MTE 494: Wathall Chapter 3 Concept Map by Holly Hoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j Chapter 3: What are Generalizations in Mathematics? en-us 2023-02-06 07:25:29 UTC 2025-09-26 15:16:36 UTC hello@padlet.com https://padlet.net/icons/png/1f469-1f3eb.png Generalizations in Math hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2469641133 Generalizations
  • Give us explanations of why and what we want our students to know in terms of the relationship between two or more concepts 
  • A.K.A enduring understandings, essential understandings, or big ideas
  • Summarize what students should be taking away from a unit

]]> 2023-02-06 07:32:19 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2469641133 Generalizations vs. Principles hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2469643334 "Generalizations are defined as two or more concepts stated as a sentence of relationship. Generalizations are truths supported by factual examples, but they may include a qualifier (often, can, may) when the idea is important but does not hold in all instances” (Erickson, 2014)

Generalizations
  • The overarching ideas we want our students to understand
  • Contain qualifiers
  • Provide goals when unit planning
  • Help learners construct and make connections when learning  
  • Example: The sum of angles in a triangle *may* add up to 180 degrees. (We use the word "may" because this does not hold true in Euclidean geometry)

Principles
  • The theorems in mathematics
  • Never contain qualifiers (may, can, often)
  • Examples: Pythagorean theorem, Fermat's theorem, etc. 
  • Principles are statements of relationship written in the Structure of Knowledge or the Structure of Process 


]]> 2023-02-06 07:34:36 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2469643334 Level 1: Students will understand that... hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474642533 Level 1 generalizations are weak and do not help learners fully understand what and why they are learning. This level of generalization contains verbs that we should try to avoid such as: is, are, have, impact, influence, and affect.]]> 2023-02-09 08:10:55 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474642533 Level 2: How or why? hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474642832 Level 2 generalizations ask the questions "How?" or "Why" and this scaffolds a level 1 generalization up to the next level. At this level, learners are given greater clarity and conceptual specificity, resulting in greater depth of knowledge. ]]> 2023-02-09 08:11:11 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474642832 Level 3: So what? What is the significance? hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474661036 At this level, we are explaining the purpose of the generalization. Critical understanding is already expressed in level 2, but in level 3 we are expressing the real-world value of mathematics and expanding level 2 with greater conceptual specificity. ]]> 2023-02-09 08:25:40 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474661036 Concept-Based Curriculum hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474662727 Inductive Teaching
  • Providing students with examples from which to draw generalizations
    • Traditional curriculum focuses on deductive approaches: students are given a generalization/ equation and then are told to study specific examples
  • Concept-based curriculum prioritizes the inductive approach. It involves guiding students through inquiry to discover principles for themselves.
]]> 2023-02-09 08:27:04 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474662727 Methods hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474668525
  • Students need to have opportunities to form conceptual understanding from learning experiences
  • Learning should involve students looking at specific examples to find patterns and then create generalizations

    • Graphic Organizers
    • Gives students visual representations of how concepts are related, making it easier for them to visually form generalizations

    Unit Webs
    • Unit webs are tools that provide strategies for including generalizations

    ]]>     2023-02-09 08:32:46 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474668525     Benefits hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474676359
  • Generalizations and principles in unit plans provide clear focus for instruction
  • Giving students the ability to generalize is the essence of math
  • Leads to inductive methods of teaching that draw out greater conceptual understandings 

    • ]]>     2023-02-09 08:40:33 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474676359     Generalization hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474677456 2023-02-09 08:41:36 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474677456 Levels and Examples hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474678877 https://us.corwin.com/sites/default/files/upm-assets/78590_book_item_78590.pdf]]> 2023-02-09 08:42:57 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474678877 Understanding the Process of Generalization in Mathematics through Activity Theory Research Article hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474680436 International Journal of Learning, Teaching and Educational Research Vol. 16, No. 12, pp. 46-69, December 2017 https://doi.org/10.26803/ijlter.16.12.4  ]]> 2023-02-09 08:44:30 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474680436 Conceptual Understanding in Math hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474682701 https://grantwiggins.wordpress.com/2014/04/23/conceptual-understanding-in-mathematics/]]> 2023-02-09 08:46:48 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474682701 Scaffolding via Levels hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474683391 2023-02-09 08:47:31 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474683391 Generalization and Advanced Thinking hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474686059 https://blogs.ams.org/mathgradblog/2016/08/21/role-generalization-advanced-mathematical-thinking/]]> 2023-02-09 08:50:04 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474686059 hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474688210 https://whatihavelearnedteaching.com/generalizations-in-math/]]> 2023-02-09 08:51:37 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474688210 hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474688572 https://math4teaching.com/teaching-making-generalizations-in-mathematics/]]> 2023-02-09 08:51:50 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474688572 Strategies for Conceptual Learning hollyhoogstra https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474690298 2023-02-09 08:53:32 UTC https://padlet.com/hollyhoogstra/z18pd1iupwef358j/wish/2474690298