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1. Universal design of a Tier 2 fraction video game.

   Hunt, J.H. ( October 2021 , North American Chapter of the International Society of the Psychology of Mathematics Education (PME-NA),)

   The 2017 National Assessment of Educational Progress highlights stagnant and declining performance in mathematics, foundational to STEM and ICT, in both 4th (17% proficient) and 8th (9% proficient) grades. At the same time, promoting diversity in the workforce is paramount for U.S. innovation in STEM and ICT fields. Individuals with disabilities are underutilized members of the STEM and ICT workforce. In this poster, we report on Model Mathematics Education (ModelME), a universally designed video game embedded into a student-centered Tier 2 (i.e., supplemental) mathematics curriculum. 

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2. The ionization fraction in OMC-2 and OMC-3

   https://doi.org/10.1051/0004-6361/202140670  

   Salas, P. ; Rugel, M. R. ; Emig, K. L. ; Kauffmann, J. ; Menten, K. M. ; Wyrowski, F. ; Tielens, A. G. ( September 2021 , Astronomy & Astrophysics)

   Context. The electron density ( n e − ) plays an important role in setting the chemistry and physics of the interstellar medium. However, measurements of n e − in neutral clouds have been directly obtained only toward a few lines of sight or they rely on indirect determinations. Aims. We use carbon radio recombination lines and the far-infrared lines of C + to directly measure n e − and the gas temperature in the envelope of the integral shaped filament (ISF) in the Orion A molecular cloud. Methods. We observed the C102 α (6109.901 MHz) and C109 α (5011.420 MHz) carbon radio recombination lines (CRRLs) using the Effelsberg 100 m telescope at ≈2′ resolution toward five positions in OMC-2 and OMC-3. Since the CRRLs have similar line properties, we averaged them to increase the signal-to-noise ratio of the spectra. We compared the intensities of the averaged CRRLs, and the 158 μm-[CII] and [ 13 CII] lines to the predictions of a homogeneous model for the C + /C interface in the envelope of a molecular cloud and from this comparison we determined the electron density, temperature and C + column density of the gas. Results. We detect the CRRLs toward four positions, where their velocity ( v LSR  ≈ 11 km s −1 ) and widths ( σ v  ≈ 1 km s −1 ) confirms that they trace the envelope of the ISF. Toward two positions we detect the CRRLs, and the 158 μm-[CII] and [ 13 CII] lines with a signal-to-noise ratio ≥5, and we find n e −  = 0.65 ± 0.12 cm −3 and 0.95 ± 0.02 cm −3 , which corresponds to a gas density n H  ≈ 5 × 10 3 cm −3 and a thermal pressure of p th  ≈ 4 × 10 5 K cm −3 . We also constrained the ionization fraction in the denser portions of the molecular cloud using the HCN(1–0) and C 2 H(1–0) lines to x (e − ) ≤ 3 × 10 −6 . Conclusions. The derived electron densities and ionization fraction imply that x (e − ) drops by a factor ≥100 between the C + layer and the regions probed by HCN(1–0). This suggests that electron collisional excitation does not play a significant role in setting the excitation of HCN(1–0) toward the region studied, as it is responsible for only ≈10% of the observed emission. 

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3. The Effect of Secondary‐Phase Fraction on the Deformation of Olivine + Ferropericlase Aggregates: 2. Mechanical Behavior

   https://doi.org/10.1029/2022JB025724  

   Wiesman, Harison S. ; Zimmerman, Mark E. ; Kohlstedt, David L. ( April 2023 , Journal of Geophysical Research: Solid Earth)

   To study the mechanical behavior of polymineralic rocks, we performed deformation experiments on two‐phase aggregates of olivine (Ol) + ferropericlase (Per) with periclase fractions (f_Per) between 0.1 and 0.8. Each sample was deformed in torsion atT = 1523 K,P = 300 MPa at a constant strain rate to a final shear strain ofγ = 6 to 7. The stress‐strain data and calculated values of the stress exponent,n, indicate that Ol in our samples deformed by dislocation‐accommodated sliding along grain interfaces while Per deformed via dislocation creep. At shear strains ofγ < 1, the strengths of samples withf_Per > 0.5 match model predictions for both phases deforming at the same stress, the lower‐strength bound for two‐phase materials, while the strengths of samples withf_Per < 0.5 are greater than predicted by models for both phases deforming at the same strain rate, the upper‐strength bound. These observations suggest a transition from a weak‐phase supported to a strong‐phase supported regime with decreasingf_Per. Aboveγ = 4, however, the strength of all two‐phase samples is greater than those predicted by either the uniform‐stress or the uniform‐strain rate bound. We hypothesize that the high strengths in the Ol + Per system are due to the presence of phase boundaries in two‐phase samples, for which deformation is rate limited by dislocation motion along interfacial boundaries. This observation contrasts with the mechanical behavior of samples consisting of Ol + pyroxene, which are weaker, possibly due to impurities at phase boundaries.

    

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4. Developing understanding of fraction equivalence

   Peperak, M. ; Austin, J. ( January 2021 , The Banneker Banner)

   In this article, we discuss how rising fourth graders' understandings of fraction equivalence developed over the course of a ten-week summer program. We found that all students steadily improved in their abilities to give conceptual explanations during the study, yet most struggled with equivalent fractions that could not be formed from a given fraction by halving or doubling the numerator and denominator. We found that the use of discrete manipulatives helped students with these difficulties. 

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5. Teachers’ Knowledge of Fraction Magnitude

   https://doi.org/10.1007/s10763-021-10173-2  

   Copur-Gencturk, Y. ( January 2021 , International journal of science and mathematical education)

   This article explores three attributes of teachers’ understanding of fraction magnitude: the accuracy and reasonableness of teachers’ estimations in response to fraction arithmetic tasks as well as the alignment of the estimation strategies they used with the concept of fraction magnitude. The data were collected from a national sample of mathematics teachers in Grades 3–7 in which fraction concepts were taught (N = 603). The results indicated the teachers’ estimations were only partially accurate and reasonable, particularly when fraction division was involved. Furthermore, teachers’ credentials and the grade level at which they taught mathematics were significantly related to teachers’ understanding of fraction magnitude. 

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