ABSTRACT
The main purpose of the study was to investigate pupils’ experiences when working on non-routine mathematics problems. This was deemed imperative to discover learners’ reactions to problems of this kind, investigate their competencies and skills necessary for successfully working out non- routine exercises and also delving deeper into how pupils felt whenever they were tasked to work on complex type problems. Again, the study sought to unearth how pupils were benefitting from useful classroom practices such as collaborative and co-operative learning, as far as the solving of non-routine problems was concerned. The research was carried out at Ghana International School (G.I.S) where a total number of 144 pupils, from Year Four to Year Six, were selected using the simple random sampling technique. With respect to the research design, the sequential mixed method was adopted where emphasis dwelt on the use of assessments and interviews as the main instruments for data collection. The data gathered was analysed statically using simple frequency tables and the responses from the participants were also transcribed into thematic areas. The results from the study revealed that most pupils exhibited inappropriate comprehension of problem solving and computational processes including inaccurate choices of strategies. Again, learners grappled with incompetence at thinking mathematically and lacked the much needed creative knowledge of how to work purposefully on non-routine problems. Some of the recommendations included teachers benefitting from more in service training, pupils making more progress through collaborative and co-operative learning and educational authorities creating more opportunities for teachers to build on their competence through more Professional Development packages in the area of mathematics.
CHAPTER ONE INTRODUCTION
Background to the Study
The significance of mathematics education cannot be over emphasised. From my years of experience as a primary school teacher, there is clear attestation that the efficient delivery of the curriculum seeks to equip learners with an array of benefits. In my opinion, the enhancement of pupils’ reasoning abilities, critical thinking skills and competence at problem solving are few of the several existing advantages associated with the in-depth study of Mathematics. In fact, Legner (2013) upholds the view that the basic arithmetic required for many jobs will be largely acquired by the end of primary school thus laying more emphasis on the essence of its fundamental teaching and learning. Based on this opinion and the research of other scholars (Kavkler, Magajna and Babuder, 2014) who have, over time, unravelled the benefits of the subject, it is prudent to say that offering young learners a meaningfully concrete mathematical knowledge remains paramount and must be the duty of all educators within this field.
Across the globe today, it is therefore not surprising that governments’ commitment to effective teaching and learning of mathematics continue to initiate several reforms within the area. Kilpatrick (1997) believes that such reform impulse especially within the United States has, over the past years, heated up so much and been led by professional organisations under the banner of raising expectations and providing mathematical literacy for all. Such growth in reforms is certainly prevalent in other developed countries; Australia’s reform for Mathematics Year 1-10, which was implemented in 2007, was an example of such excellent initiatives which aimed at representing a significant shift in the teaching of the subject from the previous syllables
(Department of Education, 1987). Bruniges (2005) believes that such rapid reforms, apart from development in teaching and learning, characterise a feature of education due to several context related developments including globalisation, new technology, knowledge of the economic and cultural diversity.
Perhaps these factors account for a number of laudable strides which have, over time, also seeped into Africa. For example, the African Mathematics Project (AMP) pursued a policy of bringing together African American and British educators in English speaking African countries to influence Mathematics education (Mereku, 1999). South Africa, for instance, has also had a taste of several radical curriculum change particularly due to the need for social, economic and political transformation; and in countries like Malawi, Ethiopia, Ghana and Nigeria, there have been other advances within the area of Mathematics (Parker, 2006; IMU, 2014). Evidently, such transformational developments remain positive leaps particularly towards the enhancement of Mathematics Education and the good news according to White, Bloomfield and Cornu (2010) is that, these contexts related developments promise opportunities for change in primary mathematics education, thus shaping the curriculum reform agenda of ‘productivity, participation and quality.’ Highlighting further on the importance of the subject, they also comment that the proper grounding of one’s knowledge in the area of Mathematics education comes with great benefits for the future. For instance, an excellent achievement in the subject often prepares the individual adequately for one’s contribution to economic productivity. The need for such global reform is therefore in the right direction.
Despite these global strides, however, pupils’ performance in Mathematics continues to be worrying. In a recent 2015 publication from Trends in International Mathematics and Science Study (TIMMS) the 2016 report disclosed that though countries like Russia, Kazakhstan and
Northern Ireland saw some improvements in the performance of students, there still appeared to be significant gaps between high performing countries and others. A good example was the gap between the United States and Singapore, the highest performer in the world (Simms, Gilmore, Sloan and McKeaveney, 2017). This was worrying considering the fact that America, for example, continues to spend more than most countries in preparing students for such external tests yet the country recurrently attains poor results from students, with 12 percent performing below the expected levels in the subject (OECD, 2012). The situation is similar in the United Kingdom where a large percentage of children fail to meet the expected levels in mathematics by the end of primary school (DE, 2015). It is also worth noting that just like the United States, the UK government also incurs a loss to poor numeracy skills which is estimated at 20.2 billion a year (Simms, Gilmore, Sloan and McKeaveney, 2017).
International Mathematics tests richly focus on non-routine problems (Marchis, 2012); this category of Mathematics is what Liljedah (2008) defines concisely: ‘Tasks that cannot be solved by direct effort and will require some creative insight to arrive at an answer.’ Lester and Kehle (2003) therefore suggest that reasoning or higher order thinking must occur during such process. Oviedo (2005) admonishes that in this peculiar task, one should be able to read, interpret and transform the stated words within their contexts into a symbolic form, before embarking on a search for manipulative or computational strategies. Unfortunately, pupils lack such skills to work capably on non-routine exercises thus aggravating cases of mathematical deficiencies among learners today. Dowker (2009) believes that such low attainment in mathematics have negative effects which are not only restricted to a small percentage of learners identified with problems such as Dyscalculia but goes beyond to include a wider proportion of children who also fail to attain the requisite mathematical skills they need to grasp for everyday life. These poor numeracy
skills, often detected at an early stage, have over years contributed to a ‘viscous cycle of disadvantage and a poverty of opportunity’ (Northern Ireland Audit Office, 2013). These revelations are indeed disturbing and stress the fact that it is high time educators adequately supported pupils to become confident and capable ‘problem-solvers’ when working on non-routine exercises.
If global scores in International mathematics tests, which are mostly non-routine problems, are not too pleasing, then there is obviously a clear indication that pupils struggle particularly in this area. Romanian pupils, for instance, have high scores, above international average on routine problems but they obtain lower scores than the average on non- routine problems (Marchis, 2012). The poor attainment in International Mathematics Assessments are perhaps even more worrying in Africa where the lukewarm performance of pupils raises questions about the effectiveness of the periodical curriculum and educational reforms in most African countries (Ndlovu and Mji, 2012). Even in countries where English is considered the official language, pupils still do not have the luxury of excelling when presented with mathematical problems in the language they know best. In the English-speaking country of Kenya, for example, where pupils and students are assessed in the official language, mathematics education has faced various challenges which have incredibly influenced poor performance in national examinations (Ndlovu and Mji, 2012). This gives an indication that to be competent especially at non-routine problems and the effective use of problem solving skills, the need to possess significant mathematical competence is necessary other than language.
Griffin and Jitendra (2009) unearth a number of these skills as one’s knowledge of sentence structure, mathematical relations, basic numerical skills and mathematical strategies to solve a number of reasoning type mathematical problems. In fact, critics like Njagi (2015) believe that
many language challenges have contributed to dismal performance such as poor mastery of specialised mathematical symbols, use of terms that have different meanings, linguistic barrier that leads to poor communication in classrooms and language challenges in solving non-routine problems. Highlighting more on the latter factor, a report by COAG (2008) acknowledged that language can provide a formidable barrier to both the understanding of mathematical concepts and the provision of students’ access to assessment items aimed at eliciting mathematical understanding. Judging from this statement, learners’ challenges at non-routine problems which Milgram (2007) explains as problems one has no straightforward solution to, could be linked to language deficiency and no wonder Moeller, Klein and Nuerk (2011) state that linguistic complexity contributes to difficulty for learners and this has over the years, contributed to pupils’ inability to work competently on such problems.
Pupils’ general poor performance on such problems have compelled experts to carry out extensive works within the area. However, most of these significant researches have been set within Western contexts thus signaling the fact that a lot more needs to be done in Africa. In Ghana, for instance, mathematics performance is poor among learners in both Primary and Junior High School, particularly due to learners’ problem solving inabilities (Mereku and Anamuah-Mensah, 2005; Adu, Acquaye, Buckle and Quansah,2007) and, obviously, the situation needs redress.
Statement of the Problem
The importance of mathematical literacy in this technological age is recognised universally, for it is a tool for developing a rational personality (Kavkler, Magajna and Babuder, 2014). Mereku (1992) believes that the subject occupies a privileged position in the school curriculum because the ability to cope with more of it improves one’s chances of social development. Undoubtedly, Mathematics education is of essence and it is not surprising that in our modern society, there are
growing needs for mathematical skills and proficiency because students must master advanced skills to stay on track for promising careers (Njagi, 2015). The desire among educators to see pupils becoming mathematically able is therefore of immense relevance though this quest remains far-fetched.
In Ghana where this study is set, pupils’ attainment in mathematics is generally low (Mereku, 2003) and the area of non-routine problems is not an exception. Evidences of such poor performances are enshrined in various reports focussing on outcomes from national assessments (NEA, EGRA & EGMA), national examinations (BECE & WASSCE) including international examinations (Mereku, 2012). Debilitating standards and records of worrying grades attained by some BECE candidates, for instance, indicate that most students underperform in the subject area; a challenge mainly caused by lack of basic concepts (Mills & Mereku, 2016). For a developing nation like Ghana which requires citizens well-grounded in numerical skills to contribute towards economic productivity, such challenges raise thought provoking questions: Are we, indeed, churning out students fit to be considered nation builders? If students continue to show such significant deficiencies, how ready is the next generation’s preparedness to cope with everyday life which undoubtedly requires a good knowledge of mathematics?