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This article explores Norwegian lower primary teachers’ views about good mathematics teaching as revealed in a focus group interview at the end of a two-year school-based professional development program. Analyses of the empirical data indicate three main categories of findings: the teachers’ facilitation of learning, the students’ thinking in and about mathematics, and the use of teaching aids in teaching. The results are discussed in relation to other Nordic studies and possible implications are also provided.

1.
Good mathematics teaching at lower primary school

level

Marianne Maugesten

To cite this version:

Marianne Maugesten. Good mathematics teaching at lower primary school level. Eleventh Congress of

the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht,

Netherlands. �hal-02430103�

HAL Id: hal-02430103

https://hal.archives-ouvertes.fr/hal-02430103

Submitted on 7 Jan 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est

archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents

entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non,

lished or not. The documents may come from émanant des établissements d’enseignement et de

teaching and research institutions in France or recherche français ou étrangers, des laboratoires

abroad, or from public or private research centers. publics ou privés.

level

Marianne Maugesten

To cite this version:

Marianne Maugesten. Good mathematics teaching at lower primary school level. Eleventh Congress of

the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht,

Netherlands. �hal-02430103�

HAL Id: hal-02430103

https://hal.archives-ouvertes.fr/hal-02430103

Submitted on 7 Jan 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est

archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents

entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non,

lished or not. The documents may come from émanant des établissements d’enseignement et de

teaching and research institutions in France or recherche français ou étrangers, des laboratoires

abroad, or from public or private research centers. publics ou privés.

2.
Good mathematics teaching at lower primary school level

Marianne Maugesten

Østfold University College, Norway; [email protected]

This paper explores Norwegian lower primary teachers’ views about good mathematics

teaching as revealed in a focus group interview at the end of a two-year school-based

professional development program. Analyses of the empirical data indicate three main

categories of findings: the teachers’ facilitation of learning, the students’ thinking in and

about mathematics and the use of teaching aids in teaching. The results are discussed in

relation to other Nordic studies and possible implications are also provided.

Keywords: Mathematics teachers' discussion, lower primary school, good mathematics

Introduction and theoretical background

This study investigates teacher views about good mathematics teaching. Several studies

attempt to identify the components of good mathematics teaching without finding a clear

answer (Cai, Kaiser, Perry, & Wong, 2009; Franke, Kazemi, & Battey, 2007; Hiebert &

Grouws, 2007; Kilpatrick, Swafford & Findell, 2001). A challenge is that cultural as well as

political differences influence mathematics teaching. Views about the role of the teacher,

about the subject in school and society, and about learning differ across cultures (Cai et al.,

2009). In the Chinese context, for instance, mathematics teaching is teacher-oriented and

exam-oriented, and teachers are more focused on the students and their learning than on

themselves and their teaching (Li, 2011). In the Nordic context, Fauskanger, Mosvold,

Valenta and Bjuland (2018) conducted a study in which upper primary school teachers’ views

on good mathematics teaching were revealed through group interviews at the start of a major

professional development project. The teachers referred to their role as teaching facilitators by

having good structure, classroom management, and the possibility to differentiate using

different types of assignments, which both motivates the students and invites more and

diverse solutions. According to Fauskanger et al. (2018), good teaching was also about

motivated, engaged, creative and curious students. In another study, Fauskanger (2016)

investigated views on the ingredients of good mathematics teaching among lower and upper

primary school mathematics teachers who participated in a professional development

program. These teachers felt that student response was the most decisive factor for high

quality teaching. They emphasized teacher qualities such as enthusiasm and attitude towards

the subject rather than the teachers’ own knowledge. Hemmi and Ryve (2015) studied

Swedish and Finnish teacher educators’ views of good mathematics teaching through focus

group interviews and individual interviews. There were many apparent similarities between

Sweden and Finland, but the Finnish teacher educators emphasized clear presentation of

mathematics for the whole class, routines for mental arithmetic and homework, and clear

learning goals for each class, while the Swedish teacher educators referred to the relationship

with each individual child, building on the students’ capabilities and finding mathematics in

everyday situations. In three studies carried out among Finnish student teachers (at lower

Marianne Maugesten

Østfold University College, Norway; [email protected]

This paper explores Norwegian lower primary teachers’ views about good mathematics

teaching as revealed in a focus group interview at the end of a two-year school-based

professional development program. Analyses of the empirical data indicate three main

categories of findings: the teachers’ facilitation of learning, the students’ thinking in and

about mathematics and the use of teaching aids in teaching. The results are discussed in

relation to other Nordic studies and possible implications are also provided.

Keywords: Mathematics teachers' discussion, lower primary school, good mathematics

Introduction and theoretical background

This study investigates teacher views about good mathematics teaching. Several studies

attempt to identify the components of good mathematics teaching without finding a clear

answer (Cai, Kaiser, Perry, & Wong, 2009; Franke, Kazemi, & Battey, 2007; Hiebert &

Grouws, 2007; Kilpatrick, Swafford & Findell, 2001). A challenge is that cultural as well as

political differences influence mathematics teaching. Views about the role of the teacher,

about the subject in school and society, and about learning differ across cultures (Cai et al.,

2009). In the Chinese context, for instance, mathematics teaching is teacher-oriented and

exam-oriented, and teachers are more focused on the students and their learning than on

themselves and their teaching (Li, 2011). In the Nordic context, Fauskanger, Mosvold,

Valenta and Bjuland (2018) conducted a study in which upper primary school teachers’ views

on good mathematics teaching were revealed through group interviews at the start of a major

professional development project. The teachers referred to their role as teaching facilitators by

having good structure, classroom management, and the possibility to differentiate using

different types of assignments, which both motivates the students and invites more and

diverse solutions. According to Fauskanger et al. (2018), good teaching was also about

motivated, engaged, creative and curious students. In another study, Fauskanger (2016)

investigated views on the ingredients of good mathematics teaching among lower and upper

primary school mathematics teachers who participated in a professional development

program. These teachers felt that student response was the most decisive factor for high

quality teaching. They emphasized teacher qualities such as enthusiasm and attitude towards

the subject rather than the teachers’ own knowledge. Hemmi and Ryve (2015) studied

Swedish and Finnish teacher educators’ views of good mathematics teaching through focus

group interviews and individual interviews. There were many apparent similarities between

Sweden and Finland, but the Finnish teacher educators emphasized clear presentation of

mathematics for the whole class, routines for mental arithmetic and homework, and clear

learning goals for each class, while the Swedish teacher educators referred to the relationship

with each individual child, building on the students’ capabilities and finding mathematics in

everyday situations. In three studies carried out among Finnish student teachers (at lower

3.
primary school level), Kaasila and Pehkonen (2009) looked at students teachers’ views of

good mathematics teaching. They believed that teachers needed to be goal-oriented, listen to

the students’ thinking and show flexibility when unexpected episodes arise. The student

teachers pointed out that teachers should have knowledge of varied work methods, base their

teaching on the students’ day-to-day experiences and have a particular focus on problem

solving. Continuous assessment and development of socio-mathematical norms were

considered important elements of good mathematics teaching.

Teaching mathematics is complex and researchers have attempted to distinguish the different

aspects to identify main practices. These are referred to as core practices (McDonald et al.,

2013) or high-leverage practices (Forzani, 2014). This study focusses on the Nordic context,

therefore core practices are not discussed further.

The study in this paper is based on a group of Norwegian teachers at lower primary school

level who, together with a teacher educator in a focus group interview, reflected on their own

mathematics teaching at the end of a two-year school-based mathematics professional

development program. The content in this development program were decided by the

headmaster in cooperation with representatives of the mathematics teachers at the school.

Among the themes were numeracy, different approaches to the four arithmetical operations

and how to lead productive mathematical discussions. This study does not measure the effect

of the program, but it can be assumed that the teachers’ descriptions of good mathematics

teaching has been influenced by them trying out exercises and activities in their own classes

and by improved research-based knowledge of mathematics didactics throughout the two-year

period. To my knowledge of research in the field, few studies have examined Norwegian

lower primary school teachers’ descriptions of good mathematics teaching. On this basis, the

study seeks to answer the following research question: What might Norwegian lower primary

school teachers’ views about good mathematics teaching look like? Teaching refers to the

interaction between teachers and students relating to subject matter. Cohen, Raudenbush and

Ball (2003) describe this interaction as the instructional triangle.

Methodological approach

The empirical data used in this study is from a focus group interview with seven lower

primary school teachers at a school that has completed a two-year professional development

program for mathematics teachers. The interview included two teachers from each of the

years one to year three and one from year four. Two were men and five were women. Two of

these were experienced preschool teachers who have worked at lower primary level for about

15 years. The others were primary and lower secondary teachers with between 15 to 30 ECTS

credits in mathematics and between four and 20 years of experience from primary and lower

secondary school. The school has three teachers on each of the four years of lower primary

The participants were informed of the topic of the focus group interview in advance. They

were asked to discuss and reflect on their own mathematics teaching on the basis of their

good mathematics teaching. They believed that teachers needed to be goal-oriented, listen to

the students’ thinking and show flexibility when unexpected episodes arise. The student

teachers pointed out that teachers should have knowledge of varied work methods, base their

teaching on the students’ day-to-day experiences and have a particular focus on problem

solving. Continuous assessment and development of socio-mathematical norms were

considered important elements of good mathematics teaching.

Teaching mathematics is complex and researchers have attempted to distinguish the different

aspects to identify main practices. These are referred to as core practices (McDonald et al.,

2013) or high-leverage practices (Forzani, 2014). This study focusses on the Nordic context,

therefore core practices are not discussed further.

The study in this paper is based on a group of Norwegian teachers at lower primary school

level who, together with a teacher educator in a focus group interview, reflected on their own

mathematics teaching at the end of a two-year school-based mathematics professional

development program. The content in this development program were decided by the

headmaster in cooperation with representatives of the mathematics teachers at the school.

Among the themes were numeracy, different approaches to the four arithmetical operations

and how to lead productive mathematical discussions. This study does not measure the effect

of the program, but it can be assumed that the teachers’ descriptions of good mathematics

teaching has been influenced by them trying out exercises and activities in their own classes

and by improved research-based knowledge of mathematics didactics throughout the two-year

period. To my knowledge of research in the field, few studies have examined Norwegian

lower primary school teachers’ descriptions of good mathematics teaching. On this basis, the

study seeks to answer the following research question: What might Norwegian lower primary

school teachers’ views about good mathematics teaching look like? Teaching refers to the

interaction between teachers and students relating to subject matter. Cohen, Raudenbush and

Ball (2003) describe this interaction as the instructional triangle.

Methodological approach

The empirical data used in this study is from a focus group interview with seven lower

primary school teachers at a school that has completed a two-year professional development

program for mathematics teachers. The interview included two teachers from each of the

years one to year three and one from year four. Two were men and five were women. Two of

these were experienced preschool teachers who have worked at lower primary level for about

15 years. The others were primary and lower secondary teachers with between 15 to 30 ECTS

credits in mathematics and between four and 20 years of experience from primary and lower

secondary school. The school has three teachers on each of the four years of lower primary

The participants were informed of the topic of the focus group interview in advance. They

were asked to discuss and reflect on their own mathematics teaching on the basis of their

4.
experiences from competence raising and what they had tried out, and their definition of good

mathematics teaching. The interview lasted one hour and was recorded and transcribed in full.

Transcripts from the focus group interviews were analyzed using conventional content

analysis (Fauskanger & Mosvold, 2015; Hsieh & Shannon, 2005) used in studies that attempt

to describe a phenomenon in order to better understand it. The phenomenon described in this

study is mathematics teaching. In conventional content analysis, inductive codes are linked to

suitable categories, as shown in a table in Figure 1. The interview subjects are referred to as

R1 to R7. The transcribed interviews were placed in a table with rows containing individual

statements, such as R3 in Figure 1, key words from these, inductive codes and categories. The

material was analyzed twice with a two-month interval to prevent categories being

overlooked. One of the challenges of conventional content analysis is not obtaining a

complete understanding of the context because of categories being left out (Hsieh & Shannon,

Category Inductive code Examples of individual comments

Communication in the R3: ‘Some years ago, if I spent much too much time on

classroom a conservation, it felt like “when are we going to do

the maths?”’

The teachers’

facilitation of Representation – R4: ‘because we’ve used manipulatives before too...

learning particularly transitions And the transition from using manipulatives to actually

between representations drawing up maths problems [...]’

The students’ thinking in R5: ‘show them that there is more than one way of

the subject of mathematics working it out, several strategies.’

The students’

thinking in and The students’ thinking R3:‘The challenge is that there are a few students in

about the subject about the subject of the class that you don’t manage to engage in the

of mathematics mathematics conversation, that only really become involved when

they are given the maths problem in the book’.

Textbook R6: ‘And then I suppose it’s very safe. You probably

very much trust that those who have written the

Subject resources

textbooks know what we need to get through and...it’s

in the facilitation

also related to time pressure sometimes, that it’s easy.’

Type of task: open, R2: ‘...to see the maths in everything around us. Grasp

explorative, tasks related to the everyday situations.’

daily activities

Table 1: Codes and categories

The analysis of the empirical data led to three main categories of findings: teachers’

facilitation of learning, students’ thinking in and about mathematics and use of teaching aids

in teaching. These three main categories are sometimes related. For example, the students’

mathematics teaching. The interview lasted one hour and was recorded and transcribed in full.

Transcripts from the focus group interviews were analyzed using conventional content

analysis (Fauskanger & Mosvold, 2015; Hsieh & Shannon, 2005) used in studies that attempt

to describe a phenomenon in order to better understand it. The phenomenon described in this

study is mathematics teaching. In conventional content analysis, inductive codes are linked to

suitable categories, as shown in a table in Figure 1. The interview subjects are referred to as

R1 to R7. The transcribed interviews were placed in a table with rows containing individual

statements, such as R3 in Figure 1, key words from these, inductive codes and categories. The

material was analyzed twice with a two-month interval to prevent categories being

overlooked. One of the challenges of conventional content analysis is not obtaining a

complete understanding of the context because of categories being left out (Hsieh & Shannon,

Category Inductive code Examples of individual comments

Communication in the R3: ‘Some years ago, if I spent much too much time on

classroom a conservation, it felt like “when are we going to do

the maths?”’

The teachers’

facilitation of Representation – R4: ‘because we’ve used manipulatives before too...

learning particularly transitions And the transition from using manipulatives to actually

between representations drawing up maths problems [...]’

The students’ thinking in R5: ‘show them that there is more than one way of

the subject of mathematics working it out, several strategies.’

The students’

thinking in and The students’ thinking R3:‘The challenge is that there are a few students in

about the subject about the subject of the class that you don’t manage to engage in the

of mathematics mathematics conversation, that only really become involved when

they are given the maths problem in the book’.

Textbook R6: ‘And then I suppose it’s very safe. You probably

very much trust that those who have written the

Subject resources

textbooks know what we need to get through and...it’s

in the facilitation

also related to time pressure sometimes, that it’s easy.’

Type of task: open, R2: ‘...to see the maths in everything around us. Grasp

explorative, tasks related to the everyday situations.’

daily activities

Table 1: Codes and categories

The analysis of the empirical data led to three main categories of findings: teachers’

facilitation of learning, students’ thinking in and about mathematics and use of teaching aids

in teaching. These three main categories are sometimes related. For example, the students’

5.
thinking in and about a mathematics exercise might be connected with the teacher’s

facilitation of learning through communication in the classroom. This is in line with the

description of teaching in the instruction triangle as an interaction between subject matter, the

students and the teacher (Cohen et al., 2003). Through the focus group interview, the teachers

emphasized increased awareness of several areas at the same time as they still had challenges

in a number of these areas. When the results are presented, both challenges and increased

awareness are shown in each category.

Teachers’ facilitation of learning

The teachers seemed to use more whole class conversations and dialogue in mathematics

teaching after participating in the professional development. They also said that it was

challenging to engage the students in subject-related talks. The teachers viewed the dialogues

with the students and between students as an aid to developing the students’ thinking: ‘Kind

of building a bridge between the terminology they have and... sort of new knowledge’ (R3).

This remark may indicate a view of learning in which the students develop new knowledge

from already established terms. The same teacher had started using learning pairs and felt that

the students gave each other ideas that were useful to the subsequent conversation with the

whole class. The teachers did not feel that the class failed if they spent time on discussion and

deviated from the class plan (R1). R6 reported that they often used to think ‘Oh no, now I

have to get the other part done,’ where the other part referred to solving exercises in the

textbook. This can mean that the teacher thought more about quality and what led to learning

than quantity, as in solving lots of math problems in the mathematics teaching. R3 described

the use of dialogues in teaching as a quantum leap in relation to before the professional

development program. In communication with the students, the teachers expressed that they

had become more precise in their use of terms, as described by R1: ‘addition and subtraction,

and not plus and minus.’

R4 specified what was meant by more dialogue in the following example. Previously, the date

and day were written on the board in the morning assembly, while the content was now more

mathematical: ‘Who’s birthday is next? How many days are there until...? How long ago was

Christmas?’ The teachers developed math problems from the information that emerged, and

the students were encouraged to develop their own problems.

Several teachers found dialogues to be challenging for both students and teachers in

mathematics classes. Students needed to practice talking and explaining their thoughts. Some

students asked (R2): ‘Can’t we just do a task?’ The teachers stated that they needed to learn

what questions to ask in order to elicit student thinking. To address some of the challenges

described by the teachers in my study, it will be necessary to develop classroom norms and

relations that are in line with several of the high-leverage practices (Forzani, 2014).

The teachers in the study taught at lower primary level and found it important to use various

representations, such as concrete manipulatives and semi-concrete manipulatives, drawings,

verbal representations and written representations in the form of math problems and numbers

intended to help more students to understand more. They expressed great awareness of the use

facilitation of learning through communication in the classroom. This is in line with the

description of teaching in the instruction triangle as an interaction between subject matter, the

students and the teacher (Cohen et al., 2003). Through the focus group interview, the teachers

emphasized increased awareness of several areas at the same time as they still had challenges

in a number of these areas. When the results are presented, both challenges and increased

awareness are shown in each category.

Teachers’ facilitation of learning

The teachers seemed to use more whole class conversations and dialogue in mathematics

teaching after participating in the professional development. They also said that it was

challenging to engage the students in subject-related talks. The teachers viewed the dialogues

with the students and between students as an aid to developing the students’ thinking: ‘Kind

of building a bridge between the terminology they have and... sort of new knowledge’ (R3).

This remark may indicate a view of learning in which the students develop new knowledge

from already established terms. The same teacher had started using learning pairs and felt that

the students gave each other ideas that were useful to the subsequent conversation with the

whole class. The teachers did not feel that the class failed if they spent time on discussion and

deviated from the class plan (R1). R6 reported that they often used to think ‘Oh no, now I

have to get the other part done,’ where the other part referred to solving exercises in the

textbook. This can mean that the teacher thought more about quality and what led to learning

than quantity, as in solving lots of math problems in the mathematics teaching. R3 described

the use of dialogues in teaching as a quantum leap in relation to before the professional

development program. In communication with the students, the teachers expressed that they

had become more precise in their use of terms, as described by R1: ‘addition and subtraction,

and not plus and minus.’

R4 specified what was meant by more dialogue in the following example. Previously, the date

and day were written on the board in the morning assembly, while the content was now more

mathematical: ‘Who’s birthday is next? How many days are there until...? How long ago was

Christmas?’ The teachers developed math problems from the information that emerged, and

the students were encouraged to develop their own problems.

Several teachers found dialogues to be challenging for both students and teachers in

mathematics classes. Students needed to practice talking and explaining their thoughts. Some

students asked (R2): ‘Can’t we just do a task?’ The teachers stated that they needed to learn

what questions to ask in order to elicit student thinking. To address some of the challenges

described by the teachers in my study, it will be necessary to develop classroom norms and

relations that are in line with several of the high-leverage practices (Forzani, 2014).

The teachers in the study taught at lower primary level and found it important to use various

representations, such as concrete manipulatives and semi-concrete manipulatives, drawings,

verbal representations and written representations in the form of math problems and numbers

intended to help more students to understand more. They expressed great awareness of the use

6.
of new representations such as sketches of blank number lines: “Blank number line. Open

number line. I think it’s almost been revolutionary. I use it in nearly every possible context,

very positive to use,” (R6). The transition from concrete representation to abstract ideas was

challenging for the students, according to several of the teachers. R4 gave an example where

she lined the students up at the front of the class to show doubles and halves. For the students

to understand what numbers represented half and double, the teachers felt that they had

improved their knowledge as to what questions to ask in order for the students to see the

connection between the practical and the written parts. The teachers believed that this

transition was important (R3). This indicated that they found it important to facilitate

students’ learning and how their current abilities could be related to what they were going to

The students’ thinking in and about the subject of mathematics

This category was also concerned with communication in mathematics teaching. When

students explain their thoughts, it takes place in a communication situation. The teachers felt

that the students must be given time to think and ask questions and that they, as teachers,

should not feel that the students should rather be solving written math problems. By letting

the students show their thoughts when solving problems, the teachers could emphasize that

mistakes can be positive in that they can help the teachers and students to understand. “And

understanding kind of how they think, and going into it and understanding a bit more why

things are wrong and why it is hard, I think is very important” (R4). This showed that

knowing about common student mistakes and ways of thinking was important for the teacher.

According to the teachers, the students also became aware of there being more than one way

of reaching the solution. They believed that the students acquired a better understanding by

explaining their thoughts since this formed a ‘bridge’ between the terminology the students

already had and new knowledge.

The teachers gave examples of their students’ remarks when thinking about the subject of

mathematics: “Oh yes, now I understand it.” This expressed a sense of mastery. However, the

teachers also described the challenges relating to students’ different understandings of the

mathematics subject. As mentioned earlier, it can be a challenge to get the students to talk in

mathematics classes precisely because they are of the impression that mathematics means

solving lots of math problems. R3 explained it in the following way: “The challenge is that

there are a few students in the class that you don’t manage to engage in the dialogue, that only

really become involved when they are given the math problem in the book.” She also believed

that this particularly applied to students who were quick at calculations and those who were

not particularly motivated in the subject of mathematics. This may indicate a view that

mathematics is about quickly solving lots of math problems.

Subject matter/resources

When the teachers described the content of their own teaching, the main topics of discussion

were the textbook and different types of tasks (often aside from the book). They expressed an

increased awareness in relation to both.

number line. I think it’s almost been revolutionary. I use it in nearly every possible context,

very positive to use,” (R6). The transition from concrete representation to abstract ideas was

challenging for the students, according to several of the teachers. R4 gave an example where

she lined the students up at the front of the class to show doubles and halves. For the students

to understand what numbers represented half and double, the teachers felt that they had

improved their knowledge as to what questions to ask in order for the students to see the

connection between the practical and the written parts. The teachers believed that this

transition was important (R3). This indicated that they found it important to facilitate

students’ learning and how their current abilities could be related to what they were going to

The students’ thinking in and about the subject of mathematics

This category was also concerned with communication in mathematics teaching. When

students explain their thoughts, it takes place in a communication situation. The teachers felt

that the students must be given time to think and ask questions and that they, as teachers,

should not feel that the students should rather be solving written math problems. By letting

the students show their thoughts when solving problems, the teachers could emphasize that

mistakes can be positive in that they can help the teachers and students to understand. “And

understanding kind of how they think, and going into it and understanding a bit more why

things are wrong and why it is hard, I think is very important” (R4). This showed that

knowing about common student mistakes and ways of thinking was important for the teacher.

According to the teachers, the students also became aware of there being more than one way

of reaching the solution. They believed that the students acquired a better understanding by

explaining their thoughts since this formed a ‘bridge’ between the terminology the students

already had and new knowledge.

The teachers gave examples of their students’ remarks when thinking about the subject of

mathematics: “Oh yes, now I understand it.” This expressed a sense of mastery. However, the

teachers also described the challenges relating to students’ different understandings of the

mathematics subject. As mentioned earlier, it can be a challenge to get the students to talk in

mathematics classes precisely because they are of the impression that mathematics means

solving lots of math problems. R3 explained it in the following way: “The challenge is that

there are a few students in the class that you don’t manage to engage in the dialogue, that only

really become involved when they are given the math problem in the book.” She also believed

that this particularly applied to students who were quick at calculations and those who were

not particularly motivated in the subject of mathematics. This may indicate a view that

mathematics is about quickly solving lots of math problems.

Subject matter/resources

When the teachers described the content of their own teaching, the main topics of discussion

were the textbook and different types of tasks (often aside from the book). They expressed an

increased awareness in relation to both.

7.
In relation to the types of tasks, R4 explained that she no longer made booklets containing

extra tasks, but used open-ended and problem-solving tasks that the students could work on

over time. She also stated that, “I hope they have become better at thinking at least, to sort of,

solve problems.” This may imply that the teachers felt that investigation and problem solving

were key elements of students’ understanding, and thereby of good mathematics teaching.

The teachers also told that they discussed mathematics teaching with colleagues more than

earlier, because the tasks were challenging. At lower primary level, the teachers gave the

students notebooks where they could draw and write problems and solutions themselves. An

open exercise for year one students was the hundred-day party where the mathematical topics

the teachers covered were even numbers, odd numbers, ten friends, bridging through ten,

counting, subtraction and addition. The teachers in my study explained that they had become

more alert to the mathematics in everything around them, which could be linked to the types

of exercises. R2 commented: “Grasp the everyday situations. And get them into what’s related

to mathematics in the class.” When the teachers in my study seem to have increased

awareness of using mathematics in all subjects, this might relate to the fact that the basic skill

of calculation in all subjects had been a theme in the professional development program. R7

summed up what she thought good mathematics teaching was in the following way: “When

the students understand when and how they can use their knowledge of mathematics in

everyday life.”

When the textbook was raised as a topic, there was some disagreement among the teachers.

R7 told that she has become “critical to the textbooks, and I don’t completely trust that the

textbooks necessarily meet all the learning goals.” R2 has become more aware of being freer

in relation to the textbook, while R6 finds the textbook safe. This shows that teachers can

disagree about the textbook’s role in mathematics teaching.

According to the research question, the teachers’ views about good mathematics teaching was

described in the three main categories of findings in this section. Some of them were related

to results in other Nordic studies, as discussed in the next session, but also to the content of

the two-year professional program. This paper does not assess the effect of the program, but

the teachers’ views about good mathematics teaching can be influenced by this content and

improved research-based knowledge. The teachers emphasized the use of open-ended and

problem solving tasks and acceptance of communication and dialogues to facilitate learning.

There were both similarities and differences between how good mathematics teaching was

described by the teachers in this study as compared with other Nordic studies (Fauskanger,

2016; Fauskanger et al., 2018; Hemmi & Ryve, 2015; Kaasila & Pehkonen, 2009) that have

examined good mathematics teaching.

The Norwegian teachers’ descriptions of dialogs in lower primary school teaching were

similar to those of other studies. Fauskanger et al. (2018) pointed out that facilitating

conversation and discussion in mathematics teaching was a key aspect of students’ learning at

the same time as such conversation could inform the teacher about the students’ thinking.

extra tasks, but used open-ended and problem-solving tasks that the students could work on

over time. She also stated that, “I hope they have become better at thinking at least, to sort of,

solve problems.” This may imply that the teachers felt that investigation and problem solving

were key elements of students’ understanding, and thereby of good mathematics teaching.

The teachers also told that they discussed mathematics teaching with colleagues more than

earlier, because the tasks were challenging. At lower primary level, the teachers gave the

students notebooks where they could draw and write problems and solutions themselves. An

open exercise for year one students was the hundred-day party where the mathematical topics

the teachers covered were even numbers, odd numbers, ten friends, bridging through ten,

counting, subtraction and addition. The teachers in my study explained that they had become

more alert to the mathematics in everything around them, which could be linked to the types

of exercises. R2 commented: “Grasp the everyday situations. And get them into what’s related

to mathematics in the class.” When the teachers in my study seem to have increased

awareness of using mathematics in all subjects, this might relate to the fact that the basic skill

of calculation in all subjects had been a theme in the professional development program. R7

summed up what she thought good mathematics teaching was in the following way: “When

the students understand when and how they can use their knowledge of mathematics in

everyday life.”

When the textbook was raised as a topic, there was some disagreement among the teachers.

R7 told that she has become “critical to the textbooks, and I don’t completely trust that the

textbooks necessarily meet all the learning goals.” R2 has become more aware of being freer

in relation to the textbook, while R6 finds the textbook safe. This shows that teachers can

disagree about the textbook’s role in mathematics teaching.

According to the research question, the teachers’ views about good mathematics teaching was

described in the three main categories of findings in this section. Some of them were related

to results in other Nordic studies, as discussed in the next session, but also to the content of

the two-year professional program. This paper does not assess the effect of the program, but

the teachers’ views about good mathematics teaching can be influenced by this content and

improved research-based knowledge. The teachers emphasized the use of open-ended and

problem solving tasks and acceptance of communication and dialogues to facilitate learning.

There were both similarities and differences between how good mathematics teaching was

described by the teachers in this study as compared with other Nordic studies (Fauskanger,

2016; Fauskanger et al., 2018; Hemmi & Ryve, 2015; Kaasila & Pehkonen, 2009) that have

examined good mathematics teaching.

The Norwegian teachers’ descriptions of dialogs in lower primary school teaching were

similar to those of other studies. Fauskanger et al. (2018) pointed out that facilitating

conversation and discussion in mathematics teaching was a key aspect of students’ learning at

the same time as such conversation could inform the teacher about the students’ thinking.

8.
International research highlights classroom discussion in mathematics teaching (e.g., Franke

et al., 2007).

Based on the empirical data, students’ thinking appeared to influence the planning of

mathematics teaching for the participating teachers. Similar to the Finnish study the findings

of this study documented that the teacher needs to listen to the students in order to understand

their way of thinking (Kaasila & Pehkonen, 2009). The student teachers in Hemmi and

Ryve’s (2015) study believed that Swedish teachers build on an extreme expression of

constructivism and were therefore more student-focused, while the Finnish referred to whole

class discussion.

Selecting open tasks and investigative activities that are motivating and give the students

opportunities to show several solutions was emphasized by both Fauskanger et al. (2018) and

Hemmi and Ryve (2015). Connecting mathematics with everyday life seemed to be important

both in the Swedish and Finnish education studies (Hemmi & Ryve, 2015; Kaasila &

Pehkonen, 2009) which are in line with the findings of my study.

In this concluding discussion, I will highlight one characteristic of good teaching that the

teachers focused on and one characteristic that previous research has highlighted, but that was

not mentioned in my focus group interview.

During the focus group interview, the teachers believed that using several different

representations and working on the transition between these both contributed to good teaching

and entailed a challenge. Using different representations has not been included in the

characteristics of good mathematics teaching in other Nordic studies. This could of course

mean that the use of representations and the transition between representations were included

in some of the other categories in these studies. However, it may also be explained by the fact

that the teachers in my study were teachers at lower primary school level and here, the need to

use different representations was greater, and the transition from concrete to abstract thinking

was more difficult than for older students. Work on expressing math problems or amounts in

numbers after using manipulatives were considered particularly difficult for this age group.

A characteristic of good mathematics teaching mentioned in both Fauskanger et al. (2018) and

Kaasila and Pehkonen’s (2009) studies, is the structure of classes, with clear learning goals

and classroom management. This was not mentioned, nor asked about, in the lower primary

school teachers’ focus group interview. This does not mean that it is not important to the

lower primary school teachers in this study; it is perhaps more important here than in other

years. However, the teachers might see classroom management and clear learning goals as

such obvious factors that they did not mention them explicitly when they described

mathematics teaching.

At the end of the professional development program the teachers felt that they had not only

become more aware of dialogues with the students, but they also had more conversations and

reflections among themselves. They feel that such discussion and reflection provide support

and inspiration for their teaching. Knowledge sharing among the teachers may therefore

et al., 2007).

Based on the empirical data, students’ thinking appeared to influence the planning of

mathematics teaching for the participating teachers. Similar to the Finnish study the findings

of this study documented that the teacher needs to listen to the students in order to understand

their way of thinking (Kaasila & Pehkonen, 2009). The student teachers in Hemmi and

Ryve’s (2015) study believed that Swedish teachers build on an extreme expression of

constructivism and were therefore more student-focused, while the Finnish referred to whole

class discussion.

Selecting open tasks and investigative activities that are motivating and give the students

opportunities to show several solutions was emphasized by both Fauskanger et al. (2018) and

Hemmi and Ryve (2015). Connecting mathematics with everyday life seemed to be important

both in the Swedish and Finnish education studies (Hemmi & Ryve, 2015; Kaasila &

Pehkonen, 2009) which are in line with the findings of my study.

In this concluding discussion, I will highlight one characteristic of good teaching that the

teachers focused on and one characteristic that previous research has highlighted, but that was

not mentioned in my focus group interview.

During the focus group interview, the teachers believed that using several different

representations and working on the transition between these both contributed to good teaching

and entailed a challenge. Using different representations has not been included in the

characteristics of good mathematics teaching in other Nordic studies. This could of course

mean that the use of representations and the transition between representations were included

in some of the other categories in these studies. However, it may also be explained by the fact

that the teachers in my study were teachers at lower primary school level and here, the need to

use different representations was greater, and the transition from concrete to abstract thinking

was more difficult than for older students. Work on expressing math problems or amounts in

numbers after using manipulatives were considered particularly difficult for this age group.

A characteristic of good mathematics teaching mentioned in both Fauskanger et al. (2018) and

Kaasila and Pehkonen’s (2009) studies, is the structure of classes, with clear learning goals

and classroom management. This was not mentioned, nor asked about, in the lower primary

school teachers’ focus group interview. This does not mean that it is not important to the

lower primary school teachers in this study; it is perhaps more important here than in other

years. However, the teachers might see classroom management and clear learning goals as

such obvious factors that they did not mention them explicitly when they described

mathematics teaching.

At the end of the professional development program the teachers felt that they had not only

become more aware of dialogues with the students, but they also had more conversations and

reflections among themselves. They feel that such discussion and reflection provide support

and inspiration for their teaching. Knowledge sharing among the teachers may therefore

9.
contribute to long-term competence raising and deserve further research. Effect studies of

professional development also need to be researched further.

Cai, J., Kaiser, G., Perry, B., & Wong, N. (Eds.). (2009). Effective mathematics teaching from

teachers’ perspektives: National and cross-national studies. Rotterdam: Sense Publishers.

Cohen, D. K., Raudenbush, S. W., & Ball, D. L. (2003). Resources, instruction, and research.

Educational Evaluation and Policy Analysis, 25(2), 119–142.

doi:10.3102/01623737025002119

Fauskanger, J. (2016). Matematikklæreres oppfatninger om ingrediensene i god

matematikkundervisning. Acta Didactica Norge, 10(3), 1–18.

Fauskanger, J., & Mosvold, R. (2015). En metodisk studie av innholdsanalyse - med analyser

av matematikklæreres undervisningskunnskap som eksempel. Nordic Studies in

Mathematics Education, 20(2), 79–96.

Fauskanger, J., Mosvold, R., Valenta, A., & Bjuland, R. (2018). Good mathematics teaching

as constructed in Norwegian teachers’ discourses. In E. Norèn, H. Palmèr, & A. Cooke

(Eds.), NORMA 17 – Nordic Research in Mathematics Education (Vol. Skrifter från

SMDF, nr 12 Göteborg: Svensk förening för matematikdidaktisk forskning, pp. 239─249).

Göteborg.

Forzani, F. M. (2014). Understanding “core practices” and “practice-based” teacher

education: Learning from the past. Journal of Teacher Education, 65(4), 357–368.

doi:10.1177/0022487114533800

Franke, M. L., Kazemi, E., & Battey, D. (2007). Mathematics teaching and classroom

practice. In F. K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics

Teaching and Learning (Vol. 1, pp. 225–256). Charlotte, NC: Information Age Publishing.

Hemmi, K., & Ryve, A. (2015). Effective mathematics teaching in Finnish and Swedish

teacher education discourses. Journal of Mathematics Teacher Education, 18(6), 501–521.

doi:10.1007/s10857-014-9293-4

Hiebert, J. S., & Grouws, D. A. (2007). The Effects of Classroom Mathematics Teaching on

Students’ Learning. In F. K. J. Lester, Jr. (Ed.), Second handbook of research on

mathematics teaching and learning (Vol. 1, pp. 371–404). Charlotte, NC: Information Age

Publishing.

Hsieh, H. F., & Shannon, S. E. (2005). Three Approaches to Qualitative Content Analysis.

Qualitative health research, 15(9), 1277–1288.

Kaasila, R., & Pehkonen, E. (2009). Effective mathematics teaching in Finland through the

eyes of elementary student teachers. In J. Cai, G. Kaiser, B. Perry, & N. Wong (Eds.),

Effective mathematics teaching from teachers’ perspectives: National and Cross-National

Studies (pp. 203–216). Rotterdam: Sense Publishers.

professional development also need to be researched further.

Cai, J., Kaiser, G., Perry, B., & Wong, N. (Eds.). (2009). Effective mathematics teaching from

teachers’ perspektives: National and cross-national studies. Rotterdam: Sense Publishers.

Cohen, D. K., Raudenbush, S. W., & Ball, D. L. (2003). Resources, instruction, and research.

Educational Evaluation and Policy Analysis, 25(2), 119–142.

doi:10.3102/01623737025002119

Fauskanger, J. (2016). Matematikklæreres oppfatninger om ingrediensene i god

matematikkundervisning. Acta Didactica Norge, 10(3), 1–18.

Fauskanger, J., & Mosvold, R. (2015). En metodisk studie av innholdsanalyse - med analyser

av matematikklæreres undervisningskunnskap som eksempel. Nordic Studies in

Mathematics Education, 20(2), 79–96.

Fauskanger, J., Mosvold, R., Valenta, A., & Bjuland, R. (2018). Good mathematics teaching

as constructed in Norwegian teachers’ discourses. In E. Norèn, H. Palmèr, & A. Cooke

(Eds.), NORMA 17 – Nordic Research in Mathematics Education (Vol. Skrifter från

SMDF, nr 12 Göteborg: Svensk förening för matematikdidaktisk forskning, pp. 239─249).

Göteborg.

Forzani, F. M. (2014). Understanding “core practices” and “practice-based” teacher

education: Learning from the past. Journal of Teacher Education, 65(4), 357–368.

doi:10.1177/0022487114533800

Franke, M. L., Kazemi, E., & Battey, D. (2007). Mathematics teaching and classroom

practice. In F. K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics

Teaching and Learning (Vol. 1, pp. 225–256). Charlotte, NC: Information Age Publishing.

Hemmi, K., & Ryve, A. (2015). Effective mathematics teaching in Finnish and Swedish

teacher education discourses. Journal of Mathematics Teacher Education, 18(6), 501–521.

doi:10.1007/s10857-014-9293-4

Hiebert, J. S., & Grouws, D. A. (2007). The Effects of Classroom Mathematics Teaching on

Students’ Learning. In F. K. J. Lester, Jr. (Ed.), Second handbook of research on

mathematics teaching and learning (Vol. 1, pp. 371–404). Charlotte, NC: Information Age

Publishing.

Hsieh, H. F., & Shannon, S. E. (2005). Three Approaches to Qualitative Content Analysis.

Qualitative health research, 15(9), 1277–1288.

Kaasila, R., & Pehkonen, E. (2009). Effective mathematics teaching in Finland through the

eyes of elementary student teachers. In J. Cai, G. Kaiser, B. Perry, & N. Wong (Eds.),

Effective mathematics teaching from teachers’ perspectives: National and Cross-National

Studies (pp. 203–216). Rotterdam: Sense Publishers.

10.
Kilpatrick, J., Swafford, J., Findell, B. (2001). Adding it up: helping children learn

mathematics. Washington, DC: National Academy Press.

Li, Y. (2011). Elementary teachers' thinking about a good mathematics lesson. International

Journal of Science and Mathematics Education, 9(4), 949–973. doi:10.1007/s10763-010-

9263-y

McDonald, M., Kazemi, E., & Kavanagh, S. S. (2013). Core practices and pedagogies of

teacher education: A call for a common language and collective activity. Journal of

Teacher Education, 64(5), 378–386. doi:10.1177/0022487113493807

mathematics. Washington, DC: National Academy Press.

Li, Y. (2011). Elementary teachers' thinking about a good mathematics lesson. International

Journal of Science and Mathematics Education, 9(4), 949–973. doi:10.1007/s10763-010-

9263-y

McDonald, M., Kazemi, E., & Kavanagh, S. S. (2013). Core practices and pedagogies of

teacher education: A call for a common language and collective activity. Journal of

Teacher Education, 64(5), 378–386. doi:10.1177/0022487113493807