De Commissie Wijnen heeft in januari van dit jaar een voorstel gedaan tot vernieuwing van de kern... more De Commissie Wijnen heeft in januari van dit jaar een voorstel gedaan tot vernieuwing van de kerndoelen. In de ogen van veel reken-wiskundedidactici vormen deze nieuwe kerndoelen een stap terug in de tijd. De doelen doen bijvoorbeeld geen recht aan de ontwikkelingen die zich de afgelopen jaren in het vakgebied rekenen-wiskunde hebben voorgedaan. De auteurs van dit artikel zijn als medewerkers van het Freudenthal Instituut nauw betrokken bij de meningsvorming rond onderwijsdoelen voor rekenen-wiskunde in de basisschool. Vanuit dit perspectief zetten zij in dit artikel een aantal argumenten tegen het rapport Wijnen op een rij. Tegelijkertijd wordt een begin gemaakt met een meer constructieve wijze van doordenken van de kerndoelen voor rekenen-wiskunde
Designing non-routine mathematical problems is a challenging task, even for high performing prosp... more Designing non-routine mathematical problems is a challenging task, even for high performing prospective teachers in elementary teacher education, especially when these non-routine problems concern knowledge at the mathematical horizon (HCK). In an experimental setting, these prospective teachers were challenged to design non-routine HCK problems. Interaction with peers, feedback from experts, analyzing HCK problems to find criteria, building a repertoire of prototypes, a cyclic design process, experts who are themselves struggling in designing problems were the most important and effective aspects of the learning environment to rise from this explorative study.
Research in mathematics education offers a considerable body of evidence that both high and low-a... more Research in mathematics education offers a considerable body of evidence that both high and low-achievers can benefit from learning mathematics in meaningful contexts. This case study offers an in-depth analysis of the learning process of a low-achieving student in the context of Realistic Mathematics Education (RME). The focus is on the use of productive and counter productive strategies in learning fractions. We found support for our idea that low-achievers do benefit from RME, but experience difficulties in the formalization process with regard to fractions. We seize upon the observed difficulties by discussing the implications of uniform standards in mathematics education.
Het NRO-project Meetkunst draaide om een verbinding tussen kunstzinnige orientatie en rekenen/wis... more Het NRO-project Meetkunst draaide om een verbinding tussen kunstzinnige orientatie en rekenen/wiskunde (onderdeel meetkunde) in de bovenbouw van het basisonderwijs. Evelyn Kroesbergen, Eveline Schoevers, Ronald Keijzer, Vincent Jonker en Monica Wijers onderzochten of deze combinatie de creativiteit bij probleemoplossing en de rekenvaardigheden van de leerlingen ten goede kwam. In dit artikel beschrijven ze de effecten van de ontwikkelde lessen en nascholing.
12th International Congress on Mathematical Education Program Name XX-YY-zz (pp. abcde-fghij) 8 J... more 12th International Congress on Mathematical Education Program Name XX-YY-zz (pp. abcde-fghij) 8 July – 15 July, 2012, COEX, Seoul, Korea
Onlangs verscheen een discussiestuk van het Tal-bovenbouwteam. In dit discussiestuk zijn verschil... more Onlangs verscheen een discussiestuk van het Tal-bovenbouwteam. In dit discussiestuk zijn verschillende standpunten verwoord over de inrichting van het onderwijs in breuken, verhoudingen, procenten en kommagetallen. Er wordt gepleit voor een verschuiving van ‘kunnen’ uitvoeren van procedures naar ‘begrijpen’ van breuken, verhoudingen, procenten en kommagetallen. In dit artikel wordt verslag gedaan van de uitkomsten van een peiling over deze standpunten. We doen verder verslag van een discussiebijeenkomst over de ideeen van het Tal-bovenbouwteam. Veel respondenten en deelnemers aan de discussie steunen de omslag van ‘kunnen’ naar ‘begrijpen’.
Teachers who engage primary school students in informal statistical inference (ISI) must themselv... more Teachers who engage primary school students in informal statistical inference (ISI) must themselves have good content knowledge of ISI (ISI-CK). However, little is known about how college education for pre-service teachers can contribute to the development of their ISI-CK. To address this shortcoming, we used a case study to investigate ISI-CK development in a class of 21 pre-service primary school teachers who participated in a short intervention (180 min). Based on qualitative and quantitative analyses of the pretest, posttest and intervention data, the results suggest that most participants acknowledged it is possible to make uncertain inferences. An assignment to search the media for inferential claims seemed to create awareness regarding inference and the need to distinguish between a sample and a population. A simulation involving random sampling and varied sample size probably increased the participants' knowledge of sampling variability and random sampling. No development was seen in the participants' knowledge about sufficient sample sizes. The statistical investigation conducted by the participants during a model lesson may have strengthened their awareness of ISI, but it also revealed that many participants continued to favour distributed sampling over random sampling. Further research on belief formation with regard to data as evidence, sampling methods and the expression of uncertainty in the context of ISI is needed.
The design brief for the contributors Each of the teams of contributors was asked to design a tas... more The design brief for the contributors Each of the teams of contributors was asked to design a task which focussed on proportional reasoning. The task had be suitable for pupils aged about 11-12 years, and it also had to be a 'stand alone' task, which could be tackled within one lesson. This condition was a significant constraint for some of the contributors, who would normally design tasks as part of a sequence. Contributors were asked to prepare their task in a form that could be presented to pupils, and were also asked to provide teachers' notes. Each of the tasks has been trialled with pairs of pupils and the papers by each of the contributing teams which follow this introduction draw on this data to illustrate the discussion of the principles which underpinned their task designs. Dirk, Wim and Lieven's task This task focuses on similarities and differences in a set of word problems, some of which require proportional reasoning, while others have a similar format, but are not, in fact, proportional problems. Yesterday, Mrs. Jones made some word problems to use in the math lessons. But they got all mixed up! Can you help Mrs. Jones to put some order in the word problems? Look at the problems very carefully and try to make groups of problems that belong together. RF01 PME29-2005 1-95 I In the hallway of our school, 2 tables stand in a line. 10 chairs fit around them. Now the teacher puts 6 tables in a line. How many chairs fit around these tables? J In the shop, 4 packs of pencils cost 8 euro. The teacher wants to buy a pack for every pupil. He needs 24 packs. How much must he pay? Now answer the following questions: • Write here the different groups of problems. (Use the letters on the sheets) • Why did you make the groups in that way? • Can you think of a different way to put the problems in groups? Explain that as well. Koeno, Frans and Ronald's task This task is based around the story of Monica and Kim making a cycle trip from Corby to Cambridge. Various resources such as a map of the route, photographs and background information (the reason for the trip, the weather conditions) are provided. After cycling for 1 hour 30 minutes, the girls reach a village called Catworth where there is a signpost showing 18 miles from Corby and 30 miles to Cambridge. "Okay", Monica says, "this is going well." 1. Could you tell why she might say this? 2. How much time has it taken them to get to Catworth? And what is the distance they have covered? So what can you say about the speed of Monica and Kim? You can use the table to judge their speed. 3. In the table, the speeds of various kinds of cyclist are given. However, if you want to compare the speeds of cyclist who are not riding the same road on the same day, conditions might be different. Could you mention the things that have to be taken into account, if we were to measure the speed of a cyclist. 4. After a short stop, Monica and Kim are moving on. They get on the road from Catworth to Cambridge, a distance of 30 miles. At about what time do you think they will arrive in Cambridge? 5. Of course, you cannot be absolutely sure about how long it will take them. Could you mention some reasons why you cannot be sure? Still, to make a sensible guess, it might be helpful to know how much time she would need if she were to keep up the same speed. 6. How much time would the ride to Cambridge take if they were to keep up the same average speed as before? Cycling at a slow speed: 8 miles per hour Cycling at a normal speed: 12 miles per hour Cycling at a fast speed: 18 miles per hour
This paper reports on a longitudinal study of teaching and learning in the subject of fractions i... more This paper reports on a longitudinal study of teaching and learning in the subject of fractions in two matched groups of ten 9-10-year-old-students. In the experimental group fractions are introduced using the bar and the number line as (mental) models, in the control group the subject is introduced by fair sharing and the circle model. In the experimental group students are invited to discuss, in the control group students work individually.. The groups are compared on several occasions during one year. After one year, the experimental students show more proficiency in fractions than those in the control group.
National standards for teaching mathematics in primary schools in the Netherlands leave little ro... more National standards for teaching mathematics in primary schools in the Netherlands leave little room for formal fractions. However,a newly developed programme in fractions aims at learning formal fractions. The starting point in the development of this curriculum is the students’ acquisition of `numeracy infractions’. In this case study we describe the growth in reasoning ability with fractions of one student
De Commissie Wijnen heeft in januari van dit jaar een voorstel gedaan tot vernieuwing van de kern... more De Commissie Wijnen heeft in januari van dit jaar een voorstel gedaan tot vernieuwing van de kerndoelen. In de ogen van veel reken-wiskundedidactici vormen deze nieuwe kerndoelen een stap terug in de tijd. De doelen doen bijvoorbeeld geen recht aan de ontwikkelingen die zich de afgelopen jaren in het vakgebied rekenen-wiskunde hebben voorgedaan. De auteurs van dit artikel zijn als medewerkers van het Freudenthal Instituut nauw betrokken bij de meningsvorming rond onderwijsdoelen voor rekenen-wiskunde in de basisschool. Vanuit dit perspectief zetten zij in dit artikel een aantal argumenten tegen het rapport Wijnen op een rij. Tegelijkertijd wordt een begin gemaakt met een meer constructieve wijze van doordenken van de kerndoelen voor rekenen-wiskunde
Designing non-routine mathematical problems is a challenging task, even for high performing prosp... more Designing non-routine mathematical problems is a challenging task, even for high performing prospective teachers in elementary teacher education, especially when these non-routine problems concern knowledge at the mathematical horizon (HCK). In an experimental setting, these prospective teachers were challenged to design non-routine HCK problems. Interaction with peers, feedback from experts, analyzing HCK problems to find criteria, building a repertoire of prototypes, a cyclic design process, experts who are themselves struggling in designing problems were the most important and effective aspects of the learning environment to rise from this explorative study.
Research in mathematics education offers a considerable body of evidence that both high and low-a... more Research in mathematics education offers a considerable body of evidence that both high and low-achievers can benefit from learning mathematics in meaningful contexts. This case study offers an in-depth analysis of the learning process of a low-achieving student in the context of Realistic Mathematics Education (RME). The focus is on the use of productive and counter productive strategies in learning fractions. We found support for our idea that low-achievers do benefit from RME, but experience difficulties in the formalization process with regard to fractions. We seize upon the observed difficulties by discussing the implications of uniform standards in mathematics education.
Het NRO-project Meetkunst draaide om een verbinding tussen kunstzinnige orientatie en rekenen/wis... more Het NRO-project Meetkunst draaide om een verbinding tussen kunstzinnige orientatie en rekenen/wiskunde (onderdeel meetkunde) in de bovenbouw van het basisonderwijs. Evelyn Kroesbergen, Eveline Schoevers, Ronald Keijzer, Vincent Jonker en Monica Wijers onderzochten of deze combinatie de creativiteit bij probleemoplossing en de rekenvaardigheden van de leerlingen ten goede kwam. In dit artikel beschrijven ze de effecten van de ontwikkelde lessen en nascholing.
12th International Congress on Mathematical Education Program Name XX-YY-zz (pp. abcde-fghij) 8 J... more 12th International Congress on Mathematical Education Program Name XX-YY-zz (pp. abcde-fghij) 8 July – 15 July, 2012, COEX, Seoul, Korea
Onlangs verscheen een discussiestuk van het Tal-bovenbouwteam. In dit discussiestuk zijn verschil... more Onlangs verscheen een discussiestuk van het Tal-bovenbouwteam. In dit discussiestuk zijn verschillende standpunten verwoord over de inrichting van het onderwijs in breuken, verhoudingen, procenten en kommagetallen. Er wordt gepleit voor een verschuiving van ‘kunnen’ uitvoeren van procedures naar ‘begrijpen’ van breuken, verhoudingen, procenten en kommagetallen. In dit artikel wordt verslag gedaan van de uitkomsten van een peiling over deze standpunten. We doen verder verslag van een discussiebijeenkomst over de ideeen van het Tal-bovenbouwteam. Veel respondenten en deelnemers aan de discussie steunen de omslag van ‘kunnen’ naar ‘begrijpen’.
Teachers who engage primary school students in informal statistical inference (ISI) must themselv... more Teachers who engage primary school students in informal statistical inference (ISI) must themselves have good content knowledge of ISI (ISI-CK). However, little is known about how college education for pre-service teachers can contribute to the development of their ISI-CK. To address this shortcoming, we used a case study to investigate ISI-CK development in a class of 21 pre-service primary school teachers who participated in a short intervention (180 min). Based on qualitative and quantitative analyses of the pretest, posttest and intervention data, the results suggest that most participants acknowledged it is possible to make uncertain inferences. An assignment to search the media for inferential claims seemed to create awareness regarding inference and the need to distinguish between a sample and a population. A simulation involving random sampling and varied sample size probably increased the participants' knowledge of sampling variability and random sampling. No development was seen in the participants' knowledge about sufficient sample sizes. The statistical investigation conducted by the participants during a model lesson may have strengthened their awareness of ISI, but it also revealed that many participants continued to favour distributed sampling over random sampling. Further research on belief formation with regard to data as evidence, sampling methods and the expression of uncertainty in the context of ISI is needed.
The design brief for the contributors Each of the teams of contributors was asked to design a tas... more The design brief for the contributors Each of the teams of contributors was asked to design a task which focussed on proportional reasoning. The task had be suitable for pupils aged about 11-12 years, and it also had to be a 'stand alone' task, which could be tackled within one lesson. This condition was a significant constraint for some of the contributors, who would normally design tasks as part of a sequence. Contributors were asked to prepare their task in a form that could be presented to pupils, and were also asked to provide teachers' notes. Each of the tasks has been trialled with pairs of pupils and the papers by each of the contributing teams which follow this introduction draw on this data to illustrate the discussion of the principles which underpinned their task designs. Dirk, Wim and Lieven's task This task focuses on similarities and differences in a set of word problems, some of which require proportional reasoning, while others have a similar format, but are not, in fact, proportional problems. Yesterday, Mrs. Jones made some word problems to use in the math lessons. But they got all mixed up! Can you help Mrs. Jones to put some order in the word problems? Look at the problems very carefully and try to make groups of problems that belong together. RF01 PME29-2005 1-95 I In the hallway of our school, 2 tables stand in a line. 10 chairs fit around them. Now the teacher puts 6 tables in a line. How many chairs fit around these tables? J In the shop, 4 packs of pencils cost 8 euro. The teacher wants to buy a pack for every pupil. He needs 24 packs. How much must he pay? Now answer the following questions: • Write here the different groups of problems. (Use the letters on the sheets) • Why did you make the groups in that way? • Can you think of a different way to put the problems in groups? Explain that as well. Koeno, Frans and Ronald's task This task is based around the story of Monica and Kim making a cycle trip from Corby to Cambridge. Various resources such as a map of the route, photographs and background information (the reason for the trip, the weather conditions) are provided. After cycling for 1 hour 30 minutes, the girls reach a village called Catworth where there is a signpost showing 18 miles from Corby and 30 miles to Cambridge. "Okay", Monica says, "this is going well." 1. Could you tell why she might say this? 2. How much time has it taken them to get to Catworth? And what is the distance they have covered? So what can you say about the speed of Monica and Kim? You can use the table to judge their speed. 3. In the table, the speeds of various kinds of cyclist are given. However, if you want to compare the speeds of cyclist who are not riding the same road on the same day, conditions might be different. Could you mention the things that have to be taken into account, if we were to measure the speed of a cyclist. 4. After a short stop, Monica and Kim are moving on. They get on the road from Catworth to Cambridge, a distance of 30 miles. At about what time do you think they will arrive in Cambridge? 5. Of course, you cannot be absolutely sure about how long it will take them. Could you mention some reasons why you cannot be sure? Still, to make a sensible guess, it might be helpful to know how much time she would need if she were to keep up the same speed. 6. How much time would the ride to Cambridge take if they were to keep up the same average speed as before? Cycling at a slow speed: 8 miles per hour Cycling at a normal speed: 12 miles per hour Cycling at a fast speed: 18 miles per hour
This paper reports on a longitudinal study of teaching and learning in the subject of fractions i... more This paper reports on a longitudinal study of teaching and learning in the subject of fractions in two matched groups of ten 9-10-year-old-students. In the experimental group fractions are introduced using the bar and the number line as (mental) models, in the control group the subject is introduced by fair sharing and the circle model. In the experimental group students are invited to discuss, in the control group students work individually.. The groups are compared on several occasions during one year. After one year, the experimental students show more proficiency in fractions than those in the control group.
National standards for teaching mathematics in primary schools in the Netherlands leave little ro... more National standards for teaching mathematics in primary schools in the Netherlands leave little room for formal fractions. However,a newly developed programme in fractions aims at learning formal fractions. The starting point in the development of this curriculum is the students’ acquisition of `numeracy infractions’. In this case study we describe the growth in reasoning ability with fractions of one student
Uploads
Papers by Ronald Keijzer