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Deaf children’s informal knowledge of multiplicative reasoning.
Terezinha Nunes, Department of Education, University of Oxford
was available from 11/23/09 until 12/13/09
to answer questions and share ideas concerning her research with Peter Bryant, Daniel Bell, Deborah Evans, Diana Burman, and Darcy Hallett
and its implications for parents of children who are deaf/hard of hearing, their teachers and other professionals who work with them.
You are encouraged to read the research summary below and review the attached discussion.
Multiplicative reasoning is required in different contexts in mathematics: it is necessary to understand the concept of multipart units, involved in learning place value and measurement, and also to solve multiplication and division problems.
Measures of hearing children’s multiplicative reasoning at school entry are reliable and specific predictors of their mathematics achievement in school. An analysis of deaf children’s informal multiplicative reasoning showed that deaf children under-perform in comparison to the hearing cohorts in their first two years of school. However, a brief training study, which significantly improved their success on these problems, suggested that this may be a performance, rather than a competence difference. Thus, it is possible and desirable to promote deaf children’s multiplicative reasoning when they start school so that they are provided with a more solid basis for learning mathematics.
When children start school, at the age of 5 or 6, they already know quite a few things that require thinking mathematically. Yet some things that seem quite obvious to adults are not so obvious to children, or not to all children at any rate. For example, think of this problem. You ask a child to share fairly 12 pretend sweets to two dolls. The child does this carefully, using a one-for-A, one-for-B procedure, and is quite certain that the sharing was done fairly and that the dolls have the same amount of sweets to eat. Then you ask the child to count A’s sweets; most 5-year-olds can count to six without making errors, so that’s an easy task. Then you ask the child to say how many sweets doll B has, without counting. To adults, the answer is obvious, but this is not obvious to many 5-year-olds.
In contrast, most 5-year olds know that if you give 2 sweets to A every time you give 1 sweet to B, this is not fair, and they know that A would have more. This may seem surprising because we tend to think that children must first learn about numbers and then learn to reason mathematically. The comparison between these two problems shows that it is not necessarily so; they can tell that A is getting more because there is a two-to-one correspondence between A’s and B’s sweets without knowing how many sweets each doll received.
Solving mathematics problems in primary school requires of children different types of reasoning. Most people recognize the importance of one-to-one correspondence for understanding number. For example, children must establish a one-to-one correspondence between the items being counted and the counting label when counting. If they fail to do so, and skip a number or count the same item twice, they will not be able to say what the correct number of items is.
Understanding one-to-many correspondence is just as important for solving practical problems, such as counting money, and for writing numbers in the way we do. If you have one 10 cents coin and I have one 1 cent coin, we have the same number of coins but we don’t have the same amount of money. Similarly, when we write the number “11”, we write the digit 1 two times but it does not represent the same value: the 1 on the right represents a unit and the 1 on the left represents one ten. These two examples illustrate the importance of one-to-many correspondence reasoning for learning mathematics from the very beginning of primary school.
One-to-many corresponding reasoning is the origin of what is known as multiplicative reasoning, which includes solving multiplication as well as division problems. Studies with hearing children show that their understanding of one-to-many correspondence (or multiplicative reasoning) is so important for learning mathematics that, if the children are assessed in how well they understand these correspondences when they start school, we can anticipate how well they will learn mathematics in the first two years of school. But not all children start school with some knowledge of one-to-many correspondence. Those who have difficulty with this concept can improve their chances of learning mathematics better if they receive teaching that helps them develop this understanding (Nunes et al., 2007).
These conclusions come from research with hearing children but they are important enough to make us want to know what deaf children’s situation is. We carried out two studies to find out how well deaf children understand one-to-many correspondence when they start school and how easily they can learn these ideas so that their mathematics learning can improve.
In the first study, we just wanted to know whether hearing children start school with an advantage in multiplicative reasoning in comparison with deaf children. We presented hearing and deaf children in the first year of school multiplicative reasoning problems that they could solve by counting, if they did not know any sums. For example, we showed them a picture, such as that in Figure 1, and told them that in each of the two houses live three dogs. We then asked how many dogs altogether live in the two houses? Children who understand the concept of one-to-many correspondence can solve this problem by counting the three dogs that they see and then pointing to the other house while counting three dogs that they imagine are in the house. They do not need to know about addition or multiplication: if they understand the idea, they can simply count.
We asked British deaf (N=28) and hearing (N=78) children in their first year in school to solve 6 multiplicative reasoning problems. The hearing children’s mean number of correct responses was 3.12 and the deaf children’s was 1.86. The answer then is that deaf children under-performed in comparison to hearing children on this task.
The second question we wanted to answer was whether this was a difference in performance that could not be easily changed. If the deaf children were taught about one-to-many correspondences in problem solving contexts, would they make progress rather quickly? So in the second study we used a teaching method that we had developed in previous work with hearing children and knew to be effective for them. We randomly assigned the deaf children (N=27) either to a taught group, that received instruction on one-to-many correspondences, or to a control group, that received instruction on visual analysis, which is irrelevant for this problem.
The teaching was carried out over two sessions in which a researcher worked individually with each child. The children were asked to solve a total of 5 multiplication and 5 division problems. The problems were presented in the child’s preferred mode of communication (oral, BSL or Sign-supported English). The children had access to manipulatives which they could use to solve the problems. For each problem, the researcher first asked the child to show with the materials what the story said and then solve the problem. For example, the children were given vans cut-out from colored paper and some small blocks. The story problem they were trying to solve was: The teacher has ordered some table for a party. Three vans are bringing the tables and each one is carrying for tables. How many tables is the teacher going to have for the party? The children were told that they could use the small blocks to be their pretend-tables. If the children did not place 4 blocks on each van, they were encouraged to do so before answering the question. An example of a division problem was:
a boy has 12 marbles; he wants to put the same number inside each of these 2 bags; how many marbles will go in each bag? The children were
give cut-out circles to represent the bags and used the blocks to be the pretend-marbles.
The children were given a pre-test, before the teaching took place, and two post-tests, one right after the teaching ended and the second one between 2 and 4 weeks later, to see whether they had entirely forgotten what they had learned. The children in the intervention group made good progress from pre-test to post-test: their mean score at pre-test was about 2 correct answers (out of 12) and at the immediate post-test it was about 6, i.e. approximately three times what it had been at pre-test. The control group’s score did not change very much at all. So it was quite clear that the taught group had learned from instruction.
There was some forgetting between the immediate post-test and the delayed post-test: the taught group’s mean score went from approximately 6 correct answers (out of 12 problems) to approximately 5 correct answers. This decrement in performance was disappointing but the children did not revert to their initial level of performance, which shows that some learning was maintained over this period.
Together, these studies have shown two things. First, deaf children do start school at a disadvantage in comparison to hearing children in their knowledge of multiplicative reasoning. Second, this difference can be minimized by appropriate teaching that helps them understand one-to-many correspondences. This latter finding is very important, both for teachers and parents. It means that if deaf children received target stimulation to develop their multiplicative reasoning, they could start school better prepared to learn mathematics.
The studies reported here were supported by the RNID (Royal National Institute for Deaf Persons). We are very grateful for their support, without which the studies could not have been carried out. We are also very thankful to all the schools, teachers, children and parents who generously contributed to this study through their participation and without any further rewards.
Terezinha Nunes, Peter Bryant, Daniel Bell, Deborah Evans, Diana Burman, Darcy Hallett
Deaf children’s informal knowledge of multiplicative reasoning
The Journal of Deaf Studies and Deaf Education 2009 14(2):260-277; doi:10.1093/deafed/enn040
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