Anna Stokke

Lattice path constructions for orthosymplectic determinantal formulas

Abstract We give lattice path proofs of determinantal formulas for orthosymplectic characters. We use the s p o ( 2 m , n ) -tableaux introduced by Benkart, Shader and Ram, which have both a semistandard symplectic part and a row-strict part. We obtain orthosymplectic Jacobi–Trudi identities and an orthosymplectic Giambelli identity by associating s p o ( 2 m , n ) -tableaux to certain families of nonintersecting lattice paths and using an adaptation of the Gessel–Viennot method.

What to Do About Canada's Declining Math Scores?

The declining performance of Canadian students on international math assessments should worry Canadians and their provincial governments. Strong mathematics knowledge is required for success in the workforce, and early achievement in math is one of the best predictors of later academic success and future career options. Between 2003 and 2012, all but two Canadian provinces showed statistically significant declines in math scores on international exams administered by the Organization for Economic Cooperation and Development. In several provinces, the percentage of students performing at the lowest levels in math significantly increased while the percentage of students performing at the highest levels significantly decreased, suggesting that more students are struggling and fewer students are excelling in math. It should be a policy priority to halt these trends and to improve math achievement for Canadian children. In this Commentary, I examine domestic and international evidence regarding three areas of provincial education programs that could play an important role in halting the downward trend in math scores. I make three main recommendations regarding best teaching practices in math, the math curriculum, and the math knowledge of future teachers. Best teaching practices in math have been at the forefront of discussions regarding declining math scores in Canada. Discovery-based instruction – also called problem-based, inquiry, experiential, and constructivist learning – has become popular in North America in recent years, pushing aside direct instruction techniques, like times table memorization, explicit teacher instruction, pencil-and-paper practice, and mastery of standard mathematical procedures. Based on international and domestic evidence, this Commentary finds that studies consistently show direct instruction is much more effective than discovery-based instruction, which leads to straightforward recommendations on how to tilt the balance toward best instructional techniques. Student fluency with particular math concepts, such as fraction arithmetic, in early and middle years has been shown to predict future math success. This Commentary recommends that provincial math curricula be rewritten to remove ineffective pedagogical directives and to stress specific topics, at appropriate grade levels, that are known to lead to later success in math. Evidence shows that teachers who are most comfortable and knowledgeable with the content they are required to teach tend to transmit that knowledge best to students. This Commentary suggests that future early and middle-years teachers be required to pass a math-content licensure exam prior to receiving certification to teach mathematics.

Jeu-de-taquin promotion and a cyclic sieving phenomenon for semistandard hook tableaux

Jeu-de-taquin promotion yields a bijection on the set of semistandard @l-tableaux with entries bounded by k. In this note, we determine the order of jeu-de-taquin promotion on the set of semistandard hook tableaux CST((n-m,1^m),k), with entries bounded by k, and on the set of semistandard hook tableaux with fixed content @a, CST((n-m,1^m),k,@a). We give a bijection between CST((n-m,1^m),k,@a) and a suitable set of standard hook tableaux that behaves nicely with respect to jeu-de-taquin promotion and use the bijection to give a cyclic sieving phenomenon for CST((n-m,1^m),k,@a).

Hook-content Formulae for Symplectic and Orthogonal Tableaux

By considering the specialisation

Determining irreducible

s_{\lambda}(1,q,q^2,...,q^{n-1})

GL(n,K)

of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape

-modules

\lambda

A Quantum Version of the Désarménien Matrix

in terms of two properties of the boxes in the diagram for

Increasing Tableaux, Narayana Numbers and an Instance of the Cyclic Sieving Phenomenon

\lambda

Codeterminants and q-Schur Algebras

. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal