Invited Lectures from the 13th International Congress on Mathematical Education pp 289306  Cite as
Powering Knowledge Versus Pouring Facts
Abstract
Many problems related to the real world admit a mathematical description (i.e., a mathematical model) based on what is studied at school. Solving the mathematical model, however, often requires a higher level of mathematics, and this is the reason for not including such problems in the curriculum. We present several problems of this kind and propose solutions to their mathematical models by means of widely available dynamic mathematics software (DMS) systems. For some of the problems, it is possible to directly use the inbuilt functionalities of the DMS and to construct a computer representation of the problem that allows exploring the situation and obtaining a solution without developing a mathematical model first. Using DMS in this way can broaden the applicability of school mathematics and increase its appeal. The ability of students to solve problems with the help of DMS has been tested by means of two types of competitions.
Keywords
Mathematical modelling Inquiry education Computational thinking17.1 Introduction
There are two partially contradicting trends in high school mathematics education. On one hand, we want mathematical knowledge to be based on a solid logical base (rigor). On the other hand, we want this knowledge to be rich both in content and applications. These two trends cannot always (and easily) be reconciled (De Lange 1996). One of the reasons for this contradiction is the fact that only a few problems related to practice allow mathematically pure and complete treatment with the traditional rigor. The demonstration of patterns of logical thinking is time consuming and often related to simplified mathematical content that does not properly reflect the unavoidable complexity of the real world. The formulation of a mathematical model for a reallife situation cannot be based on rigor only. Dropping out some features and keeping only the most essential ones in the mathematical model requires skills that have little to do with rigor, and this is an obstacle for the inclusion of complex reallife situations in the mathematics curriculum. Furthermore, there are many problems related to practice (some of which will be considered below) that can be equipped with a reasonable mathematical model based on what is studied at school. The corresponding model may be a system of equations, an optimization problem, or something else of a mathematical nature. Solving this mathematical model, however, with the traditional rigor within the frame of the school mathematics is not always possible. It may require a higher level of mathematical knowledge, for instance, advanced calculus and/or numerical methods for approximation of the exact solution. This is another reason for avoiding the consideration of genuine reallife applications within the school mathematics. However, with the appearance of powerful and widely accessible dynamic mathematics software (DMS) systems it became possible to reduce, at least partially, the mentioned contradiction between rigor and applications. Solving a model can be performed by means of DMS. As mentioned in Hegedus et al. (2017) “This leaves more time for essential mathematical skills, e.g., interpreting, reflecting, arguing and also modeling or model building for which there is mostly no time in traditional teaching” (p. 20). With the help of technology, it is possible to offer to students much more demanding mathematical content and interesting applications (Hoyles and Lagrange 2009). Such a change would drastically increase the realm of reallife problems that can be considered in school. We do not have in mind only the traditional application of computers where a mathematical model of the problem is solved by a computer; in addition, some examples will be described below where the standard inbuilt operations (“buttons”) of the DMS system can be used directly to make a computer representation of the problem without first writing the formulas of a mathematical model. This DMS representation of the problem will be called a “computer model of the problem.” By means of this model and the inbuilt functionalities of DMS (such as dragging, measuring distances, and areas), the solution of the problem can be found with a reasonable degree of precision. This direct DMS modelling of the problem as well as the mathematical modelling of the problem, followed by a DMSassisted solution, are in the focus of this paper, which is mainly oriented toward problem solving. Both types of modelling support the most natural way of knowledge acquisition: by experimenting, by formulating and verifying conjectures, by discussing with peers, and by asking more experienced people. In a nutshell, the technology provides the opportunity to learn mathematics by inquiry. This refers not only to what happens (and how it happens) in class but also to extracurricular activities that provide a fruitful playground for building mathematical literacy and cultivating elements of computational thinking (Freiman et al. 2009). Another advantage of using technology in this way is that much larger and more operational mathematical content could be given to the students at an earlier age.
Later in the paper, several problems are considered for which it is easy to assign a proper mathematical model based on school mathematics but whose solution with the necessary rigor while remaining in the frame of school mathematics is relatively difficult (or at least not easy). On the other hand, these models are easily solvable by means of DMS systems. This way of problem solving opens further opportunities for inquiry and cultivates the elementary computational thinking skills of the students, thus powering (in the sense of “adding power to”) their existing knowledge and skills. Problems such as the ones considered below and the inquirybased approach to their solving can make the mathematics studied in school more applicable and more appealing in contrast to the now prevailing pouring of mathematical facts. The ability of school students of different ages to solve such problems has been tested by means of two online competitions called VIVA Mathematics with Computer and Theme of the Month. The participants’ scores show that the use of DMS for problem solving is gradually gaining popularity in Bulgaria. The students are interested in this approach and many are capable of using it. The problems considered next have been used in these competitions.
17.2 The Sample of Problems

The Parking Entrance Problem
This problem is a further elaboration of one of the Problems of the Month used in the European Project MASCIL (http://www.mascilproject.eu/). We present both the computer modeling, which is amenable for younger students using DMS, and the pencilandpaper mathematical modeling, which requires rather advanced knowledge of mathematics.
Problem 1
The second computer model solution of this problem is completely amenable for students at earlier stages of secondary education. In contrast, as we will now see, the mathematical model of the problem requires knowledge of inverse trigonometric functions, and the classical solution uses some elements of calculus. Denote by x the length of the segment CA in Fig. 17.4. Then \( \upalpha = { \arcsin }\frac{c}{x} \) and \( \upbeta = { \arcsin }\frac{c}{b  x} \). One has to find the minimum of the function \( \updelta(x) = {\text{arcsin}}\frac{c}{x} + { \arcsin }\frac{c}{b  x} \) in the interval [c, b − c] (this is the interval where the function \( \updelta(x) \) is welldefined; we implicitly assume here that b > 2c). By finding the zeros of the derivative of \( \updelta(x) \), one can derive that the minimum of this function is attained for \( x = \frac{b}{2} \) and solve the problem.
Here are some tasks for further inquiry with the computer or the mathematical model of this problem:
Problem 1.1
What is the steepest slope (in degrees) that a baby carriage with b = 130 cm and c = 12 cm can overcome without troubles?
Problem 1.2
If the slope to the basement is 20° and the wheelbase of the car is b = 290 cm, what is the smallest radius of the wheels such that moving the car to the basement will not be a problem?
Problem 1.3

The Cylindrical Container Problem
Problem 2
This can be done in different ways. The inbuilt operations of the DMS can be used to find the area of the circular sector outlined by the segments OA, OB, and the arc from B to A (in the counterclockwise direction) and the area of the triangle AOB. The difference between the two areas is the area of the circular segment we are looking for. If the horizontal chord AB is made movable (the DMS takes care of the dynamics and automatically recalculates the areas), a position for the chord AB can be found such that the area of the circular segment is one third of the area of the entire circle. If C is the middle of the chord AB at this position, then the height of the liquid in the horizontal can is equal to the radius of the can base (5 cm) plus the length of the segment CO (which can be measured by the functionalities of the DMS). In our case, an approximate value for the height of the liquid is 6.32 cm. The computer model just developed allows exploration of similar situations with other cylindrical cans (the radius of the can could be made changeable, the part of the can volume which is filled with liquid in vertical position can change, etc.).
The length of the segment OC corresponding to this \( \upalpha \) and \( r = 5 \) can be calculated: \( OC = r\cos \frac{\upalpha}{2} = 1.32465 \). For the height of the liquid in the horizontal position of the can, we obtain 6.32465.
For further inquiries with either the computer model or with the mathematical model, one could consider the following related problems:
Problem 2.1
A horizontally laid cylindrical tank with diameter 200 cm and length 500 cm is partially filled with petrol so that the level of the petrol is 80 cm. How many liters of petrol are there in the tank?
Problem 2.2
If the height of the can from Problem 2 is 24 cm, how much additional liquid should be poured into it in a horizontal position so that the level of the liquid is elevated by 1 cm? If after the addition of the liquid the can is turned into vertical position, what is the height of the liquid level?
Problem 2.3
If the height of the can from Problem 2 is 24 cm, how much liquid should be removed from it so that in a horizontal position the liquid level drops down by 1 cm?
Problem 2.4

The Conical Container Problem
This problem is a wellknown mathematics exercise for university students. It can be settled by means of calculus or by a mathematical trick with inequalities. We present the mathematical model and demonstrate that by means of a DMS the problem can be considered and solved in school.
Problem 3
The mathematical model of this problem is based on the wellknown formula for the volume V of the cone: \( V = \frac{{\pi R^{2 } }}{3}h \). Here R is the radius of the cone base and h is cone’s height. Since \( \upalpha \) is measured in degrees, the length of the arc of the removed circular sector is \( \frac{\upalpha}{360}2\pi l \). Therefore, the length of the cone base circumference is what remains after the cutting: \( 2\pi l  \frac{\upalpha}{360}2\pi l \). Hence, \( 2\pi l  \frac{\upalpha}{360}2\pi l = 2\pi R \). It follows that the radius R can be expressed as function of \( x = \frac{\upalpha}{360}{:}R = l(1  x) \). Further, it follows from Pythagoras’s theorem that \( h^{2} = l^{2}  R^{2} = l^{2} \left( {1  (1  x)^{2} } \right) \). i.e., \( h = l\sqrt {1  (1  x)^{2} } \). Thus, the volume of the cone is \( V = \frac{1}{3}\pi l^{3} (1  x)^{2} \sqrt {1  (1  x)^{2} } \). The essence of the problem, its mathematical model, is to find a number \( x,\,0 \le x \le 1 \), for which the function \( f(x) = (1  x)^{2} \sqrt {1  (1  x)^{2} } \) attains its maximal value. Once again we see that the derivation of the mathematical model is based on school mathematics. Solving this model however requires more advanced mathematics. Using calculus one can find the extremal values of this function f by finding the zeros of its derivative. These zeros are \( x = 1  \frac{\sqrt 2 }{\sqrt 3 },\;x = 1 \) and \( x = 1 + \frac{\sqrt 2 }{\sqrt 3 } \). The last of these numbers is outside the interval [0, 1] and is not relevant for our considerations. The value \( x = 1 \) corresponds to a minimum for f because \( f(1) = 0 \). Therefore the maximum of f is attained at \( x = 1  \frac{\sqrt 2 }{\sqrt 3 } \) and the value of f at this point is equal to \( \frac{2}{3}\sqrt {\frac{1}{3}} \).
The equality will be reached when \( \frac{{(1  x)^{2} }}{2} = \left( {1  (1  x)^{2} } \right) \). This again yields \( x = 1  \frac{\sqrt 2 }{\sqrt 3 } \).

The Ice Cream Container Problem
The next problem is a challenge for pencilandpaper technology, even for university students. With the help of DMS it is completely amenable for school students.
Problem 4
 (a)Cut a circular sector of measure α (in degrees) from the plastic sheet (Fig. 17.15) and, by gluing, make from it a cone that will serve as the lower part of the ice cream container.
 (b)
Cut off from the remainder (Fig. 17.15) a full circular sector of radius t (this number t is to be specified later) and glue a cut cone (truncated cone) that will serve as the upper part of the ice cream container.
For what size of α will the ice cream container have largest volume?
The length of the arc of the circular sector of measure α is \( \frac{{2\pi l\upalpha}}{360} \). The cone made of this sector will have a radius r of the base determined from the equation \( 2\pi r = \frac{{2\pi l\upalpha}}{360} , \) i.e., \( r = lx \) where \( x = \frac{\upalpha}{360} \).
The volume \( V_{2} \) of the truncated cone is \( V_{2} = \frac{\pi }{3}h_{2} \left( {R^{2} + Rr + r^{2} } \right) \) where \( R = (1  x)l \), \( r = lx \), \( R^{2} + Rr + r^{2} = l^{2} \left( {(1  x)^{2} + (1  x)x + x^{2} } \right) = l^{2} \left( {(1  x)^{2} + x} \right) \).
Hence \( V_{2} = \frac{\pi }{3}l^{3} \frac{1  2x}{1  x}\sqrt {2x  x^{2} } \left( {1  x + x^{2} } \right) \). The volume of the ice cream container is \( V = V_{1} + V_{2} \). We note here that x must belong to the interval \( [0,\, \frac{1}{2}] \). This follows from the fact that the number \( t = \frac{r}{1  x} = \frac{lx}{1  x} \) cannot be bigger than l.
Finding the maximum of V by means of calculus is a challenge. With the help of a DMS it can be found, as in the previous problem, that the maximal value of V is attained for \( x \approx 0.23088 \), which corresponds to \( \upalpha \approx 83.12^{ \circ } \).
Here are some problems for further inquiry:
Problem 4.1
What is the minimal radius l of the initial circle from which the ice cream container is produced in the above way so that its volume is at least 200 cm^{3}?
Problem 4.2
 (a)
Cut circles centered at O (i.e., concentric with the initial circle).
 (b)
Cut from the remainder a radial segment of measure α (in degrees).
For what size of α will the volume of the bucket be the largest?
What is the largest possible volume of the bucket?

A geometrical problem
This is the last of the sample problems:
Problem 5
This problem deviates in style from the previously considered problems. It contains a researchlike component that is suitable for work on a project by the students. The computer model for this problem is easy to construct. The inbuilt operations of GeoGebra can be used to construct the orthocenter, the incenter, and the centroid of an arbitrary triangle. Using the “finding area of a polygon” command, the areas of the triangles ABC and DEF are calculated and displayed on the monitor. Due to the dynamic functionalities of GeoGebra, this computer model of Problem 5 allows to explore many triangles (by dragging some of the vertices A, B, and C). Playing with the vertices can experimentally establish that for some obtuse triangles ABC the answer to the question in Problem 5 is positive.
Note that this computer model solution of the problem does not require knowledge of more advanced mathematics (trigonometry, analytical geometry, etc.). It relies on the knowledge of the basic notions involved (orthocenter, incenter, and centroid), on acquaintance with the functionalities of GeoGebra, and on some modeling skills.
Problem 5.1
For an arbitrary triangle ABC, find the area of the triangle with vertices at the orthocenter, the circumcenter, and the centroid of ABC.
Exploring this task with the corresponding computer model can show that the required area is always zero and, therefore, the three points are collinear (they lie on the famous Euler line of the triangle ABC).
The following simplified form of Problem 5 was given as one of the tasks in the competition VIVA Mathematics with Computer.
Problem 5.2
Given is a triangle ABC (by its sides or by the coordinates of its vertices; see Fig. 17.18). Find the area of the triangle with vertices at the orthocenter D, the incenter E, and the centroid F of the triangle ABC.
17.3 The Competitions Viva Mathematics with Computer and Theme of the Month
Scores of participants in April 2016 competition VIVA MC
Grades in a group  3 and 4  5 and 6  7 and 8  9 and 10  11 and 12 

Number of participants  146  142  79  67  40 
Participants with 35–50 points  80  49  2  9  1 
Participants with 20–34 points  44  59  15  21  16 
Participants with 10–19 points  19  24  22  23  5 
Scores of participants in December 2016 competition VIVA MC
Grades in a group  3 and 4  5 and 6  7 and 8  9 and 10  11 and 12 

Number of participants  449  385  268  123  86 
Participants with 35–50 points  180  27  7  12  11 
Participants with 20–34 points  147  146  24  29  28 
Participants with 10–19 points  75  114  84  37  20 
Students’ scores in solving the problems from Sect. 17.2 were similar. Problem 5.2 from Sect. 17.2 was proposed as a last (presumably most difficult) task in the very first edition of VIVA MC (December 2014) to 207 students from Grades 8 to 12. The lack of experience with such problems and the short time to work on the problems (60 min) is clearly seen from the obtained results: About half of the students (48%) did not enter any answer for this task, 13% provided precise answer, and 2% gave an answer with satisfactory precision. The cylindrical container problem (Problem 2 from Sect. 17.2) was given to 317 students from Grade 8 to Grade 12 at the December 2015 edition of VIVA MC. An auxiliary DMS file was provided in order to facilitate the exploration of the problem. Only 13% provided an answer with sufficiently high precision. The answers of a further 37% were given with satisfactory precision. The general feeling has been that with every new edition of VIVA MC the performance of the participants improves, though rather gradually.
The other competition, TM, is conducted monthly. A theme of five tasks related to a common mathematical idea is published at the beginning of the month on the abovementioned portal (vivacognita.org). The tasks are arranged in the direction of increasing difficulty. The participants are expected to solve the problems and send responses online by the end of the month. Some of the problems are accompanied by auxiliary DMS files which allow the students to explore the mathematical problem, find suitable properties, try out different strategies, and find a (usually approximate) solution. To solve the more difficult tasks from the theme, the students have to adapt the auxiliary files from previous problems or to develop their own files for testing and solving the problem. Each problem brings at most 10 points (depending on the degree of preciseness of the answer). The maximum total score is 50 points. Usually there are hundreds of visits to the site where the theme is published. Only dozens, however, submit solutions. The theme for February 2015 was related to the parking problem (Problem 1 from Sect. 17.2). Seventeen participants submitted their solutions. Seven received between 35 and 50 points and two received between 20 and 34. Much better were the results from the theme from September 2015, which was related to conical containers (Problems 3, 4, and 4.2 from Sect. 17.2). Sixteen students submitted their solutions, with 14 scoring between 41 and 50 points and one scoring 34 points. The results of the first several runs of TM are published in Kenderov et al. (2015) and Chehlarova and Kenderov (2015).

The problems are interesting because they require logical thinking.

I like it because I could use GeoGebra for each problem.

The contest is nice since I don’t feel pressed when solving the problems.

The questions are at the right level for me.

It is interesting and helps me develop.

I find the problems entertaining.

It was easy for me to understand the formulation of the problem by means of the dynamic file I could use.

Every problem is interesting in its own way.

I like the fact that I can explore while solving the problem.

I like the parking entrance problem because it is something you could face in the real world.

This relatively modest feedback confirms the expectation that providing the students with appropriate exploration tools can increase their awareness of both the beauty and the applicability of mathematics.
Notes
Acknowledgements
The author is indebted to T. Chehlarova and E. Sendova for the longstanding collaboration in promoting the mathematics with computers idea and for the fruitful remarks and suggestions related to this paper. In particular, the final version of the title of the paper was suggested by E. Sendova. The author is indebted also to Dj. Kadijevich for pointing some relevant literature sources. Special thanks are due also to anonymous reviewers whose critical notes helped clarify and improve the paper.
References
 Chehlarova, T., & Kenderov, P. (2015). Mathematics with a computer—A contest enhancing the digital and mathematical competences of the students. In E. Kovacheva & E. Sendova (Eds.), UNESCO international workshop: Quality of education and challenges in a digitally networked world (pp. 50–62). Sofia, Bulgaria: Za Bukvite, O’Pismeneh.Google Scholar
 De Lange, J. (1996). Using and applying mathematics in education. In A. J. Bishop et al. (Eds.), International handbook of mathematics education (pp. 49–97). Dordrecht: Kluwer.Google Scholar
 Freiman, V., Kadijevich, D., Kuntz, G., Pozdnyakov, S., & Stedřy, I. (2009). Technological environments beyond the classroom. In E. J. Barbeau & P. Taylor (Eds.), Challenging mathematics in and beyond the classroom: The 16th ICMI study (pp. 97–131). New York: Springer.Google Scholar
 Hegedus, S., Laborde, C., Brady, C., Dalton, S., Siller, H.S., Tabach, M., et al. (2017). Uses of technology in upper secondary mathematics education. ICME13 topical surveys. Cham, Switzerland: Springer Open.Google Scholar
 Hoyles, C., & Lagrange, J.B. (Eds.). (2009). Mathematics education and technology—Rethinking the terrain. New York: Springer.Google Scholar
 Kenderov, P., Chehlarova, T., & Sendova. E. (2015). A mathematical theme of the month—A webbased platform for developing multiple key competences in exploratory style. Mathematics Today, 51(6), 305–309.Google Scholar
Copyright information
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.