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More reasoning and less hand calculations: how to teach mathematics in school, according to mathematicians


Mathematics teachers have identified one of the greatest obstacles for students to connect with the subject. It is the moment in which the student, pencil in hand and tired of clearing the X in the tenth operation of the day, raises his hand and asks: “And what is this for me?” For more than a decade, countries such as Canada have transformed the way mathematics is taught, spending less time repeating calculating exercises by hand to allow more space for reflection and reasoning on how to solve problems in everyday life using the math. That is one of the priorities that the Spanish Mathematics Committee (Cemat) It has been included in a document that it has sent to the Ministry of Education, with which it collaborates in the elaboration of the new school curriculum, which will reach the classrooms in the 2022-2023 academic year.

These repetitive hand exercises are usually given in primary school with the addition, subtraction, multiplication and division of a large number of numbers and in secondary school with so-called fraction castles, operations with radicals or with polynomials. “It is not that students do not have to learn how to do it, but that once they understand the mechanism they can use a calculator or application, and all those class hours that are now devoted to mechanical and repetitive operations can be used to reason about for what and why are these formulas used ”, says Luis Rodríguez, president of the education commission of the Royal Spanish Mathematical Society (RSME), member of Cemat and co-author along with 12 other experts of the report Bases for the elaboration of a curriculum of Mathematics in Non-University Education.

The document collects the current currents of the didactics of mathematics that are prescribed in the PISA report ―prepared by the Organization for Economic Cooperation and Development (OECD) -, in documents of the Ontario Ministry of Education ―which concentrates 40% of the population of Canada -, the National Council of Teachers of Mathematics (the federation of mathematics teachers) of the United States, or the Freudenthal Institute of the University of Utrecht – which owes its name to Hans Freudenthal, one of the parents of the modern mathematical didactics, the so-called realistic mathematics. The ministry will decide whether to incorporate any of the points into the primary and secondary curriculum decrees after the approval in November of the new education law (Lomloe),

“Spending less time repeating algorithms is an important change that should be considered in the curricular reform,” says the Cemat report, which brings together all the scientific and mathematical societies of the country, which for the first time have agreed to lay the foundations of how to redesign the subject. “Less time to memorize procedures in order to acquire a critical sense and be able to communicate and argue logically.” The authors continue the theoretical framework of PISA 2021, which considers that mathematical reasoning helps “make the judgments and make well-founded decisions that reflective citizens need” of the 21st century.

The learning of mathematics, the authors defend, has to be linked to experiences that are significant for the student, something that does not happen now. “In the last years of primary school we already see that they begin to not understand why we train them with so many operations at hand, and in 4th year of ESO we lose them”, indicates Iolanda Guevara, co-author of the report and teacher of Mathematics for more than 30 years. Luis Rodríguez agrees: “In elementary school an important part of the classes are unproductive, we have to ask ourselves why we crush during childhood and adolescence with mechanical procedures that demotivate them,” he adds.

Sixth grade students from the Mare Nostrum public school in Valencia, last February.
Sixth grade students from the Mare Nostrum public school in Valencia, last February. Mònica Torres / EL PAÍS

Connect with students

In the current model, classes start with definitions of new mathematical concepts and then carry out an infinite number of operations. The change proposes starting from problematic situations that connect with the interests of the students. Antonio Moreno, co-author and professor of mathematics didactics at the University of Granada gives an example. “Typically, students are asked to calculate the mean and standard deviation of quantities, without much context. If instead of doing so we ask them who they think is the best player in the NBA based on statistical data, they will connect with what they are doing. ” One student may opt for the player with the most triples per game, while another may argue that the deviation of that player is greater; He has shot more times, but has also missed more. That is also another novelty, moving from a model based on individual activities to another in which discussions and debates are generated, because “interaction evokes reflection.”

The rule of three is another example of how a process is memorized without understanding it well. “You have three numbers and you have to calculate a fourth. As it is a simple procedure, the students apply it even though there is no proportionality between the figures, it is a phenomenon that is widely described in the scientific literature, a classic error of the students ”, says Luis Rodríguez. His colleague Alfonso Gordaliza, co-author of the report and president of CEMAT, gives an example. “Most of the people, already in adulthood, look on the internet and see that to make a paella for six people you need a 50-centimeter-diameter paella pan. When they want to calculate what size they would need for 12 people, they make the rule of three and they think it’s a 100 centimeters. This is not the case because the formula to calculate the area of ​​a circle does not follow that rule ”, he explains. The solution would be to teach students to understand proportionality.

The authors believe that another widespread error in elementary school is made by restricting the teaching of probability to the contexts of card and ball games. “We have detected an abuse by some of the teachers of the examples related to the field of gambling, this gives a distorted idea because it seems that the usefulness of probability and statistics is to understand aspects related to entertainment, when they can be used real contexts of disease transmission or electoral polls ”, indicates Gordaliza.

The other major reform is that the new curriculum includes the use of computers. “Technological challenges necessarily go through connecting school mathematics with programming,” the report highlights. “There are communities of teachers who are very focused on programming who present their experiences with students in congresses, but it is not something generalized,” says Onofre Monzó, co-author and president of the Spanish Federation of Mathematics Teachers’ Societies. Use in primary programming languages ​​such as Scratch or Snap and in secondary Python are some of his suggestions. For example, when studying algorithms, students can program them or while learning geometry, algebra or statistics use programs such as Geogrebra, with which they can simulate 3D geometric figures or create graphs. “We are proposing far-reaching changes that necessarily require teacher training; if not, we are doomed to failure, “warns Monzó. His colleague Cecilia Calvo, also a co-author of the report and a secondary school mathematics teacher for 30 years, emphasizes the need to “break the classical inertia of the teaching staff”.

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