Big Ideas and Problem Solving in Junior Math Instruction

What is it all about? Mathematics is a fundamental human activity a way of making sense of the world. Children possess a natural curiosity an interest in mathematics, and come to school with an understanding of mathematical concepts and problem-solving strategies that they have discovered through explorations of the world around them (Ginsburg, 2002). http://eworkshop.on.ca/edu/resources/guides/ExpPanel_456_Numeracy.pdf

What is a big idea in mathematics? “Students are better able to see the connections in mathematics and thus to learn mathematics when it is organized in big, coherent “chunks”. In organizing a mathematics program, teachers should concentrate on the big ideas in mathematics and view the expectation in the curriculum policy documents for grade 4-6 as being clustered around these big ideas.” http://eworkshop.on.ca/edu/resources/guides/NSN_vol_1_Big_Ideas.pdf p.12

For us as teachers… The big ideas also act as a lens for: Making instructional decisions (e.g., deciding on an emphasis for a lesson or a set of lessons) Identifying prior learning Looking at students’ thinking and understanding in relation to the mathematical concepts addressed in the curriculum (e.g., making note of the strategies a student uses to count a set or to organize all possible combinations to solve a problem) Collecting observations and making anecdotal records Providing feedback to students Determining next steps Communicating concepts and providing feedback on students’ achievement to parents (e.g., in report card comments) https://faculty.nipissingu.ca/danj/DOCUMENTS/MATHEMATICS/GUIDES/GRADES%20K-6/GEIM%20K-6.pdf p.34

Problem Solving An information- and technology-based society requires individuals who are able to think critically about complex issues, people who can “analyze and think logically about new situations, devise unspecified solution procedures, and communicate their solution clearly and convincingly to others” (Baroody, 1998, p. 2-1 http://eworkshop.on.ca/edu/resources/guides/Guide_Math_K_6_Volu me_2.pdf p.3

As students engage in problem solving, they participate in a wide variety of cognitive they will encounter throughout their lives They: Learn mathematical concepts with understanding and practice skills in context Reason mathematically by exploring mathematical ideas, making conjectures, and justifying results; Reflect on and monitor their own thought processes Select appropriate tools (e.g., manipulatives, calculators, computers, communication technology) and computational strategies Connect the mathematics they learn at school with its application in their everyday lives Develop strategies that can be applied to new situations Represent mathematical ideas and model situations, using concrete materials, pictures, diagrams, graphs, tables, numbers, words, and symbols 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. http://eworkshop.on.ca/edu/resources/guides/Guide_Math_K_6_Volume_2.pdf p.4-5

4 Steps to Problem Solving In the words of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario, 2004 p.9, “instruction based on a problem- solving or investigative approach is the means by which Ontario students will most readily achieve mathematical literacy”

Classroom Structures That Support Problem Solving: A big idea, problem solving math teacher: Chooses problems that offers a range of entry points for students at different levels. Every classroom contains students with a range of understandings and prior knowledge. Teachers need to choose problems that can be accessed in some way by the students Engages the interests of students by using problems that are culturally, socially, etc. relevant Uses a variety of mathematics instruction - guided, shared, and independent Plans and integrates a variety of groupings to provide students with time to share ideas with their peers and to work independently Encourages oral communication is essential for learning math to help students clarify their ideas, get feedback for their thinking, and hear other points of view. Students learn from one another as well as from their teachers Has students communicate and investigate their thinking through ongoing discussion Allows students to make sense of the problem in their own way. To look for patterns and for connections with other problems. Different methods of solving problems are welcomed and shared

Classroom Structures That Support Problem Solving (cont’d) A big idea, problem solving math teacher: Presents math activities in real world contexts to give students access to otherwise abstract mathematical ideas Supports math learning with manipulatives, to make abstract mathematical ideas concrete. Engages student interest in mathematical ideas, even when they are not related to the curriculum Includes students as active rather than passive participants in their learning; Encourages confusion and mistakes as opportunities for learning; Has students explain their thinking, to help organize their ideas and extend their understanding. Encourages writing in math class, so students revisit their thinking and reflect on their ideas. Incorporates ongoing assessment of student understanding to guide future instruction. Wants the acquisition of mathematical knowledge by all students a top priority

Importance of Communication in Problem Solving Junior students are developing their ability to communicate their thinking to others orally and on paper http://eworkshop.on.ca/edu/resources/guides/ExpPanel_456_Numeracy.pdf Students need to talk about the problem to understand it better. Students discuss ideas with others to clarify which strategy or strategies would work best Students trying to engage and answer questions from teachers and peers in proper mathematical language e.g. “I then divided the number of marbles” makes the terminology real Students teaching their peers aloud, effectively communicates and solidifies what they have learned As important as effective communication is active listening in which students listen to the ideas and learning of a partner, group members and/or the teacher teachers is just as important in helping them develop the understanding and metacognition.

Resources and Strategies to Help Further Develop Your Understanding Teaching and Learning Mathematics: The Report of the Expert Panel in Grades 4-6 Math. in Ontariohttp://eworkshop.on.ca/edu/resources/guides/ExpPanel_456_Numeracy. pdf Big ideas in math eworkshops http://eworkshop.on.ca/cfmx/edu/core.cfm?p=guides Big ideas in number sense and numeration gr.4-6 http://www.eworkshop.on.ca/edu/resources/guides/NSN_vol_1_Big_Ideas.pdf 10 big math ideas http://www.scholastic.com/teachers/article/marilyn-burns- 10-big-math-ideas A guide to effective instruction in math, kindergarten – grade 6 https://faculty.nipissingu.ca/danj/DOCUMENTS/MATHEMATICS/GUIDES/GRADES%2 0K-6/GEIM%20K-6.pdf