Diversifying mathematics teaching: advanced educational content and methods for prospective elementary teachers
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
New Jersey
World Scientific
2017
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 268 Seiten Illustrationen |
| ISBN: | 9789813206878 9789813208902 |
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| 245 | 1 | 0 | |a Diversifying mathematics teaching |b advanced educational content and methods for prospective elementary teachers |c by Sergei Abramovich (State University of New York at Potsdam, USA) |
| 264 | 1 | |a New Jersey |b World Scientific |c 2017 | |
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Record in the Search Index
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| adam_text | Contents
Preface ....*.................................................... V
Chapter 1 ....................................................... 1
Teaching Elementary Mathematics: Standards, Recommendations
and Teacher Candidates’ Perspectives ............................ 1
LI Introduction............................................... 1
1.2 Questions as the Major Means of Learning Mathematics..... 3
1.3 Answering Questions Both Procedurally and
Conceptually ............................................ 6
1.4 Connecting Algorithmic Skills and Conceptual
Understanding............................................ 8
1.5 Developing Deep Understanding of Mathematics
Through Making Conceptual Connections .................. 11
1.6 Teaching and Learning to Think Mathematically........... 13
Chapter 2....................................................... 17
Counting Techniques .......................................... 17
2.1 Introduction............................................ 17
2.2 Rules of Sum and Product................................ 19
2.3 Tree Diagram and the Rule of Product ................... 20
2.4 Permutation of Letters in a Word........................ 21
2.5 Combinations without Repetitions........................ 24
2.6 Combinations with Repetitions......................... 27
Chapter 3........................................................ 33
Counting and Reasoning with Manipulative Materials.............. 33
3.1 Introduction............................................ 33
3.2 Constructing a Triangle out of Straws .................. 35
3.2.1 Reflection on the activity with straws ............ 38
3.2.2 A real-life application of the triangle inequality. 40
3.2.3 Modifying the S2AC2 algorithm to enable
linguistic coherency.............................. 41
3.2.4 How many triangles can be constructed? ........... 43
3.2.5 Using multicolored straws ....................... 45
3.2.5.1 An equilateral triangle................... 46
3.2.5.2 An isosceles triangle with the base being
the smaller side........................... 47
3.2.5.3 An isosceles triangle with the base being
the larger side ........................... 48
3.2.5.4 A scalene triangle ....................... 49
3.3 Two Types of Representation as Means of Transition
from Visual to Symbolic.................................. 49
3.4 Signature Pedagogy of Elementary Mathematics
Teacher Education ....................................... 51
3.5 Towards Rich Interpretations of Manipulative
Representations.......................................... 51
3.5.1 Manipulative representation as text ............... 51
3.5.2 From “brothers” to Pascal’s triangle............. 54
3.6 Learning to Move from One Type of Symbolism to
Another and Back ........................................ 56
3.7 Perimeter and Area Using Square Tiles.................... 58
3.8 The Importance of Teacher Guidance in Using
Manipulative Materials by Students ...................... 63
3.9 Conceptualizing Base-Ten System Using
Manipulative Materials .................................. 65
3.10 Modeling as a Way of Creating Isomorphic
Relationships ......................................... 68
.K
Chapter 4..................... 1-,.............................. 71
We Write What We See (W4S) Principle.............................. 71
4.1 Introduction ............................................ 71
4.2 W4S Principle and the Duality of Its Affordances ........ 73
4.3 W4S Principle in Teaching Primary School Mathematics ... 74
4.4 Comparing Non-Unit Fractions Using Area Model............ 78
4.4.1 Comparing fractions being close to each other.... 78
4.4.2 Comparing fractions that are a unit fraction
short of the whole................................. 80
4.5 From Comparison of Fractions to Arithmetical Operations
Using Area Model......................................... 81
4.5.1 Fractions as part-whole and divisor-dividend
models ............................................ 81
4.5.2 The concept of common denominator................... 83
4.5.3 Reducing a fraction to the simplest form........... 84
4.5.4 Using unit fractions as benchmark fractions ....... 84
4.5.5 Multiplying fractions using area model.............. 87
4.6 Dividing Fractions Using Area Model....................... 89
4.6.1 Partition model for division supports
contextualization................................... 89
4.6.2 The importance of unit in solving word problems
with fractions...................................... 91
4.6.3 The meaning of “invert and multiply” rule .......... 94
4.7 Ratio and Proportion ..................................... 96
4.8 Percent and Decimal as Alternative Representations
of a Fraction ............................................ 98
4.9 Multiplying and Dividing Decimal Fractions .............. 100
Chapter 5........................................................ 105
Partitioning Integers into Like Summands......................... 105
5.1 Introduction............................................. 105
5.2 Partition of Integers into Summands...................... 106
5.3 Activities with Towers Motivate Introduction of
Algebraic Notation....................................... 116
5.4 Ferrers-Young Diagrams .................................. 117
5.5 Recursive Definition of P(n,rri) Informed by
Ferrers-Young Diagrams .................................. 117
5.6 Making Mathematical Connections ......................... 120
5.7 Recursive Definition of Q(n, m) Informed by
Ferrers-Young Diagrams .................................. 122
5.8 Connection to Triangular Numbers Opens a Window
to a New Concept......................................... 125
Chapter 6........................................................ 129
Hidden Curriculum of Mathematics Teacher Education .............. 129
6.1 Introduction............................................. 129
6.2 The Basic Task........................................... 130
6.3 Background Information: Triangular and Trapezoidal
Numbers................................................. 132
6.3.1 Activities...................................... 132
6.3.2 Solutions to the tasks ........................... 133
6.3.3 Trapezoidal numbers .............................. 137
6.4 Conceptually Oriented Discussion of the Basic
Problem................................................. 139
6.4.1 Grouping the sums by the number of addends ....... 139
6.4.2 Grouping the sums by the first addend............. 142
6.4.3 Partitioning integers into the sums .............. 142
6.5 Discourse Motivated by Multiple Ways of Creating
Sums of Consecutive Natural Numbers..................... 143
6.5.1 Clarifying the meaning of the word special in the
context of arithmetic............................. 143
6.5.2 The first encounter with a special property of the
sums of two addends ............................ 144
6.5.3 Moving from novice to expert practice in
revealing special properties ..................... 146
6.5.4 Exploring the sums of four consecutive integers... 148
6.6 Proof of the Conjecture about Trapezoidal Numbers ...... 151
6.7 Sums in Pairs of Odds and Evens ...................... 152
6.7.1 Learning to generalize from special cases ........ 153
6.7.2 Comparing triangles to trapezoids with the top
row greater than two.............................. 156
6.8 Mathematical Knowledge Used for Teaching
Young Children 4........................................ 158
6.9 How Many Trapezoidal Representations Does
a Number Have and How Can One Find Them? ............... 159
Chapter 7....................................................... 163
Informal Geometry............................................... 163
7.1 Introduction............................................ 163
7.2 Geoboard Explorations .................................. 165
7.3 Towards a Double-Application Environment................ 172
7.4 Guided Exploration on a Computational Geoboard.......... 174
7.5 Transition to a Spreadsheet........................... 178
7.6 Preparing Data for Empirical Induction................. 180
7.7 Abstracting from Numbers to Equations Using
First-Order Symbols ................................... 182
7.8 Visual and Symbolic Deduction of Pick’s Formula........ 183
7.9 Moving to a New Learning Site.......................... 187
7.10 Measuring vs. Counting................................. 188
7.11 Encountering Limitation of the Environment ............ 190
7.12 From Particular to General through Visualization....... 191
7.13 Communicating about Mathematics ....................... 194
Chapter 8....................................................... 197
Probability as a Blend of Theory and Experiment................. 197
8.1 Introduction........................................... 197
8.2 Basic Concepts and Tools of the Probability Strand.....200
8.2.1 Randomness and sample space ............200
8.2.2 A more complicated example of constructing
a sample space...................................202
8.2.3 Different representations of a sample space.....202
8.3 Fractions as Tools in Measuring Chances ................205
8.4 Bernoulli Trials and the Law of Large Numbers...........207
8.5 A Problem of Chevalier De Mere .........................209
8.6 A Modification of the Problem of De Méré................210
8.7 Wagering for the Odds/Evens in a Game of Chance ........212
8.8 Paradoxes in the Theory of Probability .................214
8.8.1 Bertrand’s Paradox Box problem...................214
8.8.2 Monty Hall Dilemma ..............................215
8.9 Probabilistic Perspective on Partitioning Problems .....217
8.9.1 A problem of tossing three dice ................217
8.9.2 Unordered partitions of integers into unequal
summands.........................................218
8.10 Experimental Probability ...............................219
8.10.1 Experimental probability calls for a long
series of observations...........................219
8.10.2 Comparing experimental and theoretical
probabilities when tossing a fair coin...........220
8.10.3 Calculating relative frequencies for the problems
of De Mere ......................................222
8.11 Exploring Irreducibility of Fractions through the
Lenses of Probability ..................................224
Chapter 9........................................................227
Using Counter-Examples in the Teaching of Elementary
Mathematics......................................................227
9.1 Introduction............................................227
9.2 The Pedagogy of Using Counter-Examples..................228
9.2.1 The role of linguistic constraints...............228
9.2.2 A counter-example as a motivation for further
learning.........................................231
9.3 Providing Explanation through Counter-Examples .........231
9.4 Constructing a Counter-Example: an Illustration ........233
9.4.1 From modeling with fractions to algebraic
generalization...................................233
9.4.2 From a counter-example to its conceptualization ... 234
9.4.3 A family of jumping fractions found
by a teacher candidate...........................235
9.4.4 Conceptualizing the teacher candidate’s choice
of seven ........................................237
9.5 A Counter-Example and Empirical Induction ..............238
9.6 Counter-Example as a Tool for Conceptual Development .. 240
9.7 Counter-Example in Explaining the Meaning of
Negative Transfer ......................................241
9.7.1 Inequalities.....................................242
9.7.2 Counting matchsticks.............................244
9.8 Transition from Combinations without Repetitions to
Combinations with Repetitions...........................246
9.9 Missing Fibonacci Numbers ..............................248
Bibliography.....................................................251
Index ...........................................................265
|
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| spelling | Abramovich, Sergei Verfasser (DE-588)141706333 aut Diversifying mathematics teaching advanced educational content and methods for prospective elementary teachers by Sergei Abramovich (State University of New York at Potsdam, USA) New Jersey World Scientific 2017 XVI, 268 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Mathematik Mathematics teachers Training of Elementary school teachers Training of Mathematics Study and teaching (Elementary) Grundschule (DE-588)4022349-8 gnd rswk-swf Mathematikunterricht (DE-588)4037949-8 gnd rswk-swf Mathematikunterricht (DE-588)4037949-8 s Grundschule (DE-588)4022349-8 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029734409&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Abramovich, Sergei Diversifying mathematics teaching advanced educational content and methods for prospective elementary teachers Mathematik Mathematics teachers Training of Elementary school teachers Training of Mathematics Study and teaching (Elementary) Grundschule (DE-588)4022349-8 gnd Mathematikunterricht (DE-588)4037949-8 gnd |
| subject_GND | (DE-588)4022349-8 (DE-588)4037949-8 |
| title | Diversifying mathematics teaching advanced educational content and methods for prospective elementary teachers |
| title_auth | Diversifying mathematics teaching advanced educational content and methods for prospective elementary teachers |
| title_exact_search | Diversifying mathematics teaching advanced educational content and methods for prospective elementary teachers |
| title_full | Diversifying mathematics teaching advanced educational content and methods for prospective elementary teachers by Sergei Abramovich (State University of New York at Potsdam, USA) |
| title_fullStr | Diversifying mathematics teaching advanced educational content and methods for prospective elementary teachers by Sergei Abramovich (State University of New York at Potsdam, USA) |
| title_full_unstemmed | Diversifying mathematics teaching advanced educational content and methods for prospective elementary teachers by Sergei Abramovich (State University of New York at Potsdam, USA) |
| title_short | Diversifying mathematics teaching |
| title_sort | diversifying mathematics teaching advanced educational content and methods for prospective elementary teachers |
| title_sub | advanced educational content and methods for prospective elementary teachers |
| topic | Mathematik Mathematics teachers Training of Elementary school teachers Training of Mathematics Study and teaching (Elementary) Grundschule (DE-588)4022349-8 gnd Mathematikunterricht (DE-588)4037949-8 gnd |
| topic_facet | Mathematik Mathematics teachers Training of Elementary school teachers Training of Mathematics Study and teaching (Elementary) Grundschule Mathematikunterricht |
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