Wathall Chapter 2 by https://padlet.com/harco11/3r4ktitq38yf92sh Concept Based Mathematics en-us 2022-09-11 21:30:01 UTC 2025-10-28 13:34:58 UTC hello@padlet.com Concept Based Mathematics harco11 https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291429981 2022-09-11 21:35:03 UTC https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291429981 What is the difference between formulae and generalizations in mathematics? harco11 https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291438529 2022-09-11 21:55:54 UTC https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291438529 Formulas harco11 https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291440721
  •  Equations that use mathematical symbols or variables 
  • These equations express a relationship that represents "facts" within the Structure of Knowledge  
  • One example of a formula is the slope-intercept formula ( y = mx + b)
  • Formulas are also at the bottom of the Structure of Knowledge, as they mostly exemplify memorization and procedure, rather than deep understanding of a subject 
]]> 2022-09-11 22:00:46 UTC https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291440721 Generalizations harco11 https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291443556
  • “Statements of conceptual relationships”
  • Often demonstrate a deeper level of understanding and are at the top of the Structure of Knowledge 
  • Aligned with patterns and recognizing why or how a certain concept works
  • One examples of a generalization is trigonometry being based on similar right-angled triangles sharing acute angles and corresponding sides. 
]]> 2022-09-11 22:07:57 UTC https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291443556 How is mathematics a language of conceptual relationships made of macro, meso, and micro concepts? harco11 https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291446216 2022-09-11 22:13:36 UTC https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291446216 Structure of Knowledge harco11 https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291452614 2022-09-11 22:27:42 UTC https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291452614 A beautiful combination harco11 https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291522799 Macros:
  • Provide “breadth of understanding” 
  • Large constructs that contain a vast entity of knowledge 
  • Examples include algebra, statistics, and calculus 

Meso:
  • Smaller than Macros, but bigger than micros 
  • Can be thought of as a topic 
  • Make macros digestible and allow the formation and organization of micros 
  • One example of a meso is trigonometry

Micro:
  • Provide the “supportive depth of understanding”
  • The smallest of the scales, and often embody a small portion of the math language 
  • Examples include ratio and angles 

Ultimately, these concepts work together to establish the language of mathematics as a language of conceptual relationships by creating a logical and chain-like reaction that enhances the engagement with math, and allows people to understand and practice the language conceptually. In a sense, these concepts build and support one another to create a structure for a conceptual language, and by pushing a person to see the concept behind math, rather than the procedures.


]]> 2022-09-12 00:31:29 UTC https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291522799 A structure aiming to enhance math harco11 https://padlet.com/harco11/3r4ktitq38yf92sh/wish/2291532807
  • The structure of knowledge is a structure composed of “levels” that are based on “intellectual thinking” and demonstration of deep understanding
  • It allows one to “classify and recognize similarities, differences, and relationships” in mathematics
  • The lowest level is the “factual level,” which consists of route memorization and factual knowledge (e.g. the different name of polygons) 
  • The highest level is principle generalization, as this requires the application and understanding of mathematical concepts, and an engagement with math that goes beyond memorizing formulas and facts
  • This structure allows educators to identify whether their students are simply memorizing facts, or understanding math on a deep and conceptual level, which often provides key skills, insights, and applications.

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