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Generalization & Principle vs Generalization

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Generalization & Principle vs Generalization

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Why emphasize principles and generalizations for our math students

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Why emphasize principles and generalizations for our math students

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Opportunities provided for conceptual understanding 

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Opportunities provided for conceptual understanding 

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gives us explanations of why and what we want our students to comprehend.

Used to relate between two or more concepts

Known as the "essential understandings" or the "big ideas"

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Provoking generalization is more about tuning into students' powers and strengths to come to these generalizations rather than force feeding them. Allows for higher mathematical development and is usually accessible to all types of learners

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Difference between principle and generalization

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Difference between principle and generalization

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Principles are statements used to create generalizations, they are concrete ideas such as theorems (Pythag's theorem for example) and are taught to students, then used for generalization to gain understanding of the "big idea" or of what is really going on.

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Here we have the topic of Quadratics, where the generalization in this topic would be the significance of finding roots and discriminants. Through the use of principles such as the quadratic formula, the students will be able to generalize quadratics as a whole.

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This article, which is pretty interesting, defined generalizations into 3 categories which are

relating - forming an association between two or more problems/objects

searching - one repeats an action to locate an element of similarity

extending - where one expands a pattern or relation into a more general structure

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builds a strong foundation - by grasping fundamental generalizations and principles, students will be able to apply what they have learned across different topics, questions, and even content areas

fosters critical thinking - by understanding general concepts and principles, students will have more confidence when encountering problems they aren't familiar with, which is great for practicing problem solving and critical thinking

long term retention - Students will always remember ideas and content more if they unwrap and mess with the "why's" rather than just learning the "how's"

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collaborative group work - encouraging students to communicate their processes/steps/thinking is a great start to having them really "think out" the processes they have in their brains, which strengthens their understandings

explain their reasoning - it is important that students don't only show the right answer, but that they can explain it on their own and with practice of good math literacy

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project based learning - learning through projects is a great way to get them thinking how what their learning applies to the real world. Once they have experienced it, they will see that it is indeed applicable and will show a proper use of principles/skills

use of technology - new technology tools are coming out all the time, and many of them can do a great job at allowing you manipulate certain variables and see the real-time changes those manipulations have, great for sparking curiosity, raising questions, and exploring patterns/relationships. All equally important for gaining conceptual understanding of topics