Mathematics kindergarten to grade 9

2007·2.9 MB·English

by  Alberta. Alberta Education

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MATHEMATICS
KINDERGARTEN TO
GRADE
9
INTRODUCTION come to classrooms with varying knowledge, life
experiences and backgrounds. A key component
The Mathematics Kindergarten to Grade 9 in successfully developing numeracy is making
Program of Studies has been derived from The connections to these backgrounds and
Common Curriculum Framework for experiences.
K-9 Mathematics: Western and Northern
Canadian Protocol, May 2006 (the Common Students learn by attaching meaning to what they
Curriculum Framework). The program of studies do, and they need to construct their own meaning
incorporates the conceptual framework for of mathematics. This meaning is best developed
Kindergarten to Grade 9 Mathematics and the when learners encounter mathematical
general outcomes and specific outcomes that were experiences that proceed from the simple to the
established in the Common Curriculum complex and from the concrete to the abstract.
Framework. Through the use of manipulatives and a variety of
pedagogical approaches, teachers can address the
diverse learning styles, cultural backgrounds and
BACKGROUND
developmental stages of students, and enhance
within them the formation of sound, transferable
The Common Curriculum Framework was mathematical understandings. At all levels,
developed by the seven ministries of education students benefit from working with a variety of
(Alberta, British Columbia, Manitoba, materials, tools and contexts when constructing
Northwest Territories, Nunavut, Saskatchewan meaning about new mathematical ideas.
and Yukon Territory) in collaboration with Meaningful student discussions provide essential
teachers, administrators, parents, business links among concrete, pictorial and symbolic
representatives, post-secondary educators and representations ofmathematical concepts.
others. The framework identifies beliefs about
mathematics, general and specific student The learning environment should value and
s, and achievement indicators agreed respect the diversity of students' experiences and
the sevenjurisdictions. ways of thinking, so that students are comfortable
taking intellectual risks, asking questions and
posing conjectures. Students need to explore
FS ABOUT STUDENTS AND problem-solving situations in order to develop
EMATICS LEARNING personal strategies and become mathematically
literate. They must realize that it is acceptable to
are curious, active learners with solve problems in a variety of ways and that a
al interests, abilities and needs. They variety ofsolutions may be acceptable.
gr.K-09
Mathematics (K-9) /I
CURR GD Education, Alberta,Canada (2007)
HIST
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in 2012 with funding from
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MATHEMATICS
KINDERGARTEN TO
GRADE
9
INTRODUCTION come to classrooms with varying knowledge, life
experiences and backgrounds. A key component
The Mathematics Kindergarten to Grade 9 in successfully developing numeracy is making
Program of Studies has been derived from The connections to these backgrounds and
Common Curriculum Framework for experiences.
K-9 Mathematics: Western and Northern
Canadian Protocol, May 2006 (the Common Students learn by attaching meaning to what they
Curriculum Framework). The program of studies do, and they need to construct their own meaning
incorporates the conceptual framework for of mathematics. This meaning is best developed
Kindergarten to Grade 9 Mathematics and the when learners encounter mathematical
general outcomes and specific outcomes that were experiences that proceed from the simple to the
established in the Common Curriculum complex and from the concrete to the abstract.
Framework. Through the use of manipulatives and a variety of
pedagogical approaches, teachers can address the
diverse learning styles, cultural backgrounds and
BACKGROUND
developmental stages of students, and enhance
within them the formation of sound, transferable
The Common Curriculum Framework was mathematical understandings. At all levels,
developed by the seven ministries of education students benefit from working with a variety of
(Alberta, British Columbia, Manitoba, materials, tools and contexts when constructing
Northwest Territories, Nunavut, Saskatchewan meaning about new mathematical ideas.
and Yukon Territory) in collaboration with Meaningful student discussions provide essential
teachers, administrators, parents, business links among concrete, pictorial and symbolic
representatives, post-secondary educators and representations ofmathematical concepts.
others. The framework identifies beliefs about
mathematics, general and specific student The learning environment should value and
outcomes, and achievement indicators agreed respect the diversity of students' experiences and
upon by the sevenjurisdictions. ways of thinking, so that students are comfortable
taking intellectual risks, asking questions and
posing conjectures. Students need to explore
BELIEFS ABOUT STUDENTS AND problem-solving situations in order to develop
MATHEMATICS LEARNING personal strategies and become mathematically
literate. They must realize that it is acceptable to
Students are curious, active learners with solve problems in a variety of ways and that a
individual interests, abilities and needs. They variety ofsolutions may be acceptable.
Mathematics (K-9) /I
©AlbertaEducation,Alberta,Canada (2007)
FIRST NATIONS, METIS AND INUIT Teachers, students and parents need to recognize
PERSPECTIVES the relationship between the affective and
cognitive domains, and attempt to nurture those
First Nations, Metis and Inuit students in northern aspects of the affective domain that contribute to
and western Canada come from diverse positive attitudes. To experience success,
geographic areas with varied cultural and students must be taught to set achievable goals
linguistic backgrounds. Students attend schools in and assess themselves as they work toward these
a variety of settings, including urban, rural and goals.
isolated communities. Teachers need to
understand the diversity of students' cultures and Striving toward success and becoming
experiences. autonomous and responsible learners are ongoing,
reflective processes that involve revisiting the
First Nations, Metis and Inuit stud—ents often have setting and assessing ofpersonal goals.
a holistic view of the environment they look for
connections in learning and learn best when
mathematics is contextualized. They may come EARLY CHILDHOOD
from cultures where learning takes place through
active participation. Traditionally, little emphasis Young children are naturally curious and develop
was placed upon the written word, so oral
a variety of mathematical ideas before they enter
communication and practical applications and
Kindergarten. Children make sense of their
experiences are important to student learning and
environment through observations and interactions
understanding. By understanding and responding
at home, in daycares, in preschools and in the
to nonverbal cues, teachers can optimize student
community. Mathematics learning is embedded in
learning and mathematical understanding.
everyday activities, such as playing, reading,
A variety of teaching and assessment strategies beading, baking, storytelling and helping around
the home.
help build upon the diverse knowledge, cultures,
communication styles, skills, attitudes,
experiences and learning styles ofstudents. Activities can contribute to the development of
number and spatial sense in children. Curiosity
Research indicates that when strategies go beyond about mathematics is fostered when children are
the incidental inclusion of topics and objects engaged in, and talking about, such activities as
unique to a culture or region, greater levels of comparing quantities, searching for patterns,
understanding can be achieved (Banks and Banks, sorting objects, ordering objects, creating designs
1993). and building with blocks.
Positive early experiences in mathematics are as
AFFECTIVE DOMAIN
critical to child development as are early literacy
experiences.
A
positive attitude is an important aspect of the
affective domain and has a profound impact on
learning. Environments that create a sense of GOALS FOR STUDENTS
belonging, encourage risk taking and provide
opportunities for success help develop and The main goals of mathematics education are to
maintain positive attitudes and self-confidence
prepare students to:
within students. Students with positive attitudes
toward learning mathematics are likely to be • use mathematics confidently to solve problems
motivated and prepared to learn, participate • communicate and reason mathematically
willingly in classroom activities, persist in • appreciate and value mathematics
challenging situations and engage in reflective • make connections between mathematics and its
practices. applications
2/ Mathematics (K-9)
(2007) ©AlbertaEducation,Alberta,Canada
• commit themselves to lifelong learning
• become mathematically literate adults, using
mathematics to contribute to society.
Students who have met these goals will:
• gain understanding and appreciation of the
contributions of mathematics as a science,
philosophy and art
• exhibit a positive attitude toward mathematics
• engage and persevere in mathematical tasks
and projects
• contribute to mathematical discussions
• take risks in performing mathematical tasks
• exhibitcuriosity.
Mathematics (K-9) /3
©AlbertaEducation,Alberta,Canada (2007)
CONCEPTUAL FRAMEWORK FOR K-9 MATHEMATICS
The chart below provides an overview of how mathematical processes and the nature of mathematics
influence learning outcomes.
K123456789
~~~~~~~-------._..._^^ GRADE
^"~~~~~~-^^^
STRAND
Number
NATURE
Patternsand Relations OF
MATHEMATICS
• Patterns
• VariablesandEquations GENERALOUTCOMES
AND Change,Constancy,
ShapeandSpace SPECIFICOUTCOMES* NumberSense,
• Measurement Patterns,Relationships,
• 3-DObjectsand2-DShapes SpatialSense,
• Transformations Uncertainty
StatisticsandProbability
• DataAnalysis
• ChanceandUncertainty
MATHEMATICALPROCESSES - Communication,Connections,MentalMathematics
andEstimation,ProblemSolving,Reasoning,
Technology,Visualization
* Achievement indicators for the prescribed program of studies outcomes are provided in the companion
document TheAlberta K-9MathematicsProgram ofStudies withAchievementIndicators, 2007.
Mathematical There are critical components that students must encounterin a mathematics
Processes program in orderto achieve the goals ofmathematics education and embrace
lifelong learning in mathematics.
Students are expected to:
Communication [C] • communicate in orderto learn and express theirunderstanding
Connections [CN] • connect mathematical ideas to other concepts in mathematics, to
everyday experiences and to otherdisciplines
MentalMathematicsand • demonstrate fluency with mental mathematics and estimation
Estimation [ME]
Problem Solving [PS] • develop and apply new mathematical knowledge through problem
solving
Reasoning [R] • develop mathematical reasoning
Technology [T] • select and use technologies as tools for learning and for solving problems
Visualization [V] • develop visualization skills to assist in processing information, making
connections and solving problems.
The program of studies incorporates these seven interrelated mathematical
processes that are intended to permeate teaching and learning.
4/ Mathematics (K-9)
(2007) ©AlbertaEducation,Alberta,Canada
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COMMUNICATION [C] MENTAL MATHEMATICS AND ESTIMATION
[ME]
Students need opportunities to read about,
represent, view, write about, listen to and discuss Mental mathematics is a combination of cognitive
mathematical ideas. These opportunities allow strategies that enhance flexible thinking and
students to create links between their own number sense. It is calculating mentally without
language and ideas, and the formal language and the use ofexternal memory aids.
symbols ofmathematics.
Mental mathematics enables students to determine
Communication is important in clarifying, answers without paper and pencil. It improves
reinforcing and modifying ideas, attitudes and computational fluency by developing efficiency,
beliefs about mathematics. Students should be accuracy and flexibility.
encouraged to use a variety of forms of
communication while learning mathematics. "Even more important than performing
Students also need to communicate their learning computational procedures or using c—alculators is
using mathematical terminology. the greater f—acility that students need more than
ever before with estimation and mental math"
Communication helps students make connections (National Council of Teachers of Mathematics,
among concrete, pictorial, symbolic, oral, written May 2005).
and mental representations ofmathematical ideas.
Students proficient with mental mathematics
CONNECTIONS [CN] "become liberated from calculator dependence,
build confidence in doing mathematics, become
Contextualization and making connections to the more flexible thinkers and are more able to use
experiences of learners are powerful processes in multiple approaches to problem solving"
developing mathematical understanding. This can (Rubenstein, 2001, p. 442).
be particularly true for First Nations, Metis and
Inuit learners. When mathematical ideas are Mental mathematics "provides the cornerstone for
connected to each other or to real-world all estimation processes, offering a variety of
phenomena, students begin to view mathematics alternative algorithms and nonstandard techniques
as useful, relevant and integrated. for finding answers" (Hope, 1988, p. v).
Learning mathematics within contexts and making Estimation is used for determining approximate
connections relevant to learners can validate past values or quantities or for determining the
experiences and increase student willingness to reasonableness of calculated values. It often uses
participate and be actively engaged. benchmarks or referents. Students need to know
when to estimate, how to estimate and what
The brain is constantly looking for and making strategy to use.
connections. "Because the learner is constantly
searching for connections on many levels, Estimation assists individuals in making
educators need to orchestrate the experiences mathematical judgements and in developing
from which learners extract understanding.... useful, efficient strategies for dealing with
Brain research establishes and confirms that situations in daily life.
multiple complex and concrete experiences are
essential for meaningful learning and teaching"
(Caine and Caine, 1991, p. 5).
Mathematics (K-9) /5
©AlbertaEducation,Alberta,Canada (2007)
PROBLEM SOLVING [PS] TECHNOLOGY [T]
Learning through problem solving should be the Technology contributes to the learning of a wide
focus of mathematics at all grade levels. When range of mathematical outcomes and enables
students encounter new situations and respond to students to explore and create patterns, examine
questions of the type How wouldyou ...? or How relationships, test conjectures and solve problems.
could you ...?, the problem-solving approach is
being modelled. Students develop their own Calculators and computers can be used to:
problem-solving strategies by listening to, • explore and demonstrate mathematical
discussing and trying different strategies. relationships and patterns
• organize and display data
A problem-solving activity must ask students to • extrapolate and interpolate
determine a way to get from what is known to • assist with calculation procedures as part of
what is sought. If students have already been solving problems
pgrivoebnlemw,aybsuttporasctoilcvee. thAe tprruoeblpermo,blietmisreqnuoitresa • decrease the time spent on computations when
other mathematical learning is the focus
students to use prior learnings in new ways and
• reinforce the learning ofbasic facts
contexts. Problem solving requires and builds
• develop personal procedures for mathematical
depth of conceptual understanding and student
operations
engagement.
• create geometric patterns
Problem solving is a powerful teaching tool that • simulate situations
fosters multiple, creative and innovative solutions. • develop number sense.
Creating an environment where students openly
Technology contributes to a learning environment
look for, and engage in, finding a variety of
strategies for solving problems empowers students in which the growing curiosity of students can
to explore alternatives and develops confident, lead to rich mathematical discoveries at all grade
levels.
cognitive mathematical risktakers.
REASONING [R] VISUALIZATION [V]
Visualization "involves thinking in pictures and
Mathematical reasoning helps students think
images, and the ability to perceive, transform and
logically and make sense ofmathematics. Students
recreate different aspects of the visual-spatial
need to develop confidence in their abilities to world" (Armstrong, 1993, p. 10). The use of
reason and justify their mathematical thinking. visualization in the study of mathematics provides
High-order questions challenge students to think
students with opportunities to understand
and develop a sense ofwonder about mathematics. mathematical concepts and make connections
among them.
Mathematical experiences in and out of the
classroom provide opportunities for students to Visual images and visual reasoning are important
develop their ability to reason. Students can components of number, spatial and measurement
explore and record results, analyze observations, sense. Number visualization occurs when students
make and test generalizations from patterns, and create mental representations ofnumbers.
reach new conclusions by building upon what is
already known orassumed to be true. Being able to create, interpret and describe a
visual representation is part of spatial sense and
Reasoning skills allow students to use a logical spatial reasoning. Spatial visualization and
process to analyze a problem, reach a conclusion reasoning enable students to describe the
andjustify ordefend that conclusion. relationships among and between 3-D objects and
2-D shapes.
6/ Mathematics (K-9)
(2007) ©AlbertaEducation,Alberta,Canada
Measurement visualization goes beyond the CONSTANCY
acquisition of specific measurement skills.
Measurement sense includes the ability to Different aspects of constancy are described by
determine when to measure, when to estimate and the terms stability, conservation, equilibrium,
which estimation strategies to use (Shaw and steady state and symmetry (AAAS-Benchmarks,
Cliatt, 1989). 1993, p. 270). Many important properties in
mathematics and science relate to properties that
Visualization is fostered through the use of do not change when outside conditions change.
concrete materials, technology and a variety of Examples ofconstancy include the following:
visual representations. • The ratio of the circumference of a teepee to
its diameter is the same regardless of the
length ofthe teepee poles.
Nature ofMathematics • The sum of the interior angles of any triangle
is 180°.
Mathematics is one way of trying to understand, • The theoretical probability of flipping a coin
interpret and describe our world. There are a and getting heads is 0.5.
number of components that define the nature of
mathematics and these are woven throughout this Some problems in mathematics require students to
program of studies. The components are change, focus on properties that remain constant. The
constancy, number sense, patterns, relationships, recognition of constancy enables students to solve
spatial sense and uncertainty. problems involving constant rates ofchange, lines
with constant slope, direct variation situations or
CHANGE
the angle sums ofpolygons.
It is important for students to understand that NUMBERSENSE
mathematics is dynamic and not static. As a result,
recognizing change is a key component in Number sense, which can be thought of as
understanding and developing mathematics. intuition about numbers, is the most important
foundation of numeracy (British Columbia
Within mathematics, students encounter
Ministry ofEducation, 2000, p. 146).
conditions of change and are required to search
for explanations of that change. To make A true sense of number goes well beyond the
predictions, students need to describe and quantify skills of simply counting, memorizing facts and
their observations, look for patterns, and describe the situational rote use of algorithms. Mastery of
those quantities that remain fixed and those that number facts is expected to be attained by students
change. For example, the sequence 4, 6, 8, 10, as they develop their number sense. This mastery
12, ... can be described as: allows for facility with more complex
• the number of a specific colour of beads in computations but should not be attained at the
each row ofa beaded design expense ofan understanding ofnumber.
• skip counting by 2s, starting from 4
• an arithmetic sequence, with first term 4 and a Number sense develops when students connect
common difference of2 numbers to their own real-life experiences and
• a linear function with a discrete domain when students use benchmarks and referents. This
(Steen, 1990, p. 184). results in students who are computationally fluent
and flexible with numbers and who have intuition
about numbers. The evolving number sense
typically comes as a by-product of learning rather
than through direct instruction. However, number
sense can be developed by providing rich
Mathematics (K-9) 11
©AlbertaEducation,Alberta,Canada (2007)
mathematical tasks that allow students to make Spatial sense is developed through a variety of
connections to their own experiences and their experiences and interactions within the
previous learning. environment. The development of spatial sense
enables students to solve problems involving 3-D
PATTERNS objects and 2-D shapes and to interpret and reflect
on the physical environment and its 3-D or 2-D
Mathematics is about recognizing, describing and representations.
working with numerical and non-numerical
patterns. Patterns exist in all strands of this Some problems involve attaching numerals and
programofstudies. appropriate units (measurement) to dimensions of
shapes and objects. Spatial sense allows students
Working with patterns enables students to make to make predictions about the results of changing
connections within and beyond mathematics. these dimensions; e.g., doubling the length of the
These skills contribute to students' interaction side of a square increases the area by a factor of
with, and understanding of, theirenvironment. four. Ultimately, spatial sense enables students to
communicate about shapes and objects and to
Patterns may be represented in concrete, visual or create theirown representations.
symbolic form. Students should develop fluency
in moving fromone representation to another. UNCERTAINTY
Students must learn to recognize, extend, create In mathematics, interpretations of data and the
and use mathematical patterns. Patterns allow predictions made from data may lackcertainty.
students to make predictions and justify their
reasoning when solving routine and nonroutine Events and experiments generate statistical data
problems. that can be used to make predictions. It is
important to recognize that these predictions
Learning to work with patterns in the early grades (interpolations and extrapolations) are based upon
helps students develop algebraic thinking, which patterns that have a degree ofuncertainty.
is foundational for working with more abstract
mathematics in highergrades. The quality of the interpretation is directly related
to the quality of the data. An awareness of
RELATIONSHIPS uncertainty allows students to assess the reliability
ofdataand data interpretation.
Mathematics is one way to describe
interconnectedness in a holistic worldview. Chance addresses the predictability of the
Mathematics is used to describe and explain occurrence of an outcome. As students develop
relationships. As part of the study of their understanding ofprobability, the language of
mathematics, students look for relationships mathematics becomes more specific and describes
among numbers, sets, shapes, objects and the degree ofuncertainty more accurately.
concepts. The search for possible relationships
involves collecting and analyzing data and
describing relationships visually, symbolically, Strands
orally or in written form.
The learning outcomes in the program of studies
SPATIALSENSE are organized into four strands across the grades
K-9. Some strands are subdivided into
Spatial sense involves visualization, mental substrands. There is one general outcome per
imagery and spatial reasoning. These skills are substrand across the grades K-9.
central to the understanding ofmathematics.
8/ Mathematics (K-9)
(2007) ©AlbertaEducation,Alberta,Canada