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The remastered for less misconceptions

A textual analysis of the introduction of in education textbooks for the Swedish upper secondary school

Jesper Lindkvist

Department of Mathematics and Science Education 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 (π ≈ 3.14), is defined as the ratio between the of a circle to its . When it comes to radians, the measurement which relates the circumference of a circle to its , 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 (τ ≈ 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◦ = 2π, and then always adding the extra comment that half a turn is equal to pi, i.e. 180◦ = π. 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, π ≈ 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, τ ≈ 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 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 , 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 that mapped back to the same number in the . 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 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 , it results in representing the floating point number

1 as

τ = 2π ≈ (−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 τ = 6.283185..., to available precision. Tau is a circle constant equal to 2π, 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 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 using started with 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 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. There are several two-dimensional objects which have constant width. The defining property of a circle is its radius. The number pi is thus describing the ratio between half the circle’s circumference and its radius. To choose the constant in this way sends ripples through all mathematics and physics, and an extra constant of two showing up along for the ride.

It is argued by Hartl (2010) that when people encounter the radian angle measurement unit (one turn is two pi) for the first time, they have the unnecessary factor of two arising, yielding results such as half pi equals a quarter turn, obscuring the underlying relationship between angle measure and the circle constant. It is then stated by Hartl (2010) that to those who think it doesn’t matter, take a look at Figures 1, 2 and 3. The following statement is then given:

You will see that, from the perspective of a beginner, using pi instead of tau is a pedagogical disaster (Hartl, 2010).

This quote made me wonder if someone actually has looked into the concept of pi from a pedagogical view, especially with regard to radians. Researchers have looked into the aesthetics of mathematical proofs before (e.g., Raman Sundström (2012); Raman Sundström and Öhman (2018)) but there has not been any research where well established mathematical concepts are questioned from a pedagogical point of view

2 before. This “pedagogical disaster” needs to be looked at from several perspectives. The perspectives of focus in this paper will be aesthetics, mathematical, and usefulness in mathematics and physics.

Figure 1: in radians using pi, Figure 2: Angles given as frac- Figure 3: Angles in radians using from Hartl (2010). tions of a turn, from Hartl (2010). tau, from Hartl (2010).

Back in the start of the 21st century, professor of mathematics Bob Palais, University of California, published an opinion paper with the title of “Pi Is Wrong!” (Palais, 2001). This can be seen as the first serious scientific paper questioning the use of pi. He suggested another symbol to represent two pi that looked like a pi with three vertical lines instead of two, something like ππ, which he never gave any name and which never became popular in the scientific community.

The constant of two pi shows up at several places in mathematics (Hartl, 2010), e.g., in the circumference of a circle, b = 2πr, (4) the Gaussian (normal) distribution, 2 1 − (x−σ) √ e 2σ2 , (5) 2πσ the Fourier transform, Z ∞ f (x) = F(k)e2πikx dk, (6) −∞ Z ∞ F(k) = f (x)e−2πikx dx, (7) −∞ Cauchy’s formula, 1 I f (z) f (a) = dz, (8) 2πi γ z − a the Gauss-Bonnet formula, Z Z K dA + kg ds = 2πχ(M), (9) M ∂M special solutions to the ,

∞ 1 ζ(2n) = ∑ k2n k=1 (10) B = n (2π)2n, n = 1, 2, 3, ..., 2(2n)!

3 the n:th roots of unity,

zn = 1 (11) 2πiN/n ⇒ z = e , N ∈ Z, or simply the non-unique complex solutions to unity,

z = 1 2πiN (12) ⇒ z = e , N ∈ Z, just to name a few, and there are several more examples from both mathematics and physics. Euler’s identity is personally the biggest argument of dismissing pi for tau, and is an argument especially from an aesthetic point of view. Euler’s identity, eiπ = −1, (13) is usually never written with a minus sign and instead rewritten as

eiπ + 1 = 0, (14) to then be given a remark that the equation is beautiful because it contains the relationship of the most important numbers of mathematics: 0, 1, i, e, and π. However, the mathematics behind it says that if you rotate half a turn (pi radians) from 1, you end up at -1. The same type of equation written with tau would instead be eiτ = 1, (15) where there is no need to flip any signs around, and with an equation that shows the deep relationship that one tau corresponds to one turn around the complex plane. Since pi and tau are interchangeable one can now make the statement that the equation includes four of the most important numbers of mathematics: 1, i, e, and τ. One could of course always add a zero to any side of the equation if one wants to include that number as well: eiτ = 1 + 0, (16) which is completely unnecessary, but is exactly why the aesthetics and simplicity of using tau is more appealing in Eq. 15 than using pi in Eq. 14.

The relationship of one tau being one turn is seen when comparing the solutions of eiz = −1 against eiz = 1. The yielded solutions become z = π + 2πN and z = 2πN respectively, where N is an integer, or τ with tau notation z = 2 + τN and z = τN respectively. I would not be alone to argue (see Hartl (2010)) that the underlying beauty rather lies in the period of a full rotation in the complex plane (tau) and arrive back where one begins, rather than in the specific solution to what angle is needed to rotate half a turn in the complex plane. Using pi obscures the underlying relationship between the circle constant and the radian angle measurement.

The , πr2, is often used to advocate for pi. But one should remember that the area of a circle, A, is the result of integrating the area element (circumference times thickness), τr0dr0, from the centre of the circle out to the periphery, Z r A = dA 0 Z r = τr0 dr0 (17) 0 r2 = τ , 2

4 where the quadratic shape of the equation reminds us a lot of other equations in physics and mathematical v2 analysis (see Table 1). In physics, the momentum of a particle is mv while its kinetic energy is m 2 . For a particle at rest in a field of constant acceleration (such as the gravitational acceleration, g, close to the t2 Earth’s surface) the particle velocity at time, t, is gt, and the distance moved is g 2 . The in a linear spring is proportional to the extension, x, from equilibrium according to Hooke’s law, kx. The mechanical x2 energy stored in such a string is k 2 .

Table 1: The linear and quadratic forms of various equations in physics and mathematics.

Quantity Expression Quantity Expression v2 Momentum mv Kinetic energy m 2 t2 Velocity gained gt Distance moved g 2 x2 Spring force kx Spring energy k 2 0 00 x2 Maclaurin linear term f0 x Maclaurin quadratic term f0 2 r2 Circle vr Circle segment area v 2 r2 Circle circumference τ r Circle area τ 2 Circle circumference 2π r Circle area π r2

This quadratic shape is also seen in mathematics of the quadratic term of the Maclaurin/Taylor expansion of any function. Any continuous and smooth function, f (x), can be described by an infinite sum, i.e. a series expansion. The infinite sum and the first terms of such an expansion around x = 0 is given by

∞ xn f (x) = f (n) ∑ 0 n! n=0 (18) x2 x3 x4 = f + f 0 x + f 00 + f 000 + f 0000 + ... 0 0 0 2 0 6 0 24

The expression for the circle circumference given with pi is 2πr, and the expression for the circle area is πr2. These shapes are a factor of two off from the typical shapes seen in Table 1. The equations that contain pi are not generalised as much as they could.

I now summarize the arguments for the three perspectives, mathematics, usefulness, and aesthetics in Table 2. From a mathematics perspective, having one turn equal one tau couples the circle constant and the radian angle measurement without having a factor of two, giving the easiest and most generalised relation. This shows most prominently when Euler’s formula equals unity in Eq. 15. From a usefulness perspective, writing equations in linear and quadratic forms shows that using tau instead of pi would yield the same type of equations for the linear equations of circle arc length and circle circumference versus the quadratic equations of circle segment area and circle area. More often than not it also benefits scientific computing to use tau instead of pi. From an aesthetic perspective, most equations look simpler when using tau instead of pi and could therefore lead to less misconceptions.

5 Table 2: Short summary of arguments from three different perspectives of why tau could be superior to pi.

Mathematics Usefulness Aesthetics One turn equals one tau Linear and quadratic forms are Simple and elegant equations represented with the same shape due to a factor of two missing of equations More generalised and with less Efficient computing algorithms Less misconceptions? exceptions

Pi or tau: a pedagogical disaster?

Pi has been a constant that is used in simple geometrical problems, and is part of important mathematical ideas. However, there is a debate on whether it could made sense to keep pi in mathematics or if there could be an advantage to use tau instead. The pedagogical disaster of using pi instead of tau that Hartl (2010) mentions is that people keep on learning and using pi even though its use generates some problems in understanding and in applications.

When it comes to how pi is introduced in upper secondary mathematics, a critical topic is that of radians, because it is here where the advantage of tau over pi becomes clear, and also where problems may be generated to pupils’ understanding, i.e. misconceptions. Research suggest that pupils’ understanding of the concept of radians is central for building notions of trigonometry and for some applications:

In a study by Keçeli and Turanlı (2014) of undergraduate students related to complex numbers and misconceptions they state the following in their conclusion:

In addition, they [students] make misconceptions when calculating the n.degree roots and exponents of a z = 1 . Students lack information related to the argument concept.

In addition, students accept the π number to 180◦ and use it in equations as such.

The study discovered that misconceptions arise due to the argument concept, i.e. the angle in radians, and students simply accepted the formula π = 180◦. Another study by Topçu et al. (2006) of mathematics teachers’ concept image of radians says that

Participants did not consider radian as a although the that were given to them were explicitly defined on the set of real numbers. ... None of the participants defined the radian as a ratio of two lengths. Secondly, participants who have stronger concept images of radian established richer connections between and other concepts in trigonometry ....

These studies find that misconceptions with regards to radians are common, but also that someone with a stronger concept image of radians has a better understanding of other concepts in trigonometry. These misconceptions could be partially based on using pi as the circle constant. Therefore, it becomes of interest to start exploring whether the “pedagogical disaster” is potentially present in the Swedish upper secondary school.

6 In the current Swedish upper secondary school curriculum for the mathematics subject (Skolverket, 2018b) it is not explicitly specified that one should include pi. However, the following line is written:

Vidare ska den [undervisningen] ge eleverna utmaningar samt erfarenhet av matematikens logik, generaliserbarhet, kreativa kvaliteter och mångfacetterade karaktär.

This line is officially translated (Skolverket, 2012) to mean

In addition, it [the teaching] should provide students with challenges, as well as experience in the logic, generalisability, creative qualities and multifaceted nature of mathematics.

The generalisability part, I interpret as striving towards finding relations between various parts of known mathematics and to find easier ways of solving the same problem but in a more optimised way, e.g., a generalised formula. The multifaceted character of mathematics could be interpreted to also include the aesthetics of mathematics, and for instance mathematical formulae.

In the very first paragraph of the Swedish upper secondary curriculum (Skolverket, 2018a) it states that

Undervisningen ska vila på vetenskaplig grund och beprövad erfarenhet. which according to the official translation of the Swedish upper secondary curriculum (Skolverket, 2013) can be translated into English as

The education should be based on scientific grounds and proven experience.

Having teaching based on science would mean that in order to introduce a new concept, such as tau, we would have to conduct field studies in classrooms and find the best didactic ways to work with tau. One should assess the pupils knowledge of some teaching sequence and compare that to a group of pupils that keep using pi. If we in the end come to a scientific consensus that the circle constant could instead be changed to tau in the future, the teaching practice will no doubt change as well.

Long term aims and research questions

In the long scheme of things, I hope that we are able to answer whether the concept of tau would make mathematics simpler or not, and if so, when and why it would be better. I wish to understand that if pupils at upper secondary school mathematics fail at learning radians, is it because of the very concept of pi or how pi is being taught in the classroom? What situations could change such that tau could instead be used as a good tool in the school of today? These questions are indeed grand, but unfortunately out of the scope of this project. They are however the questions that one wishes to work towards.

The aim of this study is to do a first exploration of the introduction of the concept of radians as a key topic of upper secondary school mathematics where pi is traditionally used, and see what kinds of problems emerge in how the concept is introduced. This aim is best achieved by doing a textual analysis of the textbooks used in Swedish upper secondary school mathematics education where radians are introduced.

The overall aim of this first exploration is to think how tau could offer an alternative by constructing a mathematics textbook example on how to introduce the concept of radians using tau, and evaluate where the use of tau is better than when using pi. This is done in the discussion section.

7 To work towards these aims, and categorise the problems that emerge when radians are introduced, the following research questions will be addressed in the textual analysis:

• How is the concept of radians introduced in textbooks? Are any reasons given?

• In what part of the introduction of radians do textbooks introduce pi, and in what way? How do textbooks motivate that pi equals 180 degrees?

• What are radians used for in textbooks immediately after its introduction?

• How could tau be used instead of pi in the textbooks? What advantages can be found?

Method

In this study, I make a textual analysis of Sweden’s upper secondary school mathematics textbooks from different publishers and publication years. The topic focuses on how the concept of radians is introduced with regards to pi, and if a different approach could be suggested using a new constant, tau, that equals two pi for future generations of pupils. The textual analysis method is described by Hawkins (2018) and is closely related to what Bryman (2008) calls a qualitative content analysis.

The textbooks are such that they are in tune with the Swedish upper secondary school curricula of the time. Two such periods are chosen: The current curriculum since 2011 (Lgy11) and the one before that from 1994 to 2010 (Lpf94) (Skolverket, 2006, 2018a). From 1994 and onwards, the Swedish school has been governed according to a goal-oriented approach among pupils, instead of the earlier relative skill assessment system (Lundgren et al., 2017), which is why I choose to the study to literature published for the purpose on these curricula.

For the textual analysis I am investigating Swedish mathematics textbooks in how they introduce and use the concept of radians. This is done in the course "Matematik D" for Lpf94 and "Matematik 4" for Lgy11. I look through the mathematics textbooks to find good measures of categorisation, which are already represented in the research questions.

The books selected as data material for the study are differentiated below depending on their publisher, which curriculum they belong to, and in which chronological order during an active curriculum that publisher has a textbook for that mathematics course in this study:

• Natur och kultur, Lpf94: 1 (Björk and Brolin, 1995)

• Natur och kultur, Lpf94: 2 (Björk and Brolin, 2001)

• Natur och kultur, Lpf94: 3 (Alfredsson et al., 2009)

• Natur och kultur, Lgy11: 1 (Alfredsson et al., 2013)

• Liber, Lpf94: 1 (Holmström and Smedhamre, 2002)

• Liber, Lgy11: 1 (Holmström and Smedhamre, 2013)

• Sanoma Utbildning, Lgy11: 1 (Szabo et al., 2013)

8 • Gleerups, Lpf94: 1 (Björup et al., 1995)

• Gleerups, Lpf94: 2 (Björup et al., 2001)

This is also how these books will be referred to in the text onwards for comparison value.

I also include a university book (Dunkels et al., 2002) which purpose is to make a repetition of the upper secondary school mathematics. To also get a historical perspective, I include the compiled works of Sigurd Eriksson’s textbooks (Eriksson, 2013, 2015) from his time as teacher at – what today would be equivalent to an upper secondary school – Högre Tekniska Gymnasiet (TGÖ) in Örebro, Sweden, where he was active from 1917 to his death in 1948. Their purpose were to give a mathematics background for engineers to use in society, as well as a good introduction for further university studies. These books are included to give the study a wider range in both educational era (the old textbook for upper secondary school by Eriksson) and purpose of the book (repetition of upper secondary school by Dunkels).

Analysis and results

Here in the analysis and results section, I divide the topics into three subsections that are highly linked with the research questions of this study, and summarise them in the results subsection. All textbooks previously mentioned in the method section are studied in detail under each subsection.

Why radians are introduced The textbooks from Natur och kultur introduce radians in order to explain the of the trigonomet- ric functions, in particular the sine function. The way they do this depends largely on the authors rather than the curriculum at the time. Natur och kultur, Lpf94: 1 & 2, show with the definition of derivative,

f (x + h) − f (x) f 0 (x) = lim , (19) h→0 h that the derivative of the sine function, f (x) = sinx, is

0 f (x) = c1 sinx + c2 cosx, (20) where cosh − 1 c1 = lim , h→0 h (21) sinh c2 = lim . h→0 h When the angles are calculated with degrees, the limits become

c1 = 0, (22) c2 ≈ 0.01745329.

Now, the textbooks ask the reader if there could be an easier way of measuring angles in such a way that the constant, c2 here, would become easier [enklare, in Swedish] (only implying that the constant should equal unity for the correct angle measurement unit).

This approach is almost identical to the textbooks of Gleerups, Lpf94: 1 & 2. One difference is that

9 Gleerups make an explicit question to the reader about how to choose a new angle measurement that makes the constant equal to unity. The limit is recognised as c2 = π/180, and is the conversion factor between the new angle measurement, radians, and degrees.

Natur och kultur, Lpf94: 3 & Lgy11: 1, with new authors, start with the same reason behind why to introduce radians (i.e. the need to make the derivative of the sine function easier). However, their approach is not from the point of view of the definition of derivative, but rather what the says the derivative of the sine function is depending on if the calculator’s setting is on degrees or radian measurement.

The textbooks Liber, Lpf94: 2 & Lgy11: 1, both by the same authors, motivate the introduction of radians by saying that the radian measurement of angles is useful in practical use, giving an example of how the voltage in a power outlet varies over time.

The textbook Sanoma Utbildning, Lgy11: 1, motivate radians by their use in simplifying equations, with reasoning from the length of a circle arc, v b = · 2πr, (23) 360◦ b = vr, (24)

where v is the angle measured in degrees and radians, respectively.

This way is similar to how the old textbooks by Sigurd Eriksson introduced radians, simply stating that mathematical formulae for the circle arc length and a circle segment area become easier. Even though he introduces the concept of radians in his first book, he waits until the second book to use radians to show that the derivative of the sine function is the cosine function.

The university book by Dunkels et al. (2002) introduces radians without an explicitly given reason, defining it first and secondly stating that the book will always use radians from now on if nothing else is explicitly said.

How pi and radians are introduced The unit circle (a circle with radius equal to one) is used in almost all books to illustrate what one radian represents geometrically. The exceptions are the books by Sigurd Eriksson and Liber, Lpf94: 1 & Lgy11: 1. These books instead show a circle with radius, r, and an arc length, r, further stating that an angle giving rise to such a configuration is the definition of a radian. The four books from Natur och kultur also show this as the definition of radians geometrically, but they also include the unit circle. All books from Natur och kultur have a small highlighted where they summarize the radian to degrees conversion. In Natur och kultur, Lgy11: 1, they literally write the following:

Ett varv = 360◦ = 2π rad, d v s 180◦ = π rad

◦ π 1 = 180◦ rad ≈ 0,01745 rad

180◦ ◦ 1 rad = π ≈ 57,3

which translated to English would be something like

One turn = 360◦ = 2π rad, i.e. 180◦ = π rad

10 ◦ π 1 = 180◦ rad ≈ 0.01745 rad

180◦ ◦ 1 rad = π ≈ 57.3

The number pi is always introduced in the textbooks where radians are mentioned. Almost all books introduce pi in those sections by stating that two pi is the angle that corresponds to the length of a circle arc of a whole circle. One exception is Gleerups, Lpf94: 1 & 2, where they show that the tangent to the sine function at the origin has the value y = π when x = 180◦ (where the sine function reaches zero again), further proposing that a new angle measurement where π = 180◦ would make a tangent with slope equal to unity (i.e. the radian angle measurement). The other exception comes from the old textbooks by Sigurd Eriksson that state that if the arc would be half the periphery, the arc length would be b = πr, therefore 180◦ = π radians.

Using radians In the parts immediately following the introduction of radians in the textbooks, all of them, with the exception of the university book by Dunkels et al. (2002), start by having exercises to practise the conver- sion between degrees. The books continue with exercises where one has to use radians in trigonometric functions. By giving an angle, v, in radians, the books continue to define the length of a circle arc,

b = vr, (25)

and the area of a circle sector, vr2 A = . (26) 2

The university book by Dunkels et al. (2002) starts with a practice of using radians in trigonometric functions, but immediately continues with trigonometric identities, and how to solve trigonometric equations. It is therefore the only book that doesn’t mention the length of a circle arc or the area of a circle sector.

Summary When most textbooks introduced pi, they did so by first saying that one full circle corresponds to 360◦ = 2π and then always have to add another comment that half a turn is equal to pi, i.e. 180◦ = π. The concept of radians just got an exception, or an extra thing to remember, i.e. that half a turn equal a whole pi. This could be the reason why misconceptions occurs with pupils, since Keçeli and Turanlı (2014) mentioned that students only learned the relation 180◦ = π instead of remembering any other relations. This is where tau could make a better candidate as a new circle constant. When introducing one full turn, the book would not have to go further than to say: one turn = 360◦ = τ. The conversion factor between radians and degrees, π/180◦, would with tau be the same, but instead be given as τ/360◦.

The university book by Dunkels and the textbooks by Sanoma Utbildning, Lgy11: 1, and Gleerups, Lpf94: 1 & 2, all do not show the definition of a radian geometrically, i.e. where a circle arc length is equal to a circle radius of any size. A lack of geometrical representation could be why the misconception arises with radians not being considered a ratio between two lengths, as was mentioned by Topçu et al. (2006).

One particular textbook, Sanoma Utbildning, Lgy11: 1, explicitly says that the equations for circle arc length and circle sector area become easier to understand if radians are used. The resulting equations end

11 up with the shapes

b = vr, (27) r2 A = v . (28) 2 They are exactly in the form described earlier, i.e. the linear, r, and quadratic forms, r2/2, seen in the Taylor series expansion mentioned in the introduction section, which can also be compared in shape to the area of a circle in Eq. 17. The equations become aesthetically pleasing. Unfortunately, the relationship between those aesthetically pleasing equations and pi don’t really come to fruition. If tau would be used instead, the equations for a circle arc length and a full circle circumference would be

barc = vr, (29)

bcircle = τr, (30) b v arc = , (31) bcircle τ and the area of a circle segment compared to a whole circle area,

r2 A = v , (32) segment 2 r2 A = τ , (33) circle 2 A v segment = . (34) Acircle τ It clearly shows the underlying relationship between the circle constant and the radian angle measurement, in an aesthetically pleasing way, i.e. the linear form, r, and quadratic form, r2/2, are kept intact. If one keeps using pi the same equations as above would for a circle arc length and a full circle circumference be

barc = vr, (35)

bcircle = 2πr, (36) b v arc = , (37) bcircle 2π and the area of a circle segment compared to a whole circle area,

r2 A = v , (38) segment 2 2 Acircle = πr , (39) A v segment = . (40) Acircle 2π At first glance, the area formula looks easier with pi, but the equations do not show the underlying relationship between radians and the circle constant. The generalised equation for the circle arc length and circle segment area will not change. The choice we make for the circle constant is exactly what it is, a choice made by us. It is therefore unfortunate that the constant pi was chosen as half of a circumference to its radius, instead of a full circle circumference, i.e. tau.

12 Discussion and conclusion

The study has shown that when radians are introduced as a new concept in mathematics textbooks they say that two pi is a whole turn, to then have to redefine half a turn as pi. The pedagogical disaster mentioned throughout this study is that an extra constant of two has to be handled for the rest of trigonometry and when working with complex numbers. One textbook claims that the formulae for the circle arc length and circle segment area become easier. Using the same reasoning, I claim that the equations for the circle circumference and circle area are easier with the use of tau instead of pi, because they would take the same linear and quadratic shapes as the equations for the circle arc length and circle segment area. Instead of using pi, it would perhaps be better to use tau as the circle constant. To find out if this is indeed true in all aspects of mathematics from a perspective of learners is not in the scope of this study, but is something to investigate in the future. In this study, it has been shown that tau is both aesthetically pleasing and useful in at least some parts of mathematics. It was mentioned in the introduction section that tau is useful for physicists as well, since a multiplication of two will be left out of many equations that arise in physics as well.

Measuring angles with radians instead of degrees is a concept that those that study later mathematics courses at upper secondary school do because it makes other mathematical concepts and equations more general and easier. It is not something that is necessary to be mathematically correct however, but it does simplify things. Mathematical equations and concepts become simpler when using tau instead of pi, but this transition is never done in practice in schools yet. As mathematical concepts, degrees and pi are obsolete, and could be exchanged to radians and tau completely. However, the Swedish upper secondary school curriculum also states that education should be linked with practical use in society. And since the degrees angle measurement and the number pi are widely used in society, it is therefore necessary to keep them in the mathematical content for the time being.

If tau would indeed become more popular among scientists in their everyday work, a tau-revolution could happen. It would most likely happen by spreading the use of tau from academia down to lower ages slowly over a few generations. We could call that a tau-reform. When and if that would happen, we should be prepared on the pedagogical and didactic side on how to combine both pi and tau in the classroom at the same time for a while. Perhaps it is even our job as teachers to include this circle constant into the classroom and evaluate the outcome? The curriculum does not actually explicitly specify that pi has to be used in mathematics of the Swedish upper secondary school. This means that we already now could experiment by introducing tau at this stage of education. I would suggest having a teaching sequence containing tau, which later should be assessed and evaluated.

In the Appendix, I give a possible way to change the introduction of radians of a Swedish upper secondary mathematics textbook to instead use tau. The figure used in the Appendix belongs to Hartl (2010). It is assumed that the pupil reading that is already accustomed to the use of tau in the same way as pupils are familiar with pi. I approach the introduction of radians by posing the problem of the derivative of the sine function. The derivative becomes a constant times the cosine function. The constant depends on the way we choose to measure angles. If we introduce a new angle measurement unit, the radian, the constant equals unity. A full turn equals tau, τ. The conversion factor between radians and degrees is introduced as τ/360 ≈ 0.0174533. There is no need to make the extra statement that half a turn equals half a tau, which was always added whenever pi was concerned, i.e. that half a turn is pi. The formulae for circle arc length and circle segment area are added and have the same shape as the full circle circumference and area, but with tau exchanged for the angle, v. Using tau instead of pi could make possible misconceptions less likely to happen, which is exactly the reason why this field and topic needs to be researched in a classroom setting with pupils.

As a concluding remark I would like to mention that in today’s world, we essentially always use the

13 base ten for representing numbers. If we instead represent the numbers in a base which is a power of two, the digits of pi and two pi would actually be identical. Since we have sent digits of pi out into the universe to perhaps be observed by intelligent lifeforms, we are lucky that the binary representation of pi does not differ from for instance two pi (as discussed in the introduction), so that a possible intelligent extraterrestrial civilisation would actually consider us intelligent as well.

Do we not want to choose the most beautiful way of representing the world? I am not saying that pi is wrong, I am just saying that pi is half a tau.

14 Bibliography

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15 Palais, B. (2001). Pi is wrong! The Mathematical Intelligencer, 23(3):7–8.

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16 Appendix

A constructed textbook example on how to introduce radians with tau The concept of radians A new angle measurement Let us investigate the derivative of the sine function. From the definition of , the derivative of the sine function, f (x) = sinx, is given by

f (x + h) − f (x) f 0(x) = lim , h→0 h which is sin(x + h) − sin(x) f 0(x) = lim . h→0 h The law of addition says that sin(x + h) = sin(x) · cos(h) + cos(x) · sin(h), and the derivative can be written sin(x) · (cos(h) − 1) + cos(x) · sin(h) f 0(x) = lim . h→0 h We can write this as two terms,

 cos(h) − 1 sin(h) f 0(x) = lim sin(x) · + cos(x) · . h→0 h h

Both sin(x) and cos(x) do not depend on h, so the derivative can be written as

0 f (x) = sin(x) · c1 + cos(x) · c2,

where cos(h) − 1 c1 = lim , h→0 h sin(h) c2 = lim . h→0 h Let’s investigate the values of these limits for smaller and smaller values of h. When our are set on degrees we get the following:

17 sin(h) cos(h)−1 h h h 1 0.0174524 -0.0001523 0.1 0.0174533 -0.0000152 0.01 0.0174533 -0.0000015 0.001 0.0174533 -0.0000002

The limit containing cos(h) goes towards zero, giving c1 = 0. The limit containing sin(h) actually goes towards the value c2 = τ/360 ≈ 0.0174533. Maybe this limit could equal unity if we measure angles in some other way?

We have previously used degrees for measuring angles, where one turn equals 360◦. If a new angle measurement is defined where one turn equals τ, the constant c2 would equal unity and the derivative of the sine function, f (x) = sin(x), exactly equals the cosine function,

f 0(x) = cos(x).

The new angle measurement, 1 radian = 360◦/τ ≈ 57.3◦ (slightly smaller than one sixth of a turn), is defined as the angle that makes a circle segment with both radius and arc length equal to r. A radian is unitless. Any angle in radians, v, is given as the ratio between the circle arc length to its radius.

Definition of a radian

18 Remember that a circle circumference and a circle area are τr and τr2/2, respectively. If we have an angle, v, measured in degrees, the angle cuts the circle into a part of v/360◦. The circle arc length becomes v b = · τr, 360◦ and the circle segment area v τr2 A = · . 360◦ 2

If we instead let the angle be given in radians, we have a v/τ part of a circle. We can redefine the formula for the arc length of a circle as v b = · τr τ = vr, and the segment area as

v τr2 A = · τ 2 vr2 = . 2 Now we can use these simplified equations in our problem solving. Let’s summarise this section:

• Radian is an angle measurement unit that relates a circle arc length to the circle radius

• One turn equals τ radians

• One degree in radians: 1◦ = τ/360 ≈ 0.0174533

• One radian in degrees: 1 = 360◦/τ ≈ 57.3◦

◦ τ τ τ τ 360◦ ◦ • Conversion examples: 60 = 60 · 360 = 6 (sixth of a turn), 2 = 2 · τ = 180 (half a turn)

• The circle arc length is given by b = vr

vr2 • The circle segment area is given by A = 2

19

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