||Calculating Changes Background
a calculator is a valuable learning tool
A calculator used with manipulatives and an attitude which values children's efforts to learn is a powerful teaching package.
In the Calculating Changes classroom:
- Each student is supplied with a Poly Plug and a calculator to:
use as they wish, when they wish.
- Teachers ask How can we use the Poly Plug and/or the calculator to help us learn this? and build on:
children's ways of using the resource.
- Web-based networking:
collects and shares classroom experiences.
- The objective is to:
develop children's number sense beyond what is usually expected for the age.
Perhaps these quotes offer a challenge to our pre-conceived notions of mathematics teaching and in doing so encourage us to ask how our own reluctance to change teaching practice might appear to others 100 years from now.
- Until within a few years no studies have been permitted in the day school but spelling, reading and writing. Arithmetic was taught by a few instructors one or two evenings a week. But in spite of the most determined opposition, arithmetic is now being permitted in the day school.
Philip S. Jones, ed., A History of Mathematics Education in the United States and Canada, National Council of Teachers of Mathematics, Washington, DC, 1970. [p.13]
- Those of high social rank, theoretically above the world of getting and spending, did not deign to study the subject. The most respectable English public schools, like Eton and Harrow, did not offer any instruction in arithmetic until well into the nineteenth century.
Patricia Cline Cohen, A Calculating People: The Spread of Numeracy in Early America, University of Chicago Press, Chicago, 1982
Calculator Aware Number (C. A. N.) Project
C.A.N began in the middle 1980s with about 20 classes of children aged six in some 15 schools. The project was led by Hilary Shuard, Homerton College, Cambridge University. It is from this project that Calculating Changes draws its inspiration to encourage the free use of calculators.
Over the years the number of children involved grew from the hundreds into the thousands as more schools joined the project. Significantly, all the originating schools and Local Education Authorities stayed with the project throughout its lifetime. The following quotes are from Calculators, Children and Mathematics, the report of the Calculator Aware Number (C.A.N.) project which is now out of print. This United Kingdom project is the most extensive classroom based investigation into children using calculators that has ever taken place. Its results are well documented, far reaching in their implications for classroom teaching and have been repeated in other places, for example in the Victoria College (later Deakin University) project in Victoria, Australia.
- Page 5: This book describes a curriculum development project in England and Wales, in the period 1986-9, the aim of which was to study effects that the availability of calculators would have on the mathematics curriculum of primary schools. Thus, it was intended to develop a calculator-aware mathematics curriculum, especially in number, for children in the primary (five to 11) age range.
- Page 5: In 1985, it was very clear that the arithmetical aspects of the primary curriculum had still not been affected by calculators...
(CCh comment: despite the 1982 recommendation of the Cockroft Report to do so)
... Calculators were still not available in the majority of primary schools, and children were not usually allowed to bring their own calculators from home. Some schools seemed to regard calculator use as 'cheating'. Commercial publications provided little support for the use of calculators in mathematics, and they did not encourage teachers to use them with their classes. The number work in the published mathematics schemes which most primary schools used was still based on the traditional vertical pencil-and-paper methods of carrying out addition, subtraction, multiplication and division; these methods have been taught with little change throughout this century.
- Page 7: Before any schools were recruited, the PrIME project team discussed and clarified their own ideas about CAN. The major principle was:
Children should be allowed to use calculators in the same way as adults use them: at their own choice, whenever they wish to do so.
The following ideas (paraphrased here from pages 7 & 8) emerged as guiding principles for the implementation of this philosophy.
- Children should always have a calculator available.
The choice as to whether to use a calculator or some other method of calculation was to be the children's, not the teacher's.
- Traditional vertical pencil and paper methods for the four operations would not be taught.
They would not be needed, because children could always use calculators for calculations which they could not do mentally.
- Emphasis on good practice - practical, investigational, emphasis on language and cross curricula activities.
- Many activities not necessarily directly associated with calculators but involving apparatus and materials.
- Develop confidence in talking about numbers in gradually more precise mathematical language.
- Emphasise mental calculation and sharing mental strategies.
- 'Number' to occupy less than half of mathematics time.
Comments from C. A. N. Teachers
The C.A.N. report is liberally sprinkled with quotes from teachers, head teachers and others who evaluated the project. These samples comments suggest many of the benefits that have derived from adopting and working with the C.A.N. philosophy to enhance children's
- Page 9 - Head Teacher
The spreading of the CAN philosophy in school has been rapidly accelerated by the movement of an experienced teacher who was involved in the first year of CAN with a class of six and seven-year-olds, to a reception class of four and five-year-olds. This teacher has used the methods, and many of the activities and ideas that she developed with her six and seven-year-olds, with great success. This approach has spread sideways, as the two other reception teachers have seen how the children have developed mathematically.
(a) the children did develop mathematically (and noticeably so) when they had free access to calculators, materials and other guiding principles of the CAN project
(b) the activities and ideas the teacher used were rich enough to be applied over several ages of students
(c) good education spreads within a school by seeding experienced, enthusiastic teachers throughout the staff
- Page 9 - Teacher
It has changed my way of teaching maths, even in such a short time. I look at things differently. The CAN approach is very different from what I've done before.
Perhaps this will not be the case for all teachers. The comment is preceded by a description of the teaching style which tended to develop when the CAN philosophy was instituted:
The teaching style which has developed has given the children autonomy in their learning, allowing them to find their own ways of tackling problems, and their own mental and non-calculator methods of calculation. The teachers have been very resourceful in devising activities, and in encouraging a problem-solving and investigational approach which does not constrain the children.
Teachers who already work this way are more likely to gather the Calculating Changes approach into their teaching style than to radically alter their style.
- Page 10 - Head Teacher
The project has developed during its lifetime. From the initial concept of a calculator-based curriculum within the 'number' aspect of mathematics it has broadened, first into all the other areas of mathematics and then beyond mathematics itself.
Perhaps valuing and building on children's efforts to learn, rather than encouraging digestion of adult chosen and perfected techniques, shows children that they are respected and encourages them to show their thinking in a wider range of areas.
- Page 10 - Teacher
I'm constantly amazed at how much further on they are compared with following a traditional scheme. Previously I would never have found out what children can really do ... about their understanding of large numbers.
Teachers learning from children who have been released to construct their own learning is a key element in the professional development component of Calculating Changes.
Inclusion or Exclusion
So, the evidence from nearly 20 years of classroom research is in...
a calculator used in conjunction with manipulatives is a better use of this learning tool
a calculator used in conjunction with manipulatives and
an attitude which values children's efforts to learn is a powerful teaching
..and as a result calculating changes ...or does it?
Consider the results of research by Paul Swan and Len Sparrow reported in: Impediments to the Use of Calculators in the Primary School, Paul Swan & Len Sparrow, published in Mathematics: Creating The Future, Australian Association of Mathematics Teachers, 1997.
- 1995 survey of all schools in Western Australia
- Teachers of Years 1, 3, 5, 7
- Over 1000 responses
- Should all students use calculators?
- (The 1987 National Statement on Calculators signed by every
educational authority in Australia had already established that
- Response: Over three quarters answered "Yes"
- BUT ... "they were not referring to free access to
calculators, but rather limited use as decreed by the teacher" .
For example one teacher commented that the children are allowed to
use it to "check work, complete puzzles and to make words such as
Swan and Sparrow identified the following impediments to the
implementation of the National Statement recommendation to integrate
calculator use at all levels:
- availability of calculators
- texts and course documents did not integrate calculator use
- lack of teacher vision about how to use calculators other than
as they are used in industry
- perceiving calculators as a 'new topic' and asking how will I
have the time?
- dominance of written algorithms
- lack of 'ideas sources' for teachers
Swan and Sparrow suggested the following key changes needed to
happen if the potential learning value of calculators was to be
- development of a whole school policy
- allocating resources for calculators, professional development
and 'ideas sources'
- developing a dialogue between schools and parents to avoid
misunderstandings about calculator use
- encouraging children to develop number sense and make choices
about when to use estimation, mental methods, paper and pencil,
- The Swan & Sparrow research identified impediments to making better use of calculators.
- Better use of calculators has been shown to lead to enhanced number sense in children.
- Therefore the Swan & Sparrow research identified impediments to enhancing children's number sense.
Enhancing Children's Number Sense
Calculating Changes is structured to redress each of the
identified impediments as follows:
||Calculating Changes Action
- Calculators for every child
- Texts and course documents did not integrate
- Opportunity to review course documents and text as
part of long term PD
- PD based in the classroom experience of colleagues
- Perceiving calculators as a 'new topic' and asking
how will I have the time?
- Changing the focus from new topic to new question:
- How can we use the calculator to help us learn
- Dominance of written algorithms (designed by adults)
- Dominance of number sense (demonstrated by students
with the support of their personal calculator and
- Lack of 'ideas sources' for teachers
- PD backed up with on-going networking
And Calculating Changes offers much more because it has grown up within Mathematics Centre so sets all of its work within the context of learning to work like a mathematician. Come and join us. You're welcome.
Calculating Changes ... is a division of ... Mathematics Centre