## What is mathematics?

When a member of the general community is asked this question, they invariably focus on number and operations. However, the modern mathematics curriculum is far more complex than just arithmetic. School mathematics has changed over time. During different periods, different mathematics has been taught (Stigler & Hiebert, 1999).

A century ago, mathematics in upper primary classes involved computation of tasks involving a large amount of number, long division, square roots of non – square numbers and so on (Cockburn, 2008).

This form of curriculum remained in place until the early 1960s, when the implementation of the New Mathematics represented a considerable shift in mathematics curriculum in most western counties. New topics were included and new forms of thinking mathematically, for example the set theory were part of the new syllabuses.

Since the 1970s, other reforms have influenced the curriculum, including problem solving where students were expected to be more creative in their thinking (Suggate, Davis & Goulding, 2010).

In the 1980s and increasingly since, technology has played a strong influence in the early years of technology aided learning, it has now been replaced by other technological aids, including new software programs, spreadsheets and graphic calculators, to name a few.

Students are expected to be far more creative in their thinking, and to deal with much more knowledge and complexity than in the pre- 1960s era (Suggate, Davis & Goulding, 2010; *Study Guide CMM119 Text and Culture*, 2011).

Thus this paper seeks to evaluate some of the strategies which can be applied in the field of mathematical study. Specifically, this paper provides an analysis of some of the strategies which can be used in teaching geometry and measurements to primary school pupils.

## Effective practice in teaching mathematics in primary school

The issue of efficiency within the learning centers has been a bone of contention for many years. The politicians and the parents have had much to say about this. Essentially, there has been a general clamor by the political class that the education system should go back to basics. This calls for the system to reconsider some of the approaches which were being used in the past (Western Australia.

Curriculum Council, 1998; Bobbis, Mullican & Lowrie, 2009). What is amiss in this case is the fact that despite the fact that the politicians are seeking what seems to be the best for the pupils, it should be left to the technocrats and the teachers to develop approaches which are in sync with the dynamic 21^{st} century in which we find ourselves. Most young people are now growing up in technology – rich environments.

They do not remember a time when you had to physically get up to change the television station – remote controls do that (Haylock, 2010; Bottle & Canterbury Christ Church University College, 2005).

One of the biggest growth areas in employment is self employment which means that many young people will be creating jobs for themselves in positions that will not even exist when they exit school. Thus the skills they require in order to be able to search for and identify key information are very different from those they needed when the written word was all that was available.

The mathematics education of students of modern society must be considered in light of this.

The students need to develop mathematical ways of seeing and interpreting the world; they need to develop strong problem solving skills; they need to be numerate; and, most importantly, they must have a disposition towards using mathematics to solve the problems which they confront, school mathematics needs to adopt pedagogies that will cater for diversity within a classroom (Haylock, 1991).

The old models of seated individual work found in traditional mathematics teaching are possibly contributing to the problems that emerge as students in the upper years of primary school (Brodie, 2007). For considerable numbers in the upper years off primary school and lower secondary school, the teaching that they encounter can lead to many negative feelings and misleading learning about mathematics.

Thus if anything is to go by, it must be done in consultation with the professionals in the field in order to ascertain exactly how best this subject should be handled. It is positive if the parents feel that there need to enhance the learning approaches. Furthermore, it is right when the politicians come and state what they feel should take centre stage (Lomas, 2009).

However, what needs to be put into consideration is the fact that these concerns should be within the professional framework. Institutions such as TIMSS and PISA should be on the forefront in providing the information which is sought by the parents and the politicians especially with regard to teaching mathematics (Rowland, Thwaites & Huckstep, 2009).

## Discussion of a series of activities

Teaching primary school children is one of the interesting and challenging tasks that one can engage in. this is because children are dynamic and full of promise as they take time to learn concepts (Ryan & Williams, 2007). In this case we are going to be evaluating teaching mathematics. In this section we are going to focus on two main areas, that is, geometry and measurement. Geometry

Geometry is the science of space and extent. It deals with the position, shape and size of bodies but has nothing to do with their material or physical properties. Demonstrative geometry deals with the shape, size and position of figures by pure reasoning, based on definitions, self – evident truths, assumptions and other established geometrical truths (Booker, Sparrow & Swan, 2010; An, 2004).

Euclid, a celebrated Greek mathematician was the father of demonstrative geometry. He devised many methods for handling problems. His methods are intuitional, observational, constructive, informal, creative, and experimental and so on (National Council of Teachers of Mathematics, 2000). Practical geometry: it covers work of the subject. Of course, most of this work is also directly or indirectly based on demonstrative geometry.

## Why teach geometry?

It enables the learner to acquire a mass of geometrical facts, the geometric principles of equality, symmetry, and similarity are implanted in the very nature of things, it is important in a person’s cultural development, it develop the ability to draw accurate plans, it provides a content that is objective and non controversial, it is useful in engineering, machine shop, construction industries, landscapes architecture, interior decoration and other areas of appreciation and it demonstrates the nature and power of pure reason (Reys, Lindquist, Lambdin & Smith, 2009; Mills & Koll, 1999).

## Its function in the primary and middle school

The teachers of mathematics and curriculum – framers have now begun to realize that nature and practical arts are the primary and permanent sources of geometrical learning. The function of the middle school geometry is to systematize the information received by the pupils at the pre-school and primary school stage from nature and practical arts.

The emphasis will be on the understanding of fundamental concepts and techniques such as the meaning, drawing and use of lines, angles, triangles and polygons (McDonald, 2010; Jorgensen & Dole, 2011; Martin & Herrera, 2007). The primary object is not teaching the pupils to geometry; but rather to lead them to think geometry. The practical side of geometry, will, however, dominate at this stage.

## The systematizing stage

It is the stage of mastery in reasoning. The reasoning here will be more rigorous but properly suited to the mental age of the pupils (Koshy, Ernest & Casey, 2000). Practice in logical reasoning will be more important than convincing them that the facts are true. Dependence on axioms will also be reduced.

## When introducing reasoning

In the beginning, the child should be allowed to use logic whenever he can. Actually the child begins to use reasoning at quite an early age. The children who can see the logical connection between facts should be encouraged to do so.

But it is not to be forgotten that the main function at the early stage is to collect and experience facts (Willis & Devlin, 2004a). If the young child does not respond to logic satisfactorily, there need not be any hurry or worry about it.

## Some more teaching points

The practical work should precede and clarify logical work, the teacher should depend on firsthand experience and visual aids, the practical work should be very neat and evident, the pupils should be asked to observe things themselves by actual measurements or experiments.

There should be no oral teaching definitions and memorizing of abstract ideas, the blackboard should be sufficiently neat, clear and accurate to avoid doubts and misconceptions. The teacher should also insist upon the accuracy of language (Mills & Koll, 2003; Van de Walle et al, 2007). Colored chalks may be used to emphasize significant details analytical approach in the beginning should become a rule with the pupils.

Synthetic approach should not be applied unless and until the idea is thoroughly understood, a proper atmosphere should be created in the classroom by displaying suitable charts and models on the subject, the subject should be frequently correlated with arithmetic and algebra, the riders should not be left to be done at the end of the session (Haylock & Thangata, 2007).

They should be done side by side with thee theorems, in oral work avoid the use of letters such as A,B,C for the pupil has to strain to decide which letter was said, consequently his attention must flag.,

The teacher should always remember that the definition is the explanation of a term by means of others which are more easily understood. So the explanation should not be more difficult and confusing than the term being defined (Haylock & Thangata, 2007).

It should not be more difficult and confusing than the term being defined. It should rather be simpler and more easily understandable. Its essentials are simplicity, clarity and brevity. It must be as short as possible and be free from ambiguity. It should; dispel the existing doubts rather than create any new doubts.

## Constructions

In geometry the words “to construct” mean to draw accurately. The only instruments permissible in these constructions are the straight edge for drawing straight lines and the compasses for drawing arcs and measuring lengths of lines. Efficiency in constructions requires skillful manipulation of these instruments (Willis & Devlin, 2004b). Actual doing is very necessary.

A student may verbally know how to construct a certain figure, and yet may fail to draw it accurately, when actually required to do so. Lack of practical experience is generally responsible for this failure. Accuracy and neatness require good accurate geometrical instruments. Further constant practice is the only way to acquire self confidence in construction work.

Essentially, the teacher should see that the pupils are properly equipped in an accurate and reliable set of geometrical instruments and fine, hard and sharpened pencils.

Special drawing pencils should be used (Zevenbergen, Dole & Wright, 2004). Ordinary pencils do not give neat work. All straight lines are drawn from left to right. All lines should be uniform thickness. The lines should be drawn as thin and sharp as possible. Different steps of construction should be clearly shown.

## Applications

In this case there are several perspectives which need to be put into consideration when teaching mathematics. Give the learners an experience with map. Essentially when teaching measurements, there are some aspects which a teacher should be keen to put into consideration. First the lesson should be one ended and have as many practical examples and exercises.

These approaches capture the mind of the learner thus ensuring that the learner is able to concentrate and gain the concepts being taught. For instance, teachers should give the learners opportunities to build on what they know in groups. This has been captured in the videos clearly whereby the teacher is able to know what they have learnt.

Using practical examples is another approach which is used. Use of pattern blocks, for example when teaching about shapes, the teacher can use shapes that are common in the learner’s environment. Shapes such as rocket toys provides a good basis for the learner to be able to associate what is being taught and what is available.

In addition, it is worth noting that the learner should be be able to associate what is being taught with other learning aspects. For instance, when learning about shapes, the learner should be also be given an opportunity to learn about measurements as well as counting.

Learners should also be given the opportunity to share what they have. This should be enhanced through group work. Essentially, group work gives the learners an opportunity to share the knowledge which they individually have. This is an important tool especially in teaching abstract concepts such as geometry.

Introduction of concepts in geometry and measurement is an important aspect in learning. This is because it forms the grounds for understanding the concepts which have been learned within a given period of time.

## Shapes and squares

The teachers should use practice examples in class. Give the learners an opportunity to follow through as they do what you have given them to do. This keeps the track of thinking and helps them to verbalize what they are thinking thus gives the teacher an opportunity to know what they are doing and what they are thinking. The students will be able to learn about the various sizes and shapes of different geometrical shapes.

Use practical lessons, in this case the teacher should ensure that the methodology applied is practical in approach (Western Australia. Curriculum Council, 2005).

In this case the teacher should strive to ensure that the aspects which are taught in class are concepts which the learner can be in a position of putting into practice with ease. This is because these approaches ensure that the lessons which are taught ensure that the learner is able to relate what is taught with what exists.

The teacher should also provide the learners with ample examples enhance both individual and group knowledge. This is because the learners who are able to share knowledge are in a better position to advance their skills from an unknown state towards the known state (Western Australia. Curriculum Council, 1998).

The learners should also be provided with ample time to exercise skills and knowledge acquired. This should be done through ensuring that the learners are exposed to classroom work. In addition the learners should also be exposed to take home assignments.

In cases where the teacher has got to work with the learners towards ensuring that those from different backgrounds are able to articulate concepts which are taught in the mainstream classroom.

## Conclusion

In the current dispensation, it is worth noting that teaching has become an extremely complex craft, than it was in the 60s and 70s. This has been escalated by the technological advancement which has been witnessed in this century. In conclusion, the programmed study in mathematics has three elements, which can be summarized as follows; first, using and applying mathematics.

Pupils should learn to use and apply mathematics in practical tasks, in real – life problems and in mathematics itself (Zevenbergen, Dole & Wright, 2004). They should be taught to make decisions to solve simple problems, to begin to check their work, and to use mathematical language and to explain their thinking.

Secondly, in number work, pupils should develop flexible methods of working with numbers orally and mentally; using varied numbers and ways of recording, with practical resources, calculators and computer. They should begin to understand place value, develop methods of calculation and solving number problems.

Thirdly, in shape, space and measures, pupils should have practical experiences using various materials, electronic devices, and practical contexts for measuring. They should begin to understand and use patterns and properties of shape, position and movement, and of measures.

## References

An, S. (2004). *The middle path in math instruction:solutions for improving math education.* London: R&L Education.

Bobbis, J., Mullican, J., & Lowrie, T. (2009). *Mathematics for Children: challenging children to think mathematically* (3rd ed.). Frenchs Forest, NSW: Pearson Australia.

Booker, G. B., Sparrow, L., & Swan, P. (2010). *Teaching primary mathematics* (4th ed.). Frenchs Forest, NSW: Pearson Australia.

Bottle, G., & Canterbury Christ Church University College, P. M. (2005). *Teaching mathematics in the primary school.* London: Continuum International Publishing Group.

Brodie, A. (2007). *Supporting Maths For Ages 7-8.* London: A & C Black.

Cockburn, A. (2008). *Mathematical misconceptions:a guide for primary teachers.* New York: SAGE Publications Ltd.

Haylock, D. (1991). *Teaching mathematics to low attainers, 8-12.* New York: SAGE.

Haylock, D., & Thangata, F. (2007). *Key Concepts in Teaching Primary Mathematics.* New York: SAGE.

Haylock, D. (2010). *Mathematics Explained for Primary Teachers.* New York: SAGE Publications Ltd.

Jorgensen, R., & Dole, S. (2011). *Teaching Mathematics in Primary Schools.* Sydney: Allen & Unwin.

Koshy, V., Ernest, P., & Casey, R. (2000). *Mathematics for primary teachers.* New York: Routledge.

Lomas, G. (2009). *Mathematics Educator’s Beliefs and Their Teaching Practices.* London: Lambert Academic Publishing.

Martin, T.S., & Herrera, T. (2007). *Mathematics teaching today:improving practice, improving student learning.* New Jersey: National Council of Teachers of Mathematics.

McDonald, B. (2010). *Mathematical Misconceptions.* London: Lambert Academic Publishing.

Mills, S., & Koll, H. (1999). *Rapid Recall.* London: Folens Publishers.

Mills, S., & Koll, H. (2003). *Solving Problems.* New York: Letts Educational.

National Council of Teachers of Mathematics. (2000). *Principles and standards for school mathematics.* Michigan: University of Michigan.

Reys, R., Lindquist, M., Lambdin, D., & Smith, N. (2009). *Helping children learn mathematics* (9th ed.). New York: John Wiley & Sons.

Rowland, T., Thwaites, A., & Huckstep, P. (2009). *Developing Primary Mathematics Teaching:Reflecting on Practice with the Knowledge Quartet, Volume 2.* New York: SAGE Publications Ltd.

Ryan, J., & Williams, J. (2007). *Children’s mathematics 4-15:learning from errors and misconceptions.* California: McGraw-Hill International.

Stigler, J. W., & Hiebert, J. (1999). *The teaching gap:best ideas from the world’s teachers for improving education in the classroom.* London: Simon and Schuster.

*Study Guide CMM119 Text and Culture.* ( 2011). Brisbane: Education and Law Griffith University.

Suggate, J., Davis, A., & Goulding, M. (2010 ). *Mathematical knowledge for primary teachers.* New York: Taylor & Francis.

Van de Walle, J. et al. (2007). *Elementary and middle school mathematics: Teaching developmentally* (6th ed.). New York: Pearson.

Western Australia. Curriculum Council. (1998). *Curriculum framework for kindergarten to year 12 education in Western Australia*. Osborne Park, W.A.: Curriculum Council.

Western Australia. Curriculum Council. (2005). *Curriculum framework curriculum guide – Mathematics.* Osborne Park, W.A.: Curriculum Council.

Willis, S., & Devlin, W. (2004a). *First steps in mathematics: Measurement. Indirect measure/estimate.* Victoria: Rigby Heinemann.

Willis, S., & Devlin, W.(2004b). *First steps in mathematics: Space.* Victoria: Rigby Heinemann.

Zevenbergen, R., Dole, S., & Wright, R. J. (2004). *Teaching mathematics in primary schools.* Crows Nest, NSW: Allen & Unwin.