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,6Proposal for an Oral Presentation as Part of TSG7 at ICME 11 Mexico
Insights About Indentifying and Assisting Children Who Have Difficulty Learning Mathematics
Ann GervasoniAustralian Catholic University
Introduction
This paper explores three issues associated with assisting Australian children who have difficulty learning mathematics in regular classrooms. The first issue is how to identify these children. The second issue is the diverse instructional needs of this group of children who have difficulty. The third issue is providing assistance for children. In this regard, this paper describes one approach used in Australian Primary Schools that is based on providing children with a 20 week intervention program conducted by a specialist teacher.
Using a Clinical Interview and a Research-Based Framework of Growth-Points to Indentify Children Who Have Difficulty Learning Mathematics
A key issue for school communities is the early identification of children who have difficulty learning mathematics. This enables school to provide assistance for these children with the aim of enabling them to learn successfully. As part of the Early Numeracy Research Program (ENRP, Clarke, Cheeseman, Gervasoni, Gronn, Horne, McDonough, Montgomery, Roche, Sullivan, Clarke, & Rowley, 2002) that took place in Australia, a process was developed to identify any Grade 1 (6-year-olds) and Grade 2 children (7-year-olds) who had difficulty with number learning in the domains of Counting, Place Value, Addition and Subtraction and Multiplication and Division. This process involved examining childrens growth point profiles that were established following a clinical interview, and then prioritising childrens need for an intervention program according to these profiles (Gervasoni, 2004). For example, the ENRP Growth Points for the domain of Addition and Subtraction Strategies are:
0. Not apparent in this context
Not yet able to combine and count two collections of objects.
1. Count all (two collections)
Counts all to find the total of two collections.
2. Count on
Counts on from one number to find the total of two collections.
3. Count back/count down to/count up from
Given a subtraction situation, chooses appropriately from strategies including count back, count down to and count up from.
4. Basic strategies (doubles, commutativity, adding 10, tens facts, other known facts)
Given an addition or subtraction problem, strategies such as doubles, commutativity, adding 10, tens facts, and other known facts are evident.
5. Derived strategies (near doubles, add 9, build to 10, fact families, intuitive strategies)
Given an addition or subtraction problem, strategies such as near doubles, adding 9, build to next ten, fact families and intuitive strategies are evident.
Grade 1 children who had not reached Growth Point 1 (using count-all strategies) at the beginning of the school year were considered vulnerable in this domain because it was not apparent that these children had a successful strategy available to solve simple addition problems, and that this could preclude them from engaging in typical classroom experiences. Similarly, Grade 2 children who had not yet reached Growth Point 2 (using count-on strategies) were considered vulnerable in this domain (see Gervasoni, 2004).
A similar process was used in each of the other number domains. Overall, this process, enabled teachers to clearly identify the domains and combinations of domains for which children were vulnerable, and to identify children who might benefit from an intervention program.
The Diverse Instructional Needs of Children Who Have Difficulty Learning Mathematics
Developing effective approaches for assisting children who have difficulty learning mathematics continues to be an issue for mathematics teachers and school communities. Indeed, intervention programs for assisting children are rarely effective for all. Recent Australian research suggests that this may be because we have failed to recognise the complexity of the instructional needs of students.
Analysis of the instructional needs of 100 7-year-old and 8-year-old Australian students who were selected for a mathematics intervention program indicated that these children had diverse learning needs, and were vulnerable in a range and combination of number domains (Gervasoni, 2005). Indeed, there was no single formula that described the instructional needs of children in the domains of Counting, Place Value, Addition and Subtraction, and Multiplication and Division. Further, there were no patterns in the domains or in any combinations of domains for which the children were vulnerable. Vulnerability was widely distributed across all four domains and combinations of domains in both grade levels. However, it was found that most, but not all, Grade 2s were vulnerable in Place Value.
These findings have several implications for the structure and design of intervention programs. First, the diverse learning needs of these children call for customised instructional responses from teachers. This supports the approach advocated by other researchers in the field of mathematics learning difficulties (e.g., Ginsburg, 1997; Rivera, 1997; Wright, Martland, & Stafford, 2000). It is likely that teachers need to make individual decisions about the instructional approach for each child because there is no formula that will meet all childrens instructional needs. This does not mean that separate intervention programs are needed for individual children, but rather that teachers need to know how to customise activities and instruction so that they may focus each childs attention on salient features of their experiences so that they notice the aspects that lead to the construction of mathematical knowledge. For this to happen, it is optimal for intervention group sizes to be three or less, and for teachers to be aware of each childs current mathematical knowledge and associated zones of proximal development. This requires frequent assessment and knowledge of the pathways of childrens learning.
The diversity of childrens mathematical knowledge across the four number domains also suggests that knowledge in any one domain is not necessarily prerequisite for knowledge construction in another domain. For example, some teachers may assume that children need to be successful in Counting before they are ready for learning opportunities in Addition and Subtraction. On the contrary, the analysis of childrens instructional needs indicated that some children who had difficulty in Counting were successful in Addition and Subtraction and this pattern was maintained for the other domains also. This finding has implications for the way in which the mathematics curriculum is introduced to children. It seems likely that children may benefit from concurrent learning opportunities in all number domains, and that experiences in one domain should not be delayed until a level of mathematical knowledge is constructed in another domain.
The Extending Mathematical Understanding (EMU) Intervention Program
Knowing how best to assist children who experience difficulty learning mathematics is often a dilemma for school communities. It seems that children sometimes need assistance beyond that which the classroom teacher can provide. One classroom teacher explained the dilemma she faced when teaching David (pseudonym):
[David] needs a particular kind of support, one on one and not a few minute grabs.He needs to have that support over a period of timeintensive and regular and frequent.I wasnt able to offer David that (in the classroom).
To assist children like David who experience difficulty learning mathematics, the Extending Mathematical Understanding (EMU) Program was developed and first implemented by specialist teachers in 24 ENRP trial schools (Clarke et al., 2002). The EMU program comprises daily 30 minute sessions for between 10 and 20 weeks, depending on the progress of students. Teachers work with groups of three students. The EMU program is not remedial in nature, but is built upon constructivist learning principles. Children are engaged in experiences that required hard thinking, and children are required to reflect upon their activity and articulate what they have learnt and how they have learnt.
The EMU Program provides different learning experiences for children than are possible within the classroom setting during mathematics lessons. In particular, the specialist teachers are trained to provide intensive instruction and feedback that is directed to the particular learning needs of each child. Observations of more than 30 EMU sessions in 2000 showed that, within each 30 minute session, children and teachers engaged in more than 100 interactions focused on the mathematical ideas investigated during a session. This level of interaction between the teacher and a child is not usually possible in a classroom setting. The specialist teachers were able to constantly focus childrens attention on key ideas and aspects of their experiences so that the children were attending to the aspects that would facilitate their construction of knowledge. They helped children develop the language that would facilitate communication about mathematics, and provided manipulatives to support childrens thinking at critical moments in their learning. Teachers constantly encouraged children to describe their thinking and strategies for solving problems, and provided wait time to encourage students thinking and ability to provide extended responses. The teachers assisted children build confidence so that they would take risks, and helped the children identify their strengths and build upon and reinforce these to increase the childrens self knowledge and self confidence.
The EMU program includes further diagnosis of individual difficulties using the EMU Assessment Interview, and provides learning experiences focusing on counting, place value, addition and subtraction and multiplication and division. These activities target individuals learning needs, require the maximum involvement of each child and emphasise communication and the sharing and demonstration of the different strategies used by group members. Typically, each EMU session is structured to include:
10 minutes of activities focusing on counting and place value,
15 minutes of rich learning activities focusing on problem solving (often with an addition and subtraction, or multiplication and division focus), and
5 minutes reflection about the key aspects that were covered in the session.
Case studies and analysis of childrens number learning have shown that the EMU Program is effective for accelerating Grade 1 and Grade 2 childrens number learning, and for increasing childrens confidence (Gervasoni, 2004). However, if a choice needed to be made between running the EMU Program in Grade 1 or Grade 2, the views expressed by the parents and teachers interviewed suggest that Grade 1 is more appropriate. Indeed, it seems preferable to provide children with assistance as early as possible, before they experience failure.
References
Clarke, D., Cheeseman, J., Gervasoni, A., Gronn, D., Horne, M., McDonough, A., Montgomery, P., Roche, A., Sullivan, P., Clarke, B., & Rowley, G. (2002). Early Numeracy Research Project Final Report. Melbourne: Australian Catholic University.
Ginsburg, H. P. (1997). Mathematical learning disabilities: A view from developmental psychology. Journal of Learning Disabilities, 30(1), 20-33.
Gervasoni, A. (2004). Exploring an intervention strategy for six and seven year old children who are vulnerable in learning school mathematics. Unpublished PhD thesis, La Trobe University, Bundoora.
Gervasoni, A. (2005). The diverse learning needs of young children who were selected for an intervention program. In H. Chick & J. Vincent (Eds.), Learners and learning environments (Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Volume 3, pp. 33-40). Melbourne: University of Melbourne.
Rivera, D. P. (1997). Mathematics education and students with learning disabilities: Introduction to the special series. Journal of Learning Disabilities, 30(1), 2-19.
Wright, R., Martland, J., & Stafford, A. (2000). Early Numeracy: Assessment for teaching and intervention. London: Paul Chapman Publishing.
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