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The role of natural language when learning the symbolic language of mathematics

Research project Being able to handle the mathematical symbolic language is an important part of learning mathematics. Research shows that students often find it difficult to learn how to work with the symbolic language, especially when it comes to algebra.

An important part of knowledge about the symbolic language is to manipulate symbols, for example, from the equation x+1/x=2 to arrive at the solution x=1. Then you need to use ordinary words, that is, a natural language such as Swedish, to talk about what to do with the symbols. However, symbols can also have meaning in the same way as ordinary words, and students therefore need to use natural language to talk about the meaning of symbols. The purpose of this project is to investigate how natural language can be used to make it easier for students to learn the symbolic language.

Head of project

Magnus Österholm
Associate professor, research fellow

Project overview

Project period:

2016-09-01 2022-12-31


The Swedish Research Council, 2018-2020: SEK 4,581,000

Participating departments and units at Umeå University

Department of Science and Mathematics Education, Faculty of Science and Technology

Research subject

Educational sciences, Mathematics

Project description

In mathematics you use multiple languages. One is the natural language (e.g., Swedish), including specific words for mathematics (e.g., "differentiable" and "addition") and specific grammatical constructions (e.g., "twice as many as"). Another is the extremely specialized symbolic language used to express numbers, equations, functions and formulas (e.g., "125", "x+1/x=2", and "f(x)=e^x+3").

Being able to communicate mathematics using these different types of languages is a key part of understanding and using mathematics. Unfortunately, it has been shown that symbolic language is very difficult to master for many students. Existing research can not provide solutions to this didactic problem, partly because the few existing research frameworks that describe aspects of pupils' learning of the symbolic language have many shortcomings.

An important part of knowledge about the symbolic language means being able to manipulate symbols, for example, if you are faced with the equation x+1/x=2, being able to manipulate these symbols to arrive at the solution x=1. In this situation, teachers and students use a natural language (e.g., Swedish) in order to be able to talk about what happens. For example, they describe what they do with the symbols (e.g., move, remove or add) but also why they do so and may do so (e.g., "I can get 3x to the left by subtracting 3x on both sides and then the equality is still valid because I make the same change on both sides").

If the symbolic language is reduced to the mere act of manipulation, it can be perceived as useless to students. Symbols can also have meaning and then be used to describe things in the same way as ordinary words. For example, 3x+2x can describe the total value of Kalle's three bags of candy and Lisa's two candy bags when each candy bag costs x. Students therefore need to learn both how to use symbols to describe things and learn to translate between natural language and the mathematical symbolic language.

Natural language is thus involved in students’ learning of the symbolic language, but we do not yet know exactly how. The purpose of this project is to increase our knowledge of students’ meaning making and understanding of mathematical symbols, particularly in relation to natural language. In the project we plan for the many different types of studies using different methods and different types of data in order to gradually result in a deeper understanding of the role of natural language in the learning of the mathematical symbolic language.