Presentatie titel<br />What is <br />Realistic Mathematics Education?<br />Rotterdam, 00 januari 2007<br />Swakopmund,  Ma...
Europe<br />The Netherlands<br />Holland<br />Rotterdam<br />
Rotterdam primary schools<br />
April 12th, 1961<br />
<ul><li>Hans Freudenthal</li></ul>psychologist, mathematician, founder<br />of the Dutch New Math Movement.<br /><ul><li>M...
			‘Mathematics must be connected to 			reality, stay close to children and be 			relevant to society, in order to be of 	...
1960<br />Mechanisticmathematics<br />Education in 95% of the Dutch<br />primary schools<br />MME<br />Context problems as...
What is realistic mathematics education?<br />You enter a classroom. The teacher of this classroom gives her lessons accor...
What is Basic elements of Realistic Mathematics Education<br /><ul><li>Starting points are contexts and the learner experi...
Use of smart strategies
Learning by use of models and diagrams
Learning in interaction
Use of characteristic teaching materials</li></li></ul><li>Learner centered mathematics education<br />Realistic <br />mat...
Learning from the learners’ experiences
Learning from useful materials
Learning from pair and group activities
Learning by doing
Trial-and-error learning</li></ul>Many similarities,  small differences!<br /><ul><li>Contexts
Interaction
Models and diagrams
Strategies
Use of characteristic, structured teaching materials
Intertwining of teaching-learning trajectories</li></li></ul><li>What is Basic elements of Realistic Mathematics Education...
Strategies
Models and diagrams
Interaction
Use of characteristic materials</li></li></ul><li>Contexts in Namibian maths books<br />
Contexts in Dutch maths books<br />
What is a context for a child? (1)	<br /><ul><li>What is the childrens’ reality?
What experience do they have?
What kind of context do they understand?</li></li></ul><li>What is a context for a child? (2)	<br /><ul><li>If we know wha...
By doing so, the child will certainly be motivated to learn.
Why do we use contexts? </li></ul>	The child sees that you can use mathematics in daily life, that is to say, in your life...
What is a context for you?<br />How numerated are you?<br />
Right or wrong?<br /><ul><li>Namibia is about 20 times larger than Holland
Right
Wrong </li></li></ul><li>     Right or wrong?<br /><ul><li>Holland has 20 times as many inhabitants as</li></ul>	Namibia<b...
Wrong</li></li></ul><li>The mass of a slice of<br />bread is about …<br /><ul><li>20 gr
30 gr</li></li></ul><li>The capacity of this<br />coffee cup is about …<br /><ul><li>100 ml
200 ml</li></li></ul><li>The heigth of a grown-up<br />male giraf is at most<br />about …<br /><ul><li>5,25 meter
7,25 meter </li></li></ul><li>The mass of a<br /> grown-up dikdik is<br /> about …<br /><ul><li>	4 kg
	7 kg</li></li></ul><li>The surface of the<br />cricketfield is about …<br /><ul><li>	  7500 m2
	15000 m2</li></li></ul><li>What is Basic elements of Realistic Mathematics Education<br /><ul><li>Contexts
Strategies
Models and diagrams
Interaction
Use of characteristic materials</li></li></ul><li>Strategies (1)<br /><ul><li>We know how we teach children maths....but....
Do we know how children think?
Do we know how children really solve the maths problems? </li></ul>	Is that in the way we told them to do it?<br /><ul><li...
Strategies (4)<br /><ul><li>About the lemonade..</li></ul>	1½ liter of lemonade... let’s see: <br />	With 1 liter you can ...
Strategies (5)<br />A book with 92 pages.<br />You are reading at the bottom of page 27.<br />How many pages to read when ...
Solutions <br />Algorithm calculation:<br />Addingbystringing, the ‘cashiersstrategy’:<br />27 +   3 = 30,<br />30 + 60 = ...
Solutions <br />Stringing:<br />Splitting:<br />deficit<br />Combination of splitting and stringing:<br />
Scale strategy:<br />Jumping toofar and back:<br />Smart strategies <br />                  92 – 27 = 92 – 30 + 3 = 65<br />
Strategiescalculation up to 20 (1)<br />8 + 7<br />9, 10, 11, 12, 13, 14, 15<br />8 + 2 = 10, 10 + 5 = 15<br />5 + 3 + 5 +...
Strategies calculation up to 20 (2)<br />Maureen:<br />Thijs:<br />Luuk: 	“First, put three euro’s out of the six<br />   ...
Strategieslearning the times tables (1)<br />
Strategies learning the times tables (2)<br />
Strategies learning the times tables (3)<br />‘Stepping stones’ <br />
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What is Realistic Mathematics Education? National Mathematics Conference Swakopmund

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The basic principles of Realistic Mathematics Education; Swakopmund, May 8-11, 2011

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What is Realistic Mathematics Education? National Mathematics Conference Swakopmund

  1. 1. Presentatie titel<br />What is <br />Realistic Mathematics Education?<br />Rotterdam, 00 januari 2007<br />Swakopmund, May 2011<br />
  2. 2. Europe<br />The Netherlands<br />Holland<br />Rotterdam<br />
  3. 3.
  4. 4. Rotterdam primary schools<br />
  5. 5.
  6. 6. April 12th, 1961<br />
  7. 7. <ul><li>Hans Freudenthal</li></ul>psychologist, mathematician, founder<br />of the Dutch New Math Movement.<br /><ul><li>Main idea: each individual</li></ul>discovers mathematical structures in<br />its own living environment and creates<br />a personal concept of mathematics.<br />This is the principle of ‘Guided <br />Reinvention’.<br /><ul><li>Realistic Mathematics Education</li></ul>http://www.fisme.uu.nl/fisme/en/<br />
  8. 8. ‘Mathematics must be connected to reality, stay close to children and be relevant to society, in order to be of human value.<br />Instead of seeing mathematics as subject matter that has to be transmitted, I see the idea of mathematics as a ‘human activity’. Education should give students the “guided” opportunity to “re-invent” mathematics by doing it. <br />This means that in mathematics education, the focal point should not be on mathematics as a closed system but on the activity, on the process of mathematization, going from the world of life into the world of symbols’ <br />(Freudenthal, 1968).<br />
  9. 9. 1960<br />Mechanisticmathematics<br />Education in 95% of the Dutch<br />primary schools<br />MME<br />Context problems as a field of<br />application,<br />Procedure-focusedwayof teaching<br />in which the learning content is<br />split up in meaningless small parts,<br />Students are offered fixed solving<br />procedures to be trained by<br />exercises.<br />2011<br />Realisticmathematics<br />Education in 95% of the Dutch<br />primary schools<br />RME<br />Context problems as a sourcefor<br />the learningprocessand to apply<br />mathematicalconcepts,<br />Complex and meaningful <br />conceptualization of teaching and<br />learning,<br />Students are considered to be<br />active participants in the teaching<br />learning process, in which they<br />develop mathematical tools <br />and insights.<br />
  10. 10. What is realistic mathematics education?<br />You enter a classroom. The teacher of this classroom gives her lessons according the basic principles of RME. How do you recognize this?<br />What is the influence of teaching according RME on the learning of the learners?<br />What can you expect of the learning proceeds by teaching mathematics education according to the principles of RME?<br />Does RME make the maths lessons more interesting, more attractive for the learners?<br />What does a lessonplan look like for a maths lesson <br /> according RME?<br />
  11. 11.
  12. 12. What is Basic elements of Realistic Mathematics Education<br /><ul><li>Starting points are contexts and the learner experiences
  13. 13. Use of smart strategies
  14. 14. Learning by use of models and diagrams
  15. 15. Learning in interaction
  16. 16. Use of characteristic teaching materials</li></li></ul><li>Learner centered mathematics education<br />Realistic <br />mathematics education<br /><ul><li>Starting point is the learners’ existing knowledge
  17. 17. Learning from the learners’ experiences
  18. 18. Learning from useful materials
  19. 19. Learning from pair and group activities
  20. 20. Learning by doing
  21. 21. Trial-and-error learning</li></ul>Many similarities, small differences!<br /><ul><li>Contexts
  22. 22. Interaction
  23. 23. Models and diagrams
  24. 24. Strategies
  25. 25. Use of characteristic, structured teaching materials
  26. 26. Intertwining of teaching-learning trajectories</li></li></ul><li>What is Basic elements of Realistic Mathematics Education<br /><ul><li>Contexts
  27. 27. Strategies
  28. 28. Models and diagrams
  29. 29. Interaction
  30. 30. Use of characteristic materials</li></li></ul><li>Contexts in Namibian maths books<br />
  31. 31. Contexts in Dutch maths books<br />
  32. 32. What is a context for a child? (1) <br /><ul><li>What is the childrens’ reality?
  33. 33. What experience do they have?
  34. 34. What kind of context do they understand?</li></li></ul><li>What is a context for a child? (2) <br /><ul><li>If we know what children are interested in, it must be easier for us, the teachers, to use this environment in educating children.
  35. 35. By doing so, the child will certainly be motivated to learn.
  36. 36. Why do we use contexts? </li></ul> The child sees that you can use mathematics in daily life, that is to say, in your life.<br />
  37. 37. What is a context for you?<br />How numerated are you?<br />
  38. 38. Right or wrong?<br /><ul><li>Namibia is about 20 times larger than Holland
  39. 39. Right
  40. 40. Wrong </li></li></ul><li> Right or wrong?<br /><ul><li>Holland has 20 times as many inhabitants as</li></ul> Namibia<br /><ul><li>Right
  41. 41. Wrong</li></li></ul><li>The mass of a slice of<br />bread is about …<br /><ul><li>20 gr
  42. 42. 30 gr</li></li></ul><li>The capacity of this<br />coffee cup is about …<br /><ul><li>100 ml
  43. 43. 200 ml</li></li></ul><li>The heigth of a grown-up<br />male giraf is at most<br />about …<br /><ul><li>5,25 meter
  44. 44. 7,25 meter </li></li></ul><li>The mass of a<br /> grown-up dikdik is<br /> about …<br /><ul><li> 4 kg
  45. 45. 7 kg</li></li></ul><li>The surface of the<br />cricketfield is about …<br /><ul><li> 7500 m2
  46. 46. 15000 m2</li></li></ul><li>What is Basic elements of Realistic Mathematics Education<br /><ul><li>Contexts
  47. 47. Strategies
  48. 48. Models and diagrams
  49. 49. Interaction
  50. 50. Use of characteristic materials</li></li></ul><li>Strategies (1)<br /><ul><li>We know how we teach children maths....but....
  51. 51. Do we know how children think?
  52. 52. Do we know how children really solve the maths problems? </li></ul> Is that in the way we told them to do it?<br /><ul><li>Should each child solve the problems in the exact way we told them to do so?</li></li></ul><li>Strategies (2)<br /><ul><li>Please solve the next sum:</li></ul>I’ve got a bottle with 1½ liter of lemonade. <br /> I pour this lemonade out in glasses of 1/6 liter. <br /> How many glasses can I fill?<br /> And now:<br /><ul><li>3/4 ÷ 1/8 =</li></li></ul><li>Strategies (3)<br />How many of you solved <br />3/4 ÷ 1/8 =<br />like this: <br />3/4 ÷ 1/8 = 3/4 x 8/1 = 24/4 = 6?<br />
  53. 53. Strategies (4)<br /><ul><li>About the lemonade..</li></ul> 1½ liter of lemonade... let’s see: <br /> With 1 liter you can pour out 6 glasses of 1/6 liter. <br /> So half a liter is good for 3 glasses. <br /> That makes 9 glasses together!<br /><ul><li>Important question: </li></ul> What did they teach you at school?<br /> They told you to solve like <br /> 11/2 ÷ 1/6 =3/2 x6/1= 18/2 = 9 ?<br />
  54. 54. Strategies (5)<br />A book with 92 pages.<br />You are reading at the bottom of page 27.<br />How many pages to read when page 27 is finished?<br />
  55. 55. Solutions <br />Algorithm calculation:<br />Addingbystringing, the ‘cashiersstrategy’:<br />27 + 3 = 30,<br />30 + 60 = 90,<br />90 + 2 = 92,<br />So: 3 + 60 + 2 = 65<br />
  56. 56. Solutions <br />Stringing:<br />Splitting:<br />deficit<br />Combination of splitting and stringing:<br />
  57. 57. Scale strategy:<br />Jumping toofar and back:<br />Smart strategies <br /> 92 – 27 = 92 – 30 + 3 = 65<br />
  58. 58. Strategiescalculation up to 20 (1)<br />8 + 7<br />9, 10, 11, 12, 13, 14, 15<br />8 + 2 = 10, 10 + 5 = 15<br />5 + 3 + 5 + 2 = 10 + 5 = 15<br />7 + 7 + 1 = 14 + 1 = 15<br />8 + 8 – 1 = 16 – 1 = 15<br />etc<br />
  59. 59. Strategies calculation up to 20 (2)<br />Maureen:<br />Thijs:<br />Luuk: “First, put three euro’s out of the six<br /> to the seven euro’s; <br />thatmakes ten euro’s, <br /> and threemakesthirteen euro’s.”<br />Hannah: “Six and six is twelve; and one<br /> makes thirteen euro’s.”<br />
  60. 60. Strategieslearning the times tables (1)<br />
  61. 61. Strategies learning the times tables (2)<br />
  62. 62. Strategies learning the times tables (3)<br />‘Stepping stones’ <br />
  63. 63. Reflection <br />Do you present various strategies in maths class?<br />Do children have preferences for strategies?<br />Is the children’s numberconcept sufficient to apply various strategies?<br />Do children choose their own strategy?<br />Do all children have to learn various strategies?<br />
  64. 64. What is Basic elements of Realistic Mathematics Education<br /><ul><li>Contexts
  65. 65. Strategies
  66. 66. Models and diagrams
  67. 67. Interaction
  68. 68. Use of characteristic materials</li></li></ul><li>Models and diagrams <br />
  69. 69. Difference between € 54.- and € 37.-?<br />
  70. 70.
  71. 71. What is Basic elements of Realistic Mathematics Education<br /><ul><li>Contexts
  72. 72. Strategies
  73. 73. Models and diagrams
  74. 74. Interaction
  75. 75. Use of characteristic materials</li></li></ul><li>Verticalinteraction<br />
  76. 76. Verticalinteraction<br />
  77. 77. Interaction 1: Vertical interaction<br /><ul><li>The teacher explains - the learners ask questions – the teacher tries to activate the learners by asking stimulating and open questions.</li></ul> teacher<br /> child child<br /><ul><li>This type of interaction you see during instruction</li></ul> lessons.<br /><ul><li>We call this vertical interaction.</li></li></ul><li>HorizontalInteraction<br />
  78. 78. HorizontalInteraction<br />
  79. 79. Interaction 2: Horizontal interaction <br /><ul><li>The teacher has given a problem to the learners. </li></ul> The learners try to solve this problem by cooperation, by thinking it through, together. <br /> teacher <br /><ul><li>The teacher observes. He collects impressions. </li></ul> Later, the teacher knows what to talk about during the evaluation. <br /><ul><li>We call this horizontal interaction.</li></ul>child<br />child<br />group<br />
  80. 80. What is Basic elements of Realistic Mathematics Education<br /><ul><li>Contexts
  81. 81. Strategies
  82. 82. Models and diagrams
  83. 83. Interaction
  84. 84. Use of characteristic materials</li></li></ul><li>Use of characteristic materials (1)<br />
  85. 85. Use of characteristic materials (2)<br /><ul><li>Flash cards
  86. 86. Numeric cards
  87. 87. Number snake
  88. 88. Numbers in love
  89. 89. Sun game
  90. 90. Etcetera....</li></li></ul><li>Use of characteristic materials (3)<br /><ul><li>All these materials are meant to build structures in the minds of children. </li></ul> The children have to obtain images of quantities.<br /><ul><li>If you don’t want them to keep counting, it is necessary that the children create their own world of numbers and quantities.
  91. 91. By using these materials, children will learn to control this tremendous world of numbers and, ultimately, understand the world they live in.</li></li></ul><li><ul><li>The Construction of Realistic Mathematics Education:</li></ul>Realistic Mathematics Education always starts with<br />reality: How to solve the problem in this context?<br />This is the concrete level.<br />Realistic Mathematics Education supports the<br />thinking of children in models.<br /> This is the model level.<br />Realistic Mathematics Education gives children the opportunity to abstract from their experience. <br /> At the same time they know how it works and they can explain the action. <br /> This is the formal level. <br />
  92. 92. The levels of concept based mathematics education<br />formal level<br />model level<br />contextlevel<br />
  93. 93. Advantages of Realistic Mathematics Education<br /><ul><li>Direct link with reality. Children recognize the items in their education as subjects they meet in daily life.
  94. 94. Because of the direct link to daily life, children get the possibility to use the matter they learned at school in their environment.
  95. 95. Children will be able to solve real problems based on what they learned at school. Not only with maths!
  96. 96. Children will not forget their lessons because of the knowledge that they can use their lessons in daily life.</li></li></ul><li>Summary <br />Teaching mathematics in ‘the realistic way’ is teaching<br />mathematics by use of<br /><ul><li>Contexts
  97. 97. Strategies
  98. 98. Models and diagrams
  99. 99. Interaction
  100. 100. Structured materials</li></li></ul><li> Let’s discuss this way of teaching maths <br /> We will have to ask ourselves:<br /><ul><li>‘Do we want to teach our learners in this way?’
  101. 101. ‘How can we teach our learners in the way of Realistic Mathematics Education?’
  102. 102. ‘How can we work on a merge of Learner Centered Education and Realistic Mathematics Education, by taking the best of two visions?’</li></li></ul><li><ul><li>Thank you for your attention</li></ul>‘Working together for excellence in mathematics<br /> education in Namibia and in The Netherlands’<br /><ul><li>Jaap Griffioen, j.griffioen@hro.nl
  103. 103. Jaap de Waard, j.de.waard@hro.nl</li>

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