Importance of play
Play is seen to be important (Owens and Perry, Undated, p. 102; Holton et al., 2001) in the development of understanding of all areas of mathematics. The importance of explorative investigations in an informal context is seen to have advantages in (i) the development of understanding of concepts; (ii) the seeing relations between concepts; (iii) remembering; and (iv) bringing enjoyment to the subject.
Many researchers in the late 1980s (Olson et al, 1987; Noss, 1987) have written on the value of Logo in helping children to develop an understanding of many important principles of geometry and measurement. Logo’s use has also often been linked to the van Hiele levels, due to the fact that procedures used in Logo can be seen to be developed in a similar way to the van Hiele levels.
There are also a number of pieces of sofware that are currently in use in classrooms, including Geometer’s Sketchpad, Excel, and Omnigraph (Perks et al, 2002), all of which have value if used to help develop ideas (through play and experimentation) rather than simply as a calculating tool.
Interactive whiteboards have also been demonstrated to be a valuable tool for teaching mathematics (Ball, 2003). Because they can be used in conjunction with other computer programs they are particularly valuable when linked to programs such as Geometer’s Sketchpad.
In this section we make a number of suggestions that may appear to be outside the scope of our contract. These have nevertheless been made as they appear to us to be important aspects of the Geometry/Measurement diad.
Suggestion 1: That the merger of the Measurement and Geometry strands be considered. There is no consensus among international curricular as to whether these two areas of mathematics are best addressed individually or as one strand; The US Principles and Standards (NCTM, 2000) and the TIMSS Assessment Framework (Mullis et al, 2003) list them separately, but the UK National Numeracy Strategy (DfEE, 1998) and others (Ruddock, 1998) combine them as Space, Shape and Measures. Geometry and Measurement have a lot in common but there are also topics like Time that do not belong to Geometry but seem to be part of Measurement. It is also clear that they don’t fit together in the way that Number and Algebra do, where one area develops from the other. However, they are more closely knit than Statistics and Probability. Given our situation, where we seem to be headed for the big strands of Number to Algebra and Data Handling (Statistics and Probability), it would appear to be more natural for Geometry and Measurement to go together into one strand in order to put this strand on a near equal footing in size with the others.
Suggestion 2: In the light of Suggestion 1, it should be noted that there are some topics currently in Measurement that do not fit well into a Space, Shape and Measures strand. Money has already been discussed here and that is almost certain to be in the Number to Algebra strand in future. Time was mentioned above as also being problematic. At the school level, Time and Geometry are distinct. What’s more, classroom work on Time is essentially arithmetic. While the important Measurement skill of estimation does occur with Time, that skill is not unique to Measurement and can certainly be found in Number. We therefore suggest that Time be placed with Money in the new Number to Algebra strand.
Suggestion 3: Progressions are clearly present in Geometry and Measurement both at the Strategy and Knowledge levels and we would suggest that the new strand should emphasise such progressions. New Zealand’s Numeracy Project and Curriculum Exemplars, as well as the Count Me Into Measurement and Count Me Into Space initiatives put a great deal of emphasis on the importance of progressions within mathematics and we believe this should be reflected in the structure of the curriculum. However, in the literature, these progressions are somewhat broad in the case of the former level and implicit in curricula in the case of the latter level. Nowhere have we been able to find the kind of detail in Geometry and Measurement that has been established through the Numeracy Project. It seems clear that this kind of detail would be of significant value to the learning and teaching of Geometry and Measurement in this country (and throughout the world). It would therefore seem appropriate to suggest that a Geometry and Measurement Project might be established to work along the same lines as the Numeracy Project. This might build again on the ‘Count Me Into’ work. However, again as in the Numeracy Project, this work would require intensive classroom research and development. The timescale of the Curriculum Review is such that if this work began soon, our knowledge would have advanced sufficiently far it to be of value at the time of implementation of the new curriculum.
Suggestion 4: This literature review was not designed to look for gaps in our current curriculum. Certainly some gaps have been found through the Exemplar Project (particularly around the topic of Angle), and there may well be other important gaps in the current Achievement Objectives. It seems reasonable to suggest that, armed with at least the attached curriculum documents, possible omissions should be investigated. Focussing on the idea of progressions within key topics (Suggestion 3) is also likely to reveal cases where there are gaps present in coverage of some progressions.
Suggestion 5: Our work has brought to the fore the fact that computer technology is now able to be used in the learning of Geometry and Measurement. A number of pieces of software are currently available that enable students to be able to experiment with geometric and measurement ideas and so enhance their learning. If and when this technology is available to all classrooms, teachers will require pedagogical knowledge on how to use these new tools. We suggest that work be done in this area so that advantage can be taken of technological advances as soon as possible after the implementation of the new curriculum.