As one of the ‘big ideas’ of mathematics (Hurst & Hurrell, 2014); (Siemon et al., 2012), multiplicative thinking is characterized by a complex set of connected ideas (Hurst & Hurrell, 2016) which facilitate the learning of much of the mathematics taught from the middle primary years onwards. The Australian Curriculum, Assessment and Reporting (Curriculum & Assessment, 2020) alludes to the importance of connected mathematical thinking in describing proficiency in understanding as involving adaptable and transferable mathematical concepts upon which students can build a conceptual level of understanding. Thinking multiplicatively is highly dependent on understanding the connections between ideas and how they underpin mathematical procedures.

Researchers (Clark & Kamii, 1996)(, 2006) have noted that students who do not think multiplicatively have considerable difficulty comprehending higher-level concepts such as fractions, proportional reasoning, and algebra (Clark & Kamii, 1996)(, 2006). This link underlines the vital importance of teachers' understanding of how the various elements of multiplicative thinking are connected, so that they can make such connections explicit for students . The following definition (Hurst, 2017) is based on the work of (, 2006) and (Siemon et al., 2012). Multiplicative thinking is demonstrated through its ability to:

Skemp's 1976 seminal paper highlighted the importance of relational understanding, in which connections among aspects of mathematical structure are evident and used by children. This can be equated to the conceptual understanding described by (Rittle-Johnson & Schneider, 2015) as a linking web of relationships. This link might be concerned with connecting two previously known ideas or relating one newly learned concept to another that was already known (Rittle-Johnson et al., 2016). The links between ideas are considered to be at least as important as the aspects of the structure to which they connect (Rittle-Johnson & Schneider, 2015). (, 2009) discussed the importance of children 'sunderstanding mathematical structures by understanding how ideas are connected and explaining these connections. Hence, conceptual understanding involves more than isolated aspects of knowledge. Rather, it is about children seeing how ideas are connected with guidance from teachers, and making such connections explicit.

If students are to work flexibly with a range of numbers, explicit teaching of the many connections within the broad idea of multiplicative thinking is necessary to develop a conceptual understanding. However, (Richland et al., 2012) noted that many schools do not teach their students a conceptual understanding of mathematics, something that could support them in their capacity to transfer and generalize the mathematics they are learning. Specifically, we explore students' understanding of arrays, possible links between partitioning based on place value and the distributive property of multiplication, and the identification and generation of extended multiplication facts. In addition, we considered the students' use of written algorithms and associated methods in light of their understanding of the aforementioned factors.

Primary/junior-school students are expected to use a standard written algorithm to multiply multidigit numbers (ACARA, 2017;NGA Centre, 2010). It seems reasonable to believe that using the algorithm should be preceded by an understanding of why it works through an understanding of the mathematical structure that underpins it. There are several aspects of the mathematical structure that are relevant: the array and place value partitioning, both of which directly inform an understanding of the distributive property of multiplication, and the base ten property of place value, which informs the generation of extended number facts. The distributive property and the use of extended number facts then inform an understanding of the written algorithm.

Various mathematics researchers and educators (Haylock, 2014)(Squire et al., 2004)(The Origo handbook of mathematics education, 2007) have described the distributive property as distributing multiplication operations over the addition of a number of parts. Symbolically, this is represented as (a + b) × c = (a × c) + (b × c). Haylock noted that although people may not have realized that the distributive property underpins the multiplication algorithm, they have been unconsciously using the algorithm for many years. (, 2013) suggested that students as young as grade 3 have an intuitive understanding of area conservation when using arrays. They are readily able to understand that a 5 x 12 array can be seen as 5 x (3 + 4 + 5). This suggests that they have a conceptual understanding of the distributive property, where multiplication is distributed over addition, which is one of the key ideas for using the formal multiplication algorithm. By following a developmental progression from arrays through the grid method to a formal algorithm, students can better understand the purpose of the algorithm by understanding the component substructures on which it is built. (Matney & Daugherty, 2013) also suggested that the array is a useful prerequisite to ideas of commutativity and distributivity, and that manipulation of arrays offers the valuable experience of recognizing efficient ways of grouping numbers and links to the partial products needed for the algorithm. By offering students dot arrays for grouping, they are encouraged to seek efficient and fluent methods for counting dots.

(Matney & Daugherty, 2013) also suggested that the array is a useful prerequisite to ideas of commutativity and distributivity, and that manipulation of arrays offers the valuable experience of recognizing efficient ways of grouping numbers and links to the partial products needed for the algorithm. By offering students dot arrays for grouping, they are encouraged to seek efficient and fluent methods for counting dots.

The array can be used to develop an understanding of the distributive properties that inform the grid method for multidigit multiplication. It should be used with smaller numbers, such as two digits multiplied by one digit (e.g., 14 × 9). Once an understanding of this is developed, the grid method can be used for larger numbers. This process is illustrated in Figure 1. A standard algorithm can then be developed and used to understand the distributive properties (Figure 1).

The distributive property of multiplication can be considered the key to understanding the vertical multiplication algorithm, which is taught through various methods by teachers. In describing the importance of this property, (Kinzer & Stanford, 2013) discussed partitioning. They noted that it is critical to enable students to understand multiplication through an understanding of distributive properties. (Kinzer & Stanford, 2013) also linked the importance of the array to developing an understanding of the distributive property, noting how it helped reduce relatively complex calculations to simpler ones. Their view supports the Common Core State Standards for Mathematics (NGA Centre, 2010), which also emphasize the importance of the link between multiplication and the array and, hence, the distributive property.

Various researchers have noted the importance of the underpinning mathematics being discussed here. (Norton & Irvin, 2007) stated that concepts such as distributive law underpin algebraic thinking and must, therefore, be emphasized in the primary years. (Young-Loveridge, 2005) found that part-whole strategies for multiplicative thinking allow for flexible and efficient methods when solving, for example, 6 × 24. This can be seen as 6 × 20 added to 6 × 4, or perhaps as 6 × 25 minus 6. The known number facts (perhaps 6 × 10 or 6 × 20) can be shaded in an array, and other facts (6 × 4) can be identified from the remains. The answers for the number of facts were then combined to produce a total. Young-(Young-Loveridge & Mills, 2009) further emphasized that a quality understanding of multiplication results from knowledge about the distributive property. They stated that multiplication strategies based on partitioning and distributive properties were more advanced than those based on other ideas, such as repeated addition. Given these points, it is interesting to note that, in his discussion of the distributive property, (Haylock, 2014) did not use the term ‘partition’ but rather described it in terms of ‘breaking up’ the number. However, other mathematics educators ((Thompson, 2003); (Askew, 2012); (Cotton, n.d.); (Lemonidis, 2016)) specifically use the term ‘partitioning’ when describing the computational strategies used by children. Indeed, Cotton discusses children’s efforts to explain how they partition three-digit numbers when multiplying them. This suggests that if children are to be taught such strategies, they should be encouraged to use the appropriate terminology.

(Kaminski, 2002) studied how a group of pre-service primary teachers flexibly used the distributive property when solving multiplication exercises. He made the interesting observation that some pre-service teachers were aware of the property, and many others did not know how it was applied to multiplication. Kaminski’s sample consisted of pre-service teachers as opposed to in-service teachers, but his observation begs the question of whether the majority of in-service teachers would be clear about how distributive properties can be applied. Perhaps the same situation might apply to recognizing (or not recognizing) the connection between place-value partitioning and distributive properties (Figure 2).

Ross (2002) described the base ten properties of place value as showing how places increase by a power of ten moving from right to left. This understanding, combined with knowledge of basic multiplication facts, enables students to generate extended number facts, where one or more of the factors are increased by a power of ten. The associated understanding is that if one factor is increased by a power of ten, then the product (or multiple) must also increase accordingly. The ability to generate extended number facts helps students understand the standard multiplication algorithm (see Figure 2). In particular, they understand that the placement of a zero in the second line signifies 26 × 10 and not 26 × 1, and that the product must be ten times greater than 26. (, 2015).

suggested that algorithms be used to minimize the cognitive load, as they break up a complex calculation into several more easily managed sections. However, they also identified an interesting issue in the teaching of algorithms. If taught procedurally, a student can ‘suspend place value’ (Jazby & Pearn) and carry out the final part of the calculation in Figure 2 as 2 × 1. However, to fully understand what is happening in the algorithm, and why an answer might be correct or otherwise, a student needs to know that the ‘2 × 1’ really represents ‘2 tens × 1 ten’ or ‘2 × 1 × 10 × 10’ and that the result must be 100 times greater than 2.

There appear to be clear links between the use of the array, an understanding of place-value partitioning (, 2015), and distributive properties ((Matney & Daugherty, 2013); (Kinzer & Stanford, 2013)). Similarly, it is reasonable to suggest that the base ten property of place value underpins the use of extended number facts (Ross, 2002). Both distributive properties and extended facts underpin the understanding and efficient use of standard multiplication algorithms. (Pickreign & Rogers, 2006) noted the connection between the area or grid model for multiplication and the array as the basis for understanding the algorithm. They stated that, “Connecting the array model of multiplication to an area model for multiplication has a certain appeal” (Pickreign & Rogers, 2006). These links formed the theoretical framework of this study. This is illustrated in Figure 3.

(Hurst & Huntley, 2018) noted that developing an algorithm through the connections described takes time and effort; however, this time is well invested and can minimize the need for later remediation. (Hurst & Hurrell, 2018) described several learning pathways for developing a conceptual understanding of an algorithm. Therefore, we decided to seek an answer to this research question: To what extent can primary school students correctly use a written multiplication algorithm and demonstrate a conceptual understanding of the mathematics that underpins its use? As noted earlier, conceptual understanding is characterized by links and relationships. The study sought to learn the extent to which children had a ‘connected view’ of the mathematical structure and were able to explain what they were doing in terms of one or other aspects of the structure.

The purpose of these questions was to determine whether the children used standard place-value partitioning to determine their answers. This could be shown by 'carrying the 4' or by using partial products. We also wanted to determine whether they used the specific term 'partition. '

This question was designed to determine whether the children could correctly identify instances in which the distributive property was correctly applied. The second part of the question was designed to determine whether the participants could explain how the property worked in terms of partitioning.

The purpose of the first question was to determine whether the children could recognize that all four extended number facts could be generated from 24 × 6 = 144. The second question sought to determine whether the students could write extended multiplication and division facts. This study only considers responses to multiplication facts, as the inverse relationship between multiplication and division is not its focus. The multiplication algorithm

During the interviews, the children were asked to show how they would work out the answers to 29 × 37.

As noted above, not all the students who used partitioning to calculate the answers for 6 × 17 or 9 × 15 correctly identified the instances of the distributive property. However, every student who identified the two correct distributivity applications used partitioning. In addition, all students who explained how the distributive property works did so in terms of partitioning. In other words, some students appeared to connect the two ideas, whereas others did not. Figure 6 shows sample responses from Students IIS and DMA.

The sample from Student DMA was particularly interesting. DMA was one of the four students who correctly identified the two applications of the distributive property but found it necessary to perform the calculation to ensure that the second and fourth options gave the same answer of 89 × 3. It is also possible that students who did so worked out their answers before making their selections. If this were the case, the proportion of students who identified and trusted the distributive property would be less than 40% and 48% for Years 5 and 6, respectively.

The first question was purposefully worded, and all four options were correct. This was done to determine whether the students’ understanding was robust, because if they assumed that there had to be at least one incorrect option, their response would be wrong. The number of students who made this assumption is unknown. Responses to these two questions indicated that more students (60% and 52% for Years 5 and 6, respectively) were able to write at least three extended multiplication facts than were able to identify them (47% and 33%, respectively). In addition, fewer could explain why extended number facts worked (40% and 19%, respectively). The fact that more students could write extended facts than explain how they worked conceptually suggests that they may have been shown a procedure for writing extended facts. Figure 7 shows samples from Students IHA and OST, showing how some students explained extended facts using the idea of ‘times as many.’ It also includes samples from Students FWI and PME, illustrating how students typically recorded their extended number facts.

The majority of students in the sample (87% in Year 5 and 71% in Year 6) correctly calculated the answer to 29 × 37. Of the 15 Year 5 students, two were unable to perform the calculation, two used a grid method, one used a partial-product method based on partitioning, and ten used the standard algorithm. One student who used the algorithm also performed calculations using the grid method. Of the Year 6 students, 15 used the standard algorithm and six were unable to complete it. Both Year 5 students (GPO and MAU) who used a grid method based on partitioning also correctly used partitioning to complete the one-digit-by-two-digit multiplication calculations in the earlier MTQ question (6 × 17 or 9 × 15). Figure 8 shows samples of their work, as well as that of RCO. She used a partial-product method based on partitioning. RCO also correctly used partitioning in an earlier MTQ question.

Students who used the algorithm recorded their work correctly in a range of ways. Some students reversed the order of the numbers, others recorded the numbers they ‘carried,’ and one student (WBL) used a four-line algorithm. If students wrote a larger number at the top, they were asked why they did so. All responded, ‘I always do it that way’ or ‘Because it is easier that way.’ All who did so agreed that it made no difference to their answer, which suggests that they understood the commutative property of multiplication. Examples of students using the algorithm are shown in Figure 9.

There was considerable variation in the proportion of correct responses to the MTQ questions and students' ability to calculate the answer to questions 29 × 37. Hence, it is important to consider the structural connections between the ideas assessed in MTQ questions and how they underpin the written multiplication algorithm. The discussion now considers the students in two groups: the 28 students (13 Year 5 and 15 Year 6) who correctly completed the calculation for 29 × 37, and the eight students who did not.

Use of the array to show 4 × 3

Of the 13 Year 5 students who performed the correct calculation for 29 × 37, ten drew an array of 4 × 3, and of the 15 Year 6 students, 6 drew an array. It was noted earlier that the proportion of Year 6 students who drew an array was less than that of Year 5s and it was suggested that this might be due to a progressive decrease in the use of the array among older students. However, as discussed earlier (Benson et al., 2013;(Matney & Daugherty, 2013), the array underpins the distributive property, which, in turn, underpins the grid method and written multiplication algorithm. It is possible that these students were taught to use the algorithm without being exposed to its progressive development from the array to grid method.