Integrating computers and problem posing in mathematics teacher education

✍ Scribed by Sergei Abramovich

• Year: 2019
• Tongue: English
• Leaves: 254
• Category: Library

✦ Table of Contents

Contents
Preface
Chapter 1 On the Genesis of Problem Posing in Mathematics
1.1 Problems from the first printed arithmetic
1.1.1 Solving a 15th century problem using the modern-day pedagogy
1.1.2 Posing 15th century-like problems through conceptualization
1.1.3 Using technology for posing 15th century-like problems
1.2 From a classic problem to using the modern spreadsheet
1.2.1 The birth of the probability theory through problem posing
1.2.2 Using a spreadsheet as a problem-posing tool
1.2.3 Duality of the spreadsheet’s use
1.3 The Problem of the Grand Duke of Tuscany
1.4 Conjecturing as posing problems to find proof
1.5 Problem posing in a classic context as a springboard into experimental mathematics
1.5.1 Triangular numbers with identical digits
1.5.2 Triangular number sieves
1.6 Problem posing as setting up a research program
1.7 Summary
Chapter 2 From a Theory of Problem Posing to Classroom Practice of the Digital Era
2.1 Problem posing as educational philosophy
2.2 Problem posing in the modern educational context
2.3 Learning to ask questions about posed/solved problems
2.4 Technology as a cultural support of problem posing
2.5 Numerical coherence in problem posing
2.5.1 Using a spreadsheet to pose a numerically coherent problem
2.6 Contextual coherence in problem posing
2.7 Pedagogical coherence in problem posing
2.8 Didactical coherence in problem posing
2.9 Summary
Chapter 3 Posing Technology-Immune/Technology-Enabled (TITE) Problems
3.1 From teaching machine movement to symbolic computations
3.2 Technological advances call for the revision of mathematics curriculum
3.3 Definition of a TITE problem and a simple example
3.4 Revisiting classic problems in the digital era under the umbrella of the TITE concept
3.5 Conceptual bond and arithmetical word problems
3.5.1 Looking at the past to develop new teaching ideas
3.5.2 Posing similar problems
3.6 Revisiting mathematical problems to make them didactically coherent
3.6.1 From numerical to contextual coherence
3.6.2 Towards pedagogical coherence
3.7 From modeling data to a general formula using technology
3.8 Formulating and solving a didactically coherent problem
3.9 Maple-based mathematical induction proof
3.10 Summary
Chapter 4 Linking Algorithmic Thinking and Conceptual Knowledge through Problem Posing
4.1 On the hierarchy of two types of knowledge
4.2 A simple question leads to revealing hidden creativity
4.3 Two levels of conceptual understanding
4.4 Solving a problem seeking information
4.5 Problem posing leads to conceptual knowledge and collateral learning
4.6 Using conceptual bond in posing problems with technology
4.7 Summary
Chapter 5 Using Graphing Software for Posing Problems in Advanced High School Algebra
5.1 Introduction
5.2 Location of roots of quadratics about an interval
5.3 Digital fabrication
5.4 Connecting the coordinate plane with the plane of coefficients
5.4.1 The case RREE
5.4.2 The case RERE
5.4.3 The case REER
5.4.4 The case ERER
5.4.5 The case EERR
5.4.6 The case ERRE
5.5 Using Vieta’s Theorem
5.6 Posing TITE problems in the plane of parameters
5.7 Geometric probabilities and the partitioning diagram
5.8 Making mathematical connections
5.9 Revealing hidden concepts through collateral learning
5.10 Summary
Chapter 6 Einstellung Effect and Problem Posing
6.1 Examples of Einstellung effect
6.2 Water jar experiments and Einstellung effect
6.3 Posing and solving problems as a remediation of Einstellung effect
6.4 Posing problems for water jar experiments using a spreadsheet
6.5 Einstellung effect in finding areas on a geoboard
6.6 Einstellung effect in solving algebraic equations and inequalities
6.7 Einstellung effect in solving trigonometric inequalities
6.8 Using technology to pose problems that might lead to Einstellung effect
6.9 Einstellung effect in solving logarithmic inequalities
6.9.1 Simultaneous extension and contraction of solution set
6.9.2 Extension of solution set
6.10 Solving logarithmic inequality (6.21) in the general case
6.10.1 The case n = 2k
6.10.2 The case n = 2k + 1
6.11 Summary
Chapter 7 Explorations with Integer Sequences as TITE Problem Posing
7.1 Introduction
7.2 Exploring patterns formed by the last digits of the sums of powers of integers
7.3 Discovering patters in the last digits of the polygonal numbers
7.3.1 The triangular number sieves
7.3.2 Triangular number sieves and the last digits of their terms
7.3.3 Rises and falls in permutations
7.3.4 Connecting triangular and square numbers within the multiplication table
7.3.5 The square number sieves
7.3.6 The pentagonal number sieves
7.3.7 The general case of the m-gonal number sieves
7.4 Patterns in the behavior of the greatest common divisors of two polygonal numbers
7.5 Exploring sequences formed by the sums of powers of integers
7.6 Exploring sieves developed from the sums of powers of integers
7.7 Summary
Appendix
8.1 Spreadsheets included in Chapter 1
8.2 Spreadsheets included in Chapter 2
8.3 Spreadsheets included in Chapter 3
8.4 Spreadsheets included in Chapter 4
8.5 Spreadsheets included in Chapter 6
8.6 Spreadsheets included in Chapter 7
Bibliography
Index