Pedagogical Justifications That Zero Factorial Equals One: Making Data-Driven Instructional Choices in Education Munir Mahmood, Mathematics and Statistics Education Australia, Melbourne, Victoria, Australia Lori L. Murray, King’s University College, Western University, London, Canada Ricardas Zitikis, Western University, London, Canada Ibtihal Mahmood, University of Melbourne, Victoria, Australia Abstract Zero factorial, defined to be one, is often counterintuitive to students but nonetheless an interesting concept to convey in a classroom environment. The challenge is to delineate the concept in a simple and effective way. In this regard, the contribution of this article is two-fold: First, it reveals and makes contribution to much simpler justifications on the notion of zero factorial to be one when compared to previous studies in the area. Second, to assess the effectiveness of the proposed justifications, an online survey has been conducted at a comprehensive university and, via its statistical analysis, data-driven instructional decision- making has been illustrated. Elaborating on the first contribution, we note that the connection of zero factorial to the definition of the gamma function provides a first-hand conceptual understanding of the concept of zero factorial. But for the purpose of teaching, it is not particularly helpful from the pedagogical point of view in early years of study, as it is quite challenging to explain the rationale behind the origin of the definite integral that defines the gamma function. In this regard two algebraic and one statistical justification are presented. The “squeeze theorem” plays a pivotal role in this article. To assess the effectiveness of the justifications pedagogically, an online survey was conducted at a Canadian university. Keywords: factorial, bound, limit, squeeze theorem Introduction Quite often in introductory courses, instructors emphasize learning concepts by means of repetition and ample practice problems. Sometimes, they simply provide students with cheat 1 sheets.1 While such practices are widely accepted and unavoidable, when time and class-size permit, some elements of mathematical fun2 may be introduced to internalize those concepts (Roberts et at., 2018). This facilitates recovery of the concepts if students happen to forget their exact formulations. Another tool to facilitate internalization of concepts is to make them easier to remember by building a learning process by associating what needs to be learned with what has already been learned. For this task, the development of curricula plays a decisive role. In this regard, Chen and Zitikis (2017, 2020) have explored the effects of learning a subject due to prior subjects learned, based on real data and in-depth statistical analyses. In this article we concentrate on a specific problem facing teachers in introductory calculus- related courses. Namely, we present three intuitive pedagogical justifications – all relying on the squeeze theorem (to be recalled in Section 2) – of why zero factorial (i.e., 0!) is equal to 1, and not to 0 as many students would expect.3 These justifications are either in addition to Mahmood and Mahmood (2015) and Mahmood and Romdhane (2015) or improved versions of them. Having several choices is useful for selecting a most suitable one for the level and perceived inclinations of a given group of students (Dare, Nowicki, & Murray 2021). Based on a survey conducted at a comprehensive university, we illustrate how data and their statistical analysis can facilitate the selection. In general, surveys have played a particularly important role in educational decision making, and one of the most researched databases is the OECD Education Statistics.4 On a smaller, localized scale, educational researchers have conducted surveys on issues such as the impact of active learning in undergraduate STEMM5 education (for example Wiggins et al., 2017), assessment design (Murray & Wilson, 2021), students’ views on introductory STEM6 courses 1 An allowed cheat sheet in the exam contains concise information used for quick reference. It becomes handy for items which are difficult to remember particularly at higher levels of education. Whether allowed or not, the preparation of such a sheet also serves as an educational exercise which motivates one to figure out what needs to be known, prioritize study items for the exam, help organize the required time allocation to study those items and finally provides an overall review before the exam day. 2 Mathematical fun is described as interactive learning involving activities conducted in the classroom. 3 Quite often when teaching introductory courses, students are merely instructed that zero factorial equals one as a fact for convenience and are not provided a methodical explanation as to why (Buglear & Castell, 2019). 4 https://www.oecd-ilibrary.org/education/data/oecd-education-statistics_edu-data-en 5 Science, technology, engineering, mathematics, and medicine. 6 Science, technology, engineering, and mathematics. 2 (Meaders et al., 2020), and experiential learning perspectives (Zhai et al., 2017). A well-designed survey is crucial in achieving statistically sound and trustworthy results (Bellhouse, 1984; Thompson, 1997; Wu and Thompson, 2020). The focus of the survey in the present paper is quite distinct: we aim not at general instructional matters but at a very specific mathematical concept and, given several alternatives, at choosing a most accessible way to present the concept in a classroom environment. There are many ad hoc explanations of why zero factorial equals one, and arguments ensue whether, and to what extent, they can be considered mathematically accurate. Obviously, for the sake of facilitating intuition, at least during the first stages of explanation, we are bound to slightly depart from rigor, which could later be, and is, re-stored via the celebrated gamma function in more advanced calculus courses. It should be noted at this point, however, that there is no simple way to express the factorial function ℎ(푛) = 푛! beyond the positive integers 푛, and thus to reveal its value at 푛 = 0. In particular, for the sole purpose of conveying a meaningful concept of the zero factorial, a simpler approximation to the integral form of the gamma function to date is unavailable. Hence it is desirable to derive or present a formula that would transition the factorial function for positive integer values to non-integer values, particularly to those that are in the interval (0,1), and then approach 0 along the curve. As we have noted earlier, in more advanced calculus courses, this goal is effectively ∞ ( ) 푡−1 −푥 achieved via the well-known gamma function, which is Γ 푡 = ∫0 푥 푒 푑푥 for all real 푡 > 0. For details, see Devore (2012). The relationship Γ(푛 + 1) = 푛Γ(푛) = 푛! between the gamma and factorial functions, and the equation Γ(푡 + 1) = 푡Γ(푡) that holds for all 푡 ∈ (0, 1), create a genuine bridge between the cases 푛 = 1 and 푛 = 0. Intuitively, if we take the limit as 푡 ↓ 0 ∞ 푡 −푥 inside the integral ∫0 푥 푒 푑푥, and all limits in this paper approach 0 from the right, we are left with the exponential function, which integrates to 1. This shows that Γ(푡 + 1) converges to 1 when 푡 ↓ 0. Defining 0! as the limit of Γ(푡 + 1) when 푡 ↓ 0, we can now claim that 0! is indeed equal to 1. This explanation supports the presentations of 0! = 1 in Mahmood and Mahmood (2015) and Mahmood and Romdhane (2015) although it is not explicitly mentioned in those articles. 3 The aim of this article is to seek further simplified algebraic and statistical justifications and then let survey data choose the best7 one ranking them. Such availability of the choicest justification addresses and serves the essential purpose of effective understanding of zero factorial in classes that may not have yet encountered the concept of the gamma function, or may have seen it in other contexts such as statistics but have not associated it with real numbers other than positive integers. In this regard, we acknowledge to have greatly benefited from historical notes of Hairer and Wanner (1996) on the gamma function, factorials and Euler’s interest and achievements in interpolating factorials to non-integer values. Of course, the use of the gamma function would give the most mathematically rigorous justification of 0! = 1, whereas Mahmood and Mahmood (2015) and Mahmood and Romdhane (2015) presented particularly an intuitive explanation. The justifications in the current paper slightly reduces the intuitive appeal but adds more mathematical rigor to the justification of 0! = 1, hence exhibiting the delicate balance that we desire to achieve in a classroom setting when explaining new concepts to students. The issue is that the presence of the definite integral of the gamma function presents a tricky form, one which is not easy to absorb by students at such an early exposure stage of the concepts of 푛 factorial and zero factorial. Moreover, it is hard to explain the rationale behind the basis of such an integral in 푡 and 푥, of which 푡 ascertains the factorial calculation while the embedded exponential function assures the definite integral to produce the aim of any given factorial value. To facilitate our following arguments in a gentle and intuitive fashion, a symbolic 푡! is introduced in the next section, which allows movement from the discrete to the continuous in 푡 ∈ (0, 1). This enables taking the limits in a subtle way, aligning more with the gamma function but avoiding its extreme mathematical precision and thus its complexity. Main Results The factorial of 푛 is defined by 푛! = 푛 ∙ (푛 − 1) ∙ ∙ 3 ∙ 2 ∙ 1 (1) for integer 푛 ≥ 1, but for 푛 = 0, zero factorial, symbolically written as 0!, is defined as 0! = 1. In this section we present some easier justifications of 0! = 1, in light of Section 1, and which of 7 The “best” is of course contingent on the university and the type of students, but given the prerequisites for enrolling into courses, the notion of “best” would carry on for a given course for several years after the survey results.