Points of Departure
Starting points for mathematical investigations, suitable for a wide range of ages and abilities
Key Stage suitability • Explanation
|Points of Departure 1 - PDF
Item Ref: DNL003
|Points of Departure 2 - PDF
Item Ref: DNL004
|Points of Departure 3 - PDF
Item Ref: DNL005
|Points of Departure 4 - PDF
Item Ref: DNL006
|Primary Points of Departure - PDF
Item Ref: DNL057
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Points of Departure
A collection of starting points for mathematical investigations, suitable for a wide range of ages and abilities.
Because of the open ended nature of the investigations, they are also used widely in initial and in-service training.
Points of Departure 3 and 4 are linked directly with the National Curriculum for Mathematics: Book 3 contains activities related to Number and Algebra, while Book 4 is more likely to lead into Shape and Space, and Handling Data.
Primary Points of Departure contains over 70 starting points from Points of Departure 1 to 4 selected for pupils at Key Stage 2, and those working at similar levels.
An increasing number of teachers of mathematics at all levels are including in their work, opportunities for children to explore and investigate potentially mathematical situations. Six sessions at the ATM Conference at Nottingham in 1977 were devoted to work of this kind. These attracted considerable interest, but one difficulty expressed by many teachers was that of finding sources or examples of suitable starting points for investigations. This pamphlet is the first of a series: it is intended to provide situations for investigation which teachers have found useful with children of varying age and ability.
The investigations included in this first pamphlet draw on the experience of many teachers. None are original. We have included examples which cover a wide range of types and levels of difficulty, and we have attempted to indicate this on the following page. Categorisation of investigations in this way is inevitably subjective, but it may be helpful for those who have not used material of this sort before.
This is the second of the series of starting points for mathematical investigations and like its predecessor this pamphlet draws on the experience of many teachers and makes no claim to originality.
Points of Departure has attracted a great deal of interest and we have heard that in addition to meeting the original need for starting points for children’s mathematical investigations, the pamphlet has been widely used for the mathematical development of teachers on in-service courses. We hope that Points of Departure 2 will prove equally valuable.
Our thanks are due to those who have contributed ideas for this pamphlet and we extend an invitation to you to send material for further publications to the ATM Office.
This booklet [PoD 3] offers some starting points that are likely to lead those pursuing them into Number and Algebra. A companion booklet, Points of Departure 4, offers some starting points more likely to lead into Shape and Space, and Handling Data.
This booklet [PoD 4] offers some starting points that are likely to lead those pursuing them into Shape and Space, and Handling Data.
Within each of these booklets, similar points of departure have been placed close together, in the belief that this will make it easier for teachers who wish to use them in a structured way.
As outlined in the National Curriculum Council’s Non-Statutory Guidance for Mathematics (B.7.2, B.7.8) there is a place for offering children a base point from which to explore, and allowing them to see where they can go. This may, of course, result in different places being visited by different children, or even by the same children on different occasions.
Such an open-ended approach allows those involved great freedom, and it gives considerable opportunity for them to select their own mathematics.
An alternative approach, also outlined in the Non-Statutory Guidance (B.7.3, B.7.9), is for the teacher to choose a particular starting point because it is likely to involve those engaging in it in certain content and processes, or to lead them to perceive certain strategies and structures. It is as though, rather than taking the children to a country field from which to explore, the teacher takes them to a signpost, or perhaps to a train platform from which certain trains depart.
It would, of course, be foolish to suggest that any one point of departure will necessarily lead to particular mathematics or will lead to pupils achieving a particular Statement of Attainment; but the juxtaposition of similar starting points should help teachers to choose activities that are more likely to give their pupils opportunities to experience particular mathematics, or to build on previous mathematics, or to prepare the way for new mathematics.
Both approaches have their advantages (B.7.4), and they are not mutually exclusive (B.7.5 - B.7.8). Selecting the point for departure does not necessarily select the mathematics.- but it may select the mathematics that is likely to be selected.
The teacher’s skill lies in considering where an exploration might go and what parts of the Programmes of Study are likely to be encountered by particular pupils. Children, though, are full of surprises, and the Programmes of Study that are covered and the Statements of Attainment that are achieved may be more (or different) than anticipated.
Although these points of departure could be given to pupils directly as they stand, this is not the main intention of producing these booklets. Rather, the booklets are seen as a resource of ideas that teachers will wish to adapt and to present in forms with which they and their pupils will feel comfortable.
In order to emphasise this point, the booklets do not seek to employ a uniform structure across the presentation of the points of departure, nor is there a uniform amount of structure within each point of departure. You may wish to introduce more structure than the booklets give, or to use less structure than they provide. This is how it should be.
At times there are suggestions such as: "Try with 3 numbers. With 4 numbers...", or: "What happens with squares? With triangles?...". Some may find such hints unnecessary, even irritating; but others may find them helpful, either to answer the question "Where do we go next?", or else to see for themselves the full potentials of a starting point.
Whatever your approach, though, it is not envisaged that such suggestions for extension, (or for alternative starts); should be given to the pupils in the initial presentation of the activity. Indeed, for the teacher who listens to their pupils' own suggestions and queries, these additions will be quite unnecessary.
Many of the points of departure allow opportunities for pupils to select, not only their own mathematics, but also the materials with which to work. In particular, equipment of all kinds, consumable materials, grid papers, calculators, computers and reference material are all likely to be of help to those setting out on these explorations.
Teachers can introduce the starting points as they stand to individuals, pairs or small groups. However, this is not the main intention. Rather, the book should be seen as a resource of ideas that teachers will wish to adapt and to present in ways with which they, and their pupils, feel most comfortable. Some, for example, 24 (HALVING THE BOARD), 37 (ROUND AND ROUND) and 63 (EQUABLE TRIANGLES), could well be used for initiating a discussion which leads to the given diagrams being put on the black (or white) board. In this case the words given in the starting point can be adapted as an introduction to a class lesson. Some of the starting points require equipment such as interlocking cubes, squared or dotty paper. Sometimes, especially with younger pupils, other pieces of apparatus will be found useful.
However the points of departure are to be used, teachers are strongly advised to spend a few minutes exploring the ideas before using the points of departure with a class.
Some of the starting points may only fill a short time while others could occupy pupils all week. As all classes differ it is not possible to indicate which of these is which?
PoD 1 - 0 900095 30 X
PoD 2 - 0 900095 37 7
PoD 3 - 0 900095 80 6
PoD 4 - 0 900095 81 4
Primary - 1 898611 01 7