Students’ Possibilities to Learn From Group Discussions Integrated in Whole-class Teaching in Mathematics

ABSTRACT

The aim of this paper is to investigate how small-group and whole-class discussions during whole-class teaching contribute to students’ learning. The study includes 33 video-recorded small-group discussions in four mathematics lessons on enlarging and reducing two-dimensional geometric figures in Grade 8. The results show that different objects of learning were enacted in the different small-groups’ discussions, and that in the majority of the small-group discussions the students did not have the possibility to learn what was intended, since critical aspects were not explored. The combination of group and whole-class discussions contributed to that critical aspects of the specific subject matter came to the fore and thereby were possible to discern. The results indicate that small-group discussions without subsequent whole-class discussions may have less effect on student learning.

KEYWORDS:

• Teaching
• mathematics
• group discussion
• whole-class discussion
• variation theory

Introduction

This paper focuses on students’ possibilities to learn specific subject matter in mathematics from small-group discussions integrated in whole-class teaching. The aim of the paper is to investigate and deepen our understanding of how small-group and whole-class discussions contribute to students’ learning about a mathematical content, is this case enlarging and reducing two-dimensional geometric figures.

Research on students’ small-group discussions has most often focused on the development of social skills (Gillies, 2003). However, there is an ongoing debate concerning how collaborative activities, such as small-group discussions, can contribute to students’ learning of mathematics (Weber et al., 2008). Several studies have shown how discussions (in the whole class, in small groups, or in pairs) can promote students’ learning and understanding (e.g., Ball, 1993; Francisco, 2013; Lampert, 1990; McCrone, 2005; Ryve et al., 2013; Stein et al., 2008; Webb, 2009). Students’ possibilities to learn mathematics in collaborative activities, such as small-group discussions, have primarily been studied with focus on generic skills, for example students’ development of solution strategies, explanations, and the ability to express solutions and mathematical ideas (Webb, 2009; Yackel et al., 1991). This paper contributes to the debate by analysing what is made possible to learn in small-group and whole-class discussions in regard to the specific subject matter being taught. Previous studies have foremost studied small-group discussions and whole-class discussions separately (Francisco, 2013; Weber et al., 2008). Similarly, studies using the same theoretical framework as in this study, variation theory (Marton, 2015), have primarily analysed whole-class teaching only (e.g., Kullberg et al., 2017; Maunula, 2018; Pang & Marton, 2005) or small-group discussions as an isolated setting (e.g., in physics; see Berge & Ingerman, 2017).

The study is based on an analysis of 33 small-group discussions and twelve whole-class discussions in two Grade 8 classes. What is made possible to discern from the small-group and whole-class discussions regarding enlarging and reducing two-dimensional geometric figures, and student learning outcomes, is shown. The research questions are: What objects of learning, related to enlarging and reducing two-dimensional geometric figures, are enacted in the small-group discussions and the whole-class discussions? How do the enacted objects of learning in the small-group discussions and the whole-class discussions correspond to student learning outcomes?

Whole-class Discussions and Small-group Discussions in Mathematics Teaching

Research that focus on both small-group and whole-class discussions in the same mathematics lesson, and on the relationship between them and student learning, has been conducted to little extent (Cengiz et al., 2011). Weber et al. (2008) emphasize that studies of student learning of a specific mathematical content in small-group discussions are usually studied isolated from whole-class settings. Cengiz et al. (2011) and Francisco (2013) argue that there is a need to gain insight into how teaching with small-group and whole-class discussions contributes to students’ learning of specific subject matter. This paper is an attempt to contribute with such knowledge, by focusing on what is made possible to learn (the enacted object of learning) in small-group and whole-class discussions in the same lesson, and how they are related to what the teacher intended the students to learn (the intended object of learning), and student learning (the lived object of learning).

The purposes for using small-group and whole-class discussions in teaching vary; one example is an aim to give students the opportunity to articulate their own mathematical ideas and listen to those of others, and thereby become aware of similarities and differences. It is suggested that this helps students develop a deeper understanding of mathematical concepts, and become more explicit in expressing their mathematical thinking (Francisco, 2013; Hoyles, 1985; Mercer et al., 1999). While small-group discussions allow students to discuss mathematical ideas, studies have shown that they may have difficulty establishing productive small-group discussions in regard to the subject matter (e.g., decimal numbers, scaling, or the function of a straight line) (Weber et al., 2008).

Francisco (2013) showed that when students in a small-group discussion critically reviewed their mathematical reasoning, they could build new, more stable forms of reasoning and thus develop their mathematical understanding. It has been shown that students who were able to discuss their ideas, for instance about a task with peers or within a small group before being presented with the correct answer in the whole class learned significantly more than those who were not (Miller et al., 2006). However, having students discuss mathematical ideas does not guarantee that meaningful learning will occur (Ball, 1991; McCrone, 2005). Moreover, it has been suggested that students’ lack of communication skills (Li & Campbell, 2008), students’ lack of productive collaboration (Hämäläinen & Vähäsantanen, 2011) and teachers’ lack of designing appropriate tasks (Gillies & Boyle, 2010), lower the effect of discussions in the classroom.

When small-group discussions are used in mathematics teaching a problem is often introduced, followed by student work on the task individually or in small groups, and thereafter the teacher orchestrates a whole-class discussion (Larsson & Ryve, 2012; Miller et al., 2006). Teachers are often faced with a wide range of student responses, and it can be difficult for the teachers to know how to deal with these responses in a productive way (Forslund Frykedal & Hammar Chiriac, 2011; Gillies & Boyle, 2010). Thus, facilitating discussions based on students’ existing mathematical thinking or ideas in the classroom is a demanding pedagogical task (Cengiz et al., 2011; Lampert, 2001). Teachers play a significant role in fostering and managing fruitful small-group discussions (Webb, 2009) as well as whole-class discussions (Cengiz et al., 2011; Larsson & Ryve, 2012). Cengiz et al. (2011) suggested that students need help in focusing their mathematic reasoning on important mathematical concepts.

Teachers may be unwilling to use small-group discussions as a teaching strategy if they perceive that group work contributes more to social skills than to student learning of subject matter. One key challenge for teachers seems to be why and how they should use student input during lessons in mathematics, in a way that can enhance the students’ learning in the whole class (Ball, 1993; McCrone, 2005). Small-group discussions seem to have two functions: as objective and as means – i.e., helping the students learn to develop social skills through working together, or serving as a means through which they can obtain new subject matter knowledge (Forslund Frykedal & Hammar Chiriac, 2012). In this paper, small-group as well as whole-class discussions are related to the function of discussions as means to develop new subject matter knowledge.

Theoretical Framework

The theoretical framework used in this study is the variation theory of learning (Marton, 2015). Within this theory, learning is seen as a change in the learner’s experience of a phenomenon (Marton, 2015; Marton & Booth, 1997). A fundamental idea is that learning must be directed towards something, an object of learning. Whether the students actually learn, depends among other factors, on how they experience the object of learning. To see an object of learning in a particular way, the learner needs to focus on certain aspects that are critical to a certain way of seeing, known as critical aspects (Lo, 2012).

What is possible for students to learn in mathematics is thus related to how they have the possibility to experience the mathematical content. Different ways of handling the content (an object of learning), for instance during small-group discussions, will generate different learning opportunities (Kullberg et al., 2017; Maunula, 2018; Runesson, 1999; Runesson, 2006). Runesson (1999) showed that the five teachers she studied enacted different objects of learning in their classrooms when teaching the same mathematical content. Variation theory posits that what the learner focuses on is in the foreground of their awareness and will be discerned, while what the learner does not pay attention to will recede into the background. Thus, differences in students’ experience of an object of learning can be explained by differences in aspects they notice during the lesson (Kullberg et al., 2016; Pang & Ki, 2016; Runesson, 2005). Seeing an object of learning in a new way entails a discernment of new aspects of that object of learning, which were not previously discerned. The teacher needs to be aware of which aspects of the content could be critical, but also of how to handle the content in a way such that students can discern critical aspects. If the object of learning is the geometric shape of a circle, one of the critical aspects could be to discern “the lengths” in the circle, and the values (features) of the critical aspect are diameter, perimeter, and radius. If these different values of “lengths” are varied, the dimension of variation is opened up (Pang & Lo, 2012; Runesson, 2006; Marton & Tsui, 2004). In this study, how the object of learning is handled (in terms of which aspects are brought to the fore during the discussions) is analysed through identifying qualitatively different enacted objects of learning (see Table 3).

The object of learning in terms of its critical aspects is dynamic and specific to a group of students in a certain situation at a certain time (Pang & Ki, 2016). Marton (2015) suggests that, in order to understand an object of learning in a certain way, students need to simultaneously discern its critical aspects in order to be able to constitute the intended meaning. To discern the critical aspects of an object of learning, students need to experience patterns of variation and invariance of these aspects. In a teaching situation, it is thus important for the teacher to try to identify how students experience the object of learning, in order to understand what they need to learn in terms of critical aspects to be highlighted.

The object of learning can be seen and analysed from three different perspectives. The first is the intended object of learning, which is from the teacher’s perspective. This refers to what the teacher has planned to be learned during the lesson. The second is the enacted object of learning, which is a perspective taken by the researcher. This relates to the analysis of what is made possible to learn during the lesson regarding how the specific subject matter is handled, and what critical aspects are focused on and dimensions of variation opened up. The third and final perspective is the lived object of learning, which is seen from the students’ perspective, and is directly related to what the students have learned. The intended object of learning in the present study concerns enlarging and reducing two-dimensional geometric figures. Through the course of teaching, enacted objects of learning are jointly constituted by the teacher and students during whole-class discussions, or by the students in small-group discussions. As stated previously, studies using variation theory as their theoretical framework have foremost analysed the enacted object of learning in whole-class teaching (e.g., Kullberg et al., 2017; Maunula, 2018; Pang & Marton, 2005). The present study analyses the enacted object of learning in both whole-class and small-group discussions.

Method

This study builds on previous findings from a learning study¹ on the same topic (Svanteson Wester, 2014; Svanteson Wester & Kullberg, 2020). During the learning study, the teacher team gained insights into their students’ experience of the object of learning from analysing one video-recorded small-group discussion from the whole-class lesson. The small-group discussion turned out to be crucial for that particular group’s learning. This paper further investigates what is made possible to learn from lessons with both small-group discussions and whole-class discussions.

To answer the research questions, video-recorded lessons with 33 small-group discussions and six whole-class discussions² following up the small-group discussions were analysed. In the analysis, the focus was on what enacted objects of learning were made possible to experience during small-group discussions (with no support from the teacher) and during whole-class discussions. The chosen topic is an area of mathematics in which a majority of students aged 12–16 encounter difficulties. De Bock et al. (1998) found that there is a widespread and strong tendency among 12-16-year-old students to believe that if the lengths of a two-dimensional figure are doubled, the area and volume are doubled as well. This phenomenon is called the “illusion of linearity”, which means that when students enlarge or reduce multidimensional geometric figures they intuitively assume that if all sides are doubled, the area and volume will also be twice as great. Studies also show that the dimension of the figure is crucial and that it is the transition from one-dimensional to multidimensional figures, and not primarily the shape of the figure, that causes problems (Ayan & Bostan, 2018; De Bock et al., 2002; Van Dooren et al., 2004). Testing 10-16-year students, Hilton et al. (2013) found that fewer than 10% of the students, regardless of age, could correctly answer a question about the change in the area of a butterfly when its length and width were doubled. Between 60 and 78% of the participating students replied that the area would be doubled as well.

Lesson Design

The small-group discussions played a central role in the design of the whole-class lessons and were systematically embedded in the lessons (see Table 1). The two lessons in each class were jointly planned by the researcher (author of this paper) and the teacher team. The small-group discussions were integrated with the purpose of extending the students’ possibilities to investigate the object of learning. For each small-group discussion session during the two lessons, the teacher followed up these discussions afterwards in the whole class.

Table 1. Organization of the mix of whole-class and small-group discussions and tasks during Lessons 1 and 2.

Lesson 1					Lesson 2
Whole- class	Whole- class	Whole- class	Group	Whole- class	Group	Whole- class	Group	Whole- class
The photo	The plus sign	The square	The sheet of paper	The sheet of paper	The pizza	The pizza	The heart	The heart

The critical aspects (identified in the previous learning study) were used as means when planning the lessons. The intended object of learning was enlarging and reducing two-dimensional geometric figures. The critical aspects were: a) discern lengths in a two-dimensional geometric figure; b) discern both the change in lengths and the change in the area and its relation when enlarging and reducing the figures; and c) discern the meaning of similarity and, based on this, handle the scale factor correctly. The intended object of learning is the same for the lessons throughout all small-group tasks.

There was a progression in terms of the geometric figures used in the lessons, from a rectangle, to a circle, to an irregular figure (non-symmetrical heart) (see Table 1). A pre-test was conducted in each class one week before Lesson 1, and a post-test was conducted in each class one week after Lesson 2.

The analysis of the lessons focused on qualitative differences in what was made possible to discern related to critical aspects and dimensions of variation during small-group discussions as well as whole-class discussions. In this paper, one of the three group tasks, the “pizza task”, is used to illustrate the analysis: “Kalle and Pelle go to the pizza restaurant. They are very hungry and want as much pizza as possible. Should they have a child’s pizza each or should they share a family pizza, or does it not matter what they choose? The child’s pizza is a reduced family pizza by a scale of 1:2. Show your solution and reasoning”.

Data

Two teachers at the same school in a larger city in Sweden participated in the study. Each teacher conducted two consecutive lessons (2 × 60 min, Lessons 1 and 2) in Grade 8 (Classes A and C). Before the teaching, the classes were divided into small groups. The teachers seated the students as usual, in heterogeneous groups, with the aim of mixing three to four students with different mathematical knowledge and courage to express themselves verbally. Members of a group were seated at the same table during the lessons and worked collaboratively on three tasks. Table 2.

Table 2. Video-recorded data.

Class	N	Recorded lessons	N groups	N group tasks	Recorded group discussions	Recordings in total
A	20	2 x 60 min.	6	3	18 x 10–15 min.	360 min.
C	18	2 x 60 min.	5	3	15 x 10–15 min.	320 min.

During the lessons, one video camera captured the whole classroom and six video cameras with microphones captured the groups. The video-recorded material consists of four 60-minute lessons and 33 student-group discussions of approximately 10–15 min each, resulting in approximately 240 min of whole-class discussions and 440 min of student small-group discussions. Since the teachers planned the implementation of the specific content together, it is possible to analyse similarities and differences between how the object of learning was enacted in the whole-class and small-group discussions.

A pre-test was conducted one week before, and a post-test one week after, the lessons. The pre-test aimed to capture the students’ experience of the object of learning. The difference in learning outcomes between the pre- and post-tests was used to analyse what the students had learned during the lessons, i.e., the lived object of learning. The test consisted of nine tasks, seven of which involved enlarging different geometric figures from a given scale factor, while the final two were of a more demanding character with the students being asked to discern several critical aspects simultaneously.

Analysis of Data

The unit of analysis in this study is lesson sequences in which the intended object of learning, enlarging, and reducing geometric figures was discussed, in small-group and whole-class discussions. In order to determine what objects of learning were enacted during the discussions, the analysis was carried out in the following way: First, all small-group and whole-class discussions were transcribed verbatim. Second, the transcripts were analysed in order to search for sequences in which critical aspects of the intended object of learning were discussed and students had conflicting views. The third step involved identifying specific enacted objects of learning that related to the critical aspects elaborated and dimensions of variation opened up. Differences and similarities in how the subject matter was handled in the groups, in regards to critical aspects elicited, made it possible to identify different enacted objects of learning, (EOL1, EOL2 etc., see Table 3, Appendix 1a and Appendix 1b). For every small-group task discussion, particular enacted objects of learning were identified, as the different small-group discussions highlighted certain critical aspects of the content and opened up more or fewer dimensions of variation, which resulted in differentiated enacted objects of learning. Four enacted objects of learning (EOLs 1-4) were identified in relation to the small-groups’ different discussions about the “sheet of paper task” (rectangle) (see Appendix 1a), five (EOLs 1-5) in relation to the small-groups’ different discussions about the “pizza task” (circle) (see Table 3), and six (EOLs 1-6) in relation to the small-groups’ different discussions about the “heart task” (irregular figure) (see Appendix 1b). Only a few small-groups enacted an object of learning, during their discussions, that corresponded with the intended object of learning (see Table 4). The intended object of learning corresponded to: EOL 4 (see Appendix 1a) in the sheet of paper task; EOL 5 (see Table 3) in the pizza task; and EOL 6 (see Appendix 1b) in the heart task. The analysis was carried out repeatedly.

Table 3. Five different enacted objects of learning identified in the pizza task – a reduction of a circle by scale factor 1:2.

Enacted OL 1	Enacted OL 2	Enacted OL 3	Enacted OL 4	Enacted OL 5 = Intended OL
The shape of a circle is not in the foreground but is instead replaced by a square.	The shape of a circle is not in the foreground but is instead replaced by a square.	The shape of a circle is in the foreground initially.	The shape of a circle is in the foreground initially.	The shape of a circle is in the foreground.
The change in area, not the length, is in the foreground.	Lengths and change in lengths compared to the scale factor are focused on (also generalized by scale factor 1:4).	The circle is replaced with a square.	The change in area and partly the change in lengths are in focus.	A separation of lengths, change in lengths, and change in area in the shape of a circle appears.
The area is focused on as a quarter as large or four times smaller, by dividing the square into four equal parts with sustained proportions.	The meaning of a correct proportional image is not focused on.	The change in area is in the foreground and is focused on as four times smaller, by dividing the circle into four equal sectors with support from the square shape.	The change in area is focused on as four times smaller, by initially dividing the circle into four equal sectors with support from the square shape. Then, the circle sectors are replaced with four small circles.	The change in lengths of the circle is expressed as half the radius or half the diameter, and the change in area as a quarter as large or four times smaller.
The lengths or changes in length in a circle are not in focus; nor is the meaning of a correct proportional image.		The change in length in a circle is not in focus; nor is the meaning of a correct proportional image.	The meaning of a correct proportional image is partly in focus.	The meaning of a correct proportional image is in focus.

Table 4. The enacted objects of learning (EOLs) identified in whole-class and small-group discussions. Dark-grey marks enacted objects of learning (EOLs) in the small-group discussions that correspond to the intended object of learning (IOL) for the task.

Task	The photo IOL = EOL 4	The plus sign IOL = EOL 4	The square IOL = EOL 4	The sheet of paper (rectangle) IOL = EOL 4		The pizza (circle) IOL = EOL 5		The heart (irregular shape) IOL = EOL 6
Whole-class/Group	Whole-class	Whole-class	Whole-class	Group	Whole-class	Group	Whole-class	Group	Whole-class
W-Class A	EOL 1	EOL 4	EOL 4		EOL 4		EOL 5		EOL 6
Gr. 1A				EOL 3		EOL 4		EOL 4
Gr. 2A				EOL 4		EOL 4		EOL 4
Gr. 3A				EOL 2		EOL 4		EOL 6
Gr. 4A				EOL 2		EOL 2		EOL 1
Gr. 5A				EOL 2		EOL 1		EOL 1
Gr. 6A				EOL 3		EOL 1		EOL 3
W-class C	EOL 1	EOL 2	EOL 4		EOL 4		EOL 5		EOL 6
Gr. 7C				EOL 3		EOL 5		EOL 4
Gr. 8C				EOL 4		EOL 4		EOL 5
Gr. 9C				EOL 4		EOL 4		EOL 2
Gr. 10C				EOL 4		EOL 1		EOL 6
Gr. 11C				EOL 1		EOL 1		EOL 4

Besides the analysis of lesson sequences, pre-and post-test results were analysed to explore relationships between student learning outcomes (lived object of learning) and the enacted objects of learning during the lessons.

Results

An overview of the identified enacted objects of learning in small-group and whole-class discussions is presented first. In the second part, examples of enacted objects of learning in the small-group discussions are illustrated using excerpts from the “pizza task”. The third and fourth parts show students’ learning outcomes as measured in post-tests, and empirical examples regarding what was made possible to learn in the whole-class discussions.

The results show (see Table 4) qualitative differences between the groups concerning what objects of learning were enacted. During small-group discussions, only six of eleven groups enacted an object of learning that corresponded to the intended object of learning. However, the analysis shows that in the whole-class discussion that followed the small-group discussions, the enacted objects of learning corresponded to the intended object of learning. This was not the case in the first two whole-class discussions, however, which served as an introduction (the photo, the plus sign).

Enacted Objects of Learning in the Small-group Discussions About the Pizza Task

In this study the analysis of the second group task, the pizza task, will serve as an example of the analysis. A brief description of the analysis of the first and third group tasks is provided first.

For the first group task (the sheet of paper task), the analysis showed that four different objects of learning were enacted. In one small-group discussion (11C) (see Table 4) the students handled the subject matter in a way that lengths in the figure were highlighted, but neither change in length nor change in area was explicitly highlighted (EOL 1) (see Appendix 1a). In other groups (e.g., 2A and 8C), the students handled the subject matter in a way that highlighted changes in the lengths in two dimensions and change in area in relation to different scale factors (EOL 4). For the third task, the heart task (irregular figure), in which a heart figure was to be enlarged by scale factor four, six different objects of learning were identified. For example, in one small-group discussion (9C) the enacted object of learning (EOL 2) entailed the students enclosing the heart in a rectangle and then enlarging the length and width of this rectangle, but they did not focus on the change in area of the heart shape explicitly. One of the groups (10C) enacted an object of learning that corresponded to the intended object of learning (EOL 6), highlighting both the lengths and the area of the heart. The heart’s lengths were enlarged by a factor of four, and the students argued that a correct enlargement of the heart on the paper meant that when the total of the lengths of an object increases by a factor of four the area will be 16 times greater, like in a rectangle, even if it is hard to find the lengths of an irregular shape.

In the pizza task, five different objects of learning were enacted (see Table 3). One of them (EOL 1) was characterized by the students not having the circular shape in their foreground when trying to make a correct scaling of the pizza. Instead, they replaced the circle with a square (e.g., Groups 5A and 10C). The students did not handle the lengths in the circle at all. The change in area was expressed as four times smaller by dividing the square into four equal parts with sustained proportions. Their answer was correct: the child’s pizza was a fourth of the size of the family pizza, and the boys would get more pizza if they shared a family pizza. But, the students in this group did not handle lengths in a circle or the meaning of a proportional image, which were aspects of the intended object of learning. In another group (7C), the students did handle the subject matter in a way that corresponded to the intended object of learning (EOL 5). The shape of a circle was in focus throughout the discussion, lengths in a circle were focused on as half the diameter or half the radius, and the change in area was expressed as a quarter as large or four times smaller. Another group divided the circle into four equal sectors of the circle, and the opportunity to discern the lengths in the circle was lost (EOL 4).

The following excerpts illustrate how two different enacted objects of learning (EOL 2 and EOL 5) emerged in two different groups when solving the pizza task. The first group (4A) enacted EOL 2, replacing the pizza’s circular shape with a square. One of the students (Student 2) divided the lengths in both dimensions (length and width) by two in order to get the size of a child’s pizza, saying “This is the family pizza” and “This is the child’s pizza, so then you get more [pizza] on a family pizza”, and marked two quarters of the family pizza (see Figure 1).

Figure 1. The first part of the solution from Group 4A in the pizza task.

Student 1, on the other hand, asked why there were quarters instead of halves and expressed that it should be enough to divide the family pizza into two halves. Student 3 pointed at the picture (see Figure 1) in which a quarter of the rectangle was marked and showed that she agreed with Student 2, but did not discuss this further:

S1: But why do you have to divide it into four?

[…]

S1: You’ve divided it into four? You could just divide it into two.

[…]

S3: When you’re doing it by scale factor 1:2, you have to divide it like that.

Thereafter, Student 2 drew another pizza in the shape of a rectangle and marked four quarters of the pizza by dividing the length and width by two (see Figure 2). Student 3 again showed that she was in agreement with Student 2:

S2: If it’s four, you divide it into … here’s one, two.

S3: Because you show how much of it [measuring one of the sides].

Student 2 drew another rectangle, in which she marked a proportional image of the rectangle by scale factor 1:4 (see Figure 3). The student kept the rectangle invariant and varied the scale factor, thus making it possible to discern change in lengths in the rectangle. The contrast between the different scale factors of 1:2 and 1:4 provided an opportunity for the group to discern lengths and change in lengths, but also to separate the change in lengths from the change in area:

Figure 2. The second part of the solution from Group 4A in the pizza task.

Figure 3. The third part of the solution from Group 4A in the pizza task.

Again, Student 1 asked why Student 2 had divided the length and width when reducing the rectangle. Most likely, Student 1 did not have the length and width in the foreground when handling scale factor 1:2. Student 2, on the other hand, raised the importance of focusing on the lengths in relation to the scale factor, and varied the scale factor by showing what happened to the lengths and area of a rectangle when it was instead reduced by a scale of 1:4. However, the student group adhered to the shape of a rectangle, which means that the opportunity to discern length in a circle was lost. Although the group had a relatively rich discussion regarding the reduction of rectangles, they missed out on the intended object of learning. Two ways of seeing the scale factor emerged during their discussion: one in which the scale factor was about change in lengths in two dimensions; and another in which it was about seeing the new area in relation to the scale factor – i.e., the area was divided into two pieces.

Compared to Group 4A, described above, Group 7C, in the way they handled the content, showed that they had enacted an object of learning (EOL 5) (see Table 3) which corresponded to the intended object of learning. One of the students (Student 4) started the small-group discussion by saying that the correct answer must be that the two boys should share a family pizza to get as much pizza as possible. Another student (Student 5) asked why:

S4: They’ll have the family pizza.

S5: Why?

S4: It’s twice as big in both length and width than the child’s pizza. It has the same shape, but not the same size. It’s one size bigger.

Student 5 drew a child’s pizza (see Figure 4) and a family pizza around the child’s pizza, using lengths in the child’s pizza as the diameter and radius, and explained what she saw in the drawing. She said “Well, a family pizza is like this then. It’s double in length compared to the child’s pizza”, and pointed at the radius:

Figure 4. The solution from Group 7 C in the pizza task.

Insert Figure 4 here.

Student 5 drew the conclusion and said “Well, it will be equal, so it doesn’t matter, so two of those (points at the child’s pizza) are as much as one of those (points at the family pizza)” (see Figure 4). But a few seconds later Student 5 said, somewhat uncertainly, “It’s double, isn’t it or what?”. However, the students in the excerpt below seemed to be unsure as to whether or not the area would be double if the lengths are doubled:

S5: Well, a family pizza is like this then. It’s double in length. [compares the two pizzas’ sizes]

S4: And double in width.

S5: Then, it’s just as much. So, it doesn’t matter. Thus, two of those [points at the child’s pizza] are just as much as one of those [points at the family pizza]. Because it’s double, isn’t it or what?

S4: That half doesn’t fit in there [points at the half of the family pizza and at the whole child’s pizza].

S4: A quarter. A whole. It can fit in a fourth [points at a quarter of the family pizza and the whole child’s pizza].

Student 4, with her reply “and double in width”, indicated that she had seen two dimensions in the circle. The excerpts illustrate a contrast between two different ways of seeing the object of learning in the group. Two students in the group (Students 6 and 7) pointed out that the answer was the same if the pizza was in the shape of a rectangle, saying “it could also be a rectangle”. Student 5 drew a rectangle, and they further discussed that it is easier to see that two child pizzas fit in a family pizza when the pizza is in the form of a rectangle. During the small-group discussion, it was possible to discern all the critical aspects of the intended object of learning: the lengths in a circle, the change in lengths, the area and the change in area, as well as a proportional image of the geometric form, the circle.

In the next part, the small-groups’ learning opportunities from small-group discussions are compared with the students’ learning outcomes.

Student Learning Outcomes – the Lived Object of Learning

When students’ learning outcomes (the lived objects of learning) from the post-tests are compared to the enacted objects of learning from the small-group discussions, it appears that the students, after all, had rich possibilities to learn what was intended. On the post-test, a majority of the students showed that they had discerned the critical aspects of the intended object of learning; i.e., the students’ lived object of learning corresponded with the intended object of learning (see Appendix 1c). The results of the post-test showed that, despite different enacted objects of learning in the groups, the students’ learning outcomes were similar. They significantly increased their performance compared to the pre-test (Appendix 1c). For example, there were groups (5A, 6A, 11C) which, although they had enacted an object of learning, (e.g., EOL 1, EOL 2; see Appendix 1c) which was not the intended, performed similar at post-test to the groups (2A, 7C and 8C) which had enacted objects of learning closer to or corresponding to the intended object of learning (e.g., EOLs 4, 5, 6).

In sum, from the theoretical perspective taken, the enacted objects of learning in the groups did not reflect the lived object of learning. The learning outcomes are seemingly high, compared to the enacted objects of learning in the different groups. How can the similarities in learning outcomes be explained? The next part discusses the analysis of how the subject matter was handled during whole-class discussions, and offers examples of enacted objects of learning.

Enacted Objects of Learning in Whole-class Discussion

During the lesson, whole-class discussions followed each small-group discussion. The following excerpt illustrates how the subject matter was handled when discussing the pizza task in the whole class (Class C). The teacher listened to the groups’ different answers and solutions. All the groups had arrived at the correct answer: the child’s pizza would be a fourth of the size of the family pizza, and the boys should choose the family pizza. Some of the groups highlighted that they had used a pizza shaped like a square instead of a circle. The following excerpt illustrates that the teacher noticed this:

T2: So, if I try to draw a conclusion about what you’ve told me, it’s that most of the groups have chosen to use pizzas as rectangles; even though I told you the pizza should be a circle, you disregarded that and used a rectangle instead because it was easier. I think that’s what you did.

S11: But we started with a circle, but it was hard.

[…]

T2: Ok, I’ll draw. If this is a family pizza.

[…]

T2: Well, if this is the original picture and this is the family pizza, here’s the diameter; how long is the diameter of the family pizza then? Like this? [The teacher draws diameters]

S6: Yes.

[…]

T2: And the radius will be half. All the lengths will be half. [The teacher draws a picture]

The teacher highlighted the length in a circle when he had identified that the students had replaced the circle with a rectangle and actually lost the opportunity to discern the lengths in a circle, which was the intended object of learning. In the whole-class discussion, the subject matter was handled in such a way that the critical aspects the groups had not highlighted in their discussions were now highlighted. The teacher listened to and asked questions of the groups when they reported how they had solved the task, highlighting the difference between the students’ lived objects of learning and the intended object of learning. The teacher used the way the students had handled the subject matter during the small-group discussions to identify critical aspects not yet discerned, and from this, enacted objects of learning during the whole-class discussions that corresponded to what was intended. The whole-class discussions most likely played a crucial role in compensating for differences between the groups regarding what aspects of the intended object of learning were possible to discern.

Discussion and Conclusions

This paper investigates what is made possible to learn when small-group discussions are used as an integrated part of whole-class teaching. The aim is to contribute to the understanding of how small-group and whole-class discussions contribute to students’ learning of a specific subject matter.

The first research question concerned what objects of learning were enacted in each small-group and whole-class discussion. The analysis showed that in a majority of the small-group discussions an object of learning at a lower level than the intended one was enacted (e.g., EOLs 1, 2), and only a few groups enacted an object of learning that corresponded to the intended one (see Table 4). Hence, there was a low possibility to learn the intended object of learning from the small-group discussions only in both classes. Further, the analysis shows not only that there were different enacted objects of learning in the groups, but also what these differences consisted of. Objects of learning enacted in a discussion can be understood in relation to which critical aspects are noticed and which are elaborated. For example, in one group (4A) the students noticed that the lengths should be divided by two when reducing the pizza by scale factor 1:2, but did not elaborate the lengths in relation to the shape of a circle (only a rectangle). In Group 7C, on the other hand, the students elaborated the lengths in the circle and highlighted both the diameter and the radius. Thus, features of the critical aspect “discern lengths in a circle” were opened up as a dimension of variation in Group 7C but not in Group 4A. In comparison, the whole-class discussions enacted objects of learning that corresponded to the intended one (with one exception). The second research question concerned how the enacted objects of learning correspond to student learning outcomes. When the students’ learning outcomes on the post-test were compared to their learning opportunities from the small-group discussions (the enacted objects of learning), the learning gain was better than expected. The different enacted objects of learning during the small-group discussions did not reflect what the students had actually learned, and the post-test results of the different groups were surprisingly similar (see Appendix 1c).

A majority of the small-group discussions in the study did not enact an object of learning that corresponded to the intended object of learning. On the other hand, the study indicates that the whole-class discussions, which followed the small-group discussions, were more productive, than the small-group discussions, in relation to learning opportunities (EOL) (see Table 4). This is in line with Miller et al. (2006), who showed that students who discussed their ideas in groups before the whole-class discussion learned significantly more than those who did not, and with Ryve et al. (2013), who showed that whole-class discussions preceded by small-group discussions made the whole-class discussions more refined in relation to the subject matter. One possible explanation for the fact that the learning opportunities in this study seemed to increase in whole-class discussions may be that the teachers were aware of the critical aspects of the intended object of learning (Svanteson Wester, 2014; Svanteson Wester & Kullberg, 2020). The teachers’ awareness of critical aspects has meant that they were prepared concerning how to respond to student responses about the subject matter taught during the lesson, and how to facilitate whole-class discussions based on the students’ current thinking about the subject matter being taught. Critical aspects may have supported the teachers forming the discussion in relation to students’ responses and what was intended to be learnt. A public exchange of systematic variation of aspects (Al-Murani & Watson, 2009) seemed to be used to guide the class toward a deeper understanding regarding the subject matter. How to respond to student responses about the subject matter taught during lessons (Forslund Frykedal & Hammar Chiriac, 2011; Svanteson Wester, 2010) and how to facilitate discussions based on students’ views about the subject matter being taught (Cengiz et al., 2011; Lampert, 2001) have been emphasized in previous studies as challenging for teachers in mathematics teaching. The systematic shift between whole-class and small-group discussions during the lessons seems to have played a pivotal role with respect to the students’ learning opportunities. The small-group discussions were used by the teacher as a means to create a space in which the students had an opportunity to explore the object of learning. The whole-class discussions made it possible to further elaborate the object of learning in a public domain, and the enacted object of learning became more explicit. This elaboration, most likely, made it possible for the students to identify critical aspects they had not yet discerned (Pang & Ki, 2016).

As stated earlier, the small-group discussions generated different opportunities to learn what was intended, which may not be surprising. It has been suggested that discussions about mathematical ideas during lessons do not guarantee meaningful learning (e.g., Ball, 1991; McCrone, 2005). It seems difficult to establish productive small-group discussions related to a specific subject matter in mathematics (Weber et al., 2008). However, the results of this study add to this research by showing not only that there were different opportunities to learn what was intended but also show empirically why the small-group discussions generated different opportunities to learn (different EOLs).

From the theoretical point of view taken, critical aspects need to be brought to the fore for students. An inference that can be made is that this was not done sufficiently in the small-group discussions. Studies using variation theory as an analytical framework for analysing students’ possibilities to learn from teaching have primarily analysed whole-class teaching only (e.g., Kullberg et al., 2017; Pang & Marton, 2005; Runesson, 1999), or isolated small-group discussions (Berge & Ingerman, 2017). This study adds to this body of research, offering knowledge about learning opportunities that emerge in both small-group discussions and in whole-class discussions and thereby treating the lesson as a whole. Runesson (1999) showed that teaching the same subject matter in different classes provided different learning opportunities; i.e., different objects of learning were enacted in the classes. This study instead shows that the two teachers who planned the lessons together also succeeded in enacting the same object of learning in the whole-class discussions, and that the student learning outcomes in the two classes (Classes A and C) corresponded (see Appendix 1c). This is in line with Kullberg (2010) who showed that when the same critical aspects were made possible to experience, student learning outcomes were similar. Furthermore, the way of analysing the enacted objects of learning in small-group and whole-class discussions used in the study can be seen as a methodological contribution.

This study has implications for practice, illustrating not only what is made possible for students to learn from small-group discussions, but also what is needed in order for students to learn what is intended in lessons with small-group discussions as an integrated part of whole-class discussions. There are indeed many other factors – e.g., group composition, social norms, teacher’s support for the group, and patterns of interaction – that are of importance for whether small-group discussions will be productive (Webb, 2009), however these are not studied in this paper.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 A learning study, like a theory-based lesson study (Lewis et al., 2006; Robinson & Leikin, 2012), is a collaborative and iterative process in which teachers plan, enact, and revise a single lesson in order to improve teaching and learning, but using learning theory (variation theory) to design and enact the lessons (see Cheng & Lo, 2013 for details on learning study).

2 The whole-class discussions in the introduction part of the lessons were not analysed (see Table 4).

Appendices

Appendix 1A

Four different enacted objects of learning created in the sheet of paper task (the rectangle) are illustrated when reduced by scale factors 1:2, 1:4, and 1:3.

Table

Enacted OL 1	Enacted OL 2	Enacted OL 3	Enacted OL 4 (IOL = EOL 4)
Neither the change in length nor the change in area is discussed or in the foreground explicitly.	The scale factors’ relation to change in length in two dimensions (length and width) is discussed.	The scale factors’ relation to change in length in two dimensions (length and width) and to change in area is discussed. The different scale factors are compared, but the relationships between them are not discussed.	The scale factors’ relation to change in length in two dimensions (length and width) and to change in area is discussed. The different scale factors are compared, and the relationships between them are discussed.

Appendix 1B

Five different enacted objects of learning created in the heart task (the irregular geometric figure) when enlarging the heart by scale factor 4:1 are illustrated.

Table

Enacted OL 1	Enacted OL 2	Enacted OL 3	Enacted OL 4	Enacted OL 5	Enacted OL 6 (IOL = EOL 6)
The figure’s length and area are not separated. The heart is enclosed by a rectangle, and the rectangle’s length and width as well as the area increase by a factor of 4.	The figure’s length and area are separated. The heart is enclosed by a rectangle, and the rectangle’s length and width are discussed and increase by a factor of 4. The change in area is not explicitly discussed.	The figure’s length and area are separated. The heart is enclosed by a rectangle, and the rectangle’s length and width are discussed and increase by a factor of 4. The change in area is discussed, focusing on the fact that the area is going to be “gigantic”.	The figure’s length and area are separated. The heart is not enclosed by a rectangle. “All lengths” of the heart increase by a factor of 4. The change in area is not explicitly discussed.	The figure’s length and area are separated. The heart is not enclosed by a rectangle. “All lengths” of the heart increase by a factor of 4. The change in area is discussed, and it is highlighted that it has increased by a double scale factor; i.e., eight times greater (Svanteson Wester, 2014; Svanteson Wester & Kullberg, 2020).	The figure’s length and area are separated. The heart is not enclosed by a rectangle. “All lengths” of the heart increase by a factor of 4. The change in area is discussed, and it is highlighted that it will be 16 times greater.

Appendix 1C

Results on individual and group level, and the groups’ enacted object of learning (EOL) for each task. Maximum nine points.

Table

Group	Student	Pre-test. Correct answers.	Pre-test. Mean for the group.	Post-test. Correct answers.	Post-test. Mean for the group.	EOL for “the paper”. Intended OL = EOL 4.	EOL for “the pizza”. Intended OL = EOL 5.	EOL for “the heart”. Intended OL = OL 6.
1 A	Student 1	2	1.5	6	7.0	EOL 3	EOL 4	EOL 4
	Student 2	2		9
	Student 3	2		7
	Student 4	0		6
2 A	Student 5	0	2.0	9	7.75	EOL 4	EOL 4	EOL 4
	Student 6	3		9
	Student 7	1		7
	Student 8	4		6
3 A	Student 9	9	3.67	9	7.67	EOL 2	EOL 4	EOL 6
	Student 10	1		8
	Student 11	1		6
4 A	Student 12	1	1.67	5	6.33	EOL 2	EOL 2	EOL 1
	Student 13	4		7
	Student 14	0		7
5 A	Student 15	1	1.25	9	7.25	EOL 2	EOL 1	EOL 1
	Student 16	3		5
	Student 17	1		6
	Student 18	0		9
6 A	Student 19	0	1.67	7	7.33	EOL 3	EOL 1	EOL 3
	Student 20	3		6
	Student 21	2		9
7 C	Student 22	2	1.75	9	7.25	EOL 3	EOL 5	EOL 4
	Student 23	0		5
	Student 24	0		6
	Student 25	5		9
8 C	Student 26	3	2.75	8	8.0	EOL 4	EOL 4	EOL 5
	Student 27	3		9
	Student 28	5		9
	Student 29	0		6
9 C	Student 30	2	2.67	9	8.33	EOL 4	EOL 4	EOL 2
	Student 31	4		7
	Student 32	2		9
10 C	Student 33	2	1.67	9	7.67	EOL 4	EOL 1	EOL 6
	Student 34	0		6
	Student 35	3		8
11 C	Student 36	0	0.67	7	8.33	EOL 1	EOL 1	EOL 4
	Student 37	1		9
	Student 38	1		9