Research in the field of Cognitive Development has mapped the landscape of the important underlying concepts in mathematical learning and problem solving. The Symphony Maths scope and sequence is based upon these insights from Cognitive Development research. Each concept presented in elementary maths curricula is related to a fundamental idea or cognitive scheme. Maths understanding and learning will be more effective and meaningful if instruction and practice are explicitly connected to these fundamental ideas.

Part of what makes some of these fundamental ideas so important is that they follow a pathway throughout maths learning. These fundamental ideas repeatedly emerge in successive levels of math learning. Physical representations of the fundamental ideas provide a mechanism by which students can interact with these ideas, apply them in a variety of situations, and internalize a model that gives meaning to their operations and procedures.

Quantity

This activity helps students develop number conceptualization. While most students learn how to count, it is possible that students can learn to count and solve simple maths statements without appreciating that number is a concept that represents a quantity. A student may know that “9″ comes after “8″ but not know that “9″ represents a larger quantity than “8″.

Quantity challenges students to develop number conceptualization and understand that math is more than counting and adding on. The activity uses virtual manipulatives to develop the following concepts:

• Number
• One-to-one correspondence
• Equality
• Greater than
• Less than
• Not equal
• Not less than
• Not greater than

Quantity also emphasizes procedural skills. Once students develop a strong understanding of a concept they are challenged to accurately solve mathematical statements which incorporate that concept. The activity develops skills to solve the following types of number sentences:

• 2 = ?
• ? > 3
• 5 < ? < 8
• 5 ≠ ?

Another important component of the Quantity activity is the application of concepts and procedural skills. Word problems are presented orally for students to represent by constructing number statements. Students learn how numbers and symbols can be used to describe real situations. For example: Students hear, "Suzie has five pencils. Jamal has three pencils." The are asked to represent the problem mathematically, "5 > 3.”

Addition & Subtraction

The Addition & Subtraction activity challenges students to construct a solid understanding of the fundamental ideas of addition and subtraction. Many students learn to solve simple arithmetic number statements, but not all children develop a conceptual understanding of what these operations mean.

Many elementary maths problems can be solved by using counting as the primary strategy. Although neither fast nor efficient, students can use counting strategies to find the correct answer to math problems. While effective in the early grades, eventually these counting strategies become too cumbersome and inefficient as the complexity of the maths curriculum increases. The counting strategies also do not lend themselves to a conceptual understanding of the operations.

Addition & Subtraction emphasizes the understanding of the concept of part-to-whole relations. This is the key concept that students need to internalize to understand addition and subtraction at the conceptual level. The activity uses virtual manipulatives to develop the following concepts:

• Part-to-whole
• Missing part
• Missing parts

The activity develops procedural skills for solving addition and subtraction problems. Once students have developed a conceptual understanding of the fundamental ideas that underpin addition and subtraction, they are challenged to apply that knowledge by solving addition and subtraction problems. The activity includes tasks with addends and subtrahends up to 10 in the following formats:

• 3 + 5 = ?
• 3 + ? = 8
• ? + 5 = 8
• ? + ? = 8
• 5 + 3 + 4 = ?
• 5 + 3 = 4 + ?
• 8 - 5 = ?
• 8 - ? = 3
• ? - ? = 3
• 5 - 3 - 1 = ?
• 8 - 2 = 9 - ?

Similar to the Quantity activity, the Addition & Subtraction activity emphasizes the application of concepts and procedural skills. Word problems are presented orally for students to first represent by constructing number sentences and then solve. For example: Suzie has five pencils. Jamal has three pencils. How many pencils do they have all together. They are asked to represent and solve the problem mathematically, “5 + 3 = 8.”

Multiplication & Division

The Multiplication & Division module develops an understanding of grouping and partitioning by building upon the part-to-whole concepts established in the Addition & Subtraction module. Similar to addition, some students can learn multiplication number relationships without understanding their meaning. The Multiplication & Division module helps students develop their conceptual understanding of what these
operations mean and then helps them learn the number relationships through systematic practice and evaluation.

The Multiplication & Division module covers number relationships with products and dividends up to thirty. The activities use the repeated addition model of multiplication and the repeated subtraction model of division. This helps students understand the connection between the Addition & Subtraction module and the concepts in Multiplication & Division. The module presents tasks in the following formats:

• 3×5=?
• 3x?=15
• ?x5=15
• ?x?=15
• 15÷5=?
• 15÷?=3
• ?÷5=3
• ?÷?=3

The Multiplication & Division module offers the same five activities as the other modules. In the first activity the student needs to find equal bars that are evenly related to the whole. The second activity consists of analyzing the relationship of the bars in order to construct a corresponding number sentence. The third activity presents number sentence problems typical of traditional worksheets. The number bars appear in order to present a concrete model of what the number sentence means if the student makes an error or needs help. The fourth activity offers spoken number sentences that students must construct and solve. The fifth activity presents spoken story problems such as:

• “James has three bags of apples. In each bag there are five apples.
How many apples does James have altogether.”
• The student is asked to represent and solve the problem
mathematically (i.e. 3×5=15).

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