What are possible Sources of Common and Persistent Student Errors in Algebra and Calculus?

Research shows that college students make numerous algebra and other prerequisite content-related errors in Calculus courses. Most of these errors are common, persistent, and often observed in simple mathematical tasks. This qualitative study is an attempt to identify the potential sources of such errors. Based on our observations of student errors, we wrote a Precalculus and a Calculus test and administered them in twelve sections of four different undergraduate mathematics courses for which either Precalculus, Calculus I or both were a prerequisite. The tests were announced on the first day of the class and administered the following week. All the questions on the test were True or False questions. Based on our experience as college mathematics instructors, we assumed that many students would perceive the True answers as False and the False as True. Therefore, if students’ selected a given answer, mathematical statement, process or solution as True, they were asked to justify why that was not False and vice-versa. They were instructed to provide logical explanations and avoid plugging in numbers to check for correctness. Analysis of data using grounded theory approach resulted in the following three possible external sources of common and persistent student errors: a) Difficulty with symbols and/or lack of attendance to the meaning of those symbols, b) Instructional practices, and c) Lack of knowledge. We will provide examples to illustrate how such errors could have originated from these sources.

Action Proof: Analyzing Elementary School Students Informal Proving Stages through Counter-examples

Both female and male elementary school students have difficulty doing action proof by using manipulative objects to provide conjectures and proof of the truth of a mathematical statement. Counter-examples can help elementary school students build informal proof stages to propose conjectures and proof of the truth of a mathematical statement more precisely. This study analyzes the action proof stages through counter-examples stimulation for male and female students in elementary schools. The action proof stage in this study focuses on three stages: proved their primitive conjecture, confronted counter-examples, and re-examined the conjecture and proof. The type of research used is qualitative with a case study approach. The research subjects were two of the 40 fifth-grade students selected purposively. The research instrument used is the task of proof and interview guidelines. Data collection techniques consist of Tasks, documentation, and interviews. The data analysis technique consists of three stages: data reduction, data presentation, and concluding. The analysis results show that at the stage of proving their primitive conjecture, the conjectures made by female and male students through action proofs using manipulative objects are still wrong. At the stage of confronted counter-examples, conjectures and proof made by female and male students showed an improvement. At the stage of re-examining the conjecture and proof, the conjectures and proof by female and male students were comprehensive. It can be concluded that the stages of proof of the actions of female and male students using manipulative objects through stimulation counter-examples indicate an improvement in conjectures and more comprehensive proof.

PENGARUH PEMAHAMAN WRITING MATHEMATICS MAHASISWA DALAM MENGKOMUNIKASIKAN SUATU BUKTI PERNYATAAN DALAM MATEMATIKA

The ability of writing and communicating Mathematics text are needed to get the whole understanding of specific topic in Mathematics. But so far, it is still can be found some meaningless statement that implies some misconceptions. The qualitative data analysis is conducted in this research to find out how big the change of the writing mathematics skill after conducting a technical guidance. The result of data analysis and some evaluation show that the skill of understanding the concept of writing mathematics takes effect in communicating or creating a proof of mathematical statement.

Exploration of Student Thinking Process in Proving Mathematical Statements

A mathematical statement is not a theorem until it has been carefully derived from previously proven axioms, definitions and theorems. The proof of a theorem is a logical argument that is given deductively and is often interpreted as a justification for statements as well as a fundamental part of the mathematical thinking process. Studying the proof can help decide if and why our answers are logical, develop the habit of arguing, and make investigating an integral part of any problem solving. However, not a few students have difficulty learning it. So it is necessary to explore the student's thought process in proving a statement through questions, answer sheets, and interviews. The ability to prove is explored through 4 (four) proof schemes, namely Scheme of Complete Proof, Scheme of Incomplete Proof, Scheme of unrelated proof, and Scheme of Proof is immature. The results obtained indicate that the ability to prove is influenced by understanding and the ability to see that new theorems are built on previous definitions, properties and theorems; and how to present proof and how students engage with proof. Suggestions in this research are to change the way proof is presented, and to change the way students are involved in proof; improve understanding through routine proving new mathematical statements; and developing course designs that can turn proving activities into routine activities.

Performance Assessment to Measure Student’s Mathematical Proving Ability Based on the Abductive-Deductive Approach

This study provides an overview of performance assessment instruments to measure students’ mathematical proving abilities based on the abductive-deductive approach. This is descriptive qualitative research of performance assessment instruments and their rubrics in measuring students’ mathematical proving abilities. The research method was literature study. The performance assessment instrument consists of essays designed to identify the mathematical proving abilities of students in mathematics courses. In this article, the examples are given for the Real Analysis class. The items of performance assessment were arranged referring to the abductive-deductive reasoning approach which has a pattern containing three main questions: 1) “What conditions can be obtained from the conclusion?” which was answered with abductive reasoning, 2) “What are the consequences that can be obtained from known facts?” which can be answered with deductive reasoning, and 3) “What conditions connect the conditions of conclusions and the implications of premise?” which can be answered with a key process of the mathematical statement proving process. Keywords: performance assessment, mathematical proving ability, abductivedeductive approach

The Computer Model and the Algorithm of Designing a Target Environment of Remote Monitoring Areas Using Thermal Tomograms with Account of their Geographical Location and Meteorological Conditions

The article considers a computer model and an algorithm used for designing a target environment based on extrapolation of temperature fields of remote monitoring areas according to databases of thermal tomograms of analog objects, underlying surfaces, geographical location and meteorological conditions. A mathematical statement of the problem, concerning simulating the temperature fields of “analogue objects” and bottoming surfaces of remote monitoring areas by means of their thermal tomograms, is described. The outcomes of computer simulation of the spatial distribution of temperature fields of the remote monitoring area based on the results of the field experiment are presented

Method of Proofs in Mathematics: Higher Level Students' Understanding

The objectives of this study are to determine students’ understanding about different methods of proof in mathematics and compare their understanding with respect to their previous degree in education and other majors within the framework of a post-positivist research paradigm. Forty-two purposively selected students in the sample were selected. Their understanding of different methods of proof was determined with the help of a test paper consisting of 30 mathematical statements with proof. The study concluded that students are familiar with the methods of proofs used in verifying a mathematical statement. In addition, it is found that students with a previous degree in education and other majors differ in the level of understanding of different methods of proof in mathematics.

CARA BERPIKIR DALAM MENYELESAIKAN SOAL-SOAL SELANG, KETAKSAMAAN DAN NILAI MUTLAK

Inequality is a mathematical statement that contains one of the sequence relations <,>, ≤, ≥. This sequence relation has been known by students since they were in elementary school, junior high school, senior high school and even went to university. In Higher Education material lapse, inequality and absolute grades are found in Differential Calculus courses and become one of the prerequisites for other courses namely Differential Calculus. Based on our observations at the Mathematics Education Study Program FKIP Palembang PGRI University, the understanding of solving interval questions, inequality and absolute values cannot be understood well by students, which we conclude that students still feel confused and make many mistakes in solving problems a matter of lapse, inequality and absolute value. In this description, we present how to think in solving intervals, inaccuracies and absolute values so that students can solve these problems.Keywords: how to think with intervals, inequality, absolute value. AbstrakKetaksamaan adalah pernyataan matematik yang memuat salah satu relasi urutan <, >, . Relasi urutan tersebut sudah dikenal siswa sejak duduk di bangku Sekolah Dasar, Sekolah Menengah Pertama, Sekolah Menengah Atas bahkan sampai ke Perguruan Tinggi. Di Perguruan Tinggi materi selang, ketaksamaan dan nilai mutlak terdapat pada mata kuliah Kalkulus Diferensial dan menjadi salah satu prasyarat bagi mata kuliah yang lain yaitu Kalkulus Diferensial. Berdasarkan pengamatan kami di Program Studi Pendidikan Matematika FKIP Universitas PGRI Palembang, pemahaman tentang penyelesaian soal-soal selang, ketaksamaan dan nilai mutlak belum dapat dipahami dengan baik oleh mahasiswa, yang mana kami berkesimpulan bahwa mahasiswa masih merasa bingung dan banyak melakukan kesalahan dalam menyelesaikan soal-soal tentang selang, ketaksamaan dan nilai mutlak. Dalam uraian ini kami kemukakan bagaimana cara berpikir dalam menyelesaikan soal-soal selang, ketaksamaan dan nilai mutlak sehingga mahasiswa bisa menyelsaikan persoalan tersebut.Kata Kunci : cara berpikir dengan selang, ketaksamaan, nilai mutlak.

CHARACTERISTICS OF MATHEMATICAL REASONING IN SOLVING MATHEMATICAL PROBLEMS IN TERMS OF THE COGNITIVE STYLE OF THE ELEVENTH GRADE STUDENTS OF SMA NEGERI 3 WAJO

This research aims at describing characteristics of students mathematical reasoning in solving mathematical problem regarded from cognitive style. Based on data analysis, it can be concluded that (1) The ways of the subject reasoning of visualizer cognitive style are analyzing, synthesizing, analyzing, then generalizing. While the ways of the subject reasoning who are verbalizer cognitive are synthesizing, analyzing, then generalizing. (2) The sub indicators of subject reasoning of visualizer cognitive style are presenting mathematical statement through picture specifically and obviously, conducting a beginning asumption specifically, explaining the concept relevancy used through picture, giving reason logically in solving problem and using other alternatives with different strategies, rechecking every steps of completion, tending to conclude the result of completion with picture and tending to conclude generally based on the specific case from mathematical symptom. While the subjects who are verbalizer cognitive are presenting mathematical statement through verbal symbol specifically and obviously, less specific in doing assumption, relating the concept but ordering based on the sequences known and asked with the writing of symbol in modelling the problem, giving the relevant argument to the steps of completion with the same strategies, rechecking every steps of completion, tending to conclude the result of completion with verbal symbol, and tending to conclude generally based on the specific case and mathematical symptom.