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 Jesper Lindkvist Department of Mathematics and Science Education Degree Project in Science Education in Postgraduate programme in Education (PGCE), UM9100, 15 HE credits Mathematics Education Bridging Teacher Education Programme in Mathematics, Science and Technology for Graduates with a Third Cycle Degree, 90 HE credits Autumn term 2019 Supervisor: Professor Paola Valero Examiner: Docent Per Sund 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 Jesper Lindkvist Abstract The circle constant, the number pi (p ≈ 3:14), 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 (t ≈ 6:28), 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 360◦ = 2p, and then always adding the extra comment that half a turn is equal to pi, i.e. 180◦ = p. 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. Keywords tau, pi, radians, misconceptions, circle constant, pedagogical disaster, mathematics education Contents Contents . i Introduction . 1 Motivation . 1 Short history of pi . 2 Background of tau . 2 Pi or tau: a pedagogical disaster? . 6 Long term aims and research questions . 7 Method . 8 Analysis and results . 9 Why radians are introduced . 9 How pi and radians are introduced . 10 Using radians . 11 Summary . 11 Discussion and conclusion . 12 Bibliography . 15 Appendix: A constructed textbook example on how to introduce radians with tau . 17 i Introduction Motivation Perhaps the most interesting part of every research project is the motivation that fuels it. Ever since I was a young student myself, the concept of the number pi, p ≈ 3:14, always felt like a needle in my eye. This study explores the concept of an alternative constant, which instead equals two pi. It has previously been suggested by the theoretical physicist dr Michael Hartl, University of California, that this alternative constant should be named tau, t ≈ 6:28 (Hartl, 2010). When I was in the Swedish upper secondary school myself in the years 2002–2005, I remember when the mathematics teacher introduced us to radians (in the Matematik D course). Mathematical beauty and simplicity has always awed me, but now I instead felt a sense of dissonance with regards to pi and radians. The radian concept was not hard for me to grasp, and I realised the beauty of radians immediately, but not its combination with pi. My first instincts were that two pi was a better constant, but as I spent time philosophising about the new concepts I just learned, it was not to dismiss pi, but rather to find ways to make pi seem more special. I came to the conclusion that the period of zeroes in the sine, cosine, and tangent functions is for instance equal to pi, so my mind was satisfied for that moment. When we in the next course (Matematik E) got into contact with Euler’s formula, eix = cosx + isinx; (1) I realised that the fundamental circle constant should not be described with pi, but rather two pi, due to two pi being the rotation that mapped back to the same number in the complex plane. After having had a mental struggle for half a year, I had at least sorted things out for myself. My ambition is that no person should ever even have to face this struggle and the possible misconceptions that arise from it. Why would they have to? Mathematics should be simple, beautiful, logical, and useful. To be able to advance in our knowledge-based society, we shouldn’t be afraid to admit and change our past mistakes. As an active researcher in physics, the number two pi pops up frequently in my equations. The most common examples are the Gaussian (normal) distribution and the Fourier transform. Since I am developing my own simulation codes (Lindkvist, 2018), I have to optimise the programming in such a way to minimise calculations, but also make the code readable. Achieving both of these goals is usually done by defining a constant called something along the lines of “twopi”. People reading the code will understand its meaning, while the computer can be efficient at the same time. The unnecessary operation of multiplying with two is often not the bottleneck for optimisation of a physical simulation code, but it is still something one wants to avoid if the multiplication is done in a nested for-loop. If the same multiplication with two is done several billion times over and over again, it will cut down the running time of the program. When it comes to memory storage, two pi and pi both take the same amount of memory when stored as the same floating point data type, but they also have the same amount of precision. The reason is that a floating point number is stored in bits, either 32 (single precision) or 64 (double precision) in the IEEE 754 standard for floating-point arithmetic (IEEE Standards Association, 2019). Taking the 32 bit example, a floating point number has 1 bit deciding the sign of the number, 8 bits deciding the exponent to multiply with, and 23 bits as the mantissa, or significand (the digits after the leading one of a number in binary). Tau would for instance be stored in memory as 0 10000001 10010010000111111011011 (2) and if each of these sections are seen as binary numbers, it results in representing the floating point number 1 as t = 2p ≈ (−1)(signedbit) · 2(exponentbitsminus127) · 1:(significandbits) = (−1)0 · 2(129−127) · 1:570796251296997 (3) = 6:283185005187988 The programming language Python is one of many programming languages that has implemented tau into their numeric and mathematical modules since release 3.6 (Python Software Foundation, 2019). Tau is stored as the constant “math.tau” with the following documentation: The mathematical constant t = 6:283185:::, to available precision. Tau is a circle constant equal to 2p, the ratio of a circle’s circumference to its radius. To learn more about Tau, check out Vi Hart’s video Pi is (still) Wrong, and start celebrating Tau day by eating twice as much pie! This is a result of tau becoming more popular in the scientific community. At least one paper in the field of mathematical analysis has been published using tau instead of pi (Harremoës, 2016). Short history of pi One cannot argue that pi has great historical importance. The first way of representing the circle constant in a rigorous algorithm using polygons started with Archimedes in ancient Greece around 240 BCE (Berggren et al., 1997). It wasn’t until the mathematician William Jones used the notation for pi for the ratio between half the periphery and the radius (or the periphery and the diameter) of a circle in 1706 (Berggren et al., 1997). Pi had been used to denote the periphery of a circle with any radius before, but it was the renowned mathematician Leonard Euler who later popularized the use of pi we know today, i.e. as the ratio between the periphery and the diameter of a circle (Berggren et al., 1997). From a practical point of view, I can imagine that it is easier to measure the diameter of a circle than its radius. Background of tau The number pi is defined as the ratio between the circumference of a circle to its diameter. Unfortunately, the diameter is not the defining property of a circle.