Estimation

Appendix A: Singapore Math as a Model for this Unit

There has been a lot of research that the Singapore and Chinese models of structuring math in the early grades gives students a stronger foundation. (See Ginsburg, Geary and Ma.) Therefore, I think it is very important to examine the scope and sequence of the math curriculum in Singapore.

The Singapore curriculum splits the first grade year into two volumes of study 1A and 1B. While there are other topics of study, these units have a strong focus on number sense. Additionally, the preface of the Primary Mathematics textbooks states the following,

"The main feature of this package is the use of the Concrete à Pictorial à Abstract approach. The students are provided with the necessary learning experiences beginning with the concrete and pictorial stages, followed by the abstract stage to enable them to learn mathematics meaningfully. This package encourages active thinking processes, communication of mathematical ideas and problem solving." ¹²

1A starts with counting numbers 0-10 and exploring number bonds. The students are guided through number stories and visual representations of how you can combine numbers to make other numbers. They prompt the students to find pairs of numbers that make values up to 10. They also integrate early algebraic thinking by asking students to find missing addends, instead of just focusing on calculating sums. Then they move into subtraction, defining the operation, using word problems and pictures to give practice, and linking addition with subtraction. Once the students are proficient with number bonds to 10, they move on to counting and comparing numbers to 20. The numbers from 11-20 are introduced pictorially by showing a group of ten plus a number of separate ones. The students are then guided through exercises that emphasize that 13 is 10 and 3 or 15 is 10 and 5, etc. Then they are prompted to add using the strategy of making a 10 first and then counting on. When subtracting, they are prompted to split the larger number into a 10 and its other piece first. Subtract the smaller amount from the 10 and then add on the ones that were ignored initially. Also, they are asked to compare problems like the following 8 - 3 = 5 and 18 - 3 = 15.

1B starts with comparing numbers to 10. It quickly moves into studying numbers to 40. This is where they continue to use expanded form to represent each two-digit number. For example, 23 is 20+3 and 28 is 20+8. Then they begin to compare these larger numbers as well. I think it's important to note that with each number there is a picture that emphasizes groups of ten with some ones left over and that these pictures/diagrams take a variety of forms. Multiplication is introduced as repeated addition and division is introduced as sharing a number of items by splitting them into equal groups. By the end of 1B the students are working with numbers to 100. They start by counting by tens, 50, 60, 70, 80, 90 and 100. Then they add and subtract multiples of 10 to a number. They also do addition and subtraction using the expanded forms of numbers, teaching students to look for tens to add and subtract with first.

While this is their 1 ^st grade curriculum, I think it frames a great beginning for my 2 ^nd graders. Our Houghton Mifflin curriculum is sequenced in a very different manner, and it does not place such an emphasis on the decimal structure of our number system. I have designed this unit to bring elements of Singapore's curriculum into my classroom.