1981, page 93).

I announced to the class that they had one more minute before we would begin to discuss each groups’ findings. I was conscious of the importance of the transition:

“the transitions from one phase to another become critical points in the lesson, at which the experienced teachers give clear cues, indicating to the pupils that this is a moment when the rules change” (Costello, 1991, page 74)

When the time allocated for the task was over, I said, “Pens down, everyone look at me please”. I asked one of the pupils to collect all the shapes and I picked up a cube.

Teacher: “Pupil A, did anyone in your group know what this shape is called?”

Pupil A: “Yes sir, It’s a cube”

Teacher : “Good, and what can you tell me about cubes? Pupil D?”

Pupil D: “They have 6 sides. All the sides are the same. They have 8 corners.”

Teacher: “Excellent. Did any of the other groups notice anything else about cubes?”

Some pupils call out. This was expected:

“Young children are usually eager to call out answers, and questioning sessions can easily become disorganized and disruptive.” (Robertson, 1981, page 105)

 and I took the opportunity to deal with this behaviour. I raised my voice instructing “No calling out” I wrote this on the whiteboard as a rule. I was dogmatic in ALL lessons about this rule. Pupil G was sitting patiently with his hand up.

Teacher: “Yes, what else did you notice about cubes?”

Pupil G: “They are used to make dices [sic].”

Teacher: “Good, and can anyone guess why a cube is used to make a dice?”

This time hands are put up.

Teacher: “Yes, Pupil H”

Pupil H: “Because a dice goes from 1 to 6 and a cube has 6 sides”

Teacher: “Yes, that’s one very good reason. And what have we already noticed about all the sides”

Pupil H: “They’re all the same size.”

Teacher: “Excellent, and why do you think it is important for a dice to have all sides the same size?”

Pupil A put his hand up and responded “ Because if one side of a dice was bigger than all the rest, you would keep getting that number when you rolled the dice.”

Teacher: “Fantastic. All the sides of a dice are the same to give each number the same chance of being landed on.” The class seemed genuinely enthusiastic that they had “discovered some real-life mathematics.”  

        Whilst, these initial lessons allowed me to establish some of my expectations, I was still very conscious of the need to involve all pupils in the lesson in order to retain their interest and I tried to incorporate as much interactive learning as possible, in all subsequent lessons:

“Teaching and learning have to be interactive”. (Black and Williams, , page 2)

But, I knew that if interactive lessons were to be successful then pupils had to feel comfortable in the class environment.

“In order to protect a pupils self esteem and develop self-confidence, it is important that questioning takes place in an encouraging and supportive atmosphere.” (Kyriacou, 1986, page 64)

During a lesson on time, I ensured that everyone understood the responsibilities of creating a suitable class environment. I asked pupil Z, a less able pupil, to answer a question after she had put up her hand. She answered but gave the wrong answer. This was greeted by laughter from Pupil C. I then raised my voice asking for silence. I could see that pupil Z was genuinely upset by this.

Teacher  - “Pupil C, could you tell me why you just laughed.”

Pupil C – “Because she got it wrong.”

Teacher – “Have you ever answered a question wrong?”

Pupil C – “But that was an easy question, sir.”

Teacher – “All questions are easy if you know the answers. I think pupil Z was very brave to try to answer a question when she wasn’t sure of the answer.” (Pupil Z smiles)

Pupil C – “Sorry Sir.”

Teacher – “I wasn’t the one you laughed at. Whom should you be saying sorry to?”

Pupil C – “Sorry pupil Z.” (She smiles at me again)

Teacher – “Good, Pupil C. Can you now tell us what you think the answer is.”

Pupil C – “21.30, sir.”

Teacher – “Good. And how did you work that out?”

I felt that ensuring pupil C apologized  would encourage mutual respect in the classroom and would also ensure that pupils would not feel discouraged from answering questions in the future. I also involved Pupil C in the lesson immediately to demonstrate that the behaviour had been dealt with and would not be dwelt upon. I also felt that this was a better approach than berating or lecturing Pupil C:

“The language used should be adult and should be firm when learners react with childish behaviour. Shouting is resented by young people and should be avoided.” (Backhouse et al, 1992, page 8)

        In a subsequent lesson, I discovered that the pupils are very enthusiastic about quizzes. For subsequent lessons, I offered the incentive of having a quiz at the end of every lesson, if the class had tried hard during the lesson.

“The opportunity to engage in a preferred activity seems a sensible incentive to offer pupils, provided it is dependant upon their working satisfactorily for the rest of the time.” (Robertson, 1981, page 124)

I evolved the plenary of lessons to take the form of a quiz. To involve ALL pupils, I issued each pupil with an individual whiteboard and pen. The questions would always focus upon the main body of the lesson. Typical questions would be(units of measure lesson):-

• In what units would we measure a road? (metres would be written on the whiteboards)
• How many centimeters are there in 1metre? (100)
• Can anybody think of another word that starts with “CENT” to remind us of 100?

This question would have several answers e.g. century, centipede, centurion, cents.

(This also served as a useful means of assessment – see later). Pupils became very motivated to compete in the quiz and by organising the class into teams, it provided a great motivational tool as identified by Kyriacou:

“Individuals have a drive towards joining in with others towards achieving some objective” (Kyriacou, 1986, page 39) 

Whenever performing examples with the class, I would try to incorporate “real-life” meaning. By using information from the “characteristics cloud” I was able to maintain interest in the topics at hand. One example of this occurred when we were discussing fractions. Pupil A was restless. But by asking the following fractions question, “If a Staffordshire Bull has 8 puppies and five are girls, what fraction of the litter are boy dogs?”, I was able to maintain his interest. This technique of asking a question that I knew had a special relevance to a pupil, proved effective in preventing misbehaviour.

“The adage “prevention is better than cure” applies with particular force to dealing with misbehaviour.” (Kyriacou, 1986, page 159) 

        During latter lessons, I tried to use comments as guidance, rather than to present the answers, as Robertson points out:

“At the point where the teacher provides the answer the motivation ceases, and it is far more satisfying if one can find the answer for oneself.” (Robertson, 1981, page 104).

        This addresses one of the earlier criticisms of my lessons - I was answering my own questions. Furthermore, I found it far more satisfying to see the looks of triumph when a pupil had worked out the answer to a “rock hard” question.

        

Misconceptions and Learning difficulties

        One of the most pronounced misconceptions I encountered was while teaching fractions. Several pupils thought that 1/100 had to be bigger than 1/5 because 100 is bigger than 5. I taught this by drawing a metre line on the board and getting pupils to come out and mark various fractions on the line.

Pupil Z marked ½ in the middle of the line as expected. I reinforced this by showing how the line had been cut into 2 pieces and 2 is the bottom number of the fraction. Pupil A then came to the board to mark on the thirds. He was able to correctly identify that the line had to be cut into 3 equal parts and he marked the fractions in the correct place. When Pupil J was called to mark on the quarters, it became apparent that there was still some confusion amongst certain pupils.

I then took out a family size bar of chocolate from my case. We are going to cut this bar of chocolate into fractions. If I want to cut it into quarters, how many pieces will I get?

Pupil J: “Four sir”

Teacher: “and how do you know it is four?”

Pupil J: “Because 4 is the number on the bottom of the fraction.”

Teacher: That’s right. So how many pieces do you think I will have if I cut this bar of chocolate into eighths?” (I wrote 1/8, 2/8, 3/8….. on the board to introduce the class to the “eighth family”.)

Pupil M: “Eight sir”

Teacher: “That’s right, but how do you know?”

Pupil M: “Because 8 is on the bottom of the fraction.”

Teacher: “Excellent. So who can tell me what how many pieces I should cut this chocolate into if everyone in the class is to have a piece?” (pupils are busily counting how many pupils are in school today. A flurry of hands wave in the air.)

Teacher: “Yes, Pupil Z.”

Pupil Z: “22, sir.”

Teacher : “Hands up if you agree with her.” (The whole class put their hands in the air.)

        Children grasped the concept that the more people they had to share the chocolate with (Denominator), the smaller the piece they would receive.

        Another misconception occurred when covering line symmetry. We had covered many of the “basic shapes” and had established that there were infinite lines of symmetry in a circle. The homework task was to write their name in capital letters and then count how many lines of symmetry for each letter. The pupils could then get a total number of lines of symmetry for their name. Some pupils had an “O” in their name so had infinite lines of symmetry. One boy had two “O’s” in his name so he said “I have the most because I have “two infinity plus 4.” The concept of infinity and “two infinity” proved difficult for the pupils to grasp until I remembered the old joke about an Irishman having two wishes. He first wishes for an everlasting bottle of Guinness. When asked about his second wish, he looks at the everlasting bottle and says “I’ll have another one of these.” Whilst, providing a humorous touch to the lesson, the pupils were able to understand the concept of infinity.

        By far the most pronounced learning difficulty that I encountered was the lack of confidence that many of the pupils had. Pupils had convinced themselves, or been convinced, that they were “no good at maths.” This proved a difficult paradigm to shift. Frequently, the class would be enthusiastic during the examples and exposition phase of the lesson. But, when the tasks were given out for individual working, the pupils would panic that they could not complete, or even start, the task. I recognized this problem and one method I used to overcome this was to allow the class to work in small groups. The drawback with this was that some pupils may not contribute to the group. I tried to overcome this by making the groups consist of only two members. Another method I used was positive reinforcement and praise. I praised pupils for effort even if the answer they submitted was incorrect and I would investigate fully how they arrived at that answer:

“Even if the answer is not correct, or is not the one the teacher was expecting, it should not be ignored; exploration……..can lead to a worthwhile discussion …….. and increased awareness of specific misconceptions.” (Cockcroft, 1982, page 72).

        The most successful method I used to increase confidence was to structure my tasks to start with very simple questions. I reasoned that if pupils were able to get the first few questions “under their belt” then they would be more confident to move on to the latter questions.

Assessment and Feedback

         Assessment occurred throughout my lessons. I observed:-

• pupils who were not able to put their hands up to volunteer answers;
• answers written on individual whiteboard;
• the types of questions that pupils asked me;
• the body language of pupils when performing individual tasks.

I recognized the importance of this “informal” assessment as identified by Cockroft:

“Examinations in mathematics…… cannot assess skills of mental computation …. Qualities of perseverance and inventiveness. Work and qualities of this type can only be assessed in the classroom and such an assessment needs to be made over an extended period.” (Cockcroft (1982) as quoted in Tanner & Jones, 2002, page 10).

This assessment allowed me to give instant feedback and encouragement to pupils. Several times I observed a pupil’s body language indicating that they were uncomfortable with the task. Often they required nothing more than comments such as “that’s right, and what do you think you should do next?”

In addition, I was able to review the work that the pupils had done in their exercise books. I took books in for marking at least once per academic cycle (2 weeks). I ensured I returned the books, fully marked, the lesson after I had taken them in:

“Work should be marked and returned quickly.” (Backhouse et al, 1992, page 8)

When marking exercise books, I developed to writing informative, positive comments on each piece of work I marked, as opposed to ticks and crosses when I first started marking. I wanted not only to establish the pupil’s level of understanding, but also to provide support and guidance on how to make any necessary improvements. I wrote specific comments to achieve the requirement identified by Tanner and Jones:

“Effective feedback ….. identifies steps which the pupils can take to improve their performance.” (Tanner & Jones, 2002, page 213).

        Assessment would have been almost meaningless unless I was willing to act on the information. In later lessons, I was confident enough to change the plan for my next lesson due to the answers written on the whiteboards during the quiz phase. I felt it important to ensure that the assessment was acted upon to improve pupils understanding:

“Assessment …. Is a means to achieve educational goals.” (Webb and Coxford, 1993, Page 1)

Contribution to Lessons

        All aspects of teaching were my responsibility, although the teacher was available for discussion if I felt it necessary. I was responsible for planning of lessons, marking of books, taking and recording of register, recording of marks, construction of exercise sheets and learning aids and production of resources. The school use textbooks that were issued to the class. Whenever necessary, particularly in later lessons as my teaching methods improved, I constructed lesson handouts which were issued to the pupils to save time copying notes from the board. This also took away a “boring” part of the lesson and thus enabled the class to remain “on task”.

There are examples of the learning aids and exercise sheets I constructed in appendix 3.

Review of Pupils’ Progress

   During the lessons that I taught, one obvious forms of progression was the general behaviour during the lesson. Questions were greeted with hands up rather than shouting out. Also, the number of people who were willing to contribute to the lesson increased. During the early lessons there would be a core of 6 or 7 hands going up for most questions. During the later lessons this was at least doubled. This enthusiasm spread outside of the normal class too. Frequently, I would be joined in Maths Club by pupils who wanted me to go over a concept to ensure it had been fully understood. This demonstrated an increase in enthusiasm for the subject.

I believed this change was induced  by the supportive and encouraging manner used during lessons. I ensured this was always the case by applying the rule of silence when anybody else is addressing the whole class.

“The need for people to hear what is being said justifies the rule that there should be no talking while a teacher or learner addresses the class as a whole.” (Backhouse et al, 1992, page 139)

I feel an equally valid reason for insisting upon silence is that pupils feel that their contribution is more valued if the class are listening intently to their input.

        In order to assess whether or not the pupils met the formal objectives of the lessons, I collected in their workbooks and checked both homework and classwork. Relevant feedback was given as appropriate. However, I permitted pupils to mark their own classwork during interactive classroom discussions.  

Pupils found the work on Shapes and Symmetry enjoyable and the work tasks were performed to a more than adequate level. The only problem area was plane symmetry where some pupils had difficulty visualising the planes of symmetry.

The work on measure was very popular throughout the class. Pupils were enthusiastic about real-life applications of measurement, the only initial confusion was between imperial units and metric units. Discussions about Olympic events and World Records allowed pupils to differentiate between the measures.

        As discussed earlier the work on fractions was far more demanding. Pupils responded very positively to the colour and shading exercises and to the linking up of fractions to the drawings. (See appendix 5). Again, the pupils responded far better to repeated examples and real life applications of fractions, as Backhouse identifies:

“Learners are generally prepared to work quite hard if they see the personal relevance of the topic they are studying”. (Backhouse et al, 1992, page 11)

        In summary, I felt the class showed marked signs of improvement in their behaviour, responsiveness, enthusiasm and confidence. I regret that I will not see if this improvement is sustained and the results manifested into improved grades.

Evaluations and Modifications.

I believe I enjoyed a very open, friendly relationship with the pupils., satisfying Backhouse’s criteria :

“A teacher who is perceived by learners as working for their benefit will make them feel more comfortable. They want someone of whom they can ask questions ……. with a reasonable expectation of receiving help.” (Backhouse et al, 1992, page 61)

        I felt that the lessons and resources were always well planned and fully prepared in advance:

“All teachers need to have clear ideas about the lesson they wish to set up and have carried out the necessary preparation if it is to be successful.” (Kyriacou, 1986, page 114).

        Whilst I planned lessons in advance, I remained flexible and responsive to the needs of pupils, satisfying Andrews’ requirements:

“Weaker teachers are more didactic, have difficulty managing pupils’ responses and adhere too rigidly to their lesson plans.” (Andrews, 1997, quoted in Tanner & Jones, 2002).

        During my time teaching, I was continually identifying areas in which I could improve.

One of these areas is to explore alternative ways of explaining a concept. During some lessons, I would explain a concept and some pupils would still not fully grasp it. Sometimes I would be “thinking on my feet” about how to explain this concept from a different approach. During later lessons, I always rehearsed at least two methods of explaining each concept. Kyriacou notes:

“It is interesting to note that a teacher’s ability to explain things clearly is widely perceived to be one of the most important teaching skills.” (Kyriacou, 1986, page 61)

        Although I perceived no problems with CMC, after establishing parameters of behaviour during my early lessons, I was expecting a “backlash” that never came. But I am mindful of Robertson’s statement and would like to be fully prepared for the most difficult classes.

 “the pupils first find it necessary to explore whether or not the teacher has the tactical and managerial skills to defend the parameters he or she is seeking to establish.” (Robertson, 1981, page 50)

        Additional areas, which I have targeted for improvement, are: -

• Variety of tasks used to induce learning;
• Strive to make resources as interesting as possible;
• Improved use of anecdotes and voice to encourage enthusiasm.
• Time management

Further areas for improvement may be identified during my next placement.

        This assignment represents a candid reflection upon my teaching experience with one Year 7 class. I have, wherever appropriate, referenced relevant educational literature in a concerted attempt to explain why I acted, and pupils reacted, in the ways identified.

Bibliography

Backhouse, J., Haggarty, L., Pirie, S., and Stratton, J. (1992) Improving the Learning of Mathematics, London: Cassell.

Biggs, J. (1967) Mathematics and the Conditions of Learning, Slough: National Foundation for Educational Research.

Black, P.J. and Wiliam, D. (2001) Beyond the Black Box, Cambridge: School of Education.

Charles, C. M. (1998) Building Classroom Discipline. Harlow: Longman.

Cockcroft, W.H. (1982) Mathematics Counts, London: HMSO.

Costello, J. (1991) Teaching and Learning Mathematics 11 – 16, London: Routledge.

Denvir, B., Stolz, C., and Brown, B. (1982) Low Attainers in Mathematics 5 – 16, London: Schools Council Publications.

Kyriacou, C. (1986) Effective Teaching in Schools, Oxford: Blackwell.

Robertson, J. (1981), Effective Classroom Control, Hodder & Stoughton: London.

Skemp, R. R. (1978) The Psychology of Learning Mathematics, London: Pelican.

Tanner, H. and Jones, S. (2000) Becoming a Successful Teacher of Mathematics, London: Routledge Falmer.

Tanner, L. N. (1978) Classroom Discipline For Effective Teaching and Learning, New York: Holt, Rinehart & Winston.

Webb, N.L. & Coxford, A. F. (1993) Assessment in the Mathematics Classroom. Virginia: National Council of Teachers of Mathematics.

Appendices

Appendix 1                                 Characteristics Cloud

Appendix 2                                        Clock Used to “tell the time”

Appendix 3        to 3e                                Examples of worksheets

Appendix 4                                        Display on Symmetry

Appendix 5 to 5g                                Worksheets on fractions

Appendix 6 to 6c                                Examples of Lesson Plans

Appendix 7 to 7c                                Lesson Comments

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