Maths research internationally recognised
11 August 2004
Maths research internationally recognised
Secondary students bamboozled for decades by algebra may soon be helped by teaching approaches used in the Ministry of Education’s Numeracy Project, according to New Zealand research being published in a prestigious European journal.
Mathematics lecturers Murray Britt at the Auckland College of Education and Dr Kay Irwin at the University of Auckland describe important gains in ‘algebraic thinking’ at primary level which has implications for pupils going on to secondary school, soon to be published in Educational Studies in Mathematics, one of the top two international mathematics education journals.
“Algebra is the core of secondary school maths and is thought of as the essence of mathematical understanding. But for most, it has remained a mystery,” says Murray Britt.
A major survey in the 1990s – TIMSS (Third International Mathematics and Science Study) identified disappointing maths achievement in New Zealand pupils. In the 2003 NCEA exam 50% of students did not achieve credit for Level One algebra. Only 4.4% achieved excellence. These are worrying statistics given the central position of algebra to mathematics at the secondary level and beyond.
Better news is contained in the evaluation of the Ministry of Education’s Numeracy Project, released today by the Minister of Education.
Murray Britt and Dr Irwin have presented evidence that the project helps prepare primary students more effectively for algebra because it is teaching pupils to generalise.
“The project is giving pupils the strategies to solve numerical problems quickly in their heads, rather than slavishly writing down long calculations,” says Murray Britt. “This approach implies an early understanding of algebra – known as algebraic thinking – because it is emphasising the ability to generalise across different number operations.”
“Algebra has so far been introduced quite abruptly at secondary level. Students often panic and are stumped when they come across the literal symbols, x and y.”
In 2003, the two researchers tested the ability to generalise of 1,356 pupils in six intermediate schools in New Zealand. Three were in the Numeracy Project and three were not, and included deciles one, three and five. The test showed that children in the Numeracy Project were more able to generalise and apply a range of numerical strategies than were children who were not in the project, and also had greater success in generalising from whole number operations across to decimals.
“The test tells us that not only is numerical reasoning being improved, but that the foundations for success in algebra are being laid earlier, in the primary years. We see algebra in arithmetic; not algebra from arithmetic, as is customarily argued.”
Significantly, the test also showed that lower decile school students made gains equal to those of their higher decile counterparts. In other words, the strategies used by teachers were able to overcome disadvantages customarily relating to decile status.
“We think that this project has enormous potential to improve the link between what is taught in primary and what is taught in secondary schools,” says Dr Irwin.
“If we can demonstrate to teachers, as well as students, that this can be a smooth transition, that it’s not a new world – it’s a development of the world they’ve been working in – then we can produce better mathematicians.”
“The study of sciences as well as law, commerce, medicine, and indeed any area that demands a quantitative literacy requires an understanding of algebra for success. In America, learning algebra is described as a civil right.”
More external evaluation of the Numeracy Project will be needed to see if there is a longer-term effect, says Dr Irwin. “Preliminary piloting has begun in a longitudinal study of year 8 children (form 2), tracking them from intermediate into their first two years at secondary school.”
While some of the newer ideas in the project may be incorporated into the new mathematics curriculum due in 2006, the process of changing habits and attitudes and breaking down the traditional divide between teaching of arithmetic and algebra will take much longer, the researchers say. But it is now happening in other parts of the world.
“Algebra is more than a superficial understanding of literal equations such as 4x + 7 = x – 2. It is about expressing and using generality, locating things together in a connected way. The ability to generalise starts early, when we first begin to use language. It is a skill too important to overlook in young people’s education. It is part of preparing them for careers and success in life,” says Murray Britt.
The Numeracy Project
Children are encouraged to attempt problems mentally and efficiently using a wide range of strategies. Only when a problem is too difficult should a child turn to pencil and paper or the use of a calculator.
Within the Numeracy Project there is an expectation that a problem such as 100 002 - 99 995 would quickly be solved mentally. One efficient way to do this is to notice that 100 002 is 2 more than 100 000 and the difference between 100 000 and 99 995 is 5, so the answer to the problem is 2 + 5 = 7.
A child whose first instinct in solving this problem is to reach for pencil and paper would be diagnosed as being in need of serious teaching about mathematical reasoning.
Another example would be to find 6 x 24. Here there is a range of mental methods, all superior, in terms of thinking, to the usual pencil and paper methods. Notice that children do not write down anything. They make the calculations in their heads.
• Double 6 and halve 24, so 6 x 24 = 12 x 12. And
12 x 12 = 144 may be a known fact. So 6 x 24 = 144.
• 6 x 24 = 6 x 25 - 6. And 6 x 25 is easy since 4 x 25 = 100, so 6 x 25 = 150. So 6 x 24 = 150 - 6 = 144.
• 6 x 20 =120 and 6 x 4 = 24 so 6 x 24 = 120 +24
In a teaching situation, in which students offer these methods, there is an expectation that the teacher will follow up by asking which was the best way. Not all methods are equal.