Using Games to Engage Students in Inquiry

Abstract

This paper discusses the author’s experiences of getting advanced undergraduate math students to engage in mathematical inquiry by using games as a vehicle for exploration. The students explored the mathematics behind SET^®1, Spot it!^®2, Blokus^®3, and Six^®4. Specifically, we present the experience of the instructor and students and how the games were used to promote and incite inquiry.     

Stimulus nonequivalences in go/no-go avoidance discriminations: Sensory,drive, and response factors

Previous work with rats showed slower active avoidance (AA) and passive avoidance (PA) learning in a light-go/noise-light no-go task (L⁺NL^?) than in a noise-go/light-noise no-go task (N⁺LN^?), independently of noise intensity and of presence or absence of AA pretraining. Experiment I compared these two tasks and the corresponding discriminations with a compound AA and a simple PA signal. The latter were learned at a faster rate, with only negligible differences between the two types of stimulus arrangements (NL⁺L^? vs LN⁺N^?). Experiment II investigated the acquisition and reversal of four go/no-go tasks with a simple AA signal, with either a simple or a compound no-go signal and asymmetrical reinforcement, i.e., with an extinction instead of a PA contingency during the presentation of no-go signals. As in previous experiments with symmetrical reinforcement, the groups with a simple no-go signal (L⁺N^?, N⁺L^?) acquired AA with little or no generalization to the other stimulus. In reversal, however, the L⁺N^? group (formerly N⁺L^?) responded to the no-go signal much more frequently than the other (N⁺L^?, formerly L⁺N^?). Both the L⁺NL^? and the N⁺LN^? groups gave a large number of responses to the extinction signal, but the differences between the two groups were at least as large as in previous work with a PA contingency. These data show that perceptual or attentional phenomena cannot explain stimulus nonequivalences in situations with a compound no-go signal. Furthermore, the similar nonequivalences found in situations with and without differential fear conditioning (AA-extinction tests vs AA-PA tests) leads to complex, highly speculative models concerning possible interactions between response and drive-modulating properties of various stimuli.     

Sensitized Up—Conversion Luminescence of Ho^3＋ in Nd^3＋—Yb^3＋—Ho^3＋ Co—doped ZrF4—Based Glass

Up-conversion processes for the blue, green and red emissions were found two-photon phenomenon, known as the cooperative phenomenon. This phenomenon was assisted by Nd³⁺ → Yb³⁺ → Ho³⁺ energy transfer. The strong green emission due to the Ho³⁺ : (⁵F₄, ⁵S₂)→ ⁵I₈ transitions was observed in Nd³⁺-Ho³⁺ co-doped ZrF₄-based fluoride glasses under 800 nm excitation. As an attempt to enhance Ho³⁺ up-conversion luminescences in the Nd³⁺ – Ho³⁺ co-doped ZrF₄-based glasses, Yb³⁺ ions were added to the glasses. As a result it was found that, in 800 nm excitation of 60 ZrF₄. 30BaF2. (8-x)LaF₃. lNdF₃. xYbF₃. 1HoF₃ glasses (x=0 to 7), sensitized up-conversion luminescences are observed at around 490 nm (blue), 545 nm (green), and 650 nm (red), which correspond to the Ho³⁺ : ⁵F₃ → ⁵I₈, ( ⁵F₄, ⁵S₂)→⁵I₈ and ⁵F₅→ ⁵I₈ transitions respectively. The intensities of the green and red emissions in a 3 mol % YbF₃-containing glass were about 50 times stronger than those glasses without YbF₃. This is based on sensitization due to Yb³⁺ ions. In particular, the green emission was extremely strong and the Nd³⁺-Yb³⁺-Ho³⁺ co-doped ZrF₄-based glasses have a high possibility of realizing a green up-conversion laser glass. In this paper the up-conversion mechanism in the glasses is discussed in detail.     

Effect of lead exposure on the immune function of lymphocytes and erythrocytes in preschool children

Objective: To investigate the influence of lead exposure on the immune function of lymphocytes and erythrocytes in preschool children. Materials and methods: A group of 217 children three to six years of age from a rural area were given a thorough physical examination and the concentration of lead in blood samples taken from each subject was determined. The indices of lymphocyte immunity (CD 3CD 4, CD 3CD 8, CD 4CD 8, CD-3CD 19) and erythrocyte immunity (RBC-C3b, RBC-IC, RFER, RFIR, CD35 and its average fluorescence intensity) of 40 children with blood lead levels above 0.483 μmol/L were measured and compared with a control group. Results: The blood lead levels of the 217 children ranged from 0.11 μmol/L to 2.11 μmol/L. The CD 3CD 4and CD 4CD 8 cells were lower (P＜0.01) and the CD 3CD 8 cells were higher in the lead-poisoned subjects than those in the control group (P＜0.05). CD 3 and CD-3CD 19 did not show significant differences. Although the RBC-C3b rosette forming rate was lower and the RBC-IC rosette forming rate was higher in the lead-poisoned group, this difference could not be shown to be statistically significant (P＞0.05). RFIR was found to be lower in the lead-poisoned group (P＜0.01). Compared with the control group, the positive rate of CD35 was not found to be significantly different in a group of 25 lead-poisoned children (P＞0.05), while the average fluorescence intensity was lower in the lead-poisoned group (P＜0.05). Conclusion: Lead exposure can result in impaired immune function of T lymphocytes and erythrocytes in preschool children.     

Effects of nitrogen form on growth,CO2 assimilation,chlorophyll fluorescence,and photosynthetic electron allocation in cucumber and rice plants

Cucumber and rice plants with varying ammonium (NH₄ ⁺) sensitivities were used to examine the effects of different nitrogen (N) sources on gas exchange, chlorophyll (Chl) fluorescence quenching, and photosynthetic electron allocation. Compared to nitrate (NO₃ ⁻)-grown plants, cucumber plants grown under NH₄ ⁺-nutrition showed decreased plant growth, net photosynthetic rate, stomatal conductance, intercellular carbon dioxide (CO₂) level, transpiration rate, maximum photochemical efficiency of photosystem II, and O₂-independent alternative electron flux, and increased O₂-dependent alternative electron flux. However, the N source had little effect on gas exchange, Chl a fluorescence parameters, and photosynthetic electron allocation in rice plants, except that NH₄ ⁺-grown plants had a higher O₂-independent alternative electron flux than NO₃ ⁻-grown plants. NO₃ ⁻ reduction activity was rarely detected in leaves of NH₄ ⁺-grown cucumber plants, but was high in NH₄ ⁺-grown rice plants. These results demonstrate that significant amounts of photosynthetic electron transport were coupled to NO₃ ⁻ assimilation, an effect more significant in NO₃ ⁻-grown plants than in NH₄ ⁺-grown plants. Meanwhile, NH₄ ⁺-tolerant plants exhibited a higher demand for the reduced form of nicotinamide adenine dinucleotide phosphate (NADPH) for NO₃ ⁻ reduction, regardless of the N form supplied, while NH₄ ⁺-sensitive plants had a high water-water cycle activity when NH₄ ⁺ was supplied as the sole N source.     

Fourth Graders' Social Standing with Peers: A Prospective Study on the Role of First Grade Physical Activity,Weight Status,and Motor Proficiency

We prospectively examined the role of physical activity involvement, weight status, and motor functioning capability in 1^st grade for social status among other pupils in class in the 4^th grade. Our sample included 80 Norwegian 1^st grade pupils (Girls N = 44; Boys N = 36). 1^st grade motor proficiency and objectively measured physical activity, but not weight status was predictive of 4^th grade social standing among pupils in class as measured by socio‐metric status. Interaction findings also revealed that 1^st grade healthy weight children being proficient in the motor domain benefit more in terms of their social standing among peers in 4^th grade than do 1^st grade overweight but equally motor proficient children. Being physically active in the 1^st grade seems more important for social standing among peers in 4^th grade for 1^st grade girls than for boys. Results may inform early intervention efforts in school to enhance pupils' psychosocial development.     

A disputed evidence on obesity: comparison of the effects of <Emphasis Type="Italic">Rcan2</Emphasis><Superscript>−/−</Superscript> and <Emphasis Type="Italic">Rps6kb1</Emphasis><Superscript>−/−</Superscript> mutations on growth and body weight in C57BL/6J mice

It is widely accepted that body weight and adipose mass are tightly regulated by homeostatic mechanisms, in which leptin plays a critical role through hypothalamic pathways, and obesity is a result of homeostatic disorder. However, in C57BL/6J mice, we found that Rcan2 increases food intake and plays an important role in the development of age- and diet-induced obesity through a leptin-independent mechanism. RCAN2 was initially identified as a thyroid hormone (T3)-responsive gene in human fibroblasts. Expression of RCAN2 is regulated by T3 through the PI3K-Akt/PKB-mTOR-Rps6kb1 signaling pathway. Intriguingly, both Rcan2^?/? and Rps6kb1^?/? mutations were reported to result in lean phenotypes in mice. In this study we compared the effects of these two mutations on growth and body weight in C57BL/6J mice. We observed reduced body weight and lower fat mass in both Rcan2^?/? and Rps6kb1^?/? mice compared to the wild-type mice, and we reported other differences unique to either the Rcan2^?/? or Rps6kb1^?/? mice. Firstly, loss of Rcan2 does not directly alter body length; however, Rcan2^?/? mice exhibit reduced food intake. In contrast, Rps6kb1^?/? mice exhibit abnormal embryonic development, which leads to smaller body size and reduced food intake in adulthood. Secondly, when fed a normal chow diet, Rcan2^?/? mice weigh significantly more than Rps6kb1^?/? mice, but both Rcan2^?/? and Rps6kb1^?/? mice develop similar amounts of epididymal fat. On a high-fat diet, Rcan2^?/? mice gain body weight and fat mass at slower rates than Rps6kb1^?/? mice. Finally, using the double-knockout mice (Rcan2^?/?Rps6kb1^?/?), we demonstrate that concurrent loss of Rcan2 and Rps6kb1 has an additive effect on body weight reduction in C57BL/6J mice. Our data suggest that Rcan2 and Rps6kb1 mutations both affect growth and body weight of mice, though likely through different mechanisms.     

Very long-term retention of the control of variables strategy following a brief intervention

Middle school students (n = 354) were tested for their understanding of the control of variables strategy (CVS) 2.5 years after participating in a study comparing three different interventions for teaching CVS. The key finding was that the pattern of effects observed in the 4^th grade continued to be observed in the 6^th grade. This was because (a) students who had mastered CVS in the 4^th grade were likely to continue to perform at mastery levels in the 6^th grade whereas (b) the learning of students who had not mastered CVS in 4^th grade was independent of the teaching intervention they had received in the 4^th-grade study. These findings demonstrate that a brief intervention of direct instruction in CVS can produce long-lasting learning of the principle of controlling variables.     

Synergistic interaction of NaCl and Cd on growth and photosynthetic parameters in soybean genotypes differing in salinity tolerance

The effects of salinity(50 mmol/L NaCl) and Cd(1 μmol/L CdCl2) as sole and combined on growth and photosyn-thetic parameters were studied using two soybean genotypes,Huachun 18 and NGB.The concentrations of Cd2 ,Zn2 ,Ca2 ,Mg2 ,K and Na were also determined in seeds and pods.Huachun 18 suffered a more serious decrease than NGB in net photosynthetic rate(Pn) in the treatments of salinity stress alone and combined stress(NaCl Cd),showing that it is relatively sensitive to salinity.The decrease in Pn caused by salt stress in Huachun 18 was mainly due to the reduced total chlorophyll content and photosynthetic efficiency(the ratio of variable fluorescence to maximal fluorescence,Fv/Fm),whereas the decease in NGB was mainly related to reduced stomatal conductance(Gs).The combined stress of both Na and Cd did not induce further decrease in photosynthesis and fluorescence in the two genotypes relative to salt or Cd stress alone.Greater change in the pod concentrations of Zn2 ,Ca2 ,Mg2 ,K and Na was detected under salt stress for Huachun 18 than for NGB.The results suggested that the interactive effect of NaCl-Cd on growth and nutrient uptake differs between the two soybean genotypes.     

An elegant example of chemoselective reaction The preparation of sulfonamide drugs

In a chemoselective reaction, one of the functional groups in the reactant molecule is selectively attacked by a reagent. The present article describes this principle by demonstrating that careful alkaline hydrolysis of N¹,N⁴-diacetylsulfanilamide (7) removes only the N⁴-acetyl group from it leaving the N¹-acetyl group intact to yield N¹-acetylsulfanilamide (8).     

Study of transport behavior for Fe-doping La<Subscript>0.67</Subscript>Ca<Subscript>0.33</Subscript>MnO<Subscript>3</Subscript> perovskite manganese

Systematic studies of the transport properties of La_0.67Ca_0.33Mn_1?xFe _x O₃ (x=0–0.3) systems showed that with increasing Fe-doping contentx the resistance increases and the insulator-metal transition temperature moves to lower temperature. For small doping content, the transport property satisfies metal transport behavior below the transition temperature, and above the transition temperature it satisfies the small polaron model. This behavior can be explained by Fe³⁺ doping, which easily forms Fe³⁺?O^2??Mn⁴⁺ channel, suppressing the double exchange Mn³⁺?O^2??Mn⁴⁺ channel and enhancing the spin scattering of Mn ions induced by antiferromagnetic clusters of Fe ions.     

Performance of the Generalized S‐X2 Item Fit Index for Polytomous IRT Models

Orlando and Thissen's S‐X ² item fit index has performed better than traditional item fit statistics such as Yen's Q₁ and McKinley and Mill's G² for dichotomous item response theory (IRT) models. This study extends the utility of S‐X ² to polytomous IRT models, including the generalized partial credit model, partial credit model, and rating scale model. The performance of the generalized S‐X ² in assessing item model fit was studied in terms of empirical Type I error rates and power and compared to G². The results suggest that the generalized S‐X ² is promising for polytomous items in educational and psychological testing programs.     

Effects of ammonium application rate on uptake of soil adsorbed amino acids by rice

In recent years, excessive use of chemical nitrogen (N) fertilizers has resulted in the accumulation of excess ammonium (NH ₄ ⁺ ) in many agricultural soils. Though rice is known as an NH ₄ ⁺ -tolerant species and can directly absorb soil intact amino acids, we still know considerably less about the role of high exogenous NH ₄ ⁺ content on rice uptake of soil amino acids. This experiment examined the effects of the exogenous NH ₄ ⁺ concentration on rice uptake of soil adsorbed glycine in two different soils under sterile culture. Our data showed that the sorption capacity of glycine was closely related to soils;’ physical and chemical properties, such as organic matter and cation exchange capacity. Rice biomass was significantly inhibited by the exogenous NH ₄ ⁺ content at different glycine adsorption concentrations. A three-way analysis of variance demonstrated that rice glycine uptake and glycine nutritional contribution were not related to its sorption capacity, but significantly related to its glycine:NH ₄ ⁺ concentration ratio. After 21-d sterile cultivation, the rice uptake of adsorbed glycine accounted for 8.8%;–22.6% of rice total N uptake, which indicates that soil adsorbed amino acids theoretically can serve as an important N source for plant growth in spite of a high NH ₄ ⁺ application rate. However, further studies are needed to investigate the extent to which this bioavailability is realized in the field using the ¹³C, ¹⁵N double labeling technology.     

Effect of perioperative autologous versus allogeneic blood transfusion on the immune system in gastric cancer patients

Background: Allogeneic blood transfusion-induced immunomodulation (TRIM) and its adverse effect on the prognosis of patients treated surgically for cancer remain complex and controversial. However, the potential risk associated with allogeneic blood transfusion has heightened interest in the use of autologous blood transfusion. In the present study, the serum concentrations of neopterin, interferon-gamma (IFN-γ), T lymphocyte subsets (CD3 , CD4 , CD8 , CD4 /CD8 ) and a possible association between these variables were investigated. The purpose was to further evaluate the effect of autologous versus allogeneic blood transfusion on immunological status in patients undergoing surgery for gastric cancer. Methods: Sixty ASA I～II (American Society of Anesthesiologists) patients undergoing elective radical resection for stomach cancer were randomly allocated to receive either allogeneic blood transfusion (n=30) or autologous blood transfusion (n=30). Serum concentrations of the neopterin, IFN-γ and T lymphocyte subsets in the recipients were measured before induction of anesthesia, after operation, and on the 5th postoperative day. Results: Both two groups, serum neopterin, IFN-γ, percentages of T-cell subsets (CD3 , CD4 ), and CD4 /CD8 ratio had significantly decreased after operation, but decreased more significantly in group H (receiving allogeneic blood transfusion) than those in group A (receiving autologous whole blood transfusion) (P<0.05). On the 5th postoperative day, serum neopterin, IFN-γ, CD3 , CD4 T-cells, and CD4 /CD8 ratio returned to the baseline values in group A. In contrast, the above remain decreasing in group H, where there were no significant relations between serum neopterin and IFN-γ. Conclusion: Perioperative surgical trauma and stress have an immunosuppressive impact on gastric cancer patients. Allogeneic blood transfusion exacerbates the impaired immune response. Autologous blood transfusion might be significantly beneficial for immune-compromised patients in the perioperative period, clearly showing its superiority over allogeneic blood transfusion.     

A Comparison of Item Fit Statistics for Mixed IRT Models

In this study we examined procedures for assessing model-data fit of item response theory (IRT) models for mixed format data. The model fit indices used in this study include PARSCALE's G² , Orlando and Thissen's S − X² and S − G² , and Stone's χ²* and G²* . To investigate the relative performance of the fit statistics at the item level, we conducted two simulation studies: Type I error and power studies. We evaluated the performance of the item fit indices for various conditions of test length, sample size, and IRT models. Among the competing measures, the summed score-based indices S − X² and S − G² were found to be the sensible and efficient choice for assessing model fit for mixed format data. These indices performed well, particularly with short tests. The pseudo-observed score indices, χ²* and G²* , showed inflated Type I error rates in some simulation conditions. Consistent with the findings of current literature, the PARSCALE's G² index was rarely useful, although it provided reasonable results for long tests.     

Go-no go avoidance discriminations in rats with simple “go” and compound “no go” signals: Stimulus modality and stimulus intensity

Four groups of 16 rats were trained in a go-no go discrimination with symmetrical negative reinforcement (active-passive avoidance task). A 2 by 2 design was used with stimulus combinations as one factor [light-go, noise/light-no go, (L⁺ NL^?) vs noise-go, light/noise-no go (N⁺ LN^?)] and noise intensity as the other factor (90 dB, high, vs 70 dB, low). The two N⁺ LN^? groups learned both active and passive avoidance responses much more rapidly than the L⁺ NL^? groups. In later phases of training, high noise intensity facilitated passive avoidance in the L⁺ NL^? condition and exerted a slight effect in the opposite direction in the other condition. While the differences in active avoidance learning were the same as in previous work on light and noise CSs, the interactions between stimuli of different modalities appeared to be more important than stimulus modality or stimulus intensity per se in determining rate of passive avoidance learning.     

Projectively flat Asanov Finsler metric

In this work, we study the Asanov Finsler metric F=α(β2/α2 gβ/α 1)1/2exp{(G/2)arctan[β/(hα) G/2]}, where α=(αijyiyj)1/2 is a Riemannian metric and β=biyj is a 1-form, g∈(-2,2), h=(1-g2/4)1/2, G=g/h. We give the necessary and sufficient condition for Asanov metric to be locally projectively flat, i.e., α is projectively flat and β is parallel with respect to α. Moreover, we proved that the Douglas tensor of Asanov Finsler metric vanishes if and only if β is parallel with respect to α.     

Infrared Absorption Spectra of Serpentine Cat＇s Eye from Sichuan Province of China

Infrared （IR） absorption spectra and X-ray powder diffraction （XRD） patterrns of sexpenfine cat＇s eye from Sichuan Province have been investigated, and infrared absorption bands of the serpentine have been assigned in this paper. The results indicate that the bands near 3 600 - 3 700 cm^-1 belong to the stretching absorption band VOH, the bands in 948 - 1 100 cm^-1 axe assigned to the stretching vibraling band Usvo, and the bands at 600 570 and 440 cm^-1 respectively belong to the bending vibration band VOH, δMg-O and δSi-O. The serpentine cat＇ s eye includes two types： chrysotile and antigorite. Chrysotile has a sharp and intensive band at 3 688 cm^-1 and a weak band at 3 643 cm^-1 , while antigorite has only one intensive band in the region. At vibration band VSi-O, chrysotile and antigorite have a similar feature at 1 071 cm^-1, but the band at 980 cm^-1 in anfigorite has been split into two bands in chrysotile - a strong one at 1 027 cm^-1 and a weak one at 949 cm^-1. In addition, antigorite has a characteristic bending vibrating band δMg-O at 570 cm^-1. Based on the analysis of the feature of crystal structure, their IR differences axe well explained.     

Estimating R 2 Shrinkage in Multiple Regression: A Comparison of Different Analytical Methods

The effectiveness of various analytical formulas for estimating R ² shrinkage in multiple regression analysis was investigated. Two categories of formulas were identified: estimators of the squared population multiple correlation coefficient (ρ²) and those of the squared population cross-validity coefficient (ρ_c ²). The authors conducted a Monte Carlo experiment to investigate the effectiveness of the analytical formulas for estimating R ² shrinkage, with 4 fully crossed factors (squared population multiple correlation coefficient, number of predictors, sample size, and degree of multicollinearity) and 500 replications in each cell. The results indicated that the most widely used Wherry formula (in both SAS and SPSS) is probably not the most effective analytical formula for estimating ρ². Instead, the Pratt formula and the Browne formula outperformed other analytical formulas in estimating ρ² and ρ_c ², respectively.