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The use of Big Ideas and Problem Solving in Junior Math Instruction.The concept of Big Ideas and Problem Solving ApproachWhat are Big Ideas?“Big Ideas are really just large networks of interrelated concepts... whole chunks of information stored and retrieved as single entities rather than isolated bits."(-Van de Walle, 2001, Elementary and Middle School Mathematics: Teaching Developmentally, p. 30)They are “interrelated concepts that form a framework for learning Mathematics in a coherent way” (-The Ontario Curriculum Grade 1-8 Mathematics)In short, it is an approach that allows students “to see that mathematics is an integrated whole, and to gain a deeper understanding of the key concepts”* listed in The Ontario Curriculum Grade 1-8 Mathematics -Number Sense and NumerationMeasurementGeometry and Spatial SensePatterning and AlgebraData Management and Probability(* The Guide to Effective Instruction in Math, Kindergarten to Grade 6, Vol. 1)What is the Problem Solving Approach?The Problem Solving Approach “is based on the belief that students learn mathematics most effectively when they are given opportunities to investigate ideas and concepts through solving problems and are then guided carefully into an understanding of the mathematical principles involved”- (- The Ontario Curriculum Grade 1-8 Mathematics)Problem solving is central to the learning of mathematics; teaching through problem solving is a sense-making approach to “doing math” that focuses on big ideas. (-Module 4: Literacy / Math Programming ABQ Junior Course, Queen’s University.)The Importance of Problem Solving Approach to the development and understanding of “Big Ideas”An effective mathematics program takes the approach of clustering expectations around a big idea and investigating effective teaching strategies for that big idea.”(- Principles Underlying Effective Mathematic Instruction). Problem Solving is one technique that encourages students to understand these ‘big ideas’. It allows for students to apply prior knowledge, experience, skills, and understanding to new and unfamiliar situations in order to complete tasks. These situations will be able to provide meaningful experiences through which students are directed towards learning new concepts and skills (eworkshop.on.ca). Students need affirmation and positive reinforcement when learning new concepts. As students engage in problem solving, they participate in a wide variety of cognitive experiences that help them to prepare for the many problem solving situations they will encounter throughout their lives (- A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – Volume Two)When students learn the big ideas in math through problem solving, they-? learn mathematical concepts with understanding and practise skills in context;? reason mathematically by exploring mathematical ideas, making conjectures, and justifying results;? reflect on the nature of inquiry in the world of mathematics;? reflect on and monitor their own thought processes;? make connections between mathematical concepts;? connect the mathematics they learn at school with its application in their everyday lives;? develop strategies that can be applied to new situations;? go from one representation to another, and recognize the connections between representations;? persevere in tackling new challenges;? formulate and test their own explanations;? communicate their explanations and listen to the explanations of others;? participate in open-ended experiences that have a clear goal but a variety of solution paths;? collaborate with others to develop new strategies.(- A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – Volume Two)Marilyn Burns in her 10 Big Math ideas guides teachers towards planning more engaging and effective math classes wherein the students are comfortable with all the content areas of mathematics and are able to apply their learning to problem-solving situations. Her "big ideas" not only help children learn, understand, and enjoy math class but also-?Help students become flexible thinkersAllow students to find what “makes sense” to them Learn that Explaining helps reinforce the Big Ideas Allow students to use logic rather than memorize steps Develop the understanding that it is acceptable to be confusedEncourage different ways of thinkingConnect students to real-life problems If students can develop this resiliency, they should be able to transfer these skills to everyday tasks. Researchers, such as Carpenter et al. (1993), O’Brien (1999), and Fosnot & Dolk (2001), emphasize the importance of teaching students ‘school mathematics’, and encouraging students to ‘make sense of what they are doing’ through problem-solving techniques.Class Structures That Support Big Ideas and Problem Solving Approach “Helping students become good problem solvers is like helping them learn how to ride a bike; tips can be helpful, but it’s impossible to master the process without actually trying it” (Baroody, 1998).In order to promote problem solving skills in the classroom the teacher should introduce key concepts and encourage students to explore the same through problem solving. For this, the teacher must ensure the learning environment, the classroom setup and the lesson plan are in line with her teaching approach. Classroom Set Up- The teacher should ensure that There is a place for whole group instruction and presentationsThe desks are arranged in small groups or pairs Manipulatives available for students to useAnchor charts of mathematical concepts and problem solving skillsThe classroom environment is inclusive where students feel comfortable taking risksLesson Outline-Getting Started – Whole ClassIntroduce task to studentsEnsure students understand taskReview prior knowledge Provide manipulativesWorking on it – Small Groups or PairsStudents share their strategiesStudents reflect on their strategiesHelp students recognize their understandings and misconceptionsUse probing questionsReflecting and Connecting- Whole ClassHave a class discussion about key mathematical conceptsReflect on different methods to solve the problem(-A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6, Vol. 1)Importance of communication in problem solving approachAs mentioned earlier through studies such as Burns, and through the e-Workshop videos, communication is extremely important when students are exploring problem-solving techniques. Through effective communication, students can learn from each other, share strategies, and affirm each other’s processes. It is also very important that teachers are effective in asking conceptual questions, encouraging ‘math-talk’, and communicating concepts and providing feedback on students’ achievement to parents. According to A Guide to Effective Instruction in Mathematics, student communication will help teachers gauge students’ attitudes towards mathematics; understand student learning, including misconceptions that students have; help students make sense of what they are learning; and recognize and appreciate another perspective. Marilyn Burns in “10 Big Math Ideas” says students need to explain their reasoning to their classmates as well as the teacher becauseExplaining the reasoning behind the thinking allows students a chance to organize their ideasInteraction between students helps them clarify their own ideas as well as try another approachDiscussing ideas before writing give students a chance to organize and clarify their ideasFinally, the teacher can provide many opportunities for communication in the classroom. Some of the different strategies that can be incorporated to promote communication are-Think/Pair/Share, Word Wall, Journals, Class Discussions and so on. The teacher can model correct ways to phrase and ask questions, and encourage and challenge students to become critical thinkers and problem solvers.Resources and StrategiesTeacher Support-Provide complex problems that have a variety of entry points which do not have a solution that is immediately obviousModel how to approach problemsCreate problems that are appropriate yet challengingEncourage different approachesListen attentively as students explain their method of solving the problemQuestion or prompt students when explaining their thinkingObserve students and anticipate where they may come across obstacle(s)(- A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6, Vol. 2)Resources-A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6, Vol 1 Big Math Ideas by Marilyn Burns Sense and Numeration, Grades 4 to 6, Vol 1 Junior Mathematical Processes, Problem Solving and Big Ideas by Strand ................
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