Bridging the Mathematics Learning Gap

The teaching of Mathematics has shifted from white board learning to activity-based learning. This form of learning is inquiry-based, whereby students discover facts, concepts and relationships by themselves.
Other than problem solving skills, the following skills are also important thinking skills that the students must adopt. Some examples of such skills are:

    Observation
    Visualization
    Hand-eye co-ordination
    Analytical thinking
    Logical Reasoning
    Judgment
    Categorization
    Memory

At eimaths  students no longer learn Mathematics by memorizing, rote-learning or drilling. Instead, they must be able to connect, reason and model what they have learned in class and apply to solve problems.
Centres  at Admiralty, Aljunied, Bishan, Buangkok, Khatib and Toa Payoh.
Visit www.eimaths.com or call 6841 8148 for more information.

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-

More info about maths enrichment, click here.

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.

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Another very prominent benefit of attending this class daily is the fact that it hardly costs much. While a lot of people may think that it is expensive and unaffordable, it really is not since everybody needs it these days. That is why affordable prices are charged by people who offer appropriate sort of classes for maths to people who need it for passing entry tests, final examinations and everything as such. This way, individuals can now also get to learn maths, that too without having to spend too much in the matter. Hence, the opportunity must not be missed out.

Knowing Logic

Attending such a class is mandatory for anyone who wishes to understand logic well. Maths teaches us all there is about logic and through it, people can use logic for the purpose of knowing answers to the questions they thought they were not capable of answering. The power of reasoning and logic is something that all individuals need to understand well in the present times, which is precisely why getting the right kind of coaching for learning maths has been highly recommended to all people who have an increasing interest in maths and everything related to it in general.

The Conclusion

Since going for this class for mathematics is becoming more and more common these days and in different parts of the world on top of that, all the interested people should go for it since it really does save them a lot of time and allows them to learn a lot in the overall process. Without maths, it is hard for a lot of people to have an exceptionally bright future since it is quite required in almost every other field there is these days. Therefore, learning it in the best way possible is suggested to all those who dream of high end success.

 For more Heuristics Maths math enrichment classes in Singapore, check here.

The Purpose of Studying Concept Maths

More and more students can be seen to be going for maths as the subject allows them to explore a wide range of different careers and it also enables them to solve many things in the daily routine. This is due to the fact that our everyday life revolves around maths and almost nothing is without it. Therefore, studying it is always highly recommended to all individuals. Concept maths is the kind which really sharpens a person’s mind and also gives people the ability to understand maths in general in a much better way. A lot of schools all over the globe have included this subject into their curriculum since there is no going ahead without it. 

Enhanced Analysis 

An essential reason for studying concept maths is the fact that people can enhance their analytical skills through it. With this being possible, individuals can easily score well on critical tests that demand a lot of analysis. Since analyzing is known to be an exceptionally difficult task, the best way to polish or to acquire analytical skills is to give more importance to maths in school, college and university level. Those who succeed in understanding this are most likely to have better analytical skills than others who do not. 

Enhanced Brain Activity 

It is very important to study concept maths due to the fact that it enables people to acquire sharp thinking skills, which they cannot acquire so easily otherwise. The subject tests the thinking limits of an individual and where most people do not attempt to think outside of the box, they are most likely to do so while studying this type of maths. In short, individuals are going to be benefitting a lot in the process of increasing thinking skills on a large scale. 

Critical Thinking & Reasoning 

People study this type of maths also for the purpose of enhancing their critical thinking skills as well. Since that requires a lot of reasoning in general, this subject is known to especially help it in one way or the other. Education experts have revealed that only through maths can people increase with reasoning skills and critical thinking. While there are quite a lot of others ways to achieve this, learning mathematics is considered to be the most efficient and most commonly known ones these days. Ever since the subject was invented, it has been unique and it still is to this day. 

Understanding Concepts 

On the other hand, the subject is also taught to people in order to get their facts and concepts straight. Mathematics is all about knowing the right concepts and applying them well. Those who figure this out soon can do exceptionally good in the subject in the long run. However, people who are looking forward to know all there is about mathematical concepts are highly suggested to go for this type of maths as its main purpose is to provide the proper kind of guidance in the matter of knowing the concepts of maths. Due to this reason, more and more people can be seen to be opting for it. 

The Verdict 

All in all, learning concept mathematics has a lot of short and long-term advantages that people surely do not want to miss out. It has been included in the curriculum of many schools, colleges and even universities in different parts of the world. Due to these exclusive reasons, all the interested students are highly recommended to study maths in order to benefit from all that it has to offer. Studies also reveal that people who opt for maths are also the ones who end up having the most top notch careers as their options are countless. With maths, individuals can enter purely maths-related as well as science and IT fields if they wish. 

 More Concept Maths for Singapore Primary School, click here.

Top Pros of Maths Enrichment

As far as studying mathematics is concerned, it is most definitely recommended since it has much to teach to people from all over the world these days. The subject has been immensely special ever since it was created and since it has many short as well as long-term benefits to offer, people are highly recommended to study it in the present times. It can be rather tough and pretty boring sometimes but with increased knowledge of the subject, individuals can have a shot at bright and successful futures within a short period of time. Maths enrichment is necessary as through it, individuals get to understand everything the subject is about in the first place.

Management of Trade

With increased knowledge of the subject, people also get increased knowledge of trading, which is something that is exceptionally important these days as it allows individuals to make a lot of money. Everything in trading is based on numbers as well as predictions. Traders are known to have high end knowledge of the subject, which is precisely why they do so well in the first place. All those who want to understand trading in a much better way should definitely go on to learn maths and everything that is included in the subject.

Management of Investments

Moreover, with maths enrichment,people can easily go on to indulge in investment management. Everyone these days can be seen to be making a lot of profits through investing in different commodities such as stocks, gold etc. Had maths not existed, this would never have been possible in the first place. Individuals with sound knowledge of the subject are always the ones who do well in investment as they understand the fundamentals of the subject and use them to their advantage for the purpose of earning profit. The line between profits as well as losses also cannot be drawn without proper knowledge of maths, which is surely what makes the knowledge of this subject exceptionally important.

Management of Money

One of the most important things that are taught to people through maths is the management of money. The knowledge of currency, all the ones which exist in today’s world, is solely based on maths as they are all numbers. Everything related to numbers is involved with mathematics; therefore, people really need to have a good know-how of mathematics in order to be able to manage money more efficiently on a day to day basis. Hence, acquiring maths enrichment is something that all people should pay attention to if they want to learn the management of money in a much better way.

Management of Time

On the other hand, mathematics also allows people to be able to learn time management. This is something that individuals have to learn in order to be successful in many things these days. Those who learn to manage time and keep a good track of it have always been labeled as smart and without the knowledge of maths; time cannot be counted or even handled on a day to day basis. Therefore, the knowledge of maths is necessary for all those who wish to know all about managing time without having to face any obstacles in the matter.

The Verdict

Acquiring more mathematical knowledge is important for all those who wish to move ahead in today’s world, which is completely based around maths in one way or the other. While learning the subject in every way may take a lot of time in the future, it is completely worth it due to the fact that it enables people to become successful afterwards by choosing fields that heavily involve the subject. Even sciences as well as IT fields involve it; therefore, it is something that all people need to study these days.

More Singapore creative maths tuition classes for preschool and kindergarten, click here.

Maths Enrichment – The Necessity

Individuals are aware of how important maths is as a subject as it is what helps them solve a wide range of many issues in the short and the long run. Since many decades now,  it has held a massive amount of significance due to the fact that everything in the world has something to do with maths, which is precisely why individuals have been highly recommended to learn it as much as possible. A lot of people don’t know the significance of maths enrichment, but with proper knowledge and learning of the subject, they can definitely see how acquiring it at the earliest convenience will benefit them a lot. The subject tends to teach people a lot of things in general and that surely makes it one of the most unique subjects. 

Dealing with Investments & Profits

A lot of people should know that maths enrichment is essential due to the fact that it is also linked to making money. The fact that the knowledge of mathematics is highly required in the process of making investments in today’s world teaches people that they must study the subject in great detail in order to be able to make proper investment in the short as well as the long run. Many calculations as well as estimates have to be made before investing in something huge; therefore, maths is required in the process as without it, the outcome may seem rather unknown. 

Leaning to Calculate

One of the reasons behind studying the subject is the fact that it enables people to learn all there is about calculations. People have to calculate a lot of things on a daily basis. Whether it is regular transactions or some other little, everyone should know how to calculate certain things, most importantly money, in order to know what they are up to. Without the proper knowledge of maths, making calculations in any situation is not possible and that is precisely why the deep study of this study is important for all those who wish to learn about business as it includes profit and loss majorly.
 

High Financial Rewards

The subject is considered to be interesting by a lot of people in the present times as it is entirely related to money. Those who are interested in running successful businesses have been highly recommended to have a good knowledge of maths in the first place. For all kinds of transactions as well as many other things require a good know-how of maths. Hence, knowing it has become a necessity these days as survival without it is surely going to be a lot more difficult than people may think.

Knowing Time Management

Maths enrichment is mandatory because through it, people tend to learn about managing time. Nobody wants to lose track of time and end up wasting a lot of time or by missing important events in their lives. Therefore, maths was also created to help manage time and to know it in the first place. People with lack of knowledge of the subject are most definitely going to face a lot of trouble in the process of saving time within important exams as well as tests as in them, saving time in the main key for scoring higher than the others attempting the same thing.

The Bottom Line

Maths is a subject that a lot of people find dull and rather intricate but the most important fact is that it is one of the most beneficial subjects. Through it, many people have ended up acquiring a lot of money and high end profits without having to struggle too much. It also helps people in making day to day calculations and is also responsible for helping them in the process of saving time daily, which is quite important on a day to day basis.

Learn more about concept maths in Singapore, click here.

Maths Enrichment Classes Singapore – Why are they Important?

Everyone these days is aware of how important mathematics is as a subject for achieving enhanced learning for many aspects in life. The subject is considered to be one of the most difficult ones due to the fact that it is rather lengthy and can be pretty dull sometimes. However, that does not in any way affect its significance, which is that it tends to allow people to attain a bright future in the present times. While people may be rather good at the subject, attending maths enrichment classes Singapore is something which tends to really provide them with the necessary knowledge and edge they need for the purpose of becoming an expert at the subject in record time. 

Improved Grades

These classes are necessary for all those who have a strong desire for scoring high in the short as well as the long run. While studying maths in the school or just alone can be quite helpful, sometimes it is simply not enough in order to score really good grades in the future. Therefore, attending these classes for the subject is something that completely ensures and increases the chances of getting good grades by a long shot and that is exactly what most students want in the present times.

Improved Knowing of the Subject

Similarly, these classes are also best for anyone who is willing to go ahead for the purpose of clearing away all mathematical concepts without having to face any hindrance in the matter. The qualified teachers teaching maths focus on every little detail regarding the subject since it is their job to do so. It is exceptionally important for people to know all sorts to mathematical concepts and most importantly, to understand them, for the purpose of increasing their knowledge of maths in the long run. According to many surveys, more and more people have ended up getting better grades as well as increased motivation after attending such classes for maths at all levels.

Very Affordable

As far as these classes are concerned, people can attend them after paying reasonable prices every now and then. This is due to the fact that the demand for them is high, which has ultimately decreased the prices everywhere. Since maths enrichment classes Singapore do not cost nearly as much as most people think, it surely gives them all the more reason to go for them as they do not only cost less but also enhance their overall mathematical knowledge in a short period of time. Now individuals can go on to become experts at the subject without having to pay a lot of money in the matter.

High End Demand

Finding a place where maths enrichment classes Singapore are being given is really not as difficult as most people expect it to be. This is due to the fact that more and more people can be seen to be providing classes for this subject as the demand has drastically increased over the recent years. Since the demand has increased by a long shot, it means that more people have a desire to learn everything there is about the subject, which is what makes it a highly important subject to learn in the first place.

Maths Enrichment Classes Singapore – Are they worth it?

With all that these classes for mathematics have to offer, it is something that all people must go for if they wish to increase their knowledge in the subject, without having to spend too much money in the process. Attending these classes does not only enhance grades drastically, but it also enables people to move towards a brighter future. The best part is that that they also increase determination and that allows people to work harder in order to understand the subject in every way possible for the purpose of doing well in it.

More creative preschool maths classes, click here.