Deaf students leave school with a level of mathematical competence that seriously interferes with their future prospects. As there is no evidence for an inbuilt difficulty with mathematics among deaf people (Zarfaty et al, 2004; Bull et al, 2006), the potential for early identification of difficulties and intervention is considerable. An important longitudinal predictor of mathematics learning, shown to be independent of intelligence (Stern, 2006; Nunes et al, 2006), is the understanding of the inverse relation between addition and subtraction – or "inversion". Measures of children’s understanding of inversion assess whether they realise that a+b–b=a.

We investigated deaf children’s understanding of inversion in two studies. In Study 1, 19 deaf children were compared to 98 hearing children in their first year of school. Deaf children were significantly behind their hearing peers in inversion.

In Study 2, we developed and assessed an intervention to promote deaf children’s understanding of inversion. Participants (N=18) were randomly assigned to an intervention group or a control group. Both groups participated in a pre-test, an immediate post-test and a delayed post-test, which contained inversion tasks and control items (assessing computation skills). They also participated in two individual sessions with a researcher, where they received teaching on a number-related concept: the intervention group’s teaching was about inversion and the control groups’ about a different concept. We expected the intervention group to improve significantly more on the inversion tasks but not on the control items. The intervention group’s performance on inversion improved significantly across assessments; the control group’s did not. The difference in improvement between the groups was significant for the inversion but not for the control items.

Deaf students leave school with a relatively low level of mathematical competence. In a U.S study, approximately 80% of deaf 14 year-olds performed below basic levels in mathematics (Traxler, 2000). Yet there is no reason to think that deaf people have inbuilt mathematical difficulties. Deaf pre-school children are as good as hearing children in number representation (Zarfaty, Nunes & Bryant, 2004) and deaf adults’ response patterns in number tasks are similar to hearing adults’ (Bull, Blatto-Vallee, & Fabich, 2006).

This paper investigates deaf children’s understanding of the inverse relation between addition and subtraction, referred to as “inversion”. Stern (2006) and Nunes et al. (2006) showed that inversion is a predictor of school mathematics learning, independent of intelligence. Inversion is used implicitly when children are taught, for example, to subtract by complementary addition (e.g., in order to solve 42 – 39, what can you do to add to 39 to get to 42?). If deaf children, for any reason, lack opportunities to think about inversion before school, they will be at risk for mathematics difficulties.

Study 1

Our aim was to see whether deaf children start school with a lower performance than hearing children on inversion problems.

Participants were 19 deaf children (mean age 6.7y) and 98 hearing children (mean age 5.3y) in their first year in school.

Measures

The children answered six inversion items in which an initial quantity was presented, then something was added and something was taken away. In four items, the quantity added and subtracted was the same (e.g. 9 + 6 – 6); in two items, the quantities differed by 1 (e.g., 8 + 6 – 5). In half the items we presented a row a bricks, covered it, and then performed the transformations in front of the child. The other half was presented as a story problem, with a picture showing a box, where (the children were told) there were 9 books; two pictures then showed that 5 books were added and 5 taken away. The child was asked how many items there were after these transformations. The children were familiar with these numbers but were not expected to know the addition (e.g. 9 + 6) or the subtraction facts (15 – 6).

The children were also given the Matrix Subtest of the British Abilities Scale (BAS).

Results

Answers were scored as right or wrong. Counting-all solutions were not possible; counting on was not observed though some children attempted to count the bricks under the cloth.

Because the deaf children were significantly older, adjusted means (controlling for age and BAS scores) were calculated. The mean correct for deaf children was 1.01 (SE=0.45) and for hearing children was 2.33 (SE=0.17). An ANCOVA with age and BAS Matrix as covariates showed a significant effect of age (F_1,113=4.0; p<.05) and BAS score (F_1,113=6.46; p<.05) and a significant difference between groups (F_1,113=6.76; p<.05).

We concluded that deaf children show a delay in understanding the inversion and could benefit from an intervention aimed at improving this understanding.

Study 2

Our aim was to develop and assess and intervention to improve deaf children’s understanding of inversion.

Participants were the 18 deaf children form Study 1 (one child did not complete the post-tests).

Design

The children were randomly assigned to an intervention or control group. Both groups received a pre-test on inversion, and identical immediate and delayed post-tests. There were six inversion trials (three using blocks, three story problems) and six control items, where the addition and subtraction were unrelated (e.g. 9+9-3). We expected the intervention group to improve on the inverse but not the control items.

Both groups participated in two individual intervention sessions. The intervention group was taught about inversion and the control group about a different mathematical concept.

The intervention consisted of presenting the children with inverse problems; after they answered the question, they were allowed to count the objects. Initial problems contained non-mathematical cues (the blocks added and removed were the same and differed in colour from those in the initial row). These cues were phased out during training.

Results

The groups did not differ at pre-test on age, BAS scores or inversion. The intervention group means improved from pre-test (1.9) to immediate post-test (2.6) to delayed post-test (3.4) whereas the control group’s did not (1.6; 0.4; 1.1, respectively). A repeated measures ANOVA showed a significant effect of intervention group (F_1,14=17.1; p=.001) and a significant interaction between testing occasion and group (p<.05 for both post-tests). Neither group showed an improvement on control items. Therefore the intervention group significantly benefited from the training. This was not due to a general improvement on calculation, as there was no change in their scores in the control items.

Discussion

Young deaf children do not show difficulties in number representation but under-perform for their non-verbal intelligence in inversion tasks. As this was not known before, no provision for promoting their understanding of inversion had been attempted. We have shown that it is possible to improve their understanding of inversion. Further studies are needed to investigate whether this will positively improve their mathematics learning.

References

Bull, R., Blatto-Vallee, G., & Fabich, M. (2006). Subitizing, Magnitude Representation, and Magnitude Retrieval in Deaf and Hearing Adults. Journal of Deaf Studies and Deaf Education, 11, 289 - 302.

Nunes, T., Bryant, P., Evans, D., Bell, D., Gardner, S., Gardner, A., et al. (2006). The Contribution of Logical Reasoning to the Learning of Mathematics in Primary School. British Journal of Develomental Psychology, in press.

Stern, E. (2006). Transitions in mathematics: from intuitive quantification to symbol-based mathematics. Paper presented at the ISSBD meeting, July, Melbourne.

Traxler, C. B. (2000). The Stanford Achievement Test, 9th Edition: National norming and performance standards for deaf and hard-of-hearing students. Journal of Deaf Studies and Deaf Education, 5, 337-348.

Zarfaty, Y., Nunes, T., & Bryant, P. (2004). The performance of young deaf children in spatial and temporal number tasks. Journal of Deaf Studies and Deaf Education, 9, 315-326.