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

Figures & data

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

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.

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

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

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.

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

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.

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.

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

Svanteson Wester, J. (2014). Hur kan dubbelt så långt bli fyra gånger större? [How can twice as long be four times bigger?] Licentiate thesis. Gothenburg: Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg.

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Svanteson Wester, J., & Kullberg, A. (2020). Understanding the relationship between length and area when changing the size of a two-dimensional geometric figure. Nordic Studies in Mathematics Education, 25(1), 89–109.

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