This paper reports on the results of a four-day teaching experiment that supported two algebra teachers to develop a combinatorial meaning for algebraic structure. The purpose of the teaching episodes was to support the teachers (a) to establish a combinatorial understanding for algebraic structure (Tillema & Burch, 2022) by generalizing the cubic identity, (a + b)3 = a3 + 3a2b + 3ab2 + b3, as a symbolization of quantitative and combinatorial relationships out of a contex-tualized problem (Tillema & Gatza, 2016) and (b) to develop a combinatorial meaning as a mobilization of their understanding through a series of algebraic tasks (cf. Thompson et al., 2014). The findings from this study contribute to research literature on teachers' mathematical meanings within secondary algebra by investigating how teachers' combinatorial meanings developed and how differences in their combinatorial meanings impacted their algebraic reasoning. The findings demonstrate a combinatorial pathway for supporting the development of expanding and factoring as reversible polynomial operations (cf. Sangwin & Jones, 2017).

Children tend to prioritize whole number information over relational information in proportional reasoning tasks, such that they judge a spinner with 4/10 red pieces as more likely to land on red than a spinner with 2/3 red pieces, because 4 > 2 (e.g., Hurst & Cordes, 2018a; Jeong et al., 2007). This whole number bias is hypothesized to be a driven by fluency in verbal counting in early childhood, which is thought to promote attention to whole number information. In this study, we explored (1) the relation between verbal counting abilities and whole number biases and (2) whether distinct framing contexts - either encouraging children to maximize the number of stickers won, or minimizing the number of stickers lost -differentially impact children's proportional reasoning. Three-to nine-year olds (N = 210, M-age = 5.7 years) chose which of two spinners they preferred to spin. Children in the Gain condition learned that if the spinner landed on red, they would win a sticker and if it landed on blue, nothing would happen; children in the Loss condition learned that if the spinner landed on red, nothing would happen, but if it landed on blue, they would lose a sticker. Counter to prior work, performance of both older (6-9 year olds) and younger (3-5 year olds) children revealed whole number biases. Notably, whole number biases were not related to counting abilities. Importantly, we find framing the task in a Loss scenario lessened whole number biases, suggesting that task framing can alter children's attention to whole number information in a proportional reasoning context.

Questions of high (vs. low) cognitive demand (CD), which encourage children to engage in abstract or critical thinking (e.g., problem solve, reason about cause-and-effect relations, make inferences), may drive relations between children's language exposure and early skills. The present study adopted a micro-analytic approach to examine caregivers' high-CD questioning with their preschool-aged children while viewing a wordless picture book (n = 121) and in the moment (e.g., interaction time, child responses) and global factors (e.g., caregiver education). The probability of caregivers' high-CD questioning increased with interaction time and caregiver education. Post-hoc exploratory analyses revealed that the relation between children's responses and caregivers' high-CD questioning depended on caregivers' perceptions of children's vocabulary skills. Specifically, the probability of caregivers' subsequent high-CD questioning was greater if their child did not respond previously and if caregivers perceived them to have high vocabulary skills. In contrast, caregivers' questioning remained relatively constant for responsive children across different vocabulary skills. Thus, caregivers may employ certain types of input during brief, informal learning interactions with their children by considering their own and their child's propensities and micro-level changes that occur during their conversations.

The feelings of difficulty and familiarity (FOD and FOF) are two types of metacognitive experiences. Both may influence student engagement and the application of metacognitive strategies, but these relationships are not well understood, in part because many studies have relied on self-report measures of behaviors that may not accurately reflect students' actual behaviors. In this study, FOD and FOF were related to objective measures of off-task behaviors and metacognitive strategies. These measures were extracted from 88 sixth graders' action logs within a computer-based learning environment known as Betty's Brain. Pre- and post-tests were administered to assess learning. Results reveal that high-FOD students showed more off-task behaviors and fewer strategic behaviors than low-FOD students, particularly when this difference was measured in terms of the frequency (as opposed to proportion) of strategic behaviors. FOF was not associated with off-task behaviors and metacognitive strategies but emerged as a moderator in the relationship between FOD and learning gains. Low-FOD students learned more than high-FOD students in the low-FOF group, but such a difference was not found in the high-FOF group.

It is widely recognized that debugging is challenging for novice programmers and, as such, computing educators and researchers have called for explicit debugging instruction. Debugging requires various knowledge and skills, and different students may show different strengths and weaknesses. An understanding of such individual differences is important as it may guide personalized instruction. The current study investigated individual differences in debugging in an undergraduate introductory computer science course. We extracted variables related to debugging from students' submission traces to programming problems in the first month of the course. We applied latent profile analysis to these variables and identified three distinctive profiles. Profile A showed higher debugging accuracy and speed. Profile B showed lower debugging performance in runtime and logic errors, while profile C had lower performance in syntactic errors and tended to make large code edit every submission. Students' gender and self-rated programming ability predicted profile membership. Moreover, profile A got higher scores than the others in the first exam, and this difference persisted in the second and third exam, even controlling for background variables and score on the first exam. We investigated how students transitioned across debugging profiles over the duration of the course. From the beginning to the end of the course, a large part of students stayed in lower performance profiles. Overall, these findings support the call that debugging should be taught at an early stage and suggest that different groups may need different debugging instructions or support.

A growing body of research has examined parents' practices to support their young children's number learning at home, that is, the home numeracy environment. Many of these studies focus on formal and informal domains of numeracy activities, which are inconsistently defined and related to children's math learning. In this study, we explore dimensions of the home numeracy environment and examine their relations with children's math skills among a sample of 4-year-old children and their parents over the course of 1 year. Parents reported on the frequency of 21 numeracy activities when children were 4 and 5. Exploratory and confirmatory factor analyses revealed a 2-factors solution: number-related play activities and use of educational materials with numbers. Frequency of play with numbers was positively related to children's ability to solve applied math problems at age 5, controlling for prior number skills, child age, and socioeconomic status. In contrast, neither measure of the home numeracy environment predicted symbolic number knowledge or non-symbolic number sense when controlling for covariates. These findings underscore the need to differentiate between factors of the home numeracy environment and to develop clear theoretical definitions of these factors.(c) 2022 Elsevier Inc. All rights reserved.

Openness to Experience is most strongly related to aspects of high-level cognition, such as creativity. Yet, the role of cognitive capacities in Openness is still far from understood. We examine how individuals search their memory predicts levels of Openness. Participants (N = 163) had one minute to generate synonyms to the word hot, which operationalizes mental navigation through a multidimensional representation of the mental lexicon - a cognitive multiplex network. We find high accuracy in low- and high- Openness group classification, and good prediction of individual differences in Openness. These results support the use of computational cognitive modelling for the study of personality traits. Further, our results suggest that people high in Openness engage in a distinct style of cognitive search.

There is a need for a more robust conceptualization of engagement in mathematics education research. Investigating how teachers describe engagement can provide insight into relationships between purposes of engagement and dimensions of engagement. In this exploratory study, we examined how 28 secondary mathematics teachers in two states in the USA talked about their students' engagement. During interviews, we asked teachers to provide their definitions for engagement, describe their teaching strategies for engaging students, and describe their observations of engagement during a video clip from their own classroom. We interpreted teachers' talk to identify how they described the nature of mathematics engagement (dimensions such as behavioral, cognitive, affective, and/or social engagement) and purposes of engagement (engagement in learning or in schooling [Harris, 2011]). When teachers described the purpose of engagement as engagement in learning, they also tended to describe the nature of engagement with cognitive and social dimensions and with multiple dimensions of engagement.

Learning coding during early childhood is an effective way for children to practice computational thinking. As-pects of children's motivation can increase the likelihood that children approach computational thinking activities with enthusiasm and deep engagement. Gender inequities may interfere with children's readiness to take advan-tage of opportunities to build computational thinking skills through activities such as coding. Societal stereotypes can reduce young girls' motivation to engage with computer science, preventing them from gaining benefits from coding activities designed to support computational thinking. This study examined children's gender stereotypes as well as children's own motivation for computer coding in 363 first-through third-grade children. We assessed gender differences in both stereotypes and motivation, as well as links between the stereotypes that individual children held and their own motivation. Children generally endorsed stereotypes about interest and ability for computer coding that favored their own-gender group, although third-grade girls reported gender-egalitarian beliefs about interest in coding. There were no gender differences in children's motivation for computer cod-ing in terms of their own interest, sense of belonging, or ability self-concepts. Children's stereotypes about their own-gender group were significantly positively correlated with their own motivation for computer coding. These findings suggest that early childhood represents an excellent age for children to begin building computational thinking skills, before girls endorse negative stereotypes about their gender's interest in computer science.

Teachers' mathematical knowledge has important consequences for the quality of the learning environment they create for their students to learn mathematics. Yet relatively little is known about how teachers reason proportionally, despite the fact that proportional reasoning is foundational for several mathematics concepts and that ratios and proportional relationships constitute a major component of the middle school mathematics curriculum. In this study, we investigated how teachers reasoned proportionally on a nonroutine ratio task and the extent to which their proportional reasoning was able to predict their overall understanding of the relevant concepts: ratios and proportional relationships. Using data collected from 238 US mathematics teachers, we found that teachers' proportional reasoning could be grouped into four categories: incorrect, additive, relative, and proportional reasoning. Our results also indicated that teachers' overall knowledge of ratios and proportional relationships aligned with the way they reasoned proportionally, meaning that teachers who used incorrect reasoning on a separate task received the lowest scores on average on the ratios and proportional relationships measure, whereas those who reasoned proportionally had the highest mean scores on average. Implications of the study include the need to shift attention to the way teachers reason in relation to the two elements of proportional reasoning (covariance and invariance) to capture the nuances in their understanding of ratios and proportional relationships.