Using writing to develop conceptual understanding

Research sub-question 1 of the study addresses the support writing tasks give to the development of conceptual understanding. When learners’ initial and later use of writing was analysed and compared, it was evident they could provide more detail and refer to distinct aspects of mathematical content when they solved problems. Figure 5.1 and Figure 5.2 are examples of Learner 5’s (A) solutions to problems in the pre-test and the post-test. Figure 5.1 shows how he was able to solve the problem but did not use mathematical vocabulary to explain his strategy. In Figure 5.2, there is evidence of a detailed explanation using specific mathematical ideas although he arrived at the incorrect answer.

Figure 5.1 Learner 5 (A) Pre Q2 Figure 5.2 Learner 5 (A) Post Q1

In the literature review, conceptual understanding within problem-solving was discussed. O’Donnell (2006:349) states that problem-solving needs to encourage a higher cognitive demand where the mathematical content embedded in the problem may not be obvious to the learner. The problems presented throughout the data collection needed to encourage critical thinking and develop conceptual understanding.

Moreover, Orton and Frobisher (1996:23) suggest that “problem-solving shifts the weight from the acquisition of knowledge and skills to using and applying them”. Solving mathematical problems should encourage learners to move beyond the use of procedural knowledge and develop their own conceptual knowledge. As Sfard’s (1991:28) theory suggests, learners move between their operational and structural conceptions of mathematical ideas when they solve problems. Miller (1992:354) adds that writing is an active process that promotes students’ procedural and conceptual understanding of mathematics. Through writing, learners communicate their understanding of mathematical concepts whenever they solve mathematical problems. Heddens and Speer (2006:84) argue that the opportunity to apply conceptual knowledge is as important as understanding the concepts themselves. It provides more meaning and purpose to the knowledge and skills the learner has acquired. Learners use what they know in order to solve that which is unknown. The learner makes connections with previous knowledge and mathematical problems in order to construct new meaning.

As learners used the writing task, writing to solve mathematical problems, they had opportunities to develop and apply conceptual understanding to the problems. The problems were presented in a way that encouraged learners to connect their existing knowledge to the mathematical content. While learners solved problem 3 during the implementation stage of

data collection, they needed the skill of counting in fours as well as the related multiplication table. (Appendix G lists the differentiated mathematical problems used during the implementation of the task, writing to solve mathematical problems.) Although learners had previously engaged with this mathematical knowledge, they had not done so in that particular day’s lesson. If there was a mathematical concept that was required to solve the problem, the same concept or skill needed to be included in the mental mathematics section at the start of the lesson. It appeared that most learners required some level of scaffolding in this regard. Scaffolding will be discussed later in this chapter as one of the themes. Many learners may have had difficulty making the connection between the mathematical content embedded in the problem and their existing knowledge on their own.

Before learners solved problem 5, counting in threes was included into the mental mathematics exercises at the beginning of the lesson. This particular problem dealt with the context of tricycles for which the skill of counting in threes was required. Some learners, generally from the average and below average mathematical ability groups, had difficulty understanding the problem or finding a strategy and solution. This was significant since learners practised the skill of counting in threes before the problem was presented. Most of these learners had difficulty making a connection to their existing knowledge. As can be seen in Figure 5.3, Learner 7 (BA) needed support through written and verbal feedback to make sense of the context of this problem. The written feedback was not sufficient and he required further individual verbal guidance through scaffolding and prompts. Although he could give a verbal explanation of the strategy he had to use, he still arrived at an incorrect solution after he continued working individually. Through scaffolding, Learner 7 began to display conceptual understanding of counting in threes and related it to the mathematical content embedded in the problem. He still did not demonstrate that he fully understood the context of the problem: he circled all the tally marks rather than working with the required number of 65 wheels as stated in the problem.

Figure 5.3 Learner 7 (BA) Writing – Problem 5

Based on these findings, it is evident that the use of writing in mathematics supports the development of conceptual understanding. Throughout data collection, learners were encouraged to connect the problem they were solving to a mathematical concept or idea. Learners from the average and below average mathematical ability groups seemed to find this more challenging because they often had difficulty finding the mathematical content embedded in the problems. As the writing intervention progressed, learners were given more opportunities to use writing tasks to explain their thinking. They engaged in writing tasks in a way that encouraged them to think through their strategies and solutions in order to write a suitable explanation of their thinking. Development of their conceptual understanding was particularly evident in the post-test where learners individually wrote more detailed explanations incorporating mathematical ideas.

Figure 5.4 shows an example of the development of conceptual understanding. Learner 2 (AA) used incrementing off the decuple as a problem-solving strategy and clarified her thinking in her written explanation. Another example of the development of conceptual understanding is evident in Figure 5.5. This post-test strategy shows how Learner 7 (BA) used figurative composite grouping by adding 5 each time. His explanation showed that he could identify the mathematical content in the problem because he was able to explain his reason for subtracting 1 to reach the correct number of cups. This learner did not solve any problems that reflected higher levels of the LFIN during the pre-test. Figure 5.6 shows how Learner 5 (A) combined doubling and repeated addition in his strategy. He applied his existing knowledge to create a strategy that suited the problem. These examples demonstrate how learners combined and adapted their existing knowledge to the mathematical content in the problems. They developed their conceptual knowledge by restructuring schemas that enhanced their understanding (Skemp, 1987:28).

Figure 5.5 Learner 7 (BA) Post Q4 Figure 5.6 Learner 5 (A) Post Q4

These findings show that writing tasks support the development of conceptual understanding. As learners solve and explain mathematical problems, they critically think about the mathematical content in the problem. The majority of problem-solving strategies used by learners in the post-test reflect higher stages and levels of LFIN. These findings suggest that they were able to connect the mathematical content and context of the problem to their existing knowledge. In some instances, learners combined mathematical concepts in their strategies. This showed an improved conceptual understanding: they were connecting concepts to find a solution.