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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�
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2.
Good mathematics teaching at lower primary school level
Marianne Maugesten
Østfold University College, Norway; marianne.maugesten@hiof.no
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
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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
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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’
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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
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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.
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.
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
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.
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
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