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Students' feedback on teaching mathematics through the calculational method
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This paper describes a study conducted at the University of Nottingham, whose goal was to assess whether the students registered on the first-year module ¿Mathematics for Computer Scientists¿ appreciate the calculational method. The study consisted of two parts: ¿Proof Reading¿ and ¿Problem Solving¿. The goal of ¿Proof Reading¿ was to determine what the students think of calculational proofs, compared with more conventional ones, and which are easier to verify; we also assessed how their opinions changed during the term. The purpose of ¿Problem Solving¿ was to determine if the methods taught have influenced the students' problem-solving skills. Frequent criticisms of our approach are that we are too formal and that the emphasis on syntactic manipulation hinders students' understanding. Nevertheless, the results show that most students prefer or understand better the calculational proofs. On the other hand, regarding the problem-solving questions, we observed that, in general, the students maintained their original solutions.
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... In [11], an experiment is presented where students specify algorithmic problems in Alloy [17] and reason about problems in an algebraic and calculational way. It has been argued that students seem to prefer and understand better calculational proofs [9]. Calculational proofs are commonly used in the functional programming community to demonstrate algorithm correctness [4,16]. ...
This paper shows by examples how the Theory of Programming can be taught to first-year CS undergraduates. The only prerequisite is their High School acquaintance with algebra, geometry, and propositional calculus. The main purpose of teaching the subject is to support practical programming assignments and projects throughout the degree course. The aims would be to increase the student’s enjoyment of programming, reduce the workload, and increase the prospect of success.KeywordsAlgebraLogicGeometryTeaching formal methodsUnifying theories of programming
... In [11], an experiment is presented where students specify algorithmic problems in Alloy [17] and reason about problems in an algebraic and calculational way. It has been argued that students seem to prefer and understand better calculational proofs [9]. Calculational proofs are commonly used in the functional programming community to demonstrate algorithm correctness [4,16]. ...
This paper shows by examples how the Theory of Programming can be taught to first-year CS undergraduates. The only prerequisite is their High School acquaintance with algebra, geometry, and propositional calculus. The main purpose of teaching the subject is to support practical programming assignments and projects throughout the degree course. The aims would be to increase the student's enjoyment of programming, reduce the workload, and increase the prospect of success.
... The system should support learning, teaching, and research on calculational methods for correct-byconstruction program design. It has already been argued that calculational proofs offer some pedagogic advantages over conventional informal proofs [12,13,15] and that students prefer or understand better calculational proofs [10]. Moreover, a structure editor can assist students and teachers in learning and explaining how certain rules are applied [23]. ...
Despite great advances in computer-assisted proof systems, writing formal proofs using a traditional computer is still challenging due to mouse-and-keyboard interaction. This leads to scientists often resorting to pen and paper to write their proofs. However, when handwriting a proof, there is no formal guarantee that the proof is correct. In this paper we address this issue and present the initial steps towards a system that allows users to handwrite proofs using a pen-based device and that communicates with an external theorem prover to support the users throughout the proof writing process. We focus on calculational proofs, whereby a theorem is proved by a chain of formulae, each transformed in some way into the next. We present the implementation of a proof-of-concept prototype that can formally verify handwritten calculational proofs without the need to learn the specific syntax of theorem provers.
... This paper is part of an endeavour which aims at reinvigorating mathematical content by adopting a calculational style of reasoning [6,19,14,20,21]. As suggested by the results shown in [22], the calculational method can indeed have a positive impact on mathematics education. However, in our view, the combination of practicality with mathematical elegance that arises from an adequate focus on calculational techniques can enrich and improve, not only mathematics education, but also the process of constructing computer programs. ...
This paper proposes a calculational approach to prove properties of two well-known binary trees used to enumerate the rational numbers: the Stern–Brocot tree and the Eisenstein–Stern tree (also known as Calkin–Wilf tree). The calculational style of reasoning is enabled by a matrix formulation that is well-suited to naturally formulate path-based properties, since it provides a natural way to refer to paths in the trees.
Three new properties are presented. First, we show that nodes with palindromic paths contain the same rational in both the Stern–Brocot and Eisenstein–Stern trees. Second, we show how certain numerators and denominators in these trees can be written as the sum of two squares x2 and y2, with the rational xy appearing in specific paths. Finally, we show how we can construct Sierpiński's triangle from these trees of rationals.
... After many failed attempts there exists a procedure in a student's mind that produces the erroneous answers (Burton & Brown, 1979;Tu, Hsu, & Wu, 2002). Especially when a student is trying to learn a new math subject, because students have difficulties in employing mathematics to solve new problems (Ferreira & Mendes, 2009). Currently, most math educational software do not have the ability to give any effective assistance at the points where student makes an error. ...
ABSTRACT – Not having the proper help to pinpointing where the mistake occurred in solving math word problems can be frustrating for many students. In this paper we propose an interactive user input sensitive feedback model that assists students solving math word problems based on their own thinking pattern. This model uses a framework proposed by and fuzzy logic as a method to evaluate a user’s input to determine if a input for each step towards a solution is correct and if that student needs help correcting their mistakes. This model allows students to solve a math word problem by using a free form, and the immediate feedback at each step will be provided directly based on the student’s input, and customized towards to student’s thinking habit. This approach will be much effective in assisting building up students’ confidence in solving math word problem and stimulates learning.
... Furthermore, illustrating how logic can be used to model and solve algorithmic problems, improves the students' abilities to solve problems in general. Related research, leading to similar conclusions, is reported in [6, 1, 2, 10]. ...
Although much of mathematics is algorithmic in nature, the skills needed to formulate and solve algorithmic problems do not form an integral part of mathematics education. In particular, logic, which is central to algorithm development, is rarely taught explicitly at preuniversity level, under the justification that it is implicit in mathematics and therefore does not need to be taught as an independent topic. This paper argues in the opposite direction, describing a one-week workshop done at the University of Minho, in Portugal, whose goal was to introduce to high-school students calculational principles and techniques of algorithmic problem solving supported by calculational logic. The workshop resorted to recreational problems to convey the principles and to software tools, the Alloy
Analyzer and Netlogo, to animate models.
... (Some of the difficulties involved in the assessment are pointed out by Herbert Wilf in his essay [31].) Nevertheless, the novel results mentioned above and preliminary results on the didactical suitability of the calculation format obtained within the group (see [18]) encourage us to continue our efforts. Also, the success claimed by related work like [2] and [26], makes us believe that we can have a positive impact. ...
MathIS is a new project that aims to reinvigorate secondary- school mathematics by exploiting insights of the dynamics of algorithmic problem solving. This paper describes the main ideas that underpin the project. In summary, we propose a central role for formal logic, the de- velopment of a calculational style of reasoning, the emphasis on the algo- rithmic nature of mathematics, and the promotion of self-discoveryby the students. These ideas are discussed and the case is made, through a num- ber of examples that show the teaching style that we want to introduce, for their relevance in shaping mathematics training for the years to come. In our opinion, the education of software engineers that work effectively with formal methods and mathematical abstractions should start before university and would benefit from the ideas discussed here.
Algorithmic problem solving is a way of approaching and solving problems by using the advances that have been made in the principles of correct-by-construction algorithm design. The approach has been taught at first-year undergraduate level since September 2003 and, since then, a substantial amount of learning materials have been developed. However, the existing materials are distributed in a conventional and static way (e.g. as a textbook and as several documents in PDF format available online), not leveraging the capabilities provided by modern collaborative and open-source platforms.
We present a structure editor that aims to facilitate the presentation and manipulation of handwritten mathematical expressions. The editor is oriented to the calculational mathematics involved in algorithmic problem solving and it provides features that allow reliable structure manipulation of mathematical formulae, as well as flexible and interactive presentations. We describe some of its most important features, including the use of gestures to manipulate algebraic formulae, the structured selection of expressions, definition and redefinition of operators in runtime, gesture's editor, and handwritten templates. The editor is made available in the form of a C# class library which can be easily used to extend existing tools. For example, we have extended Classroom Presenter, a tool for ink-based teaching presentations and classroom interaction. We have tested and evaluated the editor with target users. The results obtained seem to indicate that the software is usable, suitable for its purpose and a valuable contribution to teaching and learning algorithmic problem solving.
Editorial for special edition of Sciece of Computer Programming.
Presents answers to the question "what should constitute the mathematics of program constructionand what goals should scientists endeavouring to develop it into a recognised field have".
This project aims to develop a pen-based software tool that will assist in the process of doing mathematics by providing structured manipulation of handwritten mathematical expressions. The tool will be used to support the teaching of the dynamics of problem solving in a way that combines the advantages of the traditional blackboard style of teaching with the flexibility and accuracy of computer software. It will provide not only a simpler way to input mathematics - by allowing the recognition of handwritten mathematics - but also enhance studentspsila understanding of the calculational techniques and facilitate the process of doing mathematics - by providing structure editing. Some of the most important features of this tool are the accurate selection and copy of expressions, the automatic application of algebraic rules and the use of gestures to apply them, and also the combined writing of mathematics and text. These features will have a major impact on writing, doing, and presenting mathematics. This project includes the required technical developments and also the application and testing of the tool in concrete situations, namely in mathematics and computing science courses.
Structured Derivations: a Logic Based Approach to Teaching Mathematics" In FORMED 2008: Formal Methods in Computer Science Education
- Back
- Ralph
- Linda Johan
- Patrick Mannila
- Mia Sibelius
- Petomälki
Back, Ralph-Johan, Linda Mannila, Patrick Sibelius and Mia Petomälki "Structured Derivations: a Logic Based Approach to Teaching Mathematics" In FORMED 2008: Formal Methods in Computer Science Education. 29 March 2008.
Xeqmat -Matemática 12 o
- Yolanda Lima
- Francelino Gomes
Lima, Yolanda and Francelino Gomes, "Xeqmat -Matemática 12
o ",
Editorial O Livro, 1999, p. 383
Structured Derivations: a Logic Based Approach to Teaching Mathematics
- Linda Back Ralph-Johan
- Patrick Mannila
- Mia Sibelius
- Petomälki