More High-Impact Math Instruction

Fundamentals of Great Teaching in Math

This post is a continuation of our series on fundamental components of great math instruction. We started by focusing on goals that establish what we want students to understand and know in math and continued the series by focusing on instructional moves and practices that support student learning in math. Today we will focus on three additional high-impact instructional moves that maximize student engagement and learning:

1. Teach problem solving and integrate problems into daily lessons
2. Use multiple representations of mathematics
3. Encourage students talking and writing.

Teach problem solving and integrate problems into daily lessons.

In Principles to Action: Ensuring Mathematical Success for All (2014) published by NCTM, it is stated, "Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solutions." For a single sentence, there are many implications for instruction.

Effective teachers select tasks that require a higher level of cognitive demand. These tasks allow students to engage in active inquiry and exploration.

Higher levels of demand include:
• procedures with connections such as finding an answer and explaining thinking with models, pictures, or words
• non-algorithmic thinking - it is not a well-rehearsed set of procedures
• analyzingt he task and understanding constraints that may limit the strategies and solutions
• self-monitoring of one's own cognitive processes

Lower levels of demand include:

• memorization such as recalling facts, formulas, or definitions
• procedures without connections such as using algorithms with no explanation.
• a focus on finding right answers with no connection to the understanding of the underlying math concepts

Effective teachers select tasks with more than one entry point and varied solutions. An open task allows students to approach a problem from many different fronts. The focus of the problem shifts from one right answer to showing mathematical thinking. Here are examples of an open tasks that allow students many entry points to the problems with a variety of solutions:

Create a real world situation for the following problem: 25 + 42 = ? Solve the problem and show how you solved the problem.

The answer is 1/2. What is the question?

There are 10 cars in the parking lot. Some are red and some are black. How many red cars and how many black cars could be in the parking lot?

Effective teachers help students think as they explore problem solving situations that indicate each operation. Avoid using key words as indicators of operation, because they are often confusing and misleading. Students should not approach problem solving as a rote skill, watching for certain words, but as a series of decisions based on their math understanding and the situation in the problem.

From Principles to Action: Ensuring Mathematical Success for All (2014) NCTM p. 24

Use multiple representations of mathematics.

Using multiple representations engages students in making connections between different representations in order to more deeply understand the mathematics. See the model below from Principles to Action: Ensuring Mathematical Success for All (2014) p. 24 demonstrating the connections of different representations. Tripathi was quoted as saying, "It is like examining a concept through a variety of lenses, with each lens providing a different perspective that makes the picture (concept) richer and deeper." (2008)

Encourage students talking and writing.

Susan O'Connell says, "As students talk and write, they:

• process ideas
• identify misconceptions
• rethink ideas and reconsider strategies
• make connections between ideas and methods
• develop insights."

She suggest the following ways to encourage talk in our classrooms that promotes understanding in math.

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