In a culture where teachers follow the textbook prescriptively, Malawian students perform low in mathematics, and no students reach the problem-solving levels.

To explore reasons for students’ low performance, this study aims at investigating opportunities to learn problem-solving in Malawian mathematics textbooks.

This study focuses on Malawian mathematics textbooks in the lower secondary grades focusing on the areas of linear equations and simultaneous linear equations. These areas have a particular emphasis on problem-solving.

Four textbooks from two of the most widely used series of mathematics textbooks in Malawian secondary school were analysed. The Mathematics Discourse in Instruction framework was used to analyse examples and tasks in the four textbooks.

Analysis indicates that the textbooks provide relatively few opportunities to learn problem-solving, and most of the opportunities are given through word problems. These word problems are typically presented towards the end of the chapters, and students are thus stimulated to apply already learned procedures to solve the problems rather than learn through problem-solving.

Limitations in opportunities to learn problem-solving are particularly challenging in a context like Malawi, where teacher–textbook compliance is high, where there is a shortage of qualified mathematics teachers and where few students have access to their own textbook.

This study provides an overview of impediments to learning problem-solving in Malawian mathematics textbook, and knowledge about such impediments is necessary for change.

At some point, everyone has met a mathematical task that was hard to solve – or even get started with. Such tasks are often referred to as

To answer this question, we have analysed a selection of Malawian mathematics textbooks from Grades 9 and 10. We focus on the areas of linear equations and simultaneous linear equations, which are two areas of the Malawian Mathematics curriculum for lower secondary school, in which problem-solving is presented as a suggested method of teaching and learning mathematics.

Before we approach the theoretical background, we must define two key concepts in our study:

Textbooks are resources written for teaching and learning (O’Keeffe

Although textbooks are generally considered important, mathematics textbooks have also been criticised, and many studies that we reviewed on or related to problem-solving show that textbooks provide limited opportunities to learn problem-solving. In the Malawian context, studies have found that textbooks provide limited opportunities to learn mathematics in primary (Chiyombo

Even in a context like Singapore, where problem-solving has been emphasised in the Mathematics curriculum for decades, textbooks have been found to include mostly traditional and routine tasks. For instance, Fan and Zhu (

Mathematics is a core subject in Malawi, and it is allocated more textbooks as compared to other subjects (Ministry of Education Science and Technology [MoEST]

Textbooks in Malawi are authored by private authors and published by private publishers. Yet, the content of the textbooks is determined by the curriculum that is organised by the Ministry of Education, Science and Technology. The authored textbooks are approved by the MoEST through the Malawi Institute of Education (MIE) (Maonga

Like other developing countries, mathematics textbooks are a central instructional tool in mathematics classrooms in Malawi. There are not many studies on textbooks in Malawi, and hence, the information concerning textbooks is limited. However, Mwadzaangati (

Research on problem-solving had not been extensively conducted in Malawi, as such we adopted a qualitative content analysis design to understand the problem in depth. Moreover, the text is qualitative (Krippendorff

We purposively selected two textbook series that were mostly used in Malawian classrooms. The two series were common in both Grades 9 and 10, which means we had four textbooks altogether and two series for each grade. The selected series were the most sold according to legitimate book stores. We interpreted this to mean that they are the most used textbooks in schools and by students as no statistics are available about textbooks’ use in Malawi. We further consulted some teachers who confirmed that they mostly used the same textbooks. Lastly, these textbooks had all the topics outlined in the curriculum; the topics were arranged according to the curriculum, and they had much content. The topics of equations and simultaneous linear equations for Grades 9 and 10, respectively, were the units of analysis. Firstly, problem-solving is indicated as an instructional method for these topics in the curriculum (MoEST

We employed the Mathematics Discourse in Instruction Analytic framework for Textbooks analysis (MDITx) by Ronda and Adler (

Elements of the Mathematical Discourse in Instruction framework.

Our analysis involved coding tasks and assigning levels according to the framework. The framework categorises task levels as follows. Level 1 involves tasks that demand already known procedures. Level 2 involves tasks that demand current topic procedures, while level 3 involves tasks that demand application and making connections (Adler & Ronda

This article followed all ethical standards for research without direct contact with human or animal subjects.

A total of 309 tasks were analysed across the four textbooks (see

Levels of tasks.

Task Levels | Book A (G9) |
Book B (G9) |
Book C (G10) |
Book D (G10) |
Total |
|||||
---|---|---|---|---|---|---|---|---|---|---|

% | % | % | % | % | ||||||

Level 1 tasks | 9 | 20.0 | 9 | 6.8 | 1 | 3.0 | 14 | 14.3 | 33 | 10.7 |

Level 2 tasks | 26 | 57.8 | 103 | 77.4 | 29 | 87.9 | 72 | 73.5 | 230 | 74 |

Level 3 tasks | 9 | 20.0 | 18 | 13.5 | 3 | 9.1 | 11 | 11.2 | 41 | 13.5 |

Outliers | 1 | 2.2 | 3 | 2.3 | 0 | 0.0 | 1 | 1.0 | 5 | 1.6 |

It is primarily among the level 3 tasks that we find opportunities to learn problem-solving, but such opportunities may also be found among level 2 tasks. The quantitative aspects of our analysis thus indicate that there might be opportunities to learn problem-solving in the textbooks, but we had to apply qualitative analysis to further investigate those opportunities. We elaborate on those results below, using some selected sets of examples and tasks to illustrate the findings. We first illustrate how potential opportunities to learn problem-solving might be impeded before we provide illustrations of tasks that might provide opportunities to learn problem-solving.

In order to determine the opportunities to learn problem-solving, it is necessary to analyse the connection between examples and the sets of tasks that follow. From our analysis across the textbooks, we found that the details of an examples and the similarity between a given example(s) and the tasks that followed could impede opportunities to learn problem-solving. Oftentimes, there was a strong similarity between the given examples and the following tasks. An illustration of this can be found in

Examples and tasks illustrating the substitution method, (a) example 7.3, (b) exercise 7.2.

In

When considering the example space, we noticed that there is only one example in the space, and we considered this as generalisation, because it can be interpreted as generalising the substitution method. The example space is also detailed for using a fraction equation instead of using a whole equation so that the fraction equation could challenge the students. Consequently, it highlights that students should use the method of least common denominator, thereby reducing the challenge for students. In addition, the example illustrates the substitution method, and the following prompt for tasks 9–22 (‘Use substitution method to solve the following…’) can thus be interpreted as a leading question, which points back to the given example.

As another illustration,

Introducing a textbook lesson, (a) Elimination method, (b) Exercise 7.2.

Although the tasks are normally similar to the preceding examples, there are some instances where the connection is not so strong. For instance,

Similarities between examples and tasks, (a) Example 5, (b) Example 6.

In the previous section, we reported on how tasks that vary from the given examples are more challenging as they might require students to make connections. Still, such tasks cannot necessarily be classified as genuine problems because the general approach to solving them is already given in the preceding examples. Throughout the textbooks, we found that the tasks that provide opportunities to learn problem-solving were mostly given as word problems. The following task from Book C is an illustration of this:

Two numbers are such that the sum of half of the first number and the second number is 6. But twice the second number added to three times the first number is 24. Find the two numbers. If two lines, 2

This task was coded as a level 3 task for a couple of reasons. The first part of the task requires students to generate equations before using the graphical method, and the requirement to make connections indicates level 3 (Ronda & Adler

Word problems from Book D.

From our analysis of examples and tasks in four lower secondary mathematics textbooks in Malawi – focusing on the units of linear equations and simultaneous linear equations – we have identified some opportunities to learn problem-solving, but we have also identified some impediments to learning problem-solving. A first impediment lies in the details of the examples. When worked examples are detailed, and when there is a close connection between the examples and the following tasks, opportunities to learn problem-solving are impeded. A second impediment lies in the very structure and design of the textbooks, where textbook lessons are introduced by presenting worked examples – sometimes preceded by a presentation of the learning objective and a presentation of methods to be discussed – and then providing a list of tasks. The opposite would be to introduce lessons as challenges. A third impediment is the number of leading questions in tasks. Throughout the textbooks, there were numerous examples of leading questions. These provided explicit or implicit indications of the methods to apply to solve the tasks. There were no examples of prompting questions that invited students to solve tasks in their own way. In sum, these are impediments to the students’ opportunities to learn problem-solving.

Based on our analysis of opportunities to learn problem-solving in Malawian mathematics textbooks for lower secondary school, we are going to develop three claims in the following discussion. The first claim is that opportunities to learn problem-solving are few in Malawian mathematics textbooks. Based on this, the second claim is that limited opportunities to learn problem-solving in textbooks can be an impediment in a context like Malawi, where low teacher qualifications accompany high teacher–textbook compliance (Mwadzaangati

Firstly, our study indicates that Malawian mathematics textbooks for lower secondary school provide limited opportunities to learn problem-solving. There are only 13% level 3 tasks in the units of linear equations and simultaneous linear equations. A large portion of these tasks were word problems. The textbooks seem to define problem-solving as the application of learned procedures to solve word problems. We appreciate this provision because word problems are challenging for students (Kieran

Secondly, we found that at least 74% and 11% of the tasks were level 2 and level 1, respectively. These tasks demand current topic procedures and already known procedures, respectively. Having a large portion of level 2 tasks means that the focus of the textbook is on procedural fluency which could promote rote memorisation. This tendency is not surprising, and studies of mathematics textbooks in other countries also indicate that textbooks provide few opportunities to learn problem-solving. For instance, Brehmer et al. (

Our study further found three impediments to learning problem-solving in textbooks. These constraints are foregrounded on the major finding in this study and other studies that many tasks offered are similar to the examples offered or other details given in a textbook.

The first impediment is

The second impediment is

The last constraint is

Although textbooks in many countries appear to provide few opportunities to learn problem-solving, we claim that such limitations are particularly problematic in a context like Malawi. Our claim draws on Golding’s (

Problem-solving is strongly emphasised in the Malawian curriculum, and leading mathematics textbooks for lower secondary school provide some opportunities for learning problem-solving. Yet, the textbooks also provide some limitations. In a context where the overall qualifications of teachers are low, and where teachers are prone to only rely on textbooks, these limitations might constitute impediments to learning problem-solving. We identified three impediments in the textbooks, which include detailed examples, leading questions and not introducing lessons as challenges. These strengthened the similarities between examples and tasks. Word problems offered more opportunities for problem-solving, but representing opportunities to learn problem-solving mainly in word problems provides a limited view of problem-solving. We believe that providing equally more opportunities through tasks that are not word problems would be essential for propelling problem-solving at every stage of the topic and in all forms because word problems were mostly placed at the end of the topic. The overall implication is that students in Malawi are provided with limited problem-solving opportunities through textbook tasks. However, we also believe that how teachers implement the tasks may provide different opportunities. Further research on other topics of the Malawian textbooks is needed to ascertain the extent to which the opportunities for learning problem-solving are provided and if more impediments could be discovered.

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

F.K.M. conceptualised the study and contributed to the methodology, analysis, investigation and writing. R.M. contributed to the methodology, analysis, writing and supervision.

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

The authors analysed textbooks that had already been published. Therefore, data sharing is not applicable to this article, as no new data were generated.

The views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors.