Whilst researching the relationship between learning and acquisition, a clear apprehension of Realistic Maths Education ( RME ) was difficult to accomplish. The features of RME are giving distinguished name to an bing technique that uses existent life state of affairss to mathematical job. Although it is similar to taking a real-life state of affairs and giving the students the mathematical job of it. This giving short term satisfaction, that the instructor is associating the maths aspect in real-life state of affairs and replying the very inquiry ( of mine as good ) as to how maths will assist me in ‘real-life ‘ .

My scheme into researching this topic leads me to into researching at two chief diaries these being: ‘Mathematics or Mathematizing ‘ this diary was really good debut to RME, nevertheless the diary deficiency the in-depth analysis of what RME was and how it was applied in school. Then, my researches lead me to ‘ Audrey ‘s Acquisition Of Fractions: A Case Study Into The Learning Of Formal Mathematics ‘ . This enables my apprehension of how RME is applied to lessons on fractions in much item. This gave me a practical penetration into how RME is used on a capable affair of fractions. Looking at these diaries made me reflect on the current techniques available on how fractions are taught in school.

The development of what is now known as RME started about 30 old ages ago. The foundations for it were laid by Freudenthal and his co-workers at the former IOWO, which is the oldest predecessor of the Freudenthal Institute. The existent urge for the reform motion was the origin, in 1968, of the Wiskobas undertaking, initiated by Wijdeveld and Goffree. The present signifier of RME is largely determined by Freudenthal ‘s ( 1977 ) position about mathematics.

The instruction and acquisition of mathematics has been changed as a consequence of the greater accent put on mechanical mathematics accordingly Realistic Maths Education ( RME ) has been developed and still is by the Dutch. The development of RME is 30 old ages old now, when it comes to practical schoolroom pattern much work still has to be done in ( Van den Heuvel-Panhuizen, 1998 ) .

Freudenthal believed that the human activity is ne’er considered fixed. Looking at Gardners ( 1996 ) ( et al Barrington ( 2004 ) ) thought ‘s of multiple intelligence which allowed me to understand that there are 8 intelligences which can be divided into three chief groups:

visual/spatial,

verbal/linguistic

musical/rhythmic,

logical/mathematical,

bodily/kinesthetic,

interpersonal,

intrapersonal

And realistic.

He observed that everybody had a mixture of these eight intelligence and some facet will be stronger than other, over clip and life experiences the accomplishments interchange and vary. This was something that Freudenthal really much believed in and establishing thought of mathematics as a human activity, that it can ne’er be considered a fixed and finished theory of mathematics instruction.

Concentrating on the development of students cognition and understand of mathematic was portion of the foundation of RME and something that was ever taken into consideration when looking at a subject of Mathematicss to better it ( Van den Heuvel-Panhuizen, 1998 ) . Harmonizing to Freudenthal, ‘mathematics must be connected to world, stay near to kids and be relevant to society, in order to be of human value ‘ . Alternatively of seeing mathematics as capable affair that has to be transmitted, Freudenthal emphasised that the thought of mathematics as a human activity. Education should give pupils the ‘guided ‘ chance to ‘re-invent ‘ mathematics by experimenting for themselves.

In RME, the mathematics is introduced in the context of a good thought out job. In the procedure of seeking to work out the job the kid develops mathematics. The instructor uses the method of guided reinvention, by which pupils are given the opportunity to seek and develop their ain informal methods for making mathematics. Students are encouraged to discourse with fellow pupils schemes in the schoolroom, larning from each other ‘s methods. RME is based on what we know about kid development and the development of numeracy, and that it is this organic structure of research that is driving the math instruction reform ( Eade and Dickinson )

The usage of context jobs is really important in RME compared to traditional, mechanistic attack to mathematics instruction, which contains largely basal amounts. If context jobs are used in the mechanistic attack, they are largely used to repeat learning procedure. The context jobs function merely as a field of application. By work outing context jobs the pupils can use what was learned earlier in the bare state of affairs.

In RME this is different ; Context jobs map besides as a beginning for the acquisition procedure. In other words, in RME, contexts jobs and real-life state of affairss are used both to represent and to use mathematical constructs. While working on context jobs the pupils can develop mathematical tools and apprehension.

In order to carry through the bridging map between the informal and the formal degree, theoretical accounts have to switch from a theoretical account of ” to a theoretical account for. ” It was a item that Leen Streefland developed in 1985 this important mechanism in the growing of understanding. Enabling us to derive greater apprehension of how students can do that important transportation of mathematical understanding from their informal techniques to the formal mathematics ( Van den Heuvel-Panhuizen, 1998 ) .

Fractions are my topic focal point as a feel that erstwhile fractions are n’t given the focal point that is should be given. ‘Fractions are without uncertainty the most debatable country in simple mathematics instruction ‘ ( Steefland, 2001 pg 6 ) Fractions cause trouble to most people because they involve dealingss between measures intending that A? can take many signifiers and being able to understand and see that relationship can be really hard for students ( Mathematics or mathematizing ) . Another mistake that students make with fractions is to believe that, for illustration, 1/3 of a bar is smaller than 1/5 because 3 is less than 5. Yet most kids readily recognise that a bar shared among three kids gives bigger parts than the same bar shared among five kids.

When Steefland ( 1991 ) negotiations about how the chapters of fractions in a text edition have evolved he articulates that the text edition became ‘transposition of application jobs into dressed up arithmetic amounts ‘ , he goes on to says that the orientation of the whole text edition is set out in a similar mode ( pg 9-10 ) . This came from both Bouman and Van Zelm, in add-on Diels and Nauta, where they aimed to look at alternate ways of cut downing the ‘thinking amounts ‘ to recommend mental arithmetic more.

Meaning that alternatively of the text edition being another signifier of counsel for students it was being used as a book with orientation of regulations and bare amounts ( without context ) . Through reading the history of fraction in text edition it seemed that before the thoughts of Simon Stevin in the 15th century ( that we could simplify the day-to-day pattern of fractions by utilizing denary fractions alternatively ( et Al Steefland 1991 ) ) we were so looking at fraction in more of an existing manner instead than merely the ‘bare amounts ‘ . So this construct we have in text editions of regulations and illustrations is comparatively new it was n’t until the early 19th century that this came approximately in text edition. Therefore the maths in the realistic sense is what we have all been utilizing for centuries compared to the comparatively new thought of simplification to regulations and amounts.

Detecting at schools and looking at their text editions today I can see that they have been over taken by the original thoughts of Simon Stevin. Look at SMP- Interact C1 here what detecting the pages on fractions where the aim are to ‘simply fractions ‘ , ‘put fraction in order of size ‘ and ‘add and subtract fractions ‘ .

Key constructs and thoughts of fractions are in colored boxed pulling the attending to read them instantly like these are of import in cognizing as a effect understanding these thought will state you how to work out certain inquiries on humanitarian fractions, and adding and deducting.

From looking at the chapter, I can see that there is an anticipation of the Year 7 ‘s have knowledge and understand whole number numeration graduated tables,

Clear we can see here in this text edition that Simon Stevin thoughts have truly been made a priory in the working of a mathematic text edition. Keijzer & A ; Terwel ( 2001 ) province historically we have followed these pointless regulations of mathematical methods taking students holding problem associating their apprehension of maths to formal mathematical thoughts. Particularly when it comes to larning fraction the students get confused as to how to associate their thoughts of what fractions are to how to reply inquiries on fractions. This is where the RME can assist as it giving the connexion between the students mathematical thought to how to reply inquiries or amounts on fractions ( officially )

When looking at RME in text edition signifier it follows the thought of Freudenthal ( 1973 ) ( et al Keijzer & A ; Terwel, 2001-pg 54 ) that doing that mathematical journey taking to formal mathematical thoughts comes from ‘a series of good chosen illustrations ‘ . The significance if this being that the illustration and ‘sums ‘ that we give to a student demand to be good thought out to steer the students to an thought alternatively of giving them the formal opinion foremost.

Over a class of up to 2 decennaries Streefland aid develop a new course of study in the Netherlands on fractions integrating the patterns of RME ( et al Keijzer & A ; Terwel, 2001-pg 55 ) . He thought of the chief subject of fraction activities ought to be the ‘fair-sharing ‘ and how sharing between things will give students an apprehension of fractions. He looked at fraction linguistic communication at first to give the students a assurance in utilizing fraction linguistic communication by believing of fair-sharing being pizza ‘s shared between some people. This so lead on to comparative fractions, by giving another activity which leads the students understanding that sharing three pizza ‘s with four people is the same as sharing 6 pizza ‘s with 8 people. These are activities which I have seen in the ‘Mathematics in context- Some of the parts ‘ text edition. In 1991, The University of Wisconsin, in coaction with the Freudenthal Institute, started to develop a middle-school course of study based on RME. This course of study is known as ‘Maths in Context ‘ and has now been adopted by legion schools in the US and presently being trialled in the UK.

Talk about the chapter on fraction how the patterned advance is made from informal mathematical thoughts and techniques to the formal mathematics.

Besides during an probe on division I did inquire the inquiry of sharing 3 sandwiches between 4 people, to a bright 11 twelvemonth old who was merely about to take her twelvemonth 6 SAT ‘s. here what she did:

In the 3rd inquiry I asked the person to portion 3 sandwiches with 4 people, here I presented an image of three sandwiches for them to utilize if they wanted to.

She did battle in this inquiry but did finally make an reply. First she saw that she need to divide each sandwich into 4 parts and so had 24 pieces which she could so shared between the four people. She found it in fraction signifier ‘A? + A? + A? = A? ‘ but still she was non satisfied with her reply. She so broke the sandwiches into 1/8 and so when into another page to do images of a bar split into 8 equal parts which so helped her to understand that each individual would acquire 6/8 of the sandwich. ( See appendix A )

Now looking at her response I can clearly see that how Streefland looks into just sharing as a manner of seeking to acquire students to understand tantamount fractions. Here the respondent was n’t satisfied in the reply merely being ‘A? ‘ , but to give herself a better apprehension of what A? is, she used her cognition of tantamount fraction and found that by stating 6/8 she was able to confidently understand what the reply and inquiry meant on how to portion 3 sandwiches with 4 people.

When looking through Keijzer & A ; Terwel ( 2001 ) , ‘Audrey ‘s Acquisition Of Fractions: A Case Study Into The Learning Of Formal Mathematics ‘ I found it really enlightening to the techniques used to enable the students apprehension of fractions. They talk about how alternatively of utilizing the diagrams of bars ( as circles ) to stand for fractions, they found it is more enlightening to utilize rectangular bars. This gives the students a improved visual of being able to compare fraction sizes and to be able to ‘reflect ‘ on their work. The rectangular bar is really similar to the figure this is argued by Connell and Peck ( 1993 ) ( et al Keijzer & A ; Terwel, 2001 ) . I can see visualize how the rectangular bar is the informal mathematic which will so take the students onto the formal maths which will be where the students will be able to utilize figure lines as an instrument to demo fractions.

Another thought that I truly liked, both Streefland ( 1991 ) and Keijzer & A ; Terwel ( 2001 ) realised that in order for students to understand fractions they need to hold the apprehension of ‘number sense ‘ with fractions. ‘aˆ¦teaching strategyaˆ¦aˆ¦number sense is developed ( I ) a linguistic communication of fractions ( two ) developing the figure line for fractions, ( three ) comparison fractions, ( four ) larning formal fractions. ‘ In this quotation mark I am looking at the comparison of fractions, where the tantamount fractions are the key to formal logical thinking with fractions. Meaning that if the students do non understand the construct that the ratio fraction is the same for something like A? , 2/4, 3/6aˆ¦etc so it would be difficult for them to travel onto the formal mathematics. Keijzer & A ; Terwel ( 2001 ) used the illustration of perpendicular figure line houses as fractions, where the fractions lived in each floor and the lift connected the different floors. If the edifice was 3 floors high so it had 3 Michigans ( 1/3, 2/3 and 3/3 ) if it was a 4 floor edifice so it would hold four floors ( 1/4, 2/4, 3/4 and 4/4 ) . The student would make full in every bit much information as possible on the fraction lift. Then in a treatment they will be able to do opinion on how it was easier to state A? instead than 2/4 or discourse other point excessively. This exercising will do it clear different fractions can suit in to the same place on the figure line.

This is an illustration of a different manner of doing students understand where the places are on a figure line. It was taken for the new version of the Maths in Context – Fraction Time, here the students would be provided with a tabular array and asked to make full in the fractions in the grid. I would so travel into to inquire students to happen forms and color codification any similar fraction that they can see on the grid. Once coloring material coded the students will be able to see that some fractions do hold the same place on the figure line. Although the figure line is now really clear in this exercising it will give the students a opportunity to descry form. Descrying forms is a informal scheme that pupils do like to make.

Looking at both the diaries of Streefland ( 1991 ) and Keijzer & A ; Terwel ( 2001 ) ,

My scheme into researching this topic leads me to into researching at two chief diaries these being: ‘Mathematics or Mathematizing ‘ this diary was really good debut to RME, nevertheless the diary deficiency the in-depth analysis of what RME was and how it was applied in school. Then, my researches lead me to ‘ Audrey ‘s Acquisition Of Fractions: A Case Study Into The Learning Of Formal Mathematics ‘ . This enables my apprehension of how RME is applied to lessons on fractions in much item. This gave me a practical penetration into how RME is used on a capable affair of fractions. Looking at these diaries made me reflect on the current techniques available on how fractions are taught in school.

The development of what is now known as RME started about 30 old ages ago. The foundations for it were laid by Freudenthal and his co-workers at the former IOWO, which is the oldest predecessor of the Freudenthal Institute. The existent urge for the reform motion was the origin, in 1968, of the Wiskobas undertaking, initiated by Wijdeveld and Goffree. The present signifier of RME is largely determined by Freudenthal ‘s ( 1977 ) position about mathematics.

The instruction and acquisition of mathematics has been changed as a consequence of the greater accent put on mechanical mathematics accordingly Realistic Maths Education ( RME ) has been developed and still is by the Dutch. The development of RME is 30 old ages old now, when it comes to practical schoolroom pattern much work still has to be done in ( Van den Heuvel-Panhuizen, 1998 ) .

Freudenthal believed that the human activity is ne’er considered fixed. Looking at Gardners ( 1996 ) ( et al Barrington ( 2004 ) ) thought ‘s of multiple intelligence which allowed me to understand that there are 8 intelligences which can be divided into three chief groups:

visual/spatial,

verbal/linguistic

musical/rhythmic,

logical/mathematical,

bodily/kinesthetic,

interpersonal,

intrapersonal

And realistic.

He observed that everybody had a mixture of these eight intelligence and some facet will be stronger than other, over clip and life experiences the accomplishments interchange and vary. This was something that Freudenthal really much believed in and establishing thought of mathematics as a human activity, that it can ne’er be considered a fixed and finished theory of mathematics instruction.

Concentrating on the development of students cognition and understand of mathematic was portion of the foundation of RME and something that was ever taken into consideration when looking at a subject of Mathematicss to better it ( Van den Heuvel-Panhuizen, 1998 ) . Harmonizing to Freudenthal, ‘mathematics must be connected to world, stay near to kids and be relevant to society, in order to be of human value ‘ . Alternatively of seeing mathematics as capable affair that has to be transmitted, Freudenthal emphasised that the thought of mathematics as a human activity. Education should give pupils the ‘guided ‘ chance to ‘re-invent ‘ mathematics by experimenting for themselves.

In RME, the mathematics is introduced in the context of a good thought out job. In the procedure of seeking to work out the job the kid develops mathematics. The instructor uses the method of guided reinvention, by which pupils are given the opportunity to seek and develop their ain informal methods for making mathematics. Students are encouraged to discourse with fellow pupils schemes in the schoolroom, larning from each other ‘s methods. RME is based on what we know about kid development and the development of numeracy, and that it is this organic structure of research that is driving the math instruction reform ( Eade and Dickinson )

The usage of context jobs is really important in RME compared to traditional, mechanistic attack to mathematics instruction, which contains largely basal amounts. If context jobs are used in the mechanistic attack, they are largely used to repeat learning procedure. The context jobs function merely as a field of application. By work outing context jobs the pupils can use what was learned earlier in the bare state of affairs.

In RME this is different ; Context jobs map besides as a beginning for the acquisition procedure. In other words, in RME, contexts jobs and real-life state of affairss are used both to represent and to use mathematical constructs. While working on context jobs the pupils can develop mathematical tools and apprehension.

In order to carry through the bridging map between the informal and the formal degree, theoretical accounts have to switch from a theoretical account of ” to a theoretical account for. ” It was a item that Leen Streefland developed in 1985 this important mechanism in the growing of understanding. Enabling us to derive greater apprehension of how students can do that important transportation of mathematical understanding from their informal techniques to the formal mathematics ( Van den Heuvel-Panhuizen, 1998 ) .

Fractions are my topic focal point as a feel that erstwhile fractions are n’t given the focal point that is should be given. ‘Fractions are without uncertainty the most debatable country in simple mathematics instruction ‘ ( Steefland, 2001 pg 6 ) Fractions cause trouble to most people because they involve dealingss between measures intending that A? can take many signifiers and being able to understand and see that relationship can be really hard for students ( Mathematics or mathematizing ) . Another mistake that students make with fractions is to believe that, for illustration, 1/3 of a bar is smaller than 1/5 because 3 is less than 5. Yet most kids readily recognise that a bar shared among three kids gives bigger parts than the same bar shared among five kids.

When Steefland ( 1991 ) negotiations about how the chapters of fractions in a text edition have evolved he articulates that the text edition became ‘transposition of application jobs into dressed up arithmetic amounts ‘ , he goes on to says that the orientation of the whole text edition is set out in a similar mode ( pg 9-10 ) . This came from both Bouman and Van Zelm, in add-on Diels and Nauta, where they aimed to look at alternate ways of cut downing the ‘thinking amounts ‘ to recommend mental arithmetic more.

Meaning that alternatively of the text edition being another signifier of counsel for students it was being used as a book with orientation of regulations and bare amounts ( without context ) . Through reading the history of fraction in text edition it seemed that before the thoughts of Simon Stevin in the 15th century ( that we could simplify the day-to-day pattern of fractions by utilizing denary fractions alternatively ( et Al Steefland 1991 ) ) we were so looking at fraction in more of an existing manner instead than merely the ‘bare amounts ‘ . So this construct we have in text editions of regulations and illustrations is comparatively new it was n’t until the early 19th century that this came approximately in text edition. Therefore the maths in the realistic sense is what we have all been utilizing for centuries compared to the comparatively new thought of simplification to regulations and amounts.

Detecting at schools and looking at their text editions today I can see that they have been over taken by the original thoughts of Simon Stevin. Look at SMP- Interact C1 here what detecting the pages on fractions where the aim are to ‘simply fractions ‘ , ‘put fraction in order of size ‘ and ‘add and subtract fractions ‘ .

Key constructs and thoughts of fractions are in colored boxed pulling the attending to read them instantly like these are of import in cognizing as a effect understanding these thought will state you how to work out certain inquiries on humanitarian fractions, and adding and deducting.

From looking at the chapter, I can see that there is an anticipation of the Year 7 ‘s have knowledge and understand whole number numeration graduated tables,

Clear we can see here in this text edition that Simon Stevin thoughts have truly been made a priory in the working of a mathematic text edition. Keijzer & A ; Terwel ( 2001 ) province historically we have followed these pointless regulations of mathematical methods taking students holding problem associating their apprehension of maths to formal mathematical thoughts. Particularly when it comes to larning fraction the students get confused as to how to associate their thoughts of what fractions are to how to reply inquiries on fractions. This is where the RME can assist as it giving the connexion between the students mathematical thought to how to reply inquiries or amounts on fractions ( officially )

When looking at RME in text edition signifier it follows the thought of Freudenthal ( 1973 ) ( et al Keijzer & A ; Terwel, 2001-pg 54 ) that doing that mathematical journey taking to formal mathematical thoughts comes from ‘a series of good chosen illustrations ‘ . The significance if this being that the illustration and ‘sums ‘ that we give to a student demand to be good thought out to steer the students to an thought alternatively of giving them the formal opinion foremost.

Over a class of up to 2 decennaries Streefland aid develop a new course of study in the Netherlands on fractions integrating the patterns of RME ( et al Keijzer & A ; Terwel, 2001-pg 55 ) . He thought of the chief subject of fraction activities ought to be the ‘fair-sharing ‘ and how sharing between things will give students an apprehension of fractions. He looked at fraction linguistic communication at first to give the students a assurance in utilizing fraction linguistic communication by believing of fair-sharing being pizza ‘s shared between some people. This so lead on to comparative fractions, by giving another activity which leads the students understanding that sharing three pizza ‘s with four people is the same as sharing 6 pizza ‘s with 8 people. These are activities which I have seen in the ‘Mathematics in context- Some of the parts ‘ text edition. In 1991, The University of Wisconsin, in coaction with the Freudenthal Institute, started to develop a middle-school course of study based on RME. This course of study is known as ‘Maths in Context ‘ and has now been adopted by legion schools in the US and presently being trialled in the UK.

Talk about the chapter on fraction how the patterned advance is made from informal mathematical thoughts and techniques to the formal mathematics.

Besides during an probe on division I did inquire the inquiry of sharing 3 sandwiches between 4 people, to a bright 11 twelvemonth old who was merely about to take her twelvemonth 6 SAT ‘s. here what she did:

In the 3rd inquiry I asked the person to portion 3 sandwiches with 4 people, here I presented an image of three sandwiches for them to utilize if they wanted to.

She did battle in this inquiry but did finally make an reply. First she saw that she need to divide each sandwich into 4 parts and so had 24 pieces which she could so shared between the four people. She found it in fraction signifier ‘A? + A? + A? = A? ‘ but still she was non satisfied with her reply. She so broke the sandwiches into 1/8 and so when into another page to do images of a bar split into 8 equal parts which so helped her to understand that each individual would acquire 6/8 of the sandwich. ( See appendix A )

Now looking at her response I can clearly see that how Streefland looks into just sharing as a manner of seeking to acquire students to understand tantamount fractions. Here the respondent was n’t satisfied in the reply merely being ‘A? ‘ , but to give herself a better apprehension of what A? is, she used her cognition of tantamount fraction and found that by stating 6/8 she was able to confidently understand what the reply and inquiry meant on how to portion 3 sandwiches with 4 people.

When looking through Keijzer & A ; Terwel ( 2001 ) , ‘Audrey ‘s Acquisition Of Fractions: A Case Study Into The Learning Of Formal Mathematics ‘ I found it really enlightening to the techniques used to enable the students apprehension of fractions. They talk about how alternatively of utilizing the diagrams of bars ( as circles ) to stand for fractions, they found it is more enlightening to utilize rectangular bars. This gives the students a improved visual of being able to compare fraction sizes and to be able to ‘reflect ‘ on their work. The rectangular bar is really similar to the figure this is argued by Connell and Peck ( 1993 ) ( et al Keijzer & A ; Terwel, 2001 ) . I can see visualize how the rectangular bar is the informal mathematic which will so take the students onto the formal maths which will be where the students will be able to utilize figure lines as an instrument to demo fractions.

Another thought that I truly liked, both Streefland ( 1991 ) and Keijzer & A ; Terwel ( 2001 ) realised that in order for students to understand fractions they need to hold the apprehension of ‘number sense ‘ with fractions. ‘aˆ¦teaching strategyaˆ¦aˆ¦number sense is developed ( I ) a linguistic communication of fractions ( two ) developing the figure line for fractions, ( three ) comparison fractions, ( four ) larning formal fractions. ‘ In this quotation mark I am looking at the comparison of fractions, where the tantamount fractions are the key to formal logical thinking with fractions. Meaning that if the students do non understand the construct that the ratio fraction is the same for something like A? , 2/4, 3/6aˆ¦etc so it would be difficult for them to travel onto the formal mathematics. Keijzer & A ; Terwel ( 2001 ) used the illustration of perpendicular figure line houses as fractions, where the fractions lived in each floor and the lift connected the different floors. If the edifice was 3 floors high so it had 3 Michigans ( 1/3, 2/3 and 3/3 ) if it was a 4 floor edifice so it would hold four floors ( 1/4, 2/4, 3/4 and 4/4 ) . The student would make full in every bit much information as possible on the fraction lift. Then in a treatment they will be able to do opinion on how it was easier to state A? instead than 2/4 or discourse other point excessively. This exercising will do it clear different fractions can suit in to the same place on the figure line.

This is an illustration of a different manner of doing students understand where the places are on a figure line. It was taken for the new version of the Maths in Context – Fraction Time, here the students would be provided with a tabular array and asked to make full in the fractions in the grid. I would so travel into to inquire students to happen forms and color codification any similar fraction that they can see on the grid. Once coloring material coded the students will be able to see that some fractions do hold the same place on the figure line. Although the figure line is now really clear in this exercising it will give the students a opportunity to descry form. Descrying forms is a informal scheme that pupils do like to make.

Looking at both the diaries of Streefland ( 1991 ) and Keijzer & A ; Terwel ( 2001 ) ,