Toward a set of design principles for mathematics flipped classrooms: A synthesis of research in mathematics education

On the order of magnitude of the difference between consecutive prime numbers

The present study reviewed the papers in the Scopus database relevant to flipped classrooms in all mathematics subject areas published up until 2018 from various dimensions. It was found th...

Research focuses and findings of flipping mathematics classes: a review of journal publications based on the technology-enhanced learning model

We study the action of a real reductive group G on a real submanifold X of a K"ahler manifold Z. We suppose that the action of G extends holomorphically to an action of a complex reductive group and is Hamiltonian with respect to a compatible maximal compact subgroup of the complex reductive group. There is a corresponding gradient map obtained from a Cartan decomposition of G. We obtain a Morse like function on X. Associated to its critical points are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds of X which are called pre-strata. In case that the gradient map is proper, the pre-strata form a decomposition of X and in case that X is compact they are the strata of a Morse type stratification of X. Our results are generalizations of results of Kirwan obtained in the case that X=Z is compact and the group itself is complex reductive.

/pdf/stratifications-with-respect-to-actions-of-real-reductive-2qodu9yeu4.pdf

Stratifications with respect to actions of real reductive groups

Mathematics teachers’ views about the limited utility of real analysis: A transport model hypothesis

We introduce a family of magic squares, called linear magic squares, and show that any parallel linear sudoku solution of sufficiently large order can be relabeled so that all of its subsquares are linear magic. As a consequence, we show that if n has prime factoriza- tion p k 1 1 p kt t and qD minf p k j j j 1 j tg, then there is a family of q.q 1/ mutually orthogonal magic sudoku solutions of order n 2 whenever q > 3; such an orthogonal family is complete if n is a prime power.

/pdf/magic-squares-and-sudoku-5dcpz5cd0f.pdf

Magic Squares and Sudoku

Mathematics for Secondary School Teachers

A symmetry group for Sudoku is complete if its action partitions the set of Sudoku boards into all possible orbits, and minimal if no group of smaller size would do the same. Previously, for a 4 x 4 Sudoku variation known as Shidoku, the authors used an analogous symmetry group to partition the set of Shidoku boards into so-called "nests" and then use the interplay between the physical and relabeling symmetries to find certain subgroups that were both complete and minimal. In this paper these same techniques are applied to find a minimal complete symmetry group for the modular magic Sudoku variation, as well as for another Sudoku variation called semi-magic Sudoku. The paper concludes with a simple computation which leads to the non-obvious fact that the full Sudoku symmetry group is, in fact, already minimal and complete.

/pdf/nest-graphs-and-minimal-complete-symmetry-groups-for-magic-3vro4wqmld.pdf

Nest graphs and minimal complete symmetry groups for magic Sudoku variants

While prime numbers are the fundamental building blocks of the integers, understanding how they are spread within the integers has turned out to be hard work. For example, the Prime Number Theorem stood as a conjecture for nearly a hundred years, and anyone who bags the Riemann Hypothesis first will be a million dollars richer. Modern cryptography assumes that the primes will retain their secrets for some time to come. In the presence of hard problems, it is tempting to employ models. A good model should provide an approximation of reality which is simple enough to understand, yet accurate enough to be useful. While there are several models for the primes, in this paper we tell the story of a beautiful and compelling probabilistic model known as the Hawkins primes. First introduced by David Hawkins in this MAGAZINE [11], the model is based on a simple variation of the sieve of Eratosthenes. Over the past fifty years, the Hawkins model has been used to predict the truth, in the strongest probabilistic sense, of results (both established and conjectured) concerning the distribution of the prime numbers, including the Twin Primes Conjecture and the Riemann Hypothesis. Also, the model (or generalizations thereof) has potential to shed light on interesting integer sequences other than the primes.

http://www.cs.bsu.edu/homepages/jdlorch/mathmag116-123-lorch.pdf

Primes and Probability: The Hawkins Random Sieve

We introduce a transversal method for generating orthogonal mates of sudoku solutions. We show that the method produces an orthogonal mate for every linear sudoku solution of parallel type, and we characterize the linear sudoku solutions of non-parallel type for which the method works.

/pdf/orthogonal-combings-of-linear-sudoku-solutions-1t86m6u7yc.pdf

Orthogonal combings of linear sudoku solutions.

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