scholarly journals Language Constraints in Constructing Arguments for Mathematical Proofs

2021 ◽  
Vol 2123 (1) ◽  
pp. 012046
Author(s):  
I Minggi ◽  
Alimuddin ◽  
Sabri
Keyword(s):  
Mathematical Proof ◽  
Learning Trajectory ◽  
Mathematical Proofs ◽  
Mathematical Arguments ◽  
Step Procedure

Abstract A learning trajectory for constructing mathematical proof has been developed. The trajectory is to provide the students with a step-by-step procedure in constructing arguments for proving mathematical statements. However, in proving activities, the students were found to encounter difficulties in completing a deductive axiomatic argument constituting an accepted mathematical proof. An investigation has been conducted to explore the problems the students experienced in constructing proofs. It was found that they faced language constraints in constructing mathematical arguments. They encountered challenges in how to correctly express the mathematical statements in their constructed proofs.

Graphical rules of steady-state reaction systems

1981 ◽  
Vol 59 (4) ◽  
pp. 737-755 ◽  
Author(s):  
Chou Kuo-Chen ◽  
Sture Forsen
Keyword(s):  
Steady State ◽  
Rate Constants ◽  
Mathematical Proof ◽  
Complex Reaction ◽  
Consecutive Reaction ◽  
Reaction Systems ◽  
First Order ◽  
Mathematical Proofs ◽  
Pseudo First Order ◽  
Manual Calculation

Four rules to deal with first-order or pseudo-first-order steady-state reaction systems are presented.By means of Rule 1, we can immediately write down the apparent rate constants of consecutive reaction systems. This rule is actually the same as the "Rule of Thumb" proposed by Gilbert, but here its mathematical proof is given.Rule 2 and Rule 3 may serve to derive the apparent rate constants of various complex reaction systems. In comparison with the general algebraic methods, these two rules can simplify laborious calculations that would otherwise be tedious and liable to errors.Rule 4 presents a new schematic method to calculate the concentrations of the reactants. The new method, in simplifying the calculation of complex problems, is extraordinarily efficacious in comparison with the existing schematic methods. For complex mechanisms which are too complicated to be treated with the general manual calculation method, the practical calculations show that we can easily write down the desired results by means of Rule 4.In addition, Rules 2, 3, and 4 include corresponding check formulae, by use of which we can avoid missing subgraphs to be counted. Their advantages will be manifested particularly in dealing with complex mechanisms.The mathematical proofs of these rules are given in the Appendices.

The Role of Programmable Calculators and Computers in Mathematical Proofs

1978 ◽  
Vol 71 (9) ◽  
pp. 745-750
Author(s):  
Stephen L. Snover ◽  
Mark A. Spikell
Keyword(s):  
Mathematical Proof ◽  
Mathematical Proofs

Data generated by computing devices may he used as an essential part of a mathematical proof.

MATHEMATICAL RIGOR AND PROOF

2019 ◽  
pp. 1-41 ◽  
Author(s):  
YACIN HAMAMI
Keyword(s):  
Mathematical Knowledge ◽  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Formal Proof ◽  
Major Obstacle ◽  
Standard View ◽  
Precise Formulation ◽  
Mathematical Proofs ◽  
Primary Form

Abstract Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it.

scholarly journals Learning Trajectory in Mathematical Proof

2021 ◽  
Vol 1752 (1) ◽  
pp. 012081
Author(s):  
I Minggi ◽  
U Mulabar ◽  
S F Assagaf
Keyword(s):  
Mathematical Proof ◽  
Learning Trajectory

scholarly journals Application of Discovery Learning Method in Mathematical Proof of Students in Trigonometry

2020 ◽  
Vol 3 (1) ◽  
pp. 73-82
Author(s):  
Windia Hadi ◽  
Ayu Faradillah
Keyword(s):  
Mathematical Proof ◽  
Discovery Learning ◽  
Average Score ◽  
Learning Method ◽  
Learning Methods ◽  
Test Analysis ◽  
Ability Test ◽  
Mathematical Proofs ◽  
Quasi Experimental ◽  
Discovery Method

Trigonometry is a part of mathematics in learning that is related to angles. The purpose of this study was to determine the effect of the application of discovery learning methods in students' mathematical proof ability on trigonometry. The research method used in this study is quasi-experimental. The population in this study was the second semester of 2016/2017. The sample was 66 people who were determined by purposive sampling. The instrument used in this study was a mathematically proof ability test. Analysis of the data used is the t-test. The results of this study are (1) based on an average score of mathematical proof ability The student's mathematical proof ability in applying the Discovery Learning Method to trigonometry is not higher than the mathematical proof ability of students who do not use the discovery method (2) there is no significant influence in the application of learning methods Discovery in students' mathematical proofs on trigonometry.

scholarly journals MATHEMATICAL PROOF: THE LEARNING OBSTACLES OF PRE-SERVICE MATHEMATICS TEACHERS ON TRANSFORMATION GEOMETRY

2019 ◽  
Vol 10 (1) ◽  
pp. 117-126
Author(s):  
Muchamad Subali Noto ◽  
Nanang Priatna ◽  
Jarnawi Afgani Dahlan
Keyword(s):  
Qualitative Research ◽  
Mathematics Teachers ◽  
Mathematical Proof ◽  
Research Data ◽  
Mathematical Notation ◽  
Knowing How ◽  
Mathematical Proofs ◽  
Transformation Geometry ◽  
High Level

Several studies related to mathematical proof have been done by many researchers on high-level materials, but not yet examined on the material of transformation geometry. The aim of this research is identification learning obstacles pre-service teachers on transformation geometry. This study is qualitative research; data were collected from interview sheets and test. There were four problems given to 9 pre-service mathematics teachers. The results of this research were as follows: learning obstacles related to the difficulty in applying the concept; related to visualize the geometry object; related to obstacles in determining principle; related to understanding the problem and related obstacles in mathematical proofs such as not understanding and unable to express a definition, not knowing to use the definition to construct the proof, not understanding the use of language and mathematical notation, not knowing how to start the proof.

Using Moment-by-Moment Reading Protocols to Understand Students’ Processes of Reading Mathematical Proof

2021 ◽  
Vol 52 (5) ◽  
pp. 510-538
Author(s):  
Paul Christian Dawkins ◽  
Dov Zazkis
Keyword(s):  
Undergraduate Students ◽  
Mathematical Proof ◽  
Think Aloud ◽  
Reading Behaviors ◽  
Mathematical Proofs ◽  
Think Aloud Protocols ◽  
Set Up ◽  
Novice Readers ◽  
The Way

This article documents differences between novice and experienced undergraduate students’ processes of reading mathematical proofs as revealed by moment-by-moment, think-aloud protocols. We found three key reading behaviors that describe how novices’ reading differed from that of their experienced peers: alternative task models, accrual of premises, and warranting. Alternative task models refer to the types of goals that students set up for their reading of the text, which may differ from identifying and justifying inferences. Accrual of premises refers to the way novice readers did not distinguish propositions in the theorem statement as assumptions or conclusions and thus did not use them differently for interpreting the proof. Finally, we observed variation in the type and quality of warrants, which we categorized as illustrate with examples, construct a miniproof, or state the warrant in general form.

NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

2019 ◽  
pp. 1-35
Author(s):  
WILFRIED SIEG ◽  
PATRICK WALSH
Keyword(s):  
Proof Theory ◽  
Mathematical Proof ◽  
Formal Proofs ◽  
Bernstein Theorem ◽  
Search System ◽  
Inference Mechanism ◽  
Mathematical Proofs ◽  
Computational Work ◽  
Definitional Extension

Abstract Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/. We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation. Hilbert 1918

"A Mathematical Proof Must Be Surveyable" What Wittgenstein Meant by This and What It Implies

2006 ◽  
Vol 71 (1) ◽  
pp. 57-86 ◽  
Author(s):  
Felix Mühlhölzer
Keyword(s):  
Mathematical Proof ◽  
Uniform System ◽  
Mathematical Proofs

In Part III of his Wittgenstein deals with what he calls the of proofs. By this he means that mathematical proofs can be reproduced with certainty and in the manner in which we reproduce pictures. There are remarkable similarities between Wittgenstein's view of proofs and Hilbert's, but Wittgenstein, unlike Hilbert, uses his view mainly in critical intent. He tries to undermine foundational systems in mathematics, like logicist or set theoretic ones, by stressing the unsurveyability of the proof-patterns occurring in them. Wittgenstein presents two main arguments against foundational endeavours of this sort. First, he shows that there are problems with the criteria of identity for the unsurveyable proof-patterns, and second, he points out that by making these patterns surveyable, we rely on concepts and procedures which go beyond the foundational frameworks. When we take these concepts and procedures seriously, mathematics does not appear as a uniform system, but as a mixture of different techniques.

scholarly journals Kesalahan Mahasiswa dalam Pembuktian Matematik

2019 ◽  
Vol 6 (1) ◽  
pp. 54-68
Author(s):  
Rezky Agung Herutomo
Keyword(s):  
Mathematical Proof ◽  
Mathematics Learning ◽  
Algebraic Manipulation ◽  
Mathematical Induction ◽  
Education Department ◽  
Mathematical Proofs ◽  
Proof Techniques ◽  
Reasoning Errors ◽  
And Mathematics ◽  
Academic Year

Proofs are the key component in mathematics and mathematics learning. But in reality, there are still many students who make errors when constructing mathematical proofs. Therefore this study aimed to identify common errors when the students are constructing mathematical proofs. The participant  of this study was 51 of 3rd year students of Mathematics Education Department  who enrolled in Real Analysis course in the second semester of the 2017/2018 academic year. The data of the study were obtained by conducting a test consisting of five questions and interview guidelines. The errors identified in this study were (1) proving general statements using specific examples, (2) inappropriate algebraic manipulation in mathematical induction, (3) incorrect reasoning and assumptions in proving with contradictions, and (4) reasoning errors involving natural numbers in mathematical induction. Hence, further study can be developed learning models that promote the conceptual understanding, logical reasoning, and mastery of mathematical proof techniques.

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