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A good knowledge of algebra and geometry helps schoolchildren to solve some other types of tasks, including applied ones. These are the findings made by researchers from HSE, Stanford, and Michigan State University in a joint study.

Study authors:

There is a widely-held idea that the mathematics course in high school is too complicated. Parents question whether there is a need for so much pure mathematics, algebra and geometry, and why children need those labyrinthine formulas. They’d be better off solving tasks that would be useful in real life, they believe.

Nevertheless, the focus on ‘formula’ (or in other words, ‘formal’) mathematics in the school curriculum is a sensible solution, the Russian-American research team has proved. The team consisted of Andrey Zakharov and Tatiana Khavenson, researchers at the HSE International Laboratory for Education Policy Analysis, Martin Carnoy, Vida Jacks Professor of Education at Stanford University and Academic Supervisor at the HSE International Laboratory for Education Policy Analysis, Prashant Loyalka, Leading Research Fellow at the HSE International Laboratory for Education Policy Analysis and Professor at Stanford University, and William Schmidt, Distinguished Professor at the Michigan State University. They found out that ninth-graders who solved more problems in algebra and geometry were also better at solving other tasks in science. Such schoolchildren gained better grades in the PISA (Programme for International Student Assessment) mathematics test.

The results of the study were published in the paper ‘Revisiting the Relationship Between International Assessment Outcomes and Educational Production: Evidence From a Longitudinal PISA-TIMSS Sample’ in the American Educational Research Journal (August 2016, vol. 53), which is one of the Top 10 periodicals in education studies.

Russian high schools are ‘often blamed for emphasising pure mathematics’, at the expense of applied tasks, Andrey Zakharov reasoned. Such a ‘science-focused’ approach is often considered one of the reasons why Russian schoolchildren have low results in PISA tests. This programme for student assessment pays a lot of attention to students’ ability to solve real-life tasks. The results of these tests are usually a reason to criticize the Russian high school curriculum. The course of mathematics is considered to be too abstract and out of touch with reality. Advocates of this position advise using the experience of education in Finland, which focuses on applied mathematics.

The Russian-American research team disproved such recommendations. The researchers found out that operations with formal mathematics are extremely important for success in the PISA mathematics test. ‘Schoolchildren who more often solve problems of this type in class, generally demonstrate higher results in the test, than those whose teachers focus on applied tasks’, Zakharov emphasized. The children learn to think mathematically thanks to algebra and geometry. And it’s easier for such pupils to also solve other types of task.

As part of the PISA mathematics test, the students are asked to solve various types of tasks:

- Applied: tasks from real everyday life (for example, calculate the size of a photo frame);
- Textual problems (with a lot of data, in which they have to be able to find the needed data for solution);
- Formal maths problems (for example, solve an equation).

There aren’t a lot of algebra and geometry problems in PISA, but they turn out to be the main indicator for understanding the problem. Formal mathematics is harder for comprehension, but it is also more effective for mastering sciences. It brings mathematical thinking to a new level and determines algorithms for problem solving, the researchers explained. That’s why it helps when faced with other tasks.

The ‘demonization’ of pure mathematics was partly influenced by mathematicians themselves. It looks like they’ve been teasing society with jokes about the abstractness of their formulas. Take, for example, this quote by David Hilbert, a renowned German mathematician: ‘Mathematics is a game played according to certain simple rules with meaningless marks on paper’. Grigory Perelman, a Russian mathematician and winner of the Fields Medal (‘Nobel Prize in mathematics’), defended pure mathematics: ‘The whole world is pierced by emptiness, and it is subject to formulas. This gives us endless opportunities’.

The results of the International Mathematical Olympiad also confirm that the Russian school curriculum is getting its priorities right. From 2000 to 2014, the Russian team was one of the top five. From 2000 to 2010 it took places not lower than second and third.

Russian schoolchildren’s results in the TIMSS (Trends in Mathematics and Science Study) monitoring project have also been high. In 2011, Russia demonstrated a clearly higher level of mathematical training among eight-graders, with the biggest increase among all participant countries (27 grades higher compared with 2007). The Russian students’ grades are much higher than those of students from 36 other countries, including Finland, The USA, Great Britain, and Australia.

And one more curious fact: four years ago, in Great Britain, special mathematical schools were opened, following the example of Soviet ones (one of the first such schools was founded in 1963 under the auspices of Lomonosov Moscow State University by Andrey Kolmogorov, a renowned Russian mathematician).

‘We’ve found a paradox’, Andrey Zakharov said. It would seem that training in solving applied problems should have improved the test results. But it turned out to be the opposite. ‘Training in solving applied tasks didn’t demonstrate correlation with the test results’, the researcher noted.

As Albert Einstein joked, ‘As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality’.

An increased number of applied problems decreases the students’ results in PISA, the researchers found. If the teachers place an emphasis on pure mathematical tasks, the grades of their students improve. ‘Excellent students suffer from focusing on applied mathematics the most’, Zakharov said, ‘And they are also those who gain the most grades when focusing on pure problems’.

For average students, focusing on textual problems would be a good strategy. This is how they could improve their grades in PISA. And for underachievers, such a policy is ineffective. Moreover, the abundance of applied problems in their curriculum decreases their grades more severely than those of their better achieving classmates. ‘Such students should rather pay attention not to the contents of the course in mathematics, but to the teachers’ characteristics’, Zakharov added.

The researchers looked at how schoolchildren’s achievements correlate with their teachers’ characteristics, such as education and qualifications. They determined three groups of teachers according to the level of their background: teachers of mathematics (graduates of teacher training universities), professional mathematicians (graduates of university departments of mathematics), and teachers without a degree in mathematics. As a result of the study, they revealed the following.

- For A and B students, it’s better to learn from professional mathematicians. This is also true for schoolchildren from families with a high or medium cultural level. For this audience, classes with such teachers will be the most effective;
- For children from families of low cultural level and poor academic performance, there is almost no difference (in terms of PISA grades) who the teachers are, graduates of teacher training universities or professional mathematicians. In this case, the teacher’s qualification plays a big role. It’s important for such students not to have a teacher without a degree in mathematics.

In addition to teachers’ qualification and priorities in terms of problems to be solved, the following factors also contribute to higher grades in mathematics:

- Family cultural level (kids have more opportunities for good studies, as well as more opportunities to learn to solve problems in algebra and geometry in more educated and wealthier families);
- Previous grades in the subject (achievements in studies don’t start ‘from scratch’, but are accumulative);
- The class’s cultural level. In the classes with higher cultural levels, children study better and get better-qualified teachers.

Such a school policy — ‘better class, better teacher’ — is rather popular, yet controversial, Zakharov said. The most qualified teachers are even more necessary for those classes with a lot of children from poorly-educated families. This is known as ‘compensation strategy’. If such a teacher works with these students, they can achieve more in mathematics.

The study was based on the HSE’s longitudinal study, Trajectories in Education and Careers. As part of it, the PISA 2012 test was taken by children who had participated in TIMSS 2011. The sample was over 4,000 people from more than 200 schools. TIMSS allowed the researchers to bridge the gap in the data on the schoolchildren’s previous grades and performance in their classes.

November 23, 2016