1. Big Ideas and Problem Solving
Junior Math Instruction

2. 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).

3. 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.”

4. 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)

5. Problem Solving
Teachers should encourage students to problem solve, this should be the norm.
Students will think critically and support their answers with evidence and rationale.
Students will access knowledge from exploration, use of past knowledge, collaboration with other students, manipulatives and trial & error.
Students will also begin thinking about how they solve problems by observing others solve problems.

6. How is the “Problem Solving Approach” linked to the development and understanding of “Big Ideas”?
This approach challenges students, it pushes them to become independent explorative learners. Linking small ideas they have learned to the big ideas they have been asked to tackle.
Students become active participants in learning and teaching. They find ways to solve problems, they compare their results and learning to that of other students.
Students are bound to make mistakes when challenged it is important they learn from these mistakes and build upon their knowledge.
Math becomes something tangible rather than conceptual. Students relate knowledge to real word, fables, and use manipulative.

7. Classroom structures that support a problem solving approach to learning.
Daily Challenges – daily problem solving activities that get students thinking outside the box. Problem-solving corner or bulletin board – an area where interesting problems can be place. Students can visit the area and try to solve the problem. At a later time the class goes through it together. Activity Centre – this can include multiple centers that are used by different students at different times. Students can later discuss problem solving strategies and also make connections to other questions asked.

8. What students should be doing?
Students should think, talk and write.
As well as analyze, evaluate and build on mathematical thinking and strategies of others.
Work on their own solutions, not rely on teacher to tell them everything.
Students should justify their thinking.

9. How to plan for problem-solving.
Source: Capacity Building Series – Secretariat Special Edition #13

10. Other classroom activities that can be incorporated in classroom structures/teaching/learning
Manipulative
Drama activities
Models
Diagrams
Guess-and-Check Method
Work Backwards
Logical Thinking
Tables
Lists
Similar problems
Find Patterns
Strategy Wall

11. What else can a teacher do to ensure this approach is successful?
Create an environment where students are encouraged to understand concepts rather than follow a set of steps.
Make sure students reflect on reasoning.
Have students talk to one another when solving problems, this collaboration will illuminate their learning.
Writing and communication are a part of math. Have students right down their reasoning in a math journal or something like it.

12. Present math activities in real word and fictional contexts.
Let students dictate how fast you move with the curriculum.
Make sure activities meet the needs of all students.
Make sure students know that errors are opportunities for learning. Encourage them to take risks.

13. Why is communication important in problem-solving?
It allows teacher to assess the way a student thinks rather than what he or she produces. Teachers can see how students strategize, put information together and how efficient those strategies are.
Allows student to be active in their learn. Have them express their ideas, problems, and strategies with one another and the teacher.
Allows students to work together or share their strategies with others. Teaching others is a great way to learn.

14. Mathematical communication is an essential process for learning mathematics because through communication, students reflect upon, clarify and expand their ideas and understanding of mathematical relationships and mathematical arguments. (Ontario Ministry of Education, 2005)
What can a teacher use?Use co-operative learning strategies, like Turn and Talk, Think-Pair-Share, Round Table, Think-Talk-Write and Place Mat, to organize student interaction/discussion and to provide wait time for students to formulate a response.

15. Helpful Resources… http://mathschallenge.net/
Capacity Building Series – Communication in the classroom