2012 Lower Primary School 2013 Math Handbook for Parents
Math Curriculum in the Lower Primary NAEYC Guidelines for Appropriate Practices in Primary Grades • Involve active participation with each other, adults and materials • Include varied classroom groupings • Promote materials and activities that are concrete, real and relevant • Encourage exploration, discovery and problem solving • Include projects and experiences that extend children's ideas • Respond to their questions and engage them in conversations • Integrate many subject areas • Promote social skills, inquiry, independence and choice • Develop self esteem and positive feelings toward learning and self control • Encourage family members to help in the classroom respond to and respect individual differences in ability and interests • Monitor through regular observation and narration • Record with progress reported to parents
In Lower Primary we provide an enriched, comprehensive and balanced mathematics curriculum. Our program aligns with the National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM) Principal and Standards for Teaching Mathematics. The National Association for the Education of Young Children (NAEYC) is one of the largest and most influential organizations for early childhood educators, dedicated to improving the quality of programs for children from birth through third grade with over 100,000 members and 450 affiliates throughout the United States. The National Council of Teachers of Mathematics is the world's largest mathematics education organization, with nearly 90,000 members and 250 affiliates throughout the United States and Canada. The National Council of Teachers of Mathematics Principles and Standards for School Mathematics, published in 2000, provides guidelines for excellence in mathematics education. Your child's math program is part of an elementary school mathematics curriculum developed by the University of Chicago Mathematics Project called Everyday Mathematics. This program offers students a broad background in mathematics using approaches that are based on research results, field test experiences, and the mathematics your child will need in the twenty-first century. Students engage in mathematical problems that support problem solving and critical thinking. Students learn to think and communicate mathematically. The basic skills, such as addition, subtraction, multiplication and division are an important part of this program. One feature of the Everyday Math program is that it does emphasize the necessity for memorizing the basic math facts as well as using these facts in math computation. Instruction on basic facts is taught through direct instruction and daily activities using a wide variety of hands-on manipulatives, games, flashcards, and computer programs. The Everyday Mathematics program challenges students to be active learners. Students nvestigate, explore, think and communicate while using many different materials, learning styles and strategies in both concept development and problem solving. We seek to help the children appreciate the many ways in which math affects their lives, to demonstrate the variety within the discipline, and to find the areas in which each student can excel. The Everyday Mathematics program aspires to produce students who are joyful about the subject, and who say and feel, "I can do Math!"
Source: Adapted from NAEYC, 1987 & 1990
NCTM Recommendations for Instructional Practices in Preschool to Grade Four Increased Attention to...
Decreased Attention to...
• Use of manipulative materials
• Rote Practice
• Cooperative work
• Rote Memorization of Rules
• Discussion of Mathematics
• One answer and one method
• Questioning
• Use of worksheets
• Justification of thinking
• Written practice
• Writing about mathematics
• Teaching by telling
• Problem solving approach to instruction • Content integration • Use of calculators and computers Source NCTM, 1989
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Research-Based Math Instruction
"Children will become confident 'doers' of mathematics only if mathematics makes sense to them and if they believe in their ability to make sense of it."
– Trafton and Claus, 1994
Each of the four divisions at HKIS have historically used and continue to use the National Council for Teachers of Mathematics Standards as a framework. NCTM present a common set of standards that show the growth of mathematical knowledge across the grades, rather than a different set and number of standards for each grade band. The standards are shown in four-grade bands; Reception One through Grade Two, Grade Three through Grade Five, Grade Six through Grade Eight and Grade Nine through Twelve (see the diagram below showing NCTM grade bands).
The ten standards in Principles and Standards for School Mathematics describe the mathematical knowledge, understanding, and skills students should acquire from Reception One through Grade Twelve. Five standards describe the mathematical content that students should learn to be successful, and five highlight the mathematical processes that students draw on to use their content knowledge. These ten standards define the basic mathematics that all students should have the opportunity to learn.
The content standards are:
• Number and Operations,
The process (or performance) standards are:
• Algebra,
• Problem Solving,
• Geometry,
• Reasoning and Proof,
• Measurement, and
• Communication,
• Data Analysis and Probability.
• Connections, and • Representation.
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Content Standards
Number Sense Number sense is an intuition about numbers and their relationships. This includes having a feeling for comparisons among numbers, knowledge of the various operations on numbers, the ability to represent numbers in several ways, and the skills to interpret and use numbers from real-world situations. Children with number sense pay attention to the meaning of numbers and operations to make realistic estimates of the results of computation. They possess an accurate notion of how numbers relate to each other and how those numbers provide information about the real world. A child with number sense understands both the relationship between numbers and the effects of operations on numbers.
At the Grade Two level, children apply the place value concepts, use standard numerals and compare numbers up to 1000. They begin to explore concepts of multiplication and division. They estimate solutions to addition and subtraction problems and explain their thinking. At the Grade One level, children apply the place value concept of grouping by tens using manipulatives. They associate and compare standard word names and numerals through 100. They begin using the language of "teens" (12 to 21). They build models, draw diagrams and/or act out various interpretations for addition and subtraction situations. Additionally, first grade students begin to use estimation or mental math strategies to estimate and explain results to addition and subtraction situations. At the Reception Two level, children use manipulatives to complete one to one correspondence and conservation of number tasks. Students have opportunities to experience identifying and writing numbers up to 110. They compare and estimate quantities using language experiences, manipulatives, and a variety of strategies. They use manipulatives to explore addition, subtraction, multiplication, and division. At the Reception One level, children explore numbers from one to ten through the use of manipulatives. They use estimated-related words to describe quantity (more/less, some/none, most/least). Children verbalize what will happen if the teacher takes away or adds to physical objects.
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Algebra Algebra is a way of thinking. It is a language used to express mathematical relationships. Children need to understand how quantities are related to one another and how algebra can be used to express and analyze those relationships. They need to focus on understanding the relationship between the equation and the graph and on what the graph represents in a real-life situation. Algebraic thinking should begin in Reception One with concepts such as finding patterns and guessing missing numbers and continue through adulthood. Children observe and describe many kinds of patterns in the world around them. They draw upon these experiences to explore properties of algebraic relations. The exploration of functional relationships leads to understandings of cause and effect relationships essential to solving many real-world problems. Children can model problems and find solutions based on observed patterns and relationships, expressing the process symbolically and verbally. As children develop confidence in representing and solving problems, they should extend these skills to more abstract and symbolic representations. At the Grade Two level, children describe and extend patterning schemes. They use physical objects and appropriate symbols to show the meaning of equality and inequality. They use manipulatives to solve the "unknown" in equations. At the Grade One level, children create, describe, and extend patterns. They manipulate objects to solve problem situations where one addend is unknown. They show the inverse relationship between addition and subtraction using language experiences and manipulative. At the Reception Two level, children classify and sort physical objects according to attributes such as color, shape, size or weight. They recognize, describe, and duplicate patterns based on various attributes. Additionally, they extend or create patterns. At the Reception One level, children classify and sort physical objects on the basis of one characteristic. They use concrete objects to create a simple pattern. Children use balance scales or manipulatives to show the relationship of both sides of the equation.
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Geometry and Spatial Sense Geometry is the study of objects, motions, and relationships in a spatial environment. The world of geometry is the world of patterns, shapes, and movement. Geometric activities are valuable because they not only develop spatial and geometric ideas, but they also promote exploration and reasoning. Spatial sense is often referred to as spatial perception or spatial visualization and can be characterized by a number of such spatial abilities as the ability to imagine movement or spatial displacement by mentally rotating, folding, or in some other way manipulating visual representation of objects. The fundamental ideas of sliding and turning are basic to every child's spatial explorations, and putting these notions into a geometric context should be the starting point for a child's mathematical development in the earliest grades.
At the Grade Two level, children describe, classify, and compare figures and shapes using geometric and spatial terms. They recognize and name two to three dimensional shapes presented in various orientations. They perform geometric transformations using manipulatives and drawings. At the Grade One level, children use informal geometric vocabulary to compare and contrast objects, figures and shapes. They sort objects and identify common geometric attributes used for classification. They demonstrate an understanding of a line of symmetry and use concrete materials to construct the reflection of a given shape. At the Reception Two level, children use informal geometric vocabulary to describe objects and compares similarities. They associate the name of common geometric shape with real world objects. They use concrete experiences to explore symmetry, slides and turns. At the Reception One level, children use informal geometric vocabulary to identify and sort real world shapes by attributes. They recognize that a shape is the same shape even after it is rotated. Children use materials (e.g., blocks, clay, etc.) to form geometric figures.
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Measurement Measurement activities give children many opportunities to explore, organize, and make sense of their world. The use of numbers in simple estimations can help children develop the flexible ideas about numbers. (Elementary School Mathematics, John A. Van de Walle)
At the Grade Two level, children develop an awareness of the need for standard units of measure. They estimate and use standard units and instruments to measure length, weight, capacity and temperature. They identify time concentrating on hours and half hours. They count and compare money. At the Grade One level, children use direct and indirect comparisons to order objects. They use standard and non-standard units to measure the length and weight of objects. They begin to explore the comparisons on time intervals and money. At the Reception Two level, children use oral language and concrete experiences to make direct comparisons of objects relative to length or weight. They begin to explore the concepts of time and temperature. At the Reception One level, children explore concepts of weight to determine objects that are heavy and light. They use non-standard objects to measure classroom objects and arrange items in graduated order. Children can describe concepts of time and temperature.
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Data Analysis and Probability Children learn to locate, gather, organize, manipulate, summarize, display, and analyze large quantities of information, which they can use for prediction and interpretation and/or further study. At the Grade Two level children collect and organize discrete data. They construct, describe and discuss number line plots and bar graphs. They predict which event is more likely or less likely to occur. At the Grade One level, children build and display pictographs and unifix block graphs using data collected from student activities. They discuss graph information to identify the least and most common items. They explain if an event is certain or impossible to occur.
At the Reception Two level children use familiar activities to collect data and build graphs using physical objects. They discuss and interpret collected data. At the Reception One level, children use a variety of manipulatives and materials to explore, compare, and/or graph data. They determine through class discussion if a given event is more likely, equally likely or less likely to occur.
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Process Standards
The process standards outline the methods through which children attain the mathematical knowledge, skills, and conceptual understandings set forth in the five content standards.
Problem Solving A thoughtful classroom environment promotes problem solving. All of the five content strands (numbers and operations, algebra, geometry, measurement and data and probability) engage young learners in creative persistent thinking. Matching, classifying, ordering, patterning, and thinking about numbers are a few examples of problem solving. The problem solving described in the NCTM standards is not just an important part of instruction - it is the organizing principle for a mathematics curriculum. "Problem solving is not a distinct topic but a process that should permeate the entire program and provide the context in which concepts and skills can be learned." (NCTM, 1989) The philosophy, focus, curriculum and methods for teaching mathematics have changed dramatically since the 1960s and 1970s. The heart of the "new" math curriculum is problem solving, connecting meaning to math symbols, and interrelating the various content strands of mathematics. The next few pages will describe the "old style math" with the "new style of math" taught in Lower Primary. Children will be doing tasks that involve investigations. They will be talking, writing, demonstrating and drawing explanations for their thinking. They will spend time exploring problems in depth.
Old Style Math completed as many problems as quickly as possible (Speed and quickness with basic facts has little to do with "being good" in math. Persistence and intuition are more important than speed.)
Math in Lower Primary thoughtfully work on small number of problems during a class session, sometimes working on a single problem for one or several sessions
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Reasoning and Proof Reasoning in the Lower Primary grades centers on pattern recognition and classification skills. When young children are asked to explain their reasoning and give proof as to how they got a particular answer and why they think it is correct they may say their dad showed them or their sister told them that it was correct. Later, children will develop mathematical statements about the relationships between classes of objects. Mathematical proof is a formal way of communication reasoning.
This standard requires that reasoning permeates the curriculum. This means we should have children constantly explaining their thinking and justifying their answers. Teachers and parents should ask children open ended questions. What is a pattern? Why does this work? Does this always work? How do you know this is true? Children will invent their own strategies and approaches rather than just memorizing procedures.
Old Style Math focused on getting the right answer
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Math in Lower Primary consider their own reasoning and the reasoning of other students
Communication This standard requires students to have frequent opportunities to communicate. In explaining ideas to others, students develop a clearer understanding themselves. "Students should relate physical materials, pictures, and diagrams to mathematical ideas." (NCTM, 1989)
It means finding ways to express ideas with words, diagrams, pictures, and symbols. When children talk, either with you or with their friends, it helps them think about what they are doing and makes their own thoughts clearer. As a bonus, talking with children improves their vocabulary and helps develop literacy and early reading skills as well. It is a means for supporting students' learning as they act out a situation, draw, use objects, give verbal accounts and explanations, use diagrams, write, and use mathematical symbols and numeration. Children will be collaborating to make discoveries, draw conclusions and discuss math. Children will move around the classroom as they explore mathematics in their environment and talk with their peers.
Old Style Math
Math in Lower Primary
work alone
work in a variety of groupings (whole class, individually, in pairs, in small groups)
recorded only by writing down numbers
communicate about math orally, in writing, and by using pictures, diagrams and models
used only pencil and paper, chalk and chalkboard as tools
use cubes, blocks, measuring tools, calculators and a variety of other materials
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Connections This standard requires connections within mathematics and between mathematics and other parts of the school curriculum (language arts, social studies, science, etc.). Children will be seeing that math is much more than arithmetic (knowing the facts and numbers and operations); it involves estimation, geometry, probability, statistics, and more.
Representation Representation refers to the process and product of showing a math concept. Representations help children organize their thinking. Teachers should encourage students to use variety of physical projects to represent their thinking. Representation of ideas not only helps children communicate their thinking, but also helps them reflect on their thinking. Some forms of representation are drawings, diagrams, pictures, numbers, words, equations, and symbols. Graphical displays, symbolic expressions and a variety of new forms associated with electronic technology are also types of representations.
Representations are difficult for children to develop because they require the child to have a higher level of understanding rather than just a solution to the problem presented. Children need to find meaningful ways to record and present their solutions to others and work to adequately use conventional forms of representation.
If you can do math, you can do anything!
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The Mission and Student Learning Results are the North Star for HKIS, its faculty, students, and staff. We use these as our guide in everything we do, every day.
Mission Dedicating our minds to inquiry, our hearts to compassion, and our lives to service and global understanding An American-style education grounded in the Christian faith and respecting the spiritual lives of all
Student Learning Results Academic Excellence Students will achieve their intellectual potential by striving for and attaining the highest standards of academic excellence
Spirituality Students will understand and respect Christianity and other religions and will identify and develop their own spiritual identity
Character Development Students will demonstrate respectful and caring attitudes at school and in the community, as well as the courage to stand up for what is right
Self-Motivated Learning Students willingly apply a variety of learning and motivation strategies throughout their learning process
Contributing to Society Students will develop the skills they need to form genuine relationships in our diverse society and to make contributions to our community
Chinese Culture Students will gain an understanding of China and an appreciation of the Chinese Culture
23 South Bay Close, Repulse Bay, Hong Kong
Lower Primary School
T +852 2812 5000 F +852 2812 9590 www.hkis.edu.hk