Developing a Framework for Mathematical Enrichment (Part II)

Mathematical Thinking Strategies:

Some of the mathematical thinking strategies we have identified include:

• Conjecturing/theorising;
• Being systematic;*
• Identifying common structures (isomorphisms);*
• Introducing variables;
• Generalising;*
• Specialising/clarifying/looking for specific examples;
• Considering a special case (the particular);
• Solving simpler related problems;
• Reflecting on experience - have you met something like this before?
• Multiple representations;
• Working backwards;
• Identifying and describing patterns;
• Representing information– diagram, table
• Testing ideas - guessing and testing (hypothesizing.

*  We have begun to develop curriculum resources that illustrate and support these aspects of mathematical thinking, in the form of trails.

There is still some work to do in identifying different aspects of mathematical thinking .Not all these strategies have a similar feel to them.  Currently it seems easier to implement a developmental schema for some than for others.

Problem Solving Process

There are a number of descriptions of what constitutes problem solving within the literature (Mason et al (1985), Mayer (2002), Ernest (2000), Polya (1957)). These references have many common threads and have models of the process that are broken down into a varied number of stages.  The process outlined below combines a number of the features of these existing models with our own research findings.

The C.A.P.E. model

• Comprehension

o Making sense of the problem/retelling/creating a mental image,
o Applying a model to the problem;

• Analysis and synthesis

o Identifying and accessing required pre-requisite knowledge,
o Applying facts and skills, including those listed in mathematical thinking (above),
o Conjecturing and hypothesising (what if);

• Planning and execution

o Considering novel approaches and/or solutions
o Identifying possible mathematical knowledge and skills gaps that may need addressing,
o Planning the solution/mental or diagrammatic model,
o Execute;

• Evaluation

o Reflection and review of the solution,
o Self assessment about ones own learning and mathematical tools employed,
o Communicating results.

Despite its representation, this is not a simple linear model – sometimes it is necessary to revisit and review several times – one can think of the problem solving process as a spiralling inward towards a satisfactory conclusion.

Implications for teaching for enrichment

I have discussed above the curriculum content associated with mathematical enrichment in terms of the two aspects of mathematical thinking and problem solving. For this content to have meaning, the learning (and teaching) environment needs to encourage effective use of the resources so that pupils develop the necessary skills, strategies and competence to tackle problems and use underpinning thinking skills effectively.  This has implications for the second thread of mathematical enrichment – that of the teaching approach adopted. There are a number of features of such a teaching approach, building on the work of Lerman (1999), Romberg (1993) and Ruthven (1989) and takes a view of pupils constructing their own learning in a social context, where communication and sharing are central to mathematical growth and understanding.  Aspects of such an approach include:

• The use of problems which encourage a problem solving approach that in turn supports mathematical thinking and the contextualising of the relevance of mathematical skills and facts (known or to learn).
• Employing the use of low threshold – high ceiling tasks
• Giving pupils time to engage with the problem before moving towards a solution (exploration)
• Focus on “doing mathematics” – pupils taking responsibility for tasks and identifying possible routes to and requirements of solutions rather than being led by the teacher.
• Appropriately targeted mediation that supports entry into problems and development of solutions without leading.  Building on pupil discovery and knowledge and making connections (codification)
• Transfer of knowledge which is dependent upon individuals internalising schema with the teacher identifying opportunities.

Mathematical enrichment trails

The trails are a new concept of resource management that are being developed by the NRICH team, practising teachers and mathematics educators.  They aim to combine related resources (problems, activities, games, articles, other sites) into a coherent programme of activities that have problem solving at their centre and which describe a strand of an enrichment curriculum aimed at either a particular aspect of mathematical thinking, or a particular aspect of the curriculum tackled through a problem solving approach.  They also reflect the view of teaching and learning mathematics outlined above and are being described in terms of:

• their mathematical content (standard curriculum facts and skills as well as mathematical thinking skills);
• a recommended pathway, or pathways, through the items
• prerequisite knowledge;
• anticipated learning outcomes;
• guidance notes for teachers which reflect the enrichment approach to teaching tha underpins our work
• guidance notes and hints for pupils;
• formative self-assessment mechanisms which will enable medium to long term planning and evaluation.

A trail, for example, might develop and support the work on number and problem solving through investigating Magic Squares.  For the most able students the work might lead to investigating the idea of isomorphisms and the underlying structure of some mathematical problems (looking for pattern and familiarity in problem solving contexts – “have I seen something like this before?”).  Brighter pupils may also be encouraged to consider algebraic properties and relationships in this context.  A very able student may begin to generalise and look at “higher order” mathematics, looking at articles on the subject written by established mathematicians.  Whilst students struggling with identifying patterns and relationships more generally may benefit from generalising their findings when working from one magic square context to another.

A trail on “being systematic” can offer opportunities in a range of mathematical contexts (number, geometry etc) to take a systematic approach to solving the problem.  Whilst other proof, or algebra based methods may be just as appropriate in any particular context, the aim is to use a range of systematic strategies to access, engage in, and eventually solve, a problem.  Work on the trail may extend over weeks or months or several academic years but in every case the aim is to give some structure to the development of the related skills.

The structure of a trail will enable choices concerning the routes into the resources to reflect the needs of the pupil and underlying learning theories.  Trails aim to “unpick” the opportunities being offered to pupils to use and develop their problem solving and other higher order mathematical skills in terms of content, learning theories and associated teaching styles.

Implications for Implementation

Through the intertwining of the research and development of the NRICH site, and particularly the trails, the value of this curriculum innovation is being constantly assessed.  All the work is grounded in appropriate theories as well as research and classroom experience that not only clarifies and informs the development itself but throws light on current views and practice with respect to the role, content and implementation of mathematics enrichment more generally.  As materials are developed and tested this in turn informs our theoretical framework.

Mediation

An emerging area of interest is the nature and role of mediation and how mediation can take place, or underpinning learning theory be reflected, in the materials we produce.  Current small-scale research by members of the NRICH team identifies the view of problems as rivers to be crossed rather than to be studied (the process is simply about finding the answer rather that mathematical discovery).  This view acts as a barrier to encouraging problem solving and mathematical thinking skills.  We are currently undertaking research into the role of mediation and how we can offer relevant mediation at a distance (Back, J., et al. 2004, forthcoming).

Conclusion

The clarification of the terms enrichment, mathematical thinking and problem solving have all led to a clearer understanding of the potential of NRICH to support mathematical enrichment more generally, being a vehicle for the many not simply the few.

Key outcomes:

• establishing a view of enrichment/problem solving /mathematical thinking and reflecting this view within the resources we produce.
• Placing the role of factual knowledge and skills within an enrichment framework both as a precursor and a consequence
• the identification of mediation in a “remote” environment as a key area for our future research
• continuing to reflect the importance of the social role in the construction of knowledge within an online and  remote resource
• that issues related to seeing the process and/or solution as the goal rather than the answer is key to our mediation and support work
• that there is a  role for assessment and that self and/or peer assessment is an area we need to investigate further.

Impact on the development of the NRICH site

The NRICH had the first phase of its relaunch in January 2004.  The key features of the new site that have been driven by our research findings are:

• Transparency between levels
• Range of levels and difficulty (challenge level)
• Monthly themes
• Problems also include hints and notes
• Integration of the thesaurus
• Integration of the discussion boards
• Easier access to related material within the archive.
Impact on the development of Trails
• Clear rationale for each trail
• Structure and accompanying documentation that supports learning theories and associated teaching approaches,
• Picking particular mathematical thinking and problem solving schemes as focus for each trail
• Developmental not ad-hoc organisation of resources
• Consideration of the role of mediation and developing mediation strategies.
• The choice of self-assessment as the core assessment strategy.

Bibliography

1. Back, J., Gilderdale, C., Piggott, J., 2004, forthcoming.
2. Boaler, J., Wiliam, D. et al., 2000, “Students' experiences of Ability Grouping - disaffection, polarisation and the construction of failure.” British Educational Research Journal 26(5): 631 - 648.
3. Brown, M., Millett, A. et al., 2000, “Turning our attention from the what to the how: the National Numeracy Strategy.” British Educational Research Journal 26(4): 457 – 471.
4. Cobb, P., Wood, T. and Yackel, E. (1991). 'A constructivist approach to second grade mathematics'. In von Glaserfield, E. (Ed.), Radical Constructivism in Mathematics Education, pp. 157-176. Dordrecht, The Netherlands: Kluwer Academic Publishers.
5. Ernest, P., 2000, “Teaching and Learning Mathematics”, in Koshy, V. et al, Mathematics for Primary Teachers . London Routledge.
6. Koshy, V.,2001, Teaching mathematics to able children, David Fulton.
7. Lerman, S., 1999, Culturally Situated Knowledge and the Problem of Transfer in the Learning of Mathematics, in Learning Mathematics, Burton, L., (Ed), Studies in Mathematics Education Series, Falmer Press.
8. Lester, F.K.Jr., Masingila, J.O., Mau, S.T., Lambdin, D.V., dos Santon, V.M. and Raymond, A.M., 1994.  'Learning how to teach via problem solving'. in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp. 152-166. Reston, Virginia: NCTM.
9. Mason, J., Burton, L., Stacey, K.,1985, Thinking Mathematically, Prentice Hall
10. Mayer, R 2002, Mathematical Problem solving, Mathematical Cognition, 69-72
11. Nardi, E. and Steward, S., 2002, “Part 1: 'I could be the best mathematician in the world... if I actually enjoyed it'.” Mathematics Teaching 179.
12. Nardi, E. and Steward, S., 2002, “Part 2: 'I'm 14, and I know that! Why can't some adults work it out?'.” Mathematics Teaching 180.
13. Nardi, E. and S. Stewart (2003 forthcoming). “Is Mathematics T.I.R.E.D.? A profile of quiet disaffection in the secondary mathmatics classroom.” British Educational Research Journal 28(2).
14. Polya, G., 1957, How to Solve it, Princeton Paperbacks.
15. Romberg, T., A, 1994,  Classroom instruction that fosters mathematical thinking and problem solving: Connections between theory and practice. In A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 287-304). Hillsdale, NJ: Lawrence Erlbaum Associates.
16. Schoenfeld, A., 1994. Reflections on doing and teaching mathematics. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 53-69). Hillsdale, NJ: Lawrence Erlbaum Associates.
17. Van Zoest, L., Jones, G. and Thornton, C. (1994). 'Beliefs about mathematics teaching held by pre-service teachers involved in a first grade mentorship program'. Mathematics Education Research Journal. 6(1): 37-55.
18. Watson, A., 2001,  Changes in mathematical performance of year 7 pupils who were 'boosted' for KS2 SATs. British Educational Research Association, Leeds, Education-

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Developing a Framework for Mathematical Enrichment (Part I)

Abstract
“In mathematics the ability to solve problems is not just knowing some straightforward rules”
                                                                                                                       
                                                                                                                         Polya (1957)

The NRICH Project (www.nrich.maths.org) has been in operation since 1996, when its original purpose was to support able young mathematicians whose access to opportunities in their local community was limited and often non-existent.  Since this time the resources on the web site have grown and the project has developed a reputation for creative thinking in the area of mathematics enrichment.

The most recent work of the project has centred on making more effective use of the wealth of resources we now have available to us, both in terms of access to the enormous archive and in creating meaningful frameworks within which selections of the material can be placed (enrichment trails).  As the new site and trails have developed we have questioned our understanding of mathematics enrichment and how it might be represented in classroom practice and through the NRICH site itself.   The reflection and early research findings have resulted in two key outcomes that are having a fundamental impact on our work:

• the resources are not suitable solely for the most able but have something to offer pupils of nearly all abilities.  This has resulted in the restructuring of the site and creation of the trails to facilitate a “free flow” of resources across age and ability boundaries.

• enrichment is not only an issue of content but a teaching approach that offers opportunities for exploration, discovery and communication,

• effective mediation offers a key with which to unlock the barriers to engagement and learning.
We are attempting to address the issues of the nature of enrichment, accessibility, mediation and the philosophies of learning and teaching that underpin our work both through the structure and content of the site, our work on enrichment trails and our face to face work with pupils and teachers.
This paper considers the key aspects of mathematics enrichment and how the content and design of trails (as well as the NRICH site itself) has been influenced by, and built upon, these philosophies.

Background and Rationale

The wider context

The United Kingdom Numeracy Framework offers guidance and exemplification of the mathematics curriculum giving content, structure and guidance on its implementation and delivery.  However, although there has been an overall improvement in performance in national tests, there are areas where concerns still exist in terms of performance, teaching and attitudes to mathematics:

• Concerns exist over pupil performance in algebra, geometry and problem solving (Brown, Millett et al. 2000).  These concerns have most recently resulted in changes to the national mathematics attainment tests, which will now include a problem solving section.

• Most commonly, the needs of most able pupils are met through courses of acceleration.  Pupils undertaking such courses are often taught independently (and separately) from their peers, older pupils often having to go to other schools for their lessons. These models of acceleration pose medium to long term problems of sustainability and there is no evidence of long-term benefits. Ability grouping with ‘fast track’ top sets has also been shown to cause problems in the long term (Boaler, Wiliam et al. 2000).

• Fewer pupils are choosing to study mathematics and mathematics related subjects beyond the age of 16 (Nardi and Steward 2002), (Nardi and Steward 2002) (Nardi and Stewart 2003 forthcoming).

• Evidence of lack of motivation and consequent dips in performance across KS3 is available and indicates pupils are being “turned off” mathematics. (Watson, 2001). Results in 2003 show a slight decline in performance over previous years resulting in the government adjusting long term performance targets.

Enrichment can be used:

• to support the most able alongside all children in the class; often offering differentiation by outcome,

• to promote mathematical reasoning and thinking skills, preparing pupils through breadth and experience to tackle higher level mathematics with confidence and a sense of pattern and place.

Mathematics Enrichment Materials on the NRICH Website

There have been a wealth of resources that support mathematical enrichment, most notably the NRICH online mathematics project (www.nrich.maths.org).  The resources on the NRICH site have been in “loose leaf” format; being stored with few pointers to their curriculum context and relevance.  This has left the user with issues of access to appropriate material and knowledge of the potential of, and means by which, the material can be used to support the development of high level mathematical reasoning (and other) skills.

From these points come the foci of our recent work:

• identification of key aspects of an enrichment curriculum for mathematics that makes links between content, the national frameworks, and practice explicit;
• effective presentation and structuring of resources on the NRICH site such that they will underpin an enrichment framework by offering exemplars of content and supporting material.

It is through examining the theories underpinning the development of structured content (trails) and views of teachers as users of the trails, the nature of mathematical enrichment and how it can be represented is being implemented.

Defining a Framework

Terms such as “mathematical thinking”, “mathematical problem solving” and “enrichment” are variously described in current literature.  Our work has therefore involved us in clarifying definitions of these terms.  Establishing meanings has involved a literature review, interviews with colleagues and teachers and the analysis of NRICH team discussions. In addition, the process of site and trail development has involved multiple iterations which have themselves informed the definitions. These definitions are therefore constantly being reviewed and refined as we trial and test materials and build the framework within which our work is set.  What is presented is our current view of these terms as they relate to our work.

Enrichment

In current literature, “enrichment” is almost exclusively used in the context of provision for the mathematically most able.  However, there is strong evidence from the use of the NRICH site, and our own experience working with teachers and pupils, that this fails to address the value of an enrichment approach to teaching mathematics generally.  Problems which offer suitable entry points can be used with pupils of a wide range of ability and therefore can be used within the “ordinary” classroom.  The teacher or mentor can use such materials in flexible ways that respond to the needs (and experience) of the learner.  We see enrichment as an approach to teaching and learning mathematics that is appropriate for all not simply the most able.  NRICH resources therefore continue to support the most able but this is within the context of a broad interpretation and view of enrichment not within a context of provision simply targeting the most able.  Good enrichment education is good education for all.  Good mathematics education should incorporate an approach that is an enriching and stimulating experience for all pupils.  The construction of enrichment we are adopting thus builds on two main threads:

Content

This thread describes an enrichment curriculum, which has the following components:

• Content opportunities designed to:
o develop and use problem solving strategies,*
o encourage mathematical thinking,*
o include historical cultural contexts,
o offer opportunities for mathematical extension.

* These two strands form the focus of the content discussion in this paper
Enrichment is not simply learning facts and demonstrating skills.  Mathematical skills and knowledge can be a precursors to, and also outcomes of, an enrichment curriculum (needs driven learning).  The aim of an enrichment curriculum is to support:
• a problem solving approach
• improving pupil attitudes
• a growing appreciation of mathematics
• the development of conceptual structures

                                                                                                             based on Ernest (2000)

Enrichment therefore represents an open and flexible approach to teaching mathematics which encourages experimentation and communication

Teaching approach

This places an emphasis on teaching that reflects a constructivist view of learning and which stresses:

• non-assertive mediation,
• group work, discussion, communicating …,
• varied solutions and different approaches being valued and utilised,
• exploration, making mathematical connections, extending boundaries, celebrating ideas not simply answers, flexibility… ,
• acknowledgment that maths is hard but success is all the more enjoyable when a hurdle is overcome.

Problem Solving and Mathematical Thinking
A range of  literature exists in the areas of Problem solving and Mathematical thinking.  The two terms often being used synonymously or with a lack of clarity in their inter-relationship,  As part of our framework for development we have been able to identify two distinct threads that appear in the use of the two terms and which are worthy of articulation and distinction.    These threads pull together ideas drawn from current theory (Mayer (2002); Koshy (2001); Mason, Burton and Stacey (1985); Ernest (2000), Shoenfeld (1994), Polya (1957), Lester (1994), Cobb et al (1991), Van Zoest et al (1994), and our own work in the field.

We are taking “mathematical thinking” to mean particular mathematical strategies that are employed in solving problems of different types.  Some exemplars of these strategies are given below. The aim is to identify problems where such strategies are useful and create a curriculum thread that encourages pupils to develop each strategy and identify the type of context and the ways is which such strategies can be employed.

Problem solving is reserved for the structural approach to solving problems - the overview, or steps on the journey from meeting a problem for the first time to its solution.  Problem solving identifies and developments competence in utilising the stages on the route through solving a problem.   Problem solving underpins the vast majority of NRICHs resources.

Thus mathematical thinking strategies are needed to tackle problems and will be used within the problem solving process.

Teacher knowledge: A crucial factor in supporting mathematical learning through play

This paper reports on the mathematical thinking taking place during play in a sessional kindergarten.  It identifies ways in which early childhood teachers can broaden their professional development in mathematics education, and indeed why many early childhood teachers might need to do so, in order to enhance the mathematical learning of their children.  Narratives in the form of learning stories, and photographs of the children at play, augmented and supported the findings of the investigation.

Introduction
There has always been an assumption that in the early years the initial stages of a child's mathematics learning can be seen through their play.  While the child learns by doing, however, the teacher teaches by knowing.  Therefore in order to maximise support of this early mathematical learning an early childhood teacher needs a thorough and extensive mathematical knowledge-base, coupled with theory and experience of appropriate professional pedagogy.  Too often the teacher is bereft of sufficient mathematical knowledge with which to fully employ appropriate skills and strategies needed to enhance mathematical knowledge for the child.

The significance of play in developing early mathematics understanding

Play creates a natural environment of discovery for children, allowing them to learn about themselves and the world around them.  According to Stone (1995) play is defined as an intrinsically motivated, freely chosen, process-oriented over product-oriented, non-literal, and enjoyable activity.  Play serves an important function in children’s holistic development, which includes physical, emotional, social and intellectual growth.  Through play children learn to think for themselves, to make choices and decisions, to reflect, and to tolerate uncertainty, thus enabling them to become more flexible and confident in themselves.  These are important and integral aspects of both the early childhood curriculum, Te Whaariki (Ministry of Education, 1996) and the national mathematics curriculum, Mathematics in the New Zealand curriculum (Ministry of Education, 1992).

Pound (1999) believes the thinking in action which occurs in play forms a rich foundation for the more subject-specific problem solving, mental imaging and recording, in mathematics education, that can develop from play.  Much of what young children learn is incidental, or natural, and happens through their play.  They also observe adults using mathematics for meaningful purposes, and begin to use number and other mathematical concepts themselves as part of their everyday lives.

Young children as problem solvers

Mathematical know-how is the ability to solve problems which require some degree of independence, judgement, originality and creativity, as well as the ability to solve routine problems (Polya, 1995 cited in Pound, 1999).  Mathematics, like all other human knowledge, is a consequence of social interaction.  It is a means, or framework, used to support ongoing enquiry into aspects of the world (Pateman & Johnson, 1990 cited in Steffe & Wood, 1990).

How children go about learning mathematics varies greatly from child to child according to cultural background, family orientation to mathematics, the child’s own disposition to learning, and teacher confidence.  Carr (1999) writes of children’s emerging working theories about what it is to be a learner, and about themselves as learners.  She had earlier developed the idea that the working theories were made up of packages of learning dispositions and defined such dispositions as "habits of mind", or "patterns of learning".  She further developed a framework of learning dispositions (Carr, 1998), known as learning stories, closely linked to the strands of Te Whaariki  (Ministry of Education 1996).  The framework of dispositions included courage and curiosity, trust, perseverance, confidence to express an idea, and taking responsibility for fairness and justice.  In particular, these dispositions support quality mathematics learning through children’s engagement in the problem solving nature of the mathematical processes (Ministry of Education, 1992).

Teachers supporting early mathematics learning

Early childhood teachers have a vital role in the total educative process. Alexander (1997, cited in Pound, 1999: 35), believes teachers have a responsibility to make sure that the "imperatives of early childhood" are not lost among the noisy demands for early achievement.  Meade (1997) found, when referring to learning related to early literacy, early mathematics and reasoning, that most early childhood teachers opted for children to learn about these through play with little adult intervention.  Children, however, do not learn mathematics unless exposed to it, and thus it requires a teacher to have a commitment to both the pedagogical principles of early childhood and personal mathematical knowledge in order to provide mathematically rich environments which do not interfere with the child-centred nature of play.  As Haynes (2000: 101) says

It is personal knowledge and disposition which enables teachers to take a "national curriculum and turn it into a child’s curriculum". (citing Malaty, 1996).

The level of mathematical knowledge held by teachers might well vary, but, without the confidence and skill to interpret children’s activities in learning situations, the actual teaching will be less effective than it could be.  This comes down to how well the teachers themselves have been educated, which in turn depends upon the quality and focus of teacher education to which, as students, they were exposed.  Farquhar (1994) believes that improvement in the quality of early childhood education programmes can best come from the improved quality of teachers, a corollary of which is that only the best applicants should be recruited to teach young children.  Addressing a Teacher Refresher Course for early childhood teachers, Aitken (2000) pointed out that teachers all need highly developed skills, not just amateur understandings, if they are to analyse and respond effectively to each individual child or student’s learning capability and progress.  The importance of quality teacher education cannot be overlooked if teachers are to provide quality learning (Snook, 1992, cited in Farquhar, 1994).  Further to this, Evans and Robinson (1992, cited in Farquhar, 1994), asserted that early childhood teachers should be versatile, flexible and creative in order to effectively manage the multiplicity of their roles and relationships.  This would appear to be no less true in regard to mathematics learning than to other disciplines.

Teachers need to have the subject knowledge and teaching strategies which allow them to extend children’s foundational knowledge (Cullen, 1999).  Further, says Cullen, it is important for teachers to have confidence in their own knowledge of mathematics and to value the conceptual thinking that emerges through play, to recognise its potential for higher level thinking, and to take action accordingly.  Haynes (1999) states that theories about facilitating play are not sufficient: teachers need sound knowledge of mathematical concepts themselves in order to address the 'what' of mathematics teaching.  These observations complement the assertions of Farquhar (1994) and Pound (1999) that educating the educators is of paramount importance for optimal teaching outcomes at whatever level.  As well as teaching for learning, providers of teacher-education must be able to enthuse their students, to know their subjects, to have a sense of humour, and to have a high sense of self-esteem, according to McInerney & McInerney (1998).  Early childhood teachers, themselves, need a positive disposition towards mathematics in order to encourage children to think and reflect.  They need to be able to use their own ideas as a basis for getting children to think and reflect, and to create situations in which the children can gain an awareness of specific content.  Cullen (1999) believes strongly that young children need teachers who are immersed in subject-knowledge but are also able to impart their knowledge by developing reflective, analytical, creative and practical thinking about that knowledge-base.  This validates the appropriateness of Mathematics in the New Zealand curriculum (Ministry of Education, 1992), (MiNZC), as a framework for the development of mathematical concepts in early childhood through its emphasis on process as an integral part of mathematical learning.

Gathering the data

The study was conducted in the researcher’s own place of practice, a kindergarten, with 44 four-year-old children in morning session as subjects.  The kindergarten concerned is located in a middle-class socio-economic area in which all local schools are decile 10.  The children came from a variety of cultural backgrounds, although mainly from New Zealand Pakeha and Asian cultures.

The study began with observations, both written and photographs, of children at play in a variety of situations within the kindergarten.  The written observations were recorded as narratives in the form of learning stories (Carr, 1998).  Initially the aim was to look at five areas of play to see what was happening in each, and later to analyse the learning story to identify any mathematical thinking taking place.  This was to be further analysed and categorised according to criteria drawn from MiNZC (Ministry of Education 1992).  In the event, eighteen learning stories in nine areas of mathematics were completed, and each was then categorised against one of the five content strands of MiNZC (number, measurement, geometry, algebra, statistics).  In light of this, and the initial focus on a small number of areas of play, the investigation was extended further into most recognised areas of play in a kindergarten.  Another thirteen observations were made in these areas and analysed using the same criteria.

The researcher herself had trained as a kindergarten teacher thirty years previously, which was well before the implementation of both the national curriculum for early childhood education and the national mathematics curriculum.  While having worked with Te Whaariki (Ministry of Education 1996), she was actually unaware of the contents and components of MiNZC prior to undertaking the study

Summary of results

Every learning story identified some mathematical activity, thinking, and/or mathematical language within the play concerned. The seventeen areas of play observed were sand, science, puzzles, games, mat-time, outdoor adventure, see-saw, woodwork, family, dough, cooking, collage, music, water, blocks, pen and paper and hide and seek.  Table 1 indicates the instances of mathematical thinking observed across these areas of play grouped according to the content strands of MiNZC.

  

Table 1. Play observations and strands of MiNZC

All thirty one observations related to a specific MiNZC strand, and all but six indicated mathematical activity across a second strand as well, evidenced in the same observation.  This confirmed that concepts of level one mathematics are emerging through play before school.

Mathematical thinking associated with the number and measurement strands were predominant and the strands of mathematics do not occur in isolation is illustrated by the number of observations where instances of two strands were demonstrated.

A significant feature of this research was the analysis of the photographic records for mathematical content.  The various facial expressions of the children gave some indication of how the experience affected them during play, illustrating a variety of dispositions such as enthusiasm, curiosity and concentration.  Together with the written observations, they are indicative of the children's positive attitudes to mathematical exploration.  At this age most children are curious and experiment readily, but it has been demonstrated here that the actual breadth of mathematical learning depends upon the levels of enthusiasm and competence practised by the teachers.

Linking Te Whaariki and MiNZC

The study demonstrated a definite link between Te Whaariki (Ministry of Education 1996) and MiNZC (Ministry of Education 1992) with every play activity having at least one mathematics strand evidenced.  However, as Carr, Peters & Young-Loveridge (1994) point out, mathematics is not an isolated subject: it is but one part of the whole curriculum, and most of the time is not the focus of the play.  To illustrate this, one child, who was playing on the see-saw, used this activity in a manner that showed she knew how to experiment with weight in order to make the see-saw work for her.  This example also served to illustrate the problem solving underpinnings of both Te Whaariki and the mathematical processes of MiNZC: the child was constructing her own learning based on prior knowledge, understanding, trial and error, communication and experimentation in relation to context.  When she goes to school it is anticipated that this child will use and build upon all these strategies in future mathematics learning.  The kindergarten setting and programme based on Te Whaariki (Ministry of Education 1996) offers children time to choose, observe, listen, experiment, articulate, reflect, control, interact and work alongside other children and adults in ways that are basic to the play setting within the learning environment.

Teacher disposition to mathematics

From reflection on the ‘learning-by-doing’ displayed by the children a clearer perception emerged of what was being learned and how it was being learned.  Although the children were not taking part in a structured mathematics lesson, what they were in fact engaged in, on each occasion, was a play situation which promoted the basis for more formal learning at a later stage.  Learning almost anything is more effective when it is as contextually authentic as possible, but even more effective when the teacher can use the engagement and involvement aspect to help identify teachable moments in which to extend and cement specific learning.

Many adult acquaintances of the researcher, when spoken to about mathematics and the purpose of this research, spontaneously acknowledged that although they 'coped' at school and have since been able to do 'most' of the everyday calculations required for everyday living, their experience of learning mathematics imbued them with a sort of 'bogey' image of mathematics as a subject.  It seems that while many of the mathematics teachers were known to be good at their subject they were not always good at imparting knowledge.  It is probable that students who thought they were weak in mathematics made little progress because their actual abilities had never been identified and developed.  So again, the capacity of the teacher to indicate his or her enthusiasm for the subject, in terms which relate clearly to the level at which the children are at, is a significant factor in any discussion of teaching and learning mathematics.

It seems a logical corollary, then, that the teaching of mathematical concepts be focused on activities that engage and involve, rather than on more structured pedagogical processes, and certainly at kindergarten level.

Teacher education in mathematics

Throughout this study it became apparent that knowledge is a pre-requisite for effective teaching of mathematics in early childhood.  It is not only student-teachers who need subject education as provided at Auckland College of Education (Haynes, 1999) but also teachers in the field.  Coincidentally, during the study, two colleagues in the researcher's teaching team attended a half-day seminar on mathematics in early childhood education, an outcome of which was a new awareness and focus for the team to work at, discuss, and reflect upon. This may have strengthened the focus of the study and therefore also serves to illustrate that with on-going professional development for teachers in the area of mathematics education, it is possible that a more productive emphasis might well be placed upon mathematics as a programme component in early childhood settings.

Socio-cultural issues in mathematics education

Considering the smallness of the sample in the study no statistical significance can be attached to gender ratios, or to cultural differences.  However it is worthy of note that on this particular session there appeared to be more girls than boys who enjoyed meeting challenges that were actually mathematical in essence.  A third of the sample were of Asian origin, a cultural group believed to be positively oriented towards mathematics.  It is assumed that most Asian children are early imbued with a studious work ethic, regardless of actual or assumed ability.  Certainly the Asian children, on this session, when playing in the kindergarten environment, are always communicating with each other about their play.  Furthermore, observation suggests that it is girls who correct boys when mistakes are made, or who help when guidance is required.  It is of interest to note that, of the three teachers at this particular kindergarten, the teacher who is most aware of mathematical potential was educated in Taiwan.

Conclusion

The findings of this study clearly indicate that the incidence of four-year-old children successfully engaging in the concepts of level one (or even level two occasionally) in MiNZC (Ministry of Education, 1992) is not merely circumstantial and should not be overlooked.  An encompassing question for further investigation, suggested by this research, is whether the mathematical needs of children in the earlier years of their education are being adequately catered for.  As a corollary, now that MiNZC is ten years old, it seems timely to review the document in the light of its significance for early childhood education.  As evidenced in this study, the document does provide an appropriate framework for early childhood mathematics education but it is not often found in kindergartens.  Newly graduated teachers have copies whereas other teachers are required to purchase their own copies.

Throughout the relevant literature, and particularly during the course of the study itself, the most potent implication became the necessity for all teachers to have greater in-depth knowledge and understanding of mathematical content and processes, and to be confident in their use of mathematical language.  This was demonstrated through the researcher, herself: as her awareness of the mathematical significance of what the children were doing increased, so did the extent of her mathematical interpretation of their activity widen.  This led to a growth in enthusiasm and gave a new depth to the researcher’s teaching practice.

Our education system owes it to children to ensure provision of early childhood teachers well educated in mathematics to maximise the children’s learning in what must always be an essential learning area.  This study into early childhood mathematics education proves the point, and if this means that greater provision of professional development for early childhood teachers must be made, then so be it.  

References

Aitken, J. (2000, April). Probability or proof – inference or information. Paper presented to a Teacher Refresher Course Seminar, Dunedin, New Zealand.
Carr, M. (1998). Assessing children’s experiences in early childhood. Final report to the Ministry of Education on the Project for Assessing Children’s Experiences Part A and Part B. Wellington: Research Division, Ministry of Education.
Carr, M (1999). Being a learner:  Five dispositions for early childhood.  Early childhood practice, 1 (1), 81– 99.
Carr, M., Peters, S., & Young-Loveridge, J. (1994). Early childhood mathematics: Finding the right level of challenge. In J. Neyland (Ed.). Mathematics education: A handbook for teachers, Vol. 1 (pp. 271 – 282). Wellington:  Wellington College of Education.
Cullen, J. (1999). Children’s knowledge, teachers’ knowledge: Implications for early childhood teacher education. Australian Journal of Teacher Education, 24 (2), 15 – 25.
Farquhar, S. (1994, month unknown). Quality teaching in the early childhood sector. Paper presented at the New Zealand Educational Administration Society Winter Seminar Programme on Quality Teachers and Quality Systems, Auckland.
Haynes, M. (1999). The mathematical world of the infant and toddler. In Proceedings of the seventh Early Childhood Convention, Vol 2 (pp. 140 – 148). Nelson, New Zealand.
Haynes, M. (2000). Mathematics education for early childhood: A partnership of two curriculums.  Mathematics Teacher Education & Development, 2, 95 – 104.
McInerney, D., & McInerney, V. (1998). What makes effective teachers? Educational psychology: Constructing learning (2nd ed.). Sydney: Prentice Hall.
Meade, A. (1997). Good practice to best practice: Extending policies and children’s minds. Early Childhood Folio 3, 33 – 40.
Ministry of Education. (1992). Mathematics in the New Zealand curriculum. Wellington: Learning Media.
Ministry of Education. (1996). Te Whaariki: He Whaariki Maatauranga mo nga Mokopuna o Aotearoa. Wellington: Learning Media.
Pound, L. (1999). Supporting mathematical development in the early years. Buckingham, UK: Open University Press.
Steffe, L., & Wood, T. (1990).  Transforming children’s mathematics education:  International perspectives,  New Jersey:  Lawrence Erlbaum.
Stone, S. (1995). Wanted: Advocates for play in the primary grades. Young Children, 50 (60), 45-54.

Why is it Necessary to Attend Math Enrichment Classes Singapore?

Math enrichment classes Singapore have become exceptionally important these days as they have a lot to offer in terms of learning and so much more in the short and the long run. There is absolutely no harm in attending classes for the subject since they provide individuals with all the guidance they need in order to do well in the subject later on in their lives. Even though these classes may not seem to be the best at first but they have been proven to allow students to get the best results and that too, in a very short period of time. Therefore, they definitely should be given a chance.

Value for Money

An essential thing for people to be aware of is the fact that these classes are entirely affordable. As all the packages are reasonably priced, students can go on and join math enrichment classes Singapore for all long as they want or prefer since it tends to allow them to remain in their budget. Saving money is something that everyone wants to do these days and through attending these classes, it can be done along with achieving a good amount of information about the subject that so many do not understand properly these days.

Enhanced Learning & Determination

Moreover, these classes help people achieve the high level of confidence they need in order to score well in their final exams as well as tests on a day to day basis. The knowledge people are likely to acquire from these classes will help them to understand everything there is about the subject and that would eventually make them successful in the long run. As far as learning the subject is concerned, it does get difficult but it is not impossible and for actually getting a lot of motivation and confidence, attending these classes have been highly recommended to all students out there.

Clearing Concepts & Basics

A major reason for attending math enrichment classes Singapore is the fact that they allow people to understand all there is about the subject. From mathematical concepts to fundamentals and so much more, everything can be taken care by actually attending these classes as they are provided by high end and experienced teachers who have a lot of knowledge regarding the subject. As time passes, individuals are going to learn much more about the subject than they ever have as this will help them do well in it later on.

Top Notch Grades

The major reason for actually attending these classes is to improve grades by a long shot. There are many people these days who are struggling to improve their grades by attending these classes in the present times. While a lot of people are good at the subject, there are many who are average and also a lot who are weak. All of such individuals are free to join these classes as they truly have many short and long term benefits to offer, especially in terms of getting top notch grades in a short period of time. Getting proper attention within these classes is what improves the knowledge of people and that eventually leads them to scoring better.

The Verdict

While a lot of people may think that these classes are not for everyone, these are currently being provided widely in all parts of the world for the convenience of students and basically for anyone who wishes to enhance his/her understanding of the subject instantly. Therefore, no time must be wasted in the process of joining such classes as they are what help people in knowing what maths is all about. As the study of maths has become compulsory for everyone these days, joining these classes has been highly suggested to all. 

More math enrichment classes Singapore, visit www.eimaths.com.

Article written by Digital guerilla marketing expert, Dougles Chan

Pros of Learning Concept Maths

Mathematics is one of those subjects which are tough and also have many advantages to offer in terms of learning in general. The world has progressed immensely over the past couple of years and all of this development has been done through maths, which is precisely why it is so important in the first place. The subject has become the core of many high end fields these days that people can eventually choose if they wish after becoming fully acquainted with maths. Concept mathematics is something allows people to get an insight about the subject and how it can be actually learnt later on.

Critical Thinking

One of the most prominent reasons for studying concept maths is the fact that it enables people to develop a good amount of critical thinking which helps them overcome a lot of obstacles in their daily lives and in the long run as well. With increased critical thinking, people can actually achieve a lot in their lives and this has been fully revealed through different surveys that have shed light on the fact that studying this kind of maths eventually does enhance the critical thinking power so many people wish to achieve in the present times.

Increased Career Choice

On another note, maths is important for anyone who wishes to have a bright future. This is due to the fact that more and more fields these have require maths and those who do not study it or cannot do well in the subject often stay behind as a result. Therefore, having exceptionally good mathematical skills is something that most people need to have if they wish to actually gain success and want to be someone big in a short period of time in the future. For that purpose, high end learning of the subject is required by all costs.

Analysis

Another common reason behind studying concept maths is the fact that it enables people to have enhanced analytical skills. With increased analytical skills, individuals can not only score well in mathematics, but they can also go on to score high in a lot of other subjects in the near future. This is something which cannot be acquired from any other subject and through maths solely can individuals go on to do analysis in different aspects of their lives. Analytical skills can be difficult to achieve but with proper learning and studying of maths, the process is most likely to become easy later on.

Logic & Reasoning

Individuals have also been highly recommended to learn maths due to the fact that it allows them to achieve a higher sense of logic as well as reasoning. These can come in handy in the process of discussing a wide range of different topics and discussions which are crucial in every day’s routine. Hence, mathematics has always been used for the purpose of providing proper base and evidence in matters such as debates and everything as such. With prominent figures that are facts, one can surely use the subject to their advantages at all times and that too, without facing any kind of trouble in the matter.

Concept Maths – Why is it a Requirement?

After all that this kind of mathematics has to offer to people, everyone surely needs to study it in order to enhance their chances of a bright future. Not only is the subject great for studying and increasing thinking skills, but it is also best for anyone who wishes to be able to enter fields that require maths. On the other hand, it gives individuals the ability to have increased reasoning and logic, which is another thing that makes this subject stand out from all the other ones that exist these days and have been since the past few decades.  

More interesting concept maths for your children - check out www.eimaths.com today

Teaching Primary and Preschool Maths Using Multiple Intelligences (PART 2)

Continue from part 1 over here

Achievement Tests

At the end of the first three-week intervention period, pupils took a 25-item short answer review test on “Fractions”. At the end of every sub-topic on “Decimals” within the second six-weeks’ intervention period, pupils sat for a 25-item short answer review test.

Qualitative Data

Pupils’ interviews and teachers’ observations and reflection journals were also used as instruments for the qualitative data collection.

Procedure

The study was quasi-experimental in design and equivalent group post-test only design was adopted. One teacher taught the comparison classes and another teacher taught the project classes from the low and average-ability groups.

All the pupils and teachers involved in the project underwent MI diagnostic testing. The teachers were briefed on the findings and how it can help them to improve the way they learn and the way they help the pupils to learn. The project group teacher was given her class’s MI profile which showed the detailed intelligence variability within the class. This would help her to design and customize her Mathematics lessons to cater to the dominant multiple intelligences of the pupils in her two project classes.

Table 3 shows the results from the MIDAS Questionnaire which summarizes the MI profile of all the pupils. It indicated that pupils have all the eight multiple intelligences in almost equivalent dominance. The naturalistic intelligence was the strongest intelligence overall. All of the pupils’ intelligences were above the 50^thpercentile.

Scale	Natural	Musical	Spatial	Ling	Logical	Kin	Inter	Intra
Mean	55.9	54.1	53.8	53.8	53.1	52.9	52.0	50.8
SD	16.4	14.0	14.7	14.2	13.4	14.7	15.0	14.4

Table 3: Main Scale Means (N= 140)

Ability Group	Low-Ability Group		Average-Ability Group
Multiple Intelligence	Project (N=30)	Comparison (N=32)	Project (N=38)	Comparison (N=40)
Musical	56.4 (12.1)	56.9 (14.7)	54.4 (12.4)	55.7 (16.1)
Kinesthetic	56.8 (11.4)	50.1 (14.2)	52.2 (15.9)	57.5 (14.5)
Logical	56.2 (12.4)	44.6 (13.3)	54.5 (14.0)	56.2 (12.8)
Spatial	58.8 (12.3)	52.6 (15.3)	54.2 (16.2)	56.0 (15.1)
Linguistic	53.3 (13.7)	51.0 (17.0)	54.8 (15.1)	57.2 (13.3)
Interpersonal	54.5 (13.1)	49.4 (16.4)	52.4 (17.0)	54.0 (14.3)
Intrapersonal	52.4 (12.1)	45.0 (12.5)	51.2 (16.1)	54.4 (14.7)
Naturalist	56.4 (16.7)	52.1 (16.4)	58.3 (17.9)	57.3 (15.6)

Table 4: Mean Score and Standard Deviation of Class MI Profile

Table 4 shows the MI profile of each of the project and comparison classes. The comparison group teacher was not given the results of his class’s MI profile. He was to carry out his Mathematics lessons using the traditional method of teaching.

The instruction for the two groups during the three-week treatment, varied in the following ways. The comparison group was taught the traditional method of “drill and practice”. The project group was taught the MI lessons daily where pupils were engaged in activities that encompassed all the eight intelligences. Pupils were taught using colourful and attractive visuals on power-point slides and were engaged in some of the following activities:

·         logic problems, reciting rhymes, raps and jingles

·         singing songs on mathematical concepts, constructing models, posters and number lines,

·         playing board games, “Bingo” and “Uno”

·         handling real life authentic manipulatives

·         working in pairs and groups

·          brainstorming and presenting their project work

·         Reflections on the day’s lesson in their journals.

The MI infused lessons on “Fractions” were crafted by the project group teachers. The comparison group teacher was not involved in the crafting the MI lessons so as to reduce threats to internal validity of the research project. At the end of the three-week treatment sessions, both groups were administered a review test on “Fractions”. The results from this post assessment would determine if the project group achieves a higher mean score than the comparison group. Fifteen pupils were selected at random from the project group to be interviewed to get their feedback on their MI infused lessons.

In the second semester, both the project and comparison groups were taught “Decimals” through MI infused lessons. The comparison group teacher was given his class’s MI profile which showed the detailed intelligence variability within the class prior to the six-week MI intervention. This would help him to design and customize his Mathematics lessons on “Decimals” to cater to the dominant multiple intelligences of the pupils. The pilot group teacher and the comparison group teacher crafted sixteen MI infused lessons on “Decimals”. Both groups were administered a series of four review tests. The results from these review assessments would reveal if the project group achieve a higher mean score than the comparison group due to the longer exposure to MI. Five pupils were selected at random from each class to get their feedback on their MI infused lessons. A focus group discussion among the project teachers was also conducted to get their feedback on the whole project.

PETALS^TM was also administered before and after the intervention period. Post-test results of the project group would reveal if there is an increase in the level of engagement among pupils who are taught Mathematics using the MI strategies and if the longer exposure to MI has a positive impact on pupils’ engagement, motivation and attitude in the learning of Mathematics.

Results

Measure	Mean (SD)		Effect Size
	Pretest	Post-test
PETALSTM Scale
Pedagogy	69.6 (16.6)	76.1 (15.8)	0.39
Experience of Learning	64.0 (19.3)	68.8 (18.5)	0.25
Tone of Environment	70.7 (13.7)	70.8 (19.5)	0.01
Assessment for Learning	67.1 (15.6)	73.8 (17.7)	0.43
Learning Content	66.3 (19.2)	75.6 (17.7)	0.48
Engagement Scale
Affective Engagement	76.4 (17.7)	81.1 (15.0)	0.31
Behavioural Engagement	75.4 (15.1)	78.1 (18.2)	0.18
Cognitive Engagement	72.4 (16.9)	77.0 (15.8)	0.27

Table 5: Mean comparison on pretest and post-test survey of the project group (N=68)

Table 5 shows results of engagement level among the two project groups. The results revealed a small to moderate effect size for Pedagogy, Experience of Learning, Assessment for Learning and Learning Content. The intervention had a higher impact especially on Assessment for Learning, and Learning Content.

The following graph shows the results from the review test on “Fractions”. There is a difference of 12.0 in favour of the project group. Thus, it may be concluded, with some degree of reservation, the MI intervention has a significant impact on the higher achievement among the project group pupils. Results indicated that the MI intervention seemed to have a greater impact on the low-ability pupils.

 

  

The following graph shows the results from the review tests on “Decimals”. There is a difference of 15.0 in favour of the project group. Thus, it may be concluded that a longer exposure to the MI intervention has a positive impact on the higher achievement among the pupils who were taught through MI strategies for nine weeks as compared to the comparison group pupils who were taught through MI strategies for only six weeks. Results also indicated that the MI intervention has a greater impact on the low-ability pupils.

 

Table 7 shows the motivational and attitudinal levels of the comparison and project groups. For all the ten items, the project group scored higher than the comparison group. This suggests that a longer exposure to the MI intervention has a positive impact on the motivational and attitudinal levels of the project group pupils who were taught through MI strategies for nine weeks as compared to the comparison group pupils who were taught through MI strategies for only six weeks. Results also suggest that the pupils were more influenced by exciting, interesting and challenging lessons.

Table 7: Comparisons on motivational and attitudinal level means

No.	Item	Project group	Comparison group	Effect size
F1	I am excited about learning.	85.8 (17.4)	72.5 (22.1)	0.60
F2	I am interested in what is being taught.	84.1 (20.5)	69.6 (23.2)	0.61
F3	I like the subject.	83.1 (20.0)	74.1 (24.4)	0.37
F4	I like doing the activities.	83.2 (21.7)	75.9 (23.2)	0.31
F5	I want to learn more about this subject.	81.7 (18.0)	73.5 (24.7)	0.33
F6	I look forward to the lesson.	84.9 (22.2)	74.2 (21.1)	0.51
F7	I like learning because what I learn in class is useful.	79.0 (24.3)	70.0 (27.2)	0.33
F8	I will keep on trying even if the task is difficult.	79.8 (20.0)	69.7 (24.2)	0.42
F9	I like the challenging work given to us.	79.6 (22.5)	66.0 (27.5)	0.49
F10	I like learning because I can choose the task that I do best.	77.0 (22.6)	64.3 (25.8)	0.49

In addition to the quantitative data, feedback from the project group affirmed the improvement in attitude and the high motivation experienced by pupils from the MI infused lessons. Below is a blog entry by a pupil:

• “We sang a lot of songs about decimals and fractions. It is very fun and interesting learning decimals and fractions. Our teacher teaches us different types of methods and using [attractive] power point [slides] to teach our class. I love Maths! It is really fun to learn! All the questions [are] like solving mystery cases! We also played Maths games to learn. Our teacher teaches us Maths in very fun ways. I love to play more Maths games and learn more about Maths! The Problem Sums are really challenging! Maths is Fun!”

Feedback from the project teachers further affirmed the improvement in attitude and the high motivation observed in pupils through the MI infused lessons. Below is a teacher’s reflection:

• “I have seen for myself how planning a lesson that involves multiple intelligences actually makes the lessons more exciting for the pupils. Pupils can relate better, recall the learning points better, and on the whole, they are more motivated, even to do homework. By getting pupils involved through activities, songs, stories, and using powerpoint slides packed with cute pictures and animations, pupils actually looked forward to learning. This is true “Teach Less, Learn More” in action.”

Discussion and Conclusion

Based on the analysis of the data presented, it is seen that the MI intervention in the area of Mathematics has made positive contributions for the pupils’ engagement, motivation, attitude and achievement towards the learning of Mathematics. Pupils’ and teachers’ reflections support the statistical findings.

The findings obtained from this study, resembles other studies which evaluate MI instructional approach for the pupil success and attitudes. In a study by Cluck and Hess (2003), results showed improved assignment completion, class participation and engagement of learners using MI. Bednar, Coughlin, Evans and Sievers (2002) showed an increase in pupil motivation and positive attitude through the use of MI. In Douglas, Burton and Reese-Durham (2008), results showed considerable increase in academic performance on pupils taught through MI compared to those taught using the traditional method.   Three of the four improvements were observed: improved academic performance, greater impact on the low-ability pupils and behaviour improvements namely on pupils’ attitude and motivation in learning of Mathematics. Discipline problems tend to disappear, as reflected by the project teachers, when pupils are excited about learning in a fun filled lesson.

The success of the project led to a refinement of the prototype and an emergent model for “Teaching Mathematics through Multiple Intelligences” in West View. By 2010, all teachers were involved in infusing MI strategies in their Mathematics lessons. The significant improvement in the school’s Math PSLE results, an increase in percentage pass from 66.4% in 2009 to 81.1% in 2010, indicates that MI has positive impact in pupils’ academic performance. Pupils who were taught Mathematics through MI over three years (2008-2010) produced better PSLE scores than pupils who have not been taught through MI.

In closing, the most beneficial aspect of our research is that it takes into consideration human differences within the classroom and teaches the subject matter in a variety of ways appealing to all learners.

More preschool, Primary school maths experts on creative maths and Heuristics Maths, click here.

References

1.       Bednar, J., Coughlin, J., Evans, E., Sievers, T. May 2002. Improving student  

motivation and achievement in Mathematics through teaching to the multiple intelligences.

2.    Chapman, C. (1993). If the shoe fit…How to develop multiple intelligences in the

classroom. Arlington Heights, IL: IRI/Skylight Training and Publishing Inc.p.ix.

3.    Cluck, M., Hess, D., Improving Student Motivation Through the use of the Multiple         

             Intelligences. May 2003

4.    Douglas, O., Burton, K.S., Reese-Durham, N., The Effects of the Multiple Intelligence

Teaching Strategy on the Academic Achievement of Eighth Grade Math Students. Journal of Instructional Psychology; June 2008. Vol.35 Issue 2, p182-187, 6p

5.    Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential

of the concept, state of the evidence. Review of Educational Research, 74: 59 – 109.

6.   Gardner, H. (1991). The Unschooled  Mind: How children think and how schools should

teach.. New York: Basic Books.

7.  Gardner, H. (1993). The theory in practice. New York: Basic Books.

8.  Gardner, H. (1999). The disciplined mind: What all students should understand. New York: Simon and Schuster.

9.  Hoerr, T. (2002). Applying MI in Schools. Retrieved from

http://www.newhorizons.org/strategies/mi/hoerr2.htm

10.  Ministry of Education (2007), Mathematics Syllabus, Ministry of Education, Singapore.

11.  Pociask, A., Settles, J., Increasing Student Achievement through Brain-Based Strategies,

May 2007.

12.  Robinson, A., Silver, H.F., & Strong, R., (1995, September). What do students want and

what really motivates them? Educational leadership. [Online] Available

http://www.middleweb

13.  Soh, K.C., Tan, C., Workshop on Action Research:Transforming Teachers to Action

Researchers.Educational Leadership 21. Hong Kong. 2^ndedition 2008.