Over the past two decades, there has been a declining emphasis on the importance of reiterating mathematics concepts and skills in learning this subject. The focus has shifted more towards thinking skills and “working out of the field,” which means being able to apply problem-solving skills to real-life situations. This was associated with an increase in the amount of written material in mathematics textbooks in secondary schools and a decrease in the amount of repetitive exercises where only basic mathematical skill is practiced.
This is fine in theory, but the continuing decline in mathematics standards in Australia and the United States suggests that this approach to mathematics is not as good as it seems. The problem lies in the underlying assumption that students already possess the basic skills needed to solve problems. In mathematics, it is not possible to “work out of the square” unless one is qualified with all the skills within the square. For example, a student will not be able to solve a problem about the amount of wire needed to fence a Farmer Brown farm field unless he can accurately calculate the circumference first.
The ability to complete repeated textbook exercises does not guarantee success with application problems. What it does do is provide students with the tools they need to tackle problems beyond the scope of a textbook. Trying to solve abstract problems without a solid skill base is like building a house on the sand; It is a useless exercise.
This mode can be compared to physical training. One can understand the benefits of being able to build muscle through exercise, but unless you do that you will fail when it comes to the task of lifting weights. The math works the same way. Repetition of basic skills builds the skill so that it becomes reactive. When the skill is reflexive it can be applied to other situations. Having the skill does not guarantee success in the application, but it does allow for that success.
The role of repetition of basic skills in mathematics in the context of primary and lower secondary education must be reconsidered. Without a solid foundation of arithmetic skills to build upon, students will continue to struggle with mathematics throughout their school career.