The evaluation of the Numeracy Exploratory Study at Year 9 in secondary schools in 2001 showed that a programme for assessing, teaching, and reassessing numeracy was both necessary and effective for this older age group. Students had unexpected difficulties with multiplicative concepts in particular. After a carefully graded assessment, related to specific teaching suggestions for meeting students’ needs, teachers took different steps to address these needs.

# Do Year 9 students need help with fractions and multiplying?^{1}

**Kathryn C. Irwin and Kate Niederer** University of Auckland

**The Numeracy Exploratory Study 2001**

In 2001 the New Zealand Numeracy Project, previously implemented in Years 1–3 and in Years 4–6, was extended to Years 7–10 in a limited number of schools. This numeracy project has, as its main aim, improving the numeracy skills of New Zealand school students. The project’s main components included professional development, provided by a facilitator, for teachers, and individual assessment of all students. The project emphasised both knowledge of the number system, and strategies related to addition/subtraction, multiplication/division and ratio/proportion.

The focus of this article is on one aspect of the project, the Numeracy Exploratory Study 2001, involving improvement of multiplicative strategies used by Year 9 students. The results reported here come from the initial and final assessment of students in the 10 secondary schools that participated in this project. They were not a random selection of schools, but included Decile 1, 2, 3, 4, 8, and 9 schools in five areas of New Zealand. Initial and final results were available for 1451 Year 9 students in these 10 schools.

**Assessment**

Students were assessed individually at the start and end of the project. They were assessed on their understanding of whole numbers, fractions, and the base-10 framework of our number system. They were given problems that required additive, multiplicative or proportional reasoning, and assessed on the mental strategies that they used to solve these problems.

The assessment questions were similar to those used in the Advanced Numeracy Programme. They are currently available in the Diagnostic Interview, which can be viewed on the nzmaths website.^{2}

An example of a question requiring multiplicative thinking is:

18 × 6 = 108, so what does 19 × 6 = ?

An example requiring proportional thinking is:

Of every 9 apples in a box, 2 of them are bad.

There are 45 apples in the box, how many are bad?

**Implementation**

All teachers were guided by a facilitator. This facilitator gave workshops on:

•&&the nature of the assessment and its underlying theoretical framework

•&&how to carry out the assessment

•&&how students could be grouped on the basis of their assessment results

•&&how particular topics might be taught.

In some cases, intermediate and secondary school teachers attended workshops together, exchanging ideas; in other cases, workshops were held in individual schools for their staff only. Teachers were also provided with a handbook of activities that had been tried in primary schools.

In this exploratory year, all teachers gave the standard individual assessment, for which they were released from teaching; but they were also encouraged to experiment with different ways of providing the teaching needed to enable students to advance on the Number Framework of the Numeracy Project (see Young-Loveridge, this issue). The one factor that all the Year 9 teachers had in common was their new knowledge, gained in individual interviews, of the skills and strategies that their students could use.

The study took place in the third term of the school year. The timing exacerbated differences in how teaching was implemented. Many secondary schools had year-long schemes for which mathematical topics were to be taught when, and did not alter these schemes. For most of these schools, numeracy had been taught in the first term. Teachers felt that they could not abandon other topics in the syllabus in the third term, so they found ways of including some teaching of numeracy around the edges of their other topics.

One way of implementing the goals of the Numeracy Project was through the introduction of “starters” that require students to apply mental strategies. Many of these starters came from the books of McIntosh, Reys and Reys (1997). Other schools worked on numeracy for one or two days a week, using or adapting suggestions from their facilitators and the workbook provided, but continued with their existing scheme on the other days. One school worked solely on numeracy for 11 weeks, and spent many hours preparing worksheets, providing three different levels of work for separate groups in each class. Despite this diversity, the final assessment in all schools showed that students had improved in both their knowledge and their strategies.

**Student and staff responses**

None of us were sure how secondary students would react to this numeracy project, which was initially developed for younger students. Older students could be expected to have a wide variety of experiences that might have led them to develop attitudes and practices that were hard to alter. If they had not previously seen themselves as successful in mathematics, they might be unwilling to engage with a different mathematics programme. In particular, older students have had years of practice with algorithms for calculation, and might not value mental computation.

We need not have worried. Teachers and facilitators were generally enthusiastic about the project. Examples of common responses from staff in the 10 secondary schools involved were:

This was the first time in my 20 years of teaching that I have had a chance to talk to each Year 9 student for 20 minutes. It was an eye-opener.

If nothing else happens in the whole project, the assessment in itself would be worthwhile.

We knew that our students had difficulty with numeracy, but we were shocked to find how many did not know how to read fractions or large numbers.

And, importantly,

We were very impressed with the progress they made once we planned lessons to meet their needs.

Principals and boards of trustees were also pleased, as were parents, who had the project explained to them.

The views of students were not assessed independently, but one head of department reported that while these Year 9 students had not become angels, they seemed to be happier in mathematics class. Several teachers reported that their poorer students, in particular, responded positively. These students enjoyed having mathematics tasks that they could do.

**Results of initial and final assessment**

There was improvement on all six of the scales in the assessment (see the report to the Ministry of Education, Irwin and Niederer, 2002). This article focuses on improvement in use of multiplicative (multiplication and division) strategies, as shown on three of those scales. All three scales required multiplicative thinking. These scales assessed students’ initial and final abilities in knowledge and ordering of fractions (Knowledge of Fractions); strategies for solving multiplication and division problems (Multiplicative Strategies); and strategies for solving ratio and proportion problems that usually involve two multiplicative processes (Ratio Strategies). Students had unexpected weakness in doing problems requiring multiplicative thinking.

**Knowledge of Fractions**

The lowest scored stage on Knowledge of Fractions was Stage 5. This was defined as being able to read 1/2, 1/4 and 1/3. On the initial test, 10 percent of Year 9 students could not read these fractions, so fell below this level. A further 40 percent could read these fractions, but could not order them or order decimals given to two places, as required for Stage 6. Thus half of the Year 9 students involved in the study could not order unit fractions and decimals to two places when first assessed. Many of the teachers interviewed stated that they had not realised how low their students’ level of understanding of fractions and decimals was.

This knowledge appeared to be relatively easy to teach to this age group, once the need to teach it was appreciated. On the final assessment, 24 percent of students were still at these low levels, but 76 percent were assessed as being at Stage 6 or higher, reading and ordering increasingly complex fractions, decimals and percents. Overall, students had made sound progress within the space of a term, but there was still concern that more students did not achieve at the top level of this test. Only 23 percent were able to reach Stage 8, ordering any fractions, decimals or percents.^{3}

It is worth noting that fractions, decimals and percents are domains that many adults also find difficult. It may be that the concepts underlying the multiplicative nature of this domain have never been clearly understood by many people (see Tirosh and Graeber, 1990).

There was a marked difference in the modal stages reached by students in the lower and upper decile groups of schools, for both the initial and the final tests. Students from the lower decile schools moved from an initial modal Stage 5 (reading unit fractions) to a final modal Stage 6 (ordering unit fractions and decimals to two places). In contrast, students from the higher decile schools moved from an initial modal Stage 6 to a final modal Stage 8 (ordering fractions with different numerators and denominators, together with percents and any decimal fractions). Overall, final results for students from the lower decile schools were similar to initial results for students from the higher decile schools.

Grouped scores hide individual differences in increased understanding. Of the students who were not at the top stage initially, 48 percent of the students from lower decile schools and 44 percent of the students from the upper decile schools gained one, two or three stages.

**Multiplicative Strategies**

Multiplication is introduced to students through skip counting and repeated addition. Many Year 9 students still used these counting and addition strategies for multiplication problems, and only 50 percent used multiplicative strategies (Stage 6 and 7). On the initial test of multiplicative strategies, 4 percent of the students either counted by ones or had no way of doing multiplicative problems mentally (Stage Nil), and 46 percent of students were found to be using skip counting (14 percent) or addition in order to solve multiplicative problems (Stages 4 and 5). A total of 14 percent of students at Stage 4 used skip counting and 32 percent of students at Stage 5 used repeated addition. After one term of teaching, 2 percent of students still counted by ones or had no way of doing the multiplication tasks, but the proportion using additive methods had fallen from 46 percent to 29 percent.

To be assessed at Stage 6 of the test, students needed to have a good grasp of their multiplication tables, and be able to use this knowledge to solve given multiplication or division problems, such as the first example above: “given that 18x 6 = 108, say what 19 × 6 equals”. Stage 7 required students to use their multiplication facts with a range of part-whole strategies. After a term’s teaching, 69 percent of students were assessed as being at Stage 6 or Stage 7.

Initial test results for students from the lower decile schools placed 53 percent of students at or below Stage 5, or using additive strategies at best, for multiplicative problems. In contrast, 35 percent of students from the upper decile schools were initially placed at this level.

After a term of teaching, 34 percent of students from the lower decile schools and 17 percent of students from upper decile schools were still not using multiplicative strategies (Stages 6 and 7). Conversely, 66 percent of students from lower decile schools and 83 percent of students from upper decile schools were using multiplicative strategies. A total of 40 percent of students from lower decile schools and 46 percent of students from upper decile schools gained one to four stages. As for knowledge of fractions, **final** results for students from lower decile schools tended to be similar to **initial** results for students from higher decile schools.

**Ratio Strategies**

This was the most difficult scale in the numeracy project, and the results reflected this. Ratio or proportional problems usually require two multiplication or division operations. To attain Stage 4, students had to find 1/3 of 24. Many teachers were shocked to find that 37 percent of Year 9 students could not do this on the initial test. After one term of teaching, this had been reduced to 20 percent of students. The proportion of Year 9 students who were initially unable to do this was higher than that for Year 8 students. It is interesting to note that between the initial and final tests, a slightly smaller percentage of students gained on this scale than on the other scales.

Stage 5 of this scale required students to find a fraction of a number using addition facts, while Stage 6 required students to find a fraction of a number using a range of addition and multiplication facts. Initially, 65 percent of the students were unable to solve ratio problems at Stage 6 or higher. On the final test, 51 percent of students were unable to operate at these top levels. It seems likely that both the understanding of what needs to be done on these problems, and the part-whole multiplicative skills that help in this task, take longer to learn than was available in the period of this study. Teachers may have focused on the prerequisite addition and multiplicative skills, and spent less time on ratio.

Separating the results for students from lower and higher decile schools highlights a stark contrast on this scale of the numeracy project: 43 percent of students from the lower decile schools initially failed to attain Stage 4 on this scale, compared with only 8 percent of students from the higher decile schools. This was the only scale where final test results for students from lower decile schools were not similar to initial test results for students from the higher decile schools. In terms of gains made, 40 percent of the students from lower decile schools and 46 percent of the students from the upper decile schools gained one to four stages. However, more students from upper decile schools reached the top stages in each of these scales.

**Summary**

The evaluation of this Numeracy Exploratory Study at Year 9 in secondary schools in 2001 showed that a programme for assessing, teaching, and reassessing numeracy was both necessary and effective for this older age group. The results of the initial assessment surprised and alarmed teachers and educational administrators. Yet the teachers reported that the initial assessment of each student was the single most useful aspect for them. It was a carefully graded assessment, related to specific teaching suggestions for meeting students’ needs. Once they were aware of students’ needs, the teachers took different steps to address them. All teachers and administrators who were interviewed said that they intended to use the programme in the following year, starting earlier in the year.

Students had unexpected difficulties with multiplicative concepts in particular. Teachers’ awareness of this, as a result of the initial assessment, alerted them to the fact that students found multiplicative knowledge and its application much more difficult to grasp than they had expected. Some aspects, such as knowledge and ordering of fractions and decimals, were relatively easy to teach. Other strategies, such as the double multiplicative processes involved in solving most ratio and proportion problems, were less easy to influence, although less time may have been spent on this domain.

Many reports on adult numeracy indicate poor knowledge of multiplicative concepts. It may be that many individuals never learn to use these concepts adequately.

The gains made by students in both upper and lower decile schools demonstrate that appropriate teaching can bring up the skills of students in lower decile schools, but that with intervention, the students in upper decile schools move rapidly ahead. We speculate that students in these upper decile schools already have a better underlying grasp of the fundamentals of numeracy, often called “number sense”. This number sense is likely to result from thinking about numbers, rather than concentrating on algorithmic skills as an end in themselves. Although we do not have enough evidence to support this speculation, it would be in line with other writing on the topic, which suggests that students are disadvantaged by an early emphasis on algorithms before they have a sound understanding of the additive and multiplicative composition of number (e.g. Carpenter, Franke, Jacobs, Fennema, and Epson, 1988).

In this project, understanding of the additive and multiplicative composition of number is called part-whole thinking. This exploratory research showed that Year 9 students still needed to develop this part-whole thinking and benefited from teachers helping them to do so. (For an example of a successful programme that teaches part-whole thinking skills to young students from a lower socio-economic area, see Resnick et al., 1991.) We suspect that the move to multiplicative part-whole thinking is not only more difficult than the move to part-whole thinking for younger students, but is also just as important.

**References**

Carpenter T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Epson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. *Journal for Research in Mathematics Education*, 29, pp.3–20.

Irwin, K. C., & Niederer, K. (2002). *An evaluation of the national Numeracy Exploratory Study (NEST) and the associated Numeracy Exploratory Study Assessment (NESTA) for Years 7–10: Final report prepared for the New Zealand Ministry of Education*. Wellington: Ministry of Education.

McIntosh, A., Reys. B. & Reys, R. (1997). *NumberSENSE Grades 6–8*. Parsippany NJ: Pearson Educational Inc. (Books for other levels are also available.)

Resnick, L. B., Bill, V. L., Lesgold, S. B. & Leer, M. L. (1991). Thinking in arithmetic class. In B. Means, C. Chelemer & M. S. Knapp (eds), *Teaching advanced skills to at-risk students*. San Francisco: Jossey-Bass, pp. 27–53.

Tirosh, D., & Graeber, A. O. (1990). Inconsistencies in preservice elementary teachers’ beliefs about multiplication and division. *Focus on Learning Problems in Mathematics* 12, pp.65–74.

**Notes**

1&&&We would like to thank the Ministry of Education for funding this research.

2&&&http://www.nzmaths.co.nz/Numeracy/project_material.htm

3&&&A useful resource is http://online.edfac.unimelb.edu.au/485129/DecProj/sources/welcome.htm

**KATHRYN IRWIN** (Kay) both teaches and researches in the field of mathematics education. She is a Senior Lecturer at the School of Education, University of Auckland. Her previous research has included using context to enhance understanding of decimals, and a professional development project, Teachers Researching Achievement in Mathematics (TRAM).

Email:k.irwin@auckland.ac.nz

**KATE NIEDERER** is the Deputy Director of the George Parkyn National Centre for Gifted Education. She has taught in New Zealand and England, with special interest and expertise in the teaching of mathematics and gifted education.