Research reports on problem solving in Irish primary classrooms

research reports have commented on teachers’ approaches to problem solving and

curriculum implementation in general. Inservice on the mathematics curriculum did not occur until the school year 2001-2002. Having allowed three years for the curriculum to consolidate, two reports issued in 2005. One of these was an evaluation by the Inspectorate of the Department of Education and Science entitled ‘Literacy and Numeracy in Disadvantaged Schools: Challenges for Teachers and Learners’. This publication was commonly known as the LANDS report. The report

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is implicitly constructivist in recommending the use of activity methods, concrete materials and the correct use of mathematical language. The report also recommended addressing the further development of higher-order thinking skills in both literacy and numeracy. In the middle and senior classes, the Inspectorate had identified that pupils had a poor understanding of place value and poor estimation and problem solving skills. The report recommended that pupils should be encouraged to use a range of reasoning and problem solving strategies and that problem solving tasks based on the learning needs and experiences of the pupils should be provided. It also recommended the creation of linkage between all the strands of the curriculum.

The second report issued in 2005 was also conducted by the Inspectorate but did not confine itself to disadvantaged schools. It was concerned with the subjects of English, mathematics and visual arts. The report was entitled ‘An Evaluation of Curriculum Implementation in Primary Schools’. It found that in the majority of

classrooms problem solving was a feature of the lessons observed and that pupils were provided with a range of problems which promoted the specific skills of communicating, reasoning and connecting. However, in almost a third of classrooms there were deficiencies in the use of this teaching approach. These included the non- implementation of the school plan with regard to problem solving and an over- reliance on traditional textbook problems, which did not promote specific problem- solving skills. The report noted that active involvement by pupils was very limited in a quarter of classrooms. In those classrooms the pupils were engaged solely in paper- and-pen exercises which “does not reflect the constructivist approaches that are central to the curriculum” (p. 27). In more than two-thirds of classrooms observed

there was an over-dependency on whole-class teaching, where teacher-talk prevailed and where pupils worked silently on individual tasks for excessive periods. The

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report was more optimistic on the use of linkage than the one for disadvantaged schools but did note that the data strand could be better integrated with other subjects such as Geography and Science. As in the report for disadvantaged schools, it was recommended that greater emphasis should be placed on the development of estimation strategies at different class levels.

A third report on the primary school curriculum entitled “Mathematics in Early

Childhood and Primary Education (3-8 years)” issued in 2014. Like the previous report on curriculum implementation (2005) it re-emphasises the theoretical perspectives underlying the primary school curriculum. Moreover, it brings the theoretical discourse up to date by commenting that cognitive and sociocultural perspectives provide different lenses with which to view mathematics learning and the pedagogy that can support it. It states that cognitive perspectives are helpful in focusing on individual learners while sociocultural perspectives are suitable when focusing on, for example, pedagogy. The report stresses that “learning mathematics is presented as an active process which involves meaning making, the development of understanding, the ability to participate in increasingly skilled ways in mathematically-related activities and the development of a mathematical identity” (p. 10). Learning is seen to be assisted by participation in the community of learners engaged in mathematization, in small-group and whole class dialogue. The term ‘mathematization’ is explored elsewhere in this thesis. The report states that the

processes of mathematization should permeate all learning and teaching activities. These include connecting, communicating, reasoning, argumentation, justifying, representing, problem-solving and generalizing. The language used in the report is reminiscent of Lave and Wenger’s idea of participation in communities of practice (1991) and Rogoff’s (1990) notion of apprenticeship in learning communities. I

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scant on detail concerning the theories underlying it. This report goes so far as to state that “the goal statements of a curriculum should be aligned with its underlying theory” (p. 12).

As regards higher-order thinking skills the report highlights that there are “concerns about the levels of mathematical reasoning and problem solving amongst school- going children, as evidenced in recent national and international assessments and evaluations at primary and post-primary levels” (p. 9). It suggests the use of learning paths in the teaching of mathematics. These paths are defined as sequences that apply in a general sense to children’s development in the different domains of

mathematics. In line with the ideas of the Realistic Mathematics Education movement, discussed elsewhere in this thesis, such paths are not meant to be fixed for each pupil, but rather characterised by fluidity and influenced by the role of context. The report recommends that educators should be assisted in the design and development of rich and challenging mathematical tasks that are appropriate to their children’s learning needs. This assistance needs to commence at the teacher

preparation level in colleges of education.

On the next page I present Table 2 which compares the three reports outlined above in terms of their underlying philosophy, the problem solving deficits they highlighted and the solutions which they proposed.

86 Report Underlying Philosophy Problem solving Deficits Proposed Solutions LANDS (2005) Implicitly constructivist Estimation and higher-order thinking skills  More problems based on pupils’ real life experiences

 Create more strand linkage An Evaluation of Curriculum Implementation (2005) Explicitly constructivist  Estimation skills  Disconnect between school plan and classroom practice  Over-reliance on textbook problems  Use the environment to contextualise learning  Promote active learning Mathematics in Early Childhood and Primary Education 3-8 years (2014) Both cognitive constructivist and sociocultural  Levels of higher- order thinking skills  Adaptable learning paths for pupils

 Promote

mathematization

 Design challenging mathematical tasks

Table 2:DES and NCCA reports on mathematics teaching/learning 2005-2014

In summarising the three reports, it can be seen that there is an ongoing general concern with the levels of children’s higher-order thinking skills. Solutions proposed include moving away from textbooks and basing mathematical tasks on children’s

real life experiences. It is interesting that Conway et al. (2011), referring to research by Lyons et al. (2003), state that “being able to justify solutions or displaying real

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teachers” (p. 120). The Evaluation of Curriculum Implementation Report (2005)

found that there was an over-reliance on textbook problems. Such problems are often presumed to be based on real life experiences but often wear a thin veneer of such experience. Kilpatrick (1985, p. 4) refers to one such problem: If a 7-oz. cup of cola costs 25c, what is the cost of a 12-oz. cup? Kilpatrick (1985) states that this problem is meant to simulate a real problem the pupils might face. I have two difficulties with this interpretation. Firstly, when one buys beverages the cost of a larger cup is rarely in proportion to the cost of a smaller cup (due to economies of scale). Retailers extract extra money from the consumer by providing the larger cup at a ‘discount’ to

the proportional price. I am reminded of the McDonald’s Supersize Me campaign which received a lot of criticism for encouraging consumers to overindulge. Secondly, when one does the calculation the answer turns out to be 42.857142 cents which contains a recurring decimal. This amount of money cannot exist in real life. Kilpatrick (1985) puts it well when he states that the actual calculation exercise could be termed “the computational skeleton beneath the skin” of the problem (p. 4).

The emphasis on teaching problem solving through real life contexts is discussed at length elsewhere in my critique of the Dutch RME movement in this thesis. Here it suffices to say that authors like Brown (2001), as well as Nicol and Crespo (2005) ask us to broaden our conception of what counts as real. Ross et al. (2002) state that reform mathematics should encompass student tasks which are “complex, open- ended problems, embedded in real life contexts” (p. 125). However, Brown (2001) requests that we reconceptualise what is ‘real’ in a more imaginative sense than what

exists or what we can touch and see. He states that by doing this we not only legitimise more interesting connections between mathematics and the real world, but “we also suppress the need to seek real-world connections as a slave against an otherwise ‘unreal’ world of mathematics” (p. 191). Egan (1997) and Brown (2001)

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reject the premise that making subject matter interesting and meaningful to students requires the need for it to be placed in real-life contexts. Egan (1997) suggests that school subjects need to connect with students’ fascination with the limits of reality,

their interest in heroic qualities that exceed their everyday lives, and their wish to connect with human intentions and emotions. As an example of pupils’ interest in

limitations, Egan points to their fascination with the human traits, information and numbers which abound in the Guinness Book of Records. Crespo and Nicol (2005) suggest two tasks as examples of activities that engage students’ mathematical

imaginations. The first is called the Mayan Dresden Codex. The codex artefact is a 3.5 metre band of paper which illustrates the Mayan place value system. In the activity pupils are exposed to Mayan culture and asked to work as archaeologists to discover the place value inherent in the scroll. Apparently, the Mayan numeration system operated in base twenty rather than base ten. The second activity is called Life in Flatland. It is based on a story written by Edwin Abbott in the 1800s under the pseudonym of A. Square. In this activity pupils are asked to imagine what life would be like in two dimensions. For example, pupils are asked to work in pairs whereby one pupil hides his or her eyes while the other pupil places a set of thin geometric shapes on the table. Then the first pupil bends down so that his eyes are level with the table and tries to distinguish the shapes. These activities show that although research continues into how best to solve textbook problems, other researchers as above have shifted the focus onto defining and promoting problems as tasks or activities which should engage pupils both intellectually and emotionally.

The Mathematics in Early Childhood and Primary Education Report (2014) promotes the use of fluid learning paths for pupils and includes the development of rich mathematical tasks which are challenging for pupils. These proposed solutions

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remind me of Jaworski’s Teaching Triad (1994) in that the teacher has to be

conscious of how she manages the learning in her classroom so as to be aware of pupils’ individual learning paths (sensitivity to students) and yet promote the use of

rich tasks (mathematical challenge). In these circumstances the teacher is using the Teaching Triad as a tool to develop her practice. In this thesis I also use the Triad as an analytical tool to categorise teachers’ practice as they attempt to move away from

routine textbook tasks to engage in more challenging and varied mathematical activities, as was recommended in the Evaluation of Curriculum Implementation Report (2005). I am particularly interested in tasks which can be solved using various strategies or techniques. When pupils do routine textbook problems they develop an image of mathematics as being concerned with a single procedure or algorithm to obtain a correct answer. Therefore, as part of my reform agenda, I would like teachers to experiment with problems which have different solution paths and incorporate cognitive challenge as recommended in the Mathematics in Early Childhood and Primary Education Report (2014).