Introduction

Teachers are regarded as curriculum makers who leverage their subject expertise to interpret and transcend the curriculum, thereby fostering students’ skills and abilities (Deng, 2022). Research suggests that how teachers engage with the curriculum has a greater impact on student outcomes than the curriculum materials alone, and thus, there is a need to focus on how teachers interact with the curriculum beyond the curriculum itself (Remillard, 2005; Taylor, 2016). Along the same vein, the Curricular Noticing framework proposed by Dietiker et al. (2018) delves into the three phases of noticing teachers encounter while interacting with the curriculum: attending, interpreting, and responding to curriculum materials, which play a vital role in mediating teachers’ analysis and decision-making regarding content and pedagogy in the classroom. Given the critical importance of curricular noticing in the work of teaching, it is undoubtedly necessary to address this in the training of future teachers so that they can embark on their careers with the appropriate understanding and skills.

In examining pre-service teachers’ (PSTs’) curricular noticing capacities, this study employed pattern generalization tasks. Searching for regularities or patterns is a natural human tendency in daily life, and it also takes up substantial space in the school curriculum from the early grades (Clements & Sarama, 2014; Mulligan et al., 2020; Van de Walle et al., 2018). Thus, although formal notations for algebraic or functional thinking first appear in the secondary school curriculum, experiences that foster identifying and extending patterns are foundational in early schooling.

To teach students effectively and offer interventions that support their development of pattern generalization skills, teachers from kindergarten through grade 12 need to be able to determine the appropriate cognitive demand and complexity of pattern problems. Multiple factors are involved in this ability, such as having a knowledge base that enables the teacher to recognize key mathematical ideas, being able to identify and understand students’ difficulties, and being skillful in designing appropriate tasks to promote students’ thinking (e.g., Dietiker et al., 2018; Jacobs et al., 2010; Lee, 2021a, Lee & Lee, 2021a, 2023a, b; Yang & Ricks, 2013).

In this study, we explored PSTs’ mathematical content knowledge of pattern generalization and their ability to design an appropriate instructional sequence for elementary (pre-K through 8) students by adapting Dietiker et al.’s (2018) curricular noticing framework (curricular attending, curricular interpreting, and curricular responding) to develop a task for this study. The task was designed to elicit how PSTs identified the knowledge, understanding, and skills required to solve three pattern generalization problems at distinctly different difficulty levels and how they proposed sequencing the problems for instructional purposes. In light of these considerations, this study aims to contribute to the ongoing discourse on enhancing PSTs’ curricular noticing abilities. It identifies key components and challenges in their engagement with pattern generalization tasks, thereby opening up the discussion among mathematics teacher educators.

Literature Review

Prior Research Related to PSTs’ Curricular Noticing

Teachers’ ability to construct or choose effective curricular materials and determine their appropriate order of introduction is generally recognized as a key component of teaching expertise and instructional quality (Dietiker et al., 2018; Males et al., 2015; Remillard, 2005). As not all curriculum materials adequately facilitate teachers’ efforts to support students’ conceptual understanding of mathematics (Machalow et al., 2020), an important objective of teacher education is to help PSTs develop curricular noticing abilities. In alignment with this purpose, some mathematics educators have researched PSTs’ curricular noticing abilities (Amador et al., 2017; Lee & Lee, 2021a, 2023b). For example, Amador et al. (2017) suggested that the use of nonroutine tasks may support PSTs’ curricular noticing abilities in understanding the core mathematical properties of tasks. In their study, 18 PSTs were asked to analyze two tasks, one routine, featuring an equally partitioned area model, and the other nonroutine, with an unequally partitioned area; identify the mathematical properties of the tasks; and predict students’ responses to the tasks. Similarly, in their research on PSTs’ selection, interpretation, and sequencing of fraction examples, Lee and Lee (2021a) argued that providing various examples and fractional models to represent a fraction can improve PSTs’ curricular noticing related to understanding equal partitioning.

In another study on curricular noticing in the area of fractions, Males et al. (2015) highlighted that PSTs’ experiences, beliefs, and mathematical knowledge influenced their curricular noticing abilities. The researchers examined how PSTs engaged with Grade 6 curricular materials on the division of fractions from four different curricular programs by asking them to create a lesson plan. When asked to explain why they selected or excluded particular resources, how they used them, their motivations for using them in a specific way, and their rationales for decisions about curricular materials in planning a lesson, the PSTs often referred to their own experiences and beliefs. The researchers also emphasized the importance of grappling with the content of fraction division before PSTs planned a lesson.

While these studies have explored the concept of curricular noticing by noting that PSTs’ abilities are influenced by their experiences, beliefs, and mathematical knowledge—and that nonroutine tasks, various examples, or models can enhance this knowledge—they did not explicitly use the curricular noticing framework, which consists of three sub-skills: curricular attending, curricular interpreting, and curricular responding. Additionally, these studies focused primarily on investigating PSTs’ curricular noticing in the area of fractions, even though the topic of pattern generalization is crucial for advancing PSTs’ algebraic reasoning (Kaput, 1999; Radford, 2008).

Moreover, as noted, Males et al. (2015) investigated PSTs’ curricular noticing by having them create lesson plans with curricular materials from four sources. However, PSTs in the early stages of teacher preparation may not be ready for noticing while working with such a variety of curriculum materials. They may first need to develop a lens for curricular noticing in a controlled situation with specific problems or activities (Lee, 2018). Therefore, in this study, PSTs’ curricular noticing was examined using the curricular noticing framework (Males et al., 2015) and was observed through their responses to the structured task of sequencing three pattern generation problems at graduated levels of difficulty.

Prior Research Related to Sequencing Problems for Lesson Planning

When teachers plan a mathematics lesson, one factor they need to consider is the sequencing of problems. Gagne (1973) defined two main types of sequencing as follows:

Learning sequence means the ordering of occurrences within a single learning act having a single outcome (objective), such as identifying a concept, restating a fact, and making a discrimination response. In contrast, instructional sequence refers to a succession of events occurring over longer intervals of time, whose purpose is to ensure that the necessary stimulus conditions for a single learning act (the targeted learning objective) will all be present at the proper time. (p. 4)

In this paper, sequencing refers to determining the optimal ordering of problems to teach a specific mathematics topic, finding and generalizing patterns.

Effective sequencing is critical for meaningful learning. According to Mager (1961), “the learner usually attempts to increase the meaningfulness of what he is learning by making associations and by filling in the gaps” (p. 409). Sequencing appropriate problems so as to facilitate these attempts is an important teaching practice. Rowland et al.’s (2009) Knowledge Quartet, a framework for analyzing teaching in terms of Foundation Knowledge, Transformation, Connection, and Contingency, includes sequencing as key instructional knowledge in the dimension of Connection. Mager (1961) identified ways in which teachers may sequence problems, such as chronological order and specific to general or vice versa. Among various possibilities, Mager found five features of instructional sequences that were preferred by learners: (a) grouping items with similar content, (b) progressing from the concrete to the abstract, (c) learning “how” before “why,” (d) introducing function before structure, and (e) progressing from simple wholes to more complex wholes.

Mathematics education resources (National Council of Teachers of Mathematics [NCTM], 2000; van de Walle et al., 2019) suggest the following strategies for sequencing mathematical problems: (a) start with simple problems, (b) progressively increase the level of difficulty, and (c) introduce new concepts at a comfortable pace. Beginning with straightforward problems helps students build confidence and develop a good understanding of the basics. Gradually increasing the difficulty of the problems as students progress helps them develop problem-solving skills and critical thinking. Introducing new concepts gradually and ensuring that students have a solid understanding of each concept before moving on to the next concepts help students broaden their understanding of the area being taught.

Most of the recent studies of sequencing in mathematics education (Ayalon & Rubel, 2022; Livy et al., 2017; Meikle, 2014; Rubel et al., 2024) have focused on selecting and sequencing in students’ work or on instructors’ strategies for teaching whole-class lessons rather than on providing sequencing problems for students’ practice. For example, in an investigation of how PSTs selected and sequenced students’ solutions to fraction division and hexagon pattern problems for whole-class discussions, Meikle (2014) found that some PSTs began with an erroneous solution and built up to a correct strategy while others began with visual or concrete representations and moved to an abstract representation. However, to justify their sequencing choices, PSTs used mathematical connections or concepts less frequently than pedagogical moves such as demonstrating correctness or completeness.

Livy et al. (2017) examined PSTs’ knowledge about selecting and sequencing young students’ solutions to counting chocolates in a box and found that examining an authentic learner’s work sample helped them to make connections with students’ mathematical reasoning. Ayalon and Rubel (2022) and Rubel et al. (2024) found that in classroom demonstrations, most PSTs followed a sequence that began with student work containing an error, proceeded with a direct model of the correct solution, and concluded with an inductive approach to reaching the solution.

Situating this Study

While these prior studies have shed light on how PSTs select and sequence students’ work for whole-class discussions, how PSTs sequence problems for teaching specific mathematics topics, which is an essential skill for planning mathematics lessons and guiding student learning, has rarely been addressed. If they are led through a series of problems that systematically support their understanding, students are less likely to become frustrated or disengage from a lesson, often interfering with other students’ learning as well as their own. Given the importance of effective problem sequencing in teaching mathematics concepts, therefore, we designed a two-part task for this study. In the first part, PSTs solved three pattern generalization problems. In the second part, they demonstrated the order in which they would present these problems to support student learning and explained their reasoning.

We selected the mathematical topic of pattern generalization because this area has been emphasized as important for promoting students’ development of critical thinking, problem-solving, and reasoning skills (NCTM, 2000; National Governors Association Center for Best Practices & Council of Chief State School Officers [NGA & CCSSO], 2010). Engaging in pattern generalization activities helps students learn to identify, extend, and analyze mathematical patterns, which is important for advanced mathematics (Kaput, 1999; Lee & Lee, 2023b). Moreover, pattern generalization activities help students understand mathematical concepts and principles as well as make connections among different areas of mathematics. They also provide opportunities for students to explore multiple approaches to and strategies for problem-solving, which can help them develop creativity and flexibility in their thinking (van de Walle et al., 2019).

Theoretical Framework

Our theoretical framework was based on the concept of noticing, a natural and subjective part of being aware of the world, which may or may not be consciously purposeful. Based on the relevant literature, we further conceptualized the purposeful constructs of teacher noticing and curricular noticing, which are explained below.

Teacher Noticing

To make appropriate instructional decisions that support students’ further learning, teachers need to practice intentional noticing with specified goals. Thus, teacher noticing is one of the key instructional practices that PSTs should develop in their teacher education programs. Researchers have emphasized various aspects of teacher noticing (Jacobs et al., 2010; Mason, 2002; Miller, 2011; Sherin & van Es, 2009), including what teachers attend to, how they interpret what they have attended to, and how they respond based on their interpretations, as well as the interdependence of all three constituents, attention, interpretation, and responsive action, in a comprehensive skill set.

Focusing on attending and interpreting, Mason (2002), who introduced the concept of noticing, described it as comprising two main categories: accounts-of and accounts-for. Accounts-of refers to purely descriptive observations of phenomena without interpretation or evaluation, the “what” dimension of noticing. Accounts-for refers to exploratory and interpretive observations, the “how” dimension of noticing. Sherin and van Es (2009) conceptualized a similar two-part framework for what they considered the art of noticing, composed of what teachers perceive and how they understand it.

Jacobs et al. (2010) expanded the construct of noticing to encompass three interrelated processes: attending to students’ mathematical thinking, determining its significance and implications for further learning, and formulating a concrete pedagogical rationale for subsequent instructional moves. In this framework, completing the third process requires having successfully achieved the first two, a complex engagement which constitutes the essence of noticing. Jacobs et al. characterized the seamless integration of these three processes as professional noticing, which serves as a significant source of information for making effective pedagogical decisions.

Curricular Noticing

Curriculum materials are as critical to mathematics instruction as students’ mathematical thinking (Dietiker et al., 2018; Males et al., 2015; Remillard, 2005). Thus, to extend the construct of noticing to include teachers’ interactions with curriculum materials, Males et al. (2015) coined the term curricular noticing, defined as “the process through which teachers make sense of the complexity of content and pedagogical opportunities in written or digital curricular materials” (p. 88). Like the professional noticing framework, curricular noticing consists of the three dimensions of attending, interpreting, and responding.

Dietiker et al. (2018) defined curricular attending as comprising “the skills involved in viewing information within curriculum materials prior to their interpretation” (p. 525). They emphasized that curricular attending is not limited to identifying significant or distinct features of curriculum materials. Rather, it encompasses all aspects of curriculum materials, which are broadly construed to include various forms such as “mathematical activities (e.g., tasks, games, exercises), mathematical content (e.g., definitions or theorems), and strategic teaching guidance (e.g., recommendations for pacing or grouping students” (p. 525).

Males et al. (2015) described curricular interpreting as “making sense of that to which the teacher attended;” and curricular responding as “making curricular decisions based on the interpretation (e.g., generating a lesson plan, a visualization, or an enactment)” (p. 89). They later further developed the curricular noticing construct by adding examples of actions associated with each of the three dimensions of curricular noticing, as seen in Fig. 1.

Fig. 1
figure 1

Examples of activities within the three dimensions of curricular noticing (adapted from Males et al., 2015 and Dietiker et al., 2018).

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In this study, we adopted Dietiker et al.’s (2018) model as the framework for curricular noticing. However, because PSTs have generally had few, if any, opportunities to engage in classroom teaching, we selected one activity from each component of the noticing framework, ensuring that each activity was easily adaptable to our specific task. That is, first we asked PSTs to solve three problems in order to examine how they perceived the problems (Attending), to report the reasoning required to solve them in order to probe how they comprehended the problems (Interpreting), and to sequence the three problems for instruction and provide their rationales in order to investigate how they would use the problems in lesson planning (Responding). This allowed us to maintain a clear focus and tailor the activities to best suit our goals and the framework’s requirements. By doing so, we aimed to provide a comprehensive yet concise approach that effectively integrates the key aspects of the noticing framework into our work.

Based on this theoretical framework, the research questions guiding this study were as follows:

  1. 1.

    As an activity representing one aspect of curricular attending, how do PSTs perceive the three pattern generalization problems while solving them?

  2. 2.

    As an activity representing one aspect of curricular interpreting, how do PSTs comprehend the reasoning, knowledge, and/or skills required to solve each of the three problems?

  3. 3.

    As an activity representing one aspect of curricular responding, how do PSTs sequence the three problems?

Methods

Participants

The participants were 155 PSTs enrolled in several sections of a mathematics methods course in the elementary (pre-K through 8) teacher education program of a large Southwestern state university. At the time of the study, they had completed three mathematics content courses about number and operations, geometric reasoning, and algebraic reasoning. In the mathematics pedagogy course, the PSTs learned how the topic of pattern generalization is covered in Common Core State Standards for Mathematics (NGA & CCSSO, 2010) across grades K-8. However, PSTs had no prior exposure to pattern tasks and their potential sequencing. We believe it is crucial to assess PSTs’ curricular noticing skills at this specific point because understanding their baseline is essential for developing effective teaching strategies. By knowing their starting level, we can tailor our instructional approach to better equip them with the necessary skills.

Data Sources and Collection

Data in this study comprised the participants’ responses to a two-part written task (see Fig. 2). In the first part, the PSTs were asked to solve three pattern generalization problems developed by Stump et al. (2012) and to explain the reasoning/knowledge required to solve the problems. In the second part, they were asked to determine the appropriate order in which to present the three problems to support students’ learning as well as to explain the rationale for their sequencing.

Fig. 2
figure 2

Main task of this study

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We introduced this task at the beginning of our class to help PSTs develop their curricular noticing skills, particularly in sequencing tasks. When distributing the task, we did not initially highlight any specific factors, such as cognitive demand and complexity, that should be considered when determining an ideal sequence. The task was given immediately after a review of how the topic of pattern generation progresses in the Common Core State Standards. We asked the PSTs to complete the task and then engaged them in a whole-class discussion. This discussion was aimed at helping them recognize the importance of considering cognitive demand and complexity as crucial elements in sequencing tasks effectively.

We selected these three pattern generalization problems for two reasons. First, since PSTs had already learned mathematical pattern generalization in their earlier schooling, we aimed to present nonroutine pattern generalization problems that leveraged their existing geometric knowledge, ensuring the problems would not be too easy for them. Second, because the PSTs were asked to sequence the three problems for 6th-8th grade elementary students, we wanted to include both simple (i.e., linear) and advanced (i.e., quadratic) patterns. Table 1 demonstrates the valid number sequence and the generalized expression for each problem.

Table 1 Solutions to the Three Problems
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The number-of-squares problem required the PSTs to find a linear pattern as well as to identify squares, including those hidden in each of the rectangles. The hidden-rectangles problem was similarly based on previously learned geometry, but the PSTs were prompted to think further by considering that a square is a special kind of rectangle when identifying rectangles and by finding a quadratic pattern. Finally, the herringbone-patterns problem was more transparent in that it did not require identifying any hidden component but was based on a quadratic pattern. Therefore, when considering the cognitive demand and complexity of the problems, we believed that the following sequencing would be optimal for student learning: (C) number-of-squares problem, (B) herringbone-patterns problem, and (A) hidden-rectangles problem.

Data Analysis

We organized the raw data on an Excel spreadsheet and applied an inductive content analysis approach (Grbich, 2007), which involved five processes: (a) reading each PST’s response and creating codes based on the raw data, (b) identifying the correctness and themes of the responses, (c) creating categories and subcategories based on features of the PSTs’ responses and solutions, (d) coding the data using the categories and subcategories, and (e) interpreting the data quantitatively and qualitatively. Accordingly, we first probed how the PSTs solved the three pattern generalization problems by examining whether they provided correct identification of the relationships and generalized expressions for the relationships in their solutions. Then we categorized the types of incorrect answers for each problem. To analyze the PSTs’ written explanations of the kinds of reasoning/knowledge required to solve each problem (second question of Part 1), we first read all entries focusing on the meanings of the responses. Then we transformed the initial meaning units into abridged meaning units to sort them into the following seven categories: Knowledge of formulas, Explanation/identification of relationships, Problem-solving strategies, Generic statements without specifics, Restatements of generalized expressions, Applications of geometry knowledge, and Other (refer to the details in Table 6).

To analyze the data collected from Part 2 of the task, we first identified the six possible sequences of the three problems (i.e., A-B-C, A-C-B, B-C-A, B-A-C, C-A-B, C-B-A) and sorted PSTs’ responses accordingly. Then we examined which problem was most frequently placed first, second, or third in the recommended ordering. Regarding the PSTs’ justifications for their proposed sequencing of the problem, we identified eight main categories: Difficulty, Scaffolding based on similarity, PSTs’ own understanding/skills, Knowledge of geometry, Prior knowledge, Importance, General, and No response (see the details in Table 8). To determine the reliability of the data analysis, we first independently coded some random samples, resulting in about 92% concordance between the coders. Each of us then coded the remaining data, and we resolved discrepancies between our codes through discussion until we reached 100% agreement on the coding. Finally, we calculated the number of responses in each category to identify overall tendencies.

Findings

PSTs’ Curricular Attending Demonstrated in Their Solutions

To investigate how PSTs perceived the three pattern generalization problems, we examined their curricular attending by first determining whether they correctly identified the relationships and then generalized expressions for the relationships they presented in their solutions. Here, the term “perceiving” is operationally used to indicate what features of problems the PSTs focused on in the development of a solution plan. We posited that they would develop their solution plans based on their perceptions of the features of problems. As perception is not directly observable, we inferred their perceiving of the problems from the manner in which they approached the problems.

Table 2 shows the valid number sequence for each problem, the generalized expressions for the number sequences identified, and the frequency of PSTs’ correct/incorrect responses. We considered equivalent variations of the final forms of generalized expressions as equally valid. For example, \(\:\frac{{n}^{2}+n}{2}\), 1+ \(\:\frac{{n}^{2}+n-2}{2}\), and \(\:\frac{n\:(n+1)}{2}\:\:\)were all accepted as valid solutions for the hidden-rectangles problem. We counted only the cases that presented both valid identification of number sequences and valid expressions as correct responses. The cases in which invalid or no expressions were produced, whether or not the number sequences were correctly identified, were considered incorrect. The cases in which no attempt was made to show both the identified number sequences and the generalized expressions were noted separately.

Table 2 PSTs’ Solutions to the Three Problems
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As shown in the above table, there was a wide range of differences among the PSTs in correct and incorrect responses across the three problems. Only 8.4% produced the correct answers to all three problems, indicating they had perceived them correctly. The number of incorrect or missing responses to the hidden-rectangles problem was highest, indicating it was the most difficult for PSTs to perceive correctly. To understand more about the PSTs’ work, we further examined the PSTs’ incorrect answers to each problem. Table 3 shows our analysis of the cases identified as incorrect in the hidden-rectangles problem.

Table 3 Incorrect Responses: Hidden-Rectangles Problem
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As shown in Table 3, the majority (72.8%) of incorrect solutions to the hidden-rectangles problem were misidentifications of the number sequence due to perceiving the problem incorrectly. It was also noted that 33.3% of the incorrect responses provided arithmetic sequences, and there was a match between the incorrectly identified sequence and the consequently incorrect generalized expression. In 20.2% of the false solutions, the PSTs identified the correct number sequence by counting all hidden rectangles but were unable to produce a generalized term for the identified sequence. Table 4 shows our analysis of the cases identified as incorrect in the herringbone-patterns problem.

Table 4 Incorrect Responses: Herringbone-Patterns Problem
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Of the 52 PSTs whose responses were incorrect, 39 (75%) correctly identified the fourth term in the sequence but failed to present a generalized expression. Some PSTs presented expressions that worked for only the first one or two terms, implying that they utilized a trial-and-error strategy rather than perceiving the quadratic pattern inherent in the herringbone-patterns problem. Other PSTs treated the sequence as an arithmetic sequence in which the first term is 2, and the common difference is either 4 or 2. In some cases, the PSTs wrote only the general term of the arithmetic sequence without identifying the first term and the common difference. A total of 13 PSTs (25%) did not identify the number sequence but without explanation presented generalized expressions that did not seem to be related to the herringbone-patterns problem.

As shown in Table 5, the number-of-squares problem yielded the second-highest number of incorrect responses.

Table 5 Incorrect Responses: The Number-of-Squares Problem
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As with the hidden-rectangles problem, most of the incorrect responses (more than 70%) to the number-of-squares problem involved miscounting the number of squares due to the PSTs’ incorrect perceptions of the problem (e.g., 1 × 1 squares only, a mixture of 1 × 1 squares and a rectangle, the sides of the squares, etc.), resulting in incorrect sequences. Two-thirds of the incorrect responses were arithmetic sequences, and there was a match between the incorrectly identified arithmetic sequences and the proposed generalized expressions. Except for two of these cases, the common difference in the arithmetic sequence was misidentified (e.g., 7 + (n − 1) × 3 = 3n + 4 for 7, 12, 17, 22, …).

PSTs’ Curricular Interpreting as Demonstrated in Their Identifications of Required Skills and Understanding

After solving the three problems themselves, the PSTs proposed various ideas regarding the knowledge and reasoning needed to solve each problem. Three main themes emerged across all three problems: (a) knowledge of formulas, (b) explanation/identification of the relationships, and (c) use of problem-solving strategies. In addition, they identified geometry knowledge as a prerequisite for solving the hidden-rectangles and the number-of-squares problems. Also, they produced generic statements and restatements of generalized expressions without additional explanations across all three problems. Table 6 categorizes the PSTs’ identifications of the skills and understanding required for solving the three problems.

Table 6 PSTs’ Responses to the Required Skills and Understanding
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PSTs’ Curricular Responding Demonstrated in Their Suggestion of Problem Sequences and Rationales

To examine PSTs’ curricular responding ability, we asked them to order the three problems in a sequence most likely to support students’ learning. Based on our assessment of each problem’s cognitive demand and complexity, we agreed that the following is the most appropriate sequence: C) number-of-squares problem, B) herringbone-patterns problem, and A) hidden-rectangles problem. Table 7 shows the sequences PSTs proposed.

Table 7 PSTs’ Proposed Sequences
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The most common sequence, C-A-B, accounted for 31.6% of the PSTs. Our proposed sequence, C-B-A, was also proposed by 22 PSTs (14.2%), including 13 PSTs who solved all three problems correctly and nine who did not. In general, Problem C was most often placed first in the recommended sequence, followed by Problem A in second place. Problems A and C were frequently placed together in the sequence, almost 75% of the time. Very few PSTs thought that starting with Problem B was a good idea. The justifications for the proposed sequencing of the problems by the PSTs are summarized in Table 8.

Table 8 PSTs’ Justifications of Their Proposed Sequencing
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More than half of the PSTs cited the problem’s difficulty as a criterion for its placement in the sequence. However, they construed difficulty in terms of many different aspects, such as visuality, complexity, feasibility for direct application of formulas, and connections with multiple mathematical topics. Also, 11% of the PSTs sequenced the problems based on their own understanding and skills, often mentioning what was easy or difficult for them. Although this category was also related to how the PSTs defined difficulty, we recognized this rationale as representing purely subjective interpretations based entirely on the PSTs’ own experiences in solving the problems.

Some PSTs placed Problems A and C together in the sequence because of their shared connection with geometry. It appeared that PSTs’ incorrect perceiving of the problem also influenced the sequence they proposed. For example, some PSTs wrongly attempted to solve Problem C by only counting 1 × 1 squares, resulting in simple arithmetic sequences, which they found easy. As a result, their decision to place Problem C first in their sequences was mainly influenced by their incorrect understanding of the problem (or their inability to interpret it accurately). In other words, while these PSTs may have used the same reasoning for their sequence that we would recommend (i.e., sequencing based on the complexity), their incorrect perceiving of the problem hindered their approach.

Discussion and Conclusion

In this study, we explored PSTs’ curricular noticing abilities through a series of tasks that aligned with the three dimensions of the curricular noticing framework: Attending, Interpreting, Responding. To gain an understanding of how they perceived the problems (Attending dimension), we examined how the PSTs solved three pattern generalization problems at different levels of cognitive demand and difficulty. We also probed how the PSTs comprehended the problems (Interpreting dimension) as reflected in their identification of the knowledge and skills required to solve each problem. Additionally, for insight into lesson planning (Responding dimension), we examined the PSTs’ suggested instructional sequencing of these three pattern generalization problems and their rationales. In this section, we revisit the study’s findings for further discussion.

Curricular Attending: Challenges in Perceiving the Problems

We found that only a small number of PSTs (8.4%) solved all three problems correctly. These PSTs tended to first look for the problem’s structure and then generate patterns rather than simply apply arithmetic or geometric formulas. When examining PSTs’ incorrect solutions, we found that they encountered two main challenges in the process of perceiving the problems, which warranted further discussion.

One challenge we noticed was that the majority of errors in the hidden-rectangles and the number-of-squares problems reflected a misunderstanding of the problems. Specifically, when PSTs did not count all the rectangles or squares in a configuration (e.g., counting only 1 × 1 unit squares), they ended up with straightforward arithmetic sequences, such as a sequence with a common difference of 1 or 2. Another challenge we noticed in their solutions was that many PSTs who were able to identify the correct number patterns still failed to present generalizable expressions. This result was particularly prevalent in the herringbone-patterns problem (75% of the incorrect cases). In a few instances, a generalized expression was provided that worked for only some terms, suggesting the use of a guessing-and-checking strategy. Also, although they correctly identified number patterns, some PSTs presented a generalized expression in the form of an arithmetic sequence by considering the difference between the first and the second terms as the common difference for the entire pattern, indicating the rote application of formulas.

We intentionally designed tasks that extended beyond simple arithmetic sequences, making it difficult to identify common differences and requiring the recognition of quadratic patterns that were not apparent in geometric contexts. However, despite our intentions, many PSTs still tried to solve the tasks using known formulas instead of conducting in-depth investigations. This aligns with existing research, which shows that PSTs’ teaching beliefs and decision-making are often influenced by their past learning experiences (e.g., Pajares, 1992; Stuart & Thurlow, 2000). These experiences involve relying on rule-based approaches to generalize patterns in our study’s context. In these approaches, common number patterns are typically given to illustrate the direct application of a formula, encouraging memorization of the formula rather than promoting analytical thinking. Therefore, our findings suggest that teacher educators should be mindful of this situation and consider developing tasks with atypical number patterns that require PSTs to investigate the patterns’ properties and think beyond simply applying formulas.

Moreover, the Common Core State Standards for Mathematics (NGA & CCSSO, 2010) call for students to be able to explore patterns using various modes of representations, such as physical materials, drawings, tables, words, and symbols; to make sense of the regularities they discover; and to comprehend the changes of regularities quantitively and abstractly. As PSTs’ limited previous experience in exploring mathematics could hinder their effective performance in curricular noticing, teacher educators should consider how to provide opportunities to engage in such experiences during their teacher education program.

Curricular Interpreting: The Gap between Solving Problems and Explaining the Knowledge Needed to Solve Problems

To investigate PSTs’ comprehension of the three problems, we asked them to report the skills and understandings needed to solve them. As shown in Table 6, we sorted their responses into seven categories. Formula-centric responses, exemplified in the categories “Knowledge of formula” and “Restatements of generalized expressions,” constituted 25.2%, 30.9%, and 43.8% of the knowledge identified as required across the three problems. In contrast, non-formula-focused knowledge and skills, including “Explanation/identification of relationships” and “Problem-solving strategies,” accounted for 33.6%, 25.2%, and 15.5%, respectively. This result may suggest some discrepancy between how PSTs themselves solved the problems, which revealed a heavy reliance on formula-focused approaches, and what they thought should be involved in the problem-solving process, which included both formula-focused and non-formula-focused approaches. Teacher education programs should bridge the gap between PSTs’ practices and their stated beliefs about problem-solving, encouraging them to progress toward non-formula-dependent strategies.

Additionally, we noticed that some PSTs’ responses provided only generic statements without specifying particular skills and understandings (17.4%, 34.2%, and 15.5% across the three problems). While the herringbone-patterns problem yielded the greatest number of correct solutions, it also showed the greatest number of generic statements without specifics. Although the reasons for this tendency cannot be determined from the study’s data, we infer that PSTs’ own problem-solving ability may not guarantee their ability to identify the knowledge and skills required for solving the problem, an essential curricular noticing skill. Thus, we recommend that teacher educators provide optimal tasks for addressing all dimensions of curricular noticing.

Curricular Responding: What Matters in Sequencing the Problems

Only 22 PSTs (14.2%) suggested the same instructional sequencing of the three pattern generalization problems we determined. Of these, 13 solved all three problems correctly. The PSTs who proposed simple arithmetic sequences for the hidden-rectangles problem and the number-of-squares problem apparently misunderstood the problems. This implies that while adequate content knowledge is necessary, it is not sufficient on its own for teachers to make appropriate instructional decisions.

The predominant justification PSTs gave for their proposed sequencing was “difficulty,” which is similar to prior findings in which sequencing was based on progression from simple to more complex content, an instructional sequencing strategy highly preferred by learners (Mager, 1961; NCTM, 2000; van de Walle et al., 2019). However, as noted above, the term “difficulty” was applied diversely by the PSTs. Some simply referred to it as progressing “from easier to harder” without elaboration, while others considered specific factors such as visuality, complexity, the direct applicability of arithmetic sequence formulas, and connections with other topics. This range of specificity highlights the need for PSTs to improve their ability to articulate meanings with sufficient detail.

Also, we particularly note the finding that approximately 11% of PSTs based their sequencing on their own abilities and comfort/confidence levels, seemingly assuming that their students would have similar experiences. This inclination to project their own reactions may impede PSTs from effectively eliciting and interpreting students’ thinking. This finding aligns with previous studies that emphasize the importance of conceptual mathematical knowledge for PSTs’ effectiveness as mathematics teachers, as it enables them to recognize differences in their students’ approaches and provide tailored interventions to meet the students’ needs (Lee, 2021b; Lee & Cross Francis, 2018; Lee & Lee, 2020, 2021a, b, 2023a, b, c; Lee & Lim, 2020). Thus, teacher educators need to provide more opportunities for PSTs to examine and analyze strategies different from theirs and decipher the reasoning behind those approaches. In making this recommendation, we do not suggest that there is a pre-defined strategy for instructional sequencing that PSTs need to master. Rather, we suggest that PSTs need to be guided to be mindful of various factors that must be considered when deciding on instructional sequencing. They should also cultivate the skill of recognizing which factors are more critical and which are tangential in particular contexts.

Revisiting the Curricular Noticing Framework in Light of the Findings of this Study

In this study, we examined how PSTs performed on tasks related to the three main aspects of curricular noticing in the original curricular noticing framework: attending, interpreting, and responding (see Fig. 1 above). However, considering the PSTs’ limited exposure to classroom teaching, we narrowed our focus to a more specific context by examining one activity from each component of curricular noticing. Our findings reveal a somewhat linear rather than interactive relationship among the three dimensions, as depicted in Fig. 3. Specifically, as demonstrated in their responses to the tasks, the PSTs’ curricular attending appeared to influence their interpretation of the skills and knowledge required to address the problems from a pedagogical perspective. These two dimensions, in turn, impacted their curricular responding in terms of how they sequenced the three problems.

Fig. 3
figure 3

Linear relationship among the three dimensions of curricular noticing in this study

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However, within this observed linear relationship, it was evident that the mathematical knowledge possessed by the PSTs played a pivotal role in all three dimensions of curricular noticing. This discovery resonates with prior research findings (Lee & Cross Francis, 2018; Yang et al., 2021). According to Yang et al. (2021), various components of teacher knowledge exert a significant impact on teacher noticing in distinct ways; specifically, the dimensions of interpretation and decision-making exhibit a stronger correlation with teachers’ knowledge than the dimension of perception. Similarly, Lee & Cross Francis (2018) established that the development of PSTs’ noticing skills is intricately linked to their evolving comprehension of both mathematical and pedagogical aspects inherent in the tasks at hand.

Conclusion

A methodological limitation of this study is that we depended on PSTs’ written responses to the questionnaire without conducting follow-up interviews to further probe their responses. Therefore, we suggest conducting follow-up interviews in future studies in order to enrich the findings and strengthen their validity and credibility. Also, the complexity of the tasks utilized in this study may have impacted PSTs’ sequencing decisions, as their challenges in solving these problems are likely to have affected their judgment when carrying out the pedagogical sequencing activities.

However, notwithstanding this methodological limitation, as an investigation of a specific activity within the three-dimensional curricular noticing framework, this study contributes to the current literature on curricular noticing and the knowledge base of teacher education. In particular, this study has implications for the design of mathematics education courses as well as for research on teachers’ knowledge and their pedagogical strategies. In particular, the findings of this study suggest that elementary mathematics teacher education programs should incorporate more curricular noticing activities, such as sequencing tasks and other exercises, that allow PSTs to engage in curricular noticing during the early stages of their teaching careers. This study also suggests the need for further investigation into PSTs’ curricular noticing abilities in other mathematics teaching contexts.