Similar to the larger field of mathematical cognition, researchers on children’s repeating patterning often use the terms understanding and knowledge interchangeably. For example, some prominent researchers have used the term understanding (Kidd et al., 2014; Papic et al., 2011; Pasnak et al., 2016), while others have used the term knowledge (Rittle-Johnson et al., 2015, 2017; Starkey et al., 2004), and others have used both understanding and knowledge in the same paper (Rittle-Johnson et al., 2013). Furthermore, additional terms such as abilities (Mulligan et al., 2020; Wijns et al., 2019a, 2019b, 2020, 2021, 2023), skills (Fyfe et al., 2015, 2017, 2019; Mulligan et al., 2020; Rittle-Johnson et al., 2019; Zippert et al., 2019), and competencies (Wijns et al., 2019a, 2019b) are used in the repeating patterning literature. We will use the term knowledge as an umbrella term and discuss different ways we might define understanding.

Children first learn to work with simple alternating AB patterns, such as red-blue, and then learn to work with patterns with three or four elements that repeat, called the unit of repeat or pattern unit (e.g., ABB and AABB patterns). The easiest patterning tasks are complete (add the missing item to a pattern, also called repair) and duplicate (make an exact replica of a model pattern, sometimes called copy). More complex patterning tasks include extend (continue an existing pattern), abstract (recreate a model pattern using a different set of materials, also called transfer or translate), and identifying the unit of repeat (Clements & Sarama, 2009; Papic et al., 2011; Rittle-Johnson et al., 2015; Wijns et al., 2019b). Preschool children often have the lowest accuracy on identifying the unit of repeat (e.g., Clements & Sarama, 2009; Rittle-Johnson et al., 2013; Wijns et al., 2019b).

How might we define and operationalize repeating patterning understanding? The primary definition suggested by past educational and cognitive research is recognizing and using the unit of repeat, which is the general principle underlying repeating patterns. Such a definition is in line with Crooks and Alibali (2014), who noted that general principles include rules, definitions, and aspects of domain structure. However, operationalizing this definition of understanding within repeating patterns is challenging.

One approach to measuring children’s understanding of repeating patterns has been to develop tasks that tap into children’s ability to identify the unit of repeat. Explicit measures ask children to re-create or mark the smallest unit of repeat. For example, children are shown a block tower pattern with three instances of the unit of repeat and asked to create the smallest tower that shows only the repeating unit (Clements & Sarama, 2009; Collins & Laski, 2015; Junker et al., 2025; Rittle-Johnson et al., 2013). An implicit measure asks children to re-create a pattern with the same number of items from memory (Fyfe et al., 2015; Papic et al., 2011; Rittle-Johnson et al., 2013; Wijns et al., 2019a, 2019b). Children’s verbal reports suggested that identifying the unit of repeat and how many times it repeated facilitated performance on this memory task (Papic et al., 2011), in line with measures of encoding of task structure, such as through re-creation of chess pieces on a chess board (Chi et al., 1994). In general, 4-year-old children do poorly on these unit of repeat tasks. Limited measurement validation work indicates that identifying the unit of repeat items can have adequate inter-item correlations and infit and outfit statistics with other repeating patterning items (Lüken, 2012; Rittle-Johnson et al., 2013; Tian & Huang, 2021). However, there are multiple reasons that children may not succeed on these items. Children may fail to understand the task instructions or have other general cognitive difficulties, such as working memory limitations or language comprehension difficulties. In support of this claim, children’s working memory capacity was related to their accuracy on an explicit measure of identifying the unit of repeat, but not their accuracy for duplicating and extending patterns (Collins & Laski, 2015).

A second approach is to consider children’s solution procedures and classify children who use unit of repeat procedures as understanding repeating patterns. There has long been a concern that young children succeed on repeating patterning tasks using comparison-based procedures, such as one-to-one appearance matching of individual elements in the pattern that does not require attention to the unit of repeat (Economopoulos, 1998). However, research on repeating patterning has rarely directly measured what procedures children used to complete the tasks, relying instead on the analysis of children’s response errors (Bojorque et al., 2021; Borriello et al., 2022; Collins & Laski, 2015; Junker et al., 2025; Rittle-Johnson et al., 2013). One exception used clinical interviews with a range of patterning tasks, with children prompted to describe how they solved each item (Lüken & Sauzet, 2021). Based on children’s nonverbal behavior and verbal reports, most children ages 3-6 used multiple types of procedures. The most sophisticated, unit of repeat procedures, involved explicitly naming or showing the unit of repeat (e.g., taking all elements of the unit at once out of the box or grouping them together before arranging) and was rarely used (e.g., 8% of trials by 5-year-olds; Lüken & Sauzet, 2021). Such a procedure is thought to reflect relational thinking – thinking of “expressions and equations in their entirety rather than as a process to be carried out step by step” (Carpenter et al., 2005, p. 51; as cited in Junker et al., 2025, p. 32). A more common procedure that could produce a correct answer was comparison-based procedures (e.g., 32% of trials by 5-year-olds), such as comparing and matching elements in one-to-one correspondence (called a recursive procedure by Junker et al., 2025). Children generally used less sophisticated procedures with more difficult patterning tasks, such as abstracting and unit isolation tasks, and more sophisticated procedures with less difficult patterning tasks, such as duplicating. These findings suggest that all patterning tasks used in the research literature can be solved correctly with procedures that do not require explicit attention to the unit of repeat.

A third potential operationalization for children’s understanding of repeating patterns is success with abstracting (also called transfer or translating). To recreate a model pattern using a different set of materials is often more difficult than duplicating or extending patterns, and it might require greater attention to the underlying structure of the model pattern and not just its surface features (Clements & Sarama, 2009; Collins & Laski, 2015; Rittle-Johnson et al., 2013; Warren & Cooper, 2006). Although recent research indicates that success on abstracting does not require explicit attention to the unit of repeat and can be solved correctly using comparison-based procedures (Junker et al., 2025; Lüken & Sauzet, 2021), it does at least require that children not simply match identical elements in the pattern.

We explore a fourth possibility, which is operationalizing repeating patterning understanding as spontaneous use of correct procedure(s) across multiple tasks and problem features. This definition aligns with Bisanz and LeFevre’s (1992) suggestion that understanding can be defined and measured as “using an appropriate procedure spontaneously on one particular task” or “a similar procedure on a variety of related tasks” (p. 117). Importantly, compared to other mathematical domains, such as counting, where the application of correct procedures could be “due simply to memorization of a correct procedure” (Crooks & Alibali, 2014, p. 368), preschool children are typically not taught or shown specific procedures for solving patterning tasks in many countries, including the U.S. and Türkiye (formally known as Turkey). Therefore, we contend that concerns about memorization of correct procedures without understanding is typically not an issue for repeating patterning tasks. At the same time, rather than requiring that students use a unit of repeat procedure, we could consider all correct procedures as acceptable, even if they are less sophisticated such as comparison. There is not an established threshold for what constitutes “a similar procedure on a variety of related tasks” (Bisanz & LeFevre, 1992, p. 117). In line with established patterning research, we operationalize patterning understanding through successful performance on specific task types, recognizing that explicit measurement of cognitive procedures is beyond the scope of this work. At a minimum, we suggest it should be successful performance on at least two types of tasks with at least two different units of repeat.

In the current study, we explored different potential operationalizations of repeating patterning understanding. As part of this effort, we utilized Mark Wilson’s construct modeling approach to measurement development (Wilson, 2023) to develop and test a construct map, which is a representation of the continuum of knowledge through which people are thought to progress. The construct map guides the development of a comprehensive assessment, and both the construct map and the assessment are evaluated using item-response theory (IRT) models. This construct map provided insights into different operationalizations of understanding. An additional advantage of IRT models, in contrast to using more traditional total scores, is that the difficulty of the items is considered when generating ability scores, allowing us to assess construct validity for our measure by comparing the IRT output with our hypothesized construct map. This is the approach we have used in the current paper.

Table 1 presents a previous construct map for repeating patterning based on four types of constructed-response items (Rittle-Johnson et al., 2013). Based on this construct map, understanding could be defined as success in (a) identifying pattern unit items, (b) abstracting pattern items, or (c) at least 2 patterning tasks (levels), such as duplicating and extending. In all cases, children should succeed with at least 2 different units of repeat; children first succeed with alternating AB patterns, but understanding should require flexibility when working with other units of repeat (e.g., ABB, ABC). We also consider additional pattern tasks and new potential levels to inform an expanded construct map as a result of the current study.

Importantly, most often, repeating patterning tasks were constructed-response items, with students creating their responses using physical objects (e.g., Clements & Sarama, 2009; Papic et al., 2011; Rittle-Johnson et al., 2013, 2015, 2017). However, having students create their responses using physical objects can be time-consuming and cumbersome. Selected-response tasks make patterning tasks faster and easier to administer, allow for online data collection, and allow for systematic creation of distractor response options. One repeating patterning assessment has used selected-response items for one of their four tasks (e.g., Wijns et al., 2019a) and an assessment of growing patterns (more complex patterns where a sequence changes by the same rule each unit) has used only selected-response items for complete or extend tasks (Pasnak et al., 2016; Wijns et al., 2019a). In this study, we focus on the development of an Early Patterning Assessment (EPA) – Repeating Patterns (Rittle-Johnson et al., 2020), which includes primarily selected-response items that can be administered in-person or online. In Study 1, we report on four rounds of data collection with 4- to 7-year-old children in the United States and, in Study 2, we report on data collection with 4- to 7-year-old children in Türkiye. One open question is defining and measuring repeating patterning understanding using selected-response items.

The reliability of the repeating pattern assessment across the four rounds of data collection was acceptable (α = 0.73, using multiple imputation to account for missing data across rounds). Thus, it was a reliable measure for assessing 4- to 7-year-old children’s understanding of repeating patterns in the United States. Table 3 presents the proportion correct, item-total correlations, and item difficulty organized by easier to more difficult based on item difficulty.

The Wright Map is displayed in Figure 2. This map illustrates respondent abilities and item difficulties on the same logit scale, labeled with pattern task type (colors) and specific pattern type (names). Item difficulties ranged from approximately -2.50 logits, representing the easiest items, to 1.68 logits for the most difficult items, with an overall mean difficulty of -1.32 logits (SD = 0.27). Completing pattern items were the easiest items, with the item with an AB pattern as the easiest. There was a fair amount of overlap in ID-if-pattern, extend, and abstract items. ID-Unit was the most difficult task. The complexity of the pattern unit (e.g., AB, ABC, AAB) did not seem to systematically impact item difficulty, although the AB pattern unit item was a bit easier than other items for completing and abstracting, but not extending.

We also assess construct validity by comparing the Wright Map to our previous construct map in Table 1. This comparison demonstrates mixed evidence for construct validity. Consistent with our ordering of tasks in Table 1, ID-Unit items were the most difficult while complete items were the easiest. However, although extend and abstract items were easier than ID-Unit, they overlapped in difficulty with each other and with the ID-if-pattern items, which was an item type not included in our previous construct map. Overall, the evidence suggests that although some aspects of the construct are upheld, notable inconsistencies remain that call for further refinement of the construct. Thus, in the general discussion, we introduce a revised construct map that integrates these ideas with the findings in Study 2.

The reliability of the repeating pattern assessment was good (α = 0.83). Thus, it was a reliable measure for assessing 4-7-year-old children’s understanding of repeating patterns in Türkiye. Table 4 presents the proportion correct, item-total correlations, and item difficulty organized from easier to more difficult based on item difficulty for Study 2.

The Wright Map for Study 2 is displayed in Figure 3. This map illustrates respondent abilities and item difficulties on the same logit scale, labeled with pattern task type (colors) and specific pattern type (names). Item difficulties ranged from -2.16 logits, representing the easiest items, to -0.47 logits for the most difficult items, with an overall mean difficulty of -1.32 logits (SD = 0.46). For example, Extend_AABB was the easiest item with a logit difficulty of -2.16, and Abstract_ABCC was the hardest item with a logit difficulty of -0.47. Similar to study 1, there was considerable overlap in ID-if-pattern, extend, and abstract tasks, inconsistent with our ordering of tasks in our previous construct map. Abstract AB remained the easiest among the abstract items, while Extend AB was more challenging, suggesting that AB units of repeat were not necessarily easier than other pattern units. Additionally, the complexity of the pattern unit (e.g., AB, ABC, AAB) did not appear to systematically affect item difficulty. However, while the AB pattern unit was slightly easier for abstract, this advantage did not appear for the extend task.

The Wright Map analyses revealed important findings about the difficulty of different patterning tasks on the EPA-Repeating. Overall, we found some evidence for construct validity indicated by the similarities in task difficulty in our Wright Maps in Figure 2 and our previous construct map in Table 1. Therefore, Table 5 proposes a revised construct map for children’s understanding of repeating patterns expanded from Rittle-Johnson et al. (2013) to integrate evidence from our Wright Maps from Study 1 and 2. Matching previous work (Clements & Sarama, 2009; Lüken & Sauzet, 2021), we found complete items to be the easiest task, although this task was only measured in Study 1 and thus only included U.S. participants. Because we did not include any duplicate tasks on our measure, we relied on past findings to place it as an additional easier task on our revised construct map (Clements & Sarama, 2009; Lüken & Sauzet, 2021). Previous work using constructed responses have found a clear distinction between extend and abstract tasks as moderate difficulty and higher difficulty tasks respectively (e.g., see Table 1). However, the difficulty ranking of these tasks overlapped and indicated moderate difficulty in our two studies using a selected-response format. Our results also indicated that a new task—ID-if-pattern—overlapped in difficulty ranking with extend and abstract tasks; although we had anticipated it would be an easier task. This new task does not seem to add value in assessing patterning knowledge beyond more commonly used tasks and may be removed in the future. Finally, although only included in the last round of data collection in the U.S., explicit identification of the unit of repeat was the most difficult task, matching previous work (e.g., Lüken & Sauzet, 2021; Rittle-Johnson et al., 2013).

Contrary to our expectations, there was very limited evidence that AB patterns were easier than more complex pattern units, and there were no systematic differences in the difficulty of items based on the pattern unit. Thus, our construct map does not consider pattern unit complexity. Previous research suggests that AB patterns may be particularly accessible for 2- and 3-year-old children (Clements & Sarama, 2009) and thus may be important to consider with younger samples.

The EPA-Repeating, especially without ID-Unit items, is not sufficient to distinguish between older and higher-performing children. This finding is consistent with prior research highlighting the need for more challenging tasks to better assess advanced abilities (Yitzhak et al., 2016). Future assessments could incorporate more complex and diverse items, such as multi-step or novel patterning tasks, to better target the upper range of abilities and address the ceiling effects observed in some samples. Prior research has demonstrated knowledge of repeating patterning is foundational for later math success (Fyfe et al., 2017; Rittle-Johnson et al., 2017; Wijns et al., 2019a; Zippert et al., 2019) and engaging students in identify-if-pattern, extend, and abstract patterns could support the development of algebraic reasoning and broader mathematical skills (Clements & Sarama, 2009; Papic et al., 2011). However, to further challenge students, educators may consider introducing tasks that require higher-order reasoning, such as identifying more complex or recursive patterns or growing patterns, that increase or decrease by a set amount, are also more challenging for children (e.g., Wijns et al., 2019a). An additional approach future researchers and educators may consider as a component of patterning understanding could be spontaneous focusing on patterns (SFOP; such as asking a child to build with blocks and seeing if they will create a repeating pattern). Specifically, children who demonstrated SFOP in a building task performed better on patterning and broader math tasks compared to children who randomly arranged the blocks (Wijns et al., 2020). Integrating a measure of SFOP would help assess children’s understanding of repeating patterns, specifically, in line with claims for a metacognitive component of patterning understanding.

What are the implications of these findings for operationalizing patterning understanding? We consider three operationalizations for patterning understanding proposed in the introduction and how each definition aligns with our results. A fourth operationalization, using unit of repeat procedures, could not be considered in the current study. Past research that has used strategy reports has found that a unit of repeat strategy was rarely used (e.g., 8% of trials by 5-year-olds; Lüken & Sauzet, 2021), so this operationalization for patterning understanding is setting a very high bar for children.

Several limitations warrant discussion. First, the selected-response question format might affect the cognitive demand and difficulty level of tasks as well as children’s strategies. Previous patterning instruments used constructed response items, but our assessment was designed as a selected-response multiple-choice format, thus it might include a greater tendency to use comparison-based procedures. Second, ID-Unit and complete the pattern tasks were not included in Study 2’s version of the assessment, and our assessment did not include a duplicate the pattern task, so conclusions should be made with caution, and future research including all patterning tasks is warranted. Third, the unexpected equal difficulty level of ID-if-pattern, extend, and abstract items highlights the need for further research on the difficulty of pattern tasks, with different response formats, as part of careful consideration of assessment design. Fourth, our results indicated the EPA-repeating assessment consists of relatively easy items, allowing the assessment to distinguish among low-performing students, but not high-performing students. In addition to revising the assessment to include more difficult tasks, it is important to emphasize that the EPA-repeating was designed and developed for 4- and 5-year-old children in the United States, the 6- and 7-year-old children who participated were older than the age the assessment targeted.

Finally, we must consider the instructional focus on patterning activities in early childhood and classroom practice. Based on curriculum standards and the inclusion of primary school children, the Turkish participants may have received more instructional focus on patterning activities in their curriculum and classroom practice than the U.S. participants. This exposure could have improved students’ abilities, thereby making the assessment too easy for Turkish kindergarten and primary school children.

We suggest operationalizing children’s understanding of repeating patterns as demonstrating spontaneous use of correct procedure(s) across multiple tasks and pattern units. The person scores from our IRT analyses demonstrate a continuum from low to high understanding. Our findings provide insights for informing instructional practices and assessment tools.