This study investigated (a) mathematics achievement (Test of Academic Progress) of 589 female and 644 male, predominantly white, 9th-12th grade students enrolled in mathematics courses from four schools, controlling for mathematics background and general ability (Quick Word Test); (b) relationships to mathematics achievement, and to sex-related differences in mathematics achievement, of spatial visualization (Differential Aptitude Test), eight attitudes measured by the Fennema-Sherman Mathematics Attitudes Scales, a measure of Mathematics Activities outside of school, and number of Mathematics Related Courses and Space Related Courses taken. Complex results were obtained. Few sex-related cognitive differences but many attitudinal differences were found. Analyses of variance, covariance, correlation, and principal components analysis techniques were used. The results showed important relationships between socio-cultural factors and sex-related cognitive differences.

Sex-related defferences in mathematics achievement, spatial visualization, and affective factors

Combinatory Analysis. By Major P. A. MacMohan. Vol. I. Pp. xx + 300. 15s. net. (Cam. Univ. Press)

In this paper, we define two natural $(p,q)$-analogues of the generalized Stirling numbers of the first and second kind $S^1(\alpha,\beta,r)$ and $S^2(\alpha,\beta,r)$ as introduced by Hsu and Shiue [Adv. in Appl. Math. 20 (1998), 366–384]. We show that in the case where $\beta =0$ and $\alpha$ and $r$ are nonnegative integers both of our $(p,q)$-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our $(p,q)$-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our $(p,q)$-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

/pdf/rook-theory-generalized-stirling-numbers-and-p-q-analogues-2ql2y6h8gw.pdf

Rook theory, generalized stirling numbers and (p, q)-analogues

m -Rook numbers and a generalization of a formula of Frobenius to C m ? S n

We obtain asymptotic formulas for the number of rooted 2-connected and 3-connected surface maps on an orientable surface of genus $g$ with respect to vertices and edges simultaneously. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive asymptotic formulas for the number of labelled $k$-connected graphs of orientable genus $g$ for $k\le3$.

/pdf/asymptotic-enumeration-of-labelled-graphs-by-genus-2hk3b764dr.pdf

Asymptotic enumeration of labelled graphs by genus.

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook configurations to give a combinatorial interpretation to the q-analogue of a formula of Frobenius relating the Stirling numbers of the second kind to the Eulerian polynomials. Later, Remmel and Wachs defined generalized p,q-Stirling numbers of the first and second kind in terms of rook placements. Additionally, they extended their definition to give a p,q-analogue of rook numbers for arbitrary Ferrers boards. In this paper, we use Remmel and Wach’s definition and an extension of Garsia and Remmel’s proof to give a combinatorial interpretation to a p,q-analogue of a formula of Frobenius relating the p,q-Stirling numbers of the second kind to the trivariate distribution of the descent number, major index, and comajor index over Sn. We further define a p,q-analogue of the hit numbers, and show analytically that for Ferrers boards, the p,q-hit numbers are polynomials in (p,q) with nonnegative coecients.

/pdf/a-p-q-analogue-of-a-formula-of-frobenius-54sfs58297.pdf

A $p,q$-analogue of a Formula of Frobenius

Recently, Benjamin, Plott, and Sellers proved a variety of identities involving sums of Pell numbers combinatorially by interpreting both sides of a given identity as enumerators of certain sets of tilings using white squares, black squares, and gray dominoes. In this article, we state and prove q-analogues of several Pell identities via weighted tilings.

/pdf/combinatorial-proofs-of-various-q-pell-identities-via-28ypblt9d4.pdf

Combinatorial Proofs of Various q-Pell Identities via Tilings

The authors conducted a two-semester quasi-experimental study in which each author taught a traditional lecture-based section of precalculus and a section using an inquiry-based approach called a Modified Moore Method in which the students worked through and presented the course material. A common final exam was used to compare student achievement. The results were compared for the overall population and by each instructor. Gender proved to be an important variable with the females performing significantly better in the Modified Moore Method sections than their counterparts in the traditional sections while there were no significant differences for the males.

/pdf/gender-differences-in-achievement-in-an-inquiry-based-24j9xtpl9i.pdf

Gender Differences in Achievement in an Inquiry-Based Learning Precalculus Course

The authors have completed a 2-year quasi-experimental study on the use of inquiry-based learning (IBL) in precalculus. This study included six traditional lecture-style courses and seven modified Moore method courses taught by three instructors. Both quantitative and qualitative analyses were used to investigate the attitudes and beliefs of the students in each section. In this article, we provide a summary of those results with a particular focus on the Openness to IBL survey we developed for assessing student preferences for either student-centered or teacher-centered instruction. Although the majority of the students exhibited a strong preference for direct instruction, students whose scores favored student-centered approaches according to the survey also tended to perform better on the final exam, indicating that a key element to success in an IBL course may be students’ perceptions of the method.

Assessing Student Openness to Inquiry-Based Learning in Precalculus

In this paper, we study the numbers Dn, k which are defined as the number of permu- tations σ of the symmetric group Sn such that σ has no cycles of length j for j ≤ k. In the case k = 1, Dn, 1 is simply the number of derangements of an n-element set. As such, we shall call the numbers Dn, k generalized derangement numbers. Garsia and Remmel (4) defined some nat- ural q-analogues of Dn, 1, denoted by Dn, 1(q), which give rise to natural q-analogues of the two classical recursions of the number of derangements. The method of Garsia and Remmel can be easily extended to give natural p, q-analogues Dn, 1(p, q) which satisfy natural p, q-analogues of the two classical recursions for the number of derangements. In (4), Garsia and Remmel also suggested an approach to define q-analogues of the numbers Dn, k. In this paper, we show that their ideas can be extended to give a p, q-analogue of the generalized derangements numbers. Again there are two classical recursions for generalized derangement numbers. However, the p, q-analogues of the two classical recursions are not as straightforward when k ≥ 2.

/pdf/a-p-q-analogue-of-the-generalized-derangement-numbers-573o6w7ac1.pdf

Karen S. Briggs

Papers

A $p,q$-analogue of a Formula of Frobenius

Combinatorial Proofs of Various q-Pell Identities via Tilings

Gender Differences in Achievement in an Inquiry-Based Learning Precalculus Course

Assessing Student Openness to Inquiry-Based Learning in Precalculus

A p, q-Analogue of the Generalized Derangement Numbers