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This study investigated the seeds of algebraic thinking that Kindergarten students use when engaging with function tables and graphs. Through interviews with three Kindergarteners, we explored how they reasoned about functional relationships. Our results illustrate how the Kindergarteners used seeds of algebraic thinking when using function tables and graphs to represent and reason about functional relationships. Building on the seeds of algebraic thinking and Knowledge in Pieces frameworks, we categorized these seeds as either strategies (classify, pair, and compare) or ideas (seeds of covariation). Strategy seeds were goal-oriented, and seeds of covariation were elicited without any goal and reflected a broader understanding of change between quantities. 

This study consisted of Grades 1 and 2 (ages 6-8) classroom teaching experiments (CTE) with a lesson sequence focused on graphical representations of algebraic relationships. We interviewed students before, during, and after the CTE. Here we report on the progression of one Grade 2 (age 7) student’s thinking across the CTE. Prior to the CTE, the student had not previously interacted with representations of algebraic relationships. By the end of the CTE, the student was able to generalize about the functional relationships using graphs. This study illustrates how a learning trajectory modeling students’ understandings of function graphs can be used to characterize one children’s learning and provides evidence that young students are able to reason with function graphs. 

This case study of one first grade student involves the analysis of three interviews that took place before, during, and after classroom teaching experiments (CTEs). The CTEs were designed to engage children in representing algebraic concepts using graphs. Using a knowledge-in-pieces perspective, our analysis focused on identifying students’ natural intuitions and ways of thinking algebraically about a functional relationship represented using graphs. Findings reveal four seeds, two of which were identified in prior studies, and how the activation and coordination of these seeds results in students' production of function graphs. 

This study involved a 7-lesson generalized arithmetic classroom teaching experiment (CTE) with kindergarten students. We interviewed four students individually before and after the seven weeks to explore their understandings and representations of arithmetic properties. Here, we report on students' responses to questions on the additive inverse property. Using Skemp’s framework of relational and instrumental understandings (2006), our analysis revealed that most of the interviewed kindergarteners could understand the additive inverse relationally by the end of the CTE. Our interviews revealed that tables and number lines enabled students to articulate more sophisticated understandings of the additive inverse. 

Gestures are one of the ways in which mathematical cognition is embodied and have been elevated as a potentially important semiotic device in the teaching of mathematics. As such, a better understanding of gestures used during mathematics instruction (including frequency of use, types of gestures, how they are used, and the possible relationship between gestures and student performance) would inform mathematics education. We aim to understand teachers’ gestures in the context of early algebra, particularly in the teaching of the equal sign. Our findings suggest that the equal sign is a relatively rich environment for gestures, which are used in a variety of ways. Participating teachers used gestures frequently to support their teaching about the equal sign. Furthermore, the use of gestures varied depending on the particular conception of the equal sign the instruction aimed to promote. Finally, teacher gesture use in this context is correlated with students’ high performance on an early algebra assessment. 

We present one of the largest uniform optical spectroscopic surveys of X-ray selected sources to date that were observed as a pilot study for the Black Hole Mapper (BHM) survey. The BHM program of the Sloan Digital Sky Survey (SDSS)-V is designed to provide optical spectra for hundreds of thousands of X-ray selected sources from the SRG/eROSITA all-sky survey. This significantly improves our ability to classify and characterise the physical properties of large statistical populations of X-ray emitting objects. Our sample consists of 13 079 sources in the eROSITA eFEDS performance verification field, 12 011 of which provide reliable redshifts from 0 ≲ z ≤ 5.8. The vast majority of these objects were detected as point-like sources (X-ray flux limitF_{0.5 − 2 keV} ≳ 6.5 × 10⁻¹⁵erg/s/cm²) and were observed for about 20 years with fibre-fed SDSS spectrographs. After including all available redshift information for the eFEDS sources from the dedicated SDSS-V plate programme and archival data, we visually inspected the SDSS optical spectra to verify the reliability of these redshift measurements and the performance of the SDSS pipeline. The visual inspection allowed us to recover reliable redshifts (for 99% of the spectra with a signal-to-noise ratio of > 2) and to assign classes to the sources, and we confirm that the vast majority of our sample consists of active galactic nuclei (AGNs). Only ∼3% of the eFEDS/SDSS sources are Galactic objects. We analysed the completeness and purity of the spectroscopic redshift catalogue, in which the spectroscopic completeness increases from 48% (full sample) to 81% for a cleaner, brighter (r_AB < 21.38) sample that we defined by considering a high X-ray detection likelihood, a reliable counterpart association, and an optimal sky coverage. We also show the diversity of the optical spectra of the X-ray selected AGNs and provide spectral stacks with a high signal-to-noise ratio in various sub-samples with different redshift and optical broad-band colours. Our AGN sample contains optical spectra of (broad-line) quasars, narrow-line galaxies, and optically passive galaxies. It is considerably diverse in its colours and in its levels of nuclear obscuration.