Comparison between pre-test and post-test

In the literature review, there was an overview of problem-solving and the use of mathematical problems. It was explained that problem-solving is “a process in which the learner combines previously learned elements of knowledge, rules, techniques, skills and concepts to provide a solution to a situation not encountered before” (Orton, 2004:24). A pre-test and post-test were conducted at the beginning and end of the data collection period. The purpose of doing so was to gauge the levels of problem-solving strategies learners used before and after implementing different types of writing tasks.

Table 5.1: Model for Stages of Early Arithmetic Learning (SEAL) (Wright, Martland, Stafford and

Table 5.2: Model for early multiplication and division levels (Wright, Martland, Stafford & Stanger,

2006:14)

As mentioned earlier in this chapter, the stages and levels of the different aspects of LFIN (Tables 5.1 and 5.2) provided clarity and differentiation between the strategies learners used. The results of the analysis are in tabular form below where selected learners’ strategies are listed.

Table 5.3 Analysis of pre-test and post-test strategies

LEARNER 1 (AA)

PROBLEM PRE-TEST POST-TEST

1

SEAL – Facile number sequence (breaks down into tens and ones and adds separately)

SEAL – Facile number sequence (decomposes)

2 EMD – Initial grouping

(drawing shows quotitive sharing)

EMD – Figurative composite grouping

(skip counting)

3

SEAL – Facile number sequence (breaks down tens and ones and adds separately – incorrect answer)

SEAL – Facile number sequence

4 EMD – Repeated abstract composite grouping

EMD – Figurative composite grouping

(quotitive sharing incorporating skip counting)

5

SEAL – Facile number sequence (breaks down into tens and ones and adds)

EMD – Repeated abstract composite grouping

LEARNER 2 (AA)

PROBLEM PRE-TEST POST-TEST

1 SEAL – Initial number sequence SEAL – Facile number sequence (decomposed, added separately)

2

EMD – Initial grouping

(drawing shows quotitive sharing – incorrect answer)

EMD – Initial grouping (quotitive sharing – incorrect answer)

3 SEAL – Perceptual counting CPV – increment by tens off decuple

4 EMD – Initial grouping (quotitive _{sharing – incorrect strategy)}

EMD – Repeated abstract composite grouping (repeated addition)

5 SEAL – Initial number sequence EMD – Perceptual counting in multiples

LEARNER 3 (A)

PROBLEM PRE-TEST POST-TEST

1 SEAL – Perceptual counting SEAL – Facile number sequence (only solved 1 part)

2

EMD – Initial grouping

(drawing shows quotitive sharing – adds incorrectly)

EMD – Repeated abstract composite grouping (attempt) (skip counts incorrectly) 3 SEAL – Initial number sequence

(incorrect answer) No clear strategy

4 SEAL – Facile number sequence (incorrect answer)

EMD – Repeated abstract composite grouping (incorrect answer) 5 SEAL – Initial number sequence

(incorrect answer)

SEAL – Intermediate number sequence (incorrect strategy)

LEARNER 4 (A)

PROBLEM PRE-TEST POST-TEST

1 SEAL – Perceptual counting SEAL – intermediate number sequence (incomplete strategy)

2 Combined EMD – initial grouping and SEAL – Perceptual counting

EMD – Repeated abstract composite grouping (repeated addition)

3 SEAL – Facile number sequence SEAL – Facile number sequence

4 EMD – Perceptual counting (counts all)

EMD – Repeated abstract composite grouping

(repeated addition – incorrect answer)

5 Absent EMD – Initial grouping

(quotitive sharing)

LEARNER 5 (A)

PROBLEM PRE-TEST POST-TEST

1 SEAL – Perceptual counting SEAL – Facile number sequence (decomposed – incorrect answer)

2 EMD – Initial grouping (quotitive sharing)

EMD – Repeated abstract composite grouping

3

SEAL – Combines Perceptual counting and Facile number sequence

(incorrect answer)

SEAL – Facile number sequence

4

EMD – Perceptual counting in multiples

(incorrect answer)

EMD – Repeated abstract composite grouping (repeated addition)

5 SEAL – Perceptual counting

EMD – Repeated abstract composite grouping (repeated addition)

LEARNER 6 (BA)

PROBLEM PRE-TEST POST-TEST

1 SEAL – Initial number sequence SEAL – Perceptual counting

2 Strategy didn’t fit problem type EMD – Initial grouping (quotitive sharing)

3 SEAL – Intermediate number sequence

SEAL – Perceptual counting (added instead of subtracting)

4

EMD – Initial grouping (partitive sharing – incorrect strategy)

EMD – Perceptual counting in multiples

5 SEAL – Initial number sequence (used strategy incorrectly)

EMD – Initial grouping (quotitive sharing)

LEARNER 7 (BA)

PROBLEM PRE-TEST POST-TEST

1 SEAL – Perceptual counting SEAL – Facile number sequence (didn’t complete)

2 Strategy didn’t fit problem type SEAL – Facile number sequence (correct strategy used incorrectly) 3 SEAL – Perceptual counting

(incorrect answer) SEAL – Initial number sequence

4 Absent EMD – Figurative composite

grouping

LEARNER 8 (BA)

PROBLEM PRE-TEST POST-TEST

1 SEAL – Perceptual counting

(used strategy incorrectly) No clear strategy

2

EMD – Initial grouping

(quotitive sharing – used strategy incorrectly)

EMD – Initial grouping (quotitive sharing)

3 SEAL – Facile number sequence (counting erased)

SEAL – Intermediate number sequence

(incorrect answer)

4 EMD – Initial grouping

(number sentence didn’t match)

EMD – Figurative composite grouping

(skip counting – didn’t answer problem)

5 SEAL – perceptual counting (strategy used incorrectly)

EMD – Initial grouping

(partitive sharing – used strategy incorrectly, incorrect answer)

Many learners in the selected Grade 3 class were restricted in their use of mathematical problem-solving strategies in the pre-test. Their strategies often reflected lower stages and levels of different aspects of the LFIN. Tallies were frequently used as a strategy in the pre- test and the earlier part of the writing intervention. At this stage learners were not expected to describe their thinking although they had solved mathematical problems prior to this study.

Two learners from the above average ability group already showed strategies that were more advanced during the pre-test when compared to the other learners. When these strategies were compared to those in the post-test, these learners displayed strategies at higher stages and levels where there was evidence of enriched writing to explain their strategies. For example, Learner 2 (AA) usually solved problems in the pre-test at stage 4 of the SEAL (initial number sequence) and level 1 of early multiplication and division (initial grouping) as shown in Figure 5.7 below.

Figure 5.7 Learner 2 (AA) Pre Q2

Figure 5.8 Learner 2 (AA) Post Q3 Figure 5.9 Learner 2 (AA) Post Q5

Figure 5.8 and Figure 5.9 show that the same learner solved similar problems at stage 5 of the SEAL (facile number sequence) and levels 2 and 4 of early multiplication and division (perceptual counting in multiples and repeated abstract composite grouping respectively). For one of the problems in the post-test (Figure 5.8), there was evidence of Learner 2’s strategy at level 2 of conceptual place value (incrementing by tens off the decuple). This showed that Learner 2 used increasingly sophisticated strategies when the results of the pre- test and the post-test were compared: there was no evidence of strategies of this nature in the pre-test.

Similar results were apparent in strategies used by the average and below average ability groups. There was a marked difference in the strategies Learner 5 (A) used in the post-test when compared to the pre-test. This distinction can be seen in Figure 5.10 and Figure 5.11 below. During the pre-test, his strategies were generally at level 1 of the SEAL (perceptual

counting) and there was evidence of level 2 of early multiplication and division (perceptual counting in multiples). In the post-test he solved problems at the highest stage of the SEAL (facile number sequence) and at level 4 of early multiplication and division (repeated abstract composite grouping).

Figure 5.10 Learner 5 (A) Pre Q3 Figure 5.11 Learner 5 (A) Post Q4

Writing improved problem-solving strategies of learners in the below average ability group. Learner 7, for instance, used basic strategies in the pre-test at level 1 of the SEAL (perceptual counting) to solve two of the problems. Figure 5.12 below is an example of the strategies he used. The remaining problems did not have a visible strategy or the strategy used did not match the problem type.

Figure 5.12 Learner 7 (BA) Pre Q2 Figure 5.13 Learner 7 (BA) Post Q4

The post-test reflected a significant improvement in the strategies used to solve problems. Use of writing was evident to explain how he solved the problems. The more complex strategies reflected in the post-test were at stage 5 of the SEAL (facile number sequence) and level 3 of early multiplication and division (figurative composite grouping). Learner 7’s (BA) strategy and explanation in Figure 5.13 is an example of how he used figurative composite grouping by applying his conceptual knowledge of counting in fives. He realised that he needed to subtract one in order to answer the problem correctly.

The remainder of the selected learners displayed similar tendencies when comparing the strategies used to solve the problems in the pre-test and the post-test. Throughout the data collection period, learners were encouraged to write to explain how they solved mathematical problems. Writing in this way enhanced their problem-solving strategies: they considered their strategies in detail in order to write their explanations. Some learners used mathematical language in their explanations which showed that they were able to link elements of their strategies with particular concepts they had learned previously. For example, terms such as double and decompose were used, which some learners referred to as breaking down (Figure 5.11). This usage was an example of how they used their mathematical knowledge to enhance their strategies when solving problems. This phenomenon related to Sfard’s theory of the process and object of a mathematical idea where learners could apply existing mathematical knowledge and vocabulary to the process of problem-solving. As explained in Chapter 2, the process, or operational conception, is the dynamic action where an idea is conceived at a lower level and the object, or structural conception, is conceived at higher levels that underlie relational understanding (Sfard, 1991:16).

Mathematical problem-solving requires applying existing knowledge of mathematical ideas (objects) as well as the conception and development of new ideas (process). Learners’ written explanations became more detailed in the post-test, reflecting mathematical knowledge and vocabulary. This observation suggests that concepts taught in mathematical lessons were being connected to problems being solved. Learners engaged in, and used, processes and objects of their mathematical ideas in order to find solutions.

In this study, learners used writing to solve and explain mathematical problems. When they encountered problems, either individually or corporately, learners appeared to use strategic thinking to determine how to arrive at solutions. Learners drew on their existing mathematical knowledge and applied it to their strategies. At times, mathematical problems required a reconstruction of mathematical knowledge: learners developed further invented strategies by adding to, or combining, existing mathematical ideas (Campbell et al., 1998).

In document Writing and mathematical problem-solving in grade 3 (Page 132-141)