Mathematical Modeling Thinking: Laying the Foundation for Mathematical Modeling Competency

2019 ◽  
pp. 14-19
Author(s):  
Cynthia O. Anhalt ◽  
Ricardo Cortez
Keyword(s):  
Mathematical Modeling ◽  
Problem Solving ◽  
General Problem ◽  
Multiple Representations ◽  
Sufficient Time ◽  
Building Modeling ◽  
Ways Of Thinking ◽  
Mathematical Activities

Mathematical modeling competency requires frequent practice and sufficient time to derive experience solving open-ended contextual problems. Specific ways of thinking necessary in modeling are identified by contrasting Pólya’s general problem-solving framework, which may be familiar worldwide. These ways of thinking are developed through mathematical activities that promote dispositions for eventual success in modeling. We posit that mathematical modeling thinking (MMT) is necessary for building modeling competency. This paper describes MMT and illustrates how it can be developed through a well-known problem of universal human cultural greeting exchange. While connecting to world cultures, we examine ways to promote MMT practices such as making useful simplifications, looking for patterns, utilizing multiple representations, mathematizing the situation, and reflecting on the solution. We conclude with practical ways to effect MMT as the foundation for developing mathematical modeling competency.

A Visual Approach to Solving Mixture Problems

1996 ◽  
Vol 89 (2) ◽  
pp. 108-111
Author(s):  
Albert B. Bennett ◽  
Eugene Maier
Keyword(s):  
Mathematical Modeling ◽  
Problem Solving ◽  
Mathematical Problem ◽  
Multiple Representations ◽  
School Mathematics ◽  
Mathematical Problem Solving ◽  
Evaluation Standards ◽  
Conceptual Understandings ◽  
Visual Approach

In the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), the 9–12 standards call for a shift from a curriculum dominated by memorization of isolated facts and procedures to one that emphasizes conceptual understandings, multiple representations and connections, mathematical modeling, and mathematical problem solving. One approach that affords opportunities for achieving these objectives is the use of diagrams and drawings. The familiar saying “A picture is worth a thousand words” could well be modified for mathematics to “A picture is worth a thousand numbers.” As an example of visual approaches in algebra, this article uses diagrams to solve mixture problems.

More evidence on competing cognitive systems

1985 ◽  
Vol 1 (1) ◽  
pp. 47-72 ◽  
Author(s):  
Sascha W. Felix
Keyword(s):  
Second Language ◽  
Problem Solving ◽  
General Problem ◽  
Learning Process ◽  
Native Speaker ◽  
Universal Grammar ◽  
Learning Task ◽  
Competition Model ◽  
Cognitive Systems ◽  
Specific System

This paper deals with the question of why adults, as a rule, fail to achieve native-speaker competence in a second language, whereas children appear to be generally able to acquire full command of either a first or second language. The Competition Model proposed in this paper accounts for this difference in terms of different cognitive systems or modules operating in child and adult language acquisition. It is argued that the child's learning process is guided by a language-specific module, roughly equivalent to Universal Grammar (cf. Chomsky, 1980), while adults tend to approach the learning task by utilizing a general problem-solving module which enters into competition with the language-specific system. The crucial evidence in support of the Competition Model comes from a) the availability of formal operations in different modules and b) from differences in the types of utterances produced by children and adults.

scholarly journals LOGICAL REASONING ANALYSIS BASED ON HIPPOCRATES PERSONALITY TYPES

Aksioma ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 57-73
Author(s):  
Nurdin Nurdin ◽  
Ita Sarmita Samad ◽  
Sardia Sardia
Keyword(s):  
Problem Solving ◽  
Mathematical Problem ◽  
Logical Reasoning ◽  
Personality Types ◽  
The Body ◽  
Mathematical Problem Solving ◽  
Different Types ◽  
Ways Of Thinking ◽  
The Difference ◽  
Descriptive Qualitative

Abstract: The theory distinguishes human based on four different personality types such as: sanguine, choleric, melancholic, and phlegmatic. Different types of personality caused by differences in the dominant fluid in the body. These differences will result in terms of behavior, ways of thinking and to get along. The type of this research that is descriptive qualitative which it is describing the logical reasoning based on Hippocrates personality types. The logical reasoning is analyzed through the four types of personality in relation to mathematical problem solving. The Analysis is done based on the logical reasoning indicator/ subindicator and the steps of problem solving stated by Polya. The result shows that there is a reasoning difference on each type of personalities. The difference can be terms of the strenght or the weakness. Sanguine is quicker in understanding problems and communicating results, choleric is more accelerated in work, melancholic is more perfect at work, and  phlegmatic is superior in terms of accuracy. Keywords: Logical reasoning, Hippocrates, sanguine, choleric, melancholic, phlegmatic

scholarly journals MEASURING STUDENTS’ PROBLEM-SOLVING SKILLS ON DIRECT CURRENT CIRCUITS WITH MULTIPLE REPRESENTATIONS

2020 ◽  
Vol 8 (3) ◽  
pp. 725-736
Author(s):  
Maria Dewati ◽  
A. Suparmi ◽  
Widha Sunarno ◽  
Sukarmin ◽  
C. Cari
Keyword(s):  
Problem Solving ◽  
Physics Education ◽  
Multiple Choice ◽  
Multiple Representations ◽  
Assessment Instruments ◽  
Multiple Choice Questions ◽  
Multiple Representation ◽  
Problem Solving Skills ◽  
Electrical Circuits ◽  
Physics Problems

Purpose of study: This study aims to measure the level of students' problem-solving skills, using assessment instruments in the form of multiple-choice tests based on the multiple representation approach on DC electrical circuits. Methodology: This research is a quantitative descriptive involving 46 students of physics education. Students are asked to solve the problem of DC electrical circuits on 12 multiple choice questions with open reasons, involving verbal, mathematical, and picture representations. Data were analyzed by determining means and standard deviations. Main findings: The results of the study showed that there were 3 levels of students' problem-solving skills, namely 7 (15%) students in the high category, 22 (48%) students in the medium category and 17 (37%) students in the low category. Applications of this study: The implication of this research is to continuously develop assessment instruments based on multiple representations in the form of various types of tests, to help students improve their conceptual understanding, so students can solve physics problems correctly. The novelty of this study: Researchers explain the right way to solve physics problems, 1) students are trained to focus on identifying problems, 2) students are accustomed to planning solutions using a clear approach, to build an understanding of concepts, 3) students are directed to solve problems accordingly with understanding the concepts they have built.

scholarly journals Discovering and Implementing Self-Sustaining Networks of Cooperation with General Collective Intelligence

2020 ◽  
Author(s):  
Andy E Williams
Keyword(s):  
Problem Solving ◽  
General Problem ◽  
Iterative Process ◽  
Collective Intelligence ◽  
Exponential Increase ◽  
Growing Networks ◽  
First Time

Leveraging General Collective Intelligence or GCI, a platform with the potential to achieve an exponential increase in general problem-solving ability, a methodology is defined for finding potential opportunities for cooperation, as well as for negotiating and launching cooperation. This paper explores the mechanisms by which GCI enables networks of cooperation to be formed in order to increase outcomes of cooperation and in order to make that cooperation self-sustaining. And this paper explores why implementing a GCI for the first time requires designing an iterative process that self-assembles continually growing networks of cooperation.

scholarly journals Students’ Cognitive Barrier in Problem Solving: Picture-based Problem-solving

2020 ◽  
Vol 11 (1) ◽  
pp. 73-82
Author(s):  
A.Wilda Indra Nanna ◽  
Enditiyas Pratiwi
Keyword(s):  
Problem Solving ◽  
Primary Education ◽  
Qualitative Method ◽  
Thought Processes ◽  
Structured Interviews ◽  
Mathematical Objects ◽  
New Findings ◽  
Mathematical Problems ◽  
Ways Of Thinking ◽  
Descriptive Qualitative

Pre-service teachers in primary education often have difficulty in solving mathematical problems, specifically fractions that are presented with a picture. In solving problems, some thought processes are needed by the teacher to reduce students' cognitive barriers. Therefore, this study aimed to reveal the cognitive barriers experienced by students in solving fraction problems. The cognitive barriers referred to in this study are ways of thinking about structures or mathematical objects that are appropriate in one situation and not appropriate in another situation. This study employed a descriptive-qualitative method. Furthermore, participants were followed up with in-depth semi-structured interviews to find out the cognitive barriers that occurred in solving fraction problems. This study discovers that the participants, in solving fraction problems, experienced all indicators of cognitive barrier and two cognitive obstacles are found as new findings that tend to involve mathematical calculations and violates the rules in dividing images into equal parts in the problem-solving procedure. 

scholarly journals How Indonesian Students Use the Polya’s General Problem Solving Steps

2018 ◽  
Vol 8 (1) ◽  
pp. 39-48
Author(s):  
Hari Pratikno ◽  
Endah Retnowati
Keyword(s):  
Problem Solving ◽  
Mathematics Achievement ◽  
Secondary School ◽  
General Problem ◽  
High Ability ◽  
School Students ◽  
Junior Secondary School ◽  
Junior Secondary ◽  
Mathematical Problems ◽  
Low Ability

General problem-solving steps consist of understanding the problem, developing a plan, implementing the plan and checking the result. The purpose of this study is to explore how well Indonesia junior secondary school students apply these four steps in solving mathematical problems, especially on plane geometry topics. Using a qualitative approach, with a sample of nine students, of which three students were from the low mathematics achievement category, three from the medium and three from the high category, were given a test and instructed to write the answers to each question step by step. The results were described and categorized into four groups. The first group consisted of students who used all of the four steps. The second and the third were for students who used the first three steps or the first two steps respectively. The fourth group was for those who could only show the first step. The study indicated that for this sample the level of mathematic ability corresponded to how the students applied their problem-solving steps. It was found that students with high ability were included in the first group, while those with moderate ability were in the second group. Low ability students were categorized into group four. Nevertheless, there was one student with high ability who did not to do the checking step and there was one student with low ability who was able to develop a plan.

scholarly journals Indirect Analogical Reasoning Components

2021 ◽  
Vol 4 (1) ◽  
pp. 13
Author(s):  
K Kristayulita
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Problem Solving ◽  
Word Problem ◽  
Analogical Reasoning ◽  
Quadratic Equation ◽  
Target Problem ◽  
Equation Problem ◽  
Analogical Problem ◽  
Verbal Analogies

If using different instruments obtained a different analogical reasoning component. With use  people-piece analogies, verbal analogies, and geometric analogies, have analogical reasoning component consists of encoding, inferring, mapping, and application. Meanwhile,  with use analogical problems (algebra, source problem and target problem is equal), have analogical reasoning components consist of structuring, mapping, applying, and verifying. The instrument used was analogical problems consisting of two problems where the source problem was symbolic quadratic equation problem and the target problems were trigonometric equation problem and a word problem. This study aims to provide information analogical reasoning process in solving indirect analogical problems. in addition, to identify the analogical reasoning components in solving indirect analogical problems. Using a qualitative design approach, the study was conducted at two schools in Mataram city of Nusa Tenggara Barat, Indonesia. The results of the study provide an overview of analogical reasoning of the students in solving indirect analogical problems and there is a component the representation and mathematical model in solving indirect analogical problems.  So the analogical reasoning component in solving indirect analogical problems is the representation and mathematical modeling, structuring, mapping, applying, and verifying. This means that there are additional components of analogical reasoning developed by Ruppert. Analogical reasoning components in problem-solving depend on the analogical problem is given.

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