Knowledge of mathematical equivalence, the principle that 2 sides of an equation represent the same value, is a foundational concept in algebra, and this knowledge develops throughout elementary and middle school. Using a construct-modeling approach, we developed an assessment of equivalence knowledge. Second through sixth graders (N ϭ 175) completed the assessment on 2 occasions, 2 weeks apart. Evidence supported the reliability and validity of the assessment along a number of dimensions, and the relative difficulty of items was consistent with the predictions from our construct map. By Grade 5, most students held a basic relational view of equivalence and were beginning to compare the 2 sides of an equation. This study provides insights into the order in which students typically learn different aspects of equivalence knowledge. It also illustrates a powerful but underutilized approach to measurement development that is particularly useful for developing measures meant to detect changes in knowledge over time or after intervention.

ICAP is a theory of active learning that differentiates students' engagement based on their behaviors. ICAP postulates that Interactive engagement, demonstrated by co-generative collaborative behaviors, is superior for learning to Constructive engagement, indicated by generative behaviors. Both kinds of engagement exceed the benefits of Active or Passive engagement, marked by manipulative and attentive behaviors, respectively. This paper discusses a 5-year project that attempted to translate ICAP into a theory of instruction using five successive measures: (a) teachers' understanding of ICAP after completing an online module, (b) their success at designing lesson plans using different ICAP modes, (c) fidelity of teachers' classroom implementation, (d) modes of students' enacted behaviors, and (e) students' learning outcomes. Although teachers had minimal success in designing Constructive and Interactive activities, students nevertheless learned significantly more in the context of Constructive than Active activities. We discuss reasons for teachers' overall difficulty in designing and eliciting Interactive engagement.

Knowledge of the equal sign as an indicator of mathematical equality is foundational to children's mathematical development and serves as a key link between arithmetic and algebra. The current findings reaffirmed a past finding that diverse items can be integrated onto a single scale, revealed the wide variability in children's knowledge of the equal sign assessed by different types of items, and provided empirical evidence for a link between equal-sign knowledge and success on some basic algebra items.

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Speaker eye gaze and gesture are known to help child and adult listeners establish communicative alignment and learn object labels. Here we consider how learners use these cues, along with linguistic information, to acquire abstract relational verbs. Test items were perspective verb pairs (e.g., chase/flee, win/lose), which pose a special problem for observational accounts of word learning because their situational contexts overlap very closely; the learner must infer the speaker's chosen perspective on the event. Two cues to the speaker's perspective on a depicted event were compared and combined: (a) the speaker's eye gaze to an event participant (e.g., looking at the Chaser vs. looking at the Flee-er) and (b) the speaker's linguistic choice of which event participant occupies Subject position in his utterance. Participants (3-, 4-, and 5-year-olds) were eye-tracked as they watched a series of videos of a man describing drawings of perspective events (e.g., a rabbit chasing an elephant). The speaker looked at one of the two characters and then uttered either an utterance that was referentially uninformative (He's mooping him) or informative (The rabbit's mooping the elephant/The elephant's mooping the rabbit) because of the syntactic positioning of the nouns. Eye-tracking results showed that all participants regardless of age followed the speaker's gaze in both uninformative and informative contexts. However, verbmeaning choices were responsive to speaker's gaze direction only in the linguistically uninformative condition. In the presence of a linguistically informative context, effects of speaker gaze on meaning were minimal for the youngest children to nonexistent for the older populations. Thus children, like adults, can use multiple cues to inform verb-meaning choice but rapidly learn that the syntactic positioning of referring expressions is an especially informative source of evidence for these decisions.As a general consequence of the relational semantics they express, verbs serve as the linchpins for the combinatory meaning of sentences (e.g., Carlson & Tanenhaus, 1988;Grimshaw, 2005;Jackendoff, 2002). The successful comprehension of sentences typically hinges upon having detailed knowledge of a verb's syntactic and semantic preferences, such that both sorts of information are utilized to guide parsing and interpretation by adults (Altmann & Kamide, 1999;MacDonald, Pearlmutter, & Seidenberg, 1994;Tanenhaus & Trueswell, 1995) and by children (e.g., Trueswell, Sekerina, Hill, & Logrip, 1999;Snedeker & Trueswell, 2004).In keeping with their eventual role in mature language use, the discovery procedures for verb meanings must coordinate information from the world to which the verbs refer, their distribution with respect to referring expressions, and their syntactic environments (Gleitman, 1990;Gleitman, Cassidy, Nappa, Papafragou, & Trueswell, 2005;Fisher, 1996). Moreover, other nonlinguistic sources of evidence might also be considered, including Copyright © Taylor & Francis Group, LLC Corresponde...