Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations?. Educational Studies in Mathematics

This paper analyzes the content and tasks involving quadratic equations in eight mathematics textbooks published during the period 2000-2012 at the upper-secondary level in Sweden. The study applies the theoretical hypothetical learning trajectory (HLT) framework combining conceptual and procedural knowledge. The analysis includes horizontal and vertical dimensions within an HLT. The aim is to explore embedded HLTs and learning opportunities from both dimensions in these textbooks. A total of 250 examples and 1,068 tasks have been examined. Results show that all the textbooks contain algebra identities and four different methods for solving quadratic equations as well as their applications as a core hypothetical learning trajectory but differ in how an HLT starts and ends. Geometrical representations for some algebra identities and completing the square are widely used in both theoretical presentations and tasks, which implies that conceptual learning is encouraged among the Swedish textbooks. At the same time, procedural knowledge is still emphasized as a basic but important learning process.

The original version of this article contained an editing error towards the end of section 1. The corrected passage is given below. (2011) showed that elementary school students could gain from using emphasizing brackets. In their experiment young students (grades 2 and 3 in Italy and Greece) compared simple algebraic expressions, and emphasizing brackets improved the number of correct answers in cases such as, for example, + 4 = 9 and ( + 4) = 9. Brackets can indeed change the way in which expressions are perceived. Marchini and Papadopoulos