Teaching Secondary School Mathematics: Techniques and Enrichment Units

by Posamentier, Alfred S.; Stepelman, Jay

• ISBN13: 9780136748052
• ISBN10: 0136748058
• Edition: 5th
• Format: Paperback
• Copyright: 1998-10-01
• Publisher: Pearson College Div

Summary

Using NCTM Standards as a foundation, this practical text once again leads the way in secondary mathematics instruction with unique enrichment units, technology updates, and a highly readable style. This book offers current and future instructors the nuts 'n bolts that are needed to improve mathematics instruction at the secondary level. It provides step-by step techniques on preparing lessons and tests, motivating students, designing assignments, and organizing the classroom. Also included are hands-on activities enrichment units, teaching strategies, and pre- and post-tests that are cross-referenced to methods presented earlier in the text.

Table of Contents

1 The Mathematics Teacher Today	1	(5)
A Concise History of American Mathematics Education	2	(2)
The Standards-A Vision for the Future	4	(1)
Summary	4	(2)
2 Planning Lessons, Assigning Homework, and Using Cooperative Learning	6	(45)
The Unit Plan	7	(1)
Long-Range Planning	7	(1)
Short-Range Planning	8	(4)
Performance Objectives	12	(1)
Using Performance Objectives	12	(1)
What Is Cooperative Learning?	12	(1)
How to Structure Small Learning Groups	13	(3)
The Teacher's Role in Managing Small Group Learning	16	(1)
How to Incorporate Small Group Learning into Mathematics Class	17	(1)
Assigning Homework	17	(6)
Format of the Homework Assignment	23	(1)
Reviewing the Homework Assignment	24	(2)
Checking Homework Assignments	26	(1)
Summary	27	(4)
Sample Core Lessons	31	(10)
Sample Before-Standards Lessons	41	(7)
Creating Challenging Tasks	48	(3)
3 Teaching More Effective Lessons	51	(47)
SECTION I Teaching Sensitivity	52	(2)
Concretizing Mathematics	52	(1)
Competition in Mathematics Class	52	(1)
Creating a Positive Learning Climate for All Students	52	(1)
Summary	53	(1)
SECTION 2 Strategies for Teaching More Effective Lessons	54	(21)
Using Tree Diagrams, or Branching	54	(1)
Using Paper Folding or Cutting	55	(2)
A Picture Is Worth a Thousand Words	57	(2)
Recognizing Patterns	59	(2)
Using Mathematical Models and Manipulatives	61	(6)
Using an Overhead Projector	67	(2)
Extending Familiar Concepts	69	(4)
Using a Graphing Calculator	73	(1)
Summary	74	(1)
SECTION 3 Classroom Questioning	75	(11)
Classroom Questioning Features to Develop	76	(1)
Some Precautionary Considerations for Improving Classroom Questioning	77	(2)
Ten Types of Questions to Avoid	79	(3)
Classroom Questioning as a Means to Generate Higher-Order Thinking	82	(2)
Summary	84	(2)
SECTION 4 Motivational Techniques	86	(7)
What Is Motivation?	86	(1)
Motivating Students: Eight Techniques	86	(5)
Summary	91	(2)
SECTION 5 Writing in the Mathematics Classroom	93	(5)
Student Logs	93	(1)
Student Journals	93	(1)
Exposition	93	(1)
Criteria for Evaluation Student Writing Samples	94	(1)
Benefits of Writing Activities in the Mathematics Classroom	95	(1)
Summary	96	(2)
4 The Role of Problem-Solving Strategies in Teaching Mathematics	98	(37)
A Psychological View of Problems Solving	101	(1)
Problem Solving Preliminaries	102	(2)
Problem Solving in Mathematics	104	(5)
Strategies for Solving Problems	109	(7)
Standard Algebraic Problems	116	(2)
Standard Geometric Problems	118	(3)
Nonroutine Problems	121	(4)
Creating Mathematical Problems	125	(1)
Summary	126	(9)
5 Using Technology to Enhance Mathematics Instruction	135	(39)
Using Technology to Enrich Instruction	136	(1)
Calculators	136	(5)
Enrichment in the 21st Century	141	(1)
Pseudo-Calculators	141	(1)
Computers	141	(4)
Computer Literacy	145	(2)
Computer Classroom Management	147	(6)
The Geometer's Sketchpad: Software to Explore Geometry Dynamically	153	(1)
A Brief Overview of The Geometer's Sketchpad	154	(1)
The Geometer's Sketchpad In the Classroom	155	(7)
A Quick Start with The Geometer's Sketchpad	162	(8)
The Geometer's Sketchpad and Enrichment Units	170	(1)
Summary	171	(3)
6 Alternative Student Assessments and Grading Strategies	174	(20)
Constructing a Class Test	175	(7)
Administering a Test	182	(1)
Grading a Test	183	(2)
Interpreting Test Results	185	(1)
Multiple Choice Tests	185	(1)
Student Responsibilities	186	(1)
Parental Responsibilities	187	(1)
Determining the Course Grade	187	(1)
Adjusted Classroom Tests and Quizes for Post-Standards Assessments	188	(1)
Creating New Tests from Old	188	(1)
Other Grading Schemes	189	(1)
Summary	189	(5)
7 Enriching Mathematics Instruction	194	(22)
Enriching Mathematics Instruction with a Historical Approach	195	(7)
Enrichment Techniques for All Levels	202	(1)
The Gifted Student	203	(3)
Using Calculators to Enrich Instruction	206	(2)
Models and Manipulatives that Enrich Instruction	208	(4)
Summary	212	(4)
8 Extracurricular Activities in Mathematics	216	(16)
The Mathematics Club	217	(1)
Mathematics Teams	217	(2)
Mathematics Contests	219	(1)
Mathematics Projects	219	(2)
The Mathematics Fair	221	(1)
The School Mathematics Magazine	222	(1)
The Mathematics Assembly Program	222	(1)
Guest Speakers Program	223	(1)
Class Trips of Mathematical Significance	223	(1)
Peer Teaching Program	223	(1)
The Computer	224	(1)
The Bulletin Board	224	(1)
Summary	224	(8)
9 The Learning Environment, Teacher Sensitivity, and Mathematics Teachers as Professionals	232	(14)
The Professional Teacher	233	(3)
Evaluation of the Teacher	236	(1)
Summary	237	(9)
10 Enrichment Units for the Secondary School Classroom	246	(242)
1 Constructing Odd Order Magic Squares	253	(2)
2 Constructing Even Order Magic Squares	255	(2)
3 Introduction to Alphametics	257	(2)
4 A Checkerboard Calculator	259	(2)
5 The Game of Nim	261	(2)
6 The Tower of Hanoi	263	(2)
7 What Day of the Week Was It?	265	(5)
8 Palindromic Numbers	270	(1)
9 The Fascinating Number Nine	271	(2)
10 Unusual Number Properties	273	(2)
11 Enrichment with a Hand-Held Calculator	275	(2)
12 Symmetric Multiplication	277	(2)
13 Variations on a Theme-Multiplication	279	(2)
14 Ancient Egyptian Arithmetic	281	(2)
15. Napier's Rods	283	(1)
16 Unit Pricing	284	(1)
17 Successive Discounts and Increases	285	(2)
18 Prime and Composite Factors of a Whole Number	287	(1)
19 Prime Numeration System	288	(2)
20 Repeating Decimal Expansions	290	(2)
21 Peculiarities of Perfect Repeating Decimals	292	(1)
22 Patterns in Mathematics	293	(2)
23 Googol and Googolplex	295	(2)
24 Mathematics of Life Insurance	297	(2)
25 Geometric Dissections	299	(2)
26 The Klein Bottle	301	(1)
27 The Four Color Map Problem	302	(2)
28 Mathematics on a Bicycle	304	(3)
29 Mathematics and Music	307	(2)
30 Mathematics in Nature	309	(2)
31 The Birthday Problem	311	(1)
32 The Structure of the Number System	312	(2)
33 Excursions in Number Bases	314	(2)
34 Raising Interest	316	(2)
35 Reflexive, Symmetric, and Transitive Relations	318	(2)
36 Bypassing an Inaccessible Region	320	(1)
37 The Inaccessible Angle	321	(2)
38 Triangle Constructions	323	(2)
39 The Criterion of Constructibility	325	(2)
40 Constructing Radical Lengths	327	(1)
41 Constructing a Pentagon	328	(1)
42 Investigation the Isosceles Triangle Fallacy	329	(2)
43 The Equiangular Point	331	(1)
44 The Minimum Distance Point of a Triangle	332	(2)
45 The Isosceles Triangle Revisited	334	(2)
46 Reflective Properties of the Plane	336	(2)
47 Finding the Length of a Cevian of a Triangle	338	(2)
48 A Surprising Challenge	340	(2)
49 Making Discoveries in Mathematics	342	(1)
50 Tessellations	343	(2)
51 Introducing the Pythagorean Theorem	345	(2)
52 Trisection Revisited	347	(2)
53 Proving Lines Concurrent	349	(2)
54 Squares	351	(1)
55 Proving Points Collinear	352	(2)
56 Angle Measurement with a Circle	354	(2)
57 Trisecting a Circle	356	(1)
58 Ptolemy's Theorem	357	(2)
59 Constructing PI	359	(2)
60 The Arbelos	361	(2)
61 The Nine-Point Circle	363	(1)
62 The Euler Line	364	(2)
63 The Simson Line	366	(1)
64 The Butterfly Problem	367	(2)
65 Equicircles	369	(2)
66 The Inscribed Circle and the Right Triangle	371	(2)
67 The Golden Rectangle	373	(2)
68 The Golden Triangle	375	(2)
69 Geometric Fallacies	377	(2)
70 Regular Polyhedra	379	(2)
71 An Introduction to Topology	381	(1)
72 Angles on a Clock	382	(2)
73 Averaging Rates--The Harmonic Mean	384	(2)
74 Howlers	386	(2)
75 Digit Problems Revisited	388	(1)
76 Algebraic Identities	389	(2)
77 A Method for Factoring Trinomials of the Form: ax(2) + bx + c	391	(1)
78 Solving Quadratic Equations	392	(2)
79 The Euclidean Algorithm	394	(2)
80 Prime Numbers	396	(2)
81 Algebraic Fallacies	398	(2)
82 Sum Derivations with Arrays	400	(2)
83 Pythagorean Triples	402	(2)
84 Divisibility	404	(2)
85 Fibonacci Sequence	406	(2)
86 Diophantine Equations	408	(1)
87 Continued Fractions and Diophantine Equations	409	(2)
88 Simplifying Expressions Involving Infinity	411	(2)
89 Continued Fraction Expansions of Irrational Numbers	413	(2)
90 The Farey Sequence	415	(2)
91 The Parabolic Envelope	417	(2)
92 Application of Congruence to Divisibility	419	(2)
93 Problem Solving--A Reverse Strategy	421	(3)
94 Decimals and Fractions in Other Bases	424	(1)
95 Polygonal Numbers	425	(3)
96 Networks	428	(2)
97 Angle Trisection--Possible or Impossible?	430	(1)
98 Comparing Means	431	(2)
99 Pascal's Pyramid	433	(2)
100 The Multinational Theorem	435	(2)
101 Algebraic Solution of Cubic Equations	437	(2)
102 Solving Cubic Equations	439	(2)
103 Calculating Sums of Finite Series	441	(2)
104 A General Formula for the Sum of a Series of the Form XXX t(r))	443	(3)
105 A Parabolic Calculator	446	(2)
106 Constructing Ellipses	448	(2)
107 Constructing the Parabola	450	(2)
108 Using Higher Plane Curves to Trisect an Angle	452	(2)
109 Constructing Hypocycloid and Epicycloid Circular Envelopes	454	(2)
110 The Harmonic Sequence	456	(1)
111 Transformations and Matrices	457	(3)
112 The Method of Differences	460	(1)
113 Probability Applied to Baseball	461	(2)
114 Introduction to Geometric Transformations	463	(2)
115 The Circle and the Cardioid	465	(3)
116 Complex Number Applications	468	(2)
117 Hindu Arithmetic	470	(2)
118 Proving Numbers Irrational	472	(1)
119 How to Use a Computer Spreadsheet to Generate Solutions to Certain Mathematics Problems	473	(2)
120 The Three Worlds of Geometry	475	(3)
121 XXX Mix	478	(1)
122 Graphical Iteration	479	(2)
123 The Feigenbaum Plot	481	(2)
124 The Sierpinski Triangle	483	(2)
125 Fractals	485	(3)
Index	488