Effective Pedagogy in Mathematics/Pāngarau Best Evidence Synthesis [BES]

Anthony and Walshaw (2007). Effective pedagogy in mathematics/pāngarau best evidence synthesis (BES). Wellington: Ministry of Education.

• BES: Best Evidence Synthesis Iteration
• TIMSS: The Trends in International Mathematics and Science Study (2006)
• Numeracy Development Project (NDP)
• National Education Monitoring Project (NEMP)
• Teaching and Learning Research Initiative (TLRI)

Reflect on how the metaphors of “whanau” and “friendly arguing” are used to promote productive participation in important mathematical practices.

While the context of this exemplar is mathematics, its message about the value of creating a peer environment where students feel safe to engage in constructive learning talk is of relevance across the curriculum

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eleven dimensions of quality teaching

BES Exemplar 1: Developing communities of mathematical inquiry

The teachers were working with year 4 to 6 students, most of whom were Māori or Pasifika.

• Hunter, R. (2007). Teachers developing communities of mathematical inquiry. Albany, Massey University: Unpublished doctoral thesis. Available from the New Zealand Education Theses database at www.educationcounts.govt.nz/goto/BES
• Hunter, R. (2008). Facilitating communities of mathematical inquiry. In M. Goos, R. Brown, & K. Makar (Eds.), Navigating currents and charting directions (Proceedings of the 31st annual conference of the Mathematics Education Research Group of Australasia, Vol. 1, pp. 31–39). Brisbane: MERGA.

The Trends in International Mathematics and Science Study6 shows that students in New Zealand schools, especially Māori and Pasifika students, perceive themselves as having a very low level of safety amongst their peers.

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Studying mathematics helps students develop the ability to think creatively, critically, strategically, and logically. They learn to create models, conjecture, justify and verify, and seek patterns and generalisations.

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The students actively built capability in all five key competencies:

• thinking,
• using language,
• symbols and texts,
• managing self,
• relating to others,
• participating and contributing.

Pg 3. Teacher knowledge and inquiry at the outset of the study: Ava

Ava explained that she had initially liked teaching mathematics but had lost confidence in her mathematics teaching expertise after participating in the NDP. The NDP professional development had created dissonance between her prior view of mathematics teaching as focused on rules, procedures, and routines and the new focus on students developing their mathematical thinking.

Observations revealed that Ava had tried to implement what she had learned from the NDP. But in practice, she had done what many teachers do when confronted with new curricula designed to change the nature of teaching. She had transformed the NDP learning experiences to fit with her old way of doing things. Her students listened to each other in silence and answered the teacher’s questions, but there was no process to build a mathematical learning community amongst students.

Teacher knowledge and inquiry at the outset of the study: Moana

Moana had always disliked mathematics. Her memory of her own schooling was negative: “I didn’t have the ideas that other kids had so I just never said anything. I thought they were all brighter than me.”

Before the intervention, Moana was using a learning styles approach. She saw her Māori and Pasifika learners as kinaesthetic learners needing practical activities. This approach has been found to have little benefit for student achievement and has even had negative effects on Māori and Pasifika student achievement.

In her mathematics lessons, Moana did most of the talking because she saw her role as instructing students in how to use mathematical procedures. She explained that she had adapted the NDP’s knowledge and strategy activities so that they were always at a concrete and manipulative level. She focused on students ‘learning by doing’, because she believed that they found explaining difficult.

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The use of mixed-ability groups rather than fixed-ability groups was critical to the effectiveness of this
intervention. Students were not publicly labelled as low achievers. Rather, their opportunities for mathematical argumentation were scaffolded and extended, whatever their level of prior knowledge.

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Ava builds a mathematical learning community

Ava began by developing new ‘rules for talk’. At the beginning of each maths lesson, she discussed with her students how they were to work together in a mathematical community. During the lesson, she repeatedly reviewed with them how this was going and modelled the changes she wanted to see. For example, she used collective first person pronouns when she was speaking to the students:
Can you show us with your red pen what would happen? We want to know.

Requiring mathematical reasoning

Ava sought to develop her students’ ability to engage in talk that probed mathematical ideas. She explicitly required them to listen, discuss, question, and make sense of the reasoning used by others. She also used modelling to scaffold the students into how to construct mathematical explanations that would be well-reasoned, conceptually clear, and logical:
Talk about what you are doing … so whatever number you have chosen, don’t just write them. You say, “I am going to work with …” or “I have chosen this and this because …”.

Using think time

Ava used ‘think time’ as a form of social nurturing for less confident members of the group. This involved
halting the discussion to allow them time to reflect and making it clear that their learning mattered.

Promoting constructive argumentation

Ava explicitly directed the students to argue their ideas in a productive manner, as in the following exchange with students:
Ava: Argue your maths. Explore what other people say. Listen carefully, bit by bit, and make sense of each
bit. Don’t just agree. Check it all out first. Ask a lot of questions. Make sure that you can make sense – that you understand. What’s another important thing in working in a group?
Alan: Share your ideas. Don’t just say “I can do it myself” …
Ava: That’s right. We need to use each other’s thinking … because we are very supportive and that’s the only way everyone will learn.
Ava believed that many of her students would be used to oppositional or aggressive argument and that this would have shaped negative beliefs about arguing:
Ava: I am aware that the students are growing into this behaviour now, but disagreeing can be so hard for these students, so I find I have to keep almost giving them permission to disagree or argue. This meant that teaching the skills to argue constructively was an ongoing focus:
Ava: ‘Arguing’ is not a bad word … sometimes I know that you people think to argue is … I am talking about arguing in a good way. Please feel free to say if you do not agree with what someone else has said. You can say that as long as you say it in an OK sort of way.

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Moana builds a mathematical learning community

Moana was a caring and concerned teacher, but when she looked back on the first recordings of her practice, she realised that her students had shown fear in whole-class discussions. By demanding the right answers without providing sufficient scaffolding, she had failed to provide a genuinely caring classroom environment.

When Moana first set about creating a learning community, she presented the students with another teacher’s chart of ground rules and instructed them to work collaboratively by following the rules. This did not work. Students constantly interjected and made negative comments to each other, both in the small groups and in the larger sharing sessions. Scaffolding students in working together After reflecting on the video recordings, Moana was concerned about the disengagement and silence of the girls, and especially their reluctance to question or argue with the boys.

Together with the researcher, Moana planned how to make incremental changes through scaffolding the students’ ability to work together. In a three-step process, Moana:

• stopped using small groups and returned to the previous whole-class discussion format
• asked the students to work in pairs and scaffolded the way they were to talk to and listen to each other
• required the paired students to explain each other’s reasoning (using materials to demonstrate this) as a report-back process.

Ground rules for talk

Moana developed her own classroom-specific chart of the Ground Rules for Talk: How do we kōrero in our classroom? She began each lesson by explicitly explaining how she wanted the students to work.

Considered pairings

Moana noticed that the Māori and Pasifika girls in particular were diffident in pairs with boys. She responded by putting students in single-sex pairs and putting particular girls together (for example, pairs of Pasifika girls) to scaffold a safe environment for talk. She set about creating an environment in which the students encouraged each other to be risk-takers and built up their confidence:
Moana: You don’t have to whisper. You can talk because you want to make sure that you are heard.
At the end of the first month, when the pairs were able to construct and examine their explanations more
collaboratively, Moana began to vary the number and combinations of students working together.

Moana closely monitored the less able or less confident students in the context of the heterogeneous groups in order to draw attention to how their reasoning had contributed:
Moana: Wow, Teremoana, see how you have made them think when you said that? Now they are using
your thinking.
As a result, the students’ collaborative skills developed markedly. By terms 3 and 4, the following kinds of
comments indicated a strongly functioning student learning community:
Wiremu: Don’t dis her, man, when she is taking a risk.
This 10-year-old boy’s comment exemplifies the shared responsibility students were taking for creating a peer culture that supports intellectual risk-taking. He is telling another boy to stop his disrespectful behaviour towards a female peer so she can take a risk in her public participation in a mathematical discourse. This comment illustrates the depth of the change that Moana was forging in the peer culture.

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In this excerpt, Ava uses revoicing to clarify and define mathematical terms:
Jo: Isn’t that just plussing three sticks not timesing it? You are not timesing, you’re adding.
Pania: Well, what she sort of means, it is like it is going up.
Alan: Is that timesing, going up?
Ava: When we talk about timesing, what do we actually mean?
Jo: We mean multiplying not adding. Adding is a plus [indicates + with her fingers], that sign.
Sandra: You mean when you add two more squares on, that is multiplying?
Ava: Rachel was saying she is adding 3, adding another 3, so that’s 3 plus 3 plus 3. So if you keep adding 3 all the time, what is another way of doing it?
Alan: You can just times instead of adding. It won’t take as long and it is more efficient.
Ava: Yes, you are right. Did you all hear that? Alan said you can just times it, multiply by [groups of] three,
because that is the same as adding on 3 each time. What word do we use instead of ‘timesing’?
Alan: Multiplication, multiplying.

In this excerpt, Moana scaffolds the use of mathematical language and explanations, beginning as Aporo uses counters to model how his group solved a problem:
Aporo: Two, four, six, eight, ten, twelve.
Moana: So that’s what’s called skip-counting, because you are skipping across the numbers.
Tere: We kept adding like two more. We counted in twos.
Moana: Counting in twos. Yes, that is skip counting.

Moana: What have you actually done there, Mahine?
Mahine: I have plussed 10 onto 47.
Moana: So you have added 10 onto 47? Are there any questions? Questions like “Where did you get the 10 from?”
When she saw that her students were not taking up opportunities to ask questions and dispute, Moana introduced a non-verbal scaffold – koosh balls.13 The koosh balls were placed in the middle of the discussion circle. The students picked them up to indicate that they had a question or challenge. The researcher explained that although picking up the ball indicated to the explainer that there was a challenge, their self-esteem was protected because the non-verbal signal gave time to think and then respond.

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When students had difficulties, their teachers provided time for reflection. Ava called this ‘think time’ and Moana called it ‘rethink time’. As it became safer for the students to actively participate and take risks, the teachers were increasingly able to use the students’ thinking as a resource for improving teaching and learning.

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While students were talking, the expectations for others in the group were clear. Each individual was required to:
• actively ask questions;
• follow the line of reasoning;
• take responsibility for the regulation of their own (and others’) learning;
• be accountable for their own understanding

Intellectual risk-taking

Intellectual risk-taking supports intellectual growth. Both teachers explicitly encouraged and supported their students to take intellectual risks, as in this exchange, which took place before Ava’s students began a mathematical activity:
Ava: Remember how yesterday we talked about how, in maths learning, you go almost to the edge? So
therefore I am going to move you out of your comfort zone. It’s lovely being in a comfortable, cosy
place. Even as an adult, we love to be there, too. But if you are already there, then it’s time to move on,
out a little bit … so you go out there … maybe a bit more … a bit further next time and come back in
again …
Sandra: And when you are out there, you will make that your comfort zone. Then move on and make that your comfort zone.

There was time for challenge and clarification, and there were clear expectations that ‘making sense’
was the goal.

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Positioning students as accountable for their opinions

Moana made the students aware that she held them accountable for their own understanding. She regularly halted explanations and required students to take a stance. For example, she told them:
At some point, you are going to have an opinion about it. You are going to agree with it or disagree
with it.
She ensured that they knew they needed a valid reason to support their stance, directing the students to think about what they were saying:
Make sense of it. If you don’t agree, say so but say why. If there is anything you don’t agree with, or
you would like them to explain further, or you would like to question, say so. But don’t forget that you
have to have reasons. Remember it is up to you to understand.

Validating autonomous mathematical reasoning

Aroha explains a solution strategy for adding 43, 23, 13, and 3. She records 43, 23, 13, 3, and then 3 x 4 = 12:
Aroha: I am adding 43, 23, 13, and 3, so 3 times 4 equals 12.
Kea: Why are you trying to do that with those numbers? Where did you get the 4?
Aroha: [Points at the 3 digit on each of the four numbers] These 3s, the four 3s.
Donald: All she is doing is, like, making it shorter by, like, doing 4 times 3.
Hone: Because there are only the 10s left.
Donald: Three times 4 equals 12, and she got that off all the 3s; like the 43, 23, 13, and 3. So she is just like adding the 3s all up and that equals 12.

Assessment for learning

During each lesson, the teachers’ moment-by-moment assessment enabled them to decide what questions to ask, when to intervene, and how to respond to student questions. The teachers used the information gained by observing and listening to the students to focus student attention on key examples and explanations during class discussions.

They received, and progressively internalised, multiple messages from the teacher that they needed to be able to inquire into, explain, and justify their own and others’ solutions to mathematical problems. The students became increasingly autonomous learners, able to analyse and validate their own reasoning rather than placing this responsibility on the teacher. In addition, the development of strong learning community norms meant that the students progressively took responsibility not only for their own learning but also for the learning orientation of their peers. For example, when eight-year-old Pita in Moana’s class saw another student scribbling on a recording sheet during their group work, he said: “Don’t, man. You listen or ask or you aren’t even learning, man.”

The BES section on Working in groups (pp. 64 – 68) 

…Pg 64

Working in groups

Many researchers have shown that small-group work can provide the context for social and cognitive engagement.

Thornton, Langrall, and Jones (1997) illustrate from a small study how classrooms organised for group work can provide a rich forum for diverse students to develop their mathematical thinking. They cite a study by Borasi, Kort, Leonard, and Stone (1993) in which a student who had a severe motor disability in writing, in addition to a ‘numerical’ disability, learned from his peers about how to share ideas and articulate his thinking.

Research has shown that gifted students, as well as low attainers, benefit from collaboration with peers. From a study involving six mathematically gifted students, Diezmann and Watters (2001) provide evidence that small homogeneous group collaboration significantly enhanced knowledge construction. Group participation also developed students’ sense of self-efficacy. In particular, collaborative work that was focused on solving challenging tasks produced a higher level of cognitive engagement than that produced by independent work. The supportive group provided a forum for the giving and taking of critical feedback and building upon others’ strategies and solutions.

Successful group process depends on:

• (a) the spatial configuration and interdependencies among participants,
• (b) how familiar the students are with the activity,
• (c) the rules established and the teacher’s managing skills, and (d) students’ inclinations to participate and their competencies.

It is the teacher’s responsibility to ensure that roles for participants, such as listening, writing, answering, questioning, and critically assessing, are understood and implemented.

…groups of four or five tend to be most effective They should be mixed in relation to academic achievement and any status characteristic. They need space for easy interactions and freedom from distractions.

Groups provide opportunities to work with and learn from peers

Pg 65 Advocates of grouping claim that the organisational practice gives every student the opportunity
to articulate thinking and understanding without every classroom eye and mind on what is being said.

Wood and Yackel (1990) provide examples of how, in the course of working through problems with others, students extended their own framework for thinking. Benefits accrued as they listened to what their peers were saying and tried to make sense of it and coordinate it with their own thoughts on the situation.

White (2003) found that students with limited English were more inclined to share their thinking with a friend rather than with the whole class. The teacher noted: “A lot of time they won’t share something with the whole group. But they will share it with somebody sitting next to them, or they can sometimes get ideas from other kids who are sitting next to them” (p. 42).

Pg 66 The teacher aide’s key responsibility during these discussions was to provide support for the target students to actively participate in group discussions….She made sure that

they listened to the contributions of others, that they offered their own contributions, and that they could articulate the group’s strategy for solving the problem. Baxter et al. report that the students were exposed to a wide range of ideas, strategies and solution pathways from their more academically able peers. Their peers’ more advanced cognitive levels provided richer social-emotional as well as cognitive outcomes for the target students than would have been possible in a remedial classroom setting.

Students who learned to help each other learned that to make the group work effective, communication and feedback within the group needed to be centred on mathematical explanations and justifications rather than on single answers to problems.

two-way peer tutoring “reciprocal scaffolding”

They note that students interacting need to have sufficient competence and experience to allow them to ask appropriate questions of themselves and each other

Pg 67 A New Zealand study undertaken by Higgins (1997) revealed that young students’ group work (new entrant to J2) was not as effective as their teachers believed it to be. Higgins showed that student explanations appeared to be constrained by the group process. In later research, Higgins (2000) demonstrated that teachers are often unclear about their role during student group work. However, when the mathematical intent of the group activity was articulated at the beginning and again during the feedback episode, and when student contributions were evaluated in terms of that intent, students appeared to engage more actively with the mathematics.

poor social relationships and poor communication within groups contributed to limited student mathematical engagement in an activity.

Barnes suggests that pedagogical practice that regularly includes all students in group work reinforces the norms of careful and courteous listening.

Individual thinking time

A number of studies have provided evidence of the benefits for some students of independent learning approaches….

The two teachers involved had undertaken a year-long programme to develop purposeful instructional resources and strategies for their individualised teaching sessions. They focused the level appropriately,
encouraged independent learning, and gave the students time to reflect on their thinking and methods. The success of the intervention depended crucially on the teachers’ sophisticated craftsmanship, which involved both anticipating and supporting students’ responses. The researchers report that the students increased their confidence in their ability and their mathematical understanding.

Walshaw (2004) reports on one student who advanced her learning more through independent thinking than through collaborative efforts with peers.

The researchers report on two students, Gur and Ari, who were set a task by the teacher and expected to work together towards producing a solution. Classroom observations led the researchers to believe that the students were working together but further scrutiny revealed otherwise:

While having a close look at a pair of students working together, we realised that the merits of learning-by-talking cannot be taken for granted. Our analyses compel us to conclude that if Gur did make any real progress, it was not thanks to his collaboration with Ari but rather in spite of it, and if this collaboration did, in the end, spur Gur’s development, it was probably mainly in an indirect
way, by providing him with an incentive to learn. Our experiment has shown that the interaction between the two boys was unhelpful to either of them. The present study, therefore, does not lend support to the common belief that working together can always be trusted to have a synergetic quality. It is not necessarily true that two people who join forces can do more than the sum of what each one of them can do alone. (p. 70)

‘talking to oneself’ as a form of communication …“interaction with others, with its numerous demands on one’s attention, can often be counterproductive. Indeed, it is very difficult to keep a well-focused conversation going when also trying to solve problems and be creative about them” (p. 70).

Students need some time alone to think and work quietly away from the demands of a group. Reliance on classroom grouping by ability may have a detrimental effect on the development of a mathematical disposition and students’ sense of mathematical identity. What effective teachers do is create a space for both the individual and the collective. They use a range of organisational processes to enhance students’ thinking and to engage them more fully in the creation of mathematical knowledge. More significantly, over and above establishing structures for participation, the effective teacher constantly monitors, reflects upon, and makes necessary changes to those arrangements on the basis of their inclusiveness and effectiveness for the classroom community.

Mathematics teaching for diverse learners involves respectful exchange of ideas

Mathematics teaching for diverse learners involves respectful exchange of ideas” Anthony, G., & Walshaw, M. (2007). Effective pedagogy in mathematics/pāngarau: Best evidence synthesis iteration [BES], (pp. 72-81)

IMPACT (Increasing the Mathematical Power of All Children and Teachers).

Students’ articulating thinking

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By expressing their ideas, students provide their teachers with information about what they know and what they need to learn.

Effective teachers… pick up on the critical moments in discursive interactions and take learning forward.

Reframing student talk in mathematically acceptable language provides teachers with the opportunity to enhance connections between language and conceptual understanding.

… to make classroom discourse an integral part of an overall strategy of teaching and learning (Hicks, 1998; Lampert & Blunk, 1998)… involves significantly more than developing a respectful, trusting and nonthreatening climate for discussion and problem solving

The teacher must give each child an opportunity to work through the problem under discussion while simultaneously encouraging each of them to listen to and attend to the solution paths of others, building on each others’ thinking. Yet she must also actively take a role in making certain that the class gets to the necessary goal: perhaps a particular solution or a certain formulation that will lead to the
next step … Finally, she must find a way to tie together the different approaches to a solution, taking everyone with her. At another level just as important she must get them to see themselves and each other as legitimate contributors to the problem at hand. (p. 65)

O’Connor and Michaels (1996)

Valuing and shaping students’ mathematical contributions served these important functions:

• it allowed students to see mathematics as created by communities of people;
• it supported students’ learning by involving them in the creation and validation of ideas;
• it helped students to become aware of more conceptually advanced forms of mathematical activity.

Pg 73

Classroom vignettes illustrated one of four themes that emerged from the classroom discourse:

1. (a) valuing students’ ideas,
2. (b) exploring students’ answers,
3. (c) incorporating students’ background knowledge, and
4. (d) encouraging student-to-student communication.

The teachers engaged all students in discourse by first monitoring their participation in discussions and then deciding when and how to encourage each to participate. By actively listening to students’ ideas and suggestions, they demonstrated the value they placed on each student’s contribution to the thinking of the class. The teachers encouraged their students to give critical feedback on each other’s responses and asked them to reveal their assessment of each other’s ideas by giving a ‘thumbs up’ or ‘thumbs down’ signal.

Clarifying Expectations of Classroom Discourse
The teacher’s specific pedagogical effectiveness within classroom discussion was her use of explicit
strategies to enhance the mathematical contributions of her students:
• If you don’t understand, what questions do you need to ask?
• If someone didn’t understand it though and the same thing was said to them …
• I want you to explain to the people in your group how you think you are going to go about
working it out. Then I want you to ask if they understand what you are on about and let them ask
you questions. Remember in the end you all need to be able to explain how your group did it so
think of questions you might be asked and try them out.
• Okay so I have heard lots of talking, discussing in your groups and listening to each other and
that’s good.
• Now this group is going to explain and you are going to look at what they do and how they came
up with the rule for their pattern, right? Then as they go along if you are not sure please ask them
questions. Tune in here, step by step, and as they go along if you can’t make sense of each step
remember ask those questions.
• Pen down. Have a look and think. Now has anyone got a question they want to ask of Rewa at this
point?
• Arguing is not a bad word … sometimes I know you people think to argue is … I am talking about
arguing in a good way. So please feel free if you do not agree with what someone has said as long
as you say it in an okay way. A suggestion could be that you might say I don’t actually agree
with you, could you show that to me. Do you think you could prove it mathematically, could you,
perhaps write it, or draw something to show that idea to me … and sometimes doing that the other
person thinks it wasn’t quite right so they change their idea and that’s okay.

From Hunter (2005)

Pg 74

Other New Zealand studies have found that students are not always able to elaborate on their mathematical reasoning. Meaney (2005), Anthony and Walshaw (2002), Bicknell (1998)

Jones’s (1991) classic study showed that the discursive skills and systems knowledge that are characteristic of high SES families align them favourably with the pedagogy that is operationalised within school settings. Set in the New Zealand context, Jones provided conclusive evidence that Pasifika girls were unwittingly penalised by the sorts of instructional approaches taken by classroom teachers.

Opportunities for students to explain and justify solutions

The researchers record one of many occasions during which she structured students’ mathematical practice: “Before you write it down I want you to justify it to your partner. So if you say there’s eight queens your partner needs to say, ‘How do you know that there’s eight queens?’” (p. 803). By using such pedagogical strategies, the teacher was able to develop mathematical ways of doing and being in all her students.

pg 75

McChesney notes that teachers who established classroom communities in which there was access to social, discursive, visual, and technological resources, were able to support students’ mathematical activity.

The effective teachers in this research were able to set up an environment in which conventional mathematical language migrated from the teacher to the students. Over time, students’ contributions, which were initially marked by informal understandings, began to appropriate the language and the understandings of the wider mathematical community. It was through the take-up of conventional language that mathematical ideas were seeded.

Zack and Graves (2002) have reported that teachers who make a difference are themselves active searchers and enquirers into mathematics

pg 75

Constructive feedback

In keeping with our ethic of care, we have evidence that praise, not of itself but taken together with quality feedback, can be a powerful pedagogical strategy (Hill & Hawk, 2000).

What constitutes quality feedback? Research has shown that feedback that engages learners in further purposeful knowledge construction will contribute to the development of their mathematical identities.

focus on mathematical talk and meaning enabled the students to develop mathematical reasoning in significant ways.

Teachers who provide quality feedback draw on a range of pedagogical content knowledge skills that enable them to know when to and when not to intervene.

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by itself, feedback in the shape of scores and grades was of limited value.

Feedback that is provided too early or too late can be ineffective.

a more effective pedagogical strategy would aim to support students’ mathematical thinking as well as their motivation.

Classroom research at both primary and secondary level (e.g., Ruthven, 2002; Wiliam, 1999) has shown that much of the teacher feedback that students receive is not particularly constructive.

valued teacher feedback that “pointed them in the right direction.”

“Allow the students to work on the problem

statement. As they work, your role should be one of a facilitator and observer. Avoid questions or comments that steer the students toward a particular solution”

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Too much feedback is counterproductive to learning. ‘teacher lust’ (Maddern & Court, 1989).

Cognitive space was limited by the lack of pause times for thinking, and students were occasionally ‘talked over’. Specifically, students did not have the opportunity to learn and speak the language of mathematicians.

Revoicing

‘Revoicing’ is the term used by O’Connor and Michaels (1996) to describe a subtle yet effective strategy for fine-tuning mathematical thinking. By revoicing is meant the repeating, rephrasing or expansion of student talk in order to clarify or highlight content, extend reasoning, include new ideas, or move discussion in another direction. The researchers maintain that probing into student understanding provides teachers with the opportunity to model engagement within a mathematical, multi-voiced community. According to Forman and Ansell (2001), in classrooms where revoicing is used, “[t]here is a greater tendency for students to provide the explanations … and for the teacher to repeat, expand, recast, or translate student explanations for the speaker and the rest of the class” (p. 119).

Providing cognitive structure and fine-tuning mathematical thinking

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O’Connor and Michaels (1996) provide evidence that teachers who provide cognitive structure also tend to fine-tune students’ mathematical thinking

The three key pedagogical components identified in their Advancing Children’s Thinking (ACT) framework are ‘eliciting’, ‘supporting’, and ‘extending’.

• Eliciting involves promoting and managing classroom interactions,
• supporting involves assisting individuals’ thinking, and
• extending captures those practices that work to advance students’ knowledge.

Shaping students’ mathematical thinking is a highly complex activity … they are building an understanding of what it means to think and speak mathematically” (Meyer & Turner, 2002, p. 19).

the teacher to first construct sociomathematical norms (see Yackel & Cobb, 1996) for what constitutes a mathematically acceptable, different, sophisticated, efficient, or elegant explanation. Sociomathematical norms regulate mathematical argumentation and govern the learning opportunities and ownership of knowledge made available within the classroom.

facilitate the establishment of situations in which students had to share ideas and elaborate on their thinking (e.g., Would anyone else like to add anything to S13’s explanation? Could you show that to us on the board? That is an excellent question. Does anyone want to have a shot at it?);

• help students expand the boundary of their exploration (e.g., What do you think class? Do you think that this formula would work all the time for all the rows? Why don’t you extend the sequence and see if there is a pattern);
• encourage students to make connections among different discoveries and develop a deeper understanding of the interrelationships among the patterns that students identified (e.g., I wonder if we can find out how these 2 patterns are related?);
• invite multiple representations of ideas (e.g., Is there another way of representing this?).