Analytic program derivation in type theory

1998 ◽  
Author(s):  
Petri Mäenpää
Keyword(s):  
Mathematical Method ◽  
Problem Solving ◽  
Type Theory ◽  
Mathematical Problem ◽  
Geometric Analysis ◽  
Algebraic Method ◽  
Point Of View ◽  
Program Derivation ◽  
Mathematical Problems ◽  
The Given

This work proposes a new method of deriving programs from their specifications in constructive type theory: the method of analysis-synthesis. It is new as a mathematical method only in the area of programming methodology, as it is modelled upon the most successful and widespread method in the history of exact sciences. The method of analysis-synthesis, also known as the method of analysis, was devised by Ancient Greek mathematicians for solving geometric construction problems with ruler and compass. Its most important subsequent elaboration is Descartes’s algebraic method of analysis, which pervades all exact sciences today. The present work expands this method further into one that aims at systematizing program derivation in a heuristically useful way, analogously to the way Descartes’s method systematized the solution of geometric and arithmetical problems. To illustrate the method, we derive the Boyer-Moore algorithm for finding an element that has a majority of occurrences in a given list. It turns out that solving programming problems need not be too different from solving mathematical problems in general. This point of view has been emphasized in particular by Martin-Löf (1982) and Dijkstra (1986). The idea of a logic of problem solving originates in Kolmogorov (1932). We aim to refine the analogy between programming and mathematical problem solving by investigating the mathematical method of analysis in the context of programming. The central idea of the analytic method, in modern terms, is to analyze the functional dependencies between the constituents of a geometric configuration. The aim is to determine how the sought constituents depend on the given ones. A Greek analysis starts by drawing a diagram with the sought constructions drawn on the given ones, in the relation required by the problem specification. Then the sought constituents of the configuration are determined in terms of the given ones. Analysis was the Greeks’ method of discovering solutions to problems. Their method of justification was synthesis, which cast analysis into standard deductive form. First it constructed the sought objects from the given ones, and then demonstrated that they relate as required to the given ones. In his Geometry, Descartes developed Greek geometric analysis-synthesis into the modern algebraic method of analysis.

scholarly journals Students’ suspension of sense making in problem solving

ZDM ◽  
2021 ◽  
Author(s):  
Gemma Carotenuto ◽  
Pietro Di Martino ◽  
Marta Lemmi
Keyword(s):  
Problem Solving ◽  
Qualitative Study ◽  
Word Problem ◽  
Word Problems ◽  
Mathematical Problem ◽  
Critical Issue ◽  
Mathematical Problem Solving ◽  
Sense Making ◽  
Mathematical Problems

AbstractResearch on mathematical problem solving has a long tradition: retracing its fascinating story sheds light on its intricacies and, therefore, on its needs. When we analyze this impressive literature, a critical issue emerges clearly, namely, the presence of words and expressions having many and sometimes opposite meanings. Significant examples are the terms ‘realistic’ and ‘modeling’ associated with word problems in school. Understanding how these terms are used is important in research, because this issue relates to the design of several studies and to the interpretation of a large number of phenomena, such as the well-known phenomenon of students’ suspension of sense making when they solve mathematical problems. In order to deepen our understanding of this phenomenon, we describe a large empirical and qualitative study focused on the effects of variations in the presentation (text, picture, format) of word problems on students’ approaches to these problems. The results of our study show that the phenomenon of suspension of sense making is more precisely a phenomenon of activation of alternative kinds of sense making: the different kinds of active sense making appear to be strongly affected by the presentation of the word problem.

Teaching Personal Finance Mathematical Problem Solving to Individuals With Moderate Intellectual Disability

2016 ◽  
Vol 40 (1) ◽  
pp. 5-14 ◽  
Author(s):  
Jenny Root ◽  
Alicia Saunders ◽  
Fred Spooner ◽  
Chelsi Brosh
Keyword(s):  
Problem Solving ◽  
Mathematical Problem ◽  
Functional Relation ◽  
Personal Finance ◽  
Future Research ◽  
School Students ◽  
Problem Solving Skills ◽  
Mathematical Problems ◽  
Based Instruction ◽  
Multiple Probe

The ability to solve mathematical problems related to purchasing and personal finance is important in promoting skill generalization and increasing independence for individuals with moderate intellectual disabilities (IDs). Using a multiple probe across participant design, this study investigated the effects of modified schema-based instruction (MSBI) on personal finance problem solving skills, purchasing an item on sale or leaving a tip, and using a calculator or iDevice (i.e., iPhone or iPad) for three middle school students diagnosed with a moderate ID. The results showed a functional relation between MSBI using a calculator on the participant’s ability to solve addition and subtraction personal finance word problems and generalize to iDevices. The findings of this study provide several implications for practice and offer suggestions for future research.

scholarly journals DEFRAGMENTATION THINKING STRUCTURE TO OVERCOME ERRORS IN ADDRESSING MATHEMATICAL PROBLEM

2021 ◽  
Vol 10 (1) ◽  
pp. 339
Author(s):  
Siti Puri Andriani ◽  
Triyanto Triyanto ◽  
Farida Nurhasanah
Keyword(s):  
High School ◽  
Problem Solving ◽  
Random Sampling ◽  
Mathematical Problem ◽  
Islamic State ◽  
Algebraic Functions ◽  
Process Error ◽  
Procedural Error ◽  
The Subject ◽  
The Given

This research is intended to describe students' procedural errors in solving problems derivative of algebraic functions and efforts to overcome these errors by using the defragmentation process. Error analysis is carried out based on the procedural error theory based on Elbrink which includes the following aspects of errors: 1) Mis-identification; 2) Mis-generalization; 3) Repair Theory; and 4) Overspecialization. The subjects in this study are students of class XII MIPA Islamic State Senior High School (MAN) 3 Tulungagung taken from snowball random sampling. In taking the subject, the researchers select one of the students who make procedural errors by considering the completeness of the students when solving the given problems based on the problem-solving phase according to Polya. Based on the results of this study, it is found that the procedural errors made by the students are repair theory errors and overspecialization.  The defragmenting process to correct these errors is intended to provide dis-equilibration and scaffolding. The results after the defragmenting process are the students can correct their mistakes and the structure of their thinking.Keywords: Defragmenting structure thinking; derivative algebraic functions; problem solving; procedural errors. AbstrakPenelitian ini bertujuan untuk menggambarkan kesalahan prosedural siswa dalam menyelesaikan masalah turunan fungsi aljabar dan upaya untuk mengatasi kesalahan tersebut dengan menggunakan proses defragmenting. Analisis kesalahan dilakukan berdasarkan konsep teori kesalahan prosedural menurut Elbrink yang mencakup aspek kesalahan sebagai berikut: Mis-identificstion; 2) Mis-generalization; 3) Repair Theory; dan 4) Overspecialization. Subjek dalam penelitian ini adalah siswa kelas XII MIPA MAN 3 Tulungagung yang diambil secara snowball  random sampling. Dalam pengambilan subjek dipilih salah satu siswa yang melakukan kesalahan prosedural dengan mempertimbangkan kelengkapan siswa ketika menyelesaikan masalah yang diberikan berdasarkan tahap pemecahan masalah menurut Polya. Dari hasil penelitian ini ditemukan bahwa kesalahan prosedural yang dilakukan siswa ialah kesalahan repair theory dan overspecialization. Proses defragmenting yang dilakukan untuk memperbaiki kesalahan tersebut ialah dengan memberikan dissequillibrasi dan scaffolding. Hasil yang diperoleh setelah proses defragmenting dilakukan ialah siswa mampu memperbaiki kesalahannya dan struktur berpikirnya.Kata kunci: Defragmenting struktur berpikir, kesalahan prosedural, pemecahan masalah, turunan fungsi aljabar.

scholarly journals PENDEKATAN MATEMATIKA REALISTIK UNTUK MEMBANGUN KEMAMPUAN PEMECAHAN MASALAH MATEMATIS SISWA KELAS XI IPS PADA MATERI PELUANG [REALISTIC MATHEMATICS EDUCATION IN BUILDING THE MATHEMATICS PROBLEM-SOLVING ABILITIES OF GRADE 11 SOCIAL SCIENCE TRACK STUDENTS STUDYING PROBABILITY]

2019 ◽  
Vol 2 (2) ◽  
pp. 119
Author(s):  
Susiana Juseria Tambunan ◽  
Debora Suryani Sitinjak ◽  
Kimura Patar Tamba
Keyword(s):  
Problem Solving ◽  
Social Science ◽  
Mathematical Problem ◽  
Mathematical Problem Solving ◽  
Mathematics Problem Solving ◽  
Realistic Mathematics ◽  
Mathematical Problems ◽  
Mathematic Education ◽  
Realistic Mathematic ◽  
Grade 11

<p>This research aims to build students’ abilities in mathematical problem-solving and to explain the uniqueness of the steps of realistic mathematic education in building the problem-solving abilities of a grade 11 (social science track) class in the study of probability at one of the schools in Kupang. The observation results found that every student was having difficulties to solving the mathematical problems, particularly the narrative questions. The research method is Kemmis and Taggart model of Classroom Action Research which was conducted in three cycles, from October 4 to November 3 with twenty-four students. Triangulation had been done to every instrument of variable. The data of mathematical problem-solving was obtained from the students by using test sheets, questionnaires, and student’s discussion sheets. Meanwhile, the data of realistic mathematic education’s variable was obtained from three sources: mentors, two colleagues, and students that were using test sheets, questionnaires, and student’s discussion sheets. The results showed that the fourteen-steps of Realistic Mathematic Education that had been done were able to build mathematical problem-solving abilities of the students. This was evidenced through the increase of three indicators of mathematical problem-solving in every cycle. The average increase of indicators of mathematical problem-solving of the grade 11 students from the first to the third cycle was 10%. Therefore, it can be concluded that the Realistic Mathematics Approach can build the ability of problem-solving of grade 11 students in a social science track studying probability at one of the schools in Kupang.</p><strong>BAHASA INDONESIA </strong><strong>ABSTRACT</strong>: Penelitian ini bertujuan untuk membangun kemampuan pemecahan masalah matematis siswa dan menjelaskan kekhasan langkah-langkah pendekatan matematika realistik untuk membangun kemampuan tersebut di salah satu sekolah di Kupang kelas XI IPS pada materi peluang topik kaidah pencacahan. Pada hasil pengamatan ditemukan bahwa setiap siswa kesulitan dalam memecahkan masalah matematis khususnya soal berbentuk cerita. Metode penelitian yang digunakan adalah Penelitian Tindakan Kelas model Kemmis dan Taggart yang berlangsung selama tiga siklus, yaitu 04 Oktober – 03 November kepada 24 orang siswa. Triangulasi dilakukan pada setiap instrumen variabel. Data variabel kemampuan pemecahan masalah matematis diperoleh dari siswa menggunakan lembar tes, lembar angket, dan lembar diskusi siswa. Sedangkan data variabel tingkat pelaksanaan pendekatan matematika realistik diperoleh dari tiga sumber, yaitu mentor, dua orang rekan sejawat, dan siswa menggunakan lembar observasi, lembar angket, dan lembar wawancara. Hasil penelitian menunjukkan bahwa keempat belas langkah-langkah pendekatan matematika realistik yang terlaksana dengan baik sekali mampu membangun kemampuan pemecahan masalah matematis setiap siswa kelas XI IPS di salah satu sekolah di Kupang. Hal ini dinyatakan melalui peningkatan ketiga indikator pemecahan masalah matematis di setiap siklus. Peningkatan rata-rata indikator pemecahan masalah matematis siswa kelas XI IPS dari siklus pertama sampai ketiga adalah sebesar 10%. Oleh karena itu, dapat disimpulkan bahwa pendekatan matematika realistik dapat membangun kemampuan pemecahan masalah matematis siswa kelas XI IPS di salah satu sekolah di Kupang pada materi peluang topik kaidah pencacahan.

scholarly journals Pengaruh Model Pembelajaran E-learning Berbantuan Media Pembelajaran Edmodo Terhadap Kemampuan Pemecahan Masalah Matematis Peserta Didik

2019 ◽  
pp. 31-42
Author(s):  
Hanifah Hanifah ◽  
Nanang Supriadi ◽  
Rany Widyastuti
Keyword(s):  
Problem Solving ◽  
Quantitative Research ◽  
Mathematical Problem ◽  
Learning Model ◽  
Mathematical Problem Solving ◽  
Learning Material ◽  
Learning Models ◽  
Mathematical Problems ◽  
E Learning ◽  
Initial Knowledge

Mathematical problem solving is a problem solving that uses mathematical problem solving. Students in the problem solving did not use the polya method so that students succeeded in difficulties. Educators still use conventional learning models so that students become bored, passive and reluctant to ask whether going forward working on the questions given by the educator, so that new learning models need to be applied. The e-learning learning model assisted with Edmodo learning media is an online presentation material on an Edmodo account using the mobile phone of students. PAM is the knowledge learned by students before getting learning material. This study aims to study the interaction of e-learning learning models assisted by Edmodo learning media to solve mathematical problems. This study is quantitative research. Data collection used with tests, interviews, collection and collection. The data analysis technique uses two-way anava test with cells that are not the same. From the results of the analysis, the influence of the e-learning learning model on mathematical problem solving abilities. It is necessary to question the high, medium, and low mathematical initial knowledge of Great mathematical problem solving ability, then there is no difference between assisted e-learning learning models edmodo, mathematical initial knowledge of mathematical problem solving abilities.

scholarly journals PENERAPAN I-SPRING SUITE 8 PADA MODEL PEMBELAJARAN IMPROVE UNTUK MENINGKATKAN KEMAMPUAN PEMAHAMAN DAN PEMECAHAN MASALAH MATEMATIS PESERTA DIDIK PADA POKOK BAHASAN PROGRAM LINEAR DI TINGKAT SEKOLAH MENENGAH

Gunahumas ◽  
2020 ◽  
Vol 2 (2) ◽  
pp. 357-386
Author(s):  
Yomi Chaeroni ◽  
Nizar Alam Hamdani ◽  
Akhmad Margana ◽  
Dian Rahadian
Keyword(s):  
Problem Solving ◽  
Direct Instruction ◽  
Mathematics Learning ◽  
Mathematical Problem ◽  
Learning Model ◽  
Mathematical Understanding ◽  
Learning Models ◽  
Research Subjects ◽  
Purposive Sampling ◽  
Mathematical Problems

ABSTRAK Penelitian ini dilatarbelakangi oleh fakta bahwa kemampuan pemahaman dan kemampuan pemecahan masalah matematis merupakan salah satu kemampuan matematika tingkat tinggi yang harus dimiliki oleh setiap peserta didik. Selain itu kemampuan pemahaman dan kemampuan pemecahan masalah matematis jarang diterapkan dalam pembelajaran matematika di sekolah. Salah satu model pembelajaran yang dapat menjadi alternatif bagi pembelajaran matematika dan kemampuan pemahaman dan pemecahan masalah matematis adalah model pembelajaran IMPROVE. Penelitian ini bertujuan untuk mengetahui penerapan i-spring suite 8 pada model pembelajaran IMPROVE untuk meningkatkan kemampuan pemahaman dan pemecahan masalah matematis peserta didik. Metode penelitian yang digunakan adalah quasi eksperimen karena penelitian ini menggunakan satu kelas eksperimen dan satu kelas kontrol sebagai subyek penelitian. Cara pengambilan subjek penelitian yang digunakan adalah purposive sampling. Subjek penelitian dipilih sebanyak dua kelas dari keseluruhan peserta didik kelas XI SMA Muhammadiyah Banyuresmi tahun pelajaran 2019/2020. Dari hasil penelitian dan perhitungan statistik diperoleh kesimpulan: 1) Terdapat peningkatan kemampuan pemahaman dan pemecahan masalah matematis peserta didik yang dalam pembelajarannya menggunakan i-spring suite 8 pada model pembelajaran IMPROVE; 2) Terdapat peningkatan kemampuan pemahaman dan pemecahan masalah matematis peserta didik yang dalam pembelajarannya menggunakan model pembelajaran konvensional/direct instruction; 3) Terdapat peningkatan kemampuan pemahaman dan pemecahan masalah matematis peserta didik yang dalam pembelajarannya menggunakan i-spring suite 8 pada model pembelajaran IMPROVE dibandingkan dengan peserta didik yang dalam pembelajarannya menggunakan model pembelajaran konvensional/direct instruction; 4) Tidak terdapat perbedaan kemampuan pemahaman dan pemecahan masalah matematis peserta didik yang dalam pembelajarannya menggunakan i-spring suite 8 pada model pembelajaran IMPROVE dan yang menggunakan model konvensional/direct instruction.Kata kunci: Kemampuan Pemahaman Matematis, Kemampuan Pemecahan Masalah Matematis, Model IMPROVEABSTRACT This research is motivated by the fact that the ability to understand and the ability to solve mathematical problems is one of the high-level mathematical abilities that must be possessed by every student. In addition, the ability to understand and the ability to solve mathematical problems are rarely applied in mathematics learning in schools. One learning model that can be an alternative for mathematics learning and mathematical understanding and problem solving abilities is the IMPROVE learning model. This study aims to determine the application of ispring suite 8 on the IMPROVE learning model to improve students' mathematical understanding and problem solving abilities. The research method used is quasi-experimental because this study uses one experimental class and one control class as research subjects. The method of taking the research subject used was purposive sampling. The research subjects were selected as many as two classes from all grade XI students of SMA Muhammadiyah Banyuresmi in the 2019/2020 academic year. From the results of research and statistical calculations conclusions: 1) There is an increase in the ability to understand and solve mathematical problems of students who in learning use the i-spring suite 8 on the IMPROVE learning model; 2) There is an increase in the ability of understanding and solving mathematical problems of students who in learning use conventional learning models / direct instruction; 3) There is an increase in students' mathematical understanding and problem solving abilities in learning using i-spring suite 8 in the IMPROVE learning model compared to students in learning using conventional learning models / direct instruction; 4) There is no difference in the ability to understand and solve mathematical problems of students who in learning use the i-spring suite 8 on the IMPROVE learning model and who use the conventional model / direct instruction.Keywords: Mathematical Understanding Ability, Mathematical Problem Solving Ability, IMPROVE Model

scholarly journals Pre-Service Class Teacher’ Ability in Solving Mathematical Problems and Skills in Solving Daily Problems

2016 ◽  
Vol 6 (3) ◽  
pp. 32 ◽  
Author(s):  
Nahil M. Aljaberi ◽  
Eman Gheith
Keyword(s):  
High School ◽  
Problem Solving ◽  
Daily Life ◽  
Mathematical Problem ◽  
Service Class ◽  
Problem Solving Skills ◽  
Life Issues ◽  
Mathematical Problems ◽  
Academic Year ◽  
Class Teacher

<p>This study aims to investigate the ability of pre-service class teacher at University of Petrain solving mathematical problems using Polya’s Techniques, their level of problem solving skills in daily-life issues. The study also investigates the correlation between their ability to solve mathematical problems and their level of problem solving skills in daily-life issues. The study sample consisted of 65 female students majoring in class teacher. Data were collected using two questionnaires: the mathematical problem solving test which was developed by the researchers and daily life problem solving scale which was developed by (Hamdi, 1998). The findings indicate that students had high level skills in solving daily problems; there are no statistically significant differences in daily problem solving in relation to their academic year or high-school stream. Conversely, the findings also indicate weaknesses in students’ skills in solving mathematical problems, with no statistically significant differences among students in solving mathematical problems according to Polya’s problem solving steps. However, there were statistically significant differences in students’ performance in solving mathematical problems in relation to the mathematical topic, and in favor of measurements and algebra; in addition to statistically significant differences in students’ ability to solve mathematical problems in relation to academic year and high-school stream, but no correlation between students’ abilities in solving mathematical problems and those in solving daily problems.</p>

Links to Literature: Using Representations to Explore Perimeter and Area

2001 ◽  
Vol 8 (1) ◽  
pp. 52-59
Author(s):  
Patricia S. Moyer
Keyword(s):  
Problem Solving ◽  
Word Problems ◽  
Mathematical Problem ◽  
Real Life ◽  
Problem Situation ◽  
Correct Operation ◽  
Problem Situations ◽  
Mathematical Problems ◽  
Mathematical Word Problems ◽  
Computational Processes

In an elementary school classroom, as in real life, the lines between the content areas should be blurred, particularly between mathematical problem solving and mathematical situations contextualized in good literature. For that reason, I always look for interesting books about mathematical situations. Why use children's literature to teach mathematics? A good story often places mathematical problems in the context of familiar situations and is similar to, yet a much more elaborate version of, mathematical word problems. Assertions that children's inability to solve word problems results from their inability to read or to compute effectively simply are not true. The problem is that children do not know how to choose the correct operation or sequence of operations to solve the problem. To solve a problem situation presented in words, children need to be able to connect computational processes with appropriate calculations. Their difficulties lie in the fact that children simply do not understand the mathematics well enough conceptually to make the connection with the problem- solving situation. Using books with authentic problem situations may help children see that learning computation serves a real-life purpose.

scholarly journals Pengembangan Local Instructional Theory Pada Topik Pembagian dengan Pendekatan Matematika Realistik

2020 ◽  
Vol 4 (1) ◽  
pp. 01
Author(s):  
Ahmad Fauzan ◽  
Yerizon Yerizon ◽  
Fridgo Tasman ◽  
Rendy Novri Yolanda
Keyword(s):  
Problem Solving ◽  
Mathematical Problem ◽  
Mathematical Problem Solving ◽  
School Students ◽  
Problem Solving Skills ◽  
Realistic Mathematics ◽  
Mathematical Problems ◽  
Parametric Statistics ◽  
Instruction Theory ◽  
Local Instruction Theory

This research aimed to develop local instruction theory that is valid, practical, and effective to help elementary school students developing their mathematical problem-solving skills. Therefore a sequential activityis design on dailybasis to encourage students to develop their ability to solve mathematical problems, especially on the topic division. To achieve the goal, realistic mathematics approach was implemented to grade three elementary students in the learning process. The designed activities were validated by experts on the aspects of mathematical contents, language, didactical process based on realistic mathematical approach. Data were analyzed with descriptive statistics and parametric statistics. The validation results show that the local instruction theory was valid, and the implementation shows that the local instruction theory is practical and effective in improving students' mathematical problem-solving skills.

scholarly journals STUDENTS’ CRITICAL THINKING PROFILES IN SOLVING MATHEMATICAL PROBLEMS BADES ON ADVERSITY QUOTIENT (AQ)

MATHEdunesa ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 211-220
Author(s):  
NILA NURCAHYANING KUSUMAWARDANI ◽  
RADEN SULAIMAN
Keyword(s):  
Critical Thinking ◽  
Problem Solving ◽  
Junior High School ◽  
Mathematical Problem ◽  
Good Communication ◽  
Mathematics Problem Solving ◽  
Mathematical Problems ◽  
Processing Information ◽  
Thinking Process ◽  
Mathematics Problem

Critical thinking is a thinking process in processing information logically starti from understanding, analyzing, evaluating and making precise conclusions. Critical thinking indicators are clarification, assessment, inference, and strategy that referred to Jacob and Sam. Mathematics is designed to improve students' critical thinking in a solving problem. One of the factors that affect students' critical thinking in solving a problem is AQ. This research is descriptive study with qualitative approach. The aim is to describe critical thinking profile of climber, camper, and quitter students in solving mathematical problems. The subjects were three students of VIII grade junior high school who represented each AQ category and had good communication skills. The instrument used was the ARP questionnaire, mathematics problem solving tests, and interview guidelines. The results shows that students’ critical thinking profile in understanding the problem is climber and camper student do all indicators of critical thinking in the clarification phase. Quitter student is only able mentioning known and asked information. In devising a plan, climber student implements all indicators of assessment and strategy phase. Camper student implements all indicators in assessment phase, but do not discuss the possible steps in strategy phase. Quitter student does not do both assessment and strategy phase. In carrying out the plan, climber and camper students do all indicators of inference phase, while quitter student does not. In the step of looking back, only climber student who carries out evaluating steps that have been done. Keywords: Jacob and Sam’s critical thinking, mathematical problem solving, adversity quotient

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