# Professional Investigation 1

### Introduction

The curriculum subject I am choosing interests me because the outcome may direct the course I take in my own teaching methods. The aim of the project is to contrast and compare the two approaches to the same material and assess the usefulness and shortcomings of each to try and determine the best approach.

Fundamentally I hope this investigation will give me the best approach to teaching mathematics, but even if it fails to give me an exact answer, it will give me a greater understanding of how pupils react to and learn from each method. This obviously will help a great deal in my own pedagogy and hopefully may prove useful to others who feel as strongly as I do on this topic.

The aim of the investigation is to discover the most suitable approach to teaching. A very in depth approach building from first principles to contrast against a rote learning approach. I must take into consideration that either one or both approaches will be suitable for each individual student. This may skew or give misinterpreted results as to which method is best, so I must take into consideration the student's best learning conditions. For this I could use VAK approach, so researching into this further may prove useful.

I must also cater for the situation where neither method may prove useful for a student, so researching into any possible alternatives to the two mentioned above would be a good idea.

### Literature Review

Perhaps the most important thing is to define what is truly meant by the word “learning”. We can define it like the following:-

“Learning is acquiring new knowledge, behaviours, skills, values, preferences or understanding, and may involve synthesizing different types of information.” [3].

Learning can at times occur where all of the above are acquired at once. For example, take a school child visiting a work experience placement in the latter stages of their secondary education. The job they experience may be such a job that the child understands the content of the job, e.g. a placement as an ICT technician where the child is very IT literate, and may feel that they are not learning a lot from the experience. What they may not appreciate are the subtle informal skills they are learning. These can include the formal ways to present work or how to dress in the work place, right down to simple interactions with colleagues. The skills learnt by the child informally and perhaps sub-consciously in this situation, regardless of how insignificant they may be considered, are none the less invaluable to said child's development.

A persons learning regardless of age, can be considered to fall into one of three categories [3] {taken from Blooms taxonomy}:-

- Cognitive - To recall, analyse, problem solve, etc.
- Psychomotor - To dance, swim, drive a car, ride a bike, etc.
- Affective - To like, love, hate, worship, etc.

In the example of the work experience student the cognitive skills learnt at school would need to be developed to be able to apply them in the work place. This is where the psychomotor domain of learning takes place. So it is quickly determined that often more than one domain is used before learning to be successful in a job can take place. In schools, especially in the teaching of mathematics, the primary domain of learning that is broached is that of cognitive. We often hear that in order to learn children have to enjoy the subject, or that children will learn much better when being practical in a topic, but sadly the ability to get the student to those points in a lot of subject areas are proved difficult to obtain or absent from the national curriculum. In mathematics in particular the national curriculum does not lend itself towards anything particularly practical and often the topics taught are not always necessarily interesting or relevant to most children's plans for the future. None the less teachers are expected to make the subject more “interesting” and to make it more “practical”. In certain areas of mathematics this is nearly impossible. In a lesson I observed concerning a year 10 group in a girl's grammar school, they were learning to use stem and leaf diagrams. This particular subject itself is difficult to make interesting and you cannot make it more practical beyond getting the students to practice them. As a result the girls, who are often considered to be more interested in their education to those that attend a comprehensive school, were uninterested in the topic and quickly became bored, then began to question the reasons behind learning the topic. It is likely that few of the girls in that lesson would ever use that information again when they leave school to join the work force.

So what exactly should the national curriculum do to promote learning, without changing the abilities and skills it can provide the young people in future jobs. Maybe the best way to encourage learning is to look at the very nature of mathematics and what it has to offer. Haskell B.Curry[4] writes the definition of mathematics as “ the science of formal systems”. He describes formal systems as the approach in mathematics to solving problems that arise and the use of tools available to us to achieve those solutions. So mathematics itself gives a skill set almost fundamental to any child's development to adulthood, the skills to analyse, solve and to think logically and abstractly over any problem that may arise in life. From my own observations in schools and from advice of qualified teachers the biggest problem with pupils they teach is the inability they have to use the skills stated above. When given a challenging task, their reluctance to even start is an issue that isn't being addressed fully in all schools.

One of the lessons I observed where such a task was set to pupils around the age of 13/14 was to come up with a timetable to organise a fictional table tennis tournament. They were split up into groups of 4 with no instructions but that of the task given. The majority of the class struggled instantly to look at the problem and to analyse it, preferring to ask for guidance on how to start. This wanting to lean on the teacher to guide them through the problem must of arisen through the teaching methods of old. Throughout their school life they have been taught with rote learning being their primary method of teaching, therefore creating a kind of expectation of what to expect from mathematics. None of the pupils I observed had been given a task such as this before and it quickly became obvious that it took them out of their comfort zone. The need to provide alternatives to rote learning should be paramount, to give these pupils the skills they will need in life.

Rote learning is a learning technique which avoids understanding the inner complexities and inferences of the subject that is being learned and instead focuses on memorizing the material so that it can be recalled by the learner exactly the way it was read or heard. An example of this is the reciting of the alphabet.

The method of rote learning can be considered a more outdated approach to teaching but none the less important. M.Collins[1] writes “Whilst this is useful in learning some basic things, such as times table or the alphabet, where it is the only form of teaching and learning used it is narrow and restrictive and does not teach children to think or how to apply that information.” It can be concluded from this statement that although rote learning is considered to be inferior if other methods of teaching can be applied to the topic, it is however useful on basic topics (primarily topics to be covered in Primary Schools, KS1/KS2).

So it comes down to this. Rote learning can be seen as denying pupils skills they should be gaining from studying mathematics, but on the opposite side of the fence can be seen as important in a lot of scenarios.

### With this said is there an alternative to teaching in the rote style?

Thankfully there is an alternative. You can adapt your teaching so the pupils reach the answers themselves, you can teach them first principles instead of just spoon feeding them some algorithm to solve questions that they won't remember in a week's time. Effectively your making a kind of abstract lesson where they are working towards the answers and not just being given them.

So what makes a lesson more abstract and less rote. Well initially the first thing you have to do is to make your lesson exciting. From my observations and advice from classroom teachers I have spoken to, pupils will prefer nothing else but to sit there and be given an easy method which they will write down and probably forget soon after. They prefer this approach not because it's more enjoyable, but it's because it's easier for them and doesn't require them to think independently. From my conversations with the department at my first placement, this is view that they consider is happening and are making an effort to branch out into a variety of different ways that encourages independence in the pupils own learning. One such instance of this was the class I quoted as observing above. So in order to keep it abstract they have to come up with how to answer questions themselves without being given set rules.

Now designing a lesson that encourages pupils to reach these conclusions on their own can be difficult. The aim of any abstract lesson must be to reach a set of rules that the pupils can apply and they will often need to be lead to these rules through a structured path of consideration and the clever use of words. One such lesson I observed at my placement school was that of a year 10 set 4 class coming across probability for at least the first time that year. Being a year 10 class, they would have seen probability lower down the school but due to the ability level of the class the approach used was to teach it as though seen for the first time. The lesson observed (my observations of the lesson can be seen in appendix C showed me a good approach through the use of mini whiteboards to firstly assess the level the pupils were working at and through the use of questions, aimed them towards the aims and objectives of the lesson.

### Methodology

The plan is to teach two similar ability classes a topic differently, one from an abstract point of view and one from a rote form point of view where the method is just taught. Due to my placement's choice of the classes I will teach and can create lesson plans for year 7 set 2's. Also because of the scheme of work my school is following the next topic to cover is that of adding and subtracting fractions.

Now before I design two different lesson plans a number of things have to be consistent in both lessons. We have to cover a certain amount of work and ideas to be able to solve all adding and subtracting fractions. Now looking closely at the different stages of the topic the pupils need to build up in the following way.

- Add and subtract fractions with the same denominator.
- Be able to add and subtract fractions with top heavy numerators and that of mixed numbers but still have the same denominator.
- Be able to add and subtract fractions with a different denominator, but one of the denominators being a factor of the other, e.g. 1/4 + 1/8.
- Be able to find the lowest common multiple of two numbers.
- Be able to add and subtract fractions where the denominators share a LCM.
- Able to add and subtract any two fractions aimed at their level of education.

Now to start work on planning a series of abstract lessons on adding and subtracting fractions. During my observations at a primary school I witnessed a very visual idea when teaching fractions to year 5's and 6's. The idea was to show fractions as portions of a pie or pizza. This way pupils could relate the numerical fraction to an actual real life fraction. I also witnessed the same approach when observing one of the year 7 set 2 classes when they were covering equivalent fractions. The classroom teacher had prepared some paper “pies” of varying different fractions to show the equivalences. This led to work out the difference between two equivalent fractions and formulate a method on their own. I decided to use this sort of visual aid in my lesson, but like the year 10 lesson I observed (appendix C) I decided to use mini whiteboards to not only assess the lesson but to keep it interesting for the pupils. I then decided I would display the fractions in the following way:- Where, I would get the pupils to shade in 1/5 of the shape, then shade in 2/5 of the shape. I would then get them to use their mini whiteboards to show me what they believe the fraction of the new shape was. This would not only give them a visual way of representing the adding of two fractions but allow them to come to their own conclusions about the rules of adding fractions.

I would then allow the pupils to formulate their own ideas over adding and subtracting fractions. I decided the best way for them to come up with some rules was to give them a few more examples including subtraction but keeping it with the same denominators. After these set of questions I would show the pupils a question that was wrong and ask them to tell me why it was wrong. I would then guide them into telling me the rules for adding and subtracting fractions.

Hopefully they should give me something along the lines of the following rules:-

- When adding and subtracting fractions you only add/subtract the numerators together.
- When adding and subtracting fractions you don't add/subtract the denominators.

I then believe we would be at the stage to bring in some work they had been doing recently and get them answering some questions with some top heavy and improper fractions. When dealing with mixed fractions there are multiple ways of adding fractions. One such way a pupil may decide to answer a question is to add the whole numbers and then add the fraction parts. Another way is to turn them both into improper fractions. I hope to build into my lesson some time to establish which way they worked out the answer.

The next step would be taking them to different denominators, but one of the denominators being a factor of the other. To cover this I am planning to return to the square idea as displayed above in figure 1. I would then get them to shade in the shape like before similar to the following example:-

The example shown in figure 2 shows a visual way of representing the type of question I want to get across. The pupils can then realise the connection between the two fractions. What I hope they realise at this point is the relationship between the two and a method for adding the two fractions. Like before I will ask a selection of questions and get them to comment on the method used.

### Hopefully we can then add some more rules:-

- When adding and subtracting mixed fractions you can add the whole numbers and then the fraction part.
- Alternatively you can turn mixed fractions into improper/top heavy fractions and add/subtract them that way.
- When you add and subtract fractions the denominator must be the same.
- We must use equivalent fractions to make the denominators the same.

The next step would be to move towards different denominators with a Lowest common multiple. At this point I would cover the lowest common multiple. I would then give them an example with different denominators and re-iterate the rules they should of established at this point. At this point I anticipate there may be some confusion as the rules are still fresh in their minds. Hopefully after a few more examples however we will arrive at the last rule:-

- When adding/subtracting two fractions you must find the lowest common multiple the denominators share and that will be the denominator of the answer.

At this point I plan to give them some work to do covering examples of all of the above. I plan to add some differentiation into the lesson by allowing them to choose at which point they want to start. All of the points I raised above allowed me to produce a lesson plan (appendix A) to deliver an abstract style lesson.

My next step was to produce a lesson plan that just covered a rote learning style approach. I decided to do away with mini whiteboards as this would be too similar to the abstract lesson. Initially I decided I would display all the rules at the start of the lesson, but this would be information overload and unfair on the pupils. I decided to follow a path similar to before, but instead of allowing the pupils to reach their own conclusions I would state the rules as before. I would start with writing down rules 1 and 2. At this point I would cover some examples myself on the board. I would then cover the rules 3 and 4. After establishing these rules I would move on to few examples on adding and subtracting mixed numbers, then show rules 5 and 6 moving on to some examples with different denominators, then finally show rule 7, where I will finish up with one more worked example, followed by some work from the text books or from a sheet. I will of course involve the class in my worked examples getting input from them as this is good pedagogy and not what I am testing. I will also add differentiation into this lesson as before and allow them to choose what stage they are at. The lesson I will deliver is shown in Appendix B.

I want the two lessons to follow the same format, the only thing I want to change is the way they reach the rules, one group will reach them themselves and the other will be told them from the word go.

### Ethical Considerations

I realise there are a number of issues surrounding such a plan. First of all the classes may differ in the mathematical background they would have received before reaching the secondary school, so certain individuals will have seen fractions before. This is something I cannot account for in whatever year/ability group I aim this at. Secondly I need to consider the ethics of teaching one class a fun, supposedly improved abstract way of learning while the others are subject to a more formal rote learning style or lesson. Initially this would seem unfair, but as this is my first teaching placement a classroom teacher is present in both lessons and will re-cover material if they feel it has not been taught well?. This means that neither group of children will suffer in their education. Also in my observations a lot of rote learning style lessons appeared to be a common theme so the students would be used to this anyway. The third issue is the fact that I am near the start of my teacher training and may not be able to do my lesson plans justice. Sadly this also cannot be accounted for, but I will assume that my level of teaching will be consistent over both lessons.

### Evidence

To show my results I plan to use a mixture of different results. I will assess the level of understanding between the two lessons myself and give a review of how each lesson went. I will also mark the work they are doing to establish which teaching approach proved the most successful.

### Discussion of findings

What I found from the two lessons was not as clear cut as I assumed it would be. The assessment of how I thought each lesson went, threw up a few different points for discussion.

The abstract lesson was well received by the students; they enjoyed using the mini whiteboards a lot and were eager to have the answers approved. It allowed them to get almost instant assessment and feedback for their work. It also made those students who would normally be less involved in the lesson take a step forward and explain how they got to their understanding of the problems. The pupils discovered a lot of the rules for fractions on their own. For those who where struggling, the visual representations proved very useful for them to reach the same level of understanding as their classmates around them. The lesson format also allowed for co-operation between people to get to the desired answer. In a slightly contrasting experience in the more formal rote learning lesson the pupils were also quite able to reach a level to be able to answer questions. Partially I believe this was because they could follow an example explicitly.

The abstract lesson reached a slight slow in progress when not all the pupils made the connection that the denominator must be the same to be able to add. Alternatively in the rote learning style lesson this was received by most and they were able to answer questions. What the abstract lesson was able to achieve was the ability to deal with mixed number fractions and fractions where both the denominators must be changed to the lowest common multiple. In the first instance they could formulate their own methods for answering questions and in the latter the understanding of how to answer questions came much more quickly. In the rote learning lesson working out the lowest common multiple and changing the fractions accordingly to be able to add them caused some minor headaches. I believe and what I assessed from talking to the pupils as I walked around the class, was they were forgetting important rules. This to me showed that for all its merits rote learning required some memorising before they could be implemented. It also meant a lot of practice before those methods became solid.

I also believe the abstract lesson also had its shortcomings. As previously stated when coming up with rules for themselves some of the pupils were still a bit reluctant and lacking in a bit of confidence in themselves to be able to give a definitive set of rules for adding and subtracting fractions.

I also got the chance, in the following lessons, to assess the abilities of their adding and subtracting fractions as the topic was finished up. I discovered that students of the abstract learning struggled initially to recall the methods and rules they needed to apply where as the rote learning could start at a slightly higher level initially. This quickly reversed themselves as pupils of the abstract lesson quickly got up to speed and pressed on, while the rote learning pupils struggled on harder questions.

The only other issue to note of the abstract lesson, was after using the whiteboards and getting feedback as we went along the pupils were much more confident in trying the harder questions, whereas in the rote learning lesson the pupils were lacking as much confidence and so started on easier questions.

### Conclusion

What I believe can be concluded from my research, my findings and my observations are that there is not necessarily a definitive answer to my investigation. I believed before the investigation and still do after that teaching a more abstract lesson, where getting the students to think for themselves will always be in their best interests. As I have mentioned before the ability to nurture young minds and allow them to build up valuable skills to analyse and assess, as well as building up their own confidence should be paramount in all departments in every school. As I discovered in my lesson planning and advice from colleagues was that this kind of lesson cannot always be possible. Some topics for example factorization do not lend themselves to this kind of approach. Other reasons can be the lack of time for planning that teachers have in their timetables.

Taking the abstract lesson apart and reflecting it I discovered a number of things. The abstract lesson although very useful for their understanding of the problems set forth to them did not necessarily set all the rules they need to follow in order to answer these questions. For example although they understood in a essentially visual way the process and reasoning behind why the denominator did not change, not all necessarily picked this up and got used to the idea. This is partly I believe to the way the pupils have grown up in the current educational climate where the lessons taught are very much in a Rote style. This leads them to being too reliant on the teacher to do all the explaining and rule setting and are reluctant to think about the problem and formulate their own rules. This conclusion rings true in all the discussion I have had with qualified teachers and observations I have made. Pupils are used to being told how to work a problem and when faced with a task in which they essentially investigate the rules themselves they lose interest. In a lot of examples observed they were very happy to do lots of the same question over and over.

Taking the Rote style lesson into reflection I can see the benefits and drawbacks to this type of lesson. The benefit is that they establish a set of rules and methods to be able to answer any problem. The major drawback of this is they don't necessarily understand the reasoning behind these methods. In the rote lesson I took they found it difficult to put into practice the methods taught when a slightly different question arose, whereas in the abstract they were able to progress onto harder questions easier.

### Recommendations

What I found in my conclusion is that there are benefits to teaching in an abstract way as well as teaching in a rote style. The underlying benefits of the abstract lesson is they formulate their own methods to answer the questions and therefore make the progression to harder questions easier, but the drawback is the time taken to be able to formulate their own methods. The rote style benefit is they can learn a method to answering a question very quickly and easily but quickly become stuck when they come across a question that doesn't match up exactly to the methods or worked examples they are shown. What I recommend I could change in future lesson is to try and combine these two styles.

True rote learning is considered to be outdated and a more abstract way of teaching is being promoted for learning in the present day, but the benefits of both styles are open for use. Our current educational system already contains a lot of rote learning so the pupils coming through secondary schools are already very used to this. Changing the entire educational system at this point to support an abstract style may cause more issues and as mentioned before we don't want to lose the benefits of those rote lessons. What I propose is to mix up the two lesson plans I wrote and to teach each step of my lesson in two different ways. I would start with an abstract point of view and allow them some time for those who can to formulate their own ideas, before translating this into a rote style approach in which I tell them the rules and methods to answer such questions. This allows them the chance to understand the topic before being told how to answer question on it, hopefully meaning that when the questions are different to what they are used to they can use that understanding to answer it.

If taking any further classes on this topic I would attempt to try and teach a lesson that cover both learning styles and this could be used as a point of further investigation in the future.

### References

- [Website] date: 4/11/2009, M.Collins, http://www.helium.com/items/1592076-why-you-should-say-no-to-rote-learning.
- [Book] Learning Mathematics, 3rd Edition, Issues, theory and classroom practice, Anthony Orton, ISBN: 0 8264 7113 7.
- [Website] Date: 5/11/2009, Wikipedia, http://en.wikipedia.org/wiki/Learning.
- [Book] Philosophy of Mathematics, 2nd Edition, edited by Paul Benacerraf & Hilary Putnam, ISBN: 0-521-29648-X.