Alice’s teacher would need to discuss, with her, all the ideas and concepts she holds about multiplication and division. Only by exploring the child’s understanding of underlying concepts and principles will the reason for the misconceptions become apparent. As already mentioned the teacher’s assessment of the reason for a child’s errors may not be accurate purely by studying their work.
A starting point to rectify the misconception would be to identify in the NNS (DfEE 1999) what understanding Alice should have acquired by her age. Children, according to the NNS (DfEE 1999), do not encounter multiplication or division until year 2, and confirms my previous thinking that children are to understand the two operations as repeated addition and subtraction. It is not until year 4 that children multiply and divide decimals and fractions and examine related theory, such as the commutative law.
Alice needs to be taught that multiplication and division are more than just complicated forms of addition and subtraction. There is more to multiplication and division than just computing sums. According to Nunes & Bryant (1996) “The child must learn about and understand an entirely new set of number meanings and a new set of invariants, all of which are related to multiplication and division, but not to addition and subtraction” p.144. Research conducted by Hoyles, Noss & Sutherland in 1992 showed an ingenious method to enable children to see that multiplication does not always make numbers bigger. The children were asked to reach a target number (e.g. 100) from a starting point (e.g. 13) through successive multiplications. The pupils easily overshot the target and therefore had to face the question: how do you make numbers smaller by multiplying them? The research identified this as a significant question, which helps pupils see the discontinuity between addition and multiplication. This type of activity along with one related to division could be used in school to highlight concepts children may not have realised. By addressing difficulties within topics early on children are prevented from forming certain misconceptions.
The next stage for the teacher would be to decide how to prevent these sorts of misconceptions occurring in following years. It is apparent that Alice’s previous teaching has not allowed her to access the topic in a way that she can understand it. Teachers must consider both their teaching styles and children's learning styles when approaching areas that children can easily misconstrue. The following years class should be given the opportunity to explore the concepts and theory behind multiplication and division, this may not be in line with the NNS (DfEE 1999)) order, but may provide those children with a greater comprehension of the operations they are carrying out. Once they are confident with the reasoning their manipulation and computation of numbers should be secure due to their underlying understanding.
Emily’s work shows four calculations with vulgar fractions. It is apparent that she is working horizontally to complete the sums either adding or subtracting the two numerators and the two denominators. The NNS (DfEE 1999) identifies that children from year 4 upwards should “recognise the equivalence between fractions” and at year 5 should be able to “recognise from practical work simple relationships between fractions” part 6 p.23. It appears from Emily’s work that she does not understand the relationships. In the first question (shown below) she has to add two fractions with the same denominator.
3/4 + 1/4 =
The NNS (DfEE 1999) also states that children in year 2 should be able to “recognise that three quarters and one quarter make a whole” part 5 p.23. Therefore Emily should be able to calculate this sum in her head from her knowledge of fractions. The same can be said for the second question which involves the addition of sixths, although it is slightly harder as the answer will be greater than one, she should still be able to calculate the question mentally by counting how many sixths there are in total. This theme runs throughout the questions Emily has calculated incorrectly.
Suggate et al (1998) suggests that children understand the adding and subtracting of fractions when dealing concretely, but once introduced to the procedure for multiplication which tends to be a written algorithm “some children try to use a similar one for adding fractions” p.93. To confirm this is diagnosis the child would have to complete a variety of tasks, which assess the subordinate skills for adding like and unlike fractions.
There are a variety of instructional activities which would help Emily to correct the misconception identified. Given that the error pattern is so similar to the multiplication algorithm, extensive practice on any single procedure should be avoided. The child should practice deciding which questions can be written horizontally or vertically. According to Ashlock (1986) “when adding unlike fractions it is usually best to write the example vertically so the renaming of each fraction can be recorded more easily” p.139. Counting of the fractions may be a strategy, for example showing a child how counting is appropriate when the denominators are the same, but not appropriate when they are different. Building on Skemp’s ideas (1989) of how children develop concepts by relating them to personal experience and what they already know and Bruner’s idea of iconic representation (Bigge & Shermis 1992) Emily should be given the opportunity to use diagrams, for example representing each fraction as fractional parts of a unit. This procedure should then be related to the step-by-step mechanics of calculating a written algorithm. By estimating answers before commuting Emily would have some idea of whether her final answer is right. The NNS (DfEE 1999) states that from year 2 children should be taught strategies in order to be able to check their calculations.
For the teacher there are obviously implications for teaching the following year. The children must have a solid understanding of fractions before they can begin to manipulate them. From personal experience children have a better sense of what they are dealing with if they can see it visually, although shading in of parts may be repetitive it allows the child to see a fraction represented concretely. They will also be able to see the equivalence between unlike fractions. Once the children are confident in this area the teacher can move onto mental methods, for example rapid recall of facts. The teacher can do this in the mental and oral session getting children to call out equivalent fractions. Targeting of questions to different children or groups can give the teacher a good assessment of their level of understanding before the main part of the lesson begins. The teacher needs to ensure that when the children progress to manipulation of fractions they are not working to rules they have learnt by rote, but have an understanding of what they are doing. Suydam (1984) states after examining research that, “It seems apparent that we need to shift emphasis from having students learn rules for operations on fraction to helping them develop a conceptual base for fractions” p.64.
David’s work shows three examples of written subtraction. It can be identified that David subtracts the smaller number from the larger number in each column. David may be thinking of the larger of the two numbers as the number of the set and the smaller as the number to be removed from the set, or he may be comparing the two single-digit numbers and finding the difference between them. David may also have over-generalised the commutativity for addition and assumed that subtraction is also commutative. There is a real problem identifying this misconception because many times the answer David gives will be correct, for example in the second sum; 77 – 25 the top number in each column is larger than the bottom one therefore he will complete the calculation correctly although his method and reasoning is incorrect. David is also considering each column as a separate sum, instead of calculating 77 – 25 he only considers 7 and 5 and 7 and 2. From personal experience I have seen lower ability children regularly adopting this error pattern. One child computed this way because she knew she didn’t know how to borrow and therefore even though her work was wrong she felt she had tried to solve the sum. She was also aware that sometimes her answer would turn out to be correct.
According to Suggate et al (1998), “children who have difficulty with written algorithms for subtraction are frequently encouraged to use tens and units apparatus to support their work” p.60. This idea is contradicted by Hughes (1986) who has found through research that children are often unable to transfer the use of the apparatus to the written form. Therefore tens and units apparatus would be useful to rectify subtraction misconceptions, but the teacher must ensure the children can transfer between the concrete and the abstract. David’s teacher could provide him with different strategies with which to approach this kind of calculation, he may find for example that decomposition enables him to understand the principles behind his manipulations. The NNS (DfEE 1999) shows that children from year 4 upwards “should develop and efficient standard method that can be applied generally” part 6 p.50. According to Hopkins et al (1996) “There are many forms of written methods of subtraction and children should be encouraged to use the ones they understand” p.61. Another method to help David may be to go back to horizontal calculations, this way of writing the sum may not lead him towards the situation where each column is calculated separately, but help him to think of the sum as a whole. Davis should also be provided with strategies to check his own calculations. If he can see his answer is not correct he could begin to question his methods.
The teacher must consider the language they use in the class when teaching maths. David may have heard such phrases as “Always subtract the little number from the big one” or “Work out each column separately”. The teacher may inadvertently cause these misconceptions by using types of language and vocabulary, which children will apply in their own way. Dean (1992) identifies that language used by the teacher can often inhibit a child’s learning. She offers two opportunities to overcome this, either ask the child to identify what it is they do not understand or ask another child who does understand to explain it, this is a valuable experience for both pupils. For the future the teacher must realise that children have a wide variety of difficulties with subtraction and it may only be David who always want to take the ‘little number from the big one’. Therefore the children must have access to all the different types of strategies for written subtraction and should be encouraged to use whichever suit them. Children who have a solid grounding in mental calculations will also be able to identify errors they make in their written algorithms. Above all if children cannot calculate subtraction sums using written methods they should at least be able to rely on mental strategies to begin to find an answer. No child should be left with no way of trying to complete their work.
In all the examples above teachers questioning plays a large part in rectifying misconceptions and ensuring they do not happen. Kyriacou (1998) identifies that effective questioning allows teacher to “encourage thought, understanding of ideas, phenomena, procedures and to check knowledge and skills” p.34. by questioning regularly as topics are taught any misconceptions can be identified and dealt with. When dealing with misconceptions it is important to think how grouping may help. Active learning, which is based on the discovery theory of learning advocated by Bruner, enables pupils to work in small groups on problem or investigational tasks. As identified in Kyriacou (1998) “Co-operative activities enable pupils to obtain greater insights into the conduct of learning activities through observing the performance of their peers and discussing procedures and strategies” p.39. This type of learning can allow children to learn and share ideas with their peers and gain an understanding, which may not be possible in a one-to-one situation with the teacher. There will be times when it is necessary for the teacher to focus on an individual child who needs immediate attention to be able to continue with the task. There are assessment questions available, which test important misconceptions. Pupils are given questions, which have been answered incorrectly and have to correct them and explain what the person did wrong. According to Thompson (1999) “These tests can provide useful diagnostic information” p.122.
The teaching of mathematics is the teaching of concepts, which is why many children struggle. These concepts must be taught in a way children understand in order to avoid misconceptions. Dean (1992) identifies therefore it is “important to set up appropriate situations which lead to the development of specific concepts. According to Bruner teachers must be inventive in order to translate information to levels that are appropriate to those who must understand it. He says knowledge should be ‘coded’ so it is useable by children. When it is the concepts that children understand and not just rules to apply to calculations misconceptions are less likely to occur.
Bibliography
Ashlock, R (1986) “Error Patterns in Computation” London : Merrill
Dean, J. (1992) “Organising Learning in the Primary School Classroom” London : Routledge
DfEE (1999) “The National Numeracy Strategy” Cambridge : Cambridge University Press
Hughes, M. (1986) “Children and number: difficulties in learning mathematics”. Oxford : Basil Blackwell
Kyriacou, C. (1998). “Essential Teaching Skills”. Cheltenham : Stanley Thornes
Skemp, R. (1986) “Mathematics in the Primary School” London : Routledge Falmer
Suydam, M. “Fractions” – The Arithmetic Teacher 31, no. 7 (March 1984)
Thompson, I. (1999) “Issues in Teaching Numeracy in Primary Schools” Buckingham : Open University Press
Bigge, M. & Shermis, S. (1992) “Learning Theories for Teachers” New York : Harper Collins
Nunes & Bryant (1998) “Children Doing Mathematics” Oxford : Basil Blackwell
Pollard, A. & Tann, S. (1993) “Reflective Teaching in the Primary School” London : Cassell
Suggate, J. Davis, A. Goulding, M. (1998) “Mathematical Knowledge for Primary Teachers” London : David Fulton
