Introduction - Teaching Structures
Choice of Teaching Programme
Choice of Activities
Teaching Programme
Explanation of the Teaching Program
Modifications to the Programme for Special Educational Needs
References
Teaching Program Plan
Description of Activities
Teaching Programme
This document gives a design of a teaching programme for one strand of Mathematics, for a mixed (girls and boys) school covering the age range 11 - 16 years (Years 7 - 11). This is a hypothetical school, in which pupils are setted for Mathematics (for Key Stage 3, at least) into a top set, a middle set and a bottom set. Each Maths set is timetabled for 3 or 4, 45-minute, periods of Mathematics per week.
The content of the National Curriculum and the GCSE Syllabus, together, have been compiled into 9 Mathematics strands, and the Symbolic Notation Strand has been selected here for long-term planning. Considering the total content of the 9 strands identified, there are about 22 weeks of teaching, across the five school years, available for this particular strand.
The teaching programme designed in this assignment was inspired by the Mathematics curriculum of Varndean School (1998). The Symbolic Notation strand chosen here, has been based on the corresponding strand of that document, and then divided into a number of separate, though related, units of work. Each unit of work has been assigned an (estimated) time frame, and planned for a target school year and term. A specific learning objective has been associated with each topic, and lesson activities adopted to meet each stipulated learning outcome. The resulting plan is here.
Specifically however, the programme here has been modified from the original in the following ways. Simplifying Equations has been scheduled before solving Linear Equations, and Transposing Formulae has been moved back to immediately follow them both. Consequently, finding the General Term of a sequence has been delayed until the end of Year 10. I have successfully taught this part of the programme myself, within two different courses, and on both occasions, experienced teachers have advised this latter order of development. The rationale seems to be that the techniques for solving equations are much easier to acquire if the simplification of algebraic expressions is already understood. While transposing formulae involves a similar process to the solution of equations, but is more complicated because of involving more unknowns.
Descriptions are provided of all the activities included here.
All differentiation of work across the three envisaged Maths sets, has been indicated within this description of the activities. As an example, in the unit of work Topic 7, Describe a Rule in Words all three ability sets carry out Activity 17, Make it So. The middle and bottom sets then do Activity 18, Describe the Rule, while the top set tackle Activity 19, Difference of Two Squares.
The educational implications of each activity are discussed in a later section of this document (see 'Explanation of the Teaching Programme'). Generally, classroom activities have been chosen to be basically interesting and stimulating, while at the same time, effectively achieving the intended learning objectives. Beyond this, the important considerations below have also been addressed in this design.
In that section then, each activity is justified, as to its appropriateness to the progression of children's learning in this strand. One example would be, say, Activity 14, TI Tamagotchis, which has been successfully demonstrated in a school in Portsmouth, and the results published in the literature. Other activities I have used myself and found profitable (e.g. Activity 35, Fast Food). Where the activity also consolidates basic numeracy, this is mentioned in that section too. For example, with Activity 12, Using Number Machines, it is pointed out how Number Machines can strengthen numeracy skills.
Promotion of the personal and social development of pupils, through the activities of this programme, is also emphasised in the 'Explanation' section. Many of these pursuits are intrinsically social or teamwork activities, like Activity 26, Curly Straw. Others, such as Activity 22, Rules, Rules, Rules!!!, will give the teacher opportunities to raise issues like 'when and why we obey rules in everyday life', and to discuss the personal discipline of following rules (e.g. also Activity 9, Number Chains). The particular strengths of personality of individual pupils are also fostered, in, e.g., Activity 27, Presentations, which encourages development of self-expression, and Activity 17, Make it So, which builds on pupils' skills in concentration and listening to others.
At various points in the programme, mathematical opportunities will be found for evoking a spiritual response in the class. For example, Activity 36, Spot the Similarity, reveals the amazing power of the parabola. And in the Further Sequences Activities 6 and 18, the Koch Snowflake is included, which could begin a discussion about the fascination and wonder of fractal curves. CHARIS expresses this (CHARIS Mathematics (1966a), page 47):
'It is hoped that pupils will … be awed by the complexity of the created universe and … appreciate that mathematics, far from excluding God, can be an introduction to Him'
Again, homework on the truth of mathematical statements (in Year 7), would lead to a debate about (as CHARIS Mathematics (1966b), page 57 puts it):
'… respect for truth in all its forms … (helping pupils) to see that the quest for truth applies in all fields of human discovery.'
It could therefore also spark off some thoughts about how people make up their minds about what they believe to be right and true.
Occasional cultural and multi-cultural references will also be made here. The game of Chess, for example, (Activity 10, Chess Moves, and elsewhere) can be regarded as part of our national heritage, though it belongs to many other cultures too. Such discussion points will inevitably arise during lessons analysing the game.
None of the activities planned here are differentiated by gender. It might be that some of the boys are more familiar than girls with, say using a computer, because of years of playing computer games. But IT-based activities, e.g. Activity 16, Linear Plot, will be carried out in the same way by both. There will be no assumptions made about prior computing skills brought to the classroom.
Many of the activities are social, in the sense that they will be undertaken in a group or in pairs. It may well be that some girls will have a greater facility for social co-operation and verbal expression than boys. However, both will have the same entitlement in such activities as Activity 17, Make it So. Boys and girls will work together on such co-operative tasks.
Where possible, too, an attempt has been made to render all activities and teaching material gender non-specific. For example, in Activity 34, The Mathepillar's Dilemma, King Arthur's Problem has been re-formulated in a way to remove gender stereotyping.
Throughout the programme, attention has been paid to issues of continuity and progression of learning. Topics and activities have been organised to ensure, at all times, suitability to children's present level of ability, a natural development building on pupils' current knowledge, coverage of the National Curriculum and the GCSE syllabus (Key Stage 4) and opportunities for pupils to demonstrate appropriate levels of attainment. The units of work have been referenced to their National Curriculum attainment levels. The programme has also taken some account of pupils' transitions in Mathematics education, both from primary school to Key Stage 3 and from Key Stage 4 to post-16 Mathematics.
In particular, there is considerable concern amongst teachers in the primary sector, that pupils leave them confident and capable in Mathematics, to find their abilities and knowledge are not recognised in their new secondary schools. (I noted this during my own Primary Experience Placement). Consequently, much teaching in Year 7 duplicates their primary education and they end up marking time for twelve months in their Mathematics learning. Various Year 7 activities have, therefore, been included here, which should be unfamiliar to pupils (e.g. Activity 2, 3 x 3 Magic Sum and homework on Magic Squares, and more advanced sequences in Activity 6, Further Sequences and Activity 8, More Blanks). This should maintain their interest while groundwork in symbolic notation is consolidated.
A major issue in Sixth Form colleges currently, with students starting A level Maths, is that it is possible to achieve the entry qualifications for it without knowing sufficient algebra to cope comfortably with it. Often Year 1 A level students are weak in algebraic manipulation skills, such as handling brackets and simplifying expressions. (I noticed this during my Placement at a Further Education College). This is another reason for beginning Simplifying Equations early in Year 10. The present programme also provides plenty of opportunities for pupils to practice their algebra, including Activity 40, Factorise and Activity 41, Wendy and Gavin.
The complete teaching programme design has been outlined using the pro forma as supplied. There is a separate form for each unit of work, making seventeen in all. Note that topic timings shown are rough estimates, and that the prerequisites listed for each topic imply all their own prerequisites as well.
There are links to descriptions or introductions to all the activities referred to in this section. A rationale is given here for each activity within its unit of work, and draws out the applicability and the outcomes of the various activities.
Balancing Money Bags demonstrates equals as a balance between left and right sides, and the addition and subtraction operations. This is an activity to build up basic numeracy.
3 x 3 Magic Sum exercises addition and equals. This activity also builds basic numeracy.
I think this will be a new idea to many pupils, so that this term's work is not all obvious to most of Year 7.
The Multiplication Grid shows multiplication as repeated addition and demonstrates the commutative nature of multiplication. Division is also consolidated, and, at the same time, we identify the various symbols for representing division. Again this activity supports numeracy development.
The worksheet of Diagram Sequences exercises the recognition of sequences in diagrams and their description in written words. I am indebted to Mackrell (1998) for ideas here. Picture the Pattern makes this a challenging puzzle. Working in pairs encourages co-operation.
The worksheet of Number Sequences exercises the recognition of sequences of numbers and their description in written words. Name that Sequence casts this into a mildly competitive game. Working in pairs again encourages co-operation.
The worksheet of Further Sequences offers additional, more advanced, sequences of diagrams and symbols for the top set and extension work. I am indebted to Mackrell (1998) for ideas here too.
This should also be beyond the pupils' prior, primary school work.
The Fill in the Blanks worksheet on entering missing numbers into a sequence consolidates previous work on sequences and applies it. This activity presents this as a kind of intriguing mystery to be solved. Working in pairs again provides some social activity.
The worksheet of more missing numbers in sequences is for the high ability, top set, pupils, who are exploring special sequences at this time.
Number Chains give practice in applying a rule expressed in words, generating a sequence. This idea is based on Russel (1999), where the author reports great success using it with Year 8. Number Chains strengthen numeracy skills and following rules is a good personal discipline.
Chess moves further illustrate rules and patterns. There is social activity in the game playing and demonstrating.
Magic Square generation requires rules to be followed. Pupils can self-check using the magic property.
The Using Number Machines and Number Machine Puzzles worksheets provide amusing practice in recognising functions and performing mappings. Number Machines strengthen basic numeracy too, by demanding repeated use of the four operations. I am indebted to Mackrell (1998) for these resources and for the next one.
Reverse Number Machines allows pupils to develop functions for inverse operations. It can support numeracy and provide some shared activity.
This activity idea comes from Clark-Jeavons (1999) and was conceived for a Year 8 class. That class obviously enjoyed it and produced some impressive results. It is a contemporary version of co-ordinate pictures, exercising the identification of points on a grid by their co-ordinates. It also brings in IT skills through the use of Graphic Calculators. It will additionally have social implications, as pupils will show their Tamagotchi to classmates.
Top set pupils will further demonstrate their working Tamagotchis to others.
Bottom set will also interact socially in game playing - Battleships. Battleships requires the pupils to reference grid co-ordinates verbally, giving further practice in this topic.
Plotting a linear function mapping in Excel will extend IT exposure and show co-ordinates on a graph.
Make it So is a co-operative pupil activity, where two partners try to communicate instructions only in words. It should also encourage clear verbal self-expression and concentration. It has been frequently used in Mathematics education, and was published in SMILE (19??).
The lower sets following the Further Sequences and More Blanks worksheets here, ensures all pupils have the opportunity to familiarise with the special sequences (Triangular numbers, Fibonacci Sequence etc) and will consolidate their recognition of others.
I have seen the difference of successive squares investigation carried out, reasonably profitably, with a middle set Year 8 class, using actual areas of squares. Here, at the end of Year 8, the top set will use calculators (or mental recall) to determine the squares (see worksheet later), which will stretch them and/or give them additional calculator practice.
By repeating this investigation for several different separations of the two numbers, they should be able to deduce the general rule for the difference of any two squares, and try to express this rule in their own words.
Computer graphing of the mapping will demonstrate its nature, without undue time being spent plotting the graph by hand. It will also broaden IT skills.
This is a real world example of a straight-line relationship, which will be used again later in the year. It ensures that pupils can use practical rules, expressed in words, to generate a graph.
Facility with the formula for the area of a circle is not expected at this stage, except possibly in the top set.
This worksheet gives practice in expressing, in algebraic notation, rules in words met previously (and others).
Here the investigation described in Activity 19 is given to the bottom and middle sets, except that, having deduced the rule for the difference of two successive squares, they gain practice in algebra by expressing it in symbolic form.
I have seen this investigation given to a Year 8 top set. It is an interesting problem, but it is not easy, and that Year 8 did not, in general, get very far with it. I think it would be more appropriate here, for top set Year 9.
Once some 'bounce' situations are understood, the activity provides practice in expressing pupils' own discoveries algebraically.
This is a calculator exercise, which will exercise the class in evaluating formulae and strengthen calculator (IT) skills. It also emphasises the relevance of maths to outside life. The questions are taken from Greer and Taylor (1982).
The curly straw investigation is intended to offer all learners an experience of teamwork in applying maths to solve a realistic problem. At the same time, it is an exercise using much of what has been done in this strand during the year. Only the use of water needs to be carefully managed in the classroom, especially with the bottom set.
Bottom set will be told the way to calculate the cross-sectional area. Although Varndean (1998) implies otherwise in several places, it is assumed here that the Attainment Target 3 Level 6 Description, DfEE (1995), page 27:
"They understand and use appropriate formulae for finding circumferences and areas of circles… when solving problems"
will have been achieved by the top and middle sets, by this stage - the end of Key Stage 3.
This is a way of encouraging mutual-help and personal study in these Year 10 pupils. It will also provide experience in preparing and presenting a short talk. Some pupils will be a bit reticent about this and will have smaller parts to play. However, I have seen this approach used in a Year 10 Maths class, and the pupils did very well and were warmly received by the rest of the class. So I believe, if handled sensitively, it can be a valuable experience for all the pupils.
Pupils also receive an early introduction to algebraic manipulation at this point.
This computer-based activity will provide an enjoyable way of consolidating the solution of linear equations. It will also expose the pupils to mathematical software and strengthen IT skills.
This exercise presents some intriguing situations that can be represented by linear equations in one unknown. Pupils gain practice in developing the linear equations and then in solving them. The worksheet here comes from Childs et al (1988) and the problem in the second example worksheet is taken from Gaulter and Buchanan (1992).
The Party Time coursework involves pupils in independent investigative work, forming and transposing linear equations, and solving them, in the context of a challenging puzzle. Their results will be presented, in the form of a report, for summative assessment. This will encourage clear explanations of their solutions and interpretation of their conclusions.
The challenge is again to be found in Childs et al (1988). It has been only a little extended here and cast in non-gender specific language.
This class offers pupils an opportunity to advance their coursework and to raise issues encountered with it. There will, therefore, be some time for individual teaching and some for whole class guidance in the presentation of such coursework.
Swap Around provides a social activity in which pairs of pupils check each other transposing formulae. They refer to the example given, to support learning. This should provide enjoyable practice in changing the subject of a formula.
This activity provides exercises in recognising and deriving the general term in a sequence.
The evaluation of the general term to generate a mathematical sequence is practised here. The use of a Graphic Calculator to self-check their results extends pupils' ICT capability.
This is intended to be an amusing activity, which also uses and builds on pupils’ skills in identifying mathematical patterns and representing them algebraically. It is an investigation of a mathematical situation that has appeared in maths education in various guises. I am indebted to Burnett (1998) for drawing my attention to the puzzle, embedded in a dubious story called King Arthur’s Problem. In the interests of promoting issues of equality and character development in pupils, I have cast it into a more neutral setting. A very similar challenge was set by Micromath (1999), where it was envisaged that teachers would try it with pupils, perhaps utilising ICT. It is planned here as a pencil and paper investigation, however.
It is easy enough to tabulate the last petal (child etc) and to recognise the pattern, but expressing it algebraically will probably require teacher input, especially for lower ability pupils. It is therefore planned for end of year activities for Year 10.
This tasksheet is taken directly from Roder (1993), where it is part of the SMP Mature GCSE Mathematics syllabus material. It develops the elimination process for the solution of pairs of linear equations. I have used this tasksheet with students myself, as an introduction to this subject, and found it quite successful. I think it gives a particularly clear demonstration of the procedure involved, and should ensure understanding.
This should help to link the graphical and algebraic solutions of pairs of equations, and give pupils practice in solving them.
The activity also strengthens IT capabilities, exercises pupils' understanding of simultaneous equations, and enables them to self-evaluate.
This worksheet is taken from Greer and Taylor (1982). It emphasises the relevance of the topic to real life and gives further practice in solving such problems for the higher ability. In common with other activities in Year 11, this one also exercises techniques of algebra, which will be important for any further mathematics study, post-16.
The idea here is that of a social activity, in small groups, to solve an investigative challenge. The activity should establish the shape of the quadratic function. It will also foster skills in working with others and exercise powers of observation. Use of Graphic Calculators will further consolidate IT knowledge, but the pupils will also sketch the parabola by hand, as there is something of a reversal away from a total reliance on Graphic Calculators in teaching to generate mathematical curves. The illustrations of real parabolas are from CSUSB (1999) and Childs et al (1988).
The purpose of this activity is to reinforce the method of trial and improvement for the solution of quadratic equations. There is again calculator usage and application to realistic situations.
The Factorise tasksheet comes from Roder (1993). I feel it provides a good introduction to the use of brackets and taking out factors.
This worksheet is taken from Childs et al (1988). It exercises the factorisation of quadratics. Together with the previous activity, this also ensures plenty of the practice in algebraic manipulation needed, to prepare pupils for their smooth transfer next year to A level mathematics study.
Some children with special educational needs may have difficulty with this teaching programme, as written, so consideration is given here to the particular needs of some individual pupils. Visually and hearing impaired pupils are recognised as having special needs and the all the class, naturally, respect those needs. The child concerned is invited to sit where they can most easily be involved, and during social activities, the teacher and the pupils address them carefully. But they are expected to take part in all the activities of the programme.
Dyslexic pupils are also encouraged to participate fully in the activities of the class. They may, however, experience problems with identifying written numbers and with sequencing exercises. They are advised to use a computer for textual work, and to consider voice recognition software (see, e.g. Haigh (1999)). In activities revolving around sequences (Activities 4 - 8, 18 and 32), they work in a pair where appropriate, and receive personal assistance from the teacher.
Pupils with a degree of dyspraxia are not asked to throw balls (in Activity 38, Plot the Ball). They may also have difficulty with sketching curves, e.g. in Activity 38 again, and this is accepted in the lesson. Where their dyspraxia leads to serious writing problems, they are also permitted to use Word Processing software instead.
However, the main problem I see with a Mathematics teaching programme, designed as here, to also promote pupils' social development, is that there is a growing number of children and young people in schools now with medically diagnosed social disorders. These may include autistic spectrum disorder, attention deficit/hyperactivity disorder, Asperger's syndrome, Tourette's syndrome and other pervasive development disorders. Pupils who suffer from such conditions may well have particular difficulties with co-operative activities, such as Activities 4 - 8, working on sequences in pairs, and with small group or teamwork tasks such as Activity 26, Curly Straw and Activity 27, Presentations. They are given every encouragement to join in, but they may prefer to work on their own. They may also be excused making a presentation in Activity 27. And in Activity 17, Make it So, instead of working with a partner, they can describe, in written words, a construction of their own, by which the teacher can assess their ability to communicate mathematical information.
For pupils with English as a second language, it is very important, of course, in every class, to explain English terms and conventions clearly. For example, in Activity 9, Number Chains, the teacher explains to them what a chain is and why a sequence of numbers can form a symbolic chain. This requirement also implies an additional emphasis on basic numeracy, not, perhaps, because of any difficulty with working with number, but because the vocabulary to express their thoughts may be lacking. This must be rectified before they can fully demonstrate mathematical ability.
So, for example, pupils in Year 7 who are not native English speakers, may have remedial small group work, while Activity 2 is under way. This covers the meaning and representation of concepts from Key Stage 2, such as, squares, cubes and square roots, fractions, ratios and percentages, and simple mappings.
Finally, perhaps the most challenging aspect of teaching this programme is how it can be modified to accommodate very able children in the class. What is necessary depends a lot on the pupils themselves. But, in general, it is possible with any topic, for the teacher to enable a pupil to explore it in more depth. For instance, in Year 7, when the top set encounter the commutative properties of the four operations, in Activity 3, Multiplication Grid, the very able child also tabulates which operations are associative and the distributive nature of each pair of operations.
As another example, in Activity 34, the Mathepillar investigation, mathematically gifted children may tackle all three questions on the worksheet, which leads them into advanced level Maths. Again, if the very able pupil is ready early for GCSE, they can extend the factorisation of expressions (Activity 41, Wendy and Gavin) to solving quadratic equations that do not factorise, and then to study other functions, such as cubics and reciprocal functions.
Burnett, R. (1998) Private communication
CHARIS Mathematics (1996a) 'Unit 6, Fractals' Teacher's Notes ACT, page 47
CHARIS Mathematics (1996b) 'Unit 7, The Moment of Truth' Teacher's Notes ACT, page
57
Childs, B. Emerson, C. and Manser, C. (1988) Challenging Maths for GCSE and Standard
Grade Heinemann Educational Books
Clark-Jeavons, A. (1999) 'TI-magotchi?' Micromath volume 15/1, Spring 1999 The
Association of Teachers of Mathematics
CSUSB (1999) ‘Real Parabolas’ California State University San Bernardino
Mathematics Department http://www.math.csusb.edu
Department for Education and Employment (1995) Mathematics in the National Curriculum
London HMSO
DERIVEŽ for Windows (1999) Soft Warehouse Inc., Honolulu, Hawaii http://www.derive.com/dfwset.htm
Gaulter, B. and Buchanan, L. (1992) Modular Mathematics for GCSE Oxford University
Press
Greer, A and Taylor, G.W. (1982) BTEC First Mathematics for Technicians Stanley
Thomas Ltd.
Haigh, G. (1999) 'I speak and you write' New Solutions TES Summer Term 1999, The
Times Supplements Ltd.
Mackrell, K. (1998) Various private communications
Micromath (1999) ‘Challenge’ Micromath volume 15/1, Spring 1999 The
Association of Teachers of Mathematics
Roder, P. et al (1993) ‘The language of maths’ School Mathematics Project
Modular Mathematics, Cambridge University Press
Russel, J. (1999) 'Chain Reaction' Mathematics Teaching volume 166, March 1999
SMILE (19??) 'Investigation 5' SMILE Centre, Isaac Newton Centre for Professional
Development, London W11.
Varndean School (1998) 'Varndean School Mathematics Curriculum' Mathematics
Department Handbook, Varndean School, Brighton