Towards research-based teaching of algebra - analyzing excpected student progression in the Swedish curriculum grades 1-9

13th International Congress on Mathematical Education

Hamburg, 24-31 July 2016

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TOWARDS RESEARCH-BASED TEACHING OF ALGEBRA – ANALYZING EXPECTED STUDENT PROGRESSION IN THE SWEDISH CURRICULUM

FOR GRADES 1-9

Kajsa Bråting

, Kirsti Hemmi

, Lars Madej

, Ann-Sofi Röj-Lindberg

Uppsala University

Åbo Akademi University

We investigate how the so-called big ideas in algebra are present in the national mathematics curriculum for compulsory school in Sweden and analyse the expected student progress with respect of these ideas across the Grades 1-9. The most striking result is that items connected to generalized arithmetic are largely lacking in all parts of the curriculum. We found also a heavy emphasis on problem solving connected to everyday situations. Sweden is an interesting case to study as students’ achievements in algebra in international comparisons like PISA and TIMSS have been low although algebra has been integrated in mathematics curriculum from early grades already since the end of 1960s. The study is a part of several research projects where we compare mathematics teaching in Nordic and Baltic Countries.

BACKGROUND AND AIMS

Due to the changes in the general view of child cognitive development, and critics against parallels drawn between child development and the historical development of mathematics (cf. Bråting &

Pejlare, 2015), as well as the results of international comparisons in algebra teaching (e.g. Cai et al., 2005) there has been an upswing of educational studies in the area of algebra. Recent studies show that it is beneficial to start working with algebraic ideas and generalizations in parallel with arithmetic already in early grades (e.g. Cai et al., 2005; Carraher et al., 2006; Blanton et al., 2015).

Algebra is a critical gatekeeper for more advanced studies in mathematics, and many countries have revised their curriculum attempting to integrate algebra in school mathematics from the very beginning (e.g. Prytz, 2015). In Sweden, algebra became a part of all students’ schooling after the introduction of the nine-year compulsory school in the 1960s and already the 1969 policy documents prescribe that algebra should be a part of school mathematics from Grade 2 (Prytz 2015). However, it is not possible to discern a general positive effect of these efforts on Swedish students’ learning in algebra, at least not if we consider the results of the international evaluations where the Swedish students have since the 1960s always performed below the international average in algebra. Even in the 1995 TIMSS (The Trends in International Mathematics and Science Study), where the overall result was the best ever, the algebra results were below the international average.

Therefore, we find it of the utmost interest to deepen in the Swedish case. More specifically the aim

of the present study is to characterize the Swedish national curriculum as to what algebra-related

items are identified at different school levels through Grades 1-9 and how the items are addressed

(cf. Cai et al., 2005; Hemmi et al., 2013). The present study is a part of on-going projects where we

investigate and compare mathematics teaching in Nordic and Baltic countries (e.g. Hemmi et al.,

2013; Kilhamn & Röj-Lindberg, 2013; Hemmi & Ryve, 2014; Bråting, 2015). The study will

contribute to the international research field by adding to the knowledge about incorporating

algebra in school curriculum and pointing out possible ways to improve future reform policies and

curriculum developers as well as textbook development.

Bråting, Hemmi, Madej & Röj-Lindberg

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Scholars have only recently started to theorize about possible learning progressions with respect to algebra in different school years (Cai & Knuth, 2011). There are only few international studies of the character of goals and contents of algebra formulated at the institutional level (Kendal & Stacey, 2004; Cai et al., 2005) and the Nordic countries have not been a part of these comparisons. These studies show a great diversity of algebraic contents in curricula of different countries. We have earlier investigated how proof and proof-related competences are integrated in three countries’

current mathematics curricula (Hemmi et al., 2013) and the aspect of generalization connected to the development of proof-related competences is also related to the development of algebraic thinking.

METHODOLOGY

In this study we have identified and classified the algebraic content in the current Swedish curriculum in mathematics at compulsory school level (Grades 1-9, age 7-15). Our classification is based on the framework of Blanton et. al. (2015) regarding how algebraic content can be characterized at compulsory school level. They identified the following four main areas (that they call ”big ideas”) in school algebra: 1) Equivalence, expressions, equations & inequalities; 2) Generalized arithmetic; 3) Functional thinking; 4) Variable.

The Swedish curriculum in mathematics for compulsory school is divided into three parts; Grades 1-3, Grades 4-6, and Grades 7-9. Each part comprises the mathematical content that is to be dealt in these grades and there are knowledge demands stated for the end of grade 3, 6 and 9. In our study we used the categories of Blanton et al., 2015 in our initial data analysis when identified contents and goals belonging to different aspects of algebraic learning. It appeared that we needed one more category in addition to these four categories, namely a category we call ”everyday algebra”.

Furthermore, it was sometimes difficult so separate category 3 and 4 but we decided to keep them so far although some contents were categorised to both of them. After this analysis we investigated the expected student progression within the categories across the grades in order to understand hypothetical learning trajectories (see Hemmi et al., 2013).

RESULTS AND DISCUSSION

The results show that categories 1, 3, 4 and the new 5:th category were all represented in the content of the curricula. Meanwhile, category 2 ”generalized arithmetic” was clearly under- represented or even missing in certain grades. Categories 1 and 3 had the clearest and most obvious progression trajectories. For instance, the progression line for solving equations (which is included in category 1) was the following: In grades 1-3 the pupils learn to understand the meaning of the equal sign, in grades 4-6 the pupils get acquainted to situations where there is a need to denote a number with a symbol. Finally, in grades 7-9 the equations become more complicated and the pupils learn different methods to solve them. Similar progression trajectories were found in connection with variables and functional thinking.

The absence/low occurrence of category 2, generalized arithmetic, in the curricula for compulsory

school can be one reason to the low results at TIMSS and PISA in Sweden, especially considering

that the results in algebra is, and has been, the weakest of all mathematical areas. Generalized

arithmetic is stressed as an important part of algebra by several researchers (Kieran, 2007) and it

can be seen as a bridge between arithmetic and algebraic thinking. A progression in ”algebra as

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generalized arithmetic” throughout compulsory school is necessary in order to help pupils master algebraic manipulations (cf. quasi-variables in Fujii, 2003). We assume that the exercises of the type exemplified below could contribute to students’ progress in understanding the meaning of the general patterns and the use of algebraic notation.

Example 1

Consider the following expressions:

(5 + 3) · (5 + 3) (5 + 3) · (5 – 3) (5 – 3) · (5 – 3) 5

– 3

5

+ 3

+ 2· 5 · 3 5

+ 3

– 2· 5 · 3 1. Calculate the value of each expression.

2. Which expressions have the same value?

3. Replace the numbers 5 and 3 with other numbers and calculate the values of the new expressions.

4. Which expressions have the same value?

5. What pattern does this indicate?

Generalized arithmetic is also involved in learning of proof and proving, especially in connection to number theory (Kieran, 2007). This is a part of algebra that seems to be difficult for students to develop understanding with. A task in 2003 TIMSS connected to this kind of proving activities (finding an expression for three consecutive even numbers, see Kieran, 2007) was difficult for students in most countries with only 27% of correct answers in average. Our results confirms the results presented in Hemmi et al. (2013) showing that the aspect of generalization was quite invisible in the Swedish mathematics curriculum compared to the Finnish and Estonian curricula.

Another interesting result is that there is a heavy emphasis on every-day problem solving in the Swedish curriculum. There are some indications that countries with a greater emphasis on realistic problems do not put so much effort on students’ skills concerning symbolic manipulations (Kieran, 2007). There are some other indications that this is the case in Sweden. Jakobsson-Åhl’s (2006) study about the development of algebraic content in upper secondary Swedish textbooks 1960–2000 reveals that the algebraic content had changed from being dominated by algebraic manipulations and expressions to becoming more integrated with other school subjects and thus being more anchored with reality as well as everyday activities. Particularly interesting with the Swedish case is that at the same time when a heavy emphasis has been put on problem solving connected to everyday situations Swedish students’ results in PISA, that are to measure that kind of competence, have declined. In comparison with Finland, the word everyday occurs only sporadically in the steering document and still the Finnish students’ results in PISA are outstanding compared to other Nordic countries (cf. Hemmi & Ryve, 2014). We find this aspect interesting to dig deeper in in our further studies.

The Swedish national steering document can only be considered as a general frame leaving a lot for

interpretations to textbook authors and to teachers. More investigations are needed with respect of

how Swedish and Finnish students manage with problems connected to generalized arithmetic in

PISA and TIMSS and how these items as well as every-day connections are addressed in

mathematics textbooks and in mathematics classrooms in order to learn more about the possible

reasons for the variation in the results. We have a possibility of specifically addressing these two

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aspects when analysing the material gathered within a recent international video study that looked at how instruction on early algebra was organized in four countries, among them Sweden and Finland (Kilhamn & Röj-Lindberg, 2013).

References

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