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Why Should Kids Learn to Generalize in Math?

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What does it mean to generalize? Why does generalization matter? And why do my kids need to learn how to do it?

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Level 1 Generalizations - "Students will understand that..."

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Level 1 Generalizations - "Students will understand that..."

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Typically weak and don't help learners fully grasp what and why they are learning

Ex: The coefficients of a quadratic function in a connected equation affect the quadratic formula.

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Should be used to develop teaching targets

Ex: Whether arithmetic and geometric sequences and series share a common difference or a common ratio distinguish one from another

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Level 3 Generalizations - "So what? What is the significance or effect?"

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Level 3 Generalizations - "So what? What is the significance or effect?"

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Can be used to express the real world value of mathematics

Arithmetic and geometric sequences and series describe patterns in numbers and supply algebraic tools that help to solve real-life situations

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two or more concepts stated as a sentence of relationship

transfer through time, across cultures, and situations

Supported by factual examples

Overarching - Math reveals patterns that might have remain unseen

Enduring - Statistical analysis and graphical displays often reveal patterns in seemingly random data or populations, enabling predictions

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Strategies to Draw Generalizations from Students

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Strategies to Draw Generalizations from Students

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Weighing Inquiry-Based Learning and Direct Instruction in Elementary Math

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Teachers can ask themselves three key questions in order to choose the most effective instructional approach to a topic.

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Give concepts and ask students to connect concepts

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Give concepts and ask students to connect concepts

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Example: Write the following concepts in a related sentence. This should demonstrate your understanding of how the concepts are connected:

Parallel lines, gradient, transversal, angles

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At the end of the lesson, ask students to complete the sentence, "I understand that..."

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At the end of the lesson, ask students to complete the sentence, "I understand that..."

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Example:

I understand that solving a quadratic equation by using the quadratic formula graphically displays the x-intercepts of a quadratic function

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After a lesson/unit, ask students to create a concept map which includes the concepts they've learned and how they are connected, with statements explaining them

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We need to understand how to guide students to particular principles/generalizations through inductive inquiry. This encourages students to form conceptual understandings from their own learning experiences

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How Inquiry-Based Learning Can Work in a Math Classroom

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In inquiry-based learning, questions, problems and scenarios are used to help students learn through individual thought and investigation.