Catherine A. Pearn, Marguerite Merrifield
La Trobe University, Boroondara Park Primary School
Introduction and Background to the Program
There is some debate about the best way to implement programs designed to assist children "at risk" of not being able to fully participate in the regular class mathematics program. There are three ways that assistance may be given to low achieving students: in a withdrawal group, individually, or as a group or individual within the class setting.
One example of a withdrawal program is the Mathematics Intervention program which is a collaborative project involving the Principal and staff of Boroondara Park Primary and mathematics educators from La Trobe University (Pearn, 1994; Pearn, Merrifield & Mihalic, 1994a; Pearn, Merrifield & Mihalic, 1994b). This program aims to identify, then assist, those children in Years 1 at risk of not coping with the mathematics curriculum as documented in the National Statement on Mathematics for Australian Schools (Australian Education Council, 1991). The program features elements of both Reading Recovery (Clay, 1987) and Mathematics Recovery (Wright, 1991) and offers children the chance to experience success in mathematics by developing the basic concepts of number upon which they build their understanding of mathematics. Children are withdrawn from their classes and work in small groups to assist with the development of their mathematical language skills and co-operation strategies.
While individualised instruction by a qualified teacher is extremely beneficial to the student (see for example, Wright, 1996), such instruction does not take account of the common language difficulties identified by clinical interviews, used at Boroondara Park, to assess the children at the beginning of each year. We have noted several occasions where the student-student verbal interaction was more beneficial to the student than the teacher-student interaction. For extra assistance to be made available to students experiencing difficulties in the classroom the teacher has to develop skills to be able to recognise that the student has a problem, identify the problem, and have the expertise to assist the child overcome the problem.
Mathematics Intervention incorporates learning activities based on recent research about children's early arithmetical learning (Steffe, von Glasersfeld, Richards and Cobb, 1983; Wright, 1991). The counting stages as determined by Steffe and his colleagues (1983, 1988) are summarised below:
Stage 0. Preperceptual. When attempting to count, the child is unable to coordinate number words with the items being counted.
Stage 1. Perceptual. Children are limited to counting those items they can perceive, for example, see, feel and hear.
Stage 2. Figurative. Children count from one when solving addition problems with screened collections. They appear to visualise the items and all movements are important. (Often typified by hand waving over hidden objects).
Stage 3. Initial number sequence. Children count-on to solve addition and missing addend problems with screened collections. Children no longer count from one but begin from the appropriate number.
Stage 4. Implicitly nested number sequence. Children are able to focus on the collection of unit items. They can count-on and count-down, choosing the most appropriate to solve problems. They generally count-down to solve subtraction problems.
Stage 5. Explicitly nested number sequence. Children are simultaneously aware of two number sequences and can disembed smaller composite units from the composite unit that contains it, and then compare them. That is, they can conceptualise a whole (for example 12), a part (for example, 9) and the remainder (for example, 3) simultaneously. They understand that addition and subtraction are inverse operations. They use a variety of strategies other than counting by ones, for example: using a known result, adding to ten, and commutativity.
Wright and his colleagues (1996) have documented the results from more than 200 interviews of children aged between three and eight years. The results of these interviews were used to assess young children's arithmetical knowledge. Three conclusions arising from his research include:
* Reasonable educational goals for children are to reach Arithmetical Stage 2 or 3 at the end of the Prep year and Stage 5 at the end of Grade One.
* Only a very small percentage (probably less than 5%) of beginning first grade children have not attained at least Stage 2.
* Grade one children who begin at lower than Stage 3 are less likely to advance to Stage 5 than are children who begin at Stage 3 or 4.
Identification of Students "at risk"
Both the initial assessment and the Mathematics Intervention program require the teacher to observe and interpret the child's actions as he/she works on a set task. The initial interview requires the teacher to assess the extent of the child's mathematical knowledge while the intervention program relies on the teacher's ability to interpret the child's mathematical knowledge and then design or adapt tasks and problems that enable the child to progress mathematically.
All teachers involved with the Mathematics Intervention program have attended a course in Clinical Mathematics Methods at La Trobe University (Gibson, Doig & Hunting, 1993) to develop and refine their observational and interpretative skills.
Initial Clinical Interview
Analysis of the 1993 clinical interview results showed that children requiring Mathematics Intervention were experiencing difficulties with the verbal counting sequence; specifically in counting backwards, counting forwards by numbers larger than one and were at either Stage 1 or Stage 2 of the Counting Stages. This meant that an easier and shorter test could be administered to identify the children who needed to participate in a Mathematics Intervention program. The modified interview (Pearn, Merrifield, Mihalic & Hunting, 1995) included simplified verbal counting tasks. For example,
"Can you count out loud for me, beginning at one, until I tell you to stop?"
"Can you count forwards by 10's starting with 10?"
and only two tasks based on the counting stages. The first task was designed to determine whether the child could count-back.
Ten counters are displayed.
"Here are some counters. Count them."
(Cover all the counters, remove two and display).
"How many counters are under the paper?"
The second task was designed to determine the strategy the child used: guess, count-all, count-on, or "just know it". It is an easier task than the previous one and allows the child to leave the interview feeling positive.
Six counters are displayed and three hidden.
"There are six counters on the table. Can you count them?"
"Under this paper are three counters." (Lift paper briefly).
"How many counters do I have altogether?"
Two tasks were added to the earlier 1993 interview :
"Can you count out fourteen beads?"
Cards were shown of the following numbers and the children were asked to name them: 13, 31, 15, 51, 14, 41.
Over the last three years children from both Year 1 and 2 at Boroondara Park have been individually interviewed by either the teacher or researcher associated with the project. In 1994 Year 1 and 2 children participated in the program but in 1995 and 1996 the program has been restricted to Year 1 children. At the beginning of each school year children are clinically interviewed. By carefully observing the children's solution methods the teacher ensures that she is aware of the strategies being used and if needed the following prompts are given: "How did you work that out?" or "How did you do that?" The children enjoyed coming out of class and working on a one-to-one basis with a teacher.
One of the most significant findings from three years of testing has confirmed that there appears to be a link between children needing both Mathematics Intervention and Reading Recovery. There appears to be a need for further research and the necessity for a more integrated approach in teaching mathematics.
The Intervention Program
In 1996 children are withdrawn from their classes for seven half-hour sessions per fortnight with a maximum participation of twenty weeks. Emphasis is placed on the verbal interaction between teacher and students, and between students. Each session is planned to build on previous understandings as interpreted by the teacher during the session. Many games have been adapted to ensure that concepts are presented in an informal but engaging way. Each lesson includes:
* counting activities using concrete materials such as blocks, counters, bead frames, straws.
* games designed to highlight and correct a perceived weakness.
* oral work, using concrete materials.
* questions that expected the children to reflect on their strategies.
* the expectation that all children would explain their strategies and would listen when some-one else was explaining solutions and/or strategies
* a written activity
Classroom teachers have commented on the improvement in both the attitude and skills of the children in the program. The program depends on the teacher making an instant appraisal of the child's needs and providing the appropriate activities. This is in line with the National Statement (Australian Education Council, 1991): "Whatever their particular needs or abilities, all students have the right to learn mathematics in a way that is personally challenging and stretches their capabilities. Achievable and satisfying tasks are an important prerequisite for success" (p.10).
The major difference between the 1993 program and those of subsequent years is the greater emphasis on written work. This was added specifically in response to a request from classroom teachers and assists the transition back into the classroom Mathematics program.
While we believe that children experiencing difficulties with mathematics in the early years of schooling need to be withdrawn from the mainstream classroom for lessons with a teacher who has undertaken special training we acknowledge this is not always possible. An alternative approach is to incorporate specific strategies within the context of the mainstream classroom. Hopefully these strategies will be informed and determined by the research into special programs like Mathematics Intervention. Before deciding on appropriate strategies for use in classrooms we need to determine specific problems that are experienced by children needing further assistance in mathematics.
Over the last four years there have been common problems exhibited by children considered to be mathematically "at risk". These problems have been noted both in the assessment procedure and during the Mathematics Intervention lessons and include:
* difficulty elaborating the number sequence.
* difficulty in coordinating their spoken number sequence with the actual counting of objects.
* confusion with the "teen" words and "ty " words
* difficulty in counting backwards from 20.
* bridging of the decades.
* lack of understanding of the symbols.
Classroom Teaching Strategies
Experience with the Mathematics Intervention program has highlighted several strategies that could be used by classroom teachers that will allow all children to experience success with mathematics.
To facilitate the improvement of children's counting skills time must be spent each lesson counting both orally and with structured materials. For example, counting beads on a bead frame, collections of counters, beads, bears and in fact anything countable. Emphasis must also be placed on the pronunciation of the number words. Every year Mathematics Intervention teachers have observed that children experience difficulties with the number sequence due to poor speech especially with the "teen" and "ty" words. Quite frequently the mispronunciation had been missed by classroom teachers. As Fuson (1988) wrote,
... children's ability to say the correct sequence of number words is very strongly affected by the opportunity to learn and practice this sequence. Children within a given age group show considerable variability in the length of the correct sequence they can produce. Frequent exposure to "Sesame Street" or to parents, older siblings, or teachers who provide frequent counting practice undoubtedly enables a child to say longer accurate sequences at a younger age (p. 57).
To emphasise and reinforce the difference in numbers like seventeen and seventy, a memory game was introduced (see Figure 1). This game assisted with numerical recognition, and the children became very proficient in counting by twos to determine the number of cards they had won.
Teachers need to be skilled in questioning and able to ask mathematical questions using the correct mathematical language. Skilful questioning by the teacher is imperative to ensure that the children's mathematical knowledge can be used to form a strong foundation on which to build further mathematical knowledge. Children should be expected to explain their strategies to both the teacher and other students and where necessary prompts should be given such as: "How did you do that?"
Children should be encouraged to think of and discuss different ways each task could be solved. Teachers must refrain from saying whether answer is correct or incorrect or that one procedure is better than another. Teacher should encourage children to explain their solutions and to tell each other whether or not an explanation makes sense to them.
Young children will eventually construct the algorithms that are now prematurely imposed on them. By letting them change their minds only when they are convinced that another idea makes better sense, we encourage them to build a solid foundation that will enable them to go on constructing higher-level thinking (Kamii, 1990, p. 30).
To ensure active participation in the Intervention program, games are used wherever appropriate. The variety of the games depended on the imagination and skill of the teachers. This is another activity that can be used successfully in the classroom by classroom teachers.
Games are excellent activities because children play them to please themselves rather than the teacher. They are desirable because in games children care about sums, supervise each other, and give immediate feedback. ... Games are good also because the social interaction they require contributes greatly to children's social and moral development (Kamii, 1990, p. 29).
Games using dice are used to compare numbers, add and subtract numbers and to make up their own sums. It is this ownership of the mathematics that becomes a very powerful tool in learning. Different sized dice can be used depending on the child's ability. A game called Twenty was devised to assist children to make the transition from counting all the counters (Stage 1 and 2) to counting on (Stage 3), or "counting back" which are much more powerful strategies and are necessary if the child is to succeed with addition and subtraction.
Implications for Classroom Teachers
The importance of the Mathematics intervention program to students "mathematically at risk" cannot be over-emphasised. As stated by the National Statement (Australian Education Council, 1991): "Whether a particular student gains the full benefit from mathematics may be influenced by a range of personal characteristics and circumstances. It will also depend on the quality of the mathematics offered" (p.8). Steffe and his colleagues (Steffe et al., 1983; 1988) have indicated that 6 year-old children below Stage 3 of the counting stages may require up to two years to progress to Stage 5 and even then there is no guarantee that all children will attain this level. Considered in this light the results achieved by children in a quarter of that time are a positive indication of the viability of the Mathematics Intervention program.
Strategies used by teachers in the Mathematics Intervention Program are transferable to classroom teachers. However, no matter how effectively a teacher uses these strategies, there will always be a need for a program such as Mathematics Intervention which is specifically designed to cater for those children who are "at risk". Mathematics Intervention teachers need to be confident and competent in mathematics and need to share their knowledge of these special students with the classroom teacher. Both class teacher and Intervention teacher need to be aware of the child's knowledge and strategies and able to design appropriate activities to extend their mathematical understanding together.
With the increase in Victorian class sizes, teachers are going to have even less time to spend with these children who are "at risk". If children are unable to count accurately, it will be difficult for them to succeed with other mathematical problems and processes. A clinically trained mathematics teacher, working with a small group of children of similar mathematical ability, is more likely to observe the difficulties experienced by these children and be able to work towards strengthening their basic numerical concepts.
Australian Education Council (1991). A National Statement on Mathematics for Australian Schools. Carlton: Curriculum Corporation.
Clay, M. M. (1987). Implementing Reading Recovery: Systematic adaptations to an educational innovation. New Zealand Journal of Educational Studies, 22 (1), 35-58
Directorate of School Education (1992). Mathematics Course Advice -- Primary. Melbourne: Author.
Fuson, K. (1988). Children's Counting and Concepts of Number. New York: Springer-Verlag.
Gibson, S. J., Doig, B. A., & Hunting, R. P. (1993). Inside their heads -- the clinical interview in the classroom. In J. Mousley & M. Rice (Eds.), Mathematics: Of primary importance (pp. 30-35). Brunswick: Mathematical Association of Victoria.
Kamii, C. (1990).Constructivism and beginning arithmetic. In T. J. Cooney & C. R. Hirsch (Eds) Teaching and learning mathematics in the 1990s. Reston, VA: The National Council of Teachers of Mathematics.
Pearn, C. A. (1994). A connection between mathematics and language development in early mathematics. In G. Bell, R. Wright, N. Leeson, & J. Geake (Eds.),Challenges in mathematics education: Constraints on construction (Vol 2, pp. 463-470) Lismore, NSW: Southern Cross University.
Pearn, C. A., Merrifield, M.,& Mihalic, H. (1994a) Mathematics Intervention: A Pilot Program in Mathematics Recovery Education. Paper presented at the Emilio Reggio Conference, Melbourne University. September, 1994.
Pearn, C. A., Merrifield, M., & Mihalic, H., (1994b). Intensive strategies with young children: A mathematics intervention program. In D. Rasmussen & K. Beesey (Eds), Mathematics without limits. Brunswick: Mathematical Association of Victoria.
Steffe, L. P., Von Glasersfeld, E., Richards, J.& Cobb, P. (1983). Children's counting types: Philosophy, theory, and application. New York: Praeger.
Steffe, L. P., Cobb, P., and von Glasersfeld, E. (1988). Construction of arithmetical meanings and strategies. New York: Springer-Verlag.
Wright, R. J. (1991). The role of counting in children's numerical development. The Australian Journal of Early Childhood, 16 (2), 43-48.
Wright, R.J., Stanger, G. Cowper, M. & Dyson, R. (1996). First-graders' progress in an experimental mathematics recovery program. In J. Mulligan & M. Mitchelmore (Eds.), Children's number learning(pp. ). Adelaide: The Australian Association of Mathematics Teachers.
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