Introduction
The major purpose of this report is to describe and assess mathematical and statistical investigations to single out similar and distinctive patterns. In particular, it is necessary to focus on such aspects as the validity of the proof, arguments, and presentation of quantitative and qualitative data. The first study under discussion is aimed at analyzing the angles between the hands of the clock every time their positions coincide (Henley, 2005, p 13).
In turn, a statistical study examines the influence of climate on the harvest of the beans (Stuart, 2006). These works have been chosen because they represent different approaches to mathematical thinking, which can be both numerical and qualitative. In addition to that, they demonstrate that such a concept as mathematical knowledge comprises a great number of components.
The assessment of these works should also be connected with several teaching and learning issues, for instance, the main skills which a student should acquire in the course of training, the role of educator, and the methods which the educator should employ to help learners. Overall, we can argue that mathematical activities and the investigation process in these works can be of great assistance to the learners and teachers, at least some of the examples given by the authors. All of these questions will be discussed in the following questions.
Evaluation of Investigations
Mathematical Investigation
Overall, this investigation is dedicated to one of the most interesting and complex topics in present-day geometry, namely, the problem of time and the circle geometry, associated with the position of the hands-on clock. It is stated that the hands of the clock coincide several times during the day. While everyone knows that the needles are in the same position at noon and midnight, few mentioned the next time of their coincidence. It is reasonable to suggest that they coincide every hour when the minute handle reaches the same position with the hour needle, nevertheless, the next time, when they coincide will be 01:06.
Following Calter (2008, p. 334) the following statement should be emphasized:
A general approach to such problems is to consider the rate of change of the angle in degrees per minute. The hour hand of a normal 12-hour analog clock turns 360 degrees in 12 hours. This is equivalent to 360 degrees in 720 minutes or 0.5 degrees per minute. The minute hand turns 360 degrees in 60 minutes or 6 degrees per minute.
The same situation is with 11. The hands coincide at 10:54, and then at 0:00 only
We can take t to be the time, in minutes, after 1 pm i.e. at 1 pm t = 0. We then take Ө to be the angle, in degrees, through which a hand is rotated (clockwise), measured from the 12 o’clock position. Let us assume that the hands move steadily rather than in discrete jumps. It is then possible to find a linear relationship between t and Ө for each of the hands, the minute hand, and the hour hand.
Ө = 6t
Ө = 0.5t + 30
Originally, minute and hour hands coincide every hour except the period between 00:00 and 01:06, when the hands do not place the same positions. Originally, the angles between hands are not equal to zero, the coincidence of the hands, when the angle between hands is equal to zero is 0:00 only.
The next point, when hands show 1:06, the angle is 3 degrees. Then, every coincidence presupposes the difference in positions of the hands for at least 1 degree.
Under the calculations, the positions of the hands will be the following
By this data table, it should be stated that the coincidences happen 22 times per day.
Therefore, Simon Henley tries to substantiate his hypothesis graphically. Such type of proof takes its origin in ancient Greek geometry (Kitcher 1985). It is quite difficult to evaluate the persuasiveness of such an argument and proof. Naturally, the visual representation of the clock is quite acceptable but, it should be claimed that such occurrences cannot be observed by a human eye, as the angles are miserable, and the coincidence of the hands is conditional. Consequently, this is one of the reasons why it is so difficult to give the evidence that the calculations are correct, as well as the presumptions (Rucker, 1977).
Statistical Investigation
Originally, when we look at a statistical investigation, we start with the same question of what problem is it we want to explore. We state a hypothesis, that when the hands of the clock coincide in their positions, the main consideration of the hypothesis is that the 0 angle is possible only when the hands show 0:00. Taking into consideration the fact that the angles are considered to be close to zero, the hypothesis was confirmed by the calculations. Unluckily, the visual representation of the hypothesis is impossible, as the angles between hands will not be mentioned by the human eye. Moreover, the data of the highest precision is represented in the data table.
Originally, it may be developed into several phases:
- Defining the Problem
- Gathering Relevant Information
- Presenting/Organizing Data
- Analyzing Data
- Interpreting Results
The problem is closely associated with the annual harvest of beans in South America. The data is taken from the geographic web portals, and the data, issued by the official government of Brazil.
These are the key variables in this study. The researcher’s overarching argument assumes that the data on the harvest and the weather (particularities of climate) are interconnected with one another. In this regard, we need to mention that sometimes interrelation between two variables is virtually unnoticeable. At this moment, there is a strong necessity to illustrate the basic procedures and principles to which the author followed in his analysis.
Considering the fact, that rains are not rare in this part of South America, it should be stated that droughts also happen. Thus, every 5-7 years of favorable years are featured with 1 year of drought. Originally, the harvests of beans are essentially lower. Thus, the dependence graph will be elaborated.
This research relies on longitudinal data series, gathered in the course of various surveys. These surveys cover the range of practically two decades. To some extent, this investigation can be called a systemic review because it uses the information which has already been collected by other scholars (Graham, 2006).
Thus, it should be emphasized that the dependence is not linear. Even if the precipitations are close to zero (Which is close to impossible), some level of harvest is possible due to the other factors, which influence agriculture.
Comparison of the Investigations
Certainly, one can argue that it is impossible to draw any parallels between these investigations as they differ in terms of their goals, objectives, research methods, argumentations, proof, and so forth. Yet, there are some features which they have in common. First and foremost, we need to say that both mathematicians heavily rely on such methods as proof by construction and visual proof. In other words, while trying to solve the major problem, they try to recreate a situation or to draw an example which demonstrates that their standpoint. However, the most interesting detail is the use of inductive logic.
For instance, while making their arguments both authors admit that there is a certain degree of subjective belief in their assumptions. Another detail that attracts attention: Simon Henley and Chi-Chung Chen do not use formal proofs. In their discussion, they cannot advance axioms or statements which do not require verification.
The Nature of Mathematical Knowledge
Overall, this discussion leads to the question about the peculiarities of mathematical knowledge. Mathematics as a science examines quantity, structure, space, time, and change. The knowledge of this discipline includes a great number of components, such as the ability to apply logic to find a solution to a certain problem, the ability to apply theoretical concepts to real-life situations, and most importantly (Cowan, 2006).
Mathematical knowledge cannot be reduced only to the knowledge of principles, operations, algorithms, formulas, and so forth (Leng et al 2008). Naturally, these are helpful tools but a student should have the know-how and in which situation they should be used. It is also critical that students can alternate different research methods to gather information and find a solution (Selinger, 1995). The most crucial thing to remember is that almost every branch of mathematics has practical implications; this is why learners and teachers must always try to find the connection between theory and practice, otherwise, their work may be to no purpose.
Implication for Learning and Teaching Practice
Even though these investigations may not be connected, they can be of great assistance to educators, who specialize in the teaching of mathematics. The main advantage is that they offer deep insights into mathematical reasoning, logic, and proof. The first investigation is extremely helpful because it shows the major principles of mathematical reasoning or abstract reasoning to be more exact. First, teachers can take full advantage of them while tailoring exercises for students. It may be prudent to develop similar types of tasks like constructing a four-dimensional model of an object or identifying the connection between two phenomena or processes. These investigations should not be ignored because they illustrate the use of various proofs. Students should be made acquainted with all these techniques.
However, the major benefit is that they enable the translation of theoretical knowledge of mathematics into practical skills. Learners must be able to apply their knowledge of formulas, operations, or theorems to solving real problems. Certainly, it has to be acknowledged that these investigations have several drawbacks, and these studies are by no means conclusive.
Conclusions
Therefore, we can conclude that these investigations illustrate various types of mathematical reasoning. While constructing a hypercube, Simon Henley resorts to visual proof to prove that projecting is an effective way of creating four-dimensional figures. His argument is qualitative. In turn, Chi-Chung Chen advances numerical examples to substantiate his perspective. While examining the impact of climate change on the use and cost of pesticides in the United States, he draws his conclusions from longitudinal surveys. His proof can be classified as probabilistic. These studies should be utilized by educators since on this basis many exercises to improve students analytical skills can be developed.
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