Easier Calculations for Understandable Mathematics

Jozef Hvorecký
Slovakia

The world of Mathematics Education is full of contradictions. On one side, the role of formal methods in all fields of human activities is growing. This means that more and more people are required to exploit methods of Mathematics, Statistics, and Informatics in their professions. On the other side, Mathematics is not a popular school subject around the planet. Students are reluctant to learn it. After their graduation, they are not sufficiently prepared for their positions.

Both statements also hold for Slovakia. Her situation is typical for countries of Central and Eastern Europe that recently belonged to the Soviet bloc. The quality of education is rather high (see for example the results of Olympiads in Mathematics) but the appreciation of Mathematics by general public is quite negative. In our opinion, one of the main reasons is the inability of ordinary pupils and students to grasp one of the cores of Mathematics – its irreplaceable role in our everyday life. Mathematics is taught as a highly formalized subject. Teachers and textbooks are not proposing the content in the way that would support its better – faster and deeper – understanding. Many textbooks offer surface learning and teachers feel comfortable satisfied with it because it is much easier to grade. Good students are trained to perform formal operations quickly and safely but do not comprehend why the operations are performed. Luckily, there are first initiatives to make a change.

Many of them exploit information technology. The author believes that this is a right way. Some very abstract areas can be supported by attractive practical examples. For example, most textbooks do not tell more about hyperbolic functions than their definitions

and statements like: “The derivation of the sinus hyperbolic is the cosine hyperbolic; the derivation of the cosine hyperbolic is the negation of the sinus hyperbolic”. As this is almost entire content of the particular section, most students have no idea whether they can ever meet and efficiently exploit them and where.

Classes should not start with definitions – problems and examples are more appropriate. Based on them, the students can get a picture about a relationship between the concept and the reality. Its deep theoretical background should come later when these basics are fixed.

For example: Hold the ends of a chain in each hand. The chain forms a curve. Which curve is it? Figure 1 shows its graph. To everyone’s surprise it is not a parabola but a cosine hyperbolic.

Fig. 1 Cosine hyperbolic
A graphing tool allows students to manipulate it by changing its parameter a:

Figure 2 shows graphs with a equal to 0.25, 0.5, 1 a 2. Decreasing the value of a results in the same effect as stretching hands – the curve becomes closer and closer to the straight line.

Fig. 2 The influence of a to the curve

Another method of enriching Mathematics classes is the usage the applications presented on the Internet. For example, an interesting application of cosine hyperbolic is shown in the web page at

http://mathworld.wolfram.com/Roulette.html.

This impressive animation demonstrates how “wheels” having the form of regular polygons move on “rails” consisting of identical sections of the cosine hyperbolic. Figure 3 shows its snapshot. An equilateral triangle, a square, a pentagon and a hexagon move smoothly because their centers of gravity shift along straight lines.

Fig. 3 Smoothly moving polygons on the cosine hyperbolic track

Notice that with the number of sides the sections become shorter and less bulging. A generalization of this observation can bring up students to a “surprising” conclusion. The smooth movement of a circle (which is a “limiting polygon”) requires a flat surface. Similar “funny” observations humanize Mathematics.

Mathematics is about calculations. In our opinion, they should come up on the scene only when students have seen realistic applications of the concept. Whenever possible, calculations should have their practical interpretations. In the case of the cosine hyperbolic, the calculations related to the Saint Louis Arch are such because it has the form of the inverted cosine hyperbolic specified by the formula

y = 693.8597 – 68.7672 cosh(0.0100333x)


Fig. 4 Saint Louis Arch

The following questions are very natural: What is its height? What is the distance between its pylons? Notice that without appropriate information technology, students could hardly answer them because they lead to calculations with complex formulas.

In mathematical terminology, the arch height corresponds to the maximum of the defining function. Using apt software, the problem becomes rather simple (see Figure 5). Based on Figure 4 one can conclude that the maximum is in or near the point x = 0. Within the Function Maximum operator, the user specifies the investigated function, the neighborhood in which the maximum is to be found – [-10, 10] here – and the requested precision (2 decimal places). The notation is simple and “Maths-student-friendly” because it does not require special training – just a common sense (see Figure 5.a).

a.   b.  
Fig. 5 Finding the maximum of the cosine hyperbolic
Figure 5.b shows the coordinates of the maximum: x = 0, y = 625.09. The students can quite easily verify that their result (625 feet) is correct e.g. by surfing the Internet.

From Figure 4 one can also that the arch touches the ground in the points that are the roots of the equation

693.8597 – 68.7672 cosh(0.0100333 x) = 0
The Equation Solver quickly produces the values -299.226 and 299.226. The distance between pylons is approximately 600 feet.

More difficult calculations should also benefit from information technology. Let us solve the problem: There is an elevator from the ground to the viewpoint at the top of the arch. It moves inside one of the pylons. What is its length?

To find an approximate solution, we split the arch into a broken line. From our previous calculations we know the coordinates of its ends: the top is (0, 625), the foot is (299.226, 0). Figure 6 shows the calculation in a spreadsheet. The first column contains the x-coordinates of the ends of the straight lines; the second column their y-coordinates. In the third column, the length of each section is calculated using the Pythagoras theorem. The D2 cell contains their sum – the approximate length of the elevator.

Fig. 6 The curve has been replaced by six lines

All above considerations indicate a potential way to more human-like, understandable Mathematics. Our students do not need to know a method of calculating roots of the cosine hyperbolic. But they certainly should know the relationship between the mathematical notions (like the maximum, the root, the length of a curve) and their reflexes in reality. Visualizations can substantially support it. Figure 4 can easily be interpreted in both ways – as a real structure in Saint Louis and as a curve specified by a formula. Making our students capable of finding similar relationships and exploiting them in their future professions would be our great victory.

References

[1] http://mathworld.wolfram.com/Catenary.html
[2] William V. Thayer: Owner’s manual for Saint Louis Arch, 1993.   http://www.jug.net/wt/archcgs.htm