Multifaceted assessment of children’s inversion understanding

Highlights

• Children’s inversion understanding was investigated using multifaceted assessment.

• Grade-related improvement was observed in children’s inversion understanding.

• Different inversion profiles were observed.

• Children with different inversion profiles showed different math achievement.

Abstract

The current study was aimed at examining various theoretical issues concerning children’s inversion understanding (i.e., its factor structure, development, and relation with mathematics achievement) using a multifaceted assessment. A sample of 110 fourth to sixth graders was evaluated in three different measures of inversion understanding: evaluation of examples, explicit recognition, and application of procedures. The participants were also evaluated on their mathematics achievement. A one-factor structure best explains inversion understanding involving different arithmetic operations. Grade-related improvements were observed in some facets of inversion understanding. Latent profile analysis using the three inversion measures revealed seven classes of children with different inversion profiles. Furthermore, classes with better inversion understanding also had higher mathematics achievers. The current findings provide evidence to support the multifaceted nature of inversion understanding, grade-related improvements in children’s inversion understanding as well as the relation between inversion understanding and mathematics achievement.

Introduction

Children’s conceptual knowledge in mathematics has captured an increasing amount of attention in the mathematical cognition literature (Crooks & Alibali, 2014), possibly because children who rely more on conceptual knowledge seem to be advantaged in general mathematics performance (Hallett, Nunes, & Bryant, 2010). After controlling for procedural knowledge, conceptual knowledge uniquely predicts the flexibility of using different procedures for mathematical problem solving (Schneider, Rittle-Johnson, & Star, 2011). In the current study, we focused on one particular aspect of conceptual knowledge: the understanding of the inversion principle.

The inversion principle is the arithmetic principle indicating the relations between arithmetic operations; some arithmetic operations have opposite effects on the initial quantity, and executing both operations cancels both effects. For example, adding a number y to the number x and then subtracting y from the sum returns the original number x. The same applies to multiplication and division. Children who understand inversion know that no computation is required in such conditions. The investigation of children’s understanding of the inversion principle provides a window for researchers to better understand children’s knowledge of the four arithmetical operations (Bisanz, Watchorn, Piatt, & Sherman, 2009). Piaget (1952) even suggested that the true understanding of addition and subtraction does not come until children have realized that the two arithmetic operations cancel each other out.

Despite the increasing number of studies conducted on children’s inversion understanding (Canobi, 2009, Dubé and Robinson, 2017, Rasmussen et al., 2003, Robinson et al., 2016, Robinson et al., 2018, Vilette, 2002, Watchorn et al., 2014), the designs of these studies vary along several dimensions. For example, whereas Rasmussen et al. (2003) examined the understanding of addition–subtraction inversion among kindergarteners and first graders with the use of objects, Robinson et al. (2016) investigated the understanding of multiplication–division inversion among sixth to eighth graders in a symbolic context. Furthermore, some of the studies involved rather small sample sizes (e.g., N = 48 in Rasmussen et al., 2003, N = 66 in Vilette, 2002). Because of these factors, the findings from these studies do not always agree, and further efforts are needed to obtain a more comprehensive picture concerning children’s inversion understanding.

In the current study, we focused on three aspects of inversion understanding where findings are inconsistent. The first issue is whether the understanding of the inverse relation between addition and subtraction is of a similar nature as that between multiplication and division. On the surface, these two types of inversion look similar given that both involve undoing an arithmetic operation by another arithmetic operation. Yet, a more microscopic view into the nature of these two types of inversions suggests some minor differences between the two. For the addition–subtraction inversion, the individual needs to realize that b − b = 0 and that adding 0 to any numbers results in no change. For the multiplication–division inversion, however, the insight comes from the realization that b ÷ b = 1 and that multiplying 1 by any number does not alter its value. Because of this, Robinson & LeFevre (2012) suggested that these two types of inversions are actually different in nature. This is supported by the finding that presenting one type of inversion problem does not seem to promote the use of inversion shortcuts in another type of inversion problem (Robinson, Ninowski, & Gray, 2006). However, within the same study, it was found that the individual differences in use of the inversion shortcut in the two types of problems are significantly correlated at r = .59. Therefore, it is difficult to conclude whether the two types of inversion reflect a single construct of inversion understanding.

The second issue concerns the development of children’s inversion understanding. Children as young as 4 years seem to demonstrate a sense of inversion given that they know that adding 1 and then taking away 1 returns the original quantity (Vilette, 2002). However, even up to the fourth grade, children do not seem to make good use of this understanding and apply inversion shortcuts in inverse problems (answering a to the problem a + b − b without calculating). In fact, about half of the fourth graders did not use inversion at all (Robinson & Dubé, 2009a). This leads to the question of whether inversion understanding improves with age or grade. Empirical findings are mixed on this issue. Whereas Dubé and Robinson (2017) did observe a significant grade-related improvement in inversion shortcut use from Grade 6 to Grade 8, no such grade-related improvement was observed from the study by Robinson and Dubé (2009b) among children of the same grades. A more recent study with fifth to seventh graders by Robinson et al. (2018) also failed to observe any significant grade-related improvement in the accuracy of solving an inversion problem. Even more surprising, Robinson and Dubé (2009a) actually observed a decrease in inversion shortcut use among second to fourth graders. Given the inconsistency in the existing findings, the developmental trend of inversion understanding deserves further investigation.

The third, and perhaps most puzzling, issue is whether the conceptual understanding of the inversion principle is related to better performance in mathematical problem solving. Such a relation can be explained from at least three different perspectives. First, if Piaget (1952) was correct that the understanding of inversion reflects a deeper level of understanding of arithmetic operations, then children with a more robust understanding of the inversion principle should be better at solving mathematical problems. In fact, the concept of addition–subtraction inversion is involved in carrying and borrowing during arithmetic computation (Nunes, Bryant, Evans, Bell, & Barros, 2012). For instance, when solving 108–52, the problem solver needs to understand that taking 1 away from the hundreds place and then adding 10 to the tens place does not change the value of the number. Children who understand inversion may also apply an indirect addition strategy to solve subtraction problems, which results in more efficient problem solving (Torbeyns, De Smedt, Stassens, Ghesquière, & Verschaffel, 2009). From this perspective, inversion understanding should facilitate mathematical problem solving.

Second, according to the strategy choice and discovery simulation (SCADS) model proposed by Siegler and Araya (2005), people usually start with a standard left-to-right procedure to solve many kinds of problems, including three-term inversion problems. To identify the inversion shortcut, the problem solver needs to shift attention to the relevant portion of the problem (i.e., the “b − b” part in the problem “a + b − b”). Children with greater computational skills, therefore, are at an advantageous position to discover the inversion shortcut because they can spare more of their mental capacities for other parts of the problem. From this point of view, better computational skills should predict the use of the inversion shortcut. From both aforementioned perspectives, inversion understanding is expected to be positively related to computation performance in particular or mathematics achievement in general.

However, viewing from a third perspective, it is theoretically possible that children’s inversion understanding is negatively related to their computational performance. Moreover, in the SCADS model (Siegler & Araya, 2005), it was stated that a left-to-right computation strategy was the default strategy for solving three-term problems. To use inversion shortcuts, problem solvers needed to inhibit their tendency to retrieve the relevant arithmetical facts (e.g., the “a + b” in the problem “a + b − b”) (Dubé, 2014). For problem solvers with better computational skills, such an inhibition may be more difficult because the relevant arithmetical facts may be more strongly activated in their memory. From this perspective, inversion understanding and computational performance may be negatively related.

Empirical evidence reveals a complicated picture of this relation. Robinson and Dubé (2009a) found that adolescents who have better multiplication–division inversion understanding are more accurate at solving multiplication and division problems. Ching and Nunes (2017) also showed that children’s inversion understanding in Grade 1 longitudinally predicted their mathematics achievement in Grade 2. However, the use of inversion was found to be weakly and nonsignificantly related to preschoolers’ calculation performance in Klein and Bisanz’s (2000) study. Other findings reveal a more complicated picture. Findings from Canobi (2005) suggested that children with better addition–subtraction inversion understanding tend to solve arithmetic problems faster and employ retrieval strategies more frequently, although these did not translate to higher problem-solving accuracy.

Rasmussen et al. (2003) found that performance in an arithmetic task correlated with performance in addition–subtraction inversion problems among first graders but not among preschoolers. Watchorn et al. (2014), on the contrary, did not observe any direct relation between children’s computational skills and their addition–subtraction inversion shortcut use, although the combination of good computational skills and high attention flexibility was related to more frequent use of inversion. Finally, meta-analytic findings from Gilmore and Papadatou-Pastou (2009) suggested that children’s inversion understanding can be categorized into three different clusters: those who performed well on both inversion and standard arithmetic problems, those who did not perform well on both inversion and standard arithmetic problems, and those who performed well only on the inversion problems. The existence of the third cluster suggests that good inversion understanding does not always come with good computational skills. This mixture of findings makes it difficult to conclude whether the understanding of inversion is related to mathematics achievement in general.

The aforementioned issues are further complicated by the ways of assessing inversion understanding. Within the current literature on mathematical cognition, inversion understanding is predominantly assessed by the application of procedures (Canobi, 2005, Dubé and Robinson, 2017, Gilmore and Spelke, 2008, Rasmussen et al., 2003, Torbeyns et al., 2016, Watchorn et al., 2014). Participants were given inversion problems such as “a + b − b” and “a × b ÷ b” versus standard problems such as “a + b − c” and “a × b ÷ c” to solve. Participants who understood inversion were expected to solve inversion problems more quickly and accurately than control problems (Dubé & Robinson, 2017).

Although these studies certainly provide valuable information concerning the application of inversion shortcuts by children and adolescents, they do not seem to provide a comprehensive picture concerning children’s and adolescents’ inversion understanding. This is mainly because the understanding of arithmetic principles, such as inversion principles, is multifaceted in nature, and the use of different assessment methods may reveal different facets of understanding (Bisanz et al., 2009, Prather and Alibali, 2009). For instance, people who are aware of a particular arithmetic principle might not apply the relevant shortcut themselves (Siegler & Crowley, 1994).

The existence of different levels of understanding has been demonstrated in other developmental aspects such as the theory of mind. For instance, children who are given a false belief task may give an incorrect verbal answer (the explicit measure) even though they tend to look longer at the correct answer (the implicit measure) (Clements, Rustin, & McCallum, 2000). To address these issues, Bisanz et al., 2009, Prather and Alibali, 2009 proposed the use of multifaceted knowledge assessments, or “knowledge profiles,” to summarize participants’ performance across different types of assessments. The advantage of using multifaceted assessments is that participants’ understanding of inversion is no longer categorized as either “have” or “do not have” based on their performance on a single measure. In contrast, multifaceted assessments can reveal gradations in the nature of inversion understanding (Bisanz et al., 2009). For instance, some participants may think that it is all right for another child to use an inversion shortcut to solve inversion problems (the evaluation of procedures task), but they seldom apply such shortcuts themselves (Robinson & Dubé, 2009b). Such a form of inversion understanding would not be revealed without the use of a multifaceted assessment. Unfortunately, such multifaceted assessments have seldom been used in the sphere of arithmetic principle understanding (Prather & Alibali, 2009).

In light of the measurement issue as well as the aforementioned inconsistent findings concerning children’s inversion understanding (i.e., its factor structure, development, and its relation with computational skills), the current study aimed to provide a different angle of looking at inversion understanding among fourth to sixth graders through the use of a multifaceted assessment. Fourth to sixth graders were selected because this is the period when children demonstrate growth in inversion understanding in both addition–subtraction inversion and multiplication–division inversion (Robinson et al., 2017, Robinson et al., 2018).

To assess the different facets of inversion understanding, three measures that had been employed by different studies to assess the understanding of different arithmetic principles (Dixon et al., 2001, Torbeyns et al., 2016, Wong, 2017) were modified to assess inversion understanding in the current study. The measures were (a) the evaluation of examples, (b) explicit recognition, and (c) application of procedures. In the evaluation of examples task, as modified from Dixon et al. (2001), participants rated the smartness of two hypothetical students by examining whether the students had applied the inversion principle in arithmetic problem solving. In explicit recognition tasks, modified from Dixon et al., 2001, Wong, 2017, the inversion principle was expressed in the form of logical statements (e.g., if a + b = c, then c − b = a) and participants were asked to judge whether the statements were correct. A modified version of the application of procedures task was also employed so that participants’ task performance, instead of their self-report, was used as the index of inversion understanding. Furthermore, two-term inversion problems (e.g., 36 + 95 = 131, 131–95 = ?), rather than three-term inversion problems (e.g., 36 + 95–95 = ?), were used because the former appeared to be more sensitive in detecting changes in inversion understanding (Nunes, Bryant, Hallett, Bell, & Evans, 2009).

These assessments were selected because they capture different aspects of principle understanding. Among the three, the application of procedures measure appears to require the highest level of inversion understanding because activation of such knowledge needs to be strong enough to inhibit existing problem-solving procedures (Siegler and Araya, 2005, Watchorn et al., 2014). This is perhaps why the application of procedures task usually reveals a lower level of inversion understanding than other measures (Prather & Alibali, 2009). However, given the multifaceted nature of knowledge as observed in other arithmetic principle understanding as well as other developmental domains, different inversion profiles are expected.

Various theoretical issues concerning children’s understanding were examined in the current study. First, the factor structure underlying children’s inversion understanding was examined by comparing a one-factor structure (including all four operations) with a two-factor structure (addition–subtraction vs. multiplication–division). A better fit for the one-factor model was expected because of the conceptual similarities between the two inversion pairs. Second, the age-related improvements in children’s inversion understanding were examined by comparing the performance of different inversion tasks across fourth to sixth graders. Significant grade effects indicating better inversion understanding among children at higher grades were expected. Furthermore, because the effect of grade may further interact with gender (Dubé & Robinson, 2017), participants’ gender was further included in the examination of grade effects. Third, latent profile analysis was employed to classify participants into different inversion knowledge profiles. This enabled us to examine the potential pathways of understanding inversion, on the one hand, and to investigate the relation between inversion understanding and mathematics achievement, on the other hand. Because children with better inversion profiles had better conceptual knowledge about arithmetic, they were expected to demonstrate a higher level of mathematics achievement.