Problem Solving and the Common Core

Objectives

The Common Core concentrates on a clear set of math skills and concepts. Students learn concepts in an organized way during the school year as well as across grades. The standards encourage students to solve real-world problems. ²

The Common Core calls for greater focus in mathematics. Rather than racing to cover many topics in a mile-wide, inch-deep curriculum, the standards ask math teachers to significantly narrow and deepen the way time and energy are spent in the classroom. This means focusing sharply on the major work of each grade, which for grades Kindergarten through second grade includes concepts, skills, and problem solving related to addition and subtraction.

The New Haven Public School district uses the Math in Focus Singapore Approach, a Common Core-based curriculum for students from Kindergarten to Fifth Grade. The student books and workbooks follow an instructional pathway that includes learning concepts and skills through visual lessons and teacher instruction for understanding the how and why; consolidating concepts and skills through practice, activities and math journals for deep math understanding, hands-on work in pairs and in small groups; and, applying concepts and skills through extensive problem solving practice and challenges to build real world problem solvers. ³

This approach embeds problem solving throughout each lesson and encourages frequent practice in both computation and problem solving. The word problems appear throughout each chapter and progress from 1-step to 2-step to multi-step. Each chapter concludes with a challenging problem or set of problems that require students to solve some non-routine questions. To solve these problems, the students need to draw on their deep prior knowledge as well as recently acquired concepts and skills, combining problem solving strategies with critical thinking skills, including classifying, comparing, sequencing, identifying parts and whole, identifying patterns and relationships, induction and deduction and spatial visualization.

The second grade text begins with numbers to 1000. Students begin by expressing numbers in standard form (231), expanded form (200 + 30 + 1), and word form (two hundred thirty-one). This is accompanied by concrete representations via base ten blocks, and, for two digit numbers and a few three digit numbers, representation by trains of rods, of lengths 1, 10 and 100. This initial chapter also includes sequencing numbers and comparing using greater than and less than terminology, and then moving right into addition and subtraction of two- and three-digits numbers. Here the take-away should be, if you have more hundreds, the tens and ones don’t make any/much difference; and if you have the same number of hundreds, but more tens, then the ones don’t make any/much difference. Most of my students (if not all) struggle from the start! They do not seem to have a solid foundation of understanding numbers to 100 or the concept of place value in general. This unit is designed to get ahead of the frustration that the students feel when pushed too quickly before they have a firm understanding of principles of place value and the properties of operations.

This unit launches the school year with 1-step addition and subtraction problems of all types using numbers to 10. The goal is to spend time practicing basic computations with numbers that student can work comfortably with before jumping right into the district curriculum. Once there is a level of understanding with these problem sets (numbers to 10), students will move on to solving 1-step problems using teen numbers and then onto numbers to 100. Most of the curriculum problems at the start of the year require addition and subtraction of 3-digit numbers. Some students will move quickly through the problem sets with numbers to 100 and will be ready to work with the regular curriculum.

For the duration of the unit, the focus will be steadily on solving and later constructing a collection of word problems that provide robust and balanced practice. Problems sets will be based on a scenario which will provide the substance of the story. Each scenario will allow us to extract several problems, changing the numbers and ensuring each set of numbers makes a reasonable problem. This idea looks like the following: John has 8 crayons in his box. He shares 3 with Sam. How many crayons does John have left in the box? John has some crayons in his box. He shares 3 with Sam. John has 5 crayons left in his box. How many crayons did John start with? John has 5 crayons. Sam has 2 fewer than John. How many crayons does Sam have? John and Sam are sharing crayons. John has 5 and Sam has 3. How many crayons do the friends have together? The two students participate in several crayon-sharing stories that use the same set of numbers but in slightly different situations. Some situations are more obvious and direct while others take more thinking. It is important to provide opportunities for students to work with and solve the different problem types that can be created from one set of numbers. ⁴