Transcript MATHEMATICS STANDARDS AND THEIR IMPLEMENTATION

How to Make Standards
Operational in the Curriculum
Dr. Don S. Balka
Saint Mary’s College
Notre Dame, Indiana
Director, National Council of Teachers of Mathematics
Macmillan/McGraw-Hill
Education in Saudi Arabia
• Primary School (6 - 12): Mathematics
• Intermediate School (12 - 15): Mathematics
• General Secondary School (15 - 18):
Mathematics
• Technical Secondary School (15 - 18):
• Vocational/Technical: Mathematics
• Commercial: Mathematics
• Agriculture: Mathematics
© 2008 Don Balka. All rights reserved.
Mathematics in a Changing
World
• Mathematics for life
• Mathematics as a part of cultural heritage
• Mathematics for the workplace
• Mathematics for the scientific and technical
community
© 2008 Don Balka. All rights reserved.
Historical Background
• All students should learn important
mathematical concepts with understanding
• Standards are critical for those who are
decision-makers
• Curriculum and Evaluation Standards for
School Mathematics
• Professional Standards for Teaching
Mathematics
• Assessment Standards for School
Mathematics
© 2008 Don Balka. All rights reserved.
Principles and Standards
for School Mathematics
2000
© 2008 Don Balka. All rights reserved.
Purpose
• Set forth a comprehensive and coherent set of
goals for mathematics for all students, PK - 12
• Serve as a resource for teachers, education
leaders, and policymakers to use in examining
and improving the quality of mathematics
instruction programs
© 2008 Don Balka. All rights reserved.
Purpose
• Guide the development of curriculum
Purposeand instructional
frameworks, assessments,
materials
• Stimulate ideas and ongoing conversations at
the national, provincial or state, and local levels
about how best to help students gain a deep
understanding of important mathematics
© 2008 Don Balka. All rights reserved.
Principles for School Mathematics
Equity
High expectations and strong
support
Curriculum
Coherent, focused, articulated
Teaching
Understanding, challenging,
supporting
Learning
Understanding, building new
knowledge from experience
Assessment
Supporting the learning and
furnishing information
Technology
Essential; enhancing student
learning
© 2008 Don Balka. All rights reserved.
Principles and Standards for School Mathematics
NCTM Curriculum Principle
“…a well-articulated curriculum gives
teachers guidance regarding important ideas
or major themes, which receive special attention at
different points in time. It also gives guidance about
the depth of study warranted at particular times and
when closure is expected for particular skills or
concepts.”
(PSSM)
© 2008 Don Balka. All rights reserved.
Standards for School Mathematics
Content Standards
Number and Operations
Algebra
Geometry
Measurement
Data Analysis and Probability
© 2008 Don Balka. All rights reserved.
Standards for School Mathematics
Process Standards
Problem Solving
Reasoning and Proof
Communication
Connections
Representation
© 2008 Don Balka. All rights reserved.
A Framework for Mathematical Proficiency
•Conceptual understanding: comprehension of
mathematical concepts, operations, and relations
•Procedural fluency: skill in carrying out procedures
flexibly, accurately, efficiently, and appropriately
•Strategic competence: ability to formulate, represent,
and solve mathematical problems
•Adaptive reasoning: capacity for logical thought,
reflection, explanation, and justification
•Productive disposition: habitual inclination to see
mathematics as sensible, useful, and worthwhile
National Research Council, 2001
© 2008 Don Balka. All rights reserved.
Number of Fourth-Grade GLE per State by Strand
© 2008 Don Balka. All rights reserved.
© 2008 Don Balka. All rights reserved.
What Are Curriculum Focal Points?
Concept of a Focal Point
Each topic begins in the background
of a particular grade as a context for
exploration, then emerges into the
foreground at a later grade level for
intense instruction, and then recedes
into the background again.
© 2008 Don Balka. All rights reserved.
The Content of Curriculum Focal Points
•Three per grade level, PreK-8
•Often represent multiple content strands
•Describe the majority of instruction for a
specific grade level
•Taken together across grade levels,
provide the major components of a
mathematically sound, coherent, and
cohesive preK- 8 curriculum
© 2008 Don Balka. All rights reserved.
How Will Curriculum Focal Points
Be Used?
•To design instruction around the question,
“What are the most important ideas at my
grade level?”
•To provide information about how ideas
at one grade level fit with the important
ideas in previous and following grades
•To prioritize uses of activities,
assessments, and other published
materials
© 2008 Don Balka. All rights reserved.
© 2008 Don Balka. All rights reserved.
Curriculum Focal Points
Algebra and Its Connections
GRADE 3 CONNECTIONS
Algebra
•
Understanding properties of multiplication and the relationship
between multiplication and division is a part of algebra readiness
that develops at grade 3
•
The creation and analysis of patterns and relationships involving
multiplication and division should occur at this grade level
•
Students build a foundation for later understanding of functional
relationships by describing relationships in context with such
statements as, “The number of legs is 4 times the number of
chairs.”
© 2008 Don Balka. All rights reserved.
Hawaii Objectives for Algebra
Grade Cluster Benchmarks
2-3
Create and describe spatial and numerical patterns
State a general rule that describes a given pattern
Represent the same pattern in different forms
Use concrete, pictorial, and verbal representations of
numerical situations
Symbolize mathematical situations
Quantify comparisons and describe change in familiar
situations
© 2008 Don Balka. All rights reserved.
Arizona Objectives for Algebra
PERFORMANCE OBJECTIVES
Concept 1: Patterns
(PO 1) Communicate a grade-level appropriate iterative pattern, using
symbols or numbers (3 - 5)
Concept 2: Functions and Relationships
(PO 1) Describe the rule used in a simple grade-level appropriate
function (K - 8)
© 2008 Don Balka. All rights reserved.
Arizona Objectives for Algebra
PERFORMANCE OBJECTIVES
Concept 3: Algebraic Representations
(PO 1) Evaluate expressions involving the four basic operations by
substituting given whole numbers (decimals, fractions, rational
values) for the variable(s) (4 - 6, 8)
Concept 3: Analysis of Change
(PO 1) Identify the change in a variable over time (1 - 4)
© 2008 Don Balka. All rights reserved.
EXAMPLES
Making Connections
in Algebra
Curriculum Focal Points
Algebra and Its Connections
ARROW DIAGRAMS
0
10
20
30
40
50
60
70
80
90
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
© 2008 Don Balka. All rights reserved.
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
Curriculum Focal Points
Algebra and Its Connections
ARROW DIAGRAMS
Go one place to the right!
Go one place to the left!
Go one place up!
Go one place down!
© 2008 Don Balka. All rights reserved.
Curriculum Focal Points
Algebra and Its Connections
34
0
10
20
30
40
50
60
70
80
90
1
11
21
31
41
51
61
71
81
91
© 2008 Don Balka. All rights reserved.
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
Curriculum Focal Points
Algebra and Its Connections
68
12
GUESS MY RULE!
© 2008 Don Balka. All rights reserved.
Curriculum Focal Points
Algebra and Its Connections
68
GUESS MY RULE!
ADD ONE
(0,1), (1,2), (3,4), (99,100)
y=x+1
© 2008 Don Balka. All rights reserved.
Curriculum Focal Points
Algebra and Its Connections
7
•
6
•
5
•
4
•
3
•
2
•
1 •
0
0
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1
2
3
4
5
6
7
Curriculum Focal Points
Algebra and Its Connections
HOW MANY WAYS CAN YOU MAKE CHANGE FOR
100 HALALAS USING 5 AND 10 HALALA COINS?
An Equation: 5X + 10Y = 100
5 Halala
0
2
4
?
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10 Halala
Total
10
9
8
?
100
100
100
100
MAKE ME AN L!
Stage 1 Stage 2 Stage 3 Stage 4
How many squares are needed for Stage 5?
How many squares are needed for Stage 10?
How many squares are needed for Stage N?
Where is the Algebra?
© 2008 Don Balka. All rights reserved.
MAKE ME AN L!
Stage 1
Stage
1
2
3
4
5
Number of
S quares
1
3
5
7
9
Stage 2 Stage 3
Stage 4
What do you notice about the
number of squares for each stage?
Functions!
Graphing!
f(N) = 2N - 1
20
18
16
14
12
10
8
6
4
2
0
1
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2
3
4
5
6
7
8
9
10
MAKE ME AN L!
Extending the Problem
Stage 1
Stage 2 Stage 3
Stage 4
What is the total number of squares needed to construct the
first 10 stages?
1
4
9
16
Where’s the Algebra?
© 2008 Don Balka. All rights reserved.
MAKE ME AN L!
Extending the Problem
How many squares in the first four stages?
Stage 1
Stage 2 Stage 3
1+3+5+7
7+5+3+1
8+8+8+8
Stage 4
4 x 8 = 32
32 ÷ 2 = 16 = 42
For 10 Stages, 102
For N Stages, N2
The sum of the first N odd numbers is N2.
© 2008 Don Balka. All rights reserved.
Skills for the 21st Century
• Computational Skills Needed by Middle Grades
Students Preparing to Live in the 21st Century
•Intelligent use of calculators, with a clear
understanding of their shortcomings
•Good operational knowledge of basic facts
•Clear, practical understanding of place value
and the system of numeration
•Understanding of simple fractions, decimals,
and percents, including their interrelatedness
© 2008 Don Balka. All rights reserved.
Skills for the 21st Century
• Computational Skills Needed by Middle Grades
Students Preparing to Live in the 21st Century
•Self-confidence in, and ability to use, a range
of techniques for computing
•Inclination and the ability to use thinking
strategies first when checking calculations,
estimating results, or performing various
calculations
© 2008 Don Balka. All rights reserved.
Skills for the 21st Century
• Computational Skills Needed by Middle Grades
Students Preparing to Live in the 21st Century
•Instinct and willingness to think about
numbers in natural, comfortable, and flexible
ways when computational results (where
exact answers of estimates) are produced.
•Instinct and willingness to reflect on
numerical results so as to judge their
reasonableness
•McIntosh, Reys, and Reys, 1997
© 2008 Don Balka. All rights reserved.
Considerations
Who will write the standards for mathematics?
Ministry of Education
Teachers
University Mathematicians
Business/Industry Leaders
What are the major content strands for the
mathematics curriculum?
What are the major topics for the mathematics
curriculum at each level: Primary, Intermediate,
Secondary?
© 2008 Don Balka. All rights reserved.
Considerations
Will the mathematics curriculum include process
standards: Problem Solving, Reasoning and
Proof, Communication, Connections,
Representation?
How will professional development for teachers
be offered?
Nationally, local districts
How will mathematics standards impact teacher
preparation for the primary, intermediate, and
secondary levels?
© 2008 Don Balka. All rights reserved.
SUMMARY
•All students should learn important
mathematical concepts with understanding
•The purpose of standards is to provide a
comprehensive and coherent set of goals for
mathematics for all students
•Standards serve as a resource for teachers,
education leaders, and policymakers to use
in examining and improving the quality of
mathematics instruction programs
•Standards guide the development of curriculum
frameworks, assessments, and instructional
materials
© 2008 Don Balka. All rights reserved.
MATHEMATICS CLASSROOM OF THE FUTURE
•Teachers are empowered.
•The curriculum is implemented as designed.
•Multiple instructional strategies are employed.
•Students are actively engaged.
•Assessment is varied.
Balka, Hull, and Harbin-Miles, 2008
© 2008 Don Balka. All rights reserved.
Thank you.