09 Dec Hands (and minds) on Maths
Hands (and minds) on Maths
Can life be explained through mathematics? Is the universe entirely mathematical? And what is the best way to learn this subject? Renowned researcher and full professor Nuno Crato shares his particular vision on these intricate mysteries.
MATHS AND LIFE: AN INTERSECTION
“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?” This quote from Albert Einstein, written somewhere on a wall in one of Nuno Crato’s classrooms when he was at secondary school, has stayed with him to this day because, for him, it encompasses one of the great mysteries of mathematics: “That question still gives me the shivers. There are lots of answers, and mathematicians and philosophers are divided over this question.”
For the Pythagoras school, the world is an embodiment of this exact science, but Professor Crato is hesitant: “Basically, mathematics is a language of things. But it’s a language that has a life of its own. It’s almost as if the world had another parallel life. I believe there is an amazing symbiosis between the two, which is fascinating. Here’s an example: today, we can discover planets outside our solar system without seeing them because we apply mathematical principles to analyse variations in the radial velocity of a distant star.
Mathematics helps us to conclude that there has to be a planet in that place.” But while some people enjoy dealing with complex problems, many (most?) feel tense, unsure and incapable, often when faced with the simplest of questions. Is there something wrong with maths for it to be seen as a boogie man by so many school-goers around the world? Once again, this is a complex issue. Into this equation, Nuno Crato brings a study published in 2020 by three Italian psychologists from Università di Trieste, who tested the efficacy of more cognitive or more emotional strategies in the management of the anxiety of 224 primary school pupils in 4th grade.
Nuno Crato has written several best-selling science books, playing a leading role in bringing mathematics to a wider audience.
For the study “Math anxiety and math achievement: The effects of emotional and math strategy training”, the researchers distributed the children randomly in three groups, two for intervention and one for control purposes. The children in one of the groups joined a maths teaching programme, another group learnt techniques to combat anxiety, while the control group stayed in the classroom reading comics and drawing. Each weekly training programme consisted of eight 60-minute sessions.
The conclusion reached was that the programme to combat anxiety and the maths teaching programme had the same results in combating anxiety. But the maths teaching programme had an added advantage: “not only were the pupils less anxious during tests, but they also learnt maths”, says Crato, who is also President of Iniciativa Educação (the Education Initiative).
Crato says that it all comes down to the approach: “There are general techniques for combating anxiety. For example, people who are afraid of heights shouldn’t sit near the window in a helicopter. But if we want to fight this fear of heights, we can do this little by little: first, stand on a chair, stand on a table, then look out of a window, climb a mountain. I mean, one of a series of approaches that can be done progressively and that helps people to combat that anxiety. But don’t avoid studying, to study is the essential thing.” In the case of maths, testing is a good approach strategy. And this is one of the reasons for the success of the Singapore teaching model, one of the ones that “has achieved the best results in the entire world. It’s a progressive teaching model, going from the concrete to the abstract, going from simpler to more complex tasks.
“Mathematics is a language of things. But it’s a language that has a life of its own. It’s almost as if the world had another parallel life.”
Going from one step in reasoning to two steps, going from two steps in reasoning to three steps. And always consolidating the knowledge being gained before moving on to the next phase”. Nuno Crato believes that “everybody can get there, if they’re taught properly”. He admits that some cultures may start out with certain advantages for some specific elementary things and gives the example of China: “Some people claim that the number designation system used by the Chinese has some advantages, as it is more transparent and facilitates certain simple arithmetic operations.” And at elementary level, sums can be done on your fingers – the first calculator of primitive civilisations: “For a long time, it was said that counting on your fingers was wrong and I remember at school, the teachers would say ‘children, hands on the table’, to see if the children were counting on their fingers to answer simple arithmetic questions.
Nowadays, there are several psychological studies that have come to the conclusion that fingers can be useful to help children understand the sense of numerical progression.” And tables? Should you know them by heart? Here, the mechanism is similar to that of reading: “When we adults are reading a text, we don’t read it syllable by syllable or letter by letter. We don’t even realise we’re reading; reading has become automatic in our brains. With basic arithmetic, everything needs to be automatic too, so that people don’t say ‘1+1, how much is that? Let me get my calculator’.
Being able to do things without thinking is beneficial because it lets us think about other things.” Nuno Crato takes issue with excessive dependence on calculators: “Of course young people should use a calculator if they want to know the square root of 427, but they shouldn’t use a calculator to work out 3 x 12 or for small sums and fractions.” As a researcher, he likes “fighting things to see if I can find a solution”, even if they’re only modest problems, “and I really enjoy working with others to do this. Like doing scientific work with somebody who’s very good at computing or physics”.
He remembers one project in particular, on fishing in the North Atlantic. “It was great fun because we had biologists, geographers, people in the climate area and one statistician (me), and we found that there was a relationship between the difference in atmospheric pressures at a certain time of the year and sardine fishing the following year.” For Nuno Crato, scientific divulgation of studies like this one has the power to show how science has evolved in its constant questioning, contributing to society acquiring “a more democratic and tolerant view of things.”
Nuno Crato is a researcher and a professor at the ISEG Lisbon School of Economics and Management, University of Lisbon. Now leads Iniciativa Educação.
INICIATIVA EDUCAÇÃO FOCUSES ON ACADEMIC SUCCESS
The original idea came from its promoters, Teresa and Alexandre Soares dos Santos, a couple for whom education is the key to progress in Portugal. Alexandre Soares dos Santos led the Jerónimo Martins Group and also the Santos family, the Group’s controlling shareholder, for over 45 years. The ultimate goal of Iniciativa Educação is to contribute towards real training and qualifications for young people, especially those who, due to economic, family, educational or social difficulties, experience the drama of academic failure.
There are three basic programmes: “AaZ – Ler Melhor, Saber Mais” (A to Z – Read Better, Know More), “SerPro” (Being a pro) and “ED_ON” (Education online). In its first years of operation, the “AaZ” programme helped 124 first- and second-grade pupils (aged between 6 and 7) to overcome their reading and writing difficulties. According to OECD, 19% of European 15-year-olds have serious difficulty in reading and understanding basic texts. One of the tools provided by “AaZ” to deal with this problem is access to videos where children’s stories are read slowly, with the words coming up on the screen a fraction of a second before the voice of the narrator is heard.
Teresa and Alexandre Soares dos Santos have invested personal and family funds to create Iniciativa Educação.
Its goal is to help ensure young people achieve educational success.
This facilitates the connection between the visual stimulus that the brain receives and the auditory stimulus that comes shortly afterwards. As the programme advances, the children become more and more at ease in recognising and connecting words and sounds. The earlier something is done, the better, as otherwise the gap between pupils with difficulties and other pupils has a tendency to increase, not decrease. The “SerPro” programme is aimed at supporting young people to complete secondary education, while giving them practical training and vocational qualifications.
This is a programme based on cooperation between schools, companies, municipalities and professional associations for the development of vocational training courses of interest to students and that are necessary for the economy. Finally, “ED_ON” provides articles, statistics and other tools to help teachers, parents, students and the public in general to have access to information, so as to allow for informed debate in Portuguese society.
Scientific articles on educational topics are a valuable resource, not only for teachers, but also for researchers and other professionals in the area. Iniciativa Educação believes that school failure can seriously compromise the future of young people. If provided with adequate support, pupils will be able to achieve academic success, moving one step closer to achieving a promising future.