Exploring Some Of The Difficulties Children Encounter When Faced With Word Problems

Exploring some of the difficulties children encounter when faced with word problems

For the purpose if this assignment I will be discussing children's mathematical understanding. I will look at how the teaching of word problems can develop mathematical understanding and possible obstacles children may face when presented with problems Childrens understanding develops from birth. Primarily, children will learn from their parents, they learn how to use a knife and fork, how to walk and how to talk. A child who is talked to a lot will learn lots of words and a child who frequently visits the park will learn about swings and slides, birds and trees. A child who plays with a lot of toys will have good hand-eye co-ordination. (NHS, 2007) Children will start school with varying levels of development and it is then upto the class teacher to further develop their understanding, using pedagogy tailored to individual needs. 'Each child is different because each is an individual' (NHS, 2007) Skemp recognizes two types of understanding - relational and instrumental. He defines relational understanding as 'knowing what to do and why' and instrumental understanding as 'rules without reasons'. Instrumental understanding in mathematics is wide spread for numerous reasons and pupils can prefer this method because 'all they want is some kind of rule for getting the right answer' (Skemp, R. 1989). Instrumental understanding and teaching can be seen in the majority of classrooms. It is easy to understand a specific rule that is given to us and this can quickly and reliably produce a page of correct answers leading to a self-esteem boost for our children. Some topics are difficult to understand relationally and an instrumental method is the only one acceptable. Even relational mathematicians occasionally use instrumental thinking. (Skemp, R.1989) However, the widespread instances of Instrumental teachers may be simply 'teaching to the test'. With more pressure being placed on teachers and schools to perform well in SAT's and other tests it has become easier to simply teach children what they need in order to produce a correct answer, with little understanding behind how they produced it. Instrumental thinking also requires children to know which rule to use and when, if they become overwhelmed by information and different rules they can become confused. Relational understanding is easier to remember and is more adaptable to new tasks, children with relational understanding of mathematics will find it easier to take what they know and apply it to new problems, instrumental rules however, do not always transfer effectively to problems requiring relational understanding. With pressure building on schools to achieve well, teachers may believe that relational understanding of mathematics will take too long to achieve. What is a Word Problem? "A problem, as opposed to something that is merely an exercise for practising a mathematical skill, is a situation in which we have some givens and we have a goal, but the route from the givens to the goal is not immediately apparent." (Haylock, 2006, pg. 317) 'In mathematics education, the term word problem is often used to refer to any math exercise where significant background information on the problem is presented as text rather than in mathematical notation' (L Verschaffel, B Greer, E De Corte, 2000). Word problems can be as daunting for some teachers to present as they are for learners to attempt. A teacher's negative attitude towards maths can affect the children's learning experience. As teachers, we are encouraged to be enthusiastic about teaching and learning, and to plan and deliver lessons using all of our subject knowledge in order to educate the children in our class. If a teacher is not confident in their own mathematical ability this will be detrimental to the quality of their lessons. 'It is widely recognised that a teacher's own enthusiasm for, and knowledge of, mathematics, as well as their beliefs about teaching and learning, will impact on their classroom practice, regardless of the external constraints on curriculum and lesson design.' (Williams, P. 2008)

National Guidelines and their implications In his article, Koshy (Koshy, V. 2004) talks about five different children in the same class using completely different methods for subtraction. Some are quite abstract and some are learnt from the teacher. One of these children had the correct answer but was under pressure to set his working out in his book correctly, as taught by the teacher. However, the current curriculum until 2014 says that in Key Stage One children should '1b. Develop flexible approaches to problem solving and look for ways to overcome difficulties. (DfES, 2013). The pupil was clearly able to solve the problems he was set, but was not able to communicate his working out very well, which is also a requirement of the curriculum. '1e. use the correct language, symbols and vocabulary associated with number and data. 1f. communicate in spoken, pictorial and written form, at first using informal language and recording, then mathematical language and symbols' (DfES, 2013). Whilst the pupil was using flexible approaches and successfully solving problems, he was unable to communicate how and why. To work on this the teacher could sit with him and work through the sum together, slowly on paper and try to help the pupil understand how to use the correct maths vocabulary. Demonstrating this again step-by-step to the class beforehand.

Whilst many children enter school knowing a range of basic vocabulary there are children that will only be saying a few words. This can be down to social factors, or different languages being spoken at home, amongst other reasons. The national curriculum is calling for children to be using correct mathematical language and symbols in both spoken and written form, it will be difficult to accomplish this with little underlying English vocabulary. This particular section of the curriculum puts emphasis on communicating the language with no reference to whether or not the meaning of these words is being understood. For example, the teaching of phonics is the teaching of word sounds and children may not be understanding the meaning, this will create obstacles in mathematics when trying to teach the correct vocabulary. I have encountered this difficulty whilst teaching word problems, a child stood at the front of the class and read a word problem aloud, beautifully sounding out all words using her phonics knowledge. When I asked her about what she had read, she did not know. The same was evident during mathematics sessions involving the reading of questions with a majority of pupils.

Children's preconceived ideas about word problems. One problem facing children approaching word problems can be their preconceived ideas. For example, if they have taken home a worksheet that's too difficult, away from the classroom setting with no support from teaching staff or peers they could find themselves muddled and feeling lost. Children with English as an Additional Language may be unable to ask parents for help deciphering the word problem. Some parents may not be confident mathematicians and be unable to offer assistance. Fraser and Honeyford (2000:84) devoted an entire chapter of their book to the issue of 'sum stress' and how to deal with it. They defined sum stress as "the symptoms some pupils suffer when faced with mathematical problems or even just the prospect of a mathematics lesson". (p. 79). They went on to explain that sum stress might manifest itself in different ways including behavioural problems. Some typical reasons for suffering sum stress included: failure in the past; parental pressure; low status given to mathematics by parents; parent suffers from 'sum stress'; one very bad experience in a mathematics lesson; lack of confidence; relationship with current or previous mathematics teacher; and physical problem, such as dyslexia. Swan, P. 2004) Whilst carrying out research in to problem solving and word problems in particular I began to think about children's preconceived ideas coming from media influences. While many children's television shows are educational and show mathematics in a positive light there are some that do not. For instance, 'The Simpsons' often pokes fun at innumeracy and there are references to mathematics throughout all its seasons. In one episode 'Bart' is attempting a word problem and gets very confused, and begins to daydream, children may be able to relate to this or they may become anxious. Bart's reaction can be found in classrooms everywhere. Effective pedagogy to aid children's understanding of word problems. During my teaching practice placement I taught a number of sessions on word problems to a mixed ability year three class. My first word problems sessions were unsuccessful; the children were unable to reach the correct solutions. I decided to use a method that would be immediately in line with the key stage two mathematics curriculum '1. Pupils should be taught to: break down a more complex problem or calculation into simpler steps before attempting a solution; identify the information needed to carry out the tasks' I gave the children the following problem; if one lady bird has 6 legs, how many legs will 3 ladybirds have? To help them understand what kind of answer we could expect I had them draw out 3 lady birds. They counted up all the legs and came to the answer 18. 'A picture referring to a context situation can elicit an action that has the potential to lead to a solution.' (Van den Heuval-Panhulzen, M.1999). The children liked this method and it was especially effective with one child who had English as an Additional Language, he was able to reach the correct answer by counting the legs. I asked the children; how could we write this in a 'number sentence'? I was then able to introduce the 'clouds method'. One of the clouds puts the children in a situation whereby they must decide with mathematical method they need to use. There were four clouds (see Apendix) and each was on the white board in a specific order to help the children break down the problem. Mrs Benfall has 3 pencils which she carries around in her beautiful handbag that is made from tea leaves. The pencils are 17cm, 12cm and 9cm long. What is the total length of all the pencils? What do we need to find out? How long are the pencils? Can we cross out any useless information? The lady's name doesn't matter, neither does her handbag. Which operations do we need to use? Addition. Do we have an answer that makes sense? Yes. Our answer seems reasonable. The method of breaking down the problem greatly aided many of the children in getting the right answer. However, there were some who still struggled to achieve the correct solution, on further exploration of their misconceptions I realised that words with 'double meanings' were to blame. For example, 'The word 'more', because it is naturally associated with addition, will act as a miscue, and prompt the pupils to add the numbers in the problem' (Haylock, D. 2007). This was evident in one particular case when teaching and I asked children to write their own word problems. (See Appendix) One pupil wrote 'Jessica has 2 Pokémon cards and La'Shantay has 20. How many more does La'Shantay have?' The child answered '22' Instead of 18. Evidently, the word 'more' was still being associated with addition. This is also a good example of a previous point I made that some topics are harder to understand relationally than instrumentally. This child had an instrumental understanding of the word 'more' relating to addition, when in this case a subtraction was needed.

How do word problems contribute to the development of mathematical understanding? Teachers should ensure to take into account prior learning in order for learners to progress effectively; building on prior learning can enhance ideas and motivate children. Cultural and life experience should also be taken into account where appropriate. If children can relate their own worldly experience to their problem solving strategies they are more likely to come to the correct solutions. As highlighted by Haylock when considering decimals in terms of money problems 'The pupils who worked on the contextualized problems worked much better together, because the less able students were able to contribute from their everyday knowledge and experience of decimals in the world outside the classroom.' (Haylock, D. 2006). Some children will already be handling money and be more aware of its value than others. For instance, whilst teaching I set the problem; 5 x £0.50= A child who was not used to handling money did 5 x 50 = 250 and was unsure of where to put the decimal point. A child with prior knowledge of money from going to the local shops with her older siblings knew that five fifty pence's would not make £25.00 or £0.25 and helped their classmate reach the conclusion of £2.50. This is a great example of a child learning monetary values through working with peers on a word problem. They will have learnt a new rule when it comes to decimals and money.

On the other hand, Askew (2003:78-85) has argued 'they can do the calculation when it is presented to them in numbers, but why put it into a context when it simply confuses them?' Going back to the ladybird example I used earlier, if I presented my class with the problem '6 x 3 =' they would surely give me the correct answer. It was only within adding words into the equation that confusion arose. If children are already confident with the underlying mathematical 'number sentence' then it seems there would be no need to add anything to it.

To conclude, whilst there is debate as to whether or not word problems are always beneficial it is clear that they do develop children's mathematical understanding. Children acquiring these skills at an early age will be well equipped for secondary school and beyond, taking their problem solving skills along with them.