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  1. Teaching Negatives with Progressive Autonomous Self- Scaffolding P A S S Doctor Kofman Quality Engineering Education, Inc. PASS www.QUEED.org DrKofman@QUEED.org

  2. PASS Progressive Autonomous Self -Scaffolding PASS helps students learn In the Zone of Proximal Development via process oriented problem solving and form the experiences that are necessary for acquiring further knowledge PASS Z P D Zone of Proximal Development problem solving experiences necessary

  3. ZPDand Scaffolding Prof. Vygotsky’s viewsExample – riding bicycles Two students are proficient with their tricycles. (same test results) The parents help them to learn to ride bicycles. This process is calledscaffolding. Student A learned fast. Student B needed much more time. We say that these students have different Zones of Proximal Development.

  4. ZPD ZPD of the student A ZPD of the student B ZPD from within Zone of CertainKnowledge Rapid Progress! Need more help! Student has the Necessary Background Experiences Some Background Is missing What is the difference between ZPDs?

  5. Student C Background Background Background Background Background Background Background Background Background Background Knowledge is a complexentity Knowledge about a Particular Concept Requires Multiple Background Experiences Student Chas some experiences, but does not have others

  6. Student C Background No Background No Background No Background A ZPD has a Complex Profile Knowledge about a Particular Concept Knowledge about some aspects of the concept Background

  7. Student C The Level of Preparedness The green linereflects the level of Prerequisite background knowledge ZPD Profile defines the Level of Preparedness

  8. Whole Class Student C Prerequisite Teaching a Concept Student C needs a helper to scaffold him in his ZPD. Each student in this class has a different level of preparedness. Each student needs a helper. Help Needed! How can the teacher handle this alone? The level of preparedness

  9. Student C Guidance and Experience are Needed Need Guidance Need Experience And Guidance

  10. PASS Prerequisite Progressive Autonomous Self-Scaffolding PASS provides and PASS Guidance Experience The student chooses what steps of the PASS to use!

  11. PASS Why Progressive Autonomous Self –Scaffolding? Students solve problems using AutonomousScaffolding instead of using the help of a teachers, peers or the parents. We call our method Self-Scaffolding, because students decide by themselves when to use it. We create the Autonomous Self-Scaffolding that has several Progressive Scaffolding layers. These layers allow students to progress smoothly from one concept to another.

  12. Concept of Negatives. Prerequisites. Some prerequisites are tested Other prerequisites are usually not tested The ability to Perform Basic Arithmetic Operations Concepts of the Same and Opposite Directions The ability to Visualize Measurable Numbers and Operations Fluency with Commutative Property when working with subtraction 13 - 5 = 8 7 x 3 = 7 + 7 + 7 = 21 5 + 3 = 8 5 3 Students have to understand their own hand-writing! Students develop these skills by solving Specially designed multi-step problems. Students have to understand their teachers and their textbooks, and communicate their thoughts and ideas! 37 – 15 = 30 + 7– 10 – 5 = 30 – 10+ 7 – 5 = 20 + 2 = 22 8 The grade-appropriate study skills form another set of prerequisites. Comprehensive handwriting Communication Skills The ability to concentrate On solving 2-3 steps problems

  13. PASS 23 x 5 ? 8 28 Measurable Numbers The ability to visualize measurable numbers and arithmetic operations Students visualize the problems Students solve the equations 1) 8 – 5 = ? 2) 23 + x = 28 8 – 5 = 3 x = 28 – 23 = 5

  14. PASS d c a e b i f g h The Same and Opposite Directions Concept of the same and opposite directions A segment with arrow defines a translation. 1. Find the translation equivalent to the translation a 2. Find the translation opposite to the translation b d g Answer: a)g; b)d

  15. PASS The student keeps the negative sign with the number. He uses the negative number in its concrete form. Commutative Property and Subtraction Fluency with the commutative property When working with subtraction 37 – 15 = 30 + 7– 10 – 5 = 30 – 10+ 7 – 5 = 20 + 2 = 22 This process oriented approach develops student’s study-skills.

  16. PASS PASS strategies PASS provides students with the set of problems that helps them create the prerequisite background The learning is process oriented and fosters students’ study skills. When the students have formed all the Prerequisite experiences, they start learning Negatives.

  17. PASS Learning Negatives with PASS Positive and Negative Numbers as Translations The Addition of the Translations -2 -8 0 6 -10 -8 3 -2 + (-8) = -10 -2

  18. PASS Learning Negatives with PASS For Big numbers and Non-integers Use the non-scaled representation 0 354 -854 -500 354 + (-854) = -500

  19. PASS -10 -10 -10 -10 -20 -10 is added to itself 20 10 10 -20 10 is added to itself and then flipped Learning Negatives with PASS Multiplication through visual representation 3) -(2 x (-10)) = 20 1) (-10) x 2 = -20 2) -(10 x 2) = -20 -10 is added to itself and then flipped Conclusion: Negative x Negative = Positive

  20. PASS Learning Negatives with PASS After students understand the concept of negatives, they learn that to Subtract the Number is the same as to add the Opposite Number. 60 – 80 = 60 + (-80) Solving Math problems involving negatives, students use PASS every time they need to visualize their problems

  21. PASS Conclusion. • As we have seen, PASS helps students to • learn in the Zone Of Proximal Development • acquire the necessary prerequisite experiences • apply higher order thinking • get “visual” support every time they need it • develop study skills • acquire deep conceptual understanding • PASS helps students to success

  22. Thanks and Credits Special thanks to my family and friends who Helped me by discussing my work. A lot of thanks to my students who had inspired me to develop the PASS. The photo-pictures were downloaded from the websites below: www.cdc.gov/ncipc/ bike/images/tricycle.gif http://win.niddk.nih.gov/publications/energize/frstbk.jpg http://freepages.family.rootsweb.com/~bobofwa/1951_Bob_riding_tricycle.jpg http://www.weblogsinc.com/common/images/3486140958426705.JPG?0.45871028096262767 www.humanities.uci.edu Port 80 Index of/users/vfolkenflik/VRF Sources