Wednesday, 25 February 2015

Empowered Mathematical Mindset

Learning Mathematics by heart is a thing of the past. Nowadays, children are expected to work out calculations using creative problem-solving skills

It is without a doubt that primary school maths is no longer limited to simple arithmetic sums. By the end of Primary one, children are expected to tackle problem sums involving topics on money, measurements, fractions and graphs (to name a few) - on top of basic addition, subtraction, multiplication and division. The key is to develop an understanding of mathematical concepts, and to acquire analytical skills that will enable the children to apply the mathematical concepts to problem-solving scenarios.

With a growing emphasis on encouraging creative thinking and problem-solving abilities in our school curriculum, it is not surprising that children need to be exposed to more challenging approaches to learning mathematics – and parents feel an increasing pressure to keep up with the latest trends.

One of the latest teaching methods, adopted in many primary schools, is the use of learning centres in lower primary classroom, as part of the Strategies for Effective and Engaged Development (SEED) programme. In brief, the SEED programme intends to integrate learning between mathematics, science and the English language, through a thematic approach. Such integration across the curriculum should give rise to fun learning activities in primary schools, while at the same time developing language skills with mathematic and scientific knowledge.

Other than the efforts made to improve the curriculum in our primary schools, what can parents do to help advance the learning of our children? With jargons like “analytical skills” and “problem-solving techniques”, coupled with various teaching methods involving models and its variants, parents are often at a lost as to how best introduce mathematics to their pre-schooler and subsequently to sustain an active interest in mathematics.

Need For Creative Learning

In this current knowledge-based economy, where the emphasis is on creative and independent learning for children, the traditional method of spoon-feeding students with information and grading them based on the knowledge they have retained is irrelevant. What the new economy requires are students with nimble, innovative and analytical thinking skills, to challenge problems that are new and unknown. Parents need to nurture problem-solving skills in their children and not merely train them to perform like calculators.

With the Ministry of Education announcing the changes to the A-level “thinking” examination in 2000, students can expect more questions intended to test their understanding and application of concepts across topics, instead of merely applying well-memorised formulae.

Nowadays, most enrichment centres have designed their mathematic classes to keep pace with teaching trends in primary schools. Through a series of carefully crafted questions, students are gently steered into discovering for themselves a magical and lively world of numbers. It is certainly more meaningful to acquire knowledge and upgrade abilities by way of self-exploration, than by a purely teacher-orientated approach. Active learning is the trend for modern education. Children can build up self-confidence and gain knowledge at the same time, in the learning process.

Importance of understanding concepts

Most people know how to count from 1 to 10,000. Although they may not have done it before, they know that they are able to do so because they understand that 1+1=2, 2+1=3…., and so forth. This means that they understand the Concept and Method needed to count up to 10,000.

Parents should try not to intimidate their children with big numerals, as it is more important to understand the concepts and models necessary to solve a problem, instead of blindly applying the same well-used method. Counting and calculation skills are by no means the only measure of a child’s mathematics ability. Several more fundamental skills need to be developed to boost mathematical learning and understanding.

Multiplication Table – Beyond Mere Memory
Despite earnest efforts to memorize the multiplication table, it is not unusual for children (even adults) to forget portions of it. The accumulation of knowledge and abilities are best achieved by way of thorough understanding of the subject matter, instead of relying solely on memory. New teaching methods in mathematics aim to replace sole reliance on memory work with real comprehension, so that knowledge can be recollected whenever necessary.

To replace the traditional ways of memorisation and drills in Maths learning, the current Mathematics syllabus features lively maths sums to help captivate and equip children with the initiative to think, and therefore rely on their own strengths to solve problems.

Different strategies in solving Mathematics problems

Primary school children are taught a variety of ways or strategies to work out their problem sums. They can use any method they are comfortable with, as long as they end up with the correct answer. Children’s mind needs to be simulated to think in a multi-model manner. A likely scenario is where a child, out shopping with his mother, is asked, “If you have $17 and you buy an item worth $9.90, how much change are you left with?” Before the mum could elaborate, he answered, “Mummy, I can take $10 to minus $9.90; then add the balance 10 cents to the $7. Or I can take $1 to minus 90 cents and $16 to minus $9; and then you add the 2 answers together!” Such a scenario would imply that the child has learnt many ways to derive a solution from a problem, instead of relying on a fixed way or method. The child is also able to relate and apply his mathematics skills in his daily life. Every child should be encouraged to develop similar abilities, with the appropriate exposure to different mathematical reasoning.

Calculation skills verses Problem-Solving skills

During the 19th century, when Carl Friedrich Gauss (a mathematical genius) was ten years old, he was asked to add up every numeral from 1 to 100 (ie.1 + 2 + 3 +..100 = ? ). He instantly scribbled “5050” on his slate and laid it down with the proud declaration, “There it lies.” When the other students turned in their slates after painstakingly adding the numerals one at a time, no one except Gauss had the correct answer!
How did Gauss do it? He had observed that each pair of numerals - 1 and 100; 2 and 99; 3 and 98; and so on up to 50 and 51 - added up to “101”. From the resultant 50 pairs of numerals, he deduced the mathematical statement: 50 x 101 = 5050. What Gauss had exhibited was problem-solving skill, and not just calculation skill!

1 + 2 + 3 + …50 + 51 +…99 + 100 = ?
Therefore, problem-solving skills are much more important than calculation skills.

Young students are encouraged to acquire a real comprehension of Maths, not simply memorise formulae. From as young as K1, young minds should be coached to think creatively and independently for more complicated Maths concepts which they will encounter as they grow older.

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 or call 6841 8148 for more information.

Tuesday, 24 February 2015

Early Foundation in Mathematics for PreSchool and Primary

Ready to perk up the little brains? Many parents are concerned about how to inject fun and inculcate a love and interest for mathematics in their children during the early age of pre-school learning. Are rote learning and drilling the only way to learn mathematics? Is your child very good in mathematics if they are able to perform the 4 operations of “+, -, x, /” at an early age? The answer is “No”! What we need from our children are problem-solving skills. We should not train our child to perform like a calculator! 

In this current knowledge-based economy where the Ministry of Education has been emphasizing Creative and Independent learning for our children the traditional method of spoon-feeding students with information and grading them based on knowledge they have retained is irrelevant and biased. What the new economy require are students with thinking, innovative and analyzing skills to challenge problems that are new and unknown. As for learning mathematics, there will be more questions testing students’ understanding and application of concepts across topics instead of the mere ability to apply formulae. This is exactly what eiMaths focuses on!

What is eiMaths?
Originated and developed in Singapore by a team of experienced practitioners, eiMaths is not to be mistaken as just another Maths program in the local scene. Closely following the MOE syllabus, and through years of experience gaining a deep understanding of the difficulties faced by children when studying Mathematics in school, our programs build on a step by step approach, utilizing a personalized and spiral learning system, guaranteed to build, sustain and grow your child’s confidence & interest in Maths and challenge themselves to achieve greater heights.

How will your child benefit from eiMaths?
Discover Mathematical facts, concepts and relationships
Adoption of correct strategies to apply for different types of problems
Strategic exploration, self-feedback and reflection on errors
Independent thinking skills through innovative teaching methods
Motivation and Enthusiasm in exploring Mathematics
Inspire interest and increase confidence in solving mathematical problems

eiMaths’s  Methodologies
Our learning journey comprises of 3 phases of learning:

Phase One
Understanding Maths concepts and theories through colourful workbook and enhancing the knowledge by completing it at home. 

Topics covered:
Data analysis

Phase Two
Acquiring skill through self-discovery, fun filled activities and games. 
Skills covered:
Numerical calculation
Logic reasoning
Life application
Combination thinking

Phase Three
Applying the knowledge acquired to solving heuristic problems. 
Processes covered:

Objectives of eiMaths’s curriculum 

K1 to K2… 
create interest and fun in learning Mathematics. Set the stage for cognitive development in number sense, sorting, patterning and measuring. Through hands-on activities, actively engage children in the learning process.

Numbering system
Counting, patterning, sorting
Colours, shapes
Measuring, comparing, matching
Paths, directions
Placing, mirror image
Sequencing, observing

P1 to P3… 
enhance foundation in basic Mathematics theories, methods and concepts. Focus on developing creative thinking and a deep understanding of Maths reasoning (rather than on rote memory) to approach problem solving.

Numbering system, fractions
Number operations
Mental calculations
Length, mass, volume
Time, money
Area, perimeter, parallel, perpendicular
Geometry, patterns, 2D, 3D
Data analysis, graphs

P4 to P5… 
excite students’ thinking and arouse their curiosity in mastery and application of problem solving heuristics through question-posing techniques, children learn and improve their skill in solving challenging Maths problems.

Problem solving heuristics
Factors and multiples
Complex fractions problems
Angle, symmetry, tessellation
Percentage, ratio
Area, volume

Mathematics should not be routine drilling and memorizing tables, but fun and enriching. The knowledge gained by our students should enable them to solve problems in any situation. Students with the eiMaths program will ultimately develop problem-solving skills which they use in real-life situations and which remain with them for years to come This improved cognitive and critical thinking skill will be your best gift to the students. 

eiMaths’s students are creative, independent, confident and initiating. We are here to equip our students with a smarter way to learn Mathematics such that they can be winners in the changing world!

More details, do visit, a Singapore Maths project by Scotts DIGITAL - One of the top branding agency Singapore that specialised in branding your business using SEO.

Sunday, 22 February 2015

Mighty Math Tuition Centre #Reviews

Mighty Math takes the famed mathematics curriculum of Singapore and amps it up for adapting to international schooling standards in the best possible way. Their operations began in Singapore and their fame took them beyond the shores to Malaysia. Operating in several franchise model centres across both locations, they combine the exceptional curriculum of Singapore which has over the years built its reputation in international mathematics competitions, with modern franchise-based business management techniques of the United States to deliver high quality results.

Brief History of Mighty Math Tuition Centre

Developed in accordance with latest MOE syllabus in 2007, the Mighty Math centres take into account not just the basic mathematics and problem-solving but focus on empowering the students with greater degree of computational skills, conceptual approach to problems, building foundation techniques, thinking skills and enhancing natural problem-solving abilities. Mighty Math’s reputation has however, suffered a wee bit as their rapid franchising shows dilution of their business model.

Centre methodology

As a maths enrichment strategy, Mighty Math teaching methodology is focused on adding the skills which would train the young minds to think beyond merely exam-based scoring, making the possibility of attaining positive results higher for any mathematical exercise. Their programs are fine-tuned using feedback from educators, regular reviews and surveys of parents and students as well as seeing the general trend of , GCE and exam questioning patterns.
  • Pre-schoolers, Kindergarten Maths Programme: It is a well-known fact, the journey for academic excellent performance of a child, depends on how he or she was nurtured from the grassroots education (nursery, kindergarten etc.). Solely designed for preschool children, this program focuses on basic pattern, shape and number recognition giving them early foundations of math and calculations. The highlight of this programme is to develop the elementary reckoning ability of the students as well as prepare them ahead for primary education courses. This programme ensures that students will be cognizant with the foundations of primary arithmetic such as addition and subtraction with the help of number bond. The tutors are friendly, patient and passionate with children. The patterns, basic cognitive learning methods etc. ensures the reinforcement of the creative thinking and improves memorizing ability of the pupils.
  • Primary Math Programme: It is designed for students in post-kindergarten ranging from primary 1 to primary 6. Despite being within MOE guidelines it provides major impetus on developing confidence and critical thinking ability in tackling both routine and non-routine mathematics problems with heuristics approach. Intensive preparatory class for students that are preparing for PSLE Math paper, value-addition lessons aimed at preparing for Olympiads and competitions etc. are also part of early grooming in this course. In this programme, the students will be taught on how to utilize examination time and tackle questions in order to score high. Past PSLE math questions are solved in this programme, as well as efficient computation techniques and alternative approaches are taught.
  • Colorful manipulations and adaptive teaching - Teaching, for Mighty Math is not about blindly going through the coursework for each student, rather manipulating the course to adapt to personal needs. Several students have better visual memory and retention and teachers are passionate enough to introduce colours or cognitive patterns so the students can pick up their cues and retain the relevant information and relate later.


Primary Lower (Pre-school, P1-P4)
week (three classes per week, 1hrs) 
week (three classes per week, 1.5-2 hrs) – intensive coaching
Primary Upper (P5-P6)
$90/week (three classes per week, 1hrs) 
week (three classes per week, 1.5-2 hrs) – intensive coaching

Customer reviews

“Hayden has been enjoying his lessons since day one. He has improved tremendously in Math and has gained great confidence when solving problem sums. I reckon that good foundation is essential, as such; I have enrolled my younger son in Mighty Math beginning of the year. I will recommend anyone who has difficulty and does not enjoy Math as a school subject before, to attend Mighty math programme. You will not regret it” – Hyzel Ong (parent)
“I was happy with the Mighty Math program to help my son understand the concepts of addition and subtraction through number bonds. It will really give him a head-start in primary 1. However I must say, since I’ve moved to my new place, the closest Mighty Math centre, including their teaching and management staff is a bit below their standards in Serangoon Gardens centre. I will see out this term, as long as the results show up well.” – Mrs. Grace Phua (parent) and Brendon Phua (children)


Mighty Math is still growing as a math enrichment program in the Singapore educational domain. It does well in terms of bringing together the best of eastern curriculum and western management techniques to build a rapidly growing franchise model. It’d be important however, that they don’t lose sight of their basic educational plan, since a number of branches do show some level of disparity in terms of what they deliver in results. Overall, the vibe is good, but the verdict isn’t unequivocal yet. Their course and curriculum however, has come in for some praise.

Wednesday, 18 February 2015

Pros of Teaching Kindergarten Maths

Everyone is aware of how kindergarten maths has become exceptionally important over the past couple of years. Children tend to learn a lot from it, which is why it has been made a permanent part of every kindergarten’s curriculum out there. Where children learn a lot from it and also acquire a wide range of different benefits, teachers also attain many advantages in the process. Before teaching maths to kindergarten students, teachers need to consider many things that can be beneficial for them in the near future. Jobs for teaching maths to kindergarten students can be easily found these days; therefore, those who are interested in doing so must not waste any more time.

Easy to Teach
One of the biggest benefits to teachers for teaching kindergarten maths is the fact that it is quite easy. This means that they do not have to invest in so many efforts as this is practically basic maths that really is not difficult for a teacher who has been teaching it since a long period of time. Moreover, those who have previously taught mathematics to advance level children will find teaching kindergarten children even more easy in comparison. Therefore, it definitely makes this job worth it for all teachers.

On the other hand, kindergarten maths is rather fun to teach. Mathematics is a subject that requires a lot of day to day activities as well as tasks. This makes it rather fun due to the fact that children can acquire creative thinking through it. Such activities are not a hassle and teachers can always go on and introduce fun ideas to children in order for them to acquire enhanced thinking skills and creativity as well. Hence, the job of teaching maths to kindergarten children has been labeled as fun, which is why teachers should definitely go for it.

Kindergarten maths is interesting for both children as well as teachers. The basics of this subject enable everyone to learn all there is about maths in the first place. Everything such as problem solving and creative maths in involved in this phase of learning maths, which is what makes it exceptionally interesting for children as well as teachers for that matter. One important thing for teachers to know is that teaching this subject daily is not as boring as one may expect in the first place. Teachers can always introduce new kinds of creative activities in order to make maths in kindergarten more fun for children.

Good Salaries
Teachers should go ahead and kindergarten maths to children due to the fact that the jobs tends to pay well quite often. Mathematics is considered to be one of the most thorough and intricate subjects. Therefore, whether it is being taught on a basic level or at an advanced level, those who teach it are always paid in huge amounts in comparison with the teachers of other subjects in general. This is also due to the fact that maths has more technicalities than other subjects. This is why teaching maths to kindergarten children is a great idea.


With all the different kinds of benefits that teachers can acquire through teaching maths to kindergarten children, they should not waste any time to do so. Not only will it allow them to spend quality time with children, who are going to learn a lot from the subject, but they can also get good salary packages in the process and that surely makes the task worth it. Finding such jobs is not as difficult as one may expect in the first place since countless kindergartens exist in the present times and all of those do need permanent teachers for teaching maths. 

Tuesday, 17 February 2015

Advantages of Maths Activity Class for Children

More and more parents these days seem to be concerned about how their children are doing in school. As children are taught a wide range of different subjects these days, it is important for parents to see if their child is doing well in each and every one of them. Maths is one of the most major subjects that a lot of children may find rather interesting if teachers make maths activity class fun for them. It is through this class that children can increase their knowledge about the subject in a short period of time and go on to do exceptionally well in it in the near future.

Enhanced Creativity
As far as the benefits of maths activity class are concerned, they are many in number. One of the most prominent ones is the fact that it tends to enhance the creativity skills of children. This is due to the fact that teachers use many creative methods for teaching the subject in this class, which leaves a huge impact on children who definitely enhance their creativity skills by thinking broadly on a day to day basis. Studies have revealed that students who study maths on a day to day basis have a higher thinking capacity than those who do not.

Vast Thinking
On the other hand, the maths activity class also is the best way for helping children in thinking out of the box. Most children have limited thinking which is most likely to stay that way if teachers evade teaching maths in the right way. By introducing this class, teachers make sure that children will go on to be all aware of everything that maths is about in the first place. Out of the box thinking makes children more creative, intelligent and smart in the long run and that is precisely what all parents as well as teachers want.

Introduction to Mathematics
Maths activity class tends to introduce children to mathematics in the initial years of their school lives. While a lot of children are going to instantly love it, the others may not. In order to make all students fond of maths at the earliest convenience, teachers have been suggested to use a wide range of different creative methods for teaching the subject on a day to day basis in order to prevent the students from getting bored. Once they find it interesting, they will definitely want to know more of it by studying it thoroughly through every passing day.

Interesting & Fun
One of the most prominent advantages of maths activity class is that it is actually fun for both the students as well as the teachers. This is due to the fact that children tend to learn a lot in the process and as learning takes place in both ways, the teachers end up learning a lot from the students who have a lot of ideas and theories of their own to share. Both fun and interesting, this class can make children become rather fond of the subject in a short time period, which will eventually make them opt for it when they enter high school and above.

The Verdict
Activity class for maths is conducted in every other school these days, simply due to the fact that it provides children with increased learning and understanding of basic mathematical concepts and a lot more. Through it, children can acquire a lot of benefits in the long run as the subject holds a lot of significance in today’s world. With enhanced knowledge in the subject right from the beginning, students can go on to pursue high end careers in the near future without having to face any sort of hindrances in the matter. 

Sunday, 8 February 2015

Developing a Framework for Mathematical Enrichment (Part II)

Mathematical Thinking Strategies:

Some of the mathematical thinking strategies we have identified include:

Being systematic;*
Identifying common structures (isomorphisms);*
Introducing variables;
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


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;


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.


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).


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.


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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.
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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.
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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.
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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.

Friday, 6 February 2015

Developing a Framework for Mathematical Enrichment (Part I)

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

The NRICH Project ( 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 (  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.


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:


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.

Thursday, 5 February 2015

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.

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.


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.  


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