Weng Kin Ho

Bio: Weng Kin Ho is an academic researcher from National Institute of Education. The author has contributed to research in topics: Domain theory & Operational semantics. The author has an hindex of 9, co-authored 48 publications receiving 287 citations. Previous affiliations of Weng Kin Ho include University of Birmingham & Nanyang Technological University.

Concrete-Pictorial-Abstract: Surveying its origins and charting its future

Abstract: The Concrete-Pictorial-Abstract (CPA) approach, based on Bruner's conception of the enactive, iconic and symbolic modes of representation, is a well-known instructional heuristic advocated by the Singapore Ministry of Education since early 1980's. Despite its ubiquity as a teaching strategy throughout the entire mathematics education community in Singapore, it is somewhat surprising to see a lack of an account of its theoretical roots. This paper is an attempt to contribute to this discussion on the CPA strategy and its potential role in continuing advancement of quality mathematics education.

A Category Theoretical Argument against the Possibility of Artificial Life: Robert Rosen's Central Proof Revisited

Abstract: One of Robert Rosen's main contributions to the scientific community is summarized in his book Life itself. There Rosen presents a theoretical framework to define living systems; given this definition, he goes on to show that living systems are not realizable in computational universes. Despite being well known and often cited, Rosen's central proof has so far not been evaluated by the scientific community. In this article we review the essence of Rosen's ideas leading up to his rejection of the possibility of real artificial life in silico. We also evaluate his arguments and point out that some of Rosen's central notions are ill defined. The conclusion of this article is that Rosen's central proof is wrong.

Lattices of Scott-closed sets

Abstract: A dcpo $P$ is continuous if and only if the lattice $C(P)$ of all Scott-closed subsets of $P$ is completely distributive. However, in the case where $P$ is a non-continuous dcpo, little is known about the order structure of $C(P)$. In this paper, we study the order-theoretic properties of $C(P)$ for general dcpo's $P$. The main results are: (i) every $C(P)$ is C-continuous; (ii) a complete lattice $L$ is isomorphic to $C(P)$ for a complete semilattice $P$ if and only if $L$ is weak-stably C-algebraic; (iii) for any two complete semilattices $P$ and $Q$, $P$ and $Q$ are isomorphic if and only if $C(P)$ and $C(Q)$ are isomorphic. In addition, we extend the function $P\mapsto C(P)$ to a left adjoint functor from the category {\bf DCPO} of dcpo's to the category {\bf CPAlg} of C-prealgebraic lattices.

The Ho Zhao Problem

Abstract: Given a poset $P$, the set, $\Gamma(P)$, of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory $\mathbf{C}$ of $\mathbf{Pos}_d$ (the category of posets and Scott-continuous maps) is said to be $\Gamma$-faithful if for any posets $P$ and $Q$ in $\mathbf{C}$, $\Gamma(P) \cong \Gamma(Q)$ implies $P \cong Q$. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are $\Gamma$-faithful, while $\mathbf{Pos}_d$ is not. Ho & Zhao (2009) asked whether the category $\mathbf{DCPO}$ of dcpos is $\Gamma$-faithful. In this paper, we answer this question in the negative by exhibiting a counterexample. To achieve this, we introduce a new subcategory of dcpos which is $\Gamma$-faithful. This subcategory subsumes all currently known $\Gamma$-faithful subcategories. With this new concept in mind, we construct the desired counterexample which relies heavily on Johnstone's famous dcpo which is not sober in its Scott topology.

Computational Thinking 計算論的思考

Abstract: It represents a universally applicable attitude and skill set everyone, not just computer scientists, would be eager to learn and use.

Learning mathematics in a classroom community of inquiry: examples from a secondary mathematics classroom

Abstract: This article considers the question of what specific actions a teacher might take to create a culture of inquiry in a secondary school mathematics classroom. Sociocultural theories of learning provide the framework for examining teaching and learning practices in a single classroom over a two-year period. The notion of the zone of proximal development (ZPD) is invoked as a fundamental framework for explaining learning as increasing participation in a community of practice characterized by mathematical inquiry. The analysis draws on classroom observation and interviews with students and the teacher to show how the teacher established norms and practices that emphasized mathematical sense-making and justification of ideas and arguments and to illustrate the learning practices that students developed in response to these expectations.