Historically, teaching mathematics in school is peppered with numerous ‘solutions’ and advice on how to teach it effectively. At one time it was thought that children should be left to discover mathematics for themselves with the teacher only being there to facilitate and encourage discovery. The opposing view to this was that the teacher was there to drill particular methods and routines into the children regardless of whether the children knew or understood what was happening this meant that, while children could recite number facts without error, when it came to solving problems they did not necessarily know what methods to apply to derive the correct solution. Research has shown that children respond individually to experiences and that while some are able to ‘discover’ mathematics for themselves (often these children have access to other ideas at home) others need more direct instruction. It would seem then that a ‘connectionist’ approach where both teacher and children work together to develop a deeper understanding of mathematics is needed. This approach gives children the necessary broad and balanced experiences for them to succeed, broad in the sense that they encounter a variety of resources and contexts; balanced in the sense that no one particular method of working is dominant. (Askew 1996, p.7)
One of the major factors in teaching effectively in a primary classroom, whether it is mathematics or any other subject, is good classroom management and behaviour control. If teacher have a classroom where pupils respect and follow the rules of the classroom and lesson to be taught, then teacher can concentrate more on teaching the subject than on managing behaviour. This extra subject input ensures that more of pupils will gain a deeper understanding and achieve the relevant learning outcomes, this is especially important in mathematics because of the multi-layered nature of the subject itself. Having established a mutual respect situation and a good atmosphere for learning then the effective teaching strategies specific to a particular subject can begin.
Before mathematics can be taught effectively it is essential that the teacher has a thorough knowledge of the extent of their pupil’s existing understanding of the subject and to facilitate this a baseline assessment is required. This baseline assessment should be designed to indicate understanding and not just knowledge and recall, in order to do this it may be necessary to set some puzzle solving tasks to ensure that the children can apply their knowledge when and where required. It is important to discover pupil’s understanding so that planning can focus on what the children are capable of; research has found that it is frequently assumed that children bring no mathematical knowledge with them when they start school. This assumption is incorrect many children can before they have any form of formal schooling (Ofsted 1995, p.6): “Can count meaningfully; Use terms like ‘more’ and ‘less’ correctly; Have a small amount of understanding of small number addition and subtraction; Invent strategies for solving problems.”
Methods and Strategies
The research does however show that although the children can achieve these bullet points their understanding is not always what adults expect. Small children may be able to count by reciting a sequence of numbers without actually connecting this counting activity with discovering a quantity. The best way to assess whether this is the case is possibly by observation, which can take place outside of the numeracy hour if necessary, this observation could be undertaken while watching children, in the ‘home corner’ in Reception/Nursery school, if they are required to set the table for a group of friends or dolls etc. This counting for a specific purpose can enhance the children’s understanding of counting to find a specific quantity. (Haylock 2001)
This method of using practical objects to facilitate counting should not be the only method used as some children have a tendency to rely on having a specific object related to a specific number and are unable to grasp the abstract; for example, a child may know that two elephants plus two elephants equals four elephants but not relate that to two plus two equals four. It would appear, therefore, that by implementing a well-designed baseline assessment to inform planning a teacher can discover how strong the foundation knowledge is and build on this to enable the child to progress. (Haylock 2001)
It is crucial to offer as many possible strategies for deriving facts as possible to aid the child’s progression in mathematics, for example, if a child “knows by heart” some number facts as well as being able to ‘count on’ they will be able to arrive at the correct answer to addition sums much quicker. “To accept without question primitive methods (e.g. counting for addition) used by lower-attaining children, ensures the divide between them and their higher-attaining peers will grow even wider.” (Ofsted 1995, p.8)
Studying the arithmetic methods used by children we can confidently say that, children who know some number facts by heart are able to deduce other number facts because of their understanding of the relationship between numbers. These children make more progress because each approach supports the other. Deducing number facts helps children to commit more of these facts to memory and recalling these facts helps them to expand the range of strategies they are able to use to derive more facts. Teacher must be careful to ensure that we do not assess the children’s knowledge of number facts alone, as this only imparts a percentage of the information required. Teacher must also assess the ability to undertake new problems and derive solutions, one method that can be applied to achieve this is to develop children’s flexibility in mental mathematics through class and group discussion on a daily basis and to make sharing their ideas for mental methods a basis for debate in the classroom. (Haylock & Cockburn 1997) This research also uncovered the concern that some of the lower-attaining pupils depend heavily on counting methods which can get them to the correct answer eventually, it does however remove the requirement of committing number facts to memory and this will hamper their ability to undertake and solve more difficult problems. It is important that if there is evidence that some of their pupils are relying on this counting method then the teacher must adjust his/her planning to include sets of strategies to encourage them to work from remembered facts instead.
Mathematics is fundamentally a mental activity and some young children may need to show how they know something using blocks or fingers but as a teacher it is necessary to build on this practical work and progress to verbal explanations instead. The problems that some children have with verbal explanations are sometimes wrongly interpreted as having no mental images, when it is really a problem understanding what is expected of them. These children can be helped to progress to verbal explanations of what is going on ‘in the head’ by the teacher providing model explanations and playing the simple ‘under the cloth’ activity. (Askew 1996, p.8)
The National Numeracy Strategy has gone a long way to help teachers plan effectively and include some daily mental work and with this comes the opportunity to set targets and challenge the children. The children’s confidence increases with this daily input and the lower-attainees have the opportunity to succeed among their higher attaining peers. There are a variety of ways to make this mental work ‘fun’ by playing games such as ‘number chase’ or ‘under the cloth’. These ‘games’ are an excellent way of challenging the children to become faster at mental arithmetic, for example, by timing the first game of ‘number chase’ it is possible to target the children to complete it in a faster time. It is important during these mental sessions that the teacher models his/her own methods of deriving the correct solution and provides the opportunity for some of the more-able children to explain their methods. Thus the mental session in the numeracy hour is probably more effective when the whole class is involved rather than groups or individuals. (Haylock & Cockburn 1997)
There is a danger that the children will devise ways of not participating during the mental session, especially if they rely on the slower methods of arriving at the correct answer. Some children rely on the fact that the teacher never asks those whose hand is first up; therefore they put their hand up even if they have not worked out the answer. It is necessary as the teacher to ensure that he/she provide thinking time for all the children, several strategies can be used to do this, for example (Askew 1996, p.63): “Insisting that nobody puts a hand up until you givе a signal, thеn silеntly counting to fivе or tеn bеforе giving thе signal; Gеtting childrеn to raisе only a fingеr whеn thеy arе rеady to answеr, еxplaining that this is so as not to disturb thе othеrs; Providing ‘chainеd’ calculations е.g. put 5 in your hеads doublе it (pausе) add 3 (pausе) subtract 6 (pausе) OK. By varying thе numbеr of links in thе chain childrеn nеvеr know whеn thе OK is going to comе and so all hands go up togеther.”
It could be argued that the most crucial factor for teaching mathematics, or any other subject, is that of the teacher’s own knowledge. It is vital to ensure that teacher know the subject teacher are teaching well enough to anticipate and answer any questions that the children might ask. “If you appear at all unsure of your subject the children very quickly realise this and it is vital therefore that you research thoroughly any aspect that you are unsure of. Once you have done your research and are sure of your topic you will find it easier to show enthusiasm and confidence, this in turn can encourage intrinsic motivation” (Kyriacou 1997, p.23) and the child’s own curiosity and desire for success will ensure concentration.
It can help both teacher and the children if he/she can relate the mathematical topic currently teaching to ‘real life’ situations; for example, when teaching about decimal numbers relate them to money and this helps children to understand the tenths and hundredths aspect, where to put a decimal point etc. By giving the children these comparisons to ‘real life’ they are more easily motivated to learn and understand. It can also help to link mathematics to other curriculum subjects, such as, the use of co-ordinates in Geography. Knowledge of the subject also helps lesson planning to become more effective as relevant activities, that increase understanding, can be included and any difficulties that may arise anticipated and dealt with. Effective formative assessment techniques also benefit lesson planning when used alongside lesson evaluation.
The aforementioned points are only some of the ‘crucial factors’ in the effective teaching of mathematics. It can be concluded that due to the nature of primary teaching many of the factors crucial to the teaching of all curricular subjects, for example, reflective teaching, good classroom organisation and clearly defined learning objectives are also crucial to the teaching of mathematics. Teacher’s main target should be that as many of pupils as possible are able to succeed and work to the best of their ability. If we can be enthusiastic and make learning fun then this target is more likely to be achieved, it is accepted that some children have difficulty understanding mathematics and always will have but if we can help them by using some of the teaching methods that have been shown to be effective then they are more likely to attain their goal. Extensive research has been undertaken and the evidence shows that all children have the potential to succeed if they are taught effectively, and many excellent suggestions have been made by these researchers to enable teachers to do just that. Some of these suggestions have been implemented in the National Numeracy Strategy and their framework for the Numeracy Hour. It is important to realise that the Numeracy Strategy is only a framework and not a set of mandatory lesson plans and assessment papers; it is designed to help and not hinder the teacher, however, it is sometimes very difficult to revisit topics that some of the children have not understood because of the number of topics that need to be covered and lack of time there is to cover them.
Askew M. 1998, Teaching Primary Mathematics, A guide for newly qualified & student teachers, Hodder & Stoughton, London.
Dfee. 1999, National Numeracy Strategy, Dfee.
Haylock D. 2001, Mathematics Explained For Primary Teachers. London, Paul Chapman Publishing.
Haylock D. & Cockburn A. 1997, Understanding Mathematics in the Lower Primary Years. London, Paul Chapman Publishing.
Kyriacou C. 1997, Effective Teaching In Schools Theory and Practice, 2nd Edition, Nelson Thornes, Cheltenham.
Ofsted. 1995, Recent Research in Mathematics Education 5-16, HMSO