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Answer the questions of "what is the significance or effect ?" which would explain the reason for the generalization. Often used used to explain or express the real-life application of mathematics or to extend the understanding from the level 2 generalization.

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Level 2 generalizations provide are clear states of conceptual understanding. They answer the "how or why?" a concept relates to the generalization. Level 2 should be the teaching target to aim for.

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Level 1 generalizations are the lowest and weakest and give little for the students to understand what they are learning. Often include qualifiers that stunt the understanding such as "as, is, are, have, affect, influence".

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Why we want student to understand Principles and Generalizations?

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Why we want student to understand Principles and Generalizations?

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The reason for this is so that students are able to make the profound connections to mathematics, that there are beautiful patterns that are all interconnected. When students are able to make the jump from just learning concepts as distinct separate pieces to that they are generalizations and principles that are connected, they are more likely to succeed in making connections to not only (discipline) mathematics but to our world around us.

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Like in this diagram using the Structure of Knowledge the generalizations for different equations (which are the qualifiers i.e. if it is a degree of 1 it has a single root). For principle there are no qualifiers because they are fundamentals truths drawn from the collections of concepts such as the Fundamental Theorem of Algebra (Any polynomial of degree n has n roots).

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Foundational Laws of mathematics

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Generalizations are what we want our students to comprehend. They are the relationships two or more concepts.

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Representation of the relationships  between topics and facts. From those topics and fact concepts are drawn from them and finally generalizations and principles.

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The Structure of Process is the complement to the Structure of Knowledge. it shoes the relationship between processes, strategy skills, and concept generalization.

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Formulas are based upon facts in the structure of Knowledge. This is equivalent to the algebra cheat sheets that are given in front of math textbooks which are the collections of facts that are useful for the concepts inside the material.

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Theorems are represented by the Principles on the Structure of Knowledge diagram. Theorems are the big connections for conceptual understanding.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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Strategies, Skills, and Mathematical Processes.

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Strategies, Skills, and Mathematical Processes.

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From Stanford Online's "How To Learn Math for Teachers and Parents": Number Talks

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Strategies are the number of skills that learners use to support their learning. From the Boaler video we see different strategies students use to solve the question. From decomposing the numbers, over estimating, etc.

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Skills are the small operations that make the strategies work. Such in the video a student's overestimation of 18 x 5 did not make sense so he used his math skills to make a judgement and correct for the error.

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Mathematical Processes are complex methods that are made from the prior processes. Such as the student knowing that 18 x 5 is equivalent to the multiplication of 9 * 10 for the complex strategies of factoring a 2 from 18 and applying it to 5 to make 10.