"Pupils should be given more challenging maths tasks"
This is what mathematics researcher Jorunn Reinhardtsen says. She says that learning to think is more important than imitating the teacher.
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"Simple assignments are quickly solved using patterns learned, and the pupils do not learn to think mathematically", says Assistant Professor Jorunn Reinhardtsen, at the Department of Mathematical Sciences, University of Agder (UiA).
She recently defended her mathematics doctorate, which focused on how pupils of age 11-13 make sense of algebra.
Facts
Jorunn Reinhardtsen defended her PhD with the thesis 'Student meaning making in elementary algebra teaching: An in-depth study of classrooms in four countries' in January 2020. The study is part of the international research project VIDEOMAT.
The researcher has followed the doctoral programme at the Faculty of Engineering and Science with specialisation in mathematics education. One year of the fellowship period was funded by NOS-HS – The Joint Committee for Nordic Research Councils in the Humanities and Social Sciences (grant 2135-08-210321).
The doctoral dissertation is part of an international research project in which researchers from Finland, Norway, Sweden and the US have studied how schoolchildren aged 11-13 learn algebra. She has analysed data from 16 classrooms in these four countries.
The groups in the four countries used the same kind of techniques to solve tasks. They used drawing and counting techniques and more sophisticated methods like formulating algebraic expressions. The research found that the pupils used their knowledge of arithmetic to solve problems and not the algebraic ideas they had been introduced to in four previous lessons.
Jorunn Reinhardtsen is currently an assistant professor at the Department of Mathematical Sciences at the University of Agder.
How do pupils talk about numbers, shapes, and algebra in the classroom? How do they use tools such as objects, drawings, hand gestures and discussions to create meaning in the mathematical conversation?
These are questions the researcher has addressed in her dissertation.
"It is more important to learn to think than to imitate the teacher. By discussing mathematics in the classroom, various aspects come up. The pupils talk among themselves, they ask and they answer. They try to understand. And by doing that, they not only learn how to solve a task but to think mathematically", she says.
"Algebraic symbols are powerful tools for problem solving, generalisation and modelling because they enable clear and precise expression of mathematical ideas", says Assistant Professor Jorunn Reinhardtsen.
The teacher plays a key role
At the same time, the researcher emphasises that the teacher has a crucial role. The teacher should show the pupils how to explore mathematics and help them complete the assignments on their own.
"This means that the teacher must be good at maths. Without a deep understanding of the subject, the teacher cannot improve pupils' mathematics comprehension. Pupils should be given the opportunity to develop mathematical ways of thinking that make them flexible problem solvers with deep knowledge of the subject", she says.
Thinking algebraically
The researcher knows it is provocative to suggest more challenging tasks for learners, especially when it comes to algebra that many pupils find difficult to grasp. But it is precisely understanding and thinking teachers should emphasise.
"If the pattern to solve problems is given, and the tasks are too easy, pupils might find it hard to understand the power and point of algebra. The most important takeaway from my research is that pupils learn a lot from good assignments that are not too easy", she says.
Reinhardtsen emphasises that the teacher should not give out assignments arbitrarily. The teacher must have a well thought out plan of what the pupils are supposed to learn through solving the various tasks.
From the concrete to the abstract
Her study shows how students use their knowledge of addition, subtraction, multiplication, and division (arithmetic) to solve algebraic expressions.
"Arithmetic skills are a resource for algebra but also a challenge because algebra problems require more abstract and analytical thinking", she says.
In her dissertation, she gives examples of how pupils understand various aspects of mathematics when they learn to think algebraically. One example is how pupils understand how the line on a graph goes up or down (slope) by learning algebra.
"You have not only understood what graphs and slopes are, but you have learned to think using graphs and slopes", Reinhardtsen says.
New teaching methods
At UiA, she is one of several researchers at the priority research centre MERGA (Mathematics Education Research Group Agder) who work to develop new teaching methods in mathematics.
While educators were previously preoccupied with the rules of mathematics, the new trend is that one spends more time discussing the subject and accepts that problems can be solved in many different ways. One important inspiration for the researchers at MERGA is Lev Vygotsky's educational thinking.
While Jean Piaget believed that children learn best when they discover things for themselves, Lev Vygotsky believed that learning occurs through social interactions and conversations. In his theory, Vygotsky emphasises that language and community are educational tools for learning mathematics.
Change in mathematics education
Reinhardtsen thinks we are in the middle of a time shift when it comes to teaching methods in mathematics. In Norway, the shift has been in the offing for some time. 1994 was a milestone when problem-solving was introduced as a principle in the national curriculum. But according to Reinhardtsen, it is only now that problem-solving pedagogy is seriously introduced in Norwegian schools.
She looks forward to the academic innovation and new national curriculum that will be introduced in the autumn of 2020. Emphasis will be placed on what she thinks is important to learn in mathematics class from first grade onwards.
Five of the six core elements of mathematics teaching in the new curriculum are about the mathematical process and emphasise what Reinhardtsen and MERGA have focused on for many years: argumentation, generalisation, symbolisation and explanation.
"In the new curriculum, it is all about process and method, mathematical thinking in other words. The teacher should enable pupils to argue for the various mathematical solutions", she says.
Renewing the curriculum
Subject renewal means that children can be introduced to algebra already in the first grade.
"Algebra is hard but also a basic branch of mathematics. And starting with the basics is a good principle of teaching", the researcher says.
She is pleased that pupils learn to think analytically from the first grade but realises that it can present challenges.
"The challenge is that many teachers are educated in a tradition where cramming of rules was essential. The change the innovation brings will benefit the pupils and improve their maths skills in the long term", Reinhardtsen says.
After a while, teacher education students from UiA will enter schools. And they will have completed the new course Assistant Professor Reinhardtsen has developed for the mathematics teacher education, 'Algebra in school'.