Abstract This article investigates abilities of university students in making assumptions in a ballistics model. The students were asked to make reasonable assumptions in a ballistics model often encountered in mechanics. The objects ranged from stones thrown by catapults in ancient times to launching projectiles and missiles from recent history. Students’ responses to the questionnaire regarding their assumptions are presented and analysed in the article. 1. Introduction There are many diagrams created by mathematics education researchers and authors of mathematics textbooks that illustrate a mathematical modelling process. The diagrams produced by mathematics education researchers tend to be rather complex, like the one presented in Table 1. Table 1. Mathematical modelling process by Niss (2010) Table 1. Mathematical modelling process by Niss (2010) Authors of mathematics textbooks tend to use more practical diagrams that are easier to follow like the one presented in Table 2. Table 2. Mathematical modelling process by Stewart (2006) Table 2. Mathematical modelling process by Stewart (2006) Some elaborate diagrams specifically indicate the ‘making assumptions’ step, although in most diagrams it is ‘hidden’ under the formulation (construction/mathematization) stage of a mathematical modelling process. Many researchers draw attention to the importance of assumptions in a mathematical modelling process while teaching mathematical modelling to students. Seino (2005) regards assumptions as the foundation of the proposed model that defines the balance between adequacy and complexity of the model: ‘the setting up of appropriate assumptions can be considered as the most important thing in performing mathematical modelling’ (p. 664). He proposed ‘the awareness of assumptions’ as a teaching principle to make students understand the importance of setting up assumptions and examine particular assumptions closely. It has a dual role: first, as a bridge to connect the real world to the mathematical world; secondly, as promotion of activities that reflect on the formulation step of the mathematical modelling process. Shugan (2007) made more general comments about the importance of assumptions in science: ‘Virtually all scholarly research, benefiting from mathematical models or not, begins with both implicit and explicit assumptions … The assumptions are the foundation of proposed models, hypotheses, theories, forecasts, and so on. They dictate which variables to observe, not to observe, and the relationship between them’ (p. 450). Peters (2015) argued that ‘in order for learners to be effective problem solvers they must be able to make sensible assumptions’. Galbraith (1996) claimed that assumptions are important in all stages of the modelling process: ‘In the formulation phase much is made of assumptions that need to be made in setting up the model for solution. However, what is often not emphasized is that assumptions need to be invoked at all stages of the modelling process, i.e., during solution, interpretation and evaluation as well as during formulation. Furthermore, the assumptions are of different types and play different roles’ (p. 76). However, some researchers express concern that ‘the role of assumptions in modelling activity has been over-simplified’ (Galbraith & Stillman, 2001). Teaching experience shows that often students skip the assumptions stage and rush to ‘do the sums’. Sometimes assumptions are based on the modeller’s intuition and common sense and not supported by calculations or experiments, especially in the education settings. Grigoras (2011) pointed out: The emerging hypotheses in students’ work are essentially of different nature than hypotheses in scientific research; like in physics, for example, where stated hypotheses are followed by experiments, and afterwards evaluated, therefore sustained or rejected. Here the students do not check many of these hypotheses, but simply state them and take them as granted. They are either led by intuition or the use of their background knowledge. (p. 1020).Peters (2015) commented on the lack of conceptual knowledge by entering university students in their ability to recognize for the need of making assumptions: The majority of learners come to university with predominantly procedural knowledge, they know how to apply a procedure to a set of variables and constants and obtain a result but with little understanding of what the result implies or means. They do not seem to possess the conceptual knowledge necessary to be able to make assumptions. (p. 53). In the case study with first-year undergraduate students, Peters (2015) reported that: it was evident from the start of the investigation that the students found it extremely difficult to form assumptions and to implement a problem solving strategy…They were not comfortable with making assumptions and constantly asked the investigator to tell them where they should start. (p. 58). The purpose of the modelling exercise described below was to illustrate the importance of making appropriate assumptions to the first-year university students. The students were not required to make their own assumptions—only to reflect upon assumptions presented to them in a modelling context. The research question was to check students’ intuition, common sense and competence to reflect and decide about the relevance of different assumptions to the modelling of different types of ballistic phenomena. 2. The study 2.1 A modelling exercise A group of university students were given a modelling exercise on a ballistics model, a typical problem from mechanics. The group consisted of 65 first-year science students majoring in applied mathematics from an Australian university. The exercise was inspired by the modelling example from the book on applied mathematics (Tichonov & Kostomarov, 1984). The exercise considered four objects thrown with the initial velocity at the given angle to the surface of the Earth. We started talking about stones thrown by catapults in ancient times and proceeded to discussing firing balls from cannons in medieval times and launching projectiles (shells from colossal field guns) and missiles in recent history. In each of the four cases—a stone, ball, projectile and missile—the distance from the starting point to the landing point was given. Also provided was the maximum height reached by a projectile and missile. For each case, students were challenged to think about the appropriateness of the following four assumptions: The Earth is flat; The Earth is an inertial system; Air resistance can be ignored; Acceleration due to gravity is constant. These assumptions were given and discussed in the modelling example (Tichonov & Kostomarov, 1984, pp. 11–25). Other assumptions (air pressure, the relationship between the air density and height, speed and direction of wind, influence of random factors like small deviations in the size of the ‘identical’ objects, and other possible assumptions) were not considered. Some students asked what the inertial system meant and it was explained to them. As in the modelling example (Tichonov & Kostomarov, 1984, pp. 11–25), it was agreed that a relative error of less than 3% was not significant in the sense that if, say due to air resistance, a stone from a catapult lands up to 3 m before the target which is 100 m away the assumption of ignoring the air resistance is still reasonable in that case Without doing any calculations the students were asked to indicate which of the above assumptions were reasonable and which were not for each of the four cases: a stone, ball, projectile and missile. They were asked to fill Table 3 by inserting ‘+’ in the box if the assumption was reasonable and ‘−‘ if not. After completing the modelling exercise, the model solutions (Appendix) were presented and discussed with the students. Table 3. Solutions to the ballistic modelling exercise Object assumption Stone from catapult Ball from Cannon Projectile Missile h = 20 km h = 200 km l = 100 m l = 1 km l = 200 km l = 8000 km Earth is flat + + − − Earth is an inertial System + + − − Ignore air resistance + (2–3%) − (15%) − + g is constant + + + Object assumption Stone from catapult Ball from Cannon Projectile Missile h = 20 km h = 200 km l = 100 m l = 1 km l = 200 km l = 8000 km Earth is flat + + − − Earth is an inertial System + + − − Ignore air resistance + (2–3%) − (15%) − + g is constant + + + Table 3. Solutions to the ballistic modelling exercise Object assumption Stone from catapult Ball from Cannon Projectile Missile h = 20 km h = 200 km l = 100 m l = 1 km l = 200 km l = 8000 km Earth is flat + + − − Earth is an inertial System + + − − Ignore air resistance + (2–3%) − (15%) − + g is constant + + + Object assumption Stone from catapult Ball from Cannon Projectile Missile h = 20 km h = 200 km l = 100 m l = 1 km l = 200 km l = 8000 km Earth is flat + + − − Earth is an inertial System + + − − Ignore air resistance + (2–3%) − (15%) − + g is constant + + + The percentages in the brackets in Table 3 represent the relative percentage error in the calculations assuming that we ‘ignore air resistance’. They were given only in the model solution to illustrate the effect of one of the assumptions in the two cases. The calculations of the relative errors are presented in the Appendix. 2.2 The questionnaire After the discussion of the correct assumptions, the following anonymous questionnaire was given to each student: Question 1. How many correct assumptions out of 16 did you make? Question 2. Is common sense and intuition enough to make correct assumptions? Why? Question 3. Would special knowledge in physics help you to make correct assumptions? If yes, in which way? If no, why not? Question 4. Which case (stone, ball, projectile or missile) was easiest to answer for you? Why? Question 5. Which case (stone, ball, projectile or missile) was hardest to answer for you? Why? Question 6. Which assumption out of 4 was the most difficult to estimate? Why? 2.3 Students’ responses to the questionnaire The participation in the study was voluntary. The response rate was 100%. Below are summaries of the students’ responses and their typical comments. Question 1: How many correct assumptions out of 16 did you make? The distribution of correct assumptions made by the students is presented in Figure 1. Fig. 1. View largeDownload slide The distribution for the number of students’ giving the correct assumptions. Fig. 1. View largeDownload slide The distribution for the number of students’ giving the correct assumptions. Question 2: Is common sense and intuition enough to make correct assumptions? Why? In total, 54% of students agreed that common sense and intuition are sufficient to identify the correct assumptions. The most common reasons to support their assertions were: ‘Yes, because when you don’t quite know the correct answer that applies then common sense will usually provide the most accurate answer’; ‘Yes, these are simple, practical things we can relate to’; ‘To a point – but the rules bend in some scenarios and it is these previously learned tricks that are needed. However, some students expressed some scepticism about their own reliance on common sense and intuition in the exercise: ‘No, you can never make assumptions unless it is completely universal knowledge’. Question 3: Would special knowledge in physics help you to make correct assumptions? If yes, in which way? If no, why not? As the surveyed group are applied mathematics students and many (but not all) also enrolled in physics, it is perhaps not surprising that 86% of the students commented that special knowledge of physical principles would aid in identifying correct assumptions. The typical response made an explicit reference to the use of a formula or concepts taken from their first-year physics courses: ‘Yes, I used a formula for gravitation in my head to help answer’. For those students who did not agree (14%) with the sentiment of the question said: ‘No, physics tends to overcomplicate problems’. Question 4: Which case (stone, ball, projectile or missile) was easiest to answer for you? Why? The easiest case for the students was stone (58%) followed by missile (22%), ball (7%), projectile (6%) and ‘no difference’ (6%). The most common responses given reflect their familiarity with every day experiences: ‘Stone, it made sense and was more relevant to my everyday experiences’. ‘Stone, we have all thrown one once in our life’.Question 5: Which case (stone, ball, projectile or missile) was hardest to answer for you? Why? The hardest objects for the students were projectile (42%) and missile (25%) followed by stone (18%), ball (9%) and ‘no difference’ (6%). The most frequent comments were: ‘Missile, because it needs to consider about projectile force and its movement as it is fired by force that not happen in nature’ [sic]; ‘Projectile - too many unknown factors’; ‘Projectile, it was in the middle range of the other objects’. Question 6: Which assumption out of 4 was the most difficult to estimate? Why? The hardest assumptions to make for the students were: ‘ignore air resistance’ (32%) and ‘Earth is inertial system’ (31%) followed by ‘no difference’ (20%), ‘g is constant’ (14%) and ‘Earth is flat’ (3%). The most common comments were: ‘Didn’t know initially what “inertial” meant in this context’; ‘Ignoring air resistance – it was hard to visualise such scenarios when they do not exist in real life’. 3. Discussion and conclusions The distribution of students’ correct assumptions was clearly skewed to the left. Only 2 students out of 65 made all 16 correct assumptions. Another 24 students made one or two mistakes. So, 39 students out of 65 (or 60%) made three or more mistakes. Students’ responses on the role of common sense and intuition for identifying the correct assumptions were polarized—almost a 50/50 split. The vast majority of the students (86%) reported that special knowledge in physics helped them to make correct assumptions. Most of the students relied on the practical and familiar experiences in answering the questions about the easiest/hardest object. For most students (63%), the hardest assumptions to estimate across all four objects were ‘Earth is an inertial system’ and ‘Ignore air resistance’ for the reasons related to lack of knowledge (e.g., definition of the inertial system) and everyday experiences (difficult to visualize). Students’ feedback was similar to the findings from a study by Flegg et al. (2013) on investigating students’ approaches to learning a new mathematical model: ‘Students were required to give their understanding of the assumptions of the model as well as their novel thoughts on the limitations and flaws, allowing them to explore how they would change the model’ (p. 30). Discussions and observations in class and informal interviews with selected students revealed that the modelling exercise did increase ‘the awareness of assumptions’ (Seino, 2005) and their important role in the mathematical modelling process. In particular, it was consistent with Seino’s claim that it was possible (after the discussion of the correct solution) to develop an awareness of assumptions by making students recognize conditions of the problem as assumptions, by helping students realize how assumptions affect selection of formulas and functions, and whether there are other assumptions to consider. Students found that ‘what-if’ questions were especially helpful in the modelling exercise on making assumptions. Appendix The suggested solutions were based on the calculations presented in (Tichonov & Kostomarov, 1984) and consultations with experts. As an example, the calculations of the relative errors of the distances in the case of a stone and a ball if air resistance is included in the model are presented below. The following formulae are applicable for when an object is thrown with an initial velocity v0 at an angle α to the horizontal. The x(t) and y(t) are the horizontal and vertical distance travelled, while l is the horizontal distance from the starting point to the landing point. x=t v0 cos α; y=t v0 sin α−gt22y=x tan α−x2g2(v0)2 cos 2αl=(v0)2gsin 2α A. Stone from catapult l≈100m; v≈30m/s Including air resistance into the model gives: F=CπR2ρv22; C−drag coefficient (≈0.15), R≈0.1m−radius of the stone, ρ=1.3kg/m3−density of air The force due to gravity: P=mg=4π3R3ρ0g; ρ0=2.3×103kg/m3−density of stone The relative error: Δll≈FP≈0.03 B. Cannon ball l≈1km, R≈0.07m, ρ0≈7×103kg/m3, v0≈100m/sΔll≈0.15 As one can see there were many assumptions made in the calculations. They included estimations of distances from the starting to the landing points and the size of the stone and ball, the constant value of the velocity of the objects throughout the flight, the constant value of the drag coefficient C for the estimated velocities (for high velocities C depends on v). As we emphasized earlier, the students neither asked nor given time to do calculations. The above calculations were presented to the students merely as an illustration during the discussions of the model solutions. References Flegg J. , Mallet D. , Lupton M. ( 2013 ) Students’ approaches to learning a new mathematical model . Teaching Mathematics and Its Applications , 32 , 28 – 37 . Google Scholar Crossref Search ADS Galbraith P. ( 1996 ) Modelling comparative performance: some Olympic examples . Teach. Math. Appl ., 15 , 67 – 77 . Galbraith P. , Stillman G. ( 2001 ) Assumptions and context: pursuing their role in modelling activity. Modelling and Mathematics Education ( Matos J. F. , Blum W. , Houston S. K. , Carreira S. P. eds), Chichester, UK : Horwood Publishing, pp . 300 – 310 . Grigoras R. ( 2011 ) Hypothesis and assumptions by modelling—a case study. Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education ( Pytlak M. et al., eds). Rzeszów, Poland , pp. 1020 – 1029 . Niss M. ( 2010 ) Modeling a crucial aspect of students’ mathematical modelling. Modelling Students’ Mathematical Competences ( Lesh R. et al., eds). New York : Springer , pp. 43 – 59 . Peters M. ( 2015 ) Using cognitive load theory to interpret student difficulties with a problem-based learning approach to engineering education: a case study . Teach. Math. Appl ., 34 , 53 – 62 . Seino T. ( 2005 ). Understanding the role of assumptions in mathematical modelling: analysis of lessons with emphasis on the ‘awareness of assumptions’. Building Connections: Theory, Research and Practice, Proceedings of the 28th Annual Conference of the Mathematics Education Research Group of Australasia (P. Clarkson et al. eds), Vol. 2. Melbourne, Australia: MERGA, pp. 664 – 671 . Shugan S. M. ( 2007 ) Its the findings, stupid, not the assumptions . Market. Sci ., 26 , 449 – 459 . Google Scholar Crossref Search ADS Stewart J. ( 2006 ) Calculus: Concepts & Contexts . Belmont, CA: Thomson Brooks/Cole . Tichonov A. N. , Kostomarov D. P. ( 1984 ) Introductory Lectures on Applied Mathematics . Moscow : Nauka (in Russian ). © The Author 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
Teaching Mathematics and Its Applications: International Journal of the IMA – Oxford University Press
Published: Dec 3, 2018
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