Category:Assessment

From OER in Education
Revision as of 11:49, 22 August 2012 by SimonKnight (talk | contribs)

Assessment may take many forms, including whole class, and individual. Readers should consider reading the pages on these approaches in addition to the guidance given below. Where appropriate links have been incorporated - if you are a wiki-contributor, please do add further internal links, and if of high quality, external too.

The Typical Structure of Dialogue

This resource is licenced under an Open Government Licence (OGL).


This resource is adapted from an Initiatl Teacher Education - English resource available: http://www.ite.org.uk/ite_topics/speaking_listening/001.html

Lyn Dawes Research Associate at the University of Cambridge’s Faculty of Education

Neil Mercer Professor of Education, Fellow of Hughes Hall Editor, International Journal of Educational Research University of Cambridge

Section 1.1 - Children need direct guidance and structured practice in speaking and listening.

Throughout the curriculum there is a strong emphasis on enabling children to use language to work together effectively. One reason for both these developments is that

recent research has shown the importance of the link between spoken language, learning and cognitive development (e.g. Mercer, Wegerif & Dawes, 1999; Mercer, Dawes, Wegerif & Sams, 2004 – see below). Through using language and hearing how others use it, children become able to describe the world, make sense of life's experiences and get things done. They learn to use language as a tool for thinking, collectively and alone. However, children will not learn how to make the best use of language as a tool for communicating and thinking without guidance from their teachers. School may provide the only opportunity many children have for acquiring some extremely important speaking, listening and thinking skills.

For the research findings which underpin these claims, see:

Mercer, N., Wegerif, R. and Dawes, L. (1999) 'Children's talk and the development of reasoning in the classroom', British Educational Research Journal, 25, 1, 95-111

Mercer, N., Dawes, L., Wegerif, R., & Sams, C. (2004). Reasoning as a scientist: ways of helping children to use language to learn science. British Educational Research Journal, 30, 3, 367-385.

While many realise that children need instruction in literacy and numeracy, student teachers may not realise that spoken language skills can, and should, be directly taught to children. Despite references to the development of children's language skills, National Curriculum guidance does not make it clear that such direct teaching may often be required. There seems to be an implicit belief that the subtle skills of active listening and reasoned speaking will develop simply through children's involvement in whole class and small group dialogues. To some degree, the children will develop their language use through practice. But all children can benefit from exposure to good models for speaking and listening. They also gain from guidance about how to communicate effectively and from taking part in structured activities for practising communicating (including, crucially, group interactions with light supervision from a teacher). It is therefore very important that student teachers become aware of this and learn how to guide children's spoken language development. This will include learning how to:

  1. assess children's language skills (see below)
  2. engage children in dialogues in which they are encouraged to develop and use spoken language skills. This means more than the capacity to provide brief answers to questions in whole class settings. Children need more of the kind of interaction which is generated by what Robin Alexander calls 'dialogic teaching' (see his publication listed below)
  3. use varied lessons (in English, the Speaking and Listening curriculum) as a basis for raising children's awareness about how talk can be used most effectively to share ideas, negotiate thinking, challenge and agree, build relationships and generally get things done
  4. design pair and group activities based on interesting problem-solving tasks or creative endeavours which will stretch children's communication skills and help them practice what they are learning about language as a tool for communicating.

Section 1.2 - Children need direct guidance and structured practice in speaking and listening.

Use of Ground Rules to establish a clear, constructive, context for talk is important – these rules can provide a structure of mutual respect, and encourage Thinking Together, and Exploratory Talk, as opposed to dismissive disputational, or ‘cumulative’ talk which just restates the already known. Teachers should explore the resources on the (freely available) Thinking Together website www.thinkingtogether.educ.cam.ac.uk/resources/

Section 1.3 - Children need direct guidance and structured practice in speaking and listening.

An activity: What makes a good discussion?

A useful awareness-raising activity is to ask the following question: if you were in a classroom and overheard a group of children who were working together having what you would consider a good, productive discussion, what exactly would they be doing, and what would you hear? (That is, describe the observable features of the children's talk, such as 'asking each other questions'.).

Write down all their responses on a flip chart and try to establish some agreement about which features are most crucial, the extent to which features might vary with particular kinds of task, and so on. Then ask:

  1. if 'good discussions' are common in classrooms (research has shown they are not)
  2. if they are effective users of language in such situations
  3. if they were ever taught how to engage in such discussions in school
  4. how they think children might best be helped to develop good discussion skills; and
  5. you might also compare their list of the features of a good discussion with the definition of Exploratory Talk (see above).

Section 2.1 - Raising student teachers' awareness of the structure of teacher-pupil talk in classrooms and how to assess its quality.

A striking insight provided by classroom research is that much talk between teachers and their pupils has the following pattern: a teacher's question, a pupil's response, and then an evaluative comment by the teacher. This is described as an Initiation-Response-Feedback exchange, or IRF. Here's an example

I Teacher - What's the capital city of Argentina?
R Pupil - Buenos Aires
F Teacher - Yes, well done

This pattern was first pointed out in the 1970s by the British researchers Sinclair and Coulthard. Their original research was reported in

Sinclair, J. and Coulthard, M. (1975) Towards an Analysis of Discourse: the English used by Teachers and Pupils. London: Oxford University Press.

Sinclair and Coulthard's research has been the basis for extended debates about whether or not teachers should ask so many questions to which they already know the answer; and further debate about the range of uses and purposes of IRF in working classrooms. Despite all this, it seems that many teachers (even those who have qualified in recent decades) have not heard of it. Is this because their training did not include any examination of the structures of classroom talk – or because even if it did, the practical value of such an examination was not made clear?

A teacher's professional development (and, indeed, the development of members of any profession) should involve the gaining of critical insights into professional practice – to learn to see behind the ordinary, the taken for granted, and to question the effectiveness of what is normally done. Recognizing the inherent structure of teacher-pupil talk is a valuable step in that direction. Student teachers need to see how they almost inevitably converge on other teachers' style and generate the conventional patterns of classroom talk. By noting this, they can begin to consider what effects this has on pupil participation in class. There is nothing wrong with the use of IRFs by teachers, but question-and-answer routines can be used both productively and unproductively. By understanding and questioning what generally happens, students can begin to construct the kind of dialogues that they can feel confident have most educational value.

Section 2.2 - Raising awareness of the structure of teacher-pupil talk in classrooms and how to assess its quality.

We have found the following resources useful for working with student teachers on this topic:

(a) Alexander, R.(2004) Towards Dialogic Teaching. Cambridge: Dialogos.

It can be obtained from: Dialogos UK Ltd, Rose Hill, Osgoodby, Thirsk, North Yorkshire YO7 2 AP (Currently £6 inc p&p.)

The subtitle of this text is 'Rethinking classroom talk'. It provides an introduction to the idea of 'learning to talk, talking to learn' and a philosophy of dialogic teaching. This is teaching and learning which is essentially based on the thoughtful questions, negotiation and discussion which generate reasoned debate amongst teachers and learners. The text provides food for thought. It also gives a range of further reading and practical suggestions for encouraging dialogic teaching. It will encourage students to identify dialogue when they encounter it, and to ensure that they organise their classrooms so that dialogue is how learning proceeds.

Readers could also refer to the page Dialogic Teaching for further links and description.

(b) Chapters 3 and 4 of The Guided Construction of Knowledge: talk amongst teachers and learners by N. Mercer, 1995. Clevedon: Multilingual Matters.

The first of these chapters describes and exemplify the different ways teachers talk to pupils: 'direct elicitations', 'cued elicitations', 'reformulations' and so on. Student teachers can be encouraged to record and transcribe their own dialogues with pupils and consider which of these teaching techniques they have used and to what purpose. The second chapter focuses more on how pupils participate in classroom talk.


(c) For a useful discussion of the educational functions of IRF exchanges, see Chapter 5 of : Wells, G. (1999) Dialogic Inquiry: towards a sociocultural practice and theory of education. Cambridge: Cambridge University Press.

Section 3.1 - Teachers should learn how to assess the quality of children's talk

If teachers are to be able to help children develop their skills in speaking and listening, they will have to assess the ways that children talk, for example when they are working together in a group. In English, the Speaking and Listening component of the National Curriculum highlights some aspects of the quality of children's talk as below. However, across all subject areas, requirements for the ability to talk, and write effectively in the genre of that subject is highlighted.

'Pupils learn to change the way they speak and write to suit different situations, purposes and audiences. They read a range of texts and respond to different layers of meaning in them. They explore the use of language in literary and nonliterary texts and learn how language works.'

'Pupils learn how to speak in a range of contexts, adapting what they say and how they say it to the purpose and the audience. Taking varied roles in groups gives them opportunities to contribute to situations with different demands. They also learn to respond appropriately to others, thinking about what has been said and the language used.'

Beginning teachers need guidance, focused activities and discussion in order to recognise key features of children's talk. They need to learn to evaluate talk and subsequently use their decisions to inform planning of further speaking and listening activities.

A main concern for assessment is to consider how well the talk suits the kind of event in which children are participating. Criteria are likely to be different, depending on whether they are talking in a group, making a presentation to the class, engaged in a drama-related activity, discussing ideas in citizenship, and so on. Helping any child improve their current competence requires some sort of assessment. Talk is difficult to assess because it is context dependent and ephemeral, but good opportunities for assessment occur regularly, especially in 'talk-focused' classrooms where both teacher and children are aware of the importance of speaking and listening for learning. This is the situation students need to be able to both recognise and create.

There are of course some aspects of evaluating children's talk where great sensitivity is needed. The ways people talk can be closely related to their identities, and student teachers may rightly worry about making evaluations of some aspects of a child's way of speaking such as their accent. Student teachers will need to appreciate the distinction between on the one hand using an assessment to help a child to become more involved in learning conversations, or to develop their presentation skills, and on the other trying to alter a child's accent or to ban the use of dialect in the classroom simply because it 'sounds wrong'.

Section 3.2 - Teachers should learn how to assess the quality of children's talk

One of the practical difficulties in teaching talk is gaining access to good models to enable investigation, analysis and reflection. Teachers may find it useful to explore some of the ORBIT resources including

Section 4.1 - Teachers need the confidence to teach speaking and listening

Teacher-led dialogue and group-based activities with minimal teacher intervention are both very important for children's learning. Students will be faced with many competing priorities during their course, and need to be completely sure that they are right to be 'on a mission' to teach speaking and listening. It is important to help them see that by giving direct attention to the development of children's spoken language skills, they will help the process of teaching and learning become more effective throughout the curriculum. Once a collaborative, articulate atmosphere is established, learning objectives for speaking and listening can (and should) inform work in curriculum area. The striving for effective dialogue and learning through talk should be a continuous feature of classroom life.

Section 4.2 - Teachers need the confidence to teach speaking and listening

We have also found the following resources useful for working with student teachers on this topic:

(a) Dawes, L (2004) Talk and Learning in Classroom Science. International Journal of Science Education 26, 6, 677 – 695. (The same research is also described in the Research Digest Reasoning as a Scientist on the DfES website).

This journal article discusses the issue of using speaking and listening in science, to elicit and address children's misconceptions, to help them articulate their ideas. Students reading this article can be asked to discuss the nature and purpose of IRF sequences; the importance of sensitive dialogue between teacher and pupil; the child's use of talk for thinking aloud with others, and what the teacher can find out from hearing such talk.

(b) Mercer, N. (2003) The Educational Value of 'dialogic talk' in 'whole class dialogue' : New Perspectives on Spoken English in the Classroom: Discussion Papers, pp 73- 76(London: QCA: Available from QCA Publications 01787 884444 ref QCA/03/0170 http://dera.ioe.ac.uk/6062/1/6111_new_perspec_in_spoken_eng_class_room.pdf

All the papers in this publication merit discussion with students. Neil Mercer's contribution clarifies the importance of talk for thinking, making clear the link between learning new, rational ways to talk and developing a more rational approach to problems and new ideas. That contribution is available on the wiki: The educational value of dialogic talk in whole-class dialogue

(c) Andrea Raiker (2003) Spoken language and mathematics, Cambridge Journal of Education, Vol. 32, No1.

By analysing the impact of talk on learning in the core subject of mathematics, Andrea Raiker emphasises the fundamental nature of good oral language development for children. Andrea Raiker shows how a mismatch between what a teacher says ( e.g. 'cuboid' , 'repeated addition') and what a child thinks can create barriers to learning. Similarly, children expected to work together in mathematics may be baffled by lack of vocabulary and, more importantly, lack of the speaking and listening skills which can help them to articulate their difficulty. Students reading this paper can be asked to analyse their own teaching of mathematics in a whole class or small group context, perhaps tape recording part of their session then attending carefully to what the children actually say. By foregrounding talk, it becomes possible to evaluate at what point mathematical concepts are understood – or not. {{#lst:The Importance of Speaking and Listening|IRF}}

The Importance of Identifying High Quality Talk

This resource is licenced under an Open Government Licence (OGL).


This resource is adapted from an Initiatl Teacher Education - English resource available: http://www.ite.org.uk/ite_topics/speaking_listening/001.html

Lyn Dawes Research Associate at the University of Cambridge’s Faculty of Education

Neil Mercer Professor of Education, Fellow of Hughes Hall Editor, International Journal of Educational Research University of Cambridge

Section 1.1 - Children need direct guidance and structured practice in speaking and listening.

Throughout the curriculum there is a strong emphasis on enabling children to use language to work together effectively. One reason for both these developments is that

recent research has shown the importance of the link between spoken language, learning and cognitive development (e.g. Mercer, Wegerif & Dawes, 1999; Mercer, Dawes, Wegerif & Sams, 2004 – see below). Through using language and hearing how others use it, children become able to describe the world, make sense of life's experiences and get things done. They learn to use language as a tool for thinking, collectively and alone. However, children will not learn how to make the best use of language as a tool for communicating and thinking without guidance from their teachers. School may provide the only opportunity many children have for acquiring some extremely important speaking, listening and thinking skills.

For the research findings which underpin these claims, see:

Mercer, N., Wegerif, R. and Dawes, L. (1999) 'Children's talk and the development of reasoning in the classroom', British Educational Research Journal, 25, 1, 95-111

Mercer, N., Dawes, L., Wegerif, R., & Sams, C. (2004). Reasoning as a scientist: ways of helping children to use language to learn science. British Educational Research Journal, 30, 3, 367-385.

While many realise that children need instruction in literacy and numeracy, student teachers may not realise that spoken language skills can, and should, be directly taught to children. Despite references to the development of children's language skills, National Curriculum guidance does not make it clear that such direct teaching may often be required. There seems to be an implicit belief that the subtle skills of active listening and reasoned speaking will develop simply through children's involvement in whole class and small group dialogues. To some degree, the children will develop their language use through practice. But all children can benefit from exposure to good models for speaking and listening. They also gain from guidance about how to communicate effectively and from taking part in structured activities for practising communicating (including, crucially, group interactions with light supervision from a teacher). It is therefore very important that student teachers become aware of this and learn how to guide children's spoken language development. This will include learning how to:

  1. assess children's language skills (see below)
  2. engage children in dialogues in which they are encouraged to develop and use spoken language skills. This means more than the capacity to provide brief answers to questions in whole class settings. Children need more of the kind of interaction which is generated by what Robin Alexander calls 'dialogic teaching' (see his publication listed below)
  3. use varied lessons (in English, the Speaking and Listening curriculum) as a basis for raising children's awareness about how talk can be used most effectively to share ideas, negotiate thinking, challenge and agree, build relationships and generally get things done
  4. design pair and group activities based on interesting problem-solving tasks or creative endeavours which will stretch children's communication skills and help them practice what they are learning about language as a tool for communicating.

Section 1.2 - Children need direct guidance and structured practice in speaking and listening.

Use of Ground Rules to establish a clear, constructive, context for talk is important – these rules can provide a structure of mutual respect, and encourage Thinking Together, and Exploratory Talk, as opposed to dismissive disputational, or ‘cumulative’ talk which just restates the already known. Teachers should explore the resources on the (freely available) Thinking Together website www.thinkingtogether.educ.cam.ac.uk/resources/

Section 1.3 - Children need direct guidance and structured practice in speaking and listening.

An activity: What makes a good discussion?

A useful awareness-raising activity is to ask the following question: if you were in a classroom and overheard a group of children who were working together having what you would consider a good, productive discussion, what exactly would they be doing, and what would you hear? (That is, describe the observable features of the children's talk, such as 'asking each other questions'.).

Write down all their responses on a flip chart and try to establish some agreement about which features are most crucial, the extent to which features might vary with particular kinds of task, and so on. Then ask:

  1. if 'good discussions' are common in classrooms (research has shown they are not)
  2. if they are effective users of language in such situations
  3. if they were ever taught how to engage in such discussions in school
  4. how they think children might best be helped to develop good discussion skills; and
  5. you might also compare their list of the features of a good discussion with the definition of Exploratory Talk (see above).

Section 2.1 - Raising student teachers' awareness of the structure of teacher-pupil talk in classrooms and how to assess its quality.

A striking insight provided by classroom research is that much talk between teachers and their pupils has the following pattern: a teacher's question, a pupil's response, and then an evaluative comment by the teacher. This is described as an Initiation-Response-Feedback exchange, or IRF. Here's an example

I Teacher - What's the capital city of Argentina?
R Pupil - Buenos Aires
F Teacher - Yes, well done

This pattern was first pointed out in the 1970s by the British researchers Sinclair and Coulthard. Their original research was reported in

Sinclair, J. and Coulthard, M. (1975) Towards an Analysis of Discourse: the English used by Teachers and Pupils. London: Oxford University Press.

Sinclair and Coulthard's research has been the basis for extended debates about whether or not teachers should ask so many questions to which they already know the answer; and further debate about the range of uses and purposes of IRF in working classrooms. Despite all this, it seems that many teachers (even those who have qualified in recent decades) have not heard of it. Is this because their training did not include any examination of the structures of classroom talk – or because even if it did, the practical value of such an examination was not made clear?

A teacher's professional development (and, indeed, the development of members of any profession) should involve the gaining of critical insights into professional practice – to learn to see behind the ordinary, the taken for granted, and to question the effectiveness of what is normally done. Recognizing the inherent structure of teacher-pupil talk is a valuable step in that direction. Student teachers need to see how they almost inevitably converge on other teachers' style and generate the conventional patterns of classroom talk. By noting this, they can begin to consider what effects this has on pupil participation in class. There is nothing wrong with the use of IRFs by teachers, but question-and-answer routines can be used both productively and unproductively. By understanding and questioning what generally happens, students can begin to construct the kind of dialogues that they can feel confident have most educational value.

Section 2.2 - Raising awareness of the structure of teacher-pupil talk in classrooms and how to assess its quality.

We have found the following resources useful for working with student teachers on this topic:

(a) Alexander, R.(2004) Towards Dialogic Teaching. Cambridge: Dialogos.

It can be obtained from: Dialogos UK Ltd, Rose Hill, Osgoodby, Thirsk, North Yorkshire YO7 2 AP (Currently £6 inc p&p.)

The subtitle of this text is 'Rethinking classroom talk'. It provides an introduction to the idea of 'learning to talk, talking to learn' and a philosophy of dialogic teaching. This is teaching and learning which is essentially based on the thoughtful questions, negotiation and discussion which generate reasoned debate amongst teachers and learners. The text provides food for thought. It also gives a range of further reading and practical suggestions for encouraging dialogic teaching. It will encourage students to identify dialogue when they encounter it, and to ensure that they organise their classrooms so that dialogue is how learning proceeds.

Readers could also refer to the page Dialogic Teaching for further links and description.

(b) Chapters 3 and 4 of The Guided Construction of Knowledge: talk amongst teachers and learners by N. Mercer, 1995. Clevedon: Multilingual Matters.

The first of these chapters describes and exemplify the different ways teachers talk to pupils: 'direct elicitations', 'cued elicitations', 'reformulations' and so on. Student teachers can be encouraged to record and transcribe their own dialogues with pupils and consider which of these teaching techniques they have used and to what purpose. The second chapter focuses more on how pupils participate in classroom talk.


(c) For a useful discussion of the educational functions of IRF exchanges, see Chapter 5 of : Wells, G. (1999) Dialogic Inquiry: towards a sociocultural practice and theory of education. Cambridge: Cambridge University Press.

Section 3.1 - Teachers should learn how to assess the quality of children's talk

If teachers are to be able to help children develop their skills in speaking and listening, they will have to assess the ways that children talk, for example when they are working together in a group. In English, the Speaking and Listening component of the National Curriculum highlights some aspects of the quality of children's talk as below. However, across all subject areas, requirements for the ability to talk, and write effectively in the genre of that subject is highlighted.

'Pupils learn to change the way they speak and write to suit different situations, purposes and audiences. They read a range of texts and respond to different layers of meaning in them. They explore the use of language in literary and nonliterary texts and learn how language works.'

'Pupils learn how to speak in a range of contexts, adapting what they say and how they say it to the purpose and the audience. Taking varied roles in groups gives them opportunities to contribute to situations with different demands. They also learn to respond appropriately to others, thinking about what has been said and the language used.'

Beginning teachers need guidance, focused activities and discussion in order to recognise key features of children's talk. They need to learn to evaluate talk and subsequently use their decisions to inform planning of further speaking and listening activities.

A main concern for assessment is to consider how well the talk suits the kind of event in which children are participating. Criteria are likely to be different, depending on whether they are talking in a group, making a presentation to the class, engaged in a drama-related activity, discussing ideas in citizenship, and so on. Helping any child improve their current competence requires some sort of assessment. Talk is difficult to assess because it is context dependent and ephemeral, but good opportunities for assessment occur regularly, especially in 'talk-focused' classrooms where both teacher and children are aware of the importance of speaking and listening for learning. This is the situation students need to be able to both recognise and create.

There are of course some aspects of evaluating children's talk where great sensitivity is needed. The ways people talk can be closely related to their identities, and student teachers may rightly worry about making evaluations of some aspects of a child's way of speaking such as their accent. Student teachers will need to appreciate the distinction between on the one hand using an assessment to help a child to become more involved in learning conversations, or to develop their presentation skills, and on the other trying to alter a child's accent or to ban the use of dialect in the classroom simply because it 'sounds wrong'.

Section 3.2 - Teachers should learn how to assess the quality of children's talk

One of the practical difficulties in teaching talk is gaining access to good models to enable investigation, analysis and reflection. Teachers may find it useful to explore some of the ORBIT resources including

Section 4.1 - Teachers need the confidence to teach speaking and listening

Teacher-led dialogue and group-based activities with minimal teacher intervention are both very important for children's learning. Students will be faced with many competing priorities during their course, and need to be completely sure that they are right to be 'on a mission' to teach speaking and listening. It is important to help them see that by giving direct attention to the development of children's spoken language skills, they will help the process of teaching and learning become more effective throughout the curriculum. Once a collaborative, articulate atmosphere is established, learning objectives for speaking and listening can (and should) inform work in curriculum area. The striving for effective dialogue and learning through talk should be a continuous feature of classroom life.

Section 4.2 - Teachers need the confidence to teach speaking and listening

We have also found the following resources useful for working with student teachers on this topic:

(a) Dawes, L (2004) Talk and Learning in Classroom Science. International Journal of Science Education 26, 6, 677 – 695. (The same research is also described in the Research Digest Reasoning as a Scientist on the DfES website).

This journal article discusses the issue of using speaking and listening in science, to elicit and address children's misconceptions, to help them articulate their ideas. Students reading this article can be asked to discuss the nature and purpose of IRF sequences; the importance of sensitive dialogue between teacher and pupil; the child's use of talk for thinking aloud with others, and what the teacher can find out from hearing such talk.

(b) Mercer, N. (2003) The Educational Value of 'dialogic talk' in 'whole class dialogue' : New Perspectives on Spoken English in the Classroom: Discussion Papers, pp 73- 76(London: QCA: Available from QCA Publications 01787 884444 ref QCA/03/0170 http://dera.ioe.ac.uk/6062/1/6111_new_perspec_in_spoken_eng_class_room.pdf

All the papers in this publication merit discussion with students. Neil Mercer's contribution clarifies the importance of talk for thinking, making clear the link between learning new, rational ways to talk and developing a more rational approach to problems and new ideas. That contribution is available on the wiki: The educational value of dialogic talk in whole-class dialogue

(c) Andrea Raiker (2003) Spoken language and mathematics, Cambridge Journal of Education, Vol. 32, No1.

By analysing the impact of talk on learning in the core subject of mathematics, Andrea Raiker emphasises the fundamental nature of good oral language development for children. Andrea Raiker shows how a mismatch between what a teacher says ( e.g. 'cuboid' , 'repeated addition') and what a child thinks can create barriers to learning. Similarly, children expected to work together in mathematics may be baffled by lack of vocabulary and, more importantly, lack of the speaking and listening skills which can help them to articulate their difficulty. Students reading this paper can be asked to analyse their own teaching of mathematics in a whole class or small group context, perhaps tape recording part of their session then attending carefully to what the children actually say. By foregrounding talk, it becomes possible to evaluate at what point mathematical concepts are understood – or not.

Pedagogical Strategies(i)