A comparison between problem solving in the 1971 and 1999 curricula

stated that the aims were “to kindle a lively interest in the ‘subject’, to give the child

a grasp of basic mathematical structure and content and to lay a foundation for further work at post-primary level” (p. 125). Leading the child to a realistic level of skill in computation was also deemed to be an important aim. The 1999 curriculum was more explicit when it stated that one of its aims was “to develop problem

solving abilities and a facility for the application of mathematics to everyday life” (p. 12). Concern with ‘everyday life’ mathematics was not the only aim as an appreciation of the “aesthetic aspects” was also mentioned (p. 12).

The 1971 mathematics curriculum was split into three sections: Number, Activities, and Language Development and Recording. Problem solving is not mentioned at all at the junior and senior infant level. The curriculum is heavily influenced by a Piagetian philosophy of learning. The presumption can only be that infants are perceived as not being at an appropriate stage of readiness to solve problems. In the curriculum for 1st/2nd and 3rd/4th classes the heading ‘Problems’ appears in the Number section. The curriculum states that problems involve the pupils in making judgements and also in applying and practising the skills discovered during number activities. These activities are meant to “give the pupils ample practice in attacking

real-life problems in a sensible manner” (p. 162). Pupils are to devise problems related to their environment. The curriculum seems time bound when it states that “problems about marbles, conkers, pennies, chickens, eggs, apples, etc. set by the

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pupil will be more meaningful to him and to his classmates than any textbook problem” (p. 163). Any curriculum is a product of its time and the previous

statement reflects the importance of agriculture to the Irish economy in the early 1970s. The type of problems envisaged concerns those which involve the use of a frame or placeholder e.g. 12 - ? = 7. The pupil is meant to translate the mathematical sentence into a word problem or story and vice-versa. It is interesting that the 1999 curriculum places such problems in the 3rd/4th class sector to allow more time for this skill to be developed. The 1971 curriculum suggests that other pupils in the class “could criticise the story and discuss its suitability” (p. 164). A better choice of word might have been ‘critique’ but at least the curriculum suggests some element of

problem reformulation or posing by the other pupils. When I delivered inservice on the 1999 curriculum during its introduction in the 2001/2002 school year I received feedback from some teachers that moving curriculum content from junior to more senior classes (as above) represented a ‘dumbing down’ of the curriculum. However, I would have to take the view that such moves were designed to allow pupils more time to gain and consolidate problem solving skills and even problem pose to some degree. Both the 1971 and 1999 curriculum agree that by 4th class pupils should be able to use all four operations in reducing practical problems to some form of mathematical statement. However, in the 1971 curriculum the example given as a practical problem is as follows; “I have 5 pieces of ribbon, each of which is 4 inches long. How far will they stretch if placed end to end along the picture rail?” (p. 190). I

would contend that such a problem is not an example of a practical real life situation at all, but is instead a contrived attempt to give validity to a multiplication exercise.

In reviewing the 1971 curriculum I found it interesting that the heading ‘Problems’

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number section, including social arithmetic, to both teacher-contrived and pupil- contrived problems. The only other guidance given is that project work, so prevalent in the 1970s, should alert pupils to mathematical possibilities in solving some problem that may arise from discussion. The example given is praiseworthy. As part of a project on transport pupils could investigate the speed factor as a cause of accidents. Bar charts, showing the stopping distance of cars travelling at various speeds, are suggested as a medium. In my opinion, this is a good example of a real life problem which transcends time. It also appeals to me as a problem that exposes pupils to a more investigative approach to mathematics.

The 1999 curriculum advocates such an approach also when it states that “the

importance of providing the child with structured opportunities to engage in exploratory activity in the context of mathematics cannot be overemphasised” (p. 5). It goes on to comment that “the teacher has a crucial role to play in guiding the child

to construct meaning, to develop mathematical strategies for solving problems, and to develop self-motivation in mathematical activities” (p. 5). Problem solving is suggested as a major means of developing higher-order thinking skills. Like the 1971 curriculum, practical applications of mathematics are emphasised but the phrase ‘real life mathematics’ is not used. Instead the suggestion is that “solving problems based

on the environment of the child can highlight the uses of mathematics in a constructive and enjoyable way” (p. 8). The splitting of the 1999 curriculum into strands (number, algebra, shape and space, measures and data) and strand units has the advantage of allowing suggestions to be made for the practical applications of mathematics throughout the document. However, one of the difficulties in discussing problem solving in the 1999 curriculum is that the term appears at three levels. My contention is that there is problem solving as content (usually solving routine

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textbook problems in the Irish context), problem solving as a methodology (teacher modelling how to solve problems as outlined on pages 35/36 of the Teacher Guidelines) and problem solving as a skill to be acquired (as outlined on pages 68/69 of the Teacher Guidelines). This is a microlevel translation of the didactic triangle which captures the interplay between teacher, student and the subject of mathematics. It can be seen that it is difficult to isolate any vertex of the triangle without considering the implications for the other two vertices. The microcase is illustrated in figure 6 below:

Microlevel Macrolevel Modelling Teacher

Skill development Problems as content Student Mathematics

3.10 Research reports on problem solving in Irish primary classrooms