Reciting numbers is a relevant way to begin learning mathematics. Research findings indicate that this skill, which is sometimes regarded as just verbal chatter, is actually essential for developing later skills. At the University of Jyväskylä, education and psychology researchers have published numerous international articles on this topic in recent years.
“Children learn the basics of mathematics also by listing numerals,” says Senior Researcher Tuire Koponen from the Department of Teacher Education. Mastery of number sequences and the number system is essential in constructing further mathematical skills.
Skills learned before school age have strong predictive value of pupils’ knowledge of arithmetic at the lower secondary level. “One, two, three and so on,” says Koponen, demonstrating the skill called number sequencing.
“When children realise patterns in the series of numerals they have learned, for example, how new numerals can be formed by combining individual numerals and how these numerals are ordered, so then they have already acquired some more advanced mathematical understanding.”
Finnish numerals difficult to learn
“At first, the Finnish numeral system is a bit complex,” says Koponen. “For example, in some other languages the numeral ’eleven’ is expressed more directly as ‘ten-and one’, which also helps understand the quantitative meaning of numbers. Only from twenty onwards does the structure of Finnish numerals follow more consistently the logic of the decimal system.” Some pupils therefore need support in order to understand the structure of numerals.
To build a proper foundation for the basic numeracy skills trained at school, it is important to develop strong number sequencing skills, which means not only knowing the order of numbers but also understanding that the distance between numbers in a number line indicates their quantitative relationships.
Therefore, at the beginning of school there should be more training to support the integration of conceptual knowledge of numbers and number sequencing.
Teaching that strictly follows a textbook only does not help strengthen pupils’ basic skills, because the textbooks are based on the curricular objectives of each class level. The developmental and hierarchical order of skills is not strongly inherent in them.
“Thus, they do not have pupils practice the underlying basic skills when necessary, so it is left up to the teacher,” Koponen explains.
Gaps in basics important to fix
Mathematics is based on a strong hierarchy of skills, where new competences are built on what has been learned earlier. Hence, when it comes to learning difficulties and supporting pupils in mathematics, it is important to understand how a pupil’s mathematical skills have been constructed and how the support should be targeted so as to achieve strong basic skills.
If a fourth-grader has trouble mastering division, it is essential to consider what underlying skills such mastery is based on. Often these problems stem from shortcomings in basic skills, in this case poor skills in multiplication or even earlier skills. Teachers can easily name the pupils for whom mathematics is difficult.
“The most important question here, however, is where support should start and what skills should be developed first,” Koponen emphasises.
It is also essential to understand that learning difficulties tend to build up and overlap. For example, according to research on overlapping learning difficulties in reading and arithmetic, up to 30 to 40 percent of pupils have problems with both reading and maths skills. It is important that they receive help as needed for both these subjects. For instance, Heidi Korpipää has investigated in her recent doctoral dissertation the interconnections and common background factors of these skills.
Maths needed in studies and working life
Koponen is worried about reports and studies showing that the maths skills of Finnish children and youth have deteriorated. “Research findings indicate that we need to start investing in mathematics for real,” she stresses.
“For many of those who complete comprehensive school, their skills are inadequate for their later studies, not to mention for success in working life,” Koponen says. Maths performance is one of the most pivotal predictors for upper secondary level studies.
And the effects are not restricted to people’s studies only. Koponen refers, for example, to Senior Lecturer Tuija Aro’s study in psychology, which found that learning difficulties in mathematics can decrease wellbeing even in adulthood.
Photo: Tuire Koponen demonstrates that basic mathematical skills can be also supported by games.
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