Concept Based Mathematics Ch 3 by Viviana Tellez Matias

Generalizations (Big Ideas) in Mathematics

vtellezm — 2024-10-01 23:10:09 UTC

-Give us explanations of why and what we want our students to comprehend in terms of the relationship between two or more concepts

-Provide clear goals for students

Principle

vtellezm — 2024-10-01 23:10:36 UTC

In Mathematics, principles are laws or foundational truths that hold all attributes of generalizations and commonly describe real-life situations.

Example: Pythagorean Theorem

Generalization

vtellezm — 2024-10-01 23:10:46 UTC

In mathematics, generalizations are:

- Defined as two or more concepts that are stated as a sentence of relationship.

-Understandings that transfer through time, across cultures, and situations

-Truths supported by factual examples. They may include a qualifier when the idea is important but does not hold in all instances.

Example: In Euclidean Geometry, triangles always add up to 180 degrees, but in non-euclidean geometry, this generalization is not true.

-Provide goals when unit planning and help learners construct and make connections with what they are learning

In inductive, inquiry-led teaching, learners are guided to discover generalizations and answer the question: "Why am I learning this, and what is the point beyond this example?"

Student Connection

vtellezm — 2024-10-01 23:13:46 UTC

Students should understand principles and generalizations in mathematics because they lay the foundation for mathematics. Students should be able to take generalizations and principles and connect them to new learnings.

Level 1 Generalizations

vtellezm — 2024-10-01 23:16:38 UTC

Include "no-no" verbs such as is, are, have, impact, affect, and influence. We don't want students to write level 1 generalizations.

-Example: The coefficients of a quadratic function in a connected equation affect the quadratic formula

Level 2 Generalizations

vtellezm — 2024-10-01 23:17:01 UTC

Level 2 generalizations showcase deeper understandings than level 1 generalizations. Teachers should ask questions like "How?" or "Why?" to help scaffold a generalization to the next level because it gives learners clarity and conceptual specificity, which results in deeper understanding.

-Ex: The quadratic formula utilizes the coefficients of a quadratic function in a connected equation and describes the roots of a quadratic function.

Level 3 Generalizations

vtellezm — 2024-10-01 23:17:33 UTC

Can be used to express the real-world value of mathematics or to mathematically extend level 2 understandings. They address the question, "So what is the significance or effect?''

-Ex: Different methods of solving quadratic equations distinguish the roots of the quadratic equation, which contribute to providing a graphical representation of the function, which, in turn, describes real-life problems.

Students' Conceptual Understandings

vtellezm — 2024-10-07 23:26:12 UTC

Students look at specific examples and seek patterns to devise and write generalizations
Graphic organizers help draw out generalizations; they give students a visual representation of how the concepts are related
Students communicate conceptual understanding verbally and nonverbally

Overarching Generalizations

vtellezm — 2024-10-07 23:57:01 UTC

Understanding beyond the specifics of a unit. They are larger, transferable insights we want students to acquire

Topical Generalizations

vtellezm — 2024-10-07 23:58:49 UTC

Subject or topic specific. They focus on a particular insight we want students to acquire within a unit of study.