These pages are based on what is generically called "DJS’ Maths at College” [Shiplake or Plymouth] and is a synthesis of the year 8 top set work as developed with their help over a period in excess of twenty years. The wonderful thing about Year 8 is they know that they don’t know, but they aren’t frightened to guess, or to try out an idea; in that sense they are particularly willing to think. In a good year with a class that reached what we might call critical mass—Critical Maths, even—at several points in the year we would discover something. On occasions, the child that voiced an idea would be credited, in the sense that we would call the idea something like ‘Tracy’s Theorem’. One such would be the special isosceles trapezium whose diagonals cross at right angles; this is Gaunt’s Trapezium. Since I never found another name for it, that is what it has been called ever since. In my classes.
The objectives for the year might be characterised thus:
• To work on number skills
To the point that tables to 12x12 are instinctive, tables to 20x20, while not completely known, are recognised; squares to 30x30 are known, as are cubes of single digits; algebraic skills should permit some apparently effortless arithmetic, particularly in multiplication. The use of 2x2 matrices is included on the grounds that this is a good mental gymnastic exercise that affirms mental arithmetic by demanding precision.
• To develop thinking skills
What do we know about a problem? What do we know that might help? Can we start from a simpler situation? Do we need some research first? (and in reserve: Do I know someone who already knows?). ‘coping’ strategies, teamwork strategies.
• To identify and extend patterns
Extending a pattern of numbers (finding the next term); development of a general formula for a series. Testing of answers. In terms of investigations, to test theories and develop skills that enable patterns to be identified.
• To establish a self-confidence in mathematics
Pupils are encouraged to develop their own strategies for organising their material, their notes from lessons and their homework. Guidance is given, but from a standpoint of persuading the pupil to demonstrate its thinking. Discovering that a clarity of presentation permits not only better communication with the reader but also permits the student to better understand a complicated problem is a fundamental lesson for the year generally not discovered early.
By the end of the year, each student should have developed:
Reliable basic arithmetic skills
The habit of checking answers for sense
Alternative routes to an answer (thus supporting the answer, the methods and the student’s faith in its ability). In short, self-confidence.
No-one claims that this is an easy path. Many claim that maths thereafter is much easier. While in every year there are some who find their confidence sorely dented during the year, all pupils finish the stronger for the experience. It has been said that if you can survive Year 8 with DJS, you can cope with any subject, any teacher, any level.
Which does not mean that any of this is particularly difficult, nor especially clever.
I assume throughout an ability to read in English that is beyond the national expectation for the age group. I have tried to write accurate precise English throughout but I will not concede to a reduction in vocabulary at the cost of reduction in learning. Where I write theory my audience is already educated – they have had the lesson, they have asked the questions that occurred to them and they probably left the room thinking they understood. Thus I make no concessions to the age of my readership.
A common complaint from pupils is that they understood while in the classroom, particularly at the moment of passing out of the room - but not when they reached home. I have always felt flattered by this statement. This complaint (there are several cures) strikes me as the basis of a good argument for doing work sooner—before it falls out of short-term memory—but also subsequently an understanding that homework serves to fix ideas in one’s head (moving learning to medium-term memory) and, not least, an encouragement to capture information.
Another regular complaint from pupils is founded on the myth that the work is the same as that of the Sixth form. There are several grounds for believing this:
a) It is true that there are some common elements; unsurprisingly they are the basis of mathematics at school.
b) It is a truth that the Sixth have chosen to pursue the subject further, so generally that can be characterised as a measure of enthusiasm or at least a perceived ability in the subject. Year 8 set 1 are fired with an enthusiasm for the subject that collectively matches that of any lower sixth set. Thus the level of curiosity and the willingness to explore is at least as high in the younger year.
c) The sixth form repeat many of the skills lessons covered in year 8 – at a significantly higher speed – not least because they need the skills, but also because not all have them and a few of those that had the skills have stopped using them. Generally the algebra skills of the sixth are higher; one would certainly hope so. Where in the lower school algebra is typically used to explain why an arithmetic ‘trick’ works, in the sixth form the situation is reversed so that the algebra is seen as a tool for arithmetic.
d) There is an argument that the exam pressures of KeyStages 3 & 4 from year 9 onwards mitigate against the freedom in the classroom to simply explore some mathematics. Time for exploring a mathematical idea can usually be found in the sixth form when the work ethic is high enough and especially where students are willing to study independently. The last best chance before then is in year 8.
e) Given that both groups have an interest in maths, and given that DJS’ lessons occur in a limited set of rooms, it is then not a surprise that on occasion the sixth and year 8 have adjacent lessons. It is surely still not a surprise that the second group enquire of the apparently interesting maths left on the board, particularly when the two groups pass in the doorway each unconsciously realising that here is another group that genuinely finds the subject interesting. Viewing this situation from the perspective of the incoming class then the younger pupils will say “but we’ve done that” meaning really “I recognise some of what is on the board” and sometimes or even “is that still maths?” and “Oh my, that looks interesting, what sort of maths is that?” This last is the question that the sixth pose, but it is phrased more like “I don’t remember that” or “Did we ever do that?” and sometimes the stunned reaction is “How could they be doing any maths that I don’t understand immediately?” Which makes the younger ones feel good when the message is passed on, and sometimes results in useful revision of technique for the older group.
