Wathall Chapter 3 by https://padlet.com/harco11/ah841ogxjp6tmldf Concept Based Mathematics en-us 2022-09-22 01:44:35 UTC 2022-09-22 03:31:00 UTC hello@padlet.com Generalizations in Mathematics harco11 https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307852089 2022-09-22 01:48:42 UTC https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307852089 What are generalizations in mathematics? What is the difference between a principle and a generalization in mathematics? harco11 https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307862120 2022-09-22 01:56:20 UTC https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307862120 Generalizations harco11 https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307886914
  • Two or more statements “stated as a sentence of relationship”
  • Transfer through “time, cultures, and across situations” 
  • May contain qualifiers (e.g. may or can) 
  • Contain no proper nouns, past tense verbs, or pronouns that "associate them" with a group or person
  • For example, the angle sums in triangles “may” add to 180 degrees
  • One example of a generalization is the quadratic formula determines the zeros of functions, the roots of equations, and the x intercepts graphically 

    • ]]>     2022-09-22 02:14:20 UTC https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307886914     Principles harco11 https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307904026
  • Laws or foundational truths 
  • Hold “all the attributes of generalizations” 
  • Never contain qualifiers (key difference!) 
  • Are often more specific and less broad than a generalization 
  • At times, can be tied to a specific person or group 
  • One example of a principle is the Pythagorean Theorem 

    • ]]>     2022-09-22 02:25:59 UTC https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307904026     Why do we wish our students to understand principles and generalizations in mathematics? harco11 https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307932014
  • To make connections and recognize patterns 
  • Deepen their understanding, and take their mathematical fluency beyond procedural understanding 
  • To “release” the natural “powers” of a student
  • Give a higher sense of purpose to their learning
  • Encourage application of mathematics in other fields 

    • ]]>     2022-09-22 02:47:08 UTC https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307932014     What is the difference between Level 1, 2, and 3 generalizations? harco11 https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307940441 2022-09-22 02:54:01 UTC https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307940441 Level 1 harco11 https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307948768
  • Are “weak” and do not aid students in understanding “what and why” they are learning 
  • Often include “no no” verbs such as is, affect, and influence 
  • An example of a Level 1 is “the coefficients of a quadratic function in a connected equation affect the quadratic formula.” 

    • ]]>     2022-09-22 03:00:28 UTC https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307948768     Level 2 harco11 https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307968092
  • Guided by the questions “Why?” and “How?” 
  • Scaffolded further than a Level 1 
  • Encompasses critical understanding and should be the main target for teachers 
  • An example of a Level 2 is “the quadratic formula utilizes the coefficients of a quadratic function in a connected equation and describes the roots of a quadratic function”

    • ]]>     2022-09-22 03:15:04 UTC https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307968092     Level 3 harco11 https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307985751
  • Takes scaffolding further and asks, “So what?”
  • Provides the significance or effect of learning something 
  • Expresses the “real-world value of mathematics” 
  • This level has the deepest extent of understanding and application 
  • “Mathematically extends” the understanding of Level 2 with “greater conceptual specificity” 
  • An example includes, “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.”

    • ]]>     2022-09-22 03:30:49 UTC https://padlet.com/harco11/ah841ogxjp6tmldf/wish/2307985751