We often think that math must be learnt by heart; that we must be able to apply some formulas or remember certain rules, so we can, automatically, arrive to the correct answer. Moreover, the faster we get to it the better. In fact, we are not the only ones with this perception: data from the PISA 2012 international assessments show that a good number of students in Latin America and the Caribbean think that way. What if instead of doing math, we start thinking in a mathematical way?
According to data from PISA 2012, 30% of Brazilian students described their approach to learning math as: “memorizing as much as I am able to”. In the rest of countries in the region that participated, such as Mexico, Peru, Colombia, Costa Rica, Chile and Uruguay, around 22% of students answered in a similar manner. In Uruguay and Chile, particularly, students responded that when they learn math they “often use strategies that involve memorization”.
Generally speaking, math performance in international assessments tends to be lower for those students who often employ memorization techniques. Among students from participating countries, those with the best results are those who start to tackle a math problem by thinking about what they need to learn, make connections about previous knowledge and what they are dealing with, reflect about different concepts and combine them when faced by a problem. Moreover, what do these students have in common? They all share traits that characterize professional mathematicians and individuals who use math at work or on a daily basis: they are able to reflect, think, and do not rush to apply a theorem or rule they may or not remember correctly.
Famous mathematicians such as Lauren Schwartz (who in 1950 was awarded the Fields Medal, known as the highest recognition given to a mathematician) have confessed their limited ability to perform quick mental calculations. Others have highlighted the need to think and ponder over mathematical problems, trying out possible solutions that might not work at first but that slowly reveal a path that leads to the right answer.
Nowadays, math instructors, such as Professor Jo Boaler from Stanford University, are working hard to develop materials, homework sets, and innovative strategies so students and teachers can use their creativity, critical thinking, and abilities to learn how to think in a mathematical way. In this manner, students can begin to understand that problems can have more than one correct answer; or that there are many strategies that can help them in finding the correct answers, thus encouraging reflection and learning. These new strategies are not trying to remove memorization completely out of the picture; instead, they are meant to explore different learning techniques to equip children and youth with a “mathematical toolkit” they can use when faced with a math problem.
These developments in math instruction are supported by new discoveries in fields such as neuroscience that demonstrate that we learn more and better when we reflect, practice, are pushed to make and effort and have meaningful experiences. Latin America’s performance in math in PISA 2012 showed that there is plenty of room to improve, but also remind us that learning can take place in a wide range of contexts and that students, even those in vulnerable and complex settings, can improve their math skills when given the appropriate tools.
A good first step to improve is to promote the development of mathematical skills and reasoning from an early age. The IDB’s publication “All Children Count” contains lessons learnt from early math and science education in countries in Latin America and the Caribbean. If we manage to change the way we approach math since students first approach it, we can succeed in ensuring that all children and youth in the region can acquire the skills they need to contribute to key areas for the future, such as math, science, and technology. So, what do you say, can we add you to the list?
This blog entry was inspired by an article published by Pablo Zoido and Jo Boaler on Scientific American about math instruction in the United States. Check it out!
The article on the PISA results of the US is unfortunately poorly informed and says half truths. The result is bad advice for lower-income students and countries.
Humans are born with a sense of numbers, but schools must develop it to an intuitive processing of mental numbers line. But calculations of any size cannot just be done on the numbers line. Working memory, what is in your mind right now, is exceedingly short. It holds a few items for a few seconds. Students must MEMORIZE the additions and multiplication tables in order to solve problems without forgetting their content. Only then do students have enough space in working memory to understand and think deeply about math.
For every math course, students must memorize a few basics and use them to process intermediate product instantly in order to concentrate on the math topics of under study. The various philosophies about avoidance of memorization simply don’t take this primary memory function into account.
So what issue did the PISA questionnaire raise? First and foremost that question to students is fuzzy. It asks about reliance on memorization vs. understand of higher-level courses. By then, 15-year olds should have memorized certain basic operations in order to understand the rest. Relying on memorization of content they should comprehend is indeed a bad sign. But it does NOT condemn memorization! To the contrary; the performing students have no other shortcut to get to understanding.
Unfortunately education colleges do not teach the memory basics, so educationists make these fundamental errors. The result is bad advice to poor countries. Hopefully the IDB staff will study the issue more and understand better.