Original Research

Exploring representations of multiplication: A study in Mozambican Grade 3 classrooms

Simone Reinhold, Albertina J.L. António

Received: 09 Oct. 2025; Accepted: 09 Jan. 2026; Published: 17 Apr. 2026

Copyright: © 2026. The Authors. Licensee: AOSIS.
This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license (https://creativecommons.org/licenses/by/4.0/).

Abstract

Background: This explorative study focuses on the representations of multiplication that Mozambican primary teachers use. With a particular focus on representations of multiplication, the article presents examples of models and strategies that encompass the broad spectrum of representations related to the concept of multiplication.

Aim: The study investigates how Mozambican Grade 3 teachers use representations when teaching multiplication, with particular attention to the diversity of representations used in classroom instruction.

Setting: The research is situated in the Mozambican educational context, where numeracy teaching faces specific challenges. Besides difficulties in basic numeracy, learning is influenced by the coexistence of Portuguese as the language of instruction and children’s mother tongue (in this study, Emakhuwa).

Methods: A qualitative case study design was applied. Entire mathematics lessons from two Grade 3 teachers were analysed, focusing on teaching methods, task design, the use of representations and manipulatives, and opportunities for students’ participation.

Results: Instruction was predominantly expository and characterised by choral repetition and reproduction of procedures. Although teachers used concrete examples and several representations of multiplication, opportunities for students to engage actively with these representations were limited.

Conclusion: Students rarely explained their reasoning or justified their answers, and experiences with representations beyond symbolic notation were largely absent.

Contribution: The study contributes to discussions on improving mathematics instruction in low-resource, bilingual contexts by highlighting the need for more diverse representations and student-centred teaching approaches.

Keywords: multiplication; representations; primary; mathematics education; teachers; Mozambique; Grade 3; classroom observation.

Introduction

Understanding the concept of multiplication is a major challenge in primary education, as the concept extends beyond basic fact memorisation and underpins key mathematical concepts such as division, fractions or exponentiation, requiring well-designed learning opportunities.

In Mozambique, this claim conflicts with many challenges, including low student performance, problems concerning the low quality of training programmes and trainers at the Teacher Training Institutes (Institutos de Formação de Professores [IFP]), working conditions or school management and issues in terms of supervision and pedagogical support (Ministério da Educação e Desenvolvimento Humano [MINEDH] 2020, Cherinda, see Schorcht 2025). In particular, results of the latest studies monitored by The Southern and Eastern Africa Consortium for Monitoring Educational Quality (The Southern and Eastern Africa Consortium for Monitoring Educational Quality [SAECMEQ] 2023) reveal that many Grade 6 students in Mozambique alarmingly underperform in basic numeracy, suggesting limitations in the organisation of primary mathematics teaching, including the use of representations.

Mathematical concepts, particularly multiplication, are closely linked to representations. For example, Bruner (1966) suggested that children construct knowledge and meaning through active interaction with the world around them, using external representations (pictures, icons, tables and symbols) as integral elements. As the teachers’ specific approaches are likely to influence the development of students’ understanding of multiplication, identifying teachers’ practices might support our understanding of Mozambican children’s struggle in numeracy (especially multiplication). However, little is known about how Mozambican primary teachers use representations when teaching multiplication. Addressing this gap, the present study is guided by the following research question: Which variety of representations is used by Mozambican Grade 3 teachers to teach multiplication?

Theoretical framework

Encountering multiplicative ideas: The intuitive approach

All over the world, and even before they start school, children experience actions related to the concept of multiplication in the real world – even without consciously recognising or naming them as ‘multiplication’. In this sense, previous studies have shown that children in Grade 1 and at the beginning of Grade 2 are capable of dealing with multiplicative (word) problems even before any formal instruction on the arithmetic operation (Kouba 1989; Mulligan 1992). This learning takes place from an early age on, using local, familiar language for communication in the children’s home environment. In our context, the Emakhuwa language plays a particularly influential role in students’ learning, as it is the most widely spoken Bantu language in Mozambique, especially in the northern region of the country.

For example, children learn the meaning of ‘doing something several times’ through phrases such as ‘sleep twice’, ‘jump 10 times’, and so on, which are very similar in Portuguese, the official language of Mozambique: ‘dormir duas vezes’, ‘saltar 10 vezes’. In Emakhuwa, children encounter similarities and differences. For example, the Emakhuwa expression for ‘twice’ is ‘vara vahili’ (with ‘vahili’ meaning ‘two – referring to actions’ and using ‘pili’ for ‘two’ referring to amounts), and phrases such as ‘to sleep twice’ or ‘to jump ten times’ can be translated into ‘orupha vara vahili’ and ‘othupha vara mulokhô’, using the word ‘vara’ to indicate repetition in everyday life. As a result, children have already associated the word ‘times’ (respectively, ‘vezes’ in Portuguese or ‘vara’ in Emakhuwa) with the repetition of an action (in a sequential manner). But regarding (mathematical) expressions in Emakhuwa related to multiplication, ‘there is some interference from the Portuguese language in the mother tongue and vice versa’ (Draisma 1996:173), which means that both children and teachers may face difficulties or at least specific challenges when they encounter multiplicative structures.

Concepts and heuristic strategies for multiplication

Repeating an action (in a sequential manner) can lead to the idea that every time this action takes place, a set of items with a certain cardinality is an integral part of this activity. In this sense, a certain sum (later: the ‘multiplier’) of several disjoint sets of the same cardinality (later: the ‘multiplicand’) can be regarded as the result of this additive and also multiplicative setting (e.g. ‘buying ten cashews on every day of the week’ leads to ‘10 + 10 + 10 + 10 + 10 + 10 + 10’ or ‘walking to the water well three times a day and carrying two buckets each time’ leads to ‘2 + 2 + 2’). Furthermore, quantities of the same size may be displayed (simultaneously) in everyday situations, such as ‘4 baskets with 3 mangos in each of the baskets’ or ‘5 bunches of bananas, each bunch consisting of 10 bananas’ on a market stand (see Figure 1).

FIGURE 1: Introducing multiplication via disjoint sets of the same cardinality in a Mozambican Grade 3 textbook used by one of the teachers in the study.

Thereby, children learn that the total number of several quantities of the same size can be achieved by using repeated addition (3 + 3 + 3 +3, respectively, 10 + 10 + 10 + 10). This notion of repeated addition is usually linked to multiplication as a ‘shortcut’, in lower primary (3 + 3 + 3 + 3 = 4 × 3, respectively, 10 + 10 + 10 + 10 + 10 = 5 × 10). Yet, in addressing different ways of interpreting a multiplicative expression, Draisma (1996) reports on findings in which these interpretations caused confusion among Mozambican teachers and children:

For some people, 3 x 4 means 4 + 4 + 4, because they think it is ‘three times four’, and for others it means 3 + 3 + 3 + 3, because they think it is ‘three repeated four times’. (p. 169)

For mathematics teaching, this variation in interpretation can lead to difficulties later in the transition to more advanced concepts, such as matrix multiplication, functions or even the multiplication of fractions (Tirosh & Graber 1989). Another internationally common introduction to multiplication in textbooks includes jumps on a number line (Harries & Sutherland 2000), which corresponds to the idea of ‘jumping several times’ (see Figure 2), whereby the ‘jumps’ cover equal distances on a standardised scale. Usually, this concept also corresponds with the idea of repeated addition.

FIGURE 2: Pictorial representations for 3 × 4 multiplication: (a) multiplication in a rectangular array, and by jumps on the number line (b) multiplication with bundles in sets of the same cardinality (irregular arrangement) and (c) multiplication in bundles of the same cardinality in linear arrangement.

However, referring to Fischbein et al. (1985), repeated addition is only one of various concepts for multiplication. Here, the number of equivalent collections (one factor) works as the multiplier (or ‘operator’), whereas the magnitude of each collection is interpreted as the multiplicand (or ‘operand’). As shown above, this understanding (e.g. in word problems) is subject to limitations. Therefore, further models for multiplication beyond the model of equivalent groups (‘4 baskets with 3 mangos in each of the baskets’) need to be considered, and Greer (1992) complementarily suggests:

• multiplicative comparison (e.g. ‘4 times as many cashews as peanuts’)
• rectangular arrays (e.g. ‘4 rows with 10 packs of folded capulanas in each row’)
• Cartesian product (e.g. ‘number of possible girl-boy pairs in a group of children’).

Over time, learners in primary school should expand their understanding of multiplication and grasp various facets of the concept. For example, regardless of the order of two factors in a term like 3 × 4 it may change to 4 × 3, while the result remains the same (commutative law). Similarly, working with three or more factors does not depend on the ordering of the used factors either, as a × (b × c) = (a × b) × c = b × (a × c) (associative law). ‘Splitting up’ the term to simplify the multiplication 4 × 3 may lead to the idea that ‘2 × 3 plus 2 × 3’ and also the term ‘(2 + 2) × 3’ have the same results (distributive law). In this case, applying relations between facts, halving is part of the strategy and we know ‘4 × 3 = 2 × (2 × 3)’. Otherwise, doubling can help to solve more difficult tasks like 8 × 3, as we know ‘8 × 3 = 2 × (4 × 3)’.

Halving and doubling the multiplier and the multiplicand at the same time, but in a contrary manner, may also occur in a sophisticated multiplication strategy, which helps to deal with higher numbers. For example, 500 × 6 = 1000 × 3 = 3000, because doubling the multiplier (2 × 500) and halving the multiplicand (6 ÷ 2) at the same time will not change the product. Multiplying one of the factors by any other number and dividing the second factor by the same number will have this effect, too, and can help to compute mentally with tens or even hundreds, for example 25 × 16 = 100 × 4 considering 4 × 25 = 100 and 16 ÷ 4 = 4. To memorise basic facts, it can also be helpful to keep in mind the strategy: add one more unit or once more the second factor (Draisma 1996:170), as the result of ‘neighbour tasks’ is sometimes easier to access. Similarly, it can be helpful to go back one unit/subtract the second factor once. It seems that 9 × 7 is a difficult task, and ‘counting in units of 7’ is prone to (counting) errors. Yet, deriving it from 10 × 7 can support, as comparing the two terms reveals:

• 10 × 7 = 7 + 7 + 7 + 7 + 7 + 7 + 7 +7 + 7 + 7 + 7
• 9 × 7 = 7 + 7 + 7 + 7 + 7 + 7 + 7 +7 + 7 + 7 (differing from 10 × 7 by one unit of 7).

Therefore, we may conclude: ‘9 × 7 = (10 × 7) – 7 = 70 – 7 = 63’, or in words: ‘10 times 7 is (one unit made of) 7 more than 9 times 7. That’s why we take away (one unit of) 7 from the result of 10 times 7 (which would have been 70), with the result 63’.

Representations for learning multiplication

The previously addressed examples of models and strategies may be addressed through a wide range of representations related to the concept of multiplication, according to Bruner’s (1966) theory of different modes for representing knowledge. Following this basic and rather traditional principle, we can represent (mathematical) content in basically three different ways: enactively (action-based), iconically (pictorial and/or image-based) or symbolically (through signs or language). With respect to representations of multiplication, this means that teachers are challenged to offer ‘further embodiment’ (beyond symbolic representations in multiplication tasks), including various learning activities that are ‘mathematically isomorphic’ ‘in different materials and with altered appearance’ (Bruner & Kenney 1965:51).

Meanwhile, and over the years, mathematics education research has elaborated this theory in various ways. In particular, it has been clarified that these modes of representation should not be misunderstood as ‘one-way streets’ in the sense of a linear development (only starting from enactive → to iconic → to symbolical, in the end). Moreover, teachers should keep in mind ‘networks’ of representations, which include intermodal transfers from one mode to any of the others. For example, the picture of Mr Menete’s market stand (in Figure 1) shows four bunches of bananas lying on the board – while Mr Menete himself and the boy Gustavo are holding two more bundles. On one hand, this is in general an iconic (pictorial) representation of a situation that the children could experience (enactively) by themselves. On the other hand, the picture itself does not provide sufficient visual information concerning the multiplicand (10 bananas), and more information from the text besides the picture is needed (symbolic representation and language), before children can transfer this to another symbolic representation (notation of the equations for addition and multiplication).

Regarding representations for multiplication, Mulligan and Mitchelmore (1997) reported on ‘intuitive models’ they observed among children in Grades 2 and 3. The authors refer to various children’s calculation strategies for one-step, whole-number and multiplicative word problems (direct counting, rhythmic counting, skip counting, additive calculation and multiplicative calculation, see Kouba 1989; Mulligan 1992), which may also include the use of ‘physical material’. Results of their longitudinal study revealed that children expanded their repertoire over time, including adaptations depending on the type of task. Against this background, teachers are challenged to foster young children’s evolving understanding of multiplication. Concerning iconic representations, findings from a study by Harries and Barmby (2007) underpin that the array representation can be a powerful tool for children’s learning of multiplication. Similarly, Cheeseman et al. (2023) suggest focusing on visualisation and drawings, as the ability to form visual images of composite unit structures in multiplicative situations is fundamental to understanding multiplication and division (Cheeseman et al. 2023:792).

Kuhnke (2013) analysed German second graders’ understanding and their capability of switching between different iconic representations in the sense of intramodal transfers (see Figure 2). Here and in international literature, or in mathematics textbooks, rectangular arrays, jumps on the number line and bundles in sets of the same cardinality (either irregular or in linear order) seem to be the most common iconic representations (Harries & Sutherland 2000).

Teachers’ use of representations for multiplication

Teachers’ ability to visually represent (mathematical) content and procedures and find suitable variations of representations is regarded to be an important element of pedagogical content knowledge (Shulman 1986) resp. subject matter knowledge (Ball, Thames & Phelps 2008).

Concerning the use of representations for teaching, research has revealed pre-service teachers’ and teachers’ difficulties with both multiplication and division. For example, Barmby and Milincović (2011) asked pre-service teachers to choose suitable iconic representations from a selection of representations to serve specific multiplicative concepts, such as demonstrating the commutative or distributive law or illustrating multiplication as repeated addition. The results of this study revealed lacunae and misconceptions in teachers’ content knowledge and concerning teachers’ use of iconic representations (Barmby & Milincović 2011):

[S]ome teachers did not understand what was meant by these laws. The analysis also highlighted some limitations in teachers’ use of representations, for example with the commutative law […] where a simplistic argument was provided for using the groups of strawberries representation. (p. 110)

In a similar study with pre-service teachers, Milincović (2012) found specific preferences in using representations that were not necessarily adapted to the characteristics of a given problem. Yet, the choice of suitable representations seemed to be connected to the level of the problems’ abstractness.

Naboomba (2017) shared findings of a study that revealed that Zambian teachers and Grades 3 and 4 learners faced difficulties in varying strategies when the teachers introduced multiplication and division as repeated addition or regrouping. Yet, some of the children were able to draw iconic representations (including units of the same cardinality, see Figure 3). Teachers’ use of representations also took the multiplication table into account.

FIGURE 3: Iconic representations in Zambian children’s drawings for multiplication: (a) rectangular arrays and bundles; (b) linear arrangements.

Framings and particularities for teaching multiplication in Mozambique

Major challenges for mathematics education in sub-Saharan countries are reported in various studies, often pointing out that these challenges have an immense impact on economic and social life (Madaki 2021:203). Among all sub-Saharan countries that participated in SAECMEQ (2023) studies, Mozambique has the lowest percentage of Grade 6 pupils (30%) with either parents having secondary education and the second-biggest proportion of Grade 6 pupils having both parents with primary school education or even less (42%) (SAECMEQ 2023:18). Only around 30% of all Grade 6 students reach level 2 of mathematical competency. This level, referred to as ‘Emergent Numeracy’, involves performing two-step addition or subtraction operations that require carrying, checking or converting pictures into numbers (SAECMEQ 2023:73).

If one returns to the significance of multiplication in everyday life, not only in Mozambique, one illustrative example is market trading. While local fruits and vegetables are generally inexpensive, prices for clothing are often multiples of 10 or even 100. As a result, mental calculation, and particularly multiplication, is highly present in informal commerce. In this context, which many children frequently encounter themselves, mathematics plays a vital role in people’s everyday social practices, including the ability to multiply by larger numbers, such as by tens or hundreds.

Regarding the acquisition of arithmetic competencies that are addressed in these and many more everyday situations, studies by SAECMEQ (2023:64) indicate a shortage of resources in Mozambican schools and show that only 33% of all Mozambican Grade 6 students have a mathematics textbook at their disposal. For lower primary students, the situation is presumably even worse, as informal observations made by the authors in primary schools of Northern Mozambique throughout the last decade suggest that there is still an immense lack of instructional material, such as mathematics textbooks. Usually, only the teachers have a pupil’s book at hand (serving as a teaching manual), whereas the students themselves take individual notes in their notebooks. Therefore, the teachers’ role is even more important in terms of offering and sharing useful representations (respectively, encouraging students to design representations). Mozambican textbooks, which are available for teachers, adopt the idea of repeated addition by linking pictorial representations (iconic representations, see above) and symbolic notation (including the mathematical equations for the addition and the corresponding multiplication, see Figure 4), but little is known about how teachers incorporate these (iconic and symbolical) representations in their teaching or how they interconnect them.

FIGURE 4: Repeated addition of (simultaneously visible) equal sets of items in an exercise for repetition in a Mozambican Grade 3 textbook.

Draisma (2018) analysed textbooks for primary school in Mozambique with a focus on examples that throw a spotlight on adding two natural numbers (smaller than 10, with the sum exceeding 10). Concerning the development of Mozambican mathematics curricula and textbooks in general, Draisma states that the Mozambican curriculum published in 2003 (Instituto Nacional do Desenvolvimento da Educação [INDE] 2003) had returned to explicitly address counting strategies, whereas there was only little demand to focus on computation strategies (in the sense of using relations, see above). The following quote underpins this observation:

This subject enables students to develop skills in counting, calculating and applying the four basic operations to solve problems. It also allows them to develop skills in locating and orientating, observing, identifying, relating, classifying, estimating and measuring, interpreting messages in symbolic and graphic language, as well as collecting, organizing and interpreting data in simple tables and graphs. Mathematics also allows you to calculate perimeters, surfaces and volumes and carry out simple geometric constructions with ruler, square, compass and protractor. It also develops basic financial literacy and responsible citizenship. (INDE 2020:20, translated by the authors)

Here, representations (and those of importance for multiplication) are addressed implicitly when mentioning the interpretation of ‘messages in symbolic and graphic language’. Furthermore, the comprehensive mathematics curriculum for Grades 1 and 2 includes detailed recommendations concerning the use of manipulatives to display multiplicative situations – in enactive, iconic and symbolical mode (INDE 2015:153).

As a result of further in-depth analyses of the correspondence of curriculum development and textbook revision throughout the years in Mozambique, Kusaka (2019) states:

However, most of the ‘number and arithmetic’ areas particularly emphasize ‘basic competencies’ and the extend to which practical competencies described in the curriculum are nurtured are largely reliant on the teacher applying the lessons in the classroom. (p. 48)

This finding underlines the notion that a great deal of responsibility remains with the teachers, who might often rely on long-established ways of teaching habits. This situation is a point of concern but may also offer opportunities concerning (traditional) ways of integrating mathematical representations into the primary classroom, as Draisma (2018) reports:

I had the pleasure to work with teachers of bilingual education and adult educators who use local language in Mathematics. We found that oral computation in the mother tongue, supported by gestures, constitutes an excellent resource for elementary arithmetic. (p. 962)

Against this background, it is particularly important that teachers in lower primary use various, but coherent representations that help to address basic aspects of multiplication in a meaningful way. Teaching methods, the choice of tasks, the use of representations and manipulatives or the chance for students to actively participate are likely to have an influence on the intensity of students’ engagement and their learning. Yet, little is known from inside Mozambican classrooms and about teachers’ use of representations that incorporate the wide range of multiplicative concepts and strategies.

Research methods and design

Research question

Against the given theoretical background, the objective of the study shared here was to explore Mozambican teachers’ use of representations when addressing multiplication in the mathematics classroom of Grade 3. Being part of a larger project with a wider range of research questions (including a larger number of participating schools, teachers and students), this study refers to the following main research question: Which variety of representations is used by Mozambican Grade 3 teachers to teach multiplication?

Data collection

Data collection took place at 15 different schools in the region of Northern Mozambique between 2023 and 2024 through a series of classroom observations. These classroom observations and school visits were conducted at the request of the local Ministry of Education, which chose the schools for the sample, meeting our request to visit both rural and urban schools. This framing was supported by an international cooperation project in the field of school pedagogy, covering different school subjects (QuEProF – Qualidade da Educação pela Qualidade na Formação de Professores-Formadores, https://www.erzwiss.uni-leipzig.de/institut-fuer-bildungswissenschaften/allgemeine-didaktik-und-schulpaedagogik-des-sekundarbereichs/queprof).

Furthermore, the sampling of lessons (and thereby the sampling of the participating teachers who conducted the lessons) was guided by the schools’ situation at the time of the visit – choosing the lessons and colleagues who stated to teach multiplication during the time of the school visits. Thereby and as a result of the given administrative and cultural framing, data collection was fairly ‘random’ because we were trying to be as less invasive as possible, not aiming at statistical representativeness but rather at qualitative, explorative insights. All participating teachers were assured of data protection, and the participating researchers committed themselves to handling all data confidentially for research purposes only. Participation in the data collection for the study was voluntary, and no teacher was at risk of psychological distress or consequences for his and/or her assignment. Pseudonyms (for both persons and locations) were used to protect the teachers’ identities.

Out of all observed lessons, 10 lessons focused on various aspects of multiplication in Grades 3 and 4. Following a qualitative paradigm, the choice for the case studies shared in this article focuses on two mathematics lessons in Grade 3. While the other observed mathematics lessons on multiplication showed similar patterns (also in Grade 2 and beyond Grade 4), these cases were deliberately chosen to cover diverse contexts, such as urban versus rural schools, teacher gender and differing classroom environments, in order to provide a broad and illustrative insight.

In 2023, we visited lessons conducted by a male teacher (we call him ‘Manoel’) in an urban school in one of the districts of a city in Northern Mozambique. Manoel, with a teaching experience of approximately 10 years, taught 54 students in his Grade 3 classroom (many of them sitting on the floor right in front of the board). One year later, and in contrast, working in a rural context (in a village of Northern Mozambique) enabled us to meet a female teacher (we call her ‘Ana’, approximately same teaching experience as Manoel) who agreed to observe her mathematics instruction in a class with 19 children (who had textbooks of different editions available).

Data analysis

The lessons were entirely video recorded, and the researchers took written notes during the lessons to prepare the analyses with regard to the research question. Additionally, both textbook samples used in class and students’ individual notes were collected. All observations included complete mathematics lessons, with an average duration of approximately 45 min each. The videos were transcribed partly, prioritising episodes directly related to the teaching of multiplication, including teacher explanations, interactions with students, use of teaching materials and moments of problem-solving.

Seeking to identify and analyse the presence of representations for multiplication, the authors used the Qualitative Content Analysis Methodology (Mayring 2015). Furthermore, repeated viewing of the videos by the authors enabled them to identify the occurrence of representations in the data and to code according to the theoretical framework.

As we followed a deductive–inductive code formation process, which is typical for the Qualitative Content Analysis Methodology (Mayring 2015), the units of analysis included utterances, episodes or other representation instances (see also findings). During our coding process, we started with codes from the literature, which directly referred to the theory of modes of representation (enactive, iconic, symbolical representations, intra- and intermodal transfers).

Enriching and sharpening these codes by differentiations from theoretical models and strategies for multiplication (e.g. regarding enactive representation without material, or addressing commutative law, Ana said: ‘You can say 2 × 10 or 10 × 2, that is the same’, see Results), we were open to complement this with special findings and inductively raised codes derived directly from the data. The data of the two case studies were independently analysed by the two research teams (Mozambique and Germany) to ensure the reliability of the coding process. The identified discrepancies were discussed until consensus was reached, contributing to the refinement of the categories and the interpretative consistency of the analysis (following consensual validation in a qualitative research paradigm; see Maier 1991).

Ethical considerations

Ethical clearance to conduct this study was obtained from Leipzig University and the Ethics Advisory Board (No. IPF: 151201).

Results

In correspondence with our analytical approach, insights into the results concerning the variety of representations, we firstly refer to the theory of modes of representation. Secondly, we zoom into incidents that are related to theoretical models and strategies for multiplication, in more detail. Although we analysed the lessons of both teachers separately, the following report summarises the activities in both Ana’s and Manoel’s orchestrations of their lessons. Examples of both teachers’ notes on the board (see Figure 5) offer a first impression.

FIGURE 5: Examples of teachers’ notes on the board with equations for repeated addition and multiplication, including blanks (a), Manoel, and equal sets of items in rows of 10, forming an (almost) rectangular array (b) Ana.

Regarding the observed activities in the lessons, Table 1 reveals that the data suggest differentiations among the representation modes, which are used to teach multiplication by the two teachers. For example, any kind of ‘enactment’ (e.g. the ‘acting’ of a situation in a word problem or pretending action in a situation from everyday life) was considered enactive representation, no matter if ‘typical’ manipulatives were used. Furthermore, distinctions among facets of iconic representations (pictorial and abstract) and symbolical representations (written language, spoken language and mathematical symbols) were made.

Both teachers intensively connected spoken language to the mathematical symbols while they wrote on the board, thereby using intramodal transfers within the mode of symbolical representations. Most of the time, the teachers inspired their students to interact with these representations via repetitions in the choir. Iconic representations (abstract) that were used and connected to symbolic representations for the students occurred only in Ana’s teaching. Manoel did not make use of diagrams, drawings, tables or concrete materials to support the children’s understanding. But most interestingly, along many representations, we identified a one-to-one correspondence between the examples from the textbooks and the matter that the teachers took up thematically (e.g. with the ‘market stand’ in Manoel’s lesson and the linear arrangement of circles in Ana’s lesson).

Among all symbolic representations, spoken language prevailed, and this approach was most often accompanied by mathematical symbols to indicate repeated addition and the corresponding notation of multiplication. In doing so, Ana also addressed commutative law (‘You can say 2 × 10 or 10 × 2, that is the same’.). For this purpose, she offered an iconic representation (identical to representations in the textbook), but this was limited to an array of two rows of 10 circles each, which almost forms a horizontally lying rectangle (without comparing it to a vertically standing rectangle with the same pattern of circles, see Figure 4).

Overall, the level of students’ engagement was moderate, but there was a chance to take individual notes or make copies from the board, resp. from the textbook for most students. Single students solved equations on the board (symbolic representation and mathematical symbols). Yet, there was an excessive use of the expository method in major parts of both lessons. There were moments of distraction in the lessons of both teachers (students outside trying to follow the class). Some students tried to leave the room, and others lost focus, but both teachers managed to keep control of the class, asking for silence when necessary.

Discussion: Contextualisation and recommendations

Contextualisation of the findings

In general, our observations resemble reports on Mozambican classroom practices by Mutemba (2012), who talks about:

[… T]raditional practice in classrooms, where the teacher is in the front of the classroom and delivers more or less in the form of a lecture general methods and more specific techniques and formulae. (p. 15)

Concerning the use of representation in our study, this corresponds with our result that children’s opportunities to learn by different modes of representation were limited. This finding might be one of the reasons for problems in Mozambican lower primary schools in understanding the concept of multiplication and for low performance in arithmetic in general.

One of the main limitations observed was the lack of opportunities for students to formulate their own explanations and justify their answers. Chances to experience modes of representation beyond symbolic (mathematical) notations were almost completely missing. Yet, as long as teaching is based solely on repetition, this method can lead to misconceptions and mechanical learning, without ensuring that students truly understand the concept of multiplication. Furthermore, the lack of peer discussions prevented students from sharing strategies or helping each other, which could have contributed to more collaborative and reflective learning.

Practical implications, recommendations and research desiderata

The findings of this study have various practical implications for mathematics learning and teaching in Mozambican primary schools and for Mozambican teacher education. Most importantly, the use of concrete objects from children’s everyday lives should play a more prominent role in enhancing the teaching and understanding of multiplication. These manipulatives should be simple, inexpensive and easily translatable into iconic representations (e.g. children’s drawings of rows of stones or bottle cans on strips of 10). This method (in terms of enactively generating bundles of equal cardinality) can help children to grasp multiplication as repeated addition. Furthermore, encouraging peer interaction around multiplicative problems and the use of a wider range of representations can support conceptual understanding.

Conclusion

Our observations suggest that enactive representations, especially the ‘enactment’ of arithmetic context, like in the case of Manoel, may be a strong feature of Mozambican teaching traditions. In Manoel’s lesson, the students appeared to benefit from his instructional approach. Furthermore, teachers’ awareness of ethnomathematical heritage can provide a valuable source for concrete activities and sustainable representations in the local context (Cherinda 2015; Gerdes 1981; Vos, Devese & Rassul 2006). In this sense, Jacinto and Jakobsen (2021) report promising examples from nearby Malawi for counting and for multiplication (grouping and counting all elements on a string with sticks). Here, the pre-service teacher addresses ‘transitions’, which challenge children to switch between the enactive exploration of multiplicative situations, iconic representations (vertical lines illustrating the sticks of the self-made abacus) and symbolic notations in the corresponding equations (see also Draisma 2018:952, for similar examples in Mozambican textbooks for addition in Grade 1).

Furthermore, conclusions from a study conducted by Macie (2018) point to a similar direction, as the results from exploring a Mozambican rural context suggest using word problems or narratives from the school’s community (e.g. from the life in the children’s village), which are related to the teaching of arithmetic operations. This technique requires an active role of the teachers who should consider students’ local knowledge – including early-childhood activities related to young children’s experience (e.g. connecting numeracy and literacy practices with story books in children’s mother tongue, Graven & Jorgensen 2023).

These insights highlight the importance of teacher professional development. Workshops and training programmes should focus on exploring diverse modes of representation, connecting them to different conceptual models of multiplication and facilitating students’ transitions among enactive, iconic and symbolic forms. Teachers should also be encouraged to design classroom activities that foster peer engagement, dialogue and active exploration, using representations that are culturally relevant and conceptually meaningful. Incorporating these elements into the curricula of the Teacher Training Institutes (Institutos de Formação de Professores) can help to bridge theory and practice, ultimately improving the quality of mathematics teaching in Mozambican primary schools.

Finally, future research should expand on these findings by investigating larger samples of teachers and students across Mozambican contexts, examining how manipulatives and multiple representations impact students’ understanding of multiplication. Such studies should assess how teachers facilitate intermodal and intramodal transfers and how these practices influence students’ learning outcomes, engagement and classroom interaction. Thus, we are convinced that the combination of targeted professional development, culturally grounded teaching strategies and a diversity of representations offers a concrete pathway to strengthen both teacher practice and student learning in multiplication.

Acknowledgements

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

Simone Reinhold: Conceptualisation, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Visualisation, Writing – original draft, Writing – review & editing. Albertina J.L. António: Investigation, Methodology, Writing – original draft. All authors reviewed the article, contributed to the discussion of results, approved the final version for submission and publication and take responsibility for the integrity of its findings.

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

As negotiated with the participants of the study, data were not publicly available, but are available on personal request from the corresponding author, Simone Reinhold.

The views and opinions expressed in this article are those of the authors and are the product of professional research. They do not necessarily reflect the official policy or position of any affiliated institution, funder, agency or that of the publisher. The authors are responsible for this article’s results, findings, and content.

References