The Concept of Big Ideas
Focusing on Important Mathematical Concepts or “Big Ideas” Much research indicates that children from diverse backgrounds can learn mathematics if it is organized into big coherent chunks and if children have opportunity and time to understand each domain deeply.... Successful countries select vital grade level topics and devote enough time so that students can gain initial understandings and mastery of those topics. They do not engage in repetitive review of those topics in the next year; they move on to new topics. (Fuson, in press)
Why the Problem-Solving Approach is Important to the Development and Understanding of Big Ideas
Teaching that uses big ideas or key concepts allows students to make connections instead of seeing mathematics as disconnected ideas. Problem solving helps to support core processes such as the use of representation, communication, and connection between and among mathematical ideas (Kilpatrick, Swafford, & Findell, 200l; NAEYC, 2002; National Research Council, 1989; NCTM, 2000. Children acquire their understanding of mathematics and develop problem-solving skills as a result of solving problems, rather than of being taught something directly.
The mathematical processes that support effective learning in Mathematics are
• Problem Solving;
• Reasoning and Proving;
• Selecting Tools and Computational Strategies;
• Representing; and
Classroom Structures that Support Problem-Solving
1. Daily challenges – teachers provide students with a meaningful math problem to solve.
2. A problem-solving corner or bulletin board
This is an area in the classroom that can be used to post weekly problems, or unique and appropriate problems for the students to solve.
3. Activity Centres
These can be completed during a math lesson. All students should have the opportunity to complete the problem. Students can rotate through this centre having the chance to solve the problem in small groups and share their findings.
The Importance of Communication in Problem-Solving
Teachers learn what students are thinking through student communication. When students communicate mathematically, either orally or in writing, they make their thinking and understanding clear to others as well as to themselves. The information that teachers gain about their students as they are actively engaged is valuable not only as a stimulus for on-the-spot minilessons but also as a factor in making decisions about the course of future instruction.
• gauge students’ attitudes towards mathematics;
• understand student learning, including misconceptions that students have;
• help students make sense of what they are learning;
• recognize and appreciate another perspective.
Four-Step Problem-Solving Model
Chart for Daily Lesson Planning
How to Observe and Assess Students as they Problem Solve
Teachers may find it helpful to think about their assessment practices in problem solving in relation to these four categories
• cognition • affect • metacognition • flexibility.
Suggestions for Learning how to use the Problem-Solving Process in the Classroom
Three instructional approaches – shared mathematics, guided mathematics, and independent mathematics – support students in learning mathematics. Within those instructional approaches, it is important that teachers:-
• providing appropriate and challenging problems;
• supporting and extending student learning;
• encouraging and accepting students’ own problem-solving strategies;
• questioning and prompting students;
• using think-alouds to model how a problem is tackled;
• observing students and assessing their work as they solve problems.
• anticipating conceptual stumbling blocks, noticing students who encounter these blocks, and helping them recognize and address their misconceptions