This article looks at writing tasks as a methodology to support learners’ mathematical problem-solving strategies in the South African Foundation Phase context. It is a qualitative case study and explores the relation between the use of writing in mathematics and development of learners’ problem-solving strategies and conceptual understanding. The research was conducted in a suburban Foundation Phase school in Cape Town with a class of Grade 3 learners involved in a writing and mathematics intervention. Writing tasks were modelled to learners and implemented by them while they were engaged in mathematical problem solving. Data were gathered from a sample of eight learners of different abilities and included written work, interviews, field notes and audio recordings of ability group discussions. The results revealed an improvement in the strategies and explanations learners used when solving mathematical problems compared to before the writing tasks were implemented. Learners were able to reflect critically on their thinking through their written strategies and explanations. The writing tasks appeared to support learners in providing opportunities to construct and apply mathematical knowledge and skills in their development of problem-solving strategies.

The mathematics curriculum currently used in South African classrooms emphasises problem solving to develop critical thinking (South Africa Department of Basic Education [DBE]

As a Foundation Phase teacher, the researcher has been observing learners for many years while solving mathematical problems. During these observations, learners reflected their lack of competence in writing coherent solutions and explaining their solutions to the teacher and/or peers. Some learners appeared to wait for instructions from the teacher giving specific methods and procedures to solve the problem. It seemed that learners generally had difficulty applying mathematical concepts they had previously learned in their problem-solving strategies.

Writing is essential in supporting the development of mathematical knowledge and its application to problem-solving strategies. It helps learners clarify, define and express their thinking as well as examine their ideas and reflect on what they have learned in order to deepen and extend their understanding of mathematical ideas (Burns

In the pre-test and post-test of a study based on Grade 2 intervention conducted by Takane (in process) (Venkat & Askew

Burns (

Five writing tasks were implemented. In ‘writing to solve mathematical problems’, Burns (

Learners were introduced to the use of writing tasks in mathematics, particularly in the area of problem solving. Mathematical problems and, in particular, word problems should form part of problem solving. Heddens and Speer (

Vygotsky’s theories of the zone of proximal development and appropriation underpinned this study. This theoretical framework suited the use of writing tasks in the mathematics classroom while learners solved mathematical problems. The zone of proximal development (Vygotsky

The distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers. (p. 86)

In this study, writing activities created the opportunity for a ZPD to be established. Initially, learners engaged in problem-solving activities where their use of strategies and written explanations was limited. Different types of writing tasks were used to guide and support mathematical problem-solving strategies and explanations within the ZPD in order for learners to decrease the distance between their potential development and actual development of their independent strategies. The more knowledgeable other (MKO), be it the peer, parent or teacher, scaffolds understanding through individually tailored pacing of the problem-solving process (Bruner & Haste

Vygotsky’s theory of appropriation, as explained by Duarte (

This study was guided by the following research question:

How do various types of writing tasks support Grade 3 learners in solving mathematical problems?

This article will focus on the extent to which Grade 3 learners are able to engage in writing tasks when solving mathematical problems. It reports on the support that writing tasks give to the development of problem-solving strategies by focusing on the nature of their representations.

This qualitative case study was a systematic, in-depth investigation of a particular instance in its context in order to generate knowledge (Rule & John

Generic mathematics problems were given to all participating learners during the pre-test, intervention and post-test. The problems related to the basic operations (addition, subtraction, multiplication and division) using whole numbers with varying number ranges to accommodate the different mathematical ability groups in the class. Although the number of test items was limited, learners had the opportunity to solve 13 problems during the intervention. These problems were included in the data collected for this study. Ability group discussions with all learners were conducted after learners solved problems during the intervention. Learners shared their strategies with their peers and were guided to think critically about their own strategies as well as the strategies of others.

The study was conducted in an English medium Foundation Phase school in a suburban area in Cape Town. This school was conveniently selected because the researcher was a Grade 3 teacher at the school. Learners predominantly spoke and understood English. One of the five Grade 3 classes was conveniently selected to keep data collection manageable for the researcher as the teacher of the selected class. The population constituted all the learners of the participating class where writing tasks (Burns

Data were captured by audio-recordings of interviews and ability group discussions, learners’ written work during the pre-test, intervention and post-test, and field notes. The sample was interviewed following the pre-test and post-test to explore how writing was used as they solved mathematical problems. Interviews were semi-structured with a flexible list of questions and key themes to allow for probing, follow-up questions and in-depth investigation. After learners solved mathematical problems during the intervention, the different mathematical ability groups in the class discussed their solutions and strategies. Field notes were used to record what learners were doing while solving mathematical problems. Dialogue and conversation were scripted during ability group discussions and collaborative writing between pairs of learners. For the purpose of this article, data collected from learners’ written work were analysed to explore the support writing tasks gave to the development of problem-solving strategies through the nature of learners’ representations.

The analysis process involved developing initial insights, coding, interpretations and drawing implications (Dana & Yendel-Hoppey

Model for Stages of Early Arithmetic Learning.

Stage 0: Emergent counting | Cannot count visible items. The child either does not know the number words or cannot coordinate the number words with items. |

Stage 1: Perceptual counting | Can count perceived items but not those in screened (that is concealed) collections. This may involve seeing, hearing or feeling items. |

Stage 2: Figurative counting | Can count the items in a screened collection but counting typically includes what adults might regard as redundant activity. For example, when presented with two screened collections, told how many in each collection, and asked how many counters in all, the child will count from ‘one’ instead of counting-on. |

Stage 3: Initial number sequence | Child uses counting-on instead of counting from ‘one’ to solve addition or missing addend tasks (e.g. 6 + |

Stage 4: Intermediate number sequence | The child counts-down-to to solve missing subtrahend tasks (e.g., 17 – 14 as 16, 15, 14 – answer 3). The child can choose the more efficient of count-down-from and count-down-to strategies. |

Stage 5: Facile number sequence | The child uses a range of what are referred to as non-count-by-ones strategies. These strategies involve procedures other than counting-by-ones but may also involve some counting-by-ones. Thus, in additive and subtractive situations, the child uses strategies such as compensation, using a known result, adding to 10, commutativity, subtraction as the inverse of addition, awareness of the ‘10’ in a teen number. |

Model for early multiplication and division levels.

Level 1: Initial grouping | Uses perceptual counting (that is, by ones) to establish the numerosity of a collection of equal groups, to share items into groups of a given size (quotitive sharing) and to share items into a given number of groups (partitive sharing). |

Level 2: Perceptual counting in multiples | Uses a multiplicative counting strategy to count visible items arranged in equal groups. |

Level 3: Figurative composite grouping | Uses a multiplicative counting strategy to count items arranged in equal groups in cases where the individual items are not visible. |

Level 4: Repeated abstract composite grouping | Counts composite units in repeated addition or subtraction, that is, uses the composite unit a specified number of times. |

Level 5: Multiplication and division as operations | Can regard both the number in each group and the number of groups as a composite unit. Can immediately recall or quickly derive many of the basic facts for multiplication and division. |

The coded data were interpreted to communicate findings and conclusions were drawn to explore the extent to which learners engage with the writing tasks and the support writing tasks give to the development of learners’ problem-solving strategies.

For the purpose of this study, permission was sought from, and granted by, the Western Cape Education Department, Cape Peninsula University of Technology, the principal of the school and the parents of all the learners in the participating Grade 3 class. An informed consent form was read and signed by the parents of the learner population granting participation in the study. Pseudonyms were used for the school and all participants to maintain confidentiality.

During the intervention, learners engaged with the five writing tasks. The following examples show how learners used the tasks and developed the mathematical understanding through writing. When introduced to ‘writing to record (keeping a journal or log)’ (Burns

Kayla (Journal).

Bevan (Explanation).

Gemma (Thinking and learning processes).

Bevan (Thinking and learning processes).

The pre-test and post-test used in this study helped to gauge the levels of problem-solving strategies learners used before and after implementing different types of writing tasks. The stages and levels of the different aspects of LFIN (Wright et al.

Kayla (Pre-test).

The results showed an improvement in the level of problem-solving strategies used as learners made less use of tallies, for example. This improvement was particularly evident among the below-average learners as shown in

Jarred (Intervention-Problem 5).

Jarred (Post-test).

The tricycle factory has 65 wheels available. How many tricycles can they assemble with the wheels?

While discussing the written feedback the following day, the learner was probed to explain what he thought this meant while the researcher circled his tallies of the wheels to make a group of three wheels. This technique was at level 1 (initial grouping) of early multiplication and division. The learner gave an appropriate verbal explanation that each group represented one tricycle with three wheels. He continued to solve the problem on his own. Later, as the researcher analysed what he had done, it became clear that he had still misinterpreted the problem. He continued circling all his tallies into groups of three without counting his tally marks. This caused him to go beyond the 65 wheels mentioned in the problem. An explanation was written by the teacher to prompt further thinking about the number of tallies needed to represent the wheels in the problem. This difficulty in understanding the complexity of symbols before mastering the conceptual understanding was expressed in the study by Mutodi and Mosimege (

The problem in the post-test (

After the parent meeting coffee will be served. One pot of coffee makes 5 cups. How many pots of coffee need to be made if each person has one cup?

Jarred represented his strategy using a drawing, numbers and words that made sense. His strategy reflected figurative composite grouping, level 3 of early multiplication and division. He used repeated addition in such a way where each group is represented as an abstract composite unit (Wright et al.

Gemma (Pre-test).

Gemma (Post-test).

There are 17 pins in a box. How many pins will there be in 6 boxes?

In this pre-test problem, Gemma used division instead of multiplication. In the post-test, she was the only learner to use conceptual place value in her problem-solving strategy to solve the following problem:

Mark and Martha packed out 81 chairs. Mark packed out 48 chairs. How many did Martha pack out?

In her strategy, she incremented by tens off the decuple to work out the difference between 48 and 81. She provided a detailed explanation of her strategy through her writing, which justified her thinking demonstrating deeper conceptual understanding. This improvement in her use of a more advanced strategy that reflected a higher level of LFIN could possibly be attributed to the writing intervention she had received.

Throughout the data collection period, learners were encouraged to connect the problem they were solving to a mathematical concept or idea. Initially, some learners, especially from the average and below-average ability groups, had difficulty finding the mathematical concept or idea within the problem. As the writing intervention progressed, learners increasingly engaged in writing tasks in a way that encouraged them to think through their strategies and solutions in order to write an explanation of their thinking (Burns

Gemma (Intervention-Problem 8).

Gemma (Intervention-Problem 1).

32 birds land on the bird table. There are now 91 birds there. How many birds were already on the table?

Learners were given an opportunity to share their problem-solving strategies and explanations in an ability group discussion. After engaging with other learners’ strategies, Gemma was able to solve the problem by combining her conceptual knowledge of place value and subtraction to find the solution. She had arrived at the same solution using a more sophisticated strategy after the discussion. At this stage of the intervention period, her written explanations of her strategies were still limited.

The majority of problem-solving strategies used by learners in the post-test reflected higher stages and levels of LFIN, suggesting that they were able to connect the mathematical content and context of the problem to their existing knowledge (Orton

Bevan (Post-test).

After the parent meeting coffee will be served. One pot of coffee makes 7 cups. How many pots of coffee need to be made if each person has one cup?

Initially, this average-ability learner used the doubling strategy to a point and incorporated this into a repeated addition sum. Bevan successfully combined two strategies from his prior knowledge, which demonstrates a deeper conceptual understanding. At this stage, learners had not encountered the concept of counting by sevens. He was able to use his knowledge of doubling numbers and adding seven each time rather than reverting to tallies and counting by ones.

These results reflect that the learners’ thinking was appropriated concretely through their strategies and explanations as they solved problems and engaged in writing tasks. Learners employed the five writing tasks in order to make sense of mathematical ideas and express their thinking: their use of writing revealed their individual development of thought.

Burns’ (

Five writing tasks were modelled to learners and implemented during the intervention. One of the writing tasks, ‘writing to solve mathematical problems’ was used during the pre-test and post-test. The learners’ use of this writing task revealed the extent to which learners were able to engage with at least one of the writing tasks. This independent use of writing suggests that the use of writing tasks may increase learners’ ability to describe the thinking behind their solution processes when they engage in mathematical problem solving. Selected learners were able to provide written explanations of their solutions in order to justify their strategies.

The aim of this article focuses on the support writing tasks give to Grade 3 learners while solving mathematical problems. It explored the extent to which learners were able to engage with the writing tasks and the support writing tasks give to the development of problem-solving strategies and the nature of learners’ representations. Although the scope of this article is limited, the use of writing methodologies such as Burns’s (

The results of this study suggest that writing is beneficial in the mathematics classroom. A more comprehensive study may be required on the use of each writing task to support the development of problem-solving strategies. Learners’ talk can be explored during the ability group discussions and collaborative work in relation to their development of problem-solving strategies. Further in-depth research can be conducted in the Foundation Phase as well as higher grades to determine the usefulness of writing in mathematics across the phases and stages of the mathematics curriculum.

This study was in part made possible through a grant by the University Research Fund of the Cape Peninsula University of Technology

The authors declare that they have no financial or personal relationships which may have inappropriately influenced them in writing this article.

C.V. and S.M. were the project supervisors. B.P., C.V. and S.M. co-designed the project. B.P. implemented the project, collected the data and analysed the data. B.P., C.V. and S.M. co-formulated the results, discussion and conclusion.