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The focus of this paper is a Secondary level school of Roman Catholic denomination of predominantly white ethnicity. The class under discussion consists of thirty, mixed gender year ten students with an age range of 14 to 15. Their level of ability is mixed varying from intermediate to high. Their attitude towards learning is unfocused and the group has issues with concentrating in the classroom. The overall objective as illustrated here was to teach the group about multiplying out brackets using surds; rationalising the surd denominator and changing recurring decimals into fractions.
No pupils in the group have English as their second language and there were five gifted and talented individuals in the class.
A review of relevant literature on teaching, learning, assessment and the aspect of mathematics
One of the key resources for referencing specific outcomes and objectives where teaching algebra and number theory to this age group can be sourced is from the National Curriculum itself.
The key concepts for Key Stage 4 pupils and Mathematics include:
Applying suitable mathematics accurately within the classroom and beyond.
Communicating mathematics effectively.
Selecting appropriate mathematical tools and methods, including ICT.
Combining understanding, experiences, imagination and reasoning to construct new knowledge.
Using existing mathematical knowledge to create solutions to unfamiliar problems.
Posing questions and developing convincing arguments.
Applications and implications of mathematics
Knowing that mathematics is a rigorous, coherent discipline.
Understanding that mathematics is used as a tool in a wide range of contexts.
Recognising the rich historical and cultural roots of mathematics.
Engaging in mathematics as an interesting and worthwhile activity.
Knowing that mathematics is essentially abstract and can be used to model, interpret or represent situations.
Recognising the limitations and scope of a model or representation.
(Sourced from: http://curriculum.qca.org.uk/key-stages-3-and-4/subjects/mathematics/keystage4/index.aspx?return=/key-stages-3-and-4/subjects/mathematics/index.aspx, date accessed, 24/03/09)
These should be referred to as regular guidelines for implementing strategic teaching tools within the framework of delivering mathematical learning at this stage.
Nickson's Teaching and Learning Mathematics: A Guide to Recent Research and Its Applications is a consolidated compendium of research from across the world covering issues such as pupils' understanding and handling of numbers, algebra, measurement and problem-solving, as well as how these need to be assessed, alongside the impact and influence of ICT within the classroom setting and mathematics. The text concentrates on reviewing theoretical studies, whilst measuring different teaching methods and looking at issues of gender where this subject is concerned.
Goos et al ascertain the link between research and practical classroom delivery where Secondary level Mathematics is concerned in Teaching Secondary School Mathematics: Research and Practice for the 21st Century.
The authors discuss the challenges that many secondary mathematics teachers have and provide tried and tested examples that illustrate to teachers how they can build on their own experiences to ensure that their students successfully develop concepts and skills in mathematical thinking, in addition to gaining a positive attitude towards the study of this subject. It incorporates methods of assessments as well as pedagogical advice.
As the group being taught had a significant number of gifted and talented pupils it seemed appropriate to gain as much of an understanding of how this could be handled in the classroom as possible. Bartovich and George have complied a number of strategies for teaching mathematics to this standard of ability in their paper Teaching the Gifted and Talented in the Mathematics Classroom. The authors review methods for identifying mathematical ability and screening pupils as well as looking at taking different approaches such as cultural and academic enrichment and fast-tracking.
They detail procedures for implementing teaching strategies like fast paced mathematics classes and individualized classrooms. Similarly Tandi May's Teaching maths to pupils with different learning styles, provides a number of important solutions and ideologies around coping with scenarios that involve teaching diverse groups of children with different learning needs. The text assumes that many pupils finding even grasping the basic concepts of Maths difficult. As a consequence it offers practising teachers a range of approaches that can help struggling students and makes suggestions to stimulate pupils including:
- Ideas for lesson activities
- Suggestions for more dynamic , visual ways to teach basic concepts
- Practical advice and guidance.
And encourages teachers to explore the use of a variety of methods to teach this subject to both primary and secondary level pupils. Tandi May is also a Principal Researcher at the National Foundation for Educational Research.
In terms of generic studies the International Handbook of Mathematics Education is the leading resource for this subject. Over 150 authors, editors and chapter reviewers were involved in collating this volume of thirty six chapters providing a range of perspectives on the study of mathematics education this century. It is the principal reference work for teachers, practitioners and aspiring mathematicians and provides a constant useful source of information in this field
There are a number of relevant journals that regularly publish new scholarly activity around the subject of teaching and learning Mathematics. These include
The Journal for Research in Mathematics EducationÂ (JRME) which 'Promotes and disseminates disciplined scholarly inquiry into the teaching and learning of mathematics at all levels, including research reports, book reviews, and commentaries.' And is available online at http://my.nctm.org/eresources/journal_home.asp?journal_id=1
The International Journal for Mathematics Teaching and Learning is only published in electronic form with the aim to enhance mathematics teaching for all ages and abilities by way of articles, reviews and informed theory from around the world. It was developed for the benefit of practitioners and educationalists to inspire innovation in mathematics teaching and learning. The journal can be accessed from: http://www.cimt.plymouth.ac.uk/journal/default.htm
JCMST or the Journal of Computers in Mathematics and Science Teaching, is a useful resource which can be visited at http://www.aace.org/pubs/jcmst/default.htm. It is a highly respected scholarly journal offering 'an in-depth forum for the interchange of information in the fields of science, mathematics, and computer science. JCMST is the only periodical devoted specifically to using information technology in the teaching of mathematics and science.'
There is a vast array of academic, theoretical and practical guidance for mathematics teachers. The above examples represent a taster of what is currently available in this field to provide assistance, ideas and support for the classroom, classroom activity and child psychology.
In respect of refining literature into that which specifically focuses on the teaching of Surds, which is the basis for this report, few books exist and most are written for the benefit of the student. However Read et al's Key Maths: GCSE contain a chapter looking at the concept and application of Surds. It is written by practising classroom teachers and was developed in consultation with students and teachers to ensure a successful practical approach to comprehensively exploring GCSE Mathematics in depth. The book also differentiates to cater to a wide ability range. Year 9 Advanced Mathematics by Lyn Baker offers a section on Surds, explaining them in their simplest terms for the benefit of a reader studying year nine mathematics. From this perspective it can assist with entering the mind of this age group and how they might better comprehend the concept of Surds.
A description, with rational, of what you did with the students, how you did this and why you did this
Most of the research into student's conceptual difficulties in algebra can be analysed in relation to Piagert's theories; that cognitive development proceeds through a series of successive psychological stages, in which each stage is dependent on previous stages of development. These stages of development are termed 'reflective abstraction' whereby actions and processes at one stage become objects of thought during the next stage. Therefore teaching a child a concept before they are ready can prove to be counter-productive and result in mis-comprehension, masked by a superficial level of understanding. (Sutherland et al, 2001: 234)
From this perspective my lesson plans were influenced by the notion that both prior learning and understanding of similar mathematical concepts needed to be taken into consideration and the way in which Surds should be taught should emerge from a basic introduction and to gauge the overall level of understanding first before progressing with the subject itself.
By Key Stage 3 the pupils should have a basic understanding of algebra to include:
Equations- How to construct and solve them.
Expressions- How to simplify algebra
Symbols- To understand symbols and their different roles
Formulae- Acquired from maths and other subjects
Sequences- Build and describe sequences
Functions and graphs- Plot graphs
In particular Students will experience difficulty in rationalising surds if they have problems multiplying and simplifying fractions, (Sourced from: http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/number/N57501P.htm, Date accessed, 24/03/09) Therefore it was advantageous to gain a preliminary understanding of their level and understanding prior to teaching this topic.
Surds are a level 7 and level 8 GCSE topic. Level 7 understanding involves simplifying surds, as well as addition and subtraction. Multiplying surd brackets with single surd numbers or a whole number and expanding the surds brackets is taught at level 8.
Overall both areas were covered as a topic across a sequence of three lessons.
The major difficulty with Algebra for most students lies in acquiring appropriate meanings for symbolic statements. (French, 2002)
In particular students can find it difficult to accept that a surd in its simplest form really is actually the simplest. For example ïƒ- 32 to a student is simpler than 4ïƒ-2. Similarly ïƒ-48 appears simpler than 4ïƒ-3.
As a consequence it is often easier to ask them to reduce this to the 'lowest' form. However in a formal examination setting most questions will request that students 'simplify'.
Doug French recalls a past student in his book Teaching and Learning Algebra as a female student, Margaret who was perplexed by the equation 3x - 7-5 and unable to solve it. She went onto confidently assert however that 3x - 6 = 2 with the solution x = z.
Claiming that it was 'daft' to attempt to calculate x + 3 =15 etc. (French, 2002)
Bearing these misconceptions and logical approach to equations that this age group typically adheres to I used prime factor decomposition to teach students to reduce surds to their simplest form. Despite being somewhat time-consuming it is extremely easy for students to apply. I was also aware that when using prime factor decomposition students initially are often unsure about the 'correct' factor pair to begin with. Consequently the group required considerable reassurances when comprehending that any start point is correct and will generate the same solution.
Essentially then Surds were introduced using a small interactive exercise to assess their previous knowledge and then students were asked to make groups of rational, irrational and square numbers. While discussing the answers in a group the difference of rational and irrational numbers was revisited. It is important to note that in terms of the group's overall knowledge they had received instruction on rational and irrational numbers which ultimately helped them to grasp the overall concept of Surds.
The simple exercises during the first lesson included matching the pairs of equivalent numbers, finding the Square number and expressing the surd as a product of the square number and another number. Typically there is less chance of error if the Square number is written before the other number. For example:
2âˆš300Â Â Â Â Â Â square number is 100
= 2âˆš(100 Ã- 3)
= 2 Ã- 10âˆš3
Following an explanation of the learning materials and objectives students were then encouraged to practices what they were learning for 25 minutes. This practice consisted of exercise book work and interactive white board activities with the aim of getting the group to calculate how to multiply surds in double brackets. Haggarty in her book Teaching Mathematics in Secondary Schools has observed pupils in the early stages of a GCSE sixth-form course who have chosen to study maths further and particularly struggling with Algebra as well as experiencing feelings of total inadequacy. Yet Haggarty maintains the solution to this can often be found through text books which can help increase their algebraic fluency (Haggarty, 2002:136). Subsequently practice through individual worksheet exercises were encouraged enabling students time to familiarise themselves with this new area of work.
At the end of this lesson students were confident in their abilities to simplifying surds.
The focus of lesson two was to ensure that the class developed their skills for rationalizing the surd denominator.
During lesson two motivational exercises including a shopping scenario were incorporated into the learning process. It is important to remember to generate new activities that respond to the learners needs (Heacox, 2001:56) particularly taking into consideration that despite the high number of achievers in the group there were a number of intermediate and gifted and talented students, all of which seek different leaning needs. This diversity of the use of learning tools was also included in lesson three whereby pupils progressed onto learning to convert recurring decimals and began the session with a power point presentation.
In order to better understand Surds students need a lot of practice with a range of problems. Subsequently all three lessons utilised a combination of interactive whiteboard sessions, work sheet tasks and practice and discussion in order to consolidate this process. After lesson three the children demonstrated a clear ability to comprehend surds and solve surd equations.
An analysis of learning
The main syllabus outcomes for grasping an understanding of Surds should include the following:
An ability to perform operations with surds and indices
To use and interpret formal definitions and generalisations when explaining solutions and /or conjectures
Be able to link mathematical ideas and make connections with any generalisations about existing knowledge and understanding in relation to previous work.
(Kalra & Stamell, 2004:28)
Initially it was imperative to make sure that I had a complete understanding of the following key fundamentals of the group in order to be able to teach them effectively to a point where they will be able to fully comprehend Surds. These being:
To know prior knowledge the pupils had or should have gained at this stage
To realise what problems and misconceptions might exist for the group overall where algebra is concerned
How to assess the class
How this assessment can be used to evaluate their learning and my teaching ability
Assessing learning enables teachers to relate what pupils are learning now to what they have learnt in the past and to pave the way for what they will learn in the future and to help students at all levels of mathematical development to recognise the links between the different aspects of mathematics and the individual topics. Typically this includes making the connections between decimals, fractions and percentages so that pupils learn to convert from one to the other as well as appreciating the relationship between them. (May, 2005:7)
In this particular case the formative assessment I chose to undertake involved continuous assessment which was achieved by assessing pupils during the lesson, checking their work throughout the lesson and asking them to reiterate as well as air any individual problems they were experiencing. This included assessment by way of mini plenaries. By clearly instructing the group at the start of each lesson with regard to the objectives ahead and desired outcomes, a clear sense of the learning expectations is communicated from the outset. As the assessment took place throughout the lessons the group worked in ways that contributed to the learning experience. The intimate relationship between instruction and assessment encourages useful plenary sessions, allowing both students and teacher to engage in challenging conversations which generate an environment where questions are more open and beneficial to both the group and individuals. (Earl, 2003: 86) Independent homework was issued after each lesson. The outcomes of these tasks demonstrated that the students were sufficiently learning the concepts of surds with majoritively A and B grades awarded across the class. Individual outstanding issues with the homework tasks were followed up during the next lesson.
Research has proven that there is significant potential learning gains to be had from engaging students in peer and self-assessment strategies and becomes crucial for feedback to be utilised effectively. (Atkin et al, 2001:17) Consequently self assessment was applied during the second round of homework and pupils were given responsibility for marking each others homework and then discussing any follow-up issues that resulted from this process. A group session was also undertaken to ascertain that any problems were shared between the group as a whole.
The final activity provided to the group involved a formal written test on the topic of surds. The results indicated a pass rate of over 80% for all pupils who sat the test. This was the final decider which clarified the students' overall understanding of the topic which was taught to them over this three lesson period.
A discussion of issues arising
Essentially the prospect of teaching such a group of high achievers appeared to be one that would prove an unchallenging process. However keeping the group focussed was just as difficult to achieve as with that of keeping a lower ability group engaged. Most gifted students require a more advanced level and pace than the top streams offer (Fetterman, 1988) and perhaps in this case it may have been advantageous to try some separate group work in a different setting with the five gifted and talent pupils, involving slightly more specialist training. This also raises the issue of whether ability grouping is beneficial in the long run. Most educational theory tells us that average students do not view exceptionally talented students as role models, rather they just view them as different. Considerable differences in ability in a classroom can promote arrogance on the part of the higher-achieving and gifted students. And if a teacher teaches a mixed-ability group at a higher level and at a faster pace the less advanced students may experience unnecessary feelings of inadequacy and pressure. Nonetheless attempting to teach extreme variations in ability can also increase the likelihood of the teacher being unable to adequately manage the group. (Assouline et al, 2005:130-131)
With a lower ability group, the teacher can easily manage the learning with two work sheets or any two resources of practice material in class. But with a higher ability group the teacher must have at least 4 different resources as pupils can easily get bored and lose interest if there isn't constant variation during the lesson.
A critique of your teaching.
Firm evidence has demonstrated that formative assessment is an essential component of classroom work and that its development can raise standards of achievement. I.e. activities undertaken by teachers and by students in assessing themselves which can provide information to be used as feedback to modify future teacher and learner activities. (Black& William, 1998)
I endeavoured therefore to be thorough in my approach at utilising a number of techniques in order to assess the group and use this as an indicator for future delivery or to modify my own teaching methods.
In hindsight I would have given more thought to the levels of ability and the skills with which I needed to develop in order to manage this. In the future preparation for this type of group might involve inclusion of a greater number of materials and learning resources, in order to ensure the stimulation levels remained consistently high. I may also have carried out some additional higher level work with the gifted and talented pupils to ensure that they were receiving the correct degree of teaching provision in line with their abilities and needs.