Introduction

The purpose of the research was to explore how multiplication and division problem-solving contexts can be used to help young children’s development of part-whole thinking. We aim to show how using design study methodology, one teacher in collaboration with the researchers, constructed word problems for a class of 5-year-old children who were then supported in the problem-solving process to develop an early understanding of multiplication and division. The study builds on a body of research on the development of children’s mathematical understanding, with a particular focus on the domains of multiplication and division. It also examines aspects of the instructional program, including task design, representations, and classroom discourse.

Theoretical framework

The research described here is based on work examining the construct of learning trajectories in children’s mathematical thinking (Clements and Sarama 2004, 2014). Learning trajectories are complex, and writers have interpreted them in different ways. Important elements of learning trajectories include the learning goal, developmental progressions of thinking and learning, and a sequence of instructional tasks. In this study, the goal was to develop students’ understanding of number and operations (in particular, the relationship of the parts to the whole), with a focus on multiplication and division problem solving. The developmental progressions were based on the work of Carpenter and colleagues (1999) who identified the transition from counting-based to derived-fact strategies (part-whole thinking) for solving problems. The instructional sequence includes the design and ordering of key tasks (Clements and Sarama 2004), the role of representations (Gravemeijer 1999), and discourse (Artzt and Armour-Thomas 1999).

Background

Developmental progressions in thinking and reasoning about number and operations have been well documented (see, for example, Bobis et al. 2005; van der Ven et al. 2013). Students at the lower stages of progression frameworks solve problems by using counting strategies (Bobis et al. 2005). As they come to appreciate the relationship of the parts to the whole (e.g. additive composition), they use strategies involving partitioning and recombining quantities (part-whole thinking). Typically, the initial focus with younger children is on addition and subtraction before introducing other operational domains, such as multiplication and division, and proportional reasoning which usually occurs in the later school years (Ministry of Education 2007).

The teaching of multiplication and division provides some particular challenges for teachers in order to help students develop a conceptual understanding of these operations (Anghileri 2001; Ball et al. 2001; Lampert 1986; Nunes and Bryant 1996). Contrasting with this is a procedural approach that relies upon physical counting or the memorisation of facts. This is a limiting approach especially when students move to working with multi-digit quantities (Lampert 1986; Smith and Smith 2006). Conceptual understanding is the “implicit or explicit understanding of the principles that govern a domain and of the interrelations between units of knowledge in a domain”, whereas procedural knowledge is “the ability to execute action sequences to solve problems” (Rittle-Johnson et al. 2001, p. 346). Rittle-Johnson and her colleagues suggest that conceptual and procedural knowledge is developed as an iterative process and this knowledge is bidirectional with experiences of each one strengthening the other. That is, once children develop some knowledge of one type, then the other type of knowledge begins to develop as well. These researchers also found that correct problem representation led to improvements in procedural knowledge.

In order to develop students’ understanding of number and operations, students must be provided with well-sequenced instructional tasks (Clements and Sarama 2004, 2014). These tasks may include word problems which can be used to promote students’ growth from one level to the next (Clements and Sarama 2014). There are many components within a word problem that need to be given consideration. These include the mathematical intent, the language, and context. There are certain problem types that represent basic multiplication and division operations. They usually include grouping and partitioning. Early multiplication problems involve discrete quantities, and instead of a student being exposed to multiplication as ‘four times five’, the word problem provides a context where the size of the group (multiplicand) and the number of groups (multiplier) are made explicit within the word problem. Early division problems may be either partitive or quotitive division, with the former requiring an equal-sharing process and the latter the repeated subtraction of the same-sized group. Division problems with remainder are also valuable (Carpenter et al. 1996) because a student needs to take into account how the remainder relates to the problem.

Setting word problems in context is viewed as important to motivate students and to encourage meaningful engagement in problem solving (Chapman 2006; van den Heuvel-Panhuizen 2005). By context, we take Borasi’s (1986) definition, meaning “the situation in which the problem is embedded” (p. 129). The context is usually found within the words of the problem or in a picture or diagram. Context can play at least five different roles (Meyer et al. 2001). These include “(1) motivating students to explore new mathematics, (2) offering students a chance to apply mathematics, (3) serving as a source of new mathematics, (4) suggesting a source of solution strategy, and (5) providing an anchor for mathematical understanding” (p. 523). One context may serve one or more of these five roles. The context may engage the students in the form of text, a visual display, or even an activity. It arouses the students’ curiosities so that they want to explore the mathematical situation.

Making accurate representations of problems is an effective tool for improving problem-solving knowledge, and it is one of the teacher’s roles to support this development (Rittle-Johnson et al. 2001). A distinction can be made between internal and external representations (Goldin and Shteingold 2001). Internal representations are the images we create in our minds for mathematical objects and processes, including visual imagery and problem-solving strategies (Arcavi 2003; Presmeg 2006). Visual images are those mental constructs depicting visual or spatial information. External representations are those that we can communicate to others by drawing, writing equations, and modelling with concrete manipulative materials. The interplay between these two types of representation is fundamental to mathematics learning (Marsh 1990; Pape and Tchoshanov 2001). As students create structured external representations, they are constructing personal internal representations.

Students should have opportunities to support their learning with the use of a variety of external representations (Goldin and Shteingold 2001). In junior classes, in particular, considerable emphasis is placed on the use of materials or manipulatives. These may be unstructured (e.g. counters) or structured manipulatives (e.g. ten frames). Manipulatives are often used in lessons where students follow teacher directives in problem solving and can serve a useful but sometimes limited purpose (Moyer 2001). The teacher’s actions are critical for the effectiveness of manipulatives in supporting learning (Marshall and Swan 2008; Puchner et al. 2008). Although teachers may assume that the use of manipulatives will aid student learning, it has been shown that they can also impede learning (Ambrose 2002; Puchner et al. 2008). What is important is the linking of pedagogy and content, and teachers’ decisions about ways of using manipulatives to ensure students make sense of their representations.

Teaching and learning mathematics provides a variety of linguistic challenges (Mercer and Sams 2006; Schleppegrell 2007; Sigley and Wilkinson 2015). All students, including those for whom English is a second language, have to acquire the vocabulary specific to mathematics, construct meanings (across everyday as well as the mathematics register), and participate in discourses (Setati and Adler 2000). Pedagogical practices can support the student development of mathematics knowledge by paying attention to the way language is used in the mathematics register where certain words and phrases take on technical and precise meanings (Moschkovich 1999). Students need to be able to understand and use language to engage in mathematical learning experiences and construct knowledge and meaning ‘in context’ (Halliday and Hassan 1991). The reading and understanding of vocabulary is merely one aspect of how language and mathematics learning intersect. Knowing mathematical vocabulary assists with reading, comprehension, and communication. Students learn to make sense of multiple meanings of words and the differences between the two registers. Moschkovich (2002) suggests that if the focus is on a student’s failure to use a specific mathematical term, the teacher might misjudge the student’s construction of a mathematical concept. The student may convey meaning for mathematical terms through gesture, drawings, and other representations.

Discourse describes the verbal exchange between and among both teachers and students. It is “the vehicle through which task engagement is facilitated for learning with understanding” (Artzt and Armour-Thomas 1999, p. 216). It includes teacher-student and student-student interactions and questioning. Discourse is a key part of participation in classroom mathematical practices (Cobb et al. 2010). One of the teacher’s roles is to guide the development of children’s skills in using language as a tool for reasoning (Mercer and Sams 2006). Whilst Mercer and Sams focused on talk-based group activities, they recognised the importance of the teacher’s instructional and modelling role as a ‘discourse guide’; that is, one who through instruction, modelling, and design and provision of tasks scaffolds the development of effective use of language in mathematics. Students are expected not only to read, comprehend, and solve word problems but also to participate in classroom mathematical practices (Cobb et al. 2003).

This study was set out to investigate the following overarching research question:

How can multiplication and division word problems be used to support young children’s understanding of number operations? The specific focus was on aspects of instructional practice that included tasks and context, representation, and discourse.

This project used design research methodology undertaken as a partnership between researchers and practitioners (Barab and Squire 2004; Cobb et al. 2016). The focus was on “evolving instructional design” (Cobb et al. 2016, p. 482) where the researchers collaborated with the mathematics teacher to investigate the process of students’ learning in the number domain (Cobb et al. 2016). Additionally, design study enabled exploration at the practice-based pedagogical level and the theoretical level within the context of the classroom (Ball and Forzani 2007). It also allowed for a generative framework about the teaching and learning of specific mathematical concepts to be elicited from the study (Kelly 2004). The findings of this design study could strengthen pedagogical practices aimed at enhancing students’ mathematical thinking and understanding in number and number operations.

The study

We report on the practice of one experienced teacher, Sara (a pseudonym), who participated in the first year of a larger 2-year study designed to explore the development of young children’s part-whole thinking, using multiplication and division problem-solving contexts (Bicknell and Young-Loveridge 2015). Sara taught year 1 students (5-year-old children) from a culturally diverse, decile 5,Footnote 1 urban school (complete data was obtained for 15 students, 3 of whom were English language learners (ELLs)). This was the first time she had included multiplication and division in her mathematics program for this year level. Part of Sara’s rhetoric during mathematics lesson was to remind the students that when they were doing mathematics, they were “working like young mathematicians”. One of the purposes of this phrase was to help students move to the mathematics register and to build positive dispositions towards mathematics. She wanted her students to have the confidence and perseverance to tackle challenging problems.

Consistent with design research, a variety of data collection methods were used. The data sources were the following: semi-structured teacher interviews, individual task-based student interviews (pre- and post-assessment), video-recorded lessons, and artefacts, including children’s individual project books, class modelling books, and teacher journals.

Video footage of Sara’s teaching during two 4-week blocks in June and October–November (eight lessons) was analysed using NVivo to answer the research question. The videos of the lessons were divided into 1-min segments. The researchers identified the themes for the data analysis based on a framework for reflective practice developed by Artzt and Armour-Thomas (1999) that focused on three lesson dimensions. The three lesson dimensions were ‘tasks’, ‘learning environment’, and ‘discourse’. The three indicators for the tasks dimension of the framework were motivational strategies, sequencing/difficulty level, and modes of representation. For the learning environment dimension, the indicators were social and intellectual climate, modes of instruction and pacing, and administrative routines. The indicators for the discourse dimension were teacher-student interaction, student-student interaction, and questioning. The NVivo analysis was used to identify key episodes in teacher practice in relation to these indicators. After the first lesson was analysed to generate nodes, the team met to discuss these and minor adjustments were made to the coding system.

The individual diagnostic interview tasks were selected and adapted primarily from tasks developed to assess progressions in numeracy (Ministry of Education 2008). Students’ responses to assessment tasks were coded according to accuracy, and the strategy the student used to solve the problem was given a weighting according to the level of sophistication.

In the project, each lesson comprised a similar format, beginning with a number knowledge activity with the whole class. This was followed by the provision of a word problem (recorded in a class modelling book) for all students to solve. Manipulatives were provided, and after 5–10 min of independent time, the teacher invited selected students to share their strategies, which were then recorded in a modelling book with a focus on the key mathematical ideas. The students then worked independently on parallel problems (with the same problem structure) that had been copied and pasted in their own project books. For the parallel problems, the students had a choice of two multipliers (or dividends) increasing in magnitude, and a self-selected quantity. Students were expected to use multiple representations that included manipulatives, drawings, and equations.

Findings

We present some of the key findings from our teacher and student interviews, and video analysis showing how multiplication and division problems were used to support young children’s understanding of number and number operations. Specifically, the focus was on aspects of instructional practice that included tasks, context, representation, and discourse.

Instructional tasks

The instructional tasks for multiplication and quotitive division word problems were developed collaboratively with the teacher and researchers. The problems matched the mathematical learning goal and instructional sequence. In the first phase of the project (May–June), students solved multiplication and quotitive division problems by making groups of two. These were followed by similar problems requiring the students to work with groups of five. In the second phase of the project (October–November), students worked with quotitive division into groups of two with remainder (odd numbers) and then quotitive division problems with multiples of ten, followed by quotitive division into groups of five and ten with a remainder.

The problems used meaningful contexts that were related to children’s own experiences in order to engage them in the tasks. The contexts for groups of ‘two’ included pairs of socks, mittens, and gumboots. For groups of ‘five’, the problems involved candles on a cake and bunches of bananas, and for groups of ‘ten’, the context was ten eggs in a carton or ten chocolates in a tray. Five candles on a cake was used as a context for the multiplication by five problems (turning 5 years old was the children’s most recent birthday). The problems included other contexts from topics being explored in science and social science studies. Sara wrote in her reflections: “I look quite critically at meaningful contexts now, something that I didn’t do well before the project began”.

The following examples illustrate problem types for both multiplication and quotitive division for groups of two, five, and ten:

  • (×2) Five children each get two socks from the bag. How many socks are there altogether?

  • (÷2) There are 14 socks in the basket. How many pairs can we make?

  • (×5) There are four bunches of bananas. Each bunch has 5 bananas. How many bananas are there altogether?

  • (÷5) We have 15 lollies. We put 5 lollies in each bag. How many bags do we have?

  • (×10) John has 4 cartons of eggs. Each carton has 10 eggs in it. How many eggs does John have altogether?

  • (÷10) There are 20 eggs. Each carton holds 10 eggs. How many full cartons are there?

The early division problems into groups of two (pairs) with remainder exposed the students to odd and even numbers by recognising there was ‘one left over’. Division with two-digit quantities into groups of ten with remainder introduced children to early place-value understanding where the ‘tens’ numeral was represented by the number of groups of ten and the ‘ones’ by the ‘leftovers’. There was a deliberate move to include problems with larger two-digit numbers for the whole-class problem so that all children were encouraged to move from counting by ones to skip counting. The following vignette is based around a quotitive division problem into groups of ten. The problem was: “There are 63 eggs. Each carton holds 10 eggs. How many full cartons are there?”

Teacher: We need to think smart. Sixty-three is a really big number to count to if we go by ones.

Children: Twos.

Teacher: Even if we go by twos.

Children: Fives.

Teacher: Even if…

Children: Tens.

The teacher provided the egg cartons and a basket of cubes and suggested the children make groups of 10. The teacher then supported the children as they counted out the cubes. After filling some egg cartons, the teacher and whole class counted in tens: 10, 20, 30, 40, 50, 60. The teacher then instructed the children to count out the rest of the eggs (61, 62, 63).

Teacher: So we have our 63 eggs. Now we need to answer the problem.

Teacher: [Afi] can you please tell me how many full cartons have we got?

Child: Six

Teacher: And how many have we got left over?

Child: Three

Teacher: So our answer to our problem is how many full cartons of eggs have we got?

Children: Six

Teacher: And?

Children: Three left over.

Teacher: Now it’s time to write that. How many eggs did we start with?

Representations

The multiple modes of representation used by the students included manipulatives, drawings, and number sentences. A variety of manipulatives were provided for the children to model the various problems. These included pairs of baby socks, mittens, card material with images showing ‘groups of” two and five, egg cartons (tens), and Unifix cubes. Ten-frame material and counters were also available for independent work. Initially, manipulatives were used by the students to model the problem posed at the start of each lesson. They were then expected to use drawings in their individual workbooks to show how they had modelled the problem. Initially, students drew solutions in quite a free form but teacher guidance was given to support students’ drawings for greater accuracy of the ten frames.

Sara tried to progress the children’s thinking from simple strategies for solving multiplication problems such as counting by ones, to a more sophisticated skip counting, repeated addition, or derived-fact strategy. The teacher recording reflected the additive thinking, followed by the multiplication equation. When recording the division equations, the teacher encouraged the students to provide the related multiplication equation to build an understanding of the inverse relationship between the two operations. The students were encouraged to be ‘mathematicians’ and record their thinking in smart ways, so words were replaced with symbols, for example ‘groups of’, by ‘x’ and ‘leftovers’, ‘loners’ (as some children called them), or ‘remainders’ by ‘r’. Sara often supported the recording of symbols with actions. For example, multiplication was modelled by crossing her arms in front of her. She would often say: “we are making ‘groups of’, so we are using the multiplication sign”, as she crossed her arms. However, the students did not always transfer the formal representations modelled by the teacher into their individual books. She commented in her reflections at the end of week 1 that:

They needed to be reminded to come back to the basics and draw the pairs of socks in their book. To hope that they could go straight to writing the number sentence was too much of a jump… Groups/pairs/how many altogether! Continues to be a challenge, they are having difficulty seeing that a pair is a group of two. This confuses the answers they are getting in their number sentences, and they often write the group number [the multiplier], e.g. 2 + 2 + 2 + 2 = 4.

The teacher used the questions ‘how many groups?’ and ‘how many in each group?’ to draw children’s attention to the distinction between the multiplier and the multiplicand in order to address the confusion evident in the preceding example.

In the following excerpt, Sara encouraged children to provide the number sentence for the following word problem after the children have modelled it with manipulatives: The problem was: “Rabbit has 23 lollies. He puts 5 lollies in each bag. How many bags of lollies are there?”Footnote 2

Teacher: How many lollies do we start with?

Children: Twenty-three. [Teacher recorded 23]

Teacher: What did we do? Did we add them, multiply them, divide them, or subtract them? What did we do?

Teacher: We made them into groups. So we did the? [Teacher made the division sign in the air]

Children: Divide. [The teacher recorded the division sign in the modelling book]

Teacher: Across, dot, dot [division symbol ÷], because we made them into groups. How many in each bag?

Children: Five.

Teacher: And to make a sentence we need?

Children: Equals. [The teacher records 23 ÷ 5 =]

Teacher: How many bags do we get?

Children: Four.

Teacher: Say it confidently. How many bags?

Children: Four. [The teacher records 4]

Teacher: Four groups of?

Children: Five.

Teacher: But there are some leftover. [Teacher points to the 3 cubes]

Children: Three

[Teacher recorded ‘and three left over’ and completed the equation ‘23 ÷ 5 = 4 and 3 left over’.]

Discourse and the mathematics register

The problems were constructed by the teacher and researchers using familiar contexts and worded in such a way that they were linguistically accessible by all students. There were some particular words and phrases in relation to multiplication and division that provided a challenge for these young learners. The additional needs of the ELLs with word problem structure, instructions, and questioning were taken into consideration. For the first problems in the series involving ‘groups of two’, it was decided to introduce the word ‘pair’ as the collective noun for a group of two. Instead of an instruction for each child to get two socks, this was replaced with asking each child to “get a pair of socks”. Sara explained:

The problems helped to consolidate the idea of a pair, but still some children are confused with the wording and the idea that ‘a pair is a group of two’. Encouraging them to draw and circle each pair did help.

Other words that Sara identified as important to focus on were ‘each’, ‘altogether’, ‘full’, ‘remainder’ and the specialised language unique to the discipline of mathematics including multiplication and division. Sara also used the words ‘times’ and ‘divided by’. Occasionally, she inadvertently talked about a ‘sharing’ process instead of making ‘groups of’ a particular size when the instructional goal was to introduce a quotitive division problem.

The whole-class discussions at the start of each lesson were focused on the mathematical goal and the solving of one problem. The discourse varied from sharing strategy talk to a more focused procedural emphasis on the action inherent in the problem structure. Opportunities were provided for children working at different levels to share their strategies in order to help them see the links between their strategy and those of others. The teacher supported their thinking by recording multiple representations such as drawings and pictures, followed by a series of matching equations (e.g. 6 ÷ 2 = 3 and 6 − 2 − 2 − 2 = 0). Sara’s questioning, as evidenced in the excerpts below, was more about acquiring correct answers to questions, with few instances of questioning for student explanation and justification. For example, the teacher introduced the egg cartons for groups of ten to the children.

Teacher: Each cartoon holds how many eggs?

Children: Ten.

Teacher: But you need to prove it to me. I am not convinced.

[The teacher gave a handful of cubes and an egg carton to a child and asked her to prove how many eggs it held.]

Teacher: [Nina]. I would really like you to prove it to me.

[Nina put one cube (‘egg’) into each compartment of the carton until it was full.]

Teacher: Okay, [Nina]. How many eggs are in the carton?

Child: Ten.

The discourse was teacher led because Sara had in mind a certain line of enquiry. She wanted to build an understanding of multiplication and division and the specific language of the problems. Her goal was to do this through modelling and guiding their use of language. The class discussions were not always extensive, but Sara made very clear the nature and purpose of the word problem and encouraged student engagement in modelling the problems using a variety of suitable manipulatives.

Sara summed up her experiences in relation to the role of language, contexts, and representations:

I can just remember looking up and getting confused with the word problem myself because there were so many words in it and so that clearly would have to be an issue for them. Some of them are learner-readers… It’s about getting the language right and making the problems that we agonised over, contextual, because if they don’t understand the context, we’re wasting our time. It’s got to be about them, they’re five, it’s all about them - how many cakes, candles, shoes, slippers, it’s about them… the role of equipment. Equipment is so important, but also exposing them to being grown-up mathematicians who would use a multiplication sign. The only problem with that is sometimes it was confused with addition later on in my lesson.

Student assessment results

The 15 children were assessed individually using a diagnostic task-based interview before (T1) and after (T2) the first year of the research project. Ten tasks focused on number operations were selected from a comprehensive diagnostic assessment used in the larger study (see Bicknell and Young-Loveridge 2015). Although the research project examined the use of multiplication and division problem-solving contexts, addition and subtraction strategies were also assessed as students often used these operations to solve the instructional word problems (see Table 1). It is evident from the table that students improved on the tasks designed to assess multiplication and division. The greatest improvement was in the number of students who could successfully solve 4 × 5 (60%). One third of students got the answer initially, but by the end of the study, almost all (93%) students were successful. Similarly, the improvement on the 3 × 10 task increased from 27 to 80%. On division tasks such as 10 ÷ 2 (quotitive) and 8 ÷ 4 (partitive), there was an increase of 47% in the number of students who were successful by the end of the study. Not only more students were successful by the end of the study but also more of them chose to use a more sophisticated strategy for solving the problem than they had initially. For example, just under half of the students (47%) changed to using a skip counting strategy to solve the 6 × 2 task, whereas just over one quarter had previously counted all the items (27%). Students also improved in their success on addition tasks such as 5 + 8 (groups of beans covered by cardboard), from 7 to 60%. Initially, no child was successful on 14 − 5 (covered), but by the end of the project, one third of the children were able to work out the answer.

Table 1 Percentages of students successful on each of the tasks before and after the intervention period (underlined are improvements of 33% or more; n = 15)
Full size table

Weighted scores were calculated for each of the addition, subtraction, and multiplication problems. A score of 1 was given for getting the correct answer by using a ‘count all’ strategy. A score of 2 was given for counting on, counting back, or skip counting to find the answer. A response was assigned a score of 3 if a recalled- or derived-fact (part-whole) strategy was used to work out the correct answer. For the division problems, a total score was calculated over the four tasks listed in Table 1. Differences between the weighted scores (or total score, in the case of division) after the sequence of lessons (T2) and corresponding scores prior to the start of the lesson sequence (T1) were used to calculate effect sizes. These ranged from just over one third of a standard deviation (0.41) for the 3 + 4 task to more than 2 standard deviations (2.17) for the division tasks. Another notably large effect size of 1.59 standard deviations was found for the 4 × 5 task. Interestingly, the improvements were mostly in moving from counting by ones to counting on/back and skip counting. Only one child showed evidence of using a part-whole (derived-fact) strategy (T2) and then only for the 4 + 3 and the 8 + 5 tasks.

Conclusion and discussion

The teaching of multiplication and division is not as straightforward as sometimes presumed. Teachers of junior classes are used to focusing on early number knowledge, counting, and addition and subtraction problems, as reflected in many curriculum documents. It is commonly accepted that addition and subtraction precede the teaching and learning of multiplication and division (e.g. Ministry of Education 2007; van der Ven et al. 2013). We wanted to challenge this practice by working alongside a junior class teacher using design research methodology to explore the use of multiplication and division word problems as instructional tasks.

The results reported here also provide support for the introduction of multiplication and division problems to young children early in their schooling. The multiplication and division word problems benefited student thinking and reasoning in addition and subtraction as well as multiplication and division. Many of the effect sizes found were reasonably large with all, but the 3 + 4 task yielding an effect size of 0.60 or larger, magnitudes that are in the moderate to large range (Fan 2001). The effect size for improvement on the four division tasks was more than three times the 0.60 magnitude that, as Hattie (2009) claims, signifies excellent progress for 1 year of schooling, and the effect size for the 4 × 5 task was 1.59, more than double the 0.60 magnitude. These results challenge the view that the teaching and learning of addition and subtraction precedes that of multiplication and division (van der Ven et al. 2013).

The instructional tasks were sequenced according to mathematical complexity from groups of two to five, to ten, and introduced multiplication followed by division. These word problems enabled students to make progress from one level to the next on a learning trajectory (Clements and Sarama 2014). For example, many students used increasingly complex strategies, progressing from counting all to skip counting. The ultimate goal, which proved to be too ambitious with these young children, was for them to use recalled or derived facts (part-whole thinking). There were challenges for the teacher in retaining a focus on developing conceptual understanding of part-whole relationships, rather than working just on procedures for solving problems. Recognising the distinction between conceptual and procedural knowledge and building on these are challenging for students and teachers (Rittle-Johnson et al. 2001).

The lessons began with carefully worded problems and used meaningful contexts that were engaging and motivated the students (Chapman 2006; Meyer et al. 2001; van den Heuvel-Panhuizen 2005). The teacher’s knowledge of the interests and experiences of her learners influenced the choice of situations in which the problems were embedded (Borasi 1986). The teacher encouraged the use of a range of representations that included appropriate manipulatives. Unlike the teachers in Puchner et al.’s (2008) study, the teacher in this study made pedagogical decisions to link her mathematical goal with the chosen manipulatives so that students made sense of their learning. However, there were challenges for the teacher when introducing other representations such as diagrams and written equations. Multiplication and division were new to the students, and they grappled with making connections between the models and the symbols. By encouraging students to “work like a mathematician”, they were exposed to particular symbols and equation structure with explicit links was also made between multiplication and repeated addition, division and repeated subtraction, and multiplication and division as inverse operations. The teacher used a variety of external representations to support students’ learning by developing their internal representations (Goldin and Shteingold 2001).

The contexts of multiplication and division provided opportunities for the teacher to build students’ understanding of particular concepts and mathematical vocabulary. The students participated in discourses that supported sense making between words, symbols, and models. For example, ‘groups of’ was explicitly linked to the multiplication symbol and ‘into groups of’ to the division symbol. The ‘leftovers’ from a quotitive division problem were re-named as ‘remainder’. This helped students to make sense of particular meanings of words when working in the mathematics register (Moschkovich 1999; Sigley and Wilkinson 2015). The teacher attempted to build productive classroom discourse through reading the word problems together, questioning for comprehension, and collaborative problem solving through modelling and representation (Cobb et al. 2003).

This design study used a methodology that was effective for classroom-based research in order to examine how multiplication and division problem-solving contexts could be used to develop young children’s understanding of number and number operations (Cobb et al. 2003). The collaboration between researchers and the teacher, a key element of design research, led to the development of instructional tasks that contributed to improvements in students’ problem solving. The findings were consistent with Clements and Sarama’s (2004, 2014) theoretical model of learning trajectories, including a clear goal, developmental progressions of thinking and learning, and a sequence of instructional tasks. The particular focus on aspects of the teaching such as the instructional tasks and their relationships to student learning, use of tools and representations, and classroom discourse was consistent with Cobb and colleagues’ (2003) model of classroom-based design research.

Whilst it is acknowledged that student ideas and competencies develop over a long period of time, the evidence from this study suggests that it can be advantageous to a student’s learning in mathematics to work with multiplication and division word problems at a young age, provided that they are meaningful and supported by key instructional practices. The finding of improvements in addition and subtraction, despite the classroom focus being on multiplication and division, challenges developmental learning progressions and curriculum documents that position the learning of multiplication and division after addition and subtraction (e.g. Ministry of Education 2007).