MTE 494: Wathall Chapter 2 Concept Map by Holly Hoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u Chapter 2: What are the levels of the structures of knowledge and process for mathematics? en-us 2023-02-06 08:42:16 UTC 2023-02-13 09:32:44 UTC hello@padlet.com https://padlet.net/icons/png/1f34e.png Structure of Knowledge Diagram hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2469720543 2023-02-06 08:46:31 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2469720543 Factual Level hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478571754
  • Lowest level in the Structure of Knowledge 
  • Rote memorization
  • Does not guarantee conceptual depth of understanding
  • Is defined by facts
    • Specific examples, locked in time, not transferable, 
    • Includes definitions, symbolic formulae (ex: y = mx + b), and polygon names
    • Facts help students understand concepts, but are meant to be building blocks as knowing facts is more indicative of memorization and less of conceptual understanding
    • ]]>     2023-02-13 08:27:51 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478571754     Formula hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478573920
  • An equation that uses math symbols or variables to shown relationships 
  • Is represented by facts in the Structure of Knowledge 
    • ]]>     2023-02-13 08:29:10 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478573920     Theorems hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478575856
  • Statements that have been proven 
  • They connect explanations to conceptual understanding
  • Represented by principles in the Structure of Knowledge
    • ]]>     2023-02-13 08:30:54 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478575856     Generalizations and Principles hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478578176
  • Statements of conceptual understanding
  • Allow students to make connections between two or more topics 
  • Principles are theorems (cornerstone truths)
  • Generalization example: Similar right-angled triangles share a common acute angle
  • Principle example: Theorems in calculus allow the evaluation of a definite integral without having to go back to the definition and take the limiting sum of the bounded area 
    • ]]>     2023-02-13 08:33:15 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478578176     Skills hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478579290
  • Skills make up algorithms 
  • They are small operations or actions embedded in strategies that allow strategies to work
  • Skills underpin strategies
  • Examples: Creating a table of values, plotting a graph, using trial and error 
    • ]]>     2023-02-13 08:34:27 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478579290     Processes hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478580764
  • Complex, sophisticated performances
  • Composed of strategies, algorithms, and skills
  • Broad techniques students draw upon when learning math 
  • Support the understanding of concepts in a unit of work
  • Example: Problem solving 
    • ]]>     2023-02-13 08:35:53 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478580764     Structure of Process for Trigonometry hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478581975 2023-02-13 08:37:01 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478581975 Algorithms hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478585496
  • Specific to mathematics
  • A set of computation rules to produce a specified outcome 
  • Example: PEMDAS
  • Consist of skills 
    • Skills allow learners to follow a set of rules to come to a specified outcome
    • ]]>     2023-02-13 08:40:18 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478585496     Problem Solving hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478590799
  • Fundamental building block of math
  • "A series of skills and strategies that form mathematical techniques that a learner employs when faced with an unfamiliar situation"
  • Problem solving should be challenging, engaging, and pique learners interest enough to encourage pursuit of the presented situation  
    • ]]>     2023-02-13 08:45:16 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478590799     6 Mathematical Processes hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478591270 2023-02-13 08:45:44 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478591270 Reasoning and Proof hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478620010
  • Reasoning 
    • The ability to make generalizations and provide explanations and justification for arguments 
    • Leads to generalizations which provide interconnectedness of math knowledge
    • Gives learners opportunities to make mistakes
    • Leads to building mathematical memory to understand concepts
    • Proof
    • Driven by critical thinking skills and asking "why"

    ]]>     2023-02-13 09:13:24 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478620010     Communicating hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478621514
  • Talking and writing about math
  • Presenting different methodologies provides learners with alternative perspectives and enables them to build a repertoire of strategies for problem solving 
  • Involves: discussion, sharing ideas, collaboration ideally in the form of paired or small group work (not teacher-led whole class discussion)
  • Example: Number talks
    • ]]>     2023-02-13 09:14:58 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478621514     Making Connections hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478623302 Refers to the learner's ability to:
    • See connections between facts and how they relate to one another
    • Make connections between symbols and procedures
    • Make connections between what they are learning and the real world
    • Connect new problems to old
    • See different concepts and how they relate to one another
    ]]>     2023-02-13 09:16:42 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478623302     Creating Representations hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478625289
  • Representations include graphs, tables, geometric figures, etc.
  • Using a variety of representation methods allows students to learn in different ways and see connections
  • Purpose of representation:
    • To help learners organize, record, and communicate mathematical ideas
    • To select, apply, and translate different forms of representation to solve problems
    • To use representations to model and interpret physical, social, and mathematical phenomena 
  • Pictorial representation
    • Includes drawings that contain irrelevant aspects of a problem 
  • Schematic representation
    • Purposeful to solving the problem and are codes/ symbols to show relationships

    • ]]>     2023-02-13 09:18:38 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478625289     Investigating hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478628178
  • Being able to explore unfamiliar math situations
  • When investigating, students ask questions of themselves, furthering mathematical understanding
  • Includes the ability to be resourceful by extracting relevant ideas from research 
    • ]]>     2023-02-13 09:21:16 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478628178     Macro Concepts hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478628890
  • Give breadth of understanding
  • Posses the greatest transferability across most disciplines 
    • ]]>     2023-02-13 09:22:02 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478628890     Meso Concepts hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478629475
  • A further breakdown  of macro concepts 
  • Example: Trigonometry
    • ]]>     2023-02-13 09:22:37 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478629475     Micro Concept hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478630026
  • Give us the supportive depth of understanding
  • Example: ratio, angles, similarity 
    • ]]>     2023-02-13 09:23:09 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478630026     Examples of Macro, Meso, and Micro Math Concepts hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478630715 2023-02-13 09:23:53 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478630715 Topic and Concept Levels hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478632622
  • Topics
    • Broader concepts that break down to specific micro concepts at the next level
    • Example: The topic "integration" is supported by the micro concepts of: limits, bounded area, summation
    • ]]>     2023-02-13 09:25:35 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478632622     Generalizations and Principles Level hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478633596 Generalizations and principles are statements of conceptual understanding that let students make connections ]]> 2023-02-13 09:26:19 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478633596 hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478635080 https://www.tes.com/magazine/archive/exploring-knowledge-structure-maths]]> 2023-02-13 09:27:40 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478635080 hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478635594 https://www.sagepub.com/sites/default/files/upm-binaries/59758_Erickson_Chapter_4.pdf]]> 2023-02-13 09:28:06 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478635594 hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478636244 https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1030&context=mathmidsummative]]> 2023-02-13 09:28:43 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478636244 hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478637108 https://www.splashlearn.com/math-vocabulary/algebra/formula]]> 2023-02-13 09:29:23 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478637108 hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478637580 https://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/]]> 2023-02-13 09:29:52 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478637580 hollyhoogstra https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478638061 https://link.springer.com/chapter/10.1007/978-1-4020-8872-8_10]]> 2023-02-13 09:30:21 UTC https://padlet.com/hollyhoogstra/6xc6g3um7hcw8g4u/wish/2478638061