MATHEMATICS INTERVENTION:

THE IDENTIFICATION OF YEAR 1 STUDENTS MATHEMATICALLY “AT RISK”.

Catherine Pearn

Australian Council for Educational Research & La Trobe University

pearn@acer.edu.au

In this paper I discuss Mathematics Intervention, a program established for students "at risk" of not succeeding with Year 1 mathematics. The program is based on current research that shows that students become numerate by progressing through five counting stages. The importance for classroom teachers to be able to identify each student's strategies and thus their counting stage will be stressed as a starting point for numeracy teaching in the early years. The presentation will highlight those strategies used in the intervention program that can be modified for classroom teachers to incorporate into their mathematics program.

Objectives

Mathematics Intervention is an ongoing research project involving Principal and staff of a state elementary school in the eastern suburbs of Melbourne, and a mathematics educator from a nearby university (Pearn & Merrifield, 1996). Developed in 1993 Mathematics Intervention was designed to identify, then assist, Year 1 students "at risk" of not coping with the mathematics curriculum as documented in the National Statement on Mathematics for Australian Schools (Australian Education Council, 1991). Mathematics Intervention features elements of both Reading Recovery (Clay, 1987) and Mathematics Recovery (Wright, 1991) and offers students the chance to experience success in mathematics by developing the basic concepts of number upon which they build their understanding of mathematics. Students are withdrawn from their classes and work in small groups with a trained specialist teacher. This paper focuses on the results of clinical interviews conducted to identify Year 1 students considered “at risk” and needing to participate in the Mathematics Intervention program.

Theoretical Framework

The theoretical framework underpinning Mathematics Intervention is based on recent research about children's early arithmetical learning (Steffe, von Glasersfeld, Richards & Cobb, 1983; 1988; Wright, 1991) and about the types of strategies used by students to demonstrate their mathematical knowledge (Gray & Tall, 1994). In particular, the Mathematics Intervention program documents and promotes students' progression through the counting stages (Steffe et al. 1983, 1988) which are summarised below:

1. Perceptual. Students are limited to counting those items they can perceive.

2. Figurative. Students count from one when solving addition problems with screened collections. They appear to visualise the items and all movements are important. (Often typified by the hand waving over hidden objects.) If required to add two collections of six and three the student must first count the six items to understand the meaning of "six", then count the three items, then count the whole collection of six and three.

3. Initial number sequence. Students can now count on to solve addition and missing addend problems with screened collections. They no longer count from one but begin from the appropriate number. If adding two collections of six and three, students commence the count at six and then count on: six, seven, eight, nine.

4. Implicitly nested number sequence. Students are able to focus on the collection of unit items as one thing, as well as the abstract unit items. They can ‘count-on’ and ‘count-down’, choosing the most appropriate counting strategy to solve problems. They generally ‘count down’ to solve subtraction.

5. Explicitly nested number sequence. Students are simultaneously aware of two number sequences and can disembed smaller composite units from the composite unit that contains it, and then compare them. They understand that addition and subtraction are inverse operations.

Gray and Tall (1994) have shown that young students who were successful with mathematics use different types of strategies from those who were struggling with mathematics. Students struggling with mathematics, were usually procedural thinkers dependent on the procedure of counting and limited to strategies such as "count-all”. Gray and Tall (1994) defined procedural thinking as when:

... the numbers are used only as concrete entities to be manipulated through a counting process. The emphasis on the procedure reduces the focus on the relationship between input and output, often leading to idiosyncratic extensions of the counting procedure that may not generalize (p. 132).

For example, when asked to give the number before a given number, students were heard to count up to each number before responding. While some students were dependent on rules and procedures other students gave instantaneous answers. When asked: "How did you do that?" they usually gave several different strategies they could have used and checked that their solutions were correct. According to Gray and Tall (1994) this use of known facts and procedures to solve problems, along with a combination of both conceptual and procedural thinking, indicated that these students were proceptual thinkers. Gray and Tall (1994) defined proceptual thinking as:

... the flexible facility to ... enable(s) a symbol to be maintained in short-term memory in a compact form for mental manipulation or to trigger a sequence of actions in time to carry out a mental process. It includes both concepts to know and processes to do (pp. 124-125).

Methods

The initial assessment for the Year 1 Mathematics Intervention program required teachers to assess the extent of the student's mathematical knowledge by observing and interpreting the student's actions as he/she worked on a set task. Researchers have advocated encouraging students to talk about their mathematical strategies as the superior method of obtaining information on students’ own mathematical constructs and knowledge (see for example, Peck, Jenks & Connell, 1989).

A clinical interview protocol was developed, administered and consequently modified by three teachers. This is called the Initial Clinical Assessment Procedure-Mathematics [ICAPM] -Level AA (Pearn, Merrifield, Mihalic, & Hunting, 1994). By carefully observing the student's solution methods, interviewers ensured that they were aware of the strategies being used and if required the following prompts were given: "How did you work that out?" or "How did you do that?"

Data sources

Since 1993, 357 Year 1 students have been clinically interviewed at the beginning of their second year at school. Teacher-clinicians used the ICAPM -Level AA protocol (Pearn, Merrifield, Mihalic, & Hunting, 1994). Each clinical interview took approximately ten minutes and included tasks that ascertained students' verbal counting skills, their knowledge of the number word sequence and tasks that helped ascertain their counting stage level. For example, the verbal counting tasks included:

"Can you count out loud for me, beginning at one, until I tell you to stop?"

"What number comes after 4?"

"What number comes before 15?"

In a counting stage task, six counters were displayed and three other counters were hidden under paper:

"There are six counters on the table. Can you count them?"

"Under this paper there are three counters." (Lift paper briefly)

"How many counters do I have altogether?"

The results from these clinical interviews were recorded by the interviewers and have been collated and analysed by the researcher.

Results

The clinical interview results shown in Table 1 indicate that most Year 1 students were successful counting forwards by ones to 20 and backwards by ones from ten, counted patterns of dots and counted out exactly 14 beads. They were less successful identifying the numbers between the numbers six and twelve or determining numbers "before" or "after" a given number.

Table 1: Year 1 clinical interview results (in percentages).

Year	ones	back 20-1	back 10-1	twos	fives	tens	6 -12	before/after	14 beads	pattern	numeral	6+3=	10= +2
1993-1999	97	60	93	38	41	64	62	77	86	83	41	76	62

Year 1 students considered mathematically "at risk" and in need of additional assistance in the Mathematics Intervention program generally exhibited the following characteristics:

1. Students had difficulties in elaborating the number sequence. For example, they:

· used the right words but in the wrong order: one, two, three, four, six, five.

· omitted a number: one, two, three, five, six.

· confused two number sequences when counting by ones: 10, 11, 12, 30, 40, 50, 60, 70, 80, 90, 20

· experienced difficulty in counting backwards from 20. This was hardly surprising as in most cases students’ forward counting sequence was not accurate.

2. Students exhibited little or no one-to-one correspondence. Their verbal number sequence was not consciously co-ordinated with the actual counting of objects.

3. Students exhibited confusion with place-value concepts. For example, 13 and 31 were both considered ‘thirteen’ as they had “a three and a one”.

4. Students were also receiving additional support for reading, that is, Reading Recovery. However, not all students receiving Reading Recovery needed assistance for mathematics.

5. Students generally had poor language skills. When asked to explain their response to a task their explanations would include expressions such as: ‘I knowed.’ ‘It was in my brain.’

6. If unsure of a response students guessed with no check as to whether the response was sensible or logical.

The interviews highlighted the differences in students’ mathematical knowledge and the types of whole number strategies they used when solving tasks in different contexts. Successful Year 1 students counted fluently by ones, twos, fives and tens from a given number, and demonstrated their ability to choose and use an efficient and appropriate strategy. These students appeared to exhibit proceptual thought (Gray & Tall, 1994) and were articulate in their responses. Year 1 students requiring Mathematics Intervention experienced difficulties with the verbal counting sequence and were at either Counting Stage 0, or 1. When unsure of an answer these students guessed with no attempt to confirm their answer. All students’ answers used minimum words and highlighted poor language skills.

Conclusion

The importance of providing additional assistance as early as possible to students "mathematically at risk" cannot be over-emphasised. There will always be a need for a program such as Mathematics Intervention that is specifically designed to cater for students "at risk" in the early years of schooling. By being aware of each student's mathematical knowledge, types of strategies usually used, and language ability, the teacher should be able to design appropriate activities to extend each student’s mathematical understanding and language.

This paper reports findings from an Australian research study designed to further a deeper and better understanding of the psychological aspects of teaching and learning mathematics of young students in their early years of schooling. The research builds on previous international research in psychology of mathematics education and implications for teaching and learning. The findings that show some students required assistance in both mathematics and reading highlights a need to promote and stimulate interdisciplinary research in the area of young students mathematically “at risk” with the cooperation of psychologists, mathematicians and mathematics educators.

References

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.

Gray, E. M. & Tall, D. O. (1994). Duality, ambiguity, and flexibility: A "proceptual" view of simple arithmetic. Journal for Research in Mathematics Education. 25(2), 116-140.

Pearn, C. A., & Merrifield, M. (1996). Strategies for classroom teachers: A lesson from Mathematics Intervention. In H. Forgasz, A. Jones, G. Leder, J. Lynch, K. Maguire, & C. Pearn (Eds.), Mathematics: Making connections. Brunswick: Mathematical Association of Victoria.

Pearn C. A., Merrifield, M., Mihalic, H., & Hunting, R. P. (1994). Initial clinical assessment procedure, Mathematics - Level A A (Years 1 & 2). Bundoora: La Trobe University.

Peck, D. M., Jenks S. M., Connell, M. L. (1989). Improving instruction through brief interviews. Arithmetic Teacher, 37(3), 15-17.

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. & 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.