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Friday, 27 February 2015

Preschool Maths Cover More Than Just Counting

Maths in all grades involve rationalizing, counting, and figuring out many aspects of nearly anything. Preschool maths involve setting the path for future learning in this regard, while maximizing on your child's earliest learning days.
It's not complicated to teach a child how to count from 1 to 10. For the best results later, having them able to make more of the numbers is the goal of preschool maths. Things such as two is more than one, and that there isn't fairness in two for them and one for me. They are quite capable of this learning at the preschool level though and it gives them a great head start on later maths.
Below are a few ideas that help children find preschool maths easier and give them an extra boost on the future. These are not only in preschool classrooms; rather they can be done at home as well.
Stars For Behavior

Many recall these charts often used to keep the class informed about who was the best in various ways. This enables the children to weight differences in quantity as a useful side effect. Counting and knowing more is a great way for learning preschool maths as well as behavioral understanding.


In addition these stars help provide the preschoolers to grow in areas where recognition among peers can be helpful. Maths at this level helps put the big picture together in the long run.
The Family Bar Graph
Preschool classes will often ask each child to bring pictures from home of their family. From there they use a simple bar graph to make it visual. The bottom often has the number of family members, and the side being how many children in the preschool class have that many. These help instill some of the difference aspects of maths. Using the pictures for the vertical bars they can easily see where the most children fall. In the end, their will be the ability to see size from simple maths.
There a number of gaming and competitive ways this preschool activity can be used to make it more interesting. When it comes to the maths, it will have them comparing sizes as well.
Friendship Mix
This concept can go a long way in preschool in the areas of sharing, maths, simple counting, and more. Just like other mixes you can buy at the stores, the preschool children are each given a bag of a different dry snack. Each is then asked to put five pieces in front of them at a time from their bag. A bag is passed of which each puts their 5 in, and passes it onto another student with something different to do the same task with. They repeat this for bags for each student in the class.
The resulting mixed backs of snacks give them practice counting, seeing sizes grow, and teamwork. In the end they take them home for a snack!
Other Practice
There are many ways to help learn preschool maths. You don't need expensive education resources. Anything can do, such as cans, bottles, boxes, and anything else that can be stacked. Building towers will enable them to see growth in numbers. Beyond preschool maths they will begin more extensive use of the skills they have developed here.
You may be interested in learning more about a preschool maths. For info you simply have to know, be sure to visit the Preschool Activity Site. Check it out today here.

Sunday, 22 February 2015

Maths Tuition - Out of School Classes to Help Your Child Develop a Solid Education

Maths tuition plays a vital part in the education of any child. One needs to provide the proper education for their kids to ensure that their future prospects are not hampered. Most of us leave no stones unturned to enroll our kids into the best possible schools in order to provide them with the best possible education. However, sometimes the classes conducted by the school are just not enough to guide your children properly... especially in subjects like maths.
There will always be occasions when a child might not understand a particular maths problem at school. There might be other students too who can face the same problem of not grasping what is being explained. But it is no use blaming the teacher. They cannot be expected to tend to so many students within the short duration of the class. We now have the option of hiring private tutors to develop a solid foundation for your children's education.
Interacting with children requires a special environment that only those who are trained in the task can offer.
Not only that, but a special routine needs to be followed while teaching maths to kids so that their interest in this subject is kindled. A child might have to be explained the same maths problems several times until they are able to grasp it. Few people have the patience to undertake this task. Ask yourself honestly if you could do it. A professional out-of-school tutor can. This is the main reason why you should consider sending your child to a tuition centre that has proved their mettle over the years.
Enrolling your kids in such maths tuition centres will ensure that they get a firm grip on the subject and are able to pass their examinations with flying colours. There are certain schools in London which specialise in teaching maths to kids in such a manner that they will enjoy what they are being taught. Once they are interested in the subject they will be able to grasp it easily.
These educational centres employ professional teachers who are well-versed in their respective academic field. By enrolling your kids at such tuition classes you can rest assured that they will receive a degree of personal attention that is impossible to receive in school. Students attending such classes will find it easier to overcome the challenges posed by maths and will be well prepared for GCSE and SAT examinations.
If you feel your school is not giving your child the attention they deserve it is high time you helped your child develop a solid understanding of maths, English or science by enrolling them onto these tuition classes. Search online for a centre located close to you and see what they can offer your child out of their school environment.
If you would like to help your child's learning and build their knowledge and confidence with maths then you should consider taking free numeracy courses available for parents and carers so you can fully support your child.

Friday, 20 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. 

Wednesday, 18 February 2015

What do Children Learn in Maths Activity Class?

Children these days are taught many subjects for the purpose of high end learning right from the beginning of their school days. However, maths has a special purpose in every child’s life and while the subject can be quite demanding at times, it is extremely beneficial in the near future. A lot of preschools and kindergartens have introduced maths activity class due to the fact that it ensures learning and also gives a child the ability to think on a large scale. Due to the importance of this subject, teachers are always highly recommended to teach it with great care and dedication to the children.

Brain Enrichment
As far as the maths activity class is concerned, it allows children to think of all sorts of possibilities. The best part is the fact that they learn to use their brain for everything and the fact that they learn to properly utilize it is something that cannot be taught easily by another subject in general. This is amongst one of the most significant reasons for teaching mathematics to children in kindergarten as well as preschool. A child’s brain has a lot of imagination and creativity hidden in it and in order to bring that out adequately, the learning of maths is definitely required.

Out of the Box Thinking
Moreover, children get the ability to be able to think out of the box in maths activity class. This is due to the fact that maths is a vast and an extremely diverse subject that opens doors to a lot of new opportunities as well as concepts. While children learn all of these concepts, they also see how they can use these in real life. This is only if a teacher pays attention in teaching them about how to do it without making it entirely difficult for them.

Learning Numbers
Maths contains numbers and these are generally new for a child who has just started to learn maths in school. It does take a lot of time for children to get used to numbers but with the help of maths activity class, it has now become quite possible since this class is all about teaching children all there is about maths and how it can be understood. Sometimes children may understand it quickly, but the other times, it may take some time for them to get used to it in the process. However, the results are most definitely positive as the main reason for holding a class dedicated to this subject is to show children how important and fun it can be.

Time Management
Maths activity class also shows children the many ways through which they can manage time. Time management should be taught to children right from the beginning so that they do not have to face any difficulties during tests and important examinations in the long run. It is through time management that people get ahead in life and it can be taught very well with maths. Therefore, the purpose of holding tis class is to also teach time management to children in a short period of time.

Is the Maths Activity Class worth it?

A lot of schools these days have introduced this class and the results have been tremendous as many children have learned a lot of things regarding maths. Many studies also show that the activity class for maths is also responsible for making children interested in mathematics right from the beginning, which is definitely a great thing as the subject becomes crucial in their lives later on. The activity class is held on a day to day basis in most schools in different parts of the world and has been made a proper part in the routine of all preschool and kindergarten students.  

More Maths activity class click here.

Tips to Help Kids with Maths – Kindergarten

As is commonly known, Maths can be a very tedious subject for many children and growing up, many people will categorize it as their least favorite subject. 

Kindergarten maths can be particularly tedious since many children think it is a dry subject. However, it is still a subject that is extremely important for practical life and one will have to make sure children understand it well. Thus, it is very important to make this subject interesting and engaging for children so that they can easily understand the complexities of the subject. There are a number of ways this can be done easily and effortlessly.

Counting is one of the basic ways through which one can engage children and make them understand a bit of kindergarten maths. There are a number of songs through which children can learn to count. While some of these songs have actions to go with them, others do not. Teachers can easily improvise a few steps to go with these songs so that children are able to learn through all five senses collectively. Teachers can also visit a number of local libraries where librarians will tell them about fun books that can be used to make counting more interesting and fun for students.

Little kids have a certain fascination with computers and they love to play on all kinds of gadgets. They specifically love laptops and tablets and thankfully, mobile devices allow a number of applications to be downloaded so that children can play fun and interesting mathematics games and still are able to learn a lot while they do so. There are also a number of computer games in the nearest library or the local computer store and one can bring in or rent some CDs to store up these games in the school’s computer properly. Kindergarten maths can be fun, interesting and engaging with the help of technology.

In kindergarten maths, people must not force children to do everything at once or keep bombarding them with things to learn and practice one after the other. Big jumps do not work in a subject like maths since it is sensitive and difficult. Instead of understanding the material being taught, children will only end up getting more confused than ever and they may just feel like giving up. Therefore, teachers must devise a step-by-step plan that properly lays out how each and every lesson is to be taught and it must be done in a sequential manner for maximum effect.

It is also a very well-known factor that children take more interest in things they are more curious about. Kindergarten maths can satisfy their curiosity as they learn. They will learn more if they are taught things they want to learn about. Thus, teachers need to keep their eyes and ears open to spot what interests their class and what they want to study further. Tapping into the class’ curiosity will pay off very well since this way, children will be more enthusiastic about the learning process. For most children, outdoor activities are a great way to get their interests. Teachers can devise activities where they tell kids to bring them five leaves, seven twigs and four flowers, for example.


Lastly, kitchens have often proven to be incredibly math zones. Children love to slice and cut things up when working in the kitchen. Teachers can decide that they will bake a cake for a math lesson and ask the kids to cut the dough in quarters, for example, or put in three spoonsful of sugar in a bowl. It may get a little messy but it is a great way to earn and children will enjoy it even more when they get cookies as rewards.  

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:

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.

Friday, 6 February 2015

Developing a Framework for Mathematical Enrichment (Part I)

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

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

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

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

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

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

Background and Rationale

The wider context

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

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

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

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

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

Enrichment can be used:

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

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

Mathematics Enrichment Materials on the NRICH Website

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

From these points come the foci of our recent work:

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

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

Defining a Framework

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

Enrichment

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

Content

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

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

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

                                                                                                             based on Ernest (2000)

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


Teaching approach

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

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

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

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

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

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

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.

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.


Tuesday, 3 February 2015

Improving Mathematical Skills With Maths Enrichment Classes in Singapore

Post11As school studies are not enough for a child to cover up all the subjects and their topics especially mathematics. So it is recommended for every child to have admissions in the best maths learning centers. To enhance the mathematical skills of a student, we must offer them some maths enrichment programs to participate in as well as a maths tuition center for regular practicing over mathematical concepts.

Maths Enrichment Programs

A maths enrichment program helps a child to enhance his mathematical skills with various maths creative games which lead them to understand the number system as well as the basic concepts addition, subtraction and multiplications. In maths enrichment program a student can participate in various games which will enhance their calculations and mathematical basic concepts. The students also get teachings for independent learning, problem solving as well as creative thinking. They are able to find the creative ways to empower their mathematical problem solving skills with the deep and simple strategies. 

There are various activities in the maths enrichment classes Singapore such as various workshops for maths, hands-on sessions, quizzes and competitions, presentations and lectures, student conferences, residential courses, online presentations and conferences. Also it includes various real world scenarios which can help students to co-relate their daily life with the concepts which supports them to better understand the topics.

Through these programs, a student can have the ability to solve the problems at their own independent thinking. Also there are mathematical games so that the students will be able to learn some new skills from the game. There are strategies in the games which leads the students to win from other participants. He gets familiar with the rules of the games and try to win according to the points and learn how the game is going on and about the rules and winning criteria of the game which sharpen the mindset of the students, With enjoyment, the student will be able to learn various concepts and logics behind the mathematical questions.

Maths Learning Centers

Maths learning centers play a vital role while improving the mathematical skills with proper intellectuality. Maths requires full practice over each and every topic. Under the supervision of a well qualified and well versed trainer, one can easily gain the knowledge about the mathematical concepts and logics. As practice makes a man perfect, it is the best strategy applied in case of the maths. The trainer will guide with the basic concept behind a chapter but the understanding and applying the concept at different places will lead to the correct solutions and correct answers.

EiMaths is the best maths learning center in Singapore grooming their students for future learning and preparing them to grasp the mathematical concepts easily for higher education levels. The skilled team makes it easy for the students to understand the concepts, acquiring skills through creative activities and solving heuristic problems. They prepare candidates for higher grades with firstly simplest teachings and afterwards with the complex ones.

See More: www.eimaths.com

Maths enrichment programs offers candidates self confidence with various fun activities so that after the enrichment session they provide proper attention towards maths with their own interest as they have found relations of our daily life with mathematical concepts such as addition, subtraction etc.