The Effects on Students' Cognitive Achievement When Using the Cooperative Learning Method in Earth Science Classrooms

This study investigated the effects of cooperative learning instruction versus traditional teaching methods on students' earth science achievement in secondary schools. A total of 770 ninth-grade students enrolled in 20 sections of a required earth science course participated in this nonequivalent control group quasi-experiment. The control groups (n= 10) received a traditional approach, while the experimental groups (n= 10) used cooperative strategies. Study results include (a) no significant differences were found between the experimental groups and the control groups when overall achievement (F= 0.13, p > .05), knowledge-level (F= 0.12, p > .05), and comprehension-level (F= 0.34, p > .05) test items were considered; and (b) students who worked cooperatively performed significantly better than students who worked alone on the application-level test items (F= 4.63, p < .05). These findings suggest that cooperative-learning strategies favor students' earth science performance at higher but not lower levels of cognitive domains in the secondary schools.     

The 3‐connectivity of a graph and the multiplicity of zero “2” of its chromatic polynomial

Let G be a graph of order n, maximum degree Δ, and minimum degree δ. Let P(G, λ) be the chromatic polynomial of G. It is known that the multiplicity of zero “0” of P(G, λ) is one if G is connected, and the multiplicity of zero “1” of P(G, λ) is one if G is 2‐connected. Is the multiplicity of zero “2” of P(G, λ) at most one if G is 3‐connected? In this article, we first construct an infinite family of 3‐connected graphs G such that the multiplicity of zero “2” of P(G, λ) is more than one, and then characterize 3‐connected graphs G with Δ + δ?n such that the multiplicity of zero “2” of P(G, λ) is at most one. In particular, we show that for a 3‐connected graph G, if Δ + δ?n and (Δ, δ₃)≠(n?3, 3), where δ₃ is the third minimum degree of G, then the multiplicity of zero “2” of P(G, λ) is at most one. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Non-recursive functions,knots “with thick ropes,” and self-clenching “thick” hyperspheres

We introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n-dimensional sphere for n ≥ 5. Sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem. This recursion-theoretic approach is applied in this paper to a class of functionals on the space of C^1.1-smooth hypersurfaces diffeomorphic to Sⁿ in Rⁿ⁺¹, where n is any fixed number ≥ 5. The simplest of these functionals k_v is defined by the formula k_v(Σⁿ) = (vol(Σⁿ))^1/n/r(Σⁿ), where r(Σⁿ) denotes the radius of injectivity of the normal exponential map for Σⁿ ? Rⁿ+l. We prove the existence of an infinite set of distinct locally minimal values of k_v on the space of C^1.1-smooth topological hyperspheres in Rⁿ⁺¹ for any n ≥ 5. The functional k_v naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of “thick” circles and, more generally, “thick” spheres into Euclidean spaces. We introduce the notion of knot “with thick rope” types. The theory of knot “with thick rope” types turns out to be quite different from the classical knot theory because of the following result: There exists an infinite set of non-trivial knot “with thick rope” types in codimension one for every dimension greater than or equal to five.     

The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real‐world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007     

The classification of irreducible admissible mod <Emphasis Type="Italic">p</Emphasis> representations of a <Emphasis Type="Italic">p</Emphasis>-adic GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Let F be a finite extension of ℚ _p . Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over  [`( \mathbbF)]<sub>p</sub>\overline{ \mathbb{F}}_{p} to be supersingular. We then give the classification of irreducible admissible smooth GL <sub>n</sub> (F)-representations over  [`( \mathbbF)]_p\overline{ \mathbb{F}}_{p} in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel–Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses.     

Eigenfunctions and Very Singular Similarity Solutions of Odd‐Order Nonlinear Dispersion PDEs: Toward a “Nonlinear Airy Function” and Others

Asymptotic properties of nonlinear dispersion equations (1) with fixed exponents n > 0 and p > n+ 1 , and their (2k+ 1) th‐order analogies are studied. The global in time similarity solutions, which lead to “nonlinear eigenfunctions” of the rescaled ordinary differential equations (ODEs), are constructed. The basic mathematical tools include a “homotopy‐deformation” approach, where the limit in the first equation in ( 1 ) turns out to be fruitful. At n= 0 the problem is reduced to the linear dispersion one: whose oscillatory fundamental solution via Airy’s classic function has been known since the nineteenth century. The corresponding Hermitian linear non‐self‐adjoint spectral theory giving a complete countable family of eigenfunctions was developed earlier in [ 1 ]. Various other nonlinear operator and numerical methods for ( 1 ) are also applied. As a key alternative, the “super‐nonlinear” limit , with the limit partial differential equation (PDE) admitting three almost “algebraically explicit” nonlinear eigenfunctions, is performed. For the second equation in ( 1 ), very singular similarity solutions (VSSs) are constructed. In particular, a “nonlinear bifurcation” phenomenon at critical values {p=p_l(n)}_l≥0 of the absorption exponents is discussed.     

Dirichlet duality and the nonlinear Dirichlet problem

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F (Hess u) = 0 on a smoothly bounded domain Ω ? ?ⁿ. In our approach the equation is replaced by a subset F ? Sym²(?ⁿ) of the symmetric n × n matrices with ?F ? { F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F‐convexity” assumption on the boundary ?Ω. We also study the topological structure of F‐convex domains and prove a theorem of Andreotti‐Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F? that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F? is F, and in the analysis the roles of F and F? are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge‐Ampère equation over ?, ?, and ?; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p‐convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc.     

The onset of dominance in balls‐in‐bins processes with feedback

Consider a balls‐in‐bins process in which each new ball goes into a given bin with probability proportional to f(n), where n is the number of balls currently in the bin and f is a fixed positive function. It is known that these so‐called balls‐in‐bins processes with feedback have a monopolistic regime: if f(x) = x^p for p > 1, then there is a finite time after which one of the bins will receive all incoming balls. Our goal in this article is to quantify the onset of monopoly. We show that the initial number of balls is large and bin 1 starts with a fraction α > 1/2 of the balls, then with very high probability its share of the total number of balls never decreases significantly below α. Thus a bin that obtains more than half of the balls at a “large time” will most likely preserve its position of leadership. However, the probability that the winning bin has a non‐negligible advantage after n balls are in the system is ～const. × n^1‐p, and the number of balls in the losing bin has a power‐law tail. Similar results also hold for more general functions f. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Minimal ‐crystals and isomorphism numbers of isosimple ‐crystals

In this paper we generalize minimal p‐divisible groups defined by Oort to minimal F‐crystals over algebraically closed fields of positive characteristic. We prove a structural theorem of minimal F‐crystals and give an explicit formula of the Frobenius endomorphism of the basic minimal F‐crystals that are the building blocks of the general minimal F‐crystals. We then use minimal F‐crystals to generalize minimal heights of p‐divisible groups and give an upper bound of the isomorphism numbers of F‐crystals, whose isogeny type are determined by simple F‐isocrystals, in terms of their ranks, Hodge slopes and Newton slopes.     

Subgraph‐avoiding coloring of graphs

Given a “forbidden graph” F and an integer k, an F‐avoiding k‐coloring of a graph G is a k‐coloring of the vertices of G such that no maximal F‐free subgraph of G is monochromatic. The F‐avoiding chromatic number ac_F(G) is the smallest integer k such that G is F‐avoiding k‐colorable. In this paper, we will give a complete answer to the following question: for which graph F, does there exist a constant C, depending only on F, such that ac_F(G) ? C for any graph G? For those graphs F with unbounded avoiding chromatic number, upper bounds for ac_F(G) in terms of various invariants of G are also given. Particularly, we prove that ${{ac}}_{{{F}}}({{G}})\le {{2}}\lceil\sqrt{{{n}}}\rceil+{{1}}Given a “forbidden graph” F and an integer k, an F‐avoiding k‐coloring of a graph G is a k‐coloring of the vertices of G such that no maximal F‐free subgraph of G is monochromatic. The F‐avoiding chromatic number ac_F(G) is the smallest integer k such that G is F‐avoiding k‐colorable. In this paper, we will give a complete answer to the following question: for which graph F, does there exist a constant C, depending only on F, such that ac_F(G) ? C for any graph G? For those graphs F with unbounded avoiding chromatic number, upper bounds for ac_F(G) in terms of various invariants of G are also given. Particularly, we prove that ${{ac}}_{{{F}}}({{G}})\le {{2}}\lceil\sqrt{{{n}}}\rceil+{{1}}$, where n is the order of G and F is not K_k or $\overline{{{K}}_{{{k}}}}$. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 300–310, 2010     

Teaching Multiple Modes of Representation in Middle‐School Science Classrooms: Impact on Student Learning and Multimodal Use

This quasi‐experimental study investigated how explicit instruction about multiple modes of representation (MMR) impacted grades 7 (n = 61) and 8 (n = 141) students’ learning and multimodal use on end‐of‐unit assessments. Half of each teacher's (n = 3) students received an intervention consisting of explicit instruction on MMR in science discourse, in addition to regular science instruction enhanced by a focus on MMR; comparison groups of students received regular science instruction. Three ordinary least squares regression models used student demographic variables and whether or not students received the intervention to predict students’ (a) gain scores on end‐of‐unit tests, (b) voluntary use of embedded MMR on unit tests, and (c) retention of science knowledge as measured by a state end‐of‐level criterion‐referenced assessment. Analyses showed that explicit instruction on MMR did not make a significant impact on student gain scores, the amount of embeddedness on unit tests, or end‐of‐level scores. However, Models 2 and 3 showed Hispanics and females used MMR more on end‐of‐unit tests than Whites or males, respectively, whether or not they received the intervention. Hispanics and females scored lower than Whites or males, respectively, on end‐of‐level, multiple‐choice assessments. Implications for classroom teachers and educational researchers in relation to these underserved populations are discussed.     

The Vector Space Kinna‐Wagner Principle is Equivalent to the Axiom of Choice

We show that the axiom of choice AC is equivalent to the Vector Space Kinna‐Wagner Principle, i.e., the assertion: “For every family 𝒱= {V_i : i ∈ k} of non trivial vector spaces there is a family ℱ = {F_i : i ∈ k} such that for each i ∈ k, F_i is a non empty independent subset of V_i”. We also show that the statement “every vector space over ℚ has a basis” implies that every infinite well ordered set of pairs has an infinite subset with a choice set, a fact which is known not to be a consequence of the axiom of multiple choice MC.     

Evaluating efficacy of environmental education programming

The Clark Fork Watershed Education Program (CFWEP) goals are: (a) increasing students’ understanding of the nature of ecological impacts within their watershed as related to historic mining damage; and (b) increasing students’ sense of stewardship of newly restored landscapes. Data from 2012 to 2016 were evaluated for student knowledge gains (46 trials representing 2,395 student pre‐surveys; 2,409 student post‐surveys). Data from 2013 to 2016 were evaluated for students’ attitudes toward science and disposition toward caring for the environment (38 trials representing 1,479 pre‐surveys; 1,460 post‐surveys). The results of this study support that the program’s goals are being achieved. Students achieved statistically significant gains on knowledge surveys with a 33.4% overall gain pre‐ to posttest (p < 0.0001). Students also moved toward greater positive responses in both attitudes toward science and disposition toward caring for the environment with Cohen’s d effect sizes of “medium effect” for caring toward the environment (d = 0.52) and “small effect” of positive disposition toward science (d = 0.24).     

Attitude toward mathematics among the students at Nazarbayev University Foundation Year Programme

This article investigates the attitude toward mathematics among the students enrolled in the Foundation Year Programme at Nazarbayev University. The study is conducted quantitatively and an inventory developed by Tapia and Marsh II is adopted in this research. The inventory consists of 40 statements on the five-point Likert scale. Gender, specialization and final high school score in mathematics are collected. The number of valid returned questionnaires is 108. There are 55 males, 53 females, 73 Mathematical-Physics (MP), 22 Biology-Chemistry (BC) and 13 International Relations-Economics (IRE) students. Generally, they have a positive attitude toward mathematics, with the score mean and standard deviation are 3.999 and 0.531 out of five, respectively. We confirm a hypothesis on a positive correlation between previous high achievement in mathematics and favorable attitude toward it. The correlation value is r = 0.300, its effect size is medium and it is extremely significant (p-value = 0.0008 < 0.001). There is no significant difference between male and female students in terms of their attitude toward mathematics (t = 0.6804, p-value = 002489 > 0.05). There is a very significant difference between students who specialize in IRE and MP in terms of their attitude toward mathematics (F(2, 105) = 5.6848, p-value = 0.0045 < 0.01).     

Three classes of smooth Banach submanifolds in B(E,F)

Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(E, F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result Let ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. ＜ ∞ or codim R(T) = const.＜ ∞} and {T ∈ R(E, F) RankT =const.＜∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry.     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

Enhancing language-responsive meaning-making processes as an epistemic catalyst for developing multiplicative reasoning in young children

Multiplicative reasoning involves the ability to coordinate bundled units on a more abstract level (“unitizing”; Lamon, 1994). As it is considered a “cutoff point” for students’ future mathematical learning, teachers must provide equitable access to mathematical conceptual understanding for all students on all mathematical achievement levels. The study presented in this paper investigates to what extent a preventive and a language-responsive instructional approach can have an effect on the outcome of students on different mathematical achievement levels. Three German second grade teachers introduced multiplication to students (n = 66, aged 7–8 years) in their classes using meaning-related phrases (e.g., “6 times 4 means 6 fours”), while teachers in the control group (n = 58) did not focus on using these phrases. Analyses of both a multiplication posttest and a follow-up test showed significant differences between the intervention and control groups on all achievement levels for both conceptual and procedural items.     

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Fitting Height of a Finite Group with a Metabelian Group of Automorphisms

Let M = FH be a finite group that is a product of a normal abelian subgroup F and an abelian subgroup H. Assume that all elements in M?F have prime order p, and F has at most one subgroup of order p. Examples of such groups are dihedral groups for p = 2 and the semidirect product of a cyclic group F by a group H of prime order p such that C _F (H) = 1 or |C _F (H)| =p and C _{F/C F (H)}(H) = 1. Suppose that M acts on a finite group G in such a manner that C _G (F) = 1. We prove that the Fitting height h(G) of G is at most h(C _G (H))+ 1. Moreover, the Fitting series of C _G (H) coincides with the intersection of C _G (H) with the Fitting series of G.     

Coloring random graphs online without creating monochromatic subgraphs

Consider the following random process: The vertices of a binomial random graph G_n,p are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number r of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph F in the process. Our first main result is that for any F and r, the threshold function for this problem is given by p₀(F,r,n) = n^‐1/m*₁(F,r), where m*₁(F,r) denotes the so‐called online vertex‐Ramsey density of F and r. This parameter is defined via a purely deterministic two‐player game, in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. Our second main result states that for any F and r, the online vertex‐Ramsey density m*₁(F,r) is a computable rational number. Our lower bound proof is algorithmic, i.e., we obtain polynomial‐time online algorithms that succeed in coloring G_n,p as desired with probability 1 ‐ o(1) for any p(n) = o(n^‐1/m*₁(F,r)). © 2012 Wiley Periodicals, Inc. Random Struct. Alg. 44, 419–464, 2014