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International School ManilaMathematical Thinking in the    Elementary SchoolTuesday 11th September 2012
Conceptual vs. Procedural• Conceptual           • Proceduralunderstanding means:   understanding means:Knowing            ...
Procedural understanding …• “Rules without           • Examples:  reasons”                   – ‘borrowing’• Multiplicity o...
Conceptual understanding …• Fewer principles• More general  applications• Adaptable to new tasks• Easier to remember• Enjo...
• Procedural = learning of an increasing  number of fixed plans… to go from  starting points to finishing points• Conceptu...
Different kinds of math?• “What constitutes Mathematics is not  the subject matter, but a particular  kind of knowledge ab...
What Does It Mean to Understand Mathematics?• Knowing ≠ Understanding• Understanding is the measure of  quality and quanti...
“Understanding is the key toremembering what is learnedand being able to use itflexibly.”Hiebert, in Lester & Charles,Teac...
Computational Fluency                                        I th o u             0                                   gh  ...
Computational Fluency       13 x 27
What is Computational Fluency?Computational fluency is having and  usingefficient and accurate methods forcomputing with u...
What is Computational Fluency?Fluency demands more of students than  does memorization of a single procedure.• An understa...
Procedural vs. Conceptual       Knowledge Objects and names of objects      are not the same asrelationships between objec...
Implications for Teaching at             ISMThe need to replace the question“Does the student know it?”with the question“H...
Concrete              Place Value                      T    U                      2    6       XXXXXX                 26P...
Conceptual Understanding withProcedural Skills•It’s not either / or•At ISM it is ‘with’ASB MCI2 ProblemSolving
The value of a tool is in its          usefulness• Being able to do pencil-paper  computation will not serve students  wit...
Models• Concrete -- Pictorial -- Abstract• Multiple models help solidify thinking  and deepen understanding
Why draw?• Computational practice, but much more• Notation helps them understand the  question.• Notation helps them inven...
Manipulatives• any concrete object which can be moved  and used in a way to represent abstract  concepts in a physical fas...
Boxing Gloves
Building Mathematical Concepts  Concrete       Pictorial     AbstractManipulatives Representation   Symbols               ...
The Bridge To           Understanding            Representation                “SEEING” StageConcrete                     ...
Concrete   Representational   Abstract
Mathematical Thinking at ISM parent workshop
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Mathematical Thinking at ISM parent workshop

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Mathematical Thinking at ISM parent workshop

  1. 1. International School ManilaMathematical Thinking in the Elementary SchoolTuesday 11th September 2012
  2. 2. Conceptual vs. Procedural• Conceptual • Proceduralunderstanding means: understanding means:Knowing Knowing – What to do – What to do and – Why
  3. 3. Procedural understanding …• “Rules without • Examples: reasons” – ‘borrowing’• Multiplicity of rules – ‘carry’ the 1• Usually easier to – ‘turn upside down and understand (to follow) multiply’• Rewards are more – ‘take it over to the immediate other side and change the sign’• Less knowledge involved• Get answer quickly
  4. 4. Conceptual understanding …• Fewer principles• More general applications• Adaptable to new tasks• Easier to remember• Enjoyable = goal in itself• Natural growth = active seeking of new areas (like tree extending its roots)
  5. 5. • Procedural = learning of an increasing number of fixed plans… to go from starting points to finishing points• Conceptual= building up a conceptual structure… go from starting to finishing points in unlimited number of ways
  6. 6. Different kinds of math?• “What constitutes Mathematics is not the subject matter, but a particular kind of knowledge about it.”• Conceptual math = Mathematics• Procedural math ≠ Mathematics
  7. 7. What Does It Mean to Understand Mathematics?• Knowing ≠ Understanding• Understanding is the measure of quality and quantity of connections between new ideas and existing ideas
  8. 8. “Understanding is the key toremembering what is learnedand being able to use itflexibly.”Hiebert, in Lester & Charles,Teaching Mathematics through Problem Solving, 2004.
  9. 9. Computational Fluency I th o u 0 gh i s 14 6 25’s - t seven x 20 8 is 5 Then that’s7 I need 175. x or 21. seand 7 0 is 196 So th ven 3’s e an + 14 is 175 + 21 = swer56 196 7 x 28 I did 7 That’s x 30 fir 210. T st. off sev hen tak en 2’s e So it’s or 14. 196.
  10. 10. Computational Fluency 13 x 27
  11. 11. What is Computational Fluency?Computational fluency is having and usingefficient and accurate methods forcomputing with understanding.
  12. 12. What is Computational Fluency?Fluency demands more of students than does memorization of a single procedure.• An understanding of the meaning of the operations and their relationships to each other• Number relationships, including facts• Understanding of the base ten number system
  13. 13. Procedural vs. Conceptual Knowledge Objects and names of objects are not the same asrelationships between objects.
  14. 14. Implications for Teaching at ISMThe need to replace the question“Does the student know it?”with the question“How does the student understand it?”
  15. 15. Concrete Place Value T U 2 6 XXXXXX 26Partitioned Words / Numbers20 + 6 = 262(10) + 6 = 26 Twenty six 26
  16. 16. Conceptual Understanding withProcedural Skills•It’s not either / or•At ISM it is ‘with’ASB MCI2 ProblemSolving
  17. 17. The value of a tool is in its usefulness• Being able to do pencil-paper computation will not serve students without the ability to interpret a problem, analyze what needs to be done, and evaluate the solution.
  18. 18. Models• Concrete -- Pictorial -- Abstract• Multiple models help solidify thinking and deepen understanding
  19. 19. Why draw?• Computational practice, but much more• Notation helps them understand the question.• Notation helps them invent new solutions.• Notation helps them undo the solution.• But most important, the idea that notation/representation is powerful!
  20. 20. Manipulatives• any concrete object which can be moved and used in a way to represent abstract concepts in a physical fashion.• commonly implies a touchable and movable object which can provide children something real to reflect on.• their importance lies in being able to represent mathematical situations which are generally abstract.
  21. 21. Boxing Gloves
  22. 22. Building Mathematical Concepts Concrete Pictorial AbstractManipulatives Representation Symbols 4+4=8 IIII 2x4=8 IIII
  23. 23. The Bridge To Understanding Representation “SEEING” StageConcrete Abstract“DOING” Stage “SYMBOLIC” Stage
  24. 24. Concrete Representational Abstract

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