Estimation

Singapore Math

In a widely accepted comparison of global math skills, the Trends in International Mathematics and Science Study, Singapore continually finishes first among participants. Singapore is a prosperous, multicultural, multilingual nation of 4 ½ million people whose fourth- and eighth-grade achieve this standing. ² Singapore's world-class mathematics system has quality components aligned to produce students who learn mathematics to mastery. These components include Singapore's highly logical national mathematics framework, mathematically rich problem-based textbooks, challenging mathematics assessments, and highly qualified mathematics teachers whose pedagogy centers on teaching to mastery. ³

Beginning math students often have difficulty grasping abstract mathematical concepts. Students in first grade, and even beyond, need visual representations of mathematical ideas. Numbers are ideas. A strategy of clear illustrations to present the idea and reinforce the skill is a critical part of this curriculum. Because the six to seven year old has not achieved abstract thought and has not had enough relative experience, the event or experience needs to be visual, auditory, and/or tactile. This seems clearer and more logical possibly when discussing literacy to assist with connecting words to pictures, such as, "The cat is in the tree." The student would be reading the sentence while looking at an illustration showing the same. We must recognize that it even applies equally to mathematics.

The Singapore illustrations feature a concrete to pictorial to abstract approach. This approach allows students to begin to learn math in a meaningful way, understanding the mathematical concepts before learning the "rules" or formulas. To give an example, the Singapore first-grade text begins with a blank rectangle and the number and word for "zero." Below that is a rectangle with a single robot in it, and the number and word for "one." Then a rectangle with two dolls, and the number and word for "two," and so on. The page is very pictorial, but it refers to something very concrete. Next to the pictures are dots. Beginning with the number six (represented by six pineapples), the dots are arranged in two rows, so that six was presented as one row of five dots and a second row with onedot. ⁴

Teaching to mastery is another key conponent of this approach. Topics are covered in detail and taught to mastery. Immediately after new concepts are taught students are engaged with a variety of mathematically rich problems to ensure their deep understanding. Singapore Math is designed to produce mathematical thinkers by walking students through component parts of a problem before asking them to solve the whole problem. Making sure that students really understand allows for more success for them when new concepts are introduced.

"The Singapore texts don't make a huge deal about the concepts, but they present them in the correct and economical form," said Roger Howe, a professor of mathematics at Yale University. "It provides the basis for a very orderly and systematic conceptual understanding of arithmetic andmathematics." ⁵

This curriculum will cover the introduction of place value after the mathematics readiness skills have been mastered, including learning number names to 10 (and their serial order) and using these numbers to count quantities, developing familiarity and facility with numbers, practicing with numbers, saying numbers, writing and reading them followed by simple addition and subtraction, or as the language of this unit will be, composing and decomposing numbers to 10.

Keeping the scope narrow will be important to mastering this skill. Students will need to draw on previous learning to be prepared to comprehend place value. Proficiency at composing and decomposing a ten will supply significant support for learning addition and subtraction within 20.

I would characterize this as more of a conceptual unit than a procedural curriculum. The classroom activities will include math facts to 20 or addition with composing and subtraction with decomposing within 20, working with manipulatives, and ultimately using this conceptual knowledge when working with two-digit numbers beyond 20. Because of the close connection between this unit and the Singapore math approach, the strategies in this unit will mirror theirs. The goal here is to promote a solid learning of this topic in a logical and sequential order.

To begin, digits at different places have different meanings or stand for different values. When students learn regular addition and subtraction, place value becomes more meaningful for them for they have to line up the digits with the same place value. After that, when learning addition with composing and subtraction with decomposing, students learn the idea of composing and decomposing to a unit of higher value. The composition and decomposition of a unit are important aspects of the concept of place value. It is important for the students to know that place value is a mathematical concept not a label for the columns for lining up digits.

One strategy to teach the students is to have them think about addition when they are doing subtraction, the inverse operation. This will help to facilitate their conceptual learning. It brings awareness to the fluid movement between these two operations. Inverse operations are one of the main principles that underlie the relationships among the operations of mathematics.

Composing and decomposing are a different way to learn addition and subtraction. These two terms allow the students to know that numbers can go together and come apart. Addition with carrying is called addition with composing - subtraction with regrouping is called subtraction with decomposing. Composition and decomposition of a unit are important aspects of the concept of place value. The goal is to emphasize that 1 ten is composed of 10 ones and can be decomposed into 10 ones. ⁶

Students should have a clear idea about the rate for composing a higher value unit so that they can better understand why a higher value unit is decomposed into 10 lower value units.