by Alfonso Echazarra
Analyst, Directorate for Education and Skills
|PISA 2012 Released Mathematics Item (Proficiency Level 6)
Some of the greatest geniuses had remarkable memories. Mozart, according to legend, sat and listened to Allegri’s “Miserere”, then transcribed the piece of music, entirely from memory, later in the day. Kim Peek, the savant who was the inspiration for the blockbuster film, Rain Man, memorised as many as 12 000 books. But unlike Mozart, who composed more than 600 works during his brief life, Peek was unable to distinguish between the relevant and the irrelevant, or discover hidden meanings and metaphors in the texts he had committed to memory.
What do these stories have to do with learning mathematics? Or, put another way: in light of these stories, how would you encourage students to learn mathematics? By understanding what mathematics concepts, procedures and formulae mean and applying them to a lot of different maths problems set in a lot of different contexts? Or by learning them by heart and applying them to a lot of similar maths problems? Sooner or later, the method matters. Students who avoid making an effort to understand mathematics concepts may succeed in some school environments; but a lack of deep, critical and creative thinking may seriously penalise these students later in life when confronted with real, complex problems. As Albert Einstein provocatively said: “Any fool can know; the point is to understand”.
A similar message is relayed in this month’s PISA in Focus
and a new OECD paper on education, “How teachers teach and students learn: Successful strategies for school
”. The analyses show that students who mainly use memorisation when they study do well on easy questions. For example, “CHARTS Q1
”, a multiple-choice question from the PISA 2012 test, refers to a simple bar chart and is considered one of the easiest questions in the mathematics assessment. Some 87% of students answered this question correctly. Students who reported that they use some type of memorisation strategy when they study mathematics, such as learning by heart, recalling work already done or going through examples again and again, had about the same success rate on this easy item as students who reported using other learning strategies.
But complex problems are a different matter; they require more than a good memory. For the most challenging question from the PISA 2012 mathematics test, “REVOLVING DOOR Q2
”, students who reported using mainly memorisation strategies were much less likely than students using other strategies, such as connecting ideas or working out exactly what is important to learn, to answer correctly. Answering “REVOLVING DOOR Q2” correctly requires substantial geometric reasoning and creativity, involves multiple steps, and draws heavily on students’ ability to translate a real situation into a mathematical problem. Only 3% of participants answered this question correctly.
The findings also show that, contrary to received wisdom, East Asian students are not necessarily the ones who use memorisation strategies the most. Memorisation is used almost everywhere, but fewer 15-year-olds in Hong Kong-China, Japan, Korea, Macao-China, Shanghai-China, Chinese Taipei and Viet Nam reported using it than students in, let’s say, English-speaking countries to whom they are often compared. For instance, 5% of students in Viet Nam, 12% of students in Japan and 17% of students in Korea reported that they learn as much as they can by heart when they study mathematics, compared to 26% of students in Canada, 28% in Ireland, 29% in the United States, 35% in Australia and New Zealand, and 37% in the United Kingdom.
In some situations, memorisation is useful, even necessary. It can give students enough concrete facts on which to reflect; it can limit anxiety by reducing mathematics to a set of simple facts, rules and procedures; and it can help to develop fluency with numbers early in a child’s development, before the child is asked to tackle more complex problems. But to perform at the very top, 15-year-olds need to learn mathematics in a more reflective, ambitious and creative way – one that involves exploring alternative ways of finding solutions, making connections, adopting different perspectives and looking for meaning. So yes, you can use your memory; just use it strategically, lest Einstein call you a fool.
PISA in Focus No. 61: Is memorisation a good strategy for learning mathematics? by Alfonso Echazarra
PISA á la loupe No. 61: La mémorisation : Une stratégie payante pour l’apprentissage des mathématiques?
How teachers teach and students learn: Successful strategies for school.
PISA Try the Test: Explore PISA 2012 Mathematics, problem solving and financial literacy test question
Source: PISA 2012 Released Mathematics Items