Quantifying elementary teachers’ foreknowledge about their students

Teachers sometimes learn information about their students prior to teaching them, a phenomenon we call “teacher foreknowledge.” This information may be useful, but it may also bias teacher expectations for students in harmful ways. To date, few studies have quantitatively documented the types of foreknowledge teachers receive. When, where, and from whom do teachers commonly acquire foreknowledge? Are some sources or types of foreknowledge perceived to be more influential than others? We explore these questions through a survey of elementary teachers (N = 414) in one U.S. state. Most respondents received foreknowledge about at least some of their students months or even years prior to teaching them, and foreknowledge was primarily transmitted within school settings. Fellow teachers and students themselves were generally characterized as the most influential sources of foreknowledge. Foreknowledge about student behavior and character was considered more influential than foreknowledge about other topics. Some of these patterns covaried with classroom-, school- and teacher-level demographic or contextual variables. We discuss implications for research and practice with a focus on educational equity.


Introduction
Many teachers learn about students before teaching them. We refer to such knowledge as teacher foreknowledge about their students, or simply teacher foreknowledge. Such knowledge can be a powerful asset; indeed, many progressive educators 1 3 (including the authors of this manuscript) argue that understanding student backgrounds, interests, and experiences is an integral part of effective teaching (Dewey, 1938;Gay, 2002). Yet teacher foreknowledge can also have a downside. Under some circumstances, teacher foreknowledge-whether accurate or inaccurate-can bias teachers' beliefs about, and even their behavior toward, certain students (Jussim & Harber, 2005;Rosenthal, 1994). Such biases can have positive impacts for some students, as famously and controversially demonstrated over half a century ago in Rosenthal and Jacobson's 1968 "Pygmalion in the Classroom" study. However, such biases can also have profoundly negative consequences for many students, particularly for racially 1 or economically marginalized youth (for reviews of such work, see Murdock-Perreira & Sedlacek, 2018;Jussim & Harber, 2005).
Hundreds of studies have examined such "teacher expectancies." Building from the work of Rosenthal (1994), we use the term "teacher expectancies" to refer to a causal chain of phenomena in which teacher beliefs about students can affect teacher-student interaction (e.g., Good & Brophy, 1974), which can in turn affect student perceptions of teacher beliefs (e.g., Tal & Babad, 1990) and students' beliefs about themselves (e.g., Kuklinski & Weinstein, 2001), which can in turn affect student behaviors and outcomes (e.g., McKown & Weinstein, 2008), which can in turn affect teachers' beliefs about students (often in self-reinforcing ways). Some studies of teacher expectancies define the term to refer to this entire causal chain, or to specific subsets of links within the chain (for various examples, see Gill & Reynolds, 1999;Rosenthal, 1994). However, any link in this chain could have important implications for racial justice and other dimensions of educational equity. For this reason, we choose to define the term teacher expectancies broadly, to refer to all situations in which teacher beliefs about students influence any link(s) in the causal chain described above.
Teacher expectancy research can have important implications for educational equity in the United States. For instance, even researchers who are cautiously skeptical of the pervasiveness of biased expectancies argue that they may be more common or impactful for some populations-e.g., students with marginalized racial or ethnic identities (Jussim & Harber, 2005). Low expectancies for these students can lead teachers to subtly or overtly ignore or further marginalize these students in their classes, curtailing their learning opportunities and potentially damaging their selfimage (Du Bois, 1935;Rist, 1970). McKown and Weinstein (2008) have demonstrated that racial bias in teacher expectancies for students with comparable prior achievement appears to predict racial achievement gaps within classrooms and students' own perceptions of biased treatment.
It is also possible that racial bias, gender bias, or other types of bias might affect not only teachers' a priori expectancies for a given student, but also the ways 1 3 Quantifying elementary teachers' foreknowledge about their… teachers interpret other types of information they receive about a student. Okonofua and Eberhardt (2015) have found evidence of anti-Black racial bias in the ways that teachers interpret and respond to students' (mis)behavior; when confronted with one instance of misbehavior, teachers responded to Black and White students in similar ways, but when confronted with two instances of misbehavior, teachers began to see Black students' actions (more than White students' actions) as part of a pattern that they believed might necessitate disciplinary action. Thus, racial bias or other types of bias might affect the ways information about a student is processed during the formation of teacher expectancies (see Glock & Krolak-Schwerdt, 2013 for a related example). However, this leaves open the crucial questions of when, where, and how teachers acquire such information in real-world settings.
In spite of half a century of studies on teacher expectancies and their implications for educational equity, a key source of real-world teacher expectancies-teacher foreknowledge-remains surprisingly under-studied. Dusek and Joseph (1983) and Wang et al. (2018) reviewed extensive bodies of research on teacher expectancies and found that such research often focuses on the association of expectancies with specific types of information (e.g. student racial or ethnic identities, designation as learning disabled, teacher perceptions of student engagement, etc.). Yet this body of literature devotes relatively little attention to the details of the processes by which teachers acquire information about students. This matters because not all potentially expectancy-forming information is created equal; for example, the timing at which teachers receive information may be extremely important. José and Cody (1971) found evidence that providing teachers with information about their students partway through the school year has little or no effect on expectancies, apparently because teachers do not believe such information if it contradicts expectancies they have already developed. Raudenbush (1984) famously demonstrated that researcherinduced teacher expectancies are unlikely to have effects unless they are induced within the first few weeks of a given teacher-student relationship. This finding raises a seemingly obvious follow-up question, one that seems to remain under-researched despite decades of invaluable work on teacher expectancies: are expectancies also being induced before (and perhaps long before) a teacher begins interacting with their students? This question might have important implications for educational equity if certain types of positive or negative information-e.g. information about low (or high) achievement or student behavioral challenges-is differentially likely to be shared with teachers at certain times, in certain contexts, or for certain populations of students.
The sources, content, and settings of foreknowledge transmission may be important as well. Shavelson and colleagues (1977) found evidence that teachers found some sources' information more compelling than others, while Timmermans and colleagues (2016) found certain types of content appeared to influence expectancies more than others. In the Netherlands, Timmermans and colleagues (2016) demonstrated that teacher expectancies-operationalized as teachers' recommendations for placing students into more or less academically challenging tracks for secondary school-appear to be largely explained by past evidence of students' academic achievement, with some additional influence from information regarding students' work habits and self-confidence. Information about achievement, work habits, or 1 3 self-confidence was more influential than information about student popularity, behavior, or student-teacher relationships. These authors hypothesized that the outsize role of work ethic and confidence in shaping track recommendations may have been due to teachers' beliefs about the importance of these specific qualities for success in high-track schools (in Dutch secondary schools, the highest track involves 6 years of schooling and directly prepares students for college, whereas many of the lower tracks involve only four years of school and are largely focused on vocational education; Timmermans et al., 2018). However, other ways of operationalizing and measuring teacher expectancies may generate different results; for instance, secondary school course assignment policies in the U.S. differ from those in the Netherlands, and such differences may drive teachers to consider certain types of information more (or less) influential. Understanding these influences seems important in the current U.S. climate of high-stakes testing, where-in our own experiencemany schools gather data on student performance and regularly share such data with teachers in subsequent school years. Yet few studies have sought to comprehensively document such sharing of information in relation to other types of foreknowledge.
Clearly, a wide variety of factors might influence the effects of teacher foreknowledge on teacher expectancies. To effectively study these influences and their implications for educational equity, we first need a thorough understanding of how these factors vary across real-world contexts. For instance, Okonofua and Eberhardt's work shows how foreknowledge about student behavior could result in racially biased expectancy formation-but how frequently do teachers in real-world settings acquire foreknowledge with this content, and under what circumstances? More generally, when, where, and from whom to do teachers typically acquire information about students prior to teaching them? Which sources of foreknowledge or types of foreknowledge content do educators view as most influential? How might these attributes of teacher foreknowledge vary across different school settings and for different populations of students and teachers? These gaps in our collective understanding currently hinder our ability to generalize and respond to the results of research on teacher expectancies in ways that could advance educational equity. The present study begins to address this gap by using survey methods to quantitatively map the characteristics of foreknowledge received by public elementary school teachers in one northeastern U.S. state.

Teacher expectancies
Much of the extant research on teacher expectancies falls into two basic paradigms: experimental and naturalistic (Rubie-Davies, 2014). Both types of research often presume that teachers sometimes have access to foreknowledge: experiments often induce expectancies by providing teachers with foreknowledge, whereas naturalistic studies sometimes observe, infer, or assume that teachers may pass down foreknowledge to their colleagues. Thus, understanding teacher foreknowledge has important 1 3 Quantifying elementary teachers' foreknowledge about their… theoretical and methodological implications for both types of work. To illuminate these connections, we turn now to a brief review of these literatures.

Experimental research on teacher expectancies and inequality
Some research in the experimental paradigm has sought to establish causal relationships between expectancies and teacher behaviors or student achievement. It does so by communicating expectancy-inducing information in some way-for example, by labeling certain randomly selected students as gifted shortly before a laboratorybound tutoring session (e.g., Rubovits & Maehr (1973) in their study of racial disparities in teacher expectancy enactment), or by providing teachers with fake test scores attributed to their students (e.g., Sorotzkin et al., 1974). Such research has indicated that this information is most likely to induce expectancies if it is shared with teachers during or prior to the first 1-2 weeks of an academic year (Raudenbush, 1984), apparently because information that is provided later in the year is often considered implausible if it contradicts teachers' now-established expectancies (José & Cody, 1971).
On the other hand, some experimental research takes the form of interventions that specifically attempt to modify or disrupt teacher expectancy effects to promote equity. Some attempt to modify teachers' behaviors in order to mitigate negative effects of low expectancies that already exist. Such efforts may include helping teachers implement rigorous curriculum for students in spite of negative teacher expectancies; see for example Weinstein and colleagues' (1991) intervention research in a school serving high numbers of racial minority students. Other efforts focus on teaching teachers to use positive behavior management techniques, engage students in goal-setting, provide frequent feedback, and give students substantial autonomy over both the logistics and the focus of classroom activities (see for example Rubie-Davies & colleagues', 2015, intervention study in ethnically diverse schools in New Zealand). Other interventions have sought to directly influence teachers to raise their expectancies for students. For example, Timperley and Phillips (2003) conduct a six-month professional development focused on increasing teachers' expectancies about student capabilities as well as their own ability to positively impact student achievement through specific pedagogical practices. Notably, this professional development explicitly forbade teachers from discussing "child characteristics or their backgrounds" (p. 632) when discussing possible explanations for students' high or low achievement.
These valuable interventions generated important positive effects. At the same time, such benefits were not always maintained long-term (Weinstein et al., 1991), appeared only in some academic disciplines and not others (Rubie-Davies et al., 2015), or were heterogenous across schools (Timperley & Phillips, 2003). A review of intervention research by de Boer et al., (2018) found that expectancy interventions can have positive effects, but was inconclusive about which types of intervention processes were essential for such benefits. Interestingly, several of the reviewed studies suggested that direct interventions on expectations were most likely to succeed when they involved sharing data about a teacher's own students, rather than sharing 1 3 abstract research about student potential. Such data might have changed teachers' expectancies directly; another possibility is that such data simply made teachers more interested and engaged in other aspects of expectancy intervention programs. Understanding how teachers already receive foreknowledge about their students might help to explain some of the variability in outcomes of existing experimental studies and might suggest useful avenues for further intervention research (e.g., mechanisms for promoting or inhibiting certain types of foreknowledge-sharing).

Naturalistic research on teacher expectancies and inequality
A second paradigm of expectancy research, which we term "naturalistic," often uses longitudinal data to examine expectancy effects in real-world classroom contexts. Some of this work is qualitative in nature; for example, Rist (1970) conducted a year-long ethnography examining teacher expectancies in a segregated kindergarten classroom serving Black students, with longitudinal follow-up visits to the same students in later years. Rist argued that differential teacher expectancies for different students within the classroom appeared to be shaped by socioeconomic class stereotypes (e.g., inferences about social class based on students' clothing) and by early interactions with students and their parents. By the end of kindergarten, these initial expectancies had shaped classroom interaction and student outcomes so as to create a 'paper trail' of achievement data that effectively passed the kindergarten teacher's expectancies for students on to subsequent teachers of the same children. In other words, Rist argued that low (or high) expectancies for students were transmitted from one teacher to another and from one year to the next. This raises the question of how often and in what ways expectancy-forming foreknowledge is commonly passed from teacher to teacher.
Naturalistic research on teacher expectancies has also used quantitative methods. A longitudinal study by Sorhagen (2013) found that "teachers' inaccurate expectations [of their First grade students] predicted students' math, reading comprehension, vocabulary knowledge, and verbal reasoning standardized test scores at age 15" (p. 465). Jamil et al. (2018) found that expectancy effects in mathematics appeared to be stronger for upper elementary students compared with younger students. Examining and analyzing teacher expectancies at multiple points from first through eighth grade, the authors found that teachers' expectancies from year to year were not significantly associated with one another, but that the association of these expectancies with future student achievement appeared to grow over time-a pattern that might reflect the existence of expectancy effects with lasting consequences, though not their transmission from one year to the next. 2 Jamil and colleagues (2018) also 1 3 Quantifying elementary teachers' foreknowledge about their… found that expectancy effects in math were stronger for minority girls and boys and for White girls than they were for White boys.
Meanwhile, in a multi-year study of elementary students by Hinnant et al. (2009), teacher expectancies for student reading performance appeared to remain more stable over time than expectancies for mathematics performance, raising key questions. Is foreknowledge about some content more likely to generate expectancies than foreknowledge about other content? Are teachers of some disciplines (or grades) more likely to transmit such foreknowledge to colleagues, or more likely to be influenced by this information once it is received? We are not the only authors calling for attention to these questions; Rubie-Davies and colleagues (2014) argued for the importance of examining the "indirect pathways through which single teachers' expectations affect subsequent student achievement, and subsequent teachers' expectations over time" (p. 83). We agree that it is important to investigate such indirect (or direct) pathways, and the current study seeks to help advance this agenda.

Teacher demographics and contextual variables as potential moderators of expectancies
Many studies and reviews suggest that teachers possess varying susceptibility to expectancy formation (for examples, see Babad et al., 1982;Kuklinski & Weinstein, 2001;Jussim & Harber, 2005;McKown & Weinstein, 2008). However, relatively few studies have sought to identify specific teacher-level correlates of this susceptibility. One notable exception was the work of Tal and Babad (1990), who found that the "teacher's pet phenomenon"-arguably a form of teacher expectancy-was more common among teachers in religious schools than among teachers in secular schools, and seemed to be somewhat correlated with teachers' authoritarian attitudes. Another was the work of Borko and Niles (1982), who found that teachers' level of professional experience appeared correlated with the types of information content they used to formulate expectancies: student teachers based expectancies for academic achievement primarily on information about student achievement, while more experienced teachers additionally 'factored in' information about students' class participation and social competence (but not, interestingly, information about student behavior). Such patterns could have important implications for educational equity-implications that may relate to foreknowledge transmission. Imagine for a moment that Borko and Niles's (1982) findings generalize to all teachers, and imagine that all teachers receive foreknowledge about both student ability and student behavior in a typical year. In the U.S. context, schools with more low-income students and more minority students tend to have less-experienced teachers compared with schools that serve primarily higher-income students and White students (Peske & Haycock, 2006). In this hypothetical scenario, all teachers might receive similar foreknowledge, but foreknowledge about achievement might disproportionately affect expectancy formation in schools serving large numbers of racially marginalized or lowincome students. On the other hand, heterogeneity in the types of foreknowledge content received by teachers or the influence they attribute to this foreknowledge might result in more complex effects. Exploring these and other demographic and contextual correlates of teacher foreknowledge could provide important insights into mechanisms by which inequity is (re)produced and by which researchers, teachers, and policymakers might intervene to promote more equitable educational outcomes.

Prior research on teacher foreknowledge
As mentioned above, we use the term teacher foreknowledge to refer to information that teachers receive about students prior to teaching them. One of the more comprehensive (though dated) works on this topic is Dusek and Joseph's (1983) metaanalysis of the sources of teacher expectancies. This work focused primarily upon associating expectancies with student demographic variables (e.g. student race, gender, social class, etc.). It only addressed the transmission of complex, nuanced information about students through a few mechanisms (e.g. learning about a child by teaching an older sibling, or by receiving a report or dossier of information about the child). Most notably, Dusek and Joseph (1983) analyzed several studies in which teachers were provided with "cumulative folder information" (p. 332) about students; this section of their work is worth quoting at length: "The investigators [in these studies] provided subjects with written descriptions, either positive or negative, of a fictitious student, including information about student behavior, estimates of academic achievement, grades, or IQ, in some studies information about psychological characteristics, in some studies family background information, and in some research a diagnostic label (e.g., learning disabled, EMR). In only two studies…were aspects of these reports separated, namely, information on home/family and academic information. Because it was not possible to determine the specific aspect of the cumulative folder information to which the subjects attended, no attempt was made to form subcategories of the retrieved studies (p. 332)".
Dusek and Joseph (1983) went on to explain that such "cumulative folder information" was among the most compelling drivers of teacher expectancies in their analyses, but that the expectancies generated in the reviewed studies appeared to have important moderators.
This fits with the finding of Raudenbush (1984) that certain attributes of potentially expectancy-forming information might moderate the influence of such information. For example, expectancies may be more likely to form when teachers receive information they find to be plausible, and this appears to be more likely if information is received with a particular timing (Raudenbush, 1984). Another important attribute might be the content of potentially expectancy-forming information; for example, Borko and Niles (1982) provided teachers with information about students' achievement, engagement, and behavior, and then asked teachers to sort students into reading groups and make predictions about their future behaviors and academic performance. Different types of information appeared to influence different dimensions of teachers' expectations and pedagogical decisions, and these relationships sometimes covaried with teacher demographics: as mentioned above, student teachers formed expectancies for student achievement based almost exclusively on information about student achievement, while more experienced teachers formed expectancies for student achievement based both on student achievement and on non-academic factors such as student behavior. These findings suggest that teacher demographics or contextual factors, such as years of teaching experience, may also play a role in the impact of expectancy-forming information.
The sources of potentially expectancy-forming information may also be important. Shavelson and colleagues (1977) found that information about family background and study habits appeared to form more extreme expectancies when attributed to parent interview, deemed a "reliable" (p. 84) source, as opposed to a peer interview, deemed an "unreliable" (p.84) source. Other sources may matter as well; for instance, how do guidance counselors shape teacher expectancies? A study of guidance counselors by Auwarter and Aruguete (2008) found evidence that some counselors form impressions of students that are biased against girls and low-SES students. If we could understand what sources of information are seen by teachers as particularly influential, this could help inform researchers' and policymakers' decisions about the most productive and equitable ways to structure the sharing of information among colleagues within a school.
We also find ourselves wondering about the settings in which information is acquired, since this may co-vary with the sources and timing of such information and may have important implications for research. For instance, information that teachers encounter during the course of their regular workday-e.g. "water cooler conversations" at work, or formalized meetings in which principals or guidance counselors share information-may be easier to study and intervene upon compared with foreknowledge received outside of a school setting. On the other hand, information received through technological means-e.g., phone calls, emails, or social media-may be even easier to study and intervene upon. Understanding the quantity of foreknowledge transmitted in each of these settings could help researchers identify fruitful settings in which to study foreknowledge transmission that is already taking place, curtail the transmission of negatively framed foreknowledge, or perhaps promote the transmission of positively framed foreknowledge.
To what extent are expectancies shaped by the source and content of foreknowledge teachers receive, and what are the common timings and settings in which such foreknowledge is received? Answering these questions could help school leaders identify specific types of information to share-and specific types of information to deliberately avoid sharing-in the interests of promoting positive expectancies and avoiding negative ones. Recent studies have examined teacher cognition during expectancy formation (see for example Glock et al., 2015); however, we have not yet found a previous study which quantifies the influence attributed by many teachers to particular sources of foreknowledge and particular foreknowledge content, nor one which quantifies other key dimensions of foreknowledge such as the timings and settings in which foreknowledge is received by teachers. In the present study, we therefore sought to quantify the ways in which a large population of teachers receive potentially expectancy-inducing information, asking: • What sources of foreknowledge do teachers perceive to be most influential? How might this vary according to demographic or contextual factors? • What types of foreknowledge content do teachers perceive to be most influential?
How might this vary according to demographic or contextual factors? • When do teachers typically receive foreknowledge about their students? How might this vary according to demographic or contextual factors? • In what settings do teachers typically receive foreknowledge about their students? How might this vary according to demographic or contextual factors?

Method
Based on themes and details in the literature described above, we designed an online Qualtrics survey asking teachers to describe attributes of the foreknowledge they received about their current students, with a focus on the source, content, timing, and setting of such foreknowledge. The survey was developed by the authors in 2015 and piloted with a group of former K-12 educators (N = 26) currently attending graduate programs in education at a large private university in the Western United States. Feedback was solicited from pilot participants, and several changes were made based on this feedback; changes included modifications to the range of timings at which foreknowledge might be received by teachers and the addition of new possible sources of foreknowledge (e.g. interactions with students themselves). Next, the survey was piloted with a group of current K-12 educators (N = 16) in a rural school district in the Southeastern United States; analysis of a low response rate and participant attrition led us to shorten the survey substantially. As our study was exploratory in nature, we did not conduct a validation of the survey instrument. The final version of the survey began with two open-ended questions that asked participants to share examples of foreknowledge with a positive and negative valence (one example of each). Each of these open-ended questions was followed by a set of seven follow-up questions asking for more detail about each example; analyses of these data will be presented in another manuscript currently in preparation. This method, though text-based rather than verbal, is similar to the procedure utilized by Jager and colleagues (2021) to briefly elicit teachers' ideas about their students.
Next, participants answered four 'big-picture' questions about teacher foreknowledge as a broad phenomenon. These included one question each about possible sources, content, timing, or settings of teacher foreknowledge; four additional open-ended questions about these attributes of foreknowledge; and a series of demographic and contextual questions about survey respondents and the classrooms, schools, and communities in which they teach. The four 'big-picture' questions and their relationship to demographic and contextual variables are the focus of our analyses in this manuscript.
In the first of these four big-picture questions, teachers were shown a list of seven potential sources of foreknowledge (e.g., Parents or Guardians, Guidance Counselors, Other Teachers) and asked to rank these sources from most influential to least influential (1 = Most influential, 2 = Second most influential, etc.), leaving individual sources out of their rankings if they never acquired foreknowledge from this source. For example, if a respondent ranked parents or guardians as the most influential source of foreknowledge, followed by administrators and then by guidance counselors, and did not provide ratings for other sources, then parents would receive a rank of 1 in our dataset, administrators would receive a rank of 2, guidance counselors a rank of 3, and all other sources would be listed as missing a ranking. The list of potential sources of foreknowledge was initially developed by the authors and then expanded and revised based on feedback from former teachers during the first round of piloting.
Next, teachers were shown a list of five different types of foreknowledge content (Student Academic Ability, Student Behavior, Student Character, Student Interests, and Other) and asked to rank these sources from most influential to least influential in the same manner (1 = Most influential, 2 = Second most influential, etc.), leaving specific types of content out of their rankings if they never acquired foreknowledge with this content.
These types of information were derived from previous teacher expectancy literature and related research, which often manipulate information about or measure perceptions of student academic ability or behavior (see for example José & Cody, 1971;Okonofua & Eberhardt, 2015;Sorhagen, 2013). We recognize that the very concept of student academic ability can itself be problematic, in part because perceptions of ability-even inaccurate perceptions, or accurate perceptions of situationally irrelevant abilities-can contribute to inequitable interactions and inequitable learning outcomes in classrooms (Cohen & Lotan, 1995, 1997. Academic ability may also be difficult or impossible to directly measure; indeed, studies of teacher expectancies typically operationalize ability using student achievement data. Thus, the categories of "information about student academic ability" and "information about student achievement" probably overlap substantially in everyday discourse. "Student academic ability" may be more all-encompassing, however, since common terms such as "smart" or "gifted" seem to explicitly refer to ability but may be less explicitly connected to achievement. We were interested in all types of teacher foreknowledge, even those we consider potentially problematic; thus, we used the term "student academic ability" and not "student achievement" as a category of foreknowledge content in our survey instrument. Many such studies also incorporate other important types of information about students, attributing qualities such as "motivated" or "industrious" to particular children (e.g. Glock & Krolak-Schwerdt, 2013, p. 116, using materials developed in earlier studies). Since these qualities are sometimes framed as attributes of students rather than as specific observable behaviors, we chose to subsume such attributes into a general category of information about "student character." Participants were also prompted to consider foreknowledge they receive related to student interests; this construct is important from a theoretical standpoint, since proactively seeking out information about student interests is key to many student-centered pedagogical practices (see for example Walkington, 2013).
Next, teachers were shown a list of different timings at which they might have received foreknowledge about their current students (e.g., during the previous school year, during the previous summer, etc.) and asked to describe how much information they received at each timing using a five-point Likert-type scale (1 = None, 2 = A little, 3 = Some, 4 = A lot, 5 = Nearly all). Since we suspected school-related information might be likely to be shared in school-related interactions, we defined timings in reference to "bounding cues" (Glasner & van der Vaart, 2009) when teachers' school-related interaction patterns change, such as the beginning and end of the school year, the beginning and end of the summer holidays, and the end-of-summer professional development days common at many schools.
Teachers were also shown a list of different settings in which they might have received foreknowledge (Inside school and during the school day, At school-related events, Outside school, or Via phone, text, email, or social media) and were asked how much foreknowledge they had received in each setting, using the same Likerttype scale as was used for the question about foreknowledge timing.
Finally, teachers were asked to provide a variety of contextual and demographic details about themselves and the classrooms, schools, and communities in which they teach. These included information about respondents themselves, such as their age, years of [teaching] experience, racial identities, gender identities, and religious affiliations. (We asked about religious affiliations because we wondered whether teachers with a religious affiliation might be more likely than unaffiliated teachers to interact with students or their families in certain outside-of-school contexts such as religious services). Teachers were also asked to share information about their classrooms and schools, including whether they taught in an elementary multiplesubjects classroom or a single-subject classroom, the number of days the respondent interacted with any student(s) outside-of-school during the week immediately prior to the survey, and estimates of their own average class size, the total number of students taught in academic year 2018-19, and the school size in academic year 2018-19. Teachers were also asked whether their school served students in a rural, urban, or suburban community.

Participants and data collection
Participants were recruited via email using a publicly available database of contact information maintained by state and local educational authorities in a single northeastern U.S. state. 9171 educators, representing a substantial fraction of all public school elementary teachers currently employed in the state, were contacted via email in mid-December 2018, at approximately the start of the winter holidays in most U.S. schools. We administered the survey at this time because we wanted to ensure (1) that the school year had started in all schools across the state, so that teachers would be able to reflect on foreknowledge received up until the first day of school, and (2) that the survey was distributed at the beginning of a lengthy holiday, so that teachers would have sufficient time to complete it. Responses were accepted until a designated cut-off date and time in early January 2019.
A total of 702 individuals opened the survey prior to the cut-off date and time. Participants gave their informed consent to have their data included in the study, using a procedure approved by the Stanford University Institutional Review Board in research protocol #35041. We removed participants from the sample who did not consent to have their responses utilised in research (N = 16) or who did not proceed with the survey after providing their consent (N = 186). We also removed participants from the sample who did not complete enough of the survey to answer demographic questions about their school context, years of experience, age, etc. (N = 86). This left us with a sample of 414 complete survey responses for analysis, a response rate of approximately 4.5% from our original email. Descriptive statistics for the sample are shown in Table 1.
Notably, our research was conducted in a state where a large majority of schools are in rural communities, and this was reflected in our sample: roughly 66% of teachers reported teaching in a rural community, while about 22% reported teaching in a suburban community and 11% in an urban community.

Analysis
Our data on the influence attributed to various foreknowledge sources and foreknowledge content consisted of interrelated ranked-choice data, but used an incomplete block design (since participants did not need to rank all sources or all types of content). For this reason, we used a Skillings-Mack test (Skillings & Mack, 1981) to test whether certain foreknowledge sources or content were ranked closer to 1 (that is, were significantly more influential than others) in teachers' responses, and conducted pairwise comparisons among all foreknowledge sources and among all types of foreknowledge content using Wilcoxon signed ranks tests, calculating effect size as Z/√N (Fritz et al., 2012).
Given the simpler, Likert-scale structure of the timing and setting data, we used one-way, repeated-measures ANOVAs to determine if different amounts of foreknowledge were received at different timings or settings. Follow-up pairwise comparisons were used to test specific differences across timings and settings.
To determine how the influence (of foreknowledge with a particular source or content) and quantity (of foreknowledge received at a particular timing or setting) might covary with teacher-, classroom-, or school-level demographic or contextual variables, we calculated Pearson's correlation coefficients between these variables. Demographic and contextual variables included: teacher age, teacher years of experience, average class size, total number of students taught in academic year 2018-2019, total school size in academic year 2018-19, teacher gender identity, 3 teacher racial/ethnic identity (classified as People of Color or non-Hispanic White), 4 teacher religious affiliation (classified as either having or lacking an affiliation), and the number of of days the respondent interacted with any student(s) outside of school during the week immediately prior to the survey.
We did not conduct an a priori power analysis since we did not have pre-existing estimates of population variance in these variables. We considered conducting a post hoc power analysis, but ultimately decided not to do so since such analyses have been criticized on methodological grounds (although most such critiques apply to the specific case of post hoc analyses conducted on negative or non-significant results; Levine & Ensom, 2001).
We used a Bonferroni correction to correct for the risk of Type I errors in our study, given the large number of tests for statistical significance that we conducted. One of the weaknesses of the Bonferroni test is an increased risk of Type II errors, particularly in a study with an extremely large number of tests (Perneger, 1998); nevertheless, we conservatively elected to use the Bonferroni method in this work. We computed 348 distinct p values in our analyses and therefore corrected our threshold for statistical significance from an alpha of .05 to .05/348 = .00014. In tables, an alpha of .01 has been likewise corrected to 0.01/348 = .0000287 and .001 has been corrected to .001/348 = .00000287.

Results
Descriptive statistics for foreknowledge variables are shown in Tables 2 and 3.  Table 2 summarizes how many teachers identified particular sources of foreknowledge (e.g. administrators, guidance counselors, etc.) or types of foreknowledge content (e.g. student academic ability, student behavior, etc.), as well as the mean rankings teachers gave to characterize these foreknowledge sources or types of content as more or less influential. Larger mean rankings (further from 1) indicate foreknowledge sources or types of content were considered relatively less influential by respondents who received any foreknowledge fitting these criteria, and lower mean rankings (closer to 1) indicate that a source was ranked as more influential by the participants who received any foreknowledge from that source. Most of these sources of foreknowledge and types of content were extremely common, reported by between 80 and 98% of respondents. Table 3 summarizes the quantity of foreknowledge that teacher reported receiving at particular timings (e.g. During the previous school year, During the previous summer, etc.) and in particular settings (e.g. Inside school and during the school day, At school-related events, etc.).

What sources of foreknowledge do teachers perceive to be most influential?
As shown in Fig. 1, we found that teachers tended to rank foreknowledge from Other Teachers as more influential than foreknowledge from other sources. Students Themselves were the second most influential source of foreknowledge, followed successively by Guidance Counselors, Parents or Guardians, Administrators, and Other Adults or Other Sources.
The Skillings-Mack test showed a statistically significant difference among the influence ascribed to different sources of foreknowledge, Q = 378.259, p < .001 with a Bonferroni correction. Post hoc comparisons using the Wilcoxon signed-ranks  Table 4, indicated that foreknowledge received from Other Teachers was ranked closer to 1 (and thus considered more influential) than foreknowledge received from Students Themselves, Guidance Counselors, Parents or Guardians, Administrators, and Other Adults, but was not ranked significantly differently from Other Sources of foreknowledge. Foreknowledge received from Students Themselves was ranked closer to 1 (and thus considered more influential) than foreknowledge received from Guidance Counselors, Parents or Guardians, Administrators, and Other Adults, but was not ranked significantly differently from Other Sources. Foreknowledge received from Guidance Counselors was ranked closer to 1 (and thus considered more influential) than foreknowledge received from Administrators or Other Adults, but was not ranked significantly differently from foreknowledge received from Parents or Guardians or that received from Other Sources. Foreknowledge received from Parents or Guardians was not ranked significantly differently from foreknowledge received from Administrators or Other Sources of foreknowledge, but was ranked closer to 1 than foreknowledge received from Other Adults. Foreknowledge received from Administrators was not ranked significantly differently from Other Sources of foreknowledge, but was ranked closer to 1 (and thus considered more influential) than foreknowledge received from Other Adults. Finally, foreknowledge received from Other Sources of foreknowledge was not ranked significantly differently from foreknowledge received from Other Adults. All p-values were computed using twotailed asymptotic tests of significance (computer memory was insufficient to run exact tests) with a Bonferroni correction applied. When we measured correlations between the influence attributed to different foreknowledge sources and our set of ten demographic and contextual variables, we found no correlations that remained significant after a Bonferroni correction was applied.

What types of foreknowledge content do teachers perceive to be most influential?
We found that teachers tended to rank foreknowledge content about Student Behavior as more influential than other content, with foreknowledge about Student Character a close second followed by information about Student Academic Ability and Student Interests (see Fig. 2). The Skillings-Mack test showed a statistically significant difference among the influence ascribed to different types of foreknowledge content, Q = 209.014, p < .001 with a Bonferroni correction. As seen in Table 5, post hoc comparisons using the Wilcoxon signed-ranks test indicated that foreknowledge about Student Behavior was not ranked significantly differently from foreknowledge about Student Character or Other Types of Information, but was ranked closer to 1 (and thus considered more influential) than foreknowledge about Student Academic Ability or Student Interests.
Foreknowledge about Student Character was also ranked closer to 1 (and thus more influential) than foreknowledge about Student Academic Ability or Student Interests. Foreknowledge about Student Academic Ability was not ranked significantly differently from foreknowledge about Student Interests. All p-values for these comparisons were computed using two-tailed asymptotic tests of significance (computer memory was insufficient to run exact tests) with a Bonferroni correction applied.
When we measured correlations between the influence attributed to different types of foreknowledge content and our set of ten demographic and contextual Fig. 1 Relative influence attributed to various sources of foreknowledge variables, we found no correlations that remained significant after a Bonferroni correction was applied.

When do teachers typically receive foreknowledge about their students?
As shown in Fig. 3, virtually all teachers reported receiving foreknowledge at numerous timings prior to teaching their students. We used a one-way, repeatedmeasures ANOVA and follow-up pairwise comparisons to determine if more foreknowledge was received at certain timings compared with others. Mauchly's test indicated a moderate violation of the sphericity assumption, χ 2 (9) = 160.30, p < .001, ε = 0.837, so Huynh-Feldt corrected results are reported. The amount of foreknowledge reported varied by timing at which this foreknowledge was received, F(3.38, 1395.80) = 124.45, p < .001 after a Bonferroni correction, η 2 = 0.232.
Post hoc analysis revealed that the amount of foreknowledge received was significantly higher On the  Table 6. All p-values for these comparisons were computed using two-tailed tests with a Bonferroni correction applied.
When we measured correlations between the amount of foreknowledge received at specific timings and our set of ten demographic and contextual variables, we found several correlations that remained significant after a Bonferroni correction was applied. The more years of experience a teacher reported, the more foreknowledge they reported Table 4 Effect sizes of Wilcoxon signed-rank tests on rankings for influence of foreknowledge sources Rows are compared with columns, and both sides are filled in for readers' convenience. For example, reading across row 2, foreknowledge received from guidance counselors (#2) was ranked closer to 1, and thus more influential, than foreknowledge received from administrators (#1), parents or guardians (#4), other adults (#5), or other sources (#7), but was ranked further from 1, and thus less influential, than foreknowledge received from other teachers (#3) or students themselves (#6). *p < .05, **p < .001, ***p < .0001 after applying a Bonferroni adjustment    Rows are compared with columns, and both sides are filled in for readers' convenience. For example, reading across row 1, foreknowledge about student academic ability (#1) was ranked, and thus less influential, than foreknowledge about student behavior (#2) or student character (#3), but was ranked closer to 1, and thus more influential, than foreknowledge about student interests (#4) or other types of information (#5). *p < .05, **p < .001, ***p < .0001 after applying a Bonferroni adjustment

In what settings do teachers typically receive foreknowledge about their students?
As shown in Fig. 4, teachers reported receiving far more foreknowledge inside school and during the school day than in other settings. We used a one-way, repeated-measures ANOVA and follow-up pairwise comparisons to determine if more foreknowledge was received in certain settings compared with others. Mauchly's test indicated a moderate violation of the sphericity assumption, χ 2 (5) = 92.11, p < .001, ε = 0.889, so Huynh-Feldt corrected results are reported. The amount of foreknowledge reported varied by setting in which it was acquired, F(2.69, 1109.58) = 546.57, p < .001, η 2 = 0.570. Post hoc analysis revealed that the amount of foreknowledge received inside school and during the school day (M = 4.05, SD = 1.06) was significantly higher than the amount received at school-related events (M = 2.49, SD = 1.08), the amount received outside school (M = 1.76, SD = 0.86), and the amount received via phone calls, texting, or social media (M = 1.83, SD = 0.96). All pairwise comparisons and their statistical significance are indicated in Table 7. All p-values for these comparisons were computed using two-tailed tests with a Bonferroni correction applied.
When we measured correlations between the amount of foreknowledge received in particular settings and our set of ten demographic and contextual variables, we found only one correlation that remained significant after a Bonferroni correction was applied. The amount of foreknowledge that a teacher reported receiving outside school was positively correlated with the number of days the respondent interacted with any student(s) outside of school during the week immediately prior to

What sources of foreknowledge do teachers perceive to be most influential?
We found what appeared to be a clear hierarchy of foreknowledge sources. Teachers were the most common source, and were widely perceived to be the most influential. The next most common and most influential source was interactions with students themselves, followed by guidance counselors. Information from parents and guardians or school administrators came next with rankings that did not differ significantly from each other, and other sources of information tended to be rarer and less influential.
Nearly half a century ago, Shavelson et al. (1977) found that information attributed to parents appeared more influential than information attributed to a student's peers. Our findings do not directly contradict this, as we considered foreknowledge from students themselves rather than foreknowledge from a student's peers. However, our findings do add nuance to Shavelson et al.'s findings and suggest that some youth may, under certain circumstances, be seen as more influential sources of foreknowledge than parents. Our findings also suggest the need to further explore role of guidance counselors as sources of foreknowledge.
While unsurprising, these findings still have important methodological implications. They show that receiving foreknowledge from other teachers is nearly ubiquitous. Previous studies of teacher expectancies have rarely quantified the passage of information between teachers, although recent work has begun to shed light on this process (see for example Friberg, 2021). Our results suggest that further research Table 6 Effect sizes (Hedge's g) of pairwise comparisons on amount of foreknowledge received at various timings Rows are compared with columns, and both sides are filled in for readers' convenience. For example, reading across row 2, more foreknowledge was received during the previous school year (#2) than was received more than one year ago (#1), during the previous summer (#3), or during end-of-summer professional development (#4), but less foreknowledge was received during the previous school year than on the first day of school (#5). *p < .05, **p < .001, ***p < .0001 after applying a Bonferroni adjustment should explore when, why, and how teachers share foreknowledge, and how this process might shape both positive and negative expectancies. Future naturalistic studies that measure expectancies over multiple years should strongly consider measuring foreknowledge passed between teachers, which might help to explain variation in the persistence or non-persistence of expectancy effects. Our findings also suggest that researchers may benefit by measuring foreknowledge derived from sources other than teachers (e.g. guidance counselors). These findings also have implications for experimental research on teacher expectancies. Research in this paradigm has often manipulated expectancies by presenting information from purported sources other than teachers (e.g., test administrators in Rosenthal & Jacobsen, 1968), and has suggested that expectancies are only  Table 7 Effect sizes (Hedge's g) of pairwise comparisons on amount of foreknowledge received in various settings Rows are compared with columns, and both sides are filled in for readers' convenience. For example, reading across row 2, less foreknowledge is received at school-related events (#2) compared with inside school and during the school day (#1), but more is received at school-related events compared with outside school (#3) or via phone, email, or social media (#4). *p < .05, **p < .001, ***p < .0001 after applying a Bonferroni adjustment generated when such information is likely to be perceived as credible by teachers (José & Cody, 1971;Raudenbush, 1984). Our findings confirm that teachers widely consider each other to be the most influential sources of foreknowledge, implying that past studies using other sources might have under-estimated the effects of teacher expectancies in real-world contexts. We suggest that future experimentse.g., interventions that seek to generate positive teacher expectancies or disrupt negative ones-might focus on teachers as the most compelling bearers of such messages.
We also find that interactions with students themselves are perceived to be a relatively common and influential source of foreknowledge. While we suspect many such interactions arise in unplanned and unstructured ways, we wonder whether carefully structured opportunities for positive interactions between teachers and their future students before the start of the school year might prove beneficial in cultivating positive expectancies. Some schools already structure such interactions (e.g., structuring an opportunity for students near the end of eighth grade to visit a local high school and meet teachers of ninth grade classes); our findings suggest that the characteristics and outcomes of such structured interactions should be investigated systematically.

What types of foreknowledge content do teachers perceive to be most influential?
We also found what appeared to be a clear hierarchy of foreknowledge content. Foreknowledge about student behavior and student character were most common and were widely perceived to be the most influential. The next most common and influential types of foreknowledge related to student academic ability and student interests, finally followed by foreknowledge with other, miscellaneous content. These findings could be interpreted as in tension with the prior findings of Timmermans and colleagues (2016), which suggested that teachers tended to base expectancies more on information about student ability, work ethic, and self-confidence rather than on classroom behavior. In our survey, student behavior and student character (the latter including attributes such as work ethic) were both considered equally influential, and were considered more influential than information about student ability. The difference in our findings may have arisen in part from the different methodologies used in our respective studies: Timmermans et al. measured teacher expectancies at the end of an academic year, when a wealth of information was available to teachers, while we measured potential expectancy-forming information in the middle of an academic year. Furthermore, Timmermans et al. operationalized these expectancies using a concrete measure (tracking recommendations) that would dramatically affect students' futures, whereas we did not operationalize or measure expectancies directly. Perhaps if we had asked our participants to think specifically about the types of foreknowledge that might influence tracking decisions, they may have ranked information about student ability to be more influential.

3
It is also possible that information about student behavior and character was not actually most influential, but merely reported to be most influential by teachers. For instance, teachers might believe it is more socially acceptable to say that they are influenced by information about student behavior or character compared with information about student ability, since decades of teacher education publications have encouraged the mentality that all children can learn (see for example Slavin & Madden, 2001)-a mentality with which we agree. However, this would not explain why information about student interests-probably seen as neutral or even desirable information to act upon-was reported to be slightly (though non-significantly) less influential than information about student ability.
Another explanation may be that many teachers may see themselves as responsible for more than just student learning; they may also see themselves as responsible for influencing students' character development (see for example Narvaez & Lapsley, 2008). These beliefs may be shaped by numerous factors both inside and outside the profession of teaching, and could explain why many teachers in our study reported information about student behavior and character to be more influential than information about student ability. Indeed, there are many possible explanations for these findings. One reviewer of this manuscript insightfully pointed out that negative-valence information tends to be more impactful than positive-valence information in many social psychological phenomena (see for example Baumeister et al., 2001). This presents another important potential explanation of our results, since our qualitative analyses of respondents' anecdotes (which we will report in a separate manuscript currently in preparation) strongly suggest that foreknowledge about student behavior tends to be overwhelmingly negative in nature. It is also possible that teachers perceive information about behavior and character to be most influential based on their own self-perceptions (Bem, 1972), which may or may not be accurate. Another possibility is that, regardless of whether information about student behavior and character is truly more influential than information about student ability, the domains of student behavior and character are more salient to teachers when considering foreknowledge. Since we did not directly measure teachers' actual expectancies, we cannot know for certain; perhaps information about student ability has the greatest impact on teacher expectancies related to academic achievement, while information about student behavior and character has the greatest impact on teacher expectancies related to students' daily classroom functioning, and the latter domain may simply have been seen by teachers as more consequential in the context of our survey.
Under nearly all of the interpretations mentioned above, our findings have important implications for researchers and practitioners-particularly those concerned with educational equity. As mentioned above, foreknowledge about student behavior disproportionately tended to be information about perceived misbehavior, and thus may have the potential to generate harmful negative expectancies. Experimental research by Okonofua and Eberhardt (2015) suggests that such negative expectancies might be more likely to form when the student(s) in question are African American, or perhaps have other marginalized racial or ethnic identities. Further research should explore the transmission of negative foreknowledge about student behavior, potential biases that might be (re)produced through this transmission, and potential strategies for disrupting such processes. 5 In the meantime, we argue that teachers and school administrators should seek to cultivate professional norms that either curb the sharing of negative behavior-related information among teachers or diminish the risk that such information will lead to negative expectancy formation. For example, educators could actively work (as many already do) to cultivate school communities in which educators largely refrain from sharing stories of student misbehavior. If and when sharing such information is absolutely necessary, educators could develop procedures to share such information using anonymized narratives that focus on teacher problem-solving and classroom management strategies rather than 'warning' colleagues about individual students. Such procedures would be reminiscent of those employed Timperley and Phillips (2003), whose expectancy intervention program explicitly forbade teachers from discussing students' backgrounds as 'explanations' for low achievement.
On the other hand, researchers and practitioners may wish to promote the sharing of foreknowledge about student interests. Our findings show that a large majority of teachers already receive such foreknowledge, but most appear to consider it less influential than other types of foreknowledge. Foreknowledge about student interests might be valuable for informing teachers' implementation of culturally responsive, culturally relevant, and culturally sustaining pedagogies (Ladson-Billings, 1995Morrison et al., 2008). 6 Thus, equity-minded researchers and practitioners may wish to design and test policies, procedures, or practices that increase the influence teachers attribute to such knowledge about student interests. Perhaps paid professional development hours could be set aside at the end of each school year for teachers to record information about individual students' interests, and additional hours could be set aside at the beginning of the following year for teachers to read this information and incorporate it into lesson planning and curriculum development. Such practices could form a valuable focus for future research.

When do teachers typically receive foreknowledge about their students?
We found that teachers reported receiving the greatest amount of foreknowledge on the first day of school, and reported receiving a smaller but still-substantial amount during the previous school year. Teachers reported receiving less during the summer prior to the current school year and during end-of-summer professional development, and reported receiving the least information more than one year in the past.
This finding has important implications for expectancy research. It is long-established that exposing teachers to potentially expectancy-forming information can have stronger effects if such information is presented within the initial weeks of a school year, rather than later in the year (Raudenbush, 1984). However, we know less about the extent to which potentially expectancy-forming information persists if it is presented days, weeks, months, or even years in advance. Future research should explore whether foreknowledge received shortly before the start of the school year-for example, during end-of-summer professional development days-can successfully form expectancies, either positive or negative in nature. If so, end-of-summer professional development might provide a useful and relatively controlled environment in which to disrupt certain kinds of foreknowledge transmission (e.g., negative behavioral information) and promote other kinds of foreknowledge transmission (e.g., information about student interests). Our findings about teacher experience further suggest that any efforts to manipulate foreknowledge transmission at the start of a school year (e.g., to inhibit the formation of negative expectancies or promote the formation of positive ones) might be most impactful if they focus on relatively inexperienced teachers, who tend to receive more potentially expectancy-forming foreknowledge during this time (compared to their more experienced colleagues).
Our finding that elementary multiple-subjects teachers were more likely than their single-subject colleagues to report receiving foreknowledge during the summer also has implications for research. For example, Timperley and Phillips (2003) provide a compelling demonstration of a six-month-long intervention to modify teachers' expectations for students. Providers of similar interventions may wonder about the most appropriate time to begin a long-term intervention of this type-after the school year has begun? At the beginning of the school year? Prior to the school year? Our findings indicated that multiple-subjects teachers (who often teach very young students) tended to receive more potentially expectancy-forming information during the summer than did their single-subject colleagues (who often teach upper elementary grades). Thus, interventions which take place during the summer may be more consequential for teachers of very young children compared with teachers of upper elementary or middle school students.

In what settings do teachers typically receive foreknowledge about their students?
We found that teachers reported receiving the greatest amount of foreknowledge in school during regular school-day activities. The next most common setting in which to receive foreknowledge was at school-related events such as extracurricular gatherings. Foreknowledge received in other settings-outside of school, for example, or over phone, text, email, or social media-was much rarer. These findings are encouraging for educators and policymakers interested in intervening upon teacher expectancy transmission, since they suggest that most sharing of teacher foreknowledge is happening within the walls of schools and during school hours-a context where educators and policymakers likely have the greatest power to intervene on such transmission processes, both to promote certain types of foreknowledge-sharing and to inhibit others. Notably, we found that the amount of foreknowledge teachers receive about their students via phone, text, email, or social media was comparable to the amount received in out-of-school settings writ large. This raises the intriguing question of which media-phone calls, text messages, etc.-are the primary source(s) of the foreknowledge reported by our participants. While our parent communication experiences as teachers primarily involved phone calls, we are aware that text messaging, educational apps, and other technological tools are increasingly being used to facilitate communication between teachers and parents. Such communication is often conceptualized as one-way sharing of information from teachers to parents; see for example the experiments conducted by Hurwitz et al. (2015) and Bergman and Chan (2021). Our findings suggest that information can flow in the opposite direction as well-from parents to teachers-with both positive and negative possible consequences. Future research could use text messaging systems or apps to facilitate particular types of positively framed foreknowledge transmission. However, researchers and educational technology developers should be wary of the potential unintended consequences of such efforts; Doss et al. (2019) found evidence that text messaging to young children's parents can actually decrease parental interaction with teachers, perhaps because parents who passively receive information related to their child's education feel less urgency in proactively communicating with teachers. Any attempt to intervene on technology-based routes of foreknowledge transmission should be undertaken with caution.

Limitations
These findings come with several key limitations. First, the data are self-reported, and may therefore be subject to substantial social desirability biases. Babad and colleagues (1991) argue, based on their own findings (e.g. Tal & Babad, 1990), that many teachers are consciously or unconsciously motivated to conceal negative affect towards particular students. As mentioned above, the fact that teachers described certain pieces of information as more influential than others may represent not an actual difference in influence, but a difference in the perceived social acceptability of admitting to influence by particular sources or types of information. On the other hand, self-reported data might be the best data that are logistically feasible to collect in some scenarios, since directly documenting all teacher foreknowledge and assessing its influence might require extensive, impractical, and overly invasive data collection.
A second limitation had to do with the way we operationalized foreknowledge content in our survey items-specifically, the way we operationalized "student character". In our study, we found that respondents who shared foreknowledge examples they identified as related to "student character" often wrote about students using adjectives such as "sweet," "kind," "positive," and "resilien [t]." However, many also included descriptions of observable student behavior, such as helping peers or family members (examples of positive foreknowledge) or stealing (an example of negative foreknowledge). Thus, the construct of "student character" appears to have been partially conflated with "student behavior" by many participants-an important limitation of the study.
A third limitation had to do with the way we asked participants about the timing of foreknowledge. Specifically, we asked teachers to quantify the amount of foreknowledge they receive in windows of time of unequal length. A pre-COVID American school year typically involved about 180 days of interactions with students (Patall et al., 2010), but "end-of-summer professional development" may last only one or two days in some school districts, or many days in others. This is an important limitation of the study, as it does not allow us to determine the rate at which foreknowledge is being transmitted during each of these periods, or whether there are key moments or events within each time period when the sharing of foreknowledge is especially common or uncommon. Future studies could address this limitation using alternative methods such as event calendars (Belli, 1998).
A fourth limitation arose from our attempt to differentiate between schoolrelated settings and outside-of-school settings. For example, if a teacher attends a school sporting event where they interact with a (future) student's parents, and are then invited by these parents to accompany them to a subsequent religious gathering, should foreknowledge from these interactions be attributed to a "school-related setting," a "non-school-related setting," or both? Outside of the normal school day, distinctions between school-related and non-school-related settings are notoriously difficult to determine; indeed, such distinctions have been the subject of numerous legal cases, up to and including Supreme Court rulings (see for example Pickering v. Board of Education, (1968), or the more recent and highly controversial Kennedy v. Bremerton School District, (2022)). On the other hand, we believe more clearcut distinctions can be drawn between foreknowledge acquired "in school during the school day" and elsewhere, and between that acquired "via phone, text, email, or social media" and elsewhere. We argue that these distinctions are also the most methodologically significant, as both the normal school workday and school-mediated electronic communications could provide useful sites for intervention research that leverages the transmission of teacher foreknowledge.
Another limitation is that while these data document the influence attributed to particular foreknowledge content (e.g., information about student behavior) and sources (e.g., teachers or parents), they do not document the frequency with which teachers receive these types or sources of foreknowledge. Thus, while the data suggest certain sources and content were particularly influential, they cannot reveal whether these same sources and content were relatively commonplace or relatively rare. By the same token, our data on the timing and setting of teacher foreknowledge tell us about the frequency with which teachers receive foreknowledge in these times or places, but it does not tell us about teacher's perceptions of the relative influence of timing or setting.
Another key limitation of our data is our sample population. Our study does not address teacher foreknowledge in schools serving large proportions of Students of Color. Participants were teachers from a single US state in which approximately 90% of students identify as non-Hispanic Whites, and survey respondents were disproportionately located in rural communities. These demographics are unusual and not representative of K-12 education across the United States as a whole, where less than 48% of K-12 students identify as non-Hispanic Whites and where generally comparable numbers of schools are located in urban, suburban, and rural communities (NCES, 2020a(NCES, , 2020b. Future research should investigate whether teacher foreknowledge (and any expectancies generated by such foreknowledge) are substantially different in schools that serve student populations more representative of the U.S. as a whole. These should include schools serving large populations of Students of Color, as well as schools that are more evenly distributed among urban, suburban, and rural communities.
Finally, we note that by asking about each of these dimensions of foreknowledge separately, we were unable to assess how answers may have covaried-how, for example, particular sources of foreknowledge may have been strongly associated with particular settings or particular types of foreknowledge content. In another forthcoming manuscript we will seek to partially overcome such limitations by using a different dataset collected from the same participants (the foreknowledge anecdotes referenced above).

Conclusion
Teacher foreknowledge appears to be a nearly ubiquitous but highly heterogeneous phenomenon. Our findings thus have specific implications for educational equity and for future research on teacher expectancies. Future experiments should attend carefully to the sources of potentially expectancy-forming information. We found extensive variability in the influence attributed to various sources; for instance, foreknowledge attributed to fellow teachers was consistently rated as much more influential than foreknowledge attributed to administrators, guidance counselors, or parents. This finding implies that any interventions seeking to strengthen or disrupt expectancy transmission (e.g., through sharing of positively framed information about student interests or abilities, or through avoiding the sharing of negatively framed information) should focus on information-sharing by teachers, rather than by other adults within the school. Meanwhile, researchers conducting naturalistic studies should consider measuring such information-sharing practices of teachers as a potential moderator of the degree to which expectancies persist across multiple years.
Researchers may also wish to explore the cognitive processes by which teachers ascribe greater or lesser importance to specific foreknowledge. For example, given the importance many teachers ascribed to foreknowledge about student behavior, initiatives to combat racial disparities in school discipline might collect data like ours in order to proactively identify particular times or settings in which to intervene upon the formation of behavior-related teacher expectancies. Such interventions could form small but useful contributions to the broader agenda of dismantling systemic racism in education and society writ large.
To the best of our knowledge, no large-scale quantitative study of teacher foreknowledge has been published prior to this work. Our findings suggest this is an important gap in the research literature, as virtually all respondents reported receiving at least some foreknowledge about their students. We share this work in the hope that readers will find teacher foreknowledge to be a useful construct for working to promote equity in education.