The circle constant remastered for less misconceptions : A textual analysis of the introduction of radians in mathematics education textbooks for the Swedish upper secondary school

Abstract: The circle constant, the number pi (), is defined as the ratio between the circumference of a circle to its diameter. When it comes to radians, the angle measurement which relates the circumference of a circle to its radius, one turn equals two pi. This extra constant of two is something everyone has to remember and practice. An extra constant of two shows up in several equations in both mathematics and physics, which still are mathematically correct, but is there due to our way of defining the circle constant. Researchers have therefore recently suggested a new circle constant, tau (), which instead equals two pi. Now one turn equals one tau, and equations become more generalised, simpler and more aesthetic, some people argue. This means that pi could potentially be the cause of misconceptions when pupils learn the concept of radians in upper secondary schools, and should be investigated further in a school setting. A first exploration of problems related to the introduction of radians is thus needed. In this study, I define the problem of using pi from three perspectives: aesthetics, mathematics, and usefulness in both mathematics and physics. A textual analysis of the introduction of radians in the textbooks used for the upper secondary school mathematics is the main topic of the study. A proposed alternative way where tau is used instead of pi is given as a constructed textbook example. The analysis of the sections introducing radians of mathematics textbooks for the Swedish upper secondary school shows that pi is often introduced by first stating that one full circle corresponds to , and then always adding the extra comment that half a turn is equal to pi, i.e. . This results in pupils having to remember an exception to a rule. Radians are always implemented in the textbooks in order to work with circle arc lengths and circle segment areas. When pi is used, these equations differ from the looks of the equations for the circle circumference and the circle area with a factor two. In a constructed textbook example that introduces radians, I use tau instead of pi, and the above problems are averted. This study shows that tau is both aesthetically pleasing and practically useful in some parts of mathematics and physics. Using tau instead of pi could make misconceptions less likely to happen, which is exactly why this field and topic need to be researched more in a classroom setting with pupils.