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RME 6, Cayman Islands, September 2018

RME 6, Cayman Islands, September 2018. “Storage Rack through the Door” On the Influence of Contextualization on Geometrical Problem Solving Solutions Bernd Wollring & Andrea Peter-Koop Bielefeld University, Germany. Problem Storage Rack through the Door.

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RME 6, Cayman Islands, September 2018

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  1. RME 6, Cayman Islands, September 2018 “Storage Rack through the Door” On the Influence of Contextualization on Geometrical Problem Solving Solutions Bernd Wollring & Andrea Peter-KoopBielefeld University, Germany

  2. Problem Storage Rack through the Door • Problem Storage Rack through the Door • An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is • 2.40 meters wide, • 2.05 meters high and • 20 cm deep. • Does it fit through the door?

  3. Problem Storage Rack through the Door An Example for Reflection … Who may react how?

  4. Problem Storage Rack through the Door Problem Storage Rack throughthe Door: Scalable Topic Kids; mytoys.de Students, Wollring 2012 Student teachers, Vogel 2007

  5. Problem Storage Rack through the Door Storage Rack throughthe Door An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? Study Task 1.Create your personal solution or an approach to a solution, if you like in a team. Is this a task in the RME spirit?

  6. Problem Storage Rack through the Door Storage Rack throughthe Door An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? Study Task 2. Add a context to the task. Create a “contextualized task”. (Doug Clarke 2018)

  7. Problem Storage Rack through the Door Study Task 2. Create a “contextualized task”. (Doug Clarke 2018) Proponents of Realistic Mathematics Education (RME) from The Netherlands advocate the use of contextualized tasks,but rather than emphasize motivation from real life contexts, they focus on providing learning situations that are experientially real for students and a springboard for advancing understanding (Gravemeijer 1997). In their view, the task must require students to “imagine the situation or event so that they can make use of their own experience and knowledge” (van den Heuvel-Panhuizen 2005, p. 3). (Clarke & Roche 2018)

  8. Problem Storage Rack through the Door Storage Rack throughthe Door An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? We will now go through different contexts and illustrate the impact on the solutions.

  9. Problem Storage Rack through the Door Storage Rack throughthe Door, Context 1 An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? Context 1: Realistic situation 1. You are standing at the apartment door with the storage rack. What exactly do you do?

  10. Problem Storage Rack through the Door Work of the Student Vanessa (Grade 11), Context 1, Pre-Advanced Course in Math. As the storage rack is too high and too broad, one cannot simply push it through. If one tried to get the storage rack diagonally through the door it would nevertheless not work, as the door is by calculation a little bit too small. In spite of that, I would try to push the storage rack diagonally through the door and if it does not work I would simply let it stand where it is.

  11. Problem Storage Rack through the Door Storage Rack throughthe Door, Context 2 An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? Context 2: Realistic situation 2. You are standing at the front door of the apartment building with the storage rack. It fits through the front door. Your apartment is on the fourth floor, there is no elevator. What do you do?

  12. Problem Storage Rack through the Door Real Life Proposal of a Student Teacher (Secondary), Context 2 There are different options to find out whether the storage rack fits through the door in the fourth floor. It is assumed that all apartment doors in the building have the same size. The bottom apartment in Munich mostly lies a few steps up from the entrance. So I could kindly ask the neighbour in the bottom apartment whether we might test whether the storage rack fits through his door. So the rack must only be carried a few steps before we are sure . Of course it is necessary for this that the neighbour in the bottom apartment is at home. If not, we might simply try at the door frame whether it fits. But it would be helpful to have three persons so that two people might tilt the rack and the third one observes, whether it fits through the door frame or not. Even if the rack fits , however, it might be useful to go up to the fourth floor without the rack and to measure both doors, the upper and the lower, to be sure that they have the same sizes.

  13. Problem Storage Rack through the Door Work of the Student Alex (Grade 11), Context 2, Pre-Advanced Course in Mathematics As the storage rack has a width of 20 cm one calculates an average loss of the lengths of the doors’ sides as 14,14 cm. If one calculates like this, the storage rack does not fit, but it could tightly fit, when the tilt is slightly changed.

  14. Problem Storage Rack through the Door Storage Rack throughthe Door, Context 3 An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? Context 3. Final Examination Setting (Paper and Pencil). You pose this task in the context of a student teacher’s written final examination. What kind of solution do you expect?

  15. Problem Storage Rack through the Door The professor‘s illusion: not too difficult Spatial imagination: Correct position, Tools: Pythagoras and similarity, FIT!

  16. Problem Storage Rack through the Door Work of a Student Teacher (Primary), Context 3, Final Examination HANDOUT: Material 1 As the diagonal of the door is greater than the diagonal of the storage rack, the storage rack might fit tilt through the door.

  17. Problem Storage Rack through the Door Work of a Student Teacher (Primary), Context 3, Final Examination HANDOUT: Material 2 Finally I calculated how long x is, i.e. how high the rack may maximally be. My result is tight below 2,05m, that means the storage rack does not fit through the door frame.

  18. Problem Storage Rack through the Door How do Student Teachers (Secondary) expect other Student Teachers (Primary) to solve the task, Context 3 The current content of their lecture “modelling” frames this response. The student teachers may suppose that behind this task there is a „Pythagoras-Task“. Therefore the student teachers will firstly model the task and transfer it from the real situation to a mathematical model. Probably a sketch is made where all the necessary measures are recorded. To check whether the storage rack will fit through the door, the student teachers will compare the smallest diagonal of the storage rack with the diagonal of the door frame. Here, as already supposed, the Theorem of Pythagoras comes into the game. The student teachers will now compute and compare the diagonals. The diagonal of the door is about 2,236 m and the diagonal of the (short) side of the storage rack is about 2,059 m. This result is now evaluated and again transferred to the real situation. The student teachers will conclude that the storage rack will fit through the door, if it is tilted.

  19. Problem Storage Rack through the Door Study Task 3. What do you think is the intention of the presented contexts?

  20. Problem Storage Rack through the Door Storage Rack throughthe Door, Context 4 An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? Context 4 GeoGebra-Experts. Pose this task to students or student teachers who are trained in GeoGebra. Please design one or two GeoGebra-Apps.

  21. Problem Storage Rack through the Door Study Task 4. Have a look at the levels in the “Scoring rubric” (Doug Clarke et al. (2006): “Rich Assessment Tasks”). Please anticipate indicators to assess the following GeoGebra solutions. Then allocate the student teachers’ solutions in the rubric.

  22. Problem Storage Rack through the Door “John Brown” Context 4 GeoGebra-experts. Pose this task to students or student teachers who are trained in GeoGebra. Please design one or two GeoGebra-Apps.

  23. Problem Storage Rack through the Door “Jackie Red” Context 4 GeoGebra-experts. Pose this task to students or student teachers who are trained in GeoGebra. Please design one or two GeoGebra-Apps.

  24. Problem Storage Rack through the Door “Lorne Green” Context 4 GeoGebra-experts. Pose this task to students or student teachers who are trained in GeoGebra. Please design one or two GeoGebra-Apps.

  25. Problem Storage Rack through the Door “Millie Blue” Context 4 GeoGebra-experts. Pose this task to students or student teachers who are trained in GeoGebra. Please design one or two GeoGebra-Apps.

  26. Problem Storage Rack through the Door “Patty Purple” Context 4 GeoGebra-experts. Pose this task to students or student teachers who are trained in GeoGebra. Please design one or two GeoGebra-Apps.

  27. Problem Storage Rack through the Door Storage Rack throughthe Door, Context 5 An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? Context 5: Students. Can primary or lower secondary students solve the task? How?

  28. Problem Storage Rack through the Door Position of a Student Teacher (Secondary), Context 5 Primary students in cannot solve this task.

  29. Problem Storage Rack through the Door Position of a Student Teacher (Secondary), Context 5 Primary students cannot solve this task. As a teacher, however, you may try to improve the spatial imagination by explaining the situation to the pupils and letting them draw a door with dimensions 20 cm x 10 cm. Then one may give the storage racks to them, which the teacher has prepared at home as drawings in the scale 1:10 and then cut out. Then the students may try out themselves whether the rack fits somehow through the door. Necessary material for the students is grid paper, a ruler and a pencil, for the teacher coloured paper, a ruler, a pencil and a pair of scissors.

  30. Problem Storage Rack through the Door Anabel‘s Solution (Fourth Grader), Context 5 ANABEL‘S SOLUTION HERE

  31. Problem Storage Rack through the Door Study Task 5. What do you think is the long-time mathematical benefit for Anabel by solving this task? ANABEL‘S SOLUTION HERE

  32. Problem Storage Rack through the Door Storage Rack throughthe Door, Contexts 6 and 7 An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? Focus: Advanced mathematics and professionals

  33. Problem Storage Rack through the Door Storage Rack throughthe Door, Contexts 6 and 7 An apartment door has the inside dimensions 2 meters high and 1 meter wide. A storage rack (not dismountable) is 2.40 meters wide, 2.05 meters high and 20 cm deep. Does it fit through the door? Focus: Advanced mathematics and professionals, included some of the solutions Context 6: May the storage rack, 2.05 m high, be even deeper than 20 cm to fit through the door? How could you find out? Context 7: Professional tools. Supposed you have a removal business. Design a tool that enables your employees to solve problems like this.

  34. Problem Storage Rack through the Door Remember “Patty Purple” and Anabel.

  35. Problem Storage Rack through the Door Remember “Patty Purple” – An appropriate tool for the real world?

  36. Problem Storage Rack through the Door Findings from examples of student work 80% of the solutions presented in context 3 (examination situation) are wrong or useless. The same student teachers’ context-4-solutions (GeoGebra) and context-5-solutions (primary students, example Anabel) were fancy and correct. The “praxis-solutions” were useful up to 80%, which was checked by people not involved in studying mathematics.

  37. Problem Storage Rack through the Door Recommendations for Researchers and Practitioners Whether a problem is perceived in the sense of Realistic Math Education (RME) depends on the given or supposed contexts, especially on the meaning which is attributed to such contexts during the teacher training unit which precedes the posing of the task.

  38. Problem Storage Rack through the Door Recommendations for Researchers and Practitioners Whether a problem is perceived in the sense of Realistic Math Education (RME) depends on the given or supposed contexts, especially on the meaning which is attributed to such contexts during the teacher training unit which precedes the posing of the task. Question Do you think that solutions of this taskby “context localisation” may occur in your classroom situation too?

  39. Problem Storage Rack through the Door Recommendations for Researchers and Practitioners Whether a problem is perceived in the sense of Realistic Math Education (RME) depends on the given or supposed contexts, especially on the meaning which is attributed to such contexts during the teacher training unit which precedes the posing of the task. If and only if there is still a little bit of time left: Do you remember Vanessa?

  40. Problem Storage Rack through the Door Study Task 6. Try to give an adaptive Feedback to Vanessa:First adaptive appreciation , then adaptive support impulse As the storage rack is too high and too broad, one cannot simply push it through. If one tried to get the storage rack diagonally through the door it would nevertheless not work, as the door is by calculation a little bit too small. In spite of that, I would try to push the storage rack diagonally through the door and if it does not work I would simply let it stand where it is.

  41. Problem Storage Rack through the Door Problem Storage Rack throughthe Door: Scalable Topic Kids; mytoys.de Student teachers, Wollring 2016

  42. Problem Storage Rack through the Door References Clarke, Dg. & Roche, A. (2018) UsingContextualizedTesks to engagestudents in meaningfulaldworthwilemathematicslearning. Journal of MathematicalBehavior 51 (2018), 95-108. KMK (2005): Bildungsstandards im Fach Mathematik für den Primarbereich. Beschluss der Kultusminister-konferenz vom 15.10.2004. München: Luchterhand, Wolters Kluwer. http://www.kmk.org Leuders, T., Hußmann, S., Barzel, B., & Prediger, S. (2011) „Das macht Sinn!“ Sinnstiftung mit Kontexten und Kernideen. Praxis der Mathematik 53, Heft 37, 2-9. Lithner, J., Jonsson, B., Carina Granberg, C., Liljekvist, Y., Norqvist, M., & Olsson, J. (2013). Designing tasks that enhance mathematics learning through creative reasoning. in: C. Margolinas, (Ed.), Task Design in Mathematics Education, 221-230. Proceedings of ICMI Study 22. Oxford 2013. Niss, M. (2003). Mathematicalcompetencies and thelearning of mathematics: The Danish KOM project. Third Mediterranean Conference on Mathematics Education, Athens, 115-124. Vogel, D. (2007). Das sperrige Regal. In: W. Herget, S. Schwehr, & R. Sommer (Hrsg.): Materialien für einen realitätsbezogenen Mathematikunterricht. ISTRON-Gruppe, Band 10, S. 141-144. Hildesheim: Franzbecker. Wollring, B., & Reimers, H. (2018). Warum ist das so? Argumentieren bei Begründungsaufgaben. Grund-schulmagazin 4/18 VERA – und dann?, 38-43. Wollring, B. (2011), Raumvorstellung, Kontextverortung und Magic Effects. In: K. Eilerts; A. Hilligus; G. Kaiser & P. Bender, Kompetenzorientierung in Schule und Lehrerbildung. Paderborner Beiträge zur Unterrichts-forschung und Lehrerbildung, Band 15, S.355 – 370. Münster: LIT Verlag.

  43. Thanks! Andrea.peter-koop@uni-bielefeld.de Wollring@mathematik.uni-kassel.de

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