How can mathematics learning in
primary school be facilitated? A recent study conducted by the University of
Geneva (UNIGE), Switzerland, had shown that our everyday knowledge strongly
influences our ability to solve problems, sometimes leading us into making
errors. This is why UNIGE, in collaboration with four research teams in France,
has now developed an intervention to promote the learning of maths in school.
Named ACE-ArithmEcole, the programme is designed to help schoolchildren surpass
their intuitions and informal knowledge, and rely instead on the use of
arithmetic principles. And the results are surprising. More than half (50.5%)
of the students who took part in the intervention were able to solve difficult
problems, as compared to only 29.8% for pupils who followed the standard course
of study. The corresponding study can be found in the journal ZDM
Mathematics Education.
From the age of
6 or 7, schoolchildren are confronted with mathematical problems involving
additions and subtractions. Instinctively, they use mental simulations of the
situations described by the problems in order to come up with solutions. But as
soon as a problem becomes complex, recourse to this representation using
imagery becomes impossible or leads the student into error. “We reflected on a
method that would enable them to detach themselves from these initial
representations and that would foster the use of abstract principles of
arithmetic,” explains Katarina Gvozdic, a researcher at the Faculty of
Psychology and Education (FPSE) at UNIGE. This approach, based on semantic
re-encoding, spurs students to achieve knowledge in arithmetic at an early age.
It was put into practice by teachers in a primary school arithmetic course
called ACE-ArithmEcole that substituted the standard arithmetic curriculum.
So that intuitive mental
representations will give way to mathematical representations
At the end of
the school year, the UNIGE team evaluated ten classes of children aged 6 to 7
in France (second grade of primary school). In five classes, known as the
control classes, the teachers had taught maths in a conventional way. In the
other five classes, they had implemented the ACE-ArithmEcole intervention which
encouraged students to favour abstraction. “To get the students to practice
semantic re-encoding, we provided them with different tools such as line
diagrams and box diagrams,” says Emmanuel Sander, professor at the Department
of Education of the FPSE at UNIGE. The idea is that when they read a problem,
such as “Luke has 22 marbles, he loses 18. How many marbles does he have
left?,” the pupils should detach themselves from the idea that subtraction
always consists in a search for what remains after a loss, and should instead
manage to see it as the calculation of a difference, or a distance that has to
be measured. It’s all about showing students how to re-encode this situation.”
After a year of
teaching based on this practice, the UNIGE researchers evaluated their
intervention by asking the pupils to solve problems that were divided into
three main categories: combine (“I have 7 blue marbles and 4 red marbles, how
many do I have in all?”), comparison (“I have a bouquet with 7 roses and 11
daisies, how many more daisies do I have than roses?”) and change problems (“I
had 4 euros and I earned some more. Now I have 11. How much did I earn?”). In
each of these categories, there were some problems that were easy to represent
mentally and to solve using informal strategies, and others that were difficult
to simulate mentally and for which it was necessary to have recourse to
arithmetic principles.
Undeniable results
At the
conclusion of the tests, researchers observed undeniable results. Amongst
students who had learned to solve mathematical problems with the
ACE-ArithmEcole method, 63.4% gave correct answers to the problems that were
easy to simulate mentally, and 50.5% found the answers to the more complex
problems. “In contrast, only 42.2% of the pupils in the standard curriculum
succeeded in solving simple problems, and only 29.8% gave the right answer to
the complex problems,” exclaims Katarina Gvozdic. “Yet their level measured on
other aspects of maths was exactly the same,” adds Emmanuel Sander.
This discrepancy
can be explained by the frequent recourse to the use of mathematical principles
rather than to mental simulations by the students who had taken part in the
ACE-ArithmEcole intervention. “Thanks to the representational tools that had
been offered to them and to the activities they had recourse to in class, the
students learned to detach themselves from informal mental simulations and
avoid the traps they lead to,” comments Katarina Gvozdic enthusiastically.
The results are
promising and they provide a foundation for promoting abstraction and breaking
away from mental simulations. “Now we want to extend this teaching method to
higher classes, incorporating multiplication and division as well,” continues
Gvozdic. “Moreover, the method could be applied to other subjects — such as
science and grammar — for which intuitive conceptions constitute obstacles,”
adds Sander. The idea is to put semantic re-encoding to widespread use in
schools and to incorporate it more amply into teaching methods.
Journal article:
https://link.springer.com/article/10.1007%2Fs11858-019-01114-z
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