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When researchers code behavior that is undetectable or falls outside of the validated ordinal scale, the resultant outcomes often suffer from informative missingness. Incorrect analysis of such data can lead to biased arguments around efficacy and effectiveness in the context of experimental and intervention research. Here, we detail a new Bayesian mixture approach that analyzes ordinal responses with undetectable/uncodable behaviors in two stages: (1) estimate a likelihood of response detection and (2) estimate an Explanatory Item Response Model for the ordinal variable conditional on detection. We present an independent random effects and correlated random effects variant of the new model and demonstrate evidence of model functionality using two simulation studies. To illustrate the utility of our proposed approach, we describe an extended application to data collected during a length measurement teaching experiment (N = 186, 56% girls, 5-6 years at preassessment). Results indicate that students assigned to a learning trajectories instructional condition were more likely to use detectable, mathematically relevant problem-solving strategies than their peers in two comparison conditions and that their problem-solving strategies were also more sophisticated. [This is the online first version of an article published in "Journal of Research on Educational Effectiveness."]

Youth bullying is a significant concern in the USA, particularly for youth with disabilities. This pilot study explores out- come data related to the DIsability Anti-BuLlying (DIAL) training program for general and special education teachers to prevent bullying generally with a particular focus on youth with disabilities. Six schools were randomly assigned to receive the DIAL intervention or delayed (control) intervention. All teachers (n = 65) responded to a self-report survey before and after module completion. Teachers completed measures on their teaching efficacy, interactions with students, and attitudes toward bullying and completed reports of social cognition, anxiety, and bullying for their K-5 students (n = 472). Teachers (n = 36) enrolled in the three intervention schools completed four online modules focused on improving attitudes, efficacy, and skills to prevent bullying/victimization for K-5 students with and without disabilities. Results indicated teachers who received the DIAL program reported higher levels of self-efficacy and lower levels of maladaptive attitudes toward bullying. [This is the online version of an article published "International Journal of Bullying Prevention."]

Although basing instruction on a learning trajectory (LT) is often recommended, there is little direct evidence to support the premise of a "LT approach"--that to be maximally meaningful, engaging, and effective, instruction is best presented one LT level beyond a child's present level of thinking. The present report serves to address the question: Is it necessary to teach each contiguous level of a LT or can instruction be similarly or more effective when skipping levels, provided the necessary exemplars are made? In a multimethod research study that included individual teaching experiments embedded inside of a quasi-experimental research design, one group of 13 kindergartners received instruction based on an empirically-validated LT for addition and subtraction (the "LT" treatment). The counterfactual, "skip" treatment (n = 12), received instruction focused mainly on levels at least two levels above their present level for the same amount of time as the LT treatment. More children in the LT treatment exhibited greater addition and subtraction learning during sessions and from pretest to posttest than children in the skip treatment. Implications for future study are discussed. [This paper was published in "ZDM Mathematics Education" v52 n4 p637-648 2020 (EJ1261937).]

Quality early STEM experiences provide a critical foundation for learning about science, technology, engineering, and mathematics (STEM) in ways that facilitate later learning. Early childhood educators have a unique opportunity to benefit all students by bringing the STEM disciplines into greater balance with literacy, reducing early inequities in access and opportunity to learn STEM, and developing skills and understandings that students can build on as they progress through school and into civic life and the workforce. This brief draws on research and development supported by the National Science Foundation to highlight important considerations about STEM educational experiences for young children and professional learning for educators who provide those experiences.

Prior research shows that short-term effects from preschool may disappear, but little research has considered which environmental conditions might sustain academic advantages from preschool into elementary school. Using secondary data from two preschool experiments, we investigate whether features of elementary schools, particularly advanced content and high-quality instruction in kindergarten and first grade, as well as professional supports to coordinate curricular instruction, reduce fadeout. Across both studies, our measures of instruction did not moderate fadeout. However, results indicated that targeted teacher professional supports substantially mitigated fadeout between kindergarten and first grade but that this was not mediated through classroom quality. Future research should investigate the specific mechanisms through which aligned preschool-elementary school curricular approaches can sustain the benefits of preschool programs for low-income children. [This paper was published in the "Journal of Research on Educational Effectiveness" v11 n3 p339-374 2018 (EJ1181585).]

Tomorrow's inventors and scientists are today's curious young children--as long as those children are given ample chances to explore and are guided by adults equipped to support them. "STEM Starts Early" is the culmination of a deep inquiry by the Joan Ganz Cooney Center at Sesame Workshop and New America embarked on an exploratory project, funded by the NSF, to: (1) better understand the challenges to and opportunities in STEM learning as documented in a review of early childhood education research, policy, and practice; (2) make recommendations to help stimulate research and policy agendas; and (3) encourage collaboration between pivotal sectors to implement and sustain needed changes. Prominent early STEM researchers, policy makers, and teacher educators were interviewed to gain perspectives from stakeholders in each of the early childhood areas. Next, two focus groups were conducted with teachers, one with child care and preschool educators and one with early elementary school teachers. The insight gained from these interviews and focus groups shaped the focus of this report. The examination of the STEM landscape and the players in it produced found five key findings: (1) Both parents and teachers appear to be enthusiastic and capable of supporting early STEM learning; however, they require additional knowledge and support to do so effectively; (2) Teachers in early childhood environments need more robust training and professional development to effectively engage young children in developmentally appropriate STEM learning; (3) Parents and technology help connect school, home, and other learning environments like libraries and museums to support early STEM learning; (4) Research and public policies play a critical role in the presence and quality of STEM learning in young children's lives, and both benefit from sustained dialogue with one another and with teachers in the classroom; and (5) An empirically-tested, strategic communications effort is needed to convey an accurate understanding of developmental science to the public, leading to support for meaningful policy change around early STEM learning. Two appendixes are included: (1) STEM in early childhood: An analysis of NSF grant awards; and (2) How reframing research can enhance STEM support: A two-science approach. [For the executive summary, see ED574401.]

Tomorrow's inventors and scientists are today's curious young children--as long as those children are given ample chances to explore and are guided by adults equipped to support them. "STEM Starts Early" is the culmination of a deep inquiry by the Joan Ganz Cooney Center at Sesame Workshop and New America embarked on an exploratory project, funded by the NSF, to: (1) better understand the challenges to and opportunities in STEM learning as documented in a review of early childhood education research, policy, and practice; (2) make recommendations to help stimulate research and policy agendas; and (3) encourage collaboration between pivotal sectors to implement and sustain needed changes. Prominent early STEM researchers, policy makers, and teacher educators were interviewed to gain perspectives from stakeholders in each of the early childhood areas. Next, two focus groups were conducted with teachers, one with child care and preschool educators and one with early elementary school teachers. The insight gained from these interviews and focus groups shaped the focus of this report. This executive summary highlights the findings and recommendations from this research. [This executive summary was written with contributions from Peggy Ashbrook and Cindy Hoisington. For the full report, see ED574402.]

Wisconsin Virtual School, a state online learning program that offers supplemental online courses to partner districts in Wisconsin, administered a survey to its teachers in fall 2014 to learn more about their training experiences and challenges. More Wisconsin Virtual School teachers reported participating in training or professional development related to online instruction while teaching online than before teaching online or during preservice education. The most frequently reported professional challenges concerned student engagement with and perseverance in online courses. Wisconsin Virtual School teachers indicated that they preferred unstructured professional development to structured professional development for addressing challenges related to students' perseverance and engagement. [For the full report, see ED561235.]

Although basing instruction on a learning trajectory (LT) is often recommended, there is little evidence regarding the premise of a LT approach--that to be maximally meaningful, engaging, and effective, instruction is best presented one LT level beyond a child's present level of thinking. This hypothesis was evaluated for an empirically-validated LT for early arithmetic, using with 291 kindergartners from four schools in a Mountain West state. Students randomly assigned to the LT condition received one-on-one instruction one level above their present level of thinking. Students in the counterfactual condition received one-on-one instruction that involved solving story problems three levels above their initial level of thinking (a teach-to-target approach). At posttest, children in the LT condition exhibited significantly greater learning, including target knowledge, than children in the teach-to-target condition, particularly those with low entry knowledge of arithmetic. Child gender and dosage were not significant moderators of the effects. [This paper was published in "Journal of Educational Psychology" v113 n7 p1323-1337 2021 (EJ1318761).]

A robust finding across research on early childhood educational interventions is that the treatment effect diminishes over time, with children not receiving the intervention eventually catching up to children who did. One popular explanation for fadeout of early mathematics interventions is that elementary school teachers may not teach the kind of advanced content that children are prepared for after receiving the intervention, so lower-achieving children in the control groups of early mathematics interventions catch up to the higher-achieving children in the treatment groups. An alternative explanation is that persistent individual differences in children's long-term mathematical development result more from relatively stable pre-existing differences in their skills and environments than from the direct effects of previous knowledge on later knowledge. We tested these two hypotheses using data from an effective preschool mathematics intervention previously known to show a diminishing treatment effect over time. We compared the intervention group to a matched subset of the control group with a similar mean and variance of scores at the end of treatment. We then tested the relative contributions of factors that similarly constrain learning in children from treatment and control groups with the same level of post-treatment achievement and pre-existing differences between these two groups to the fadeout of the treatment effect over time. We found approximately 72% of the fadeout effect to be attributable to pre-existing differences between children in treatment and control groups with the same level of achievement at post-test. These differences were fully statistically attenuated by children's prior academic achievement. [This article was published in "Developmental Psychology" v52 n9 p1457-1469 Sep 2016 (EJ1112297).]

Although some research-based educational practices have shown promise, many fail to be implemented at a scale that affects more than a small proportion of children. Further, research on interventions for young children includes mixed results, with most documenting "fadeout" of effects after several years, but some showing lasting effects. In this study, the authors evaluated the long-term impacts of a model for scaling up early interventions, in this case a successful early mathematics curriculum, testing to see whether the originally-sustained impacts persisted up to 7 years beyond baseline. The original evaluation of TRIAD (Technology-enhanced, Research-based, Instruction, Assessment, and professional Development) employed a cluster randomized trial (CRT) design to test the effectiveness of the TRIAD scale-up approach, using the Building Blocks® early mathematics curriculum (BB) in preschool. In this evaluation, 42 schools were randomly assigned to three groups. One of the groups was a business-as-usual control (CTRL). In preschool, the other two conditions were identical and entailed TRIAD scale-up (intensive training, coaching focused on both BB curriculum and linked assessment and online professional development). After preschool, one of the treatment conditions included follow-through professional development through 1st grade, including knowledge of the intervention and ways to build upon that knowledge using learning trajectories (TRIAD Follow-Through or TRIAD-FT) while the other condition implemented business-as-usual control (TRIAD Non-follow-through or TRIAD-NFT). Students in the two treatment conditions significantly outperformed their peers in the control condition through 1st grade. The primary purpose of the current investigation was to examine the long-term effects of the early mathematics curriculum, but the secondary aim was to explore whether the TRIAD scale-up approach itself, which included aligned professional development for kindergarten and 1st grade teachers, also enhanced the effects of the pre-K curriculum beyond 1st grade. Research questions were as follows: (1) What are the differences in the average math achievement of the three original study groups (TRIAD-NFT, TRIAD-FT, and CTRL) in 3rd, 4th, and 5th grade; (2) Do the differences in the 3rd, 4th, and 5th grade math achievement of the original study groups vary by the socio-economic and racial/ethnic backgrounds of the students in the groups; (3) What are the differences in the 3rd, 4th, and 5th grade math achievement of the subset of children who remained in the group to which their original school was randomly assigned throughout the intervention period (pre-K through 1st grade); and (4) Do the differences in the in the 3rd, 4th, and 5th grade math achievement of the subset of children who remained in the group to which their original school was randomly assigned throughout the intervention period (pre-K through 1st grade) vary by the students' socio-economic status and racial/ethnic backgrounds? Figure 1 presents results addressing the first research question that pertain to the differences between the three groups at the end of pre-K through 5th grade. Figures 2a and 2b inform the second research question and show the effects of the FT and NFT conditions separately for African-American, Hispanic, other minority, and white students. Figures 3a and 3b show the FT and NFT effect estimates separately for the low and high SES subgroups formed based on students' eligibility for FRPL. Figures 4, 5a, 5b, 6a, and 6b inform the 3rd and 4th research questions and show parallel results for the stayers (students who stayed in the original conditions between pre-K and 1st grade). Impacts at the end of 5th grade were statistically significant and considerably larger than impacts in 3rd and 4th grade for both conditions and all subgroups. Tables and figures are appended.

REL Midwest, in partnership with the Midwest Virtual Education Research Alliance, analyzed the results of a survey administered to Wisconsin Virtual School teachers about the training in which they participated related to online instruction, the challenges they encounter while teaching online, and the type of training they thought would help them address those challenges. REL Midwest researchers and Virtual Education Research Alliance members collaborated to develop the survey based on items from the Going Virtual! survey (Dawley et al., 2010; Rice & Dawley, 2007; Rice et al., 2008). Wisconsin Virtual School administered the survey to its 54 teachers, and 49 (91 percent) responded to the survey. The responses of the 48 teachers who indicated that they taught an online course during the 2013/14 or 2014/15 school year were analyzed for the report. Results indicate that all Wisconsin Virtual School teachers reported participating in training or professional development related to online instruction and that more teachers reported participating in training that occurred while teaching online than prior to teaching online or during preservice education. The teachers most frequently reported challenges related to students' perseverance and engagement and indicated that they preferred unstructured professional development to structured professional development to help them address those challenges. Further research is needed to determine what types of professional development and training are most effective in improving teaching practice, especially related to student engagement and perseverance. The following are appended: (1) Survey on online teacher training and challenges; (2) Data and methodology; and (3) Supplementary tables.

Background/Context: Early childhood education research has consistently found evidence of immediate, positive effects on developmental outcomes. Despite these strong initial benefits, follow-up analyses often reveal that these effects fadeout in the years following participation (e.g., Bailey et al., 2017, Clements et al., 2013; Lipsey et al. 2018). Many researchers have proposed that fadeout stems from discontinuities in the learning environment from preschool to kindergarten (e.g., Bailey et al., 2020). Within this framework, children may continue to reap the benefits of high-quality preschool efforts if the quality of the subsequent learning environment adequately promotes continued skill building. Surprisingly, evidence supporting the sustaining environments hypothesis has been mixed. Thus, it remains unknown which characteristics of the elementary school environment may contribute to the sustained benefits of early childhood interventions (e.g., Burchinal et al., 2022). However, the peer composition of children's subsequent classroom environments could be a key factor. Classrooms with a greater share of high-quality preschool attendees may be more academically similar, and require less catchup, than classrooms with a greater share of non-program attendees. Teachers may respond to the overall ability level of their classroom by altering their instruction to focus on more advanced materials. These instructional differences may not only facilitate continued academic growth for program children, but they may also promote skill-building opportunities for non-program children. Purpose/Objective/Research Question: This project examines the association between classroom composition and mathematics achievement among children who participated in the TRIAD study (see Clements et al., 2011), a scale-up evaluation of a preschool math curriculum. Specifically, we examined whether: (1) the proportion of "treated" children in each classroom predicted achievement through first grade, (2) peer mathematics achievement predicted child mathematics ability through first grade, and (3) classroom composition predicted changes in the quality and quantity of classroom mathematics instruction. Setting: The TRIAD Study was a cluster-RCT implemented in Boston, Massachusetts and Buffalo, New York. The current analyses rely on student-level data that was collected beginning in the 2006-2007 and ending in the 2008-2009. Population/Participants/Subjects: Participants were eligible for inclusion in our analyses if they completed all assessments, and neither skipped a grade nor were retained. Of the original sample (N=1305), 93% of participants met inclusion criteria in kindergarten (n=1214) and 84% of participants met the criteria for first grade (n=1095). Overall, participants may be described as predominantly low income (85%), with approximately 53% identifying as Black and 21% as Hispanic. Intervention/Program/Practice: Although this proposal does not report impact estimates, the original study examined the effectiveness of "Building Blocks," a preschool mathematics program that utilizes evidence-based learning trajectories to enhance children's understanding of mathematics. In addition to the program's curriculum, the intervention included extensive teacher coaching and professional development (Clements and Sarama, 2008). Research Design: We rely on data from a cluster-RCT, which involved 42 elementary schools in Buffalo and Boston that had public preschool programs. Schools were blocked by achievement and randomized within block to either the "preschool-only" intervention group, the "follow-through" intervention group, or the treatment-as-usual control group. Data Collection and Analysis: Participants' mathematics achievement was assessed in the fall (pretest) and spring (posttest) of preschool using the research-based early mathematics assessment (REMA; Clements et al., 2008). Mathematics achievement was measured again in the spring of kindergarten and first grade, along with classroom observation to assess the quantity and quality of mathematics instruction (Clements et al., 2008). Peer exposure variables were calculated for each condition. The proportion of treated peers was generated by counting the number of "treated" children, subtracting one to effectively remove each child from their own peer environment, and dividing by the class size. This process was replicated for the control group to create the proportion of control peers, which we treat as a type of "placebo" check for our models focusing on treated peers. We analyzed the association between the peer environment and end-of-the-year mathematics achievement using a series of regressions, whereby each specification builds on the last. The first model strictly examines the relationship between classroom composition and student math achievement (z-scored); the second model incorporates baseline covariates and fixed effects for the blocking group and site; the third model includes baseline covariates and preschool fixed effects (i.e., child's school at the time of assignment); and the fourth model includes baseline covariates, preschool fixed effects, and student math score from the prior year (i.e., end of preschool or kindergarten). Findings/Results: Table 1 presents descriptive statistics of classroom level predictors for the kindergarten and first grade samples. Statistics are provided for each of the three conditions, including the preschool only condition, the follow-through condition, and the control condition. Table 2 presents results from the within-condition regressions examining the relationship between the proportion of treatment or control peers and student math achievement. Estimates in each panel were derived from a series of four models, which were run separately for the kindergarten and first grade sample. Results indicate that having a greater share of treated peers in kindergarten was not predictive of kindergarten math achievement. However, the proportion of treated peers was significantly predictive of first grade math achievement. When baseline covariates and clustering were accounted for (columns 2-3 and 6-7), a ten-percentage point increase in treated peers equated to a 0.04-0.05 standard deviation increase in first grade math achievement. This effect held for the follow-through condition even after individual math ability was controlled (b= 0.04; p<0.05). Finally, the proportion of control peers was not significantly related to math achievement at the end kindergarten and first grade for children in the control group. Conclusions: Preliminary results indicate that the proportion of treated peers was not predictive of kindergarten math achievement. Interestingly, having a greater share of "Building Blocks" peers was associated with significant, albeit small, improvements in first grade achievement. While these results are promising, further analyses will probe the sensitivity of our findings. Additionally, we will examine the relationship between the proportion of treated peers and classroom math instruction, as well as the association between peer math achievement and individual math achievement in kindergarten and first grade.

A substantial literature documents the benefits of early childhood education and formal preschool experiences on children's school readiness, with low-income and otherwise disadvantaged children benefitting the most from these programs. However, these academic benefits often fade out as children age, and most disappear by the end of kindergarten or first grade. The authors suggest one reason for preschool fade-out might be that children's elementary school teachers continue to teach content that children already learned during preschool, thus curtailing academic growth. In this study, the authors investigate two salient approaches available to policymakers that may improve preschool participants' instructional experiences in elementary school: (1) that of advanced and challenging instruction in kindergarten and first grade; and (2) professional support in which preschool teachers interact with their kindergarten and first grade counterparts to develop a seamless transition from one grade to the next. Two experimental studies of preschool interventions (Head Start and Building Blocks) and children's elementary school environments examined whether the quality of instructional content or providing professional development supports to early grade teachers moderate the impacts on children's cognitive skills. Tables are appended.

In this Appendix, we provide details about the data used for the current study, the curricula used in the classrooms from which data were collected, and the current study's methodological approach. (Contains 14 tables and 5 footnotes.)[For full report, see ED544189.]

This brief is directed to researchers and adds to the research base about instructional practices that are related to student achievement. Additional evidence on these relationships can suggest specific hypotheses for the future study of such instructional practices, which, in turn, will provide research evidence that could inform professional development of teachers and the writing of instructional materials. The results of this study revealed a pattern of relationships largely consistent with earlier research, but not in every case. Results that are consistent with previous research include increased student achievement associated with teachers dedicating more time to whole-class instruction, suggesting specific practices in response to students' work (1st grade only), using more representations of mathematical ideas, asking the class if it agrees with a student's answer, directing students to help one another understand mathematics, and differentiating curriculum for students above grade level (2nd grade only). Less consistent results were found in three 2nd-grade results, and include lower achievement associated with teachers' higher frequency of eliciting multiple strategies and solutions; prompting a student to lead the class in a routine; and with students more frequently asking each other questions. These findings suggest that practices associated with higher achievement gains include both student-centered and teacher-directed practices; however, some student-centered practices were associated with lower achievement gains. (Contains 11 endnotes and 6 tables.) [For "Instructional Practices and Student Achievement: Correlations from a Study of Math Curricula. NCEE Technical Appendix. NCEE 2013-4020," see ED544192.]

This paper describes a retrospective analysis of data collected during a 4-year longitudinal study on children's thinking about measurement through a teaching experiment methodology. It focuses on results from individual interviews with two students on volume measurement. Data analysis was guided by Sarama and Clements' (2009) learning trajectory on volume measurement. Results indicate that (1) both students progressed through levels of the learning trajectory during the study, (2) different representations of 3-D objects (e.g., physical objects, cubes, pictorial representations) influenced their strategies, and (3) their individually constructed definitions for the term "volume" affected their decisions in volume comparisons. [For the complete proceedings, see ED584829.]

The author and her colleagues' TRIAD model (Sarama, Clements, Starkey, Klein, & Wakeley, 2008), including the "Building Blocks" curriculum, have significantly and substantially increased preschooler's mathematical competence, both in previous studies (Clements & Sarama, 2008, g = 1.07) and in their present, largest implementation (Clements, Sarama, Spitler, Lange, & Wolfe, in press, g = 0.72). The present study was the first to evaluate the effects of this model longitudinally. The authors' overarching research question was: "What are the long-range (persistence of) effects of the TRIAD intervention, with and without follow through, on achievement?" The present research included two experimental groups (and one control group). In both, pre-K teachers participated in the intervention. In the Follow-Through experimental group, teachers in grades K and 1 were taught about the pre-K intervention and ways to build upon it. Do both experimental groups outperform those in the comparison group in math achievement on the average, at the end of kindergarten and first grade? Do children in the experimental TRIAD Follow-Through (TRIAD-FT) group on the average outperform children in the TRIAD (non-follow through, TRIAD-NFT) experimental group (the value added question)? At the end of Kindergarten, both experimental groups outperformed the control group in math achievement at the end of kindergarten. However, contrary to their hypothesis, the TRIAD Follow Through did not statistically significantly outperform the TRIAD Non-Follow Through (TRIAD-NFT) group. Although it is encouraging that the gains of both TRIAD groups persisted, these findings did not support their hypothesis that Follow Through would be effective and necessary for this persistence. Nevertheless, the effect size of the TRIAD Follow Through group vs. the control group (0.55) was greater than that of the TRIAD-NFT group (0.34), so the trends were consistent with their expectations. There was no evidence that the "Building Blocks" intervention was differentially effective for schools with different percentages of students with free or reduced lunch or English Language Learners, nor for individual children with or without IEPs. There was evidence that the intervention was differentially effective for one ethnic/racial comparison: African-American children learned less than other children in the same control classrooms and African-American children learned more than other children in the same Building Blocks classrooms in Kindergarten. There are five basic recommendations: (1) Curriculum and policy should ensure that children, especially those living in poverty, should be provided with research-based, focused early mathematical interventions which can increase their knowledge of multiple mathematical concepts and skills (including, but also going beyond number); (2) It is essential that preschool mathematics interventions be continued into the primary grades; (3) The Curriculum Research Framework (Clements, 2007) upon which the curriculum was based has been repeatedly empirically supported and may serve as a useful guide to policy makers, curriculum and software developers, and administrators; (4) The learning trajectories at the core of the curriculum and TRIAD model may constitute a useful construct in future research, curriculum development, and professional development efforts; and (5) This is the first study of this kind of which they are away. They need more research on the conditions that children from early interventions enter in the primary school years. Appended are: (1) References; and (2) Tables and Figures. (Contains 2 figures.) [This paper was written with Julie Sarama, Mary Elaine Spitler and Christopher B. Wolfe.]

Background: Learning trajectories (LTs) in early mathematics curriculum and teaching have received increasing attention (Baroody et al., 2019; Clements, 2007; Clements & Sarama, 2021; Maloney et al., 2014; Sarama & Clements, 2009). For example, LTs were a core construct in the National Research Council (2009) report on early mathematics education (subtitled "Paths toward excellence and equity") and the notion of levels of thinking was a key first step in the writing of the Common Core State Standards--Mathematics (NGA/CCSSO, 2010). However, little research has directly tested the specific contributions of LTs to teaching compared to instruction provided without LTs (Frye et al., 2013). Objective: The goal of the study was to compare the learning of kindergarteners who received instruction on length measurement following an empirically-validated LT (see Table 1) to those who received an equal amount of time on the same instructional activities that were not sequenced along the LT's developmental progression. Setting: This study is a randomized control trial that took place in an urban school district in a Mountain West state. Population: The sample was composed of 186 kindergarten students (104 girls) from 16 classrooms across six schools (four public; n = 149). Table 2 presents demographic information for the public schools. Intervention: The intervention was composed of three experimental conditions: the LT group, the reverse-order (REV) group, and control group (BAU). Only the LT and REV group received ten one-on-one instructional sessions. In the LT condition each instructor had access to a set of instructional activities that aligned with each level of the developmental progression and selected activities based on the child's present level of thinking. As children demonstrated higher levels of thinking, they were encouraged to use more sophisticated strategies (e.g., Level 3, iterating with a single unit instead of Level 2, using multiple units; Table 1). The REV group received 10 length activities selected from each level of the developmental progression in reverse order (Table 1 and LearningTrajetories.org). Thus, students were exposed to similar activities as the LT condition, but began with the most sophisticated level: Level 5. REV instructors provided feedback about the correctness of children's solutions, but did not modify activities to accommodate less sophisticated levels of thinking. Research Design: We used a randomized control trial. All students for whom we acquired parental consent were administered a pre-assessment prior to random assignment. The pre- and post-assessment was composed of 26 items adapted from extant instruments (Battista, 2012; Clements et al., 2008/2021). We tested two child-level covariates: child sex (0 = girl) and school type (public/private; 0 = public) are coded as binary. Data Collection: and Analysis The research question was examined within a Bayesian hierarchical linear modeling (HLM) framework using the brms package (Bürkner, 2018) in R 3.6.2 (R Core Team, 2019). Bayesian models more accurately quantify and propagate uncertainty (e.g., Kruschke, 2014) and can be more reliable in cases where traditional HLM methods typically fail (Eager & Roy, 2017). Results: Students in the LT and REV condition outperformed their peers in the BAU condition. Contrasts further reveal that students in the LT group also outperform their REV peers = 0.32 (0.57, 0.07). Credible Intervals of 95% were estimated for child gender and whether the child attended a public or private school. However, these intervals include zero and therefore deemed to be statistically non-significant. Discussion: As one of a set of experiments rigorously testing the efficacy of the educational application of learning trajectories (LTs), this study focused on the second assumption of an LT approach: there is a sequence of learning and teaching that is determined by a research-based developmental progression. The topic of length was selected as being both important to early mathematics learning and amenable to the LT and counterfactual conditions. The first of these, intended within the research context to represent the traditional "theme" approach for choosing activities (Helm & Katz, 2016; Katz & Chard, 2000; Tullis, 2011), reversed the sequence of the LT activities to directly test the assumption (REV). The second counterfactual, business-as-usual (BAU), served as a passive control. Students in the LT group outperformed their BAU and their REV peers. The latter contrast, especially, supports the hypothesis that following the developmental progression of an empirically-validated LT promotes learning more than the same activities not in that order. Students in the REV condition also outperformed their peers in the BAU condition. This indicates that the activities, even when implemented in an order other that of the LT's developmental progression, are still effective. This result is similar to that of the studies testing the first LT assumption, those that had a "teach-to-target" counterfactual (Authors, 2019, 2020). That is, these and the present study suggest teaching each contiguous level in developmental order of a LT is more efficacious and thus useful than alternatives, but not necessary to facilitate learning in all cases. However, note that in previous studies using a teach-to-target approach to test assumption 1 (Authors, 2019, 2020) instruction was at levels n + 2 or n + 3, avoiding instruction at n + 1. In the present study, children in the REV condition experienced activities at each level considered in this study (Table 2) and thus the activities eventually crossed over the child's present level of thinking (including n - 1, n, n + 1, etc.). This supports the LT assumption that each builds hierarchically on the concepts and processes of the previous levels (e.g., Goodson, 1982; Sarama & Clements, 2009; van Hiele, 1986). That is, each level is characterized by specific concepts (e.g., mental objects) and processes (mental "actions-on-objects") (Clements et al., 2004; Steffe & Cobb, 1988) that underlie mathematical thinking at level n and serve as a foundation to support successful learning of subsequent levels (Sarama & Clements, 2009). However, the learning process is not intermittent and step-like, but rather incremental and gradually integrative. A critical mass of ideas from each level must be constructed before thinking characteristic of the subsequent level becomes ascendant in the child's thinking and behavior (Clements et al., 2001).

Background: Data on children's ability to answer assessment questions correctly paints an incomplete portrait of what they know and can do mathematically; yet, it remains a common basis for program evaluation. Indeed, pre-post-assessment correctness is necessary but insufficient evidence for making inferences about learning and program effectiveness; assessment should also evaluate the strategies developed to solve problems. There is evidence to suggest that using Learning Trajectories (LTs; Clements & Sarama, 2021) as the basis for instruction increases "correctness" in kindergarten populations. However, the impact of LTs on "strategy sophistication" has not been examined. We describe a statistical technique that models strategy sophistication and apply it to investigate the efficacy of LTs in early length-measurement (LM). Objective: The present study analyzes the "strategies" deployed by children in research project (Sarama et al., 2021) to answer two question: RQ1: Do children in the LT condition use more sophisticated strategies relative to their peers in two counterfactual conditions? RQ2: How do we model strategy sophistication in the presence of informative missing data? Setting: This study leverages data shared with us from "Evaluating the Efficacy of the Learning Trajectories in Early Mathematics," funded by the Institute of Education Sciences. This randomized control trial took place in an urban school district in a Mountain West state. Population: The sample was composed of 186 kindergarten students (104 girls) from 16 classrooms across six schools (four public; n = 149). Table 2 presents demographic information for the public schools. Intervention: The intervention was composed of three experimental conditions: the LT group, the reverse-order (REV) group, and control group (BAU). Only the LT and REV group received ten one-on-one instructional sessions. In the LT condition each instructor had access to a set of instructional activities that aligned with each level of the developmental progression and selected activities based on the child's present level of thinking. As children demonstrated higher levels of thinking, they were encouraged to use more sophisticated strategies (e.g., Level 3, iterating with a single unit instead of Level 2, using multiple units; Table 1). The REV group received 10 length activities selected from each level of the developmental progression in reverse order (Table 1 and LearningTrajetories.org). Thus, students were exposed to similar activities as the LT condition, but began with the most sophisticated level: Level 5. REV instructors provided feedback about the correctness of children's solutions, but did not modify activities to accommodate less sophisticated levels of thinking. Research Design: All students for whom we acquired parental consent were administered a preassessment prior to random assignment. The pre- and post-assessment was composed of 26 items adapted from extant instruments (Battista, 2012; Clements et al., 2008/2021) and scored for correctness and strategy sophistication. We tested two child-level covariates: child sex (0 = girl) and school type (public/private; 0 = public) are coded as binary. Data Collection and Analysis: A team of coders watched pre- and post-assessment videos and assigned their behavior into one of up to 10 research-based strategy sophistication codes (Clements et al., 2008/2021) These codes were collapsed into a 4-level ordinal outcome variable (where 0 = "Unlikely to Solve", 1 = "Low", 2 = "Medium", and 3 = "High"). Behaviors identified as "Other", "Strategy not observed", or "NA" could not be assigned into the 4-level ordinal outcome. We deem these behaviors "nondetectable" and/or "non-codable" and were assigned into the hurdle component of the statistical model (Fig. 1). Hurdle Model: In a naïve modeling approach, "non-detectable" and "non-codable" strategies would be treated as missing data and removed prior to analysis. However, this would lead to undesirable outcomes: 1) reduced sample size and diminished power; 2) biased cause-and-effect inferences due to informative missingness; and 3) exclusion of items due to missing data. Our hurdle ordinal logit model allows us to account for informative missingness and model the two data-generating processes separately: 1) detection probability (); and 2) ordinal sophistication once a strategy has been detected (). Intuitively, our model is the usual cumulative logit (McCullagh, 1980) weighted by the probability of detecting a strategy. We assign the same set of covariates using the logit link to both parts of the model: pre-sophistication Rasch score, experimental condition, child sex, and school type, in addition to random intercepts by child (), item (), and classroom () (see Fig. 2). Estimation took place in a Bayesian paradigm using No-U-Turn Hamiltonian Monte Carlo (NUTS HMC, Hoffman & Gelman, 2014) implemented in Stan (Stan Development Team, 2020). Diagnostics revealed robust convergence with no model misspecification and Effective Sample Sizes of at least 1000 on all parameters. Results: Model 6 had the smallest Bayesian information criteria (WAIC and LOOIC) among the 8 models tested (Table 3) and was selected as the preferred model. Pre-sophistication and experimental condition had a statistically significant effect on the odds of strategy detection (Table 4). LT and REV students had 2.2 (1.6, 3.0) and 1.5 (1.1, 2.1) times greater odds of exhibiting a detectable strategy, respectively, relative to the BAU peers. Comparing the two one-to-one instructional approaches, students in the LT condition were more likely to use a detectable strategy relative to those in the REV condition (posterior probability = 99.3%). Given a strategy was detected, LT students had 2.4 (1.6, 4.1) times greater odds of using a more sophisticated strategy relative to BAU students (Table 5) and were also more likely to use a more sophisticated strategy relative to REV students (posterior probability = 97.9%). Private school students had 1.7 (1.2, 2.7) times greater odds of using a more sophisticated strategy once a strategy was detected, compared to public school students. Conclusions: Novel methods in the present study enable us to construct multidimensional arguments about the efficacy of early interventions. Results indicated a greater probability that kindergarten students in the LT condition used detectable, mathematically relevant problem-solving strategies relative to their peers in counterfactual conditions. Additionally, there was a 98% probability that LT students used more sophisticated strategies. This underscores the benefits of teaching early length-measurement following a learning trajectory.

This investigation is part of a combined research/curriculum development project in which children's learning is being examined in the context of developing and testing instructional units on 3-D geometry in grades 3, 4, and 5. There are two components to the article. First, the strategies and cognitive constructions that students utilize to conceptualize and enumerate the cubes in 3-D arrays are described. Second, the change in thinking of students as they are involved in instructional tasks that help them develop more sophisticated thinking about enumerating cubes in 3-D arrays are examined. (Author/MKR)