Big Ideas in Mathematics!

The answer is 5. What is the question?

How might your students respond to this question?

(We will come back to this question later but let's explore the BIG IDEAS!)

The term big ideas was coined by Deborah Schifter and Cathy Fosnot in the course of their professional development work with teachers who were learning mathematics more deeply. Schifter and Fosnot defined big ideas as the

“central, organizing ideas of mathematics – principles that define mathematical order"

Schifter, D., & Fosnot, C. (1993). Reconstructing mathematics education: Stories of teachers meeting the challenge of reform. New York: Teachers College Press

In the Grades 1-8 Ontario Mathematics curriculum the term "Big Ideas" is defined as:

“the interrelated concepts that form a framework for learning mathematics in a coherent way.”

A "Big Idea" is a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole.

• Helps teachers make instructional decisions
• Helps teachers prioritize and organize expectations
• Helps teachers to identify prior learning
• Helps teachers to clarify what aspect of an outcome or learning goal
• Helps teachers to select rich resources and problems
• Helps teacherst to create and choose appropriate assessments

"When one understands Big Ideas, mathematics is no longer seen as a set of disconnected concepts, skills, and facts. Rather, mathematics becomes a coherent set of ideas." (Charles, 2005)

• students make richer and more meaningful connections
• students gain a deeper understanding
• transfer of learning is enhances
• is motivating for students
• promotes the development of self directed learners
• reduces the load on students memory
• is motivating and engaging for students!

Math programs that are organized around big ideas and focus on problem solving provide cohesive learning opportunities that allow students to explore mathematical concepts in depth. An emphasis on big ideas contributes to the main goal of mathematics instruction – to help students gain a deeper understanding of mathematical concepts.

A Guide to Effective Instruction on Mathematics: Kindergarten to Grade 6. Ontario Ministry of Education)

Guides to Effective Instruction in Mathematics Kindergarten to Grade 6 (Strand Specific) Ontario Ministry of Education

Math really is not a set of tiny little pieces. It is a connected whole. It is our job to help our students seen those connections and one way we can do this is through open questions and parallel tasks.

The use of open questions allows teachers to differentiate their instruction and allows students at different stages of math development to enter a task. Every student has the opportunity to become part of the math learning community.

A problem is open when it is framed in a way that a variety of responses or approaches are possible.

For example:

Closed Problem: What is the sum of 2, 4, and 6?

Open Problem: The sum of three numbers is 12. What can these numbers be?

The open problem provides an opportunity for richer conversations and all students can participate. By using open problems teachers are able to observe students current level of understanding and students gain confidence.

There are a variety of ways to open up a closed question including:

• turning the question around (start with the answer)
• asking for similarities and differences
• replacing a number with a blank
• asking for a number sentence
• changing the question

Examples

Closed Question: What number has 3 hundreds, 2 tens, 2 thousands, and 4 ones?

Open Question: You can model a number with 11 base ten blocks. What could the number be?

Closed Question: Complete the next 6 terms of the pattern: 1, 4, 9, 16,…

Open Question: Create a growing pattern where the 10th term is 100.

Closed Question: Use 3 triangles to make a trapezoid. Draw the lines of symmetry.

Open Question: Draw a design or shape made up of three shapes. The design should have symmetry.

Check out more resources from Marian at her website:

www.onetwoinfinity.ca

THE BIG IDEA: There are many ways to represent numbers.

Open Problems that address this Big Idea could include:

• The answer is 5. What is the question?
• What makes 5 a special number?
• Show the number 7 in as many different ways as you can

These questions provide all students with meaningful opportunities to participate in a math conversation.

Students can show multiple representations and use a variety of tools and resources to respond.

Big Ideas and Proportional Reasoning K-12

Big Ideas for Number

Open Task Problem Solving Math Bank

Charles, R. (2005) Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics NCSM Journal.

Small, Marian. (2012). Good Questions Great Ways to Differentiate Mathematics Instruction. Canada: Nelson.

Small, Marian. (2008). Big Ideas from Dr. Small Grades K-3. Canada: Nelson.

Asking Effective Questions (Ontario Ministry Monograph)

Beyond One Right Answer (Marian Small)