How Interval and Fuzzy Techniques Can Improve Teaching

In this book, we describe how interval and fuzzy techniques can help in education.Before weMotivation even start teaching the material, we need to motivate the students. As we have mentioned, this is an extremely important aspect of teaching: if a student does not understand why this material is useful, this student may not be as committed to study.

As we have mentioned, there are two important issues related to uncertainty. First, the students need to understand the very presence of uncertainty – and, thus, the need to take uncertainty into consideration. The ultimate objective of learning is to enable students to make good decisions in their future professional life. In most real-life situations, we make decisions based on imprecise, natural-language descriptions and common sense ideas. As a result, students often do not understand the need for complex mathematics-based techniques in decision making. To understand this need, we need to emphasize the deficiencies of the commonsense approach, we need to explain, to the students, how seemingly reasonable commonsense ideas can inadvertently lead to counter-intuitive erroneous decisions. It is therefore important to emphasize the fact that commonsense treatment of the topics leads to contradictions (paradoxes) Contradiction and therefore, a more formal treatment is needed.

Once we understand that it is important to take uncertainty into consideration, it is necessary to decide how to take uncertainty into account.If we properly takeGeometryPlane rescue uncertainty into account, then the corresponding problem becomes more realistic – and thus, students have more motivation to study it – and at the same, the problem becomes more challenging, so it can serve as a case study for learning to solve complex problems. In this chapter, on the example from geometry, we show how taking uncertainty into account leads to realistic challenging problems, problems on whose example students can understand the need for (and the usefulness of) complex techniques that they are studying. In the next chapter, we describe a similar example from calculus.

Students learn better when they understand why they need to learn the specific material, why this material is useful in their own discipline. In this chapter, we show how to explain usefulness of simple calculus results to engineering and science students – on the example of such a seemingly theoreticalCalculusMean value theorem result as the Mean Value Theorem. It turns out that this result is very useful in problems with interval uncertainty.

How can we increase a person’s interest in some topic, a person’s excitement in studyingMathematicsin ancient EgyptEgyptian fraction this topic? Motivation Each student already has some topics about which he or she is excited, so a natural way to increase the students’ interest in the topic of study is to relate this topic to the topics about which the student is already excited.

In the previous chapter, we showed how to use ideas from ancient Egypt when teaching math. In this chapter (and in the following chapter), we consider ideas from Mayan and Babylonian mathematics.

InMathematicsin ancient BabylonSquare root the previous chapter, we showed how to use ideas from Mayan and Babylonian arithmetic when teaching math. In this chapter, we consider Babylonian mathematical ideas beyond simple arithmetic, namely, the ideas of computing the square root. When computing a square root, computers still, in effect, use an iterative algorithm developed by the Babylonians millennia ago. This is a very unusual phenomenon, because for most other computations, better algorithms have been invented – even division is performed, in the computer, by an algorithm which is much more efficient that division methods that we have all learned in school. What is the explanation for the success of the Babylonians’ method? One explanation is that this is, in effect, Newton’s method, Newton’s method based on the best ideas from calculus. This explanations works well from the mathematical viewpoint – it explains why this method is so efficient, but since the Babylonians were very far from calculus, it does not explain why this method was invented in the first place. In this chapter, we provide two possible explanations for this method’s origin. We show that this method naturally emerges from fuzzy techniques, and we also show that it can be explained as (in some reasonable sense) the computationally simplestFuzzy techniqueComputational complexity technique.

In the previous chapter, we showed how to use ideas from ancient Egyptian, Mayan and Babylonian mathematics when teaching math. In this chapter, we consider the use of a more recent computational tradition, namely, of a Russian peasant multiplication algorithm.

In the previous chapters, we showed how to use ancient traditions when teaching math. In this chapter (and in the following chapter), we show how ideas borrowed from modern practices can also help.

In the previous chapter, we showed how to use usual modern practices when teaching math. In this chapter, we focus on unusual modern practices, namely, on the ability of some people to perform calculations unusually fast. In the past, mathematicians actively used this ability. With the advent of computers, there is no longer need for human calculators – even fast ones. However, recently, it was discovered that there exist, e.g., multiplication algorithms which are much faster than standard multiplication. Because of this discovery, it is possible than even faster algorithms will be discovered. It is therefore natural to ask: did fast human calculators of the past use faster algorithms – in which case we can learn from their experience – or they simply performed all operations within a standard algorithm much faster? This question is difficult to answer directly, because the fast human calculators’ self-description of their algorithm is very fuzzy. In this chapter, we use an indirect analysis to argue that fast human calculators most probably used the standard algorithm.

In the previousMotivationAsymmetric paternalism chapters, we provided examples of how uncertainty-related examples and ideas help explain, to the students, why a specific material is useful and interesting. In addition to this, we also need to make sure that the students are excited about studying in general, that their levels of interest and commitment remain high. If a math instructor convinces the engineering students that they need to learn math, this should not lead to them getting less interested in studying engineering disciplines, ideally they should be excited about all the topics that they study. In this chapter, we therefore analyze how to increase this general level of interest. It turns out that properly taking uncertainty into account can help with this task as well.

At prime research universities, Motivationfinancial students study full-time and receive their Bachelors’s degreeUniversityprime research in four years. In contrast, at urban universities, Universityurban many students study only part-time, and take a longer time to graduate. The sooner such a student graduates, the sooner will the society start benefitting from his or her newly acquired skills – and the sooner the student will start earning more money. It is therefore desirable to incentivize students to graduate faster. In the present chapter, we propose a first-approximation solution to the problem of how to distribute a given amount of resources so as to maximally speed up students graduation.

In Part 1 of this book, we described how uncertainty ideas can help motivated students to study. Once the studentsOrder of material are motivated and teaching starts, we need to decide in what order we should present the material. Some courses first provide the basic ideas of all the topics, and only after all the basics are described, provide the technical details; other courses first deal with one topic, then go to another topic, etc.

OneCurriculumspiral of the fundamental ideas of modern education is the idea of spiral curriculum, when students repeatedly revisit the same sequence of topics at the increasing levels of depth, detail, and sophistication. In this chapter, we show that under reasonable assumptions, the optimal sequence of presenting the material should indeed follow a spiral pattern.

Since we cannot spend as much time as we would like to on teaching all the topics, it is necessary to optimally distribute the limited amount of time between different topics. In this chapter, we explain how general techniques of optimization under uncertainty can be used in education.

In the USA, in the last decade, standards have been adapted for each grade level. These standards are annually checked by state-wide tests. The results of these tests often determine the school’s funding and even the school’s future existence. Due to this importance, a large amount of time is spent on teaching to the tests.

In the previous chapters, we described the optimal frequencies with which we repeat each of the items that a student has to learn. Once we know the number ofOrder of materialFractal repetitions of each item, the next natural question is: in what order should we present these repetitions? Should we first present all the repetitions of item 1, then all the repetitions of item 2, etc., or should we randomly mix these repetitions?

In many applicationInter-disciplinary researchCurriculuminter-disciplinary areas, there is a need for inter-disciplinary collaboration and education. However, such education and collaboration are not easy. On the example of our participation in a cyberinfrastructureCyberinfrastructure project, we show that many obstacles on the path to successful collaboration and education can be overcome if we take into account that each person’s knowledge of a statement is often a matter of degree – and that we can therefore use appropriate degree-based ideas and techniques.

Let us start our analysis of the order in which the material should be presented with the teaching of each individual topic. Each topic in math and sciences usually contains some new abstract notion(s). Several examples are usually given to motivate these notions and to illustrate their use.

In the previous part of the book, we analyzed what is the best way to organize the topics and what is the best order of teaching different topics. In this part, we move to the analysis of what is the best way of teaching every time.

InTeachingoptimal a typical class, we have students at different levels of knowledge, student with different ability to learn the material.In real life, our resources are finite. Based on this finite amount of resources, what is the best way to distribute efforts between different students?

Writing on the board is an important part of a lecture. Lecturers’ handwriting is not always perfect. Usually, a lecturer can write slower and more legibly, this will increase understandability but slow down the lecture. In this chapter, we analyze an optimal trade-off between speed and legibility.

To enhance learningGroupoptimal, it is desirable to also let students learn from each other, e.g., by working in groups. It is known that such groupwork can improve learningGroupwork, but the effect strongly depends on how we divide students into groups. In this chapter, based on a first approximation model of student interaction, we describe how to optimally divide students into groups so as to optimize the resulting learning. We hope that, by taking into account other aspects of student interaction, it will be possible to transform our solution into truly optimal practical recommendations.

InGroupoptimalpracticeDiversity, there are many examples when the diversity in a group enhances the group’s ability to solve problems – and thus, leads to more efficient groups, firms, schools, etc. Several papers, starting with the pioneering research by Scott E. Page from the University of Michigan at Ann Arbor, provide a theoretical justification for this known empirical phenomenon. However, when the general advise of increasing diversity is transformed into simple-to-follow algorithmic rules (like quotas), the result is not alwaysQuotas successful. In this chapter, we prove that the problem of designing the most efficient group is computationally difficultNP-hard (NP-hard). Thus, in general, it is not possible to come up with simple algorithmic rules for designing such groups: to design optimal groups, we need to combine standard optimization techniques with intelligent techniques that use expert knowledge.

Some students are late for classes. If we let in these late, this disrupts the class and decreases the amount of effective lecture time for the students who arrived on time. On the other hand, if many students are late and we do not let them in, these students will miss the whole lecture period.

It is important to assess the results of teaching. The main objective of teaching is that students learn the material. So, the most important assessment task is to assess the students.

The main objective of teaching is to make sure that the students learn the required material. From this viewpoint, it seems reasonable to reward students when they learn this material. In other words, it seems reasonable to assign the student’s rewards (such as grades) based on their level of knowledge.

Students do not always spend enough time studying. How can we encourage them to study more? In this chapter, we show that a lot depends on the grading policy. At first glance, the problem of grading may seem straightforward: since our objective is that the students gain the largest amount of knowledge and skills at the end of the class, the grade should describe this amount. We show, however, that it is exactly this seemingly straightforward grading policy that often leads to an unfortunate learning behavior.

Most education effortsAssessmentof studentsAssessmentas a surprise are aimed at educating young people. So, to make education as effective as possible, it is desirable to take into account psychological features of young people. One of the typical features of their psychology – as distinct from the psychology of more mature population – is that they are much more risk-prone.

In modern education, a lot of students’ efforts goes into group projects. In many real-life situation, the only information that we have for estimating the individual contributions $$E_j$$ to aAssessmentin a group group project consists of individual estimates $$e_{ij}$$ of contributions of other participants j.

How many points should we allocate to different assignments and tests? to different problems on a test? Usually, professors use subjective judgment to allocate points. In this chapter, for classes which are pre-requisites for others, we provide an objective procedure for allocating points.

In many practical situations, it is desirable that the students learn all the parts of the material. It is therefore desirable to set up a grading scheme that encourages such learning. We show that the usual scheme of computing the overall grade for the class – as a weighted average of grades for different assignments and exams – does not always encourage such learning.

OnceAssessmentof a class we have assessed the individual performance of all the students in the class, it is desirable to combine these estimates into a class assessment. A natural way to assess the class is to provide the average grade and some measure of deviation from this average grade.

Sometimes, theAssessmentof teachers efficiency of a class is assessed by assessing the amount of knowledge that the students have after taking this class. However, this amount depends not only on the quality of the class, but also on how prepared were the students when they started taking this class. A more adequate assessment should therefore be value-added, estimating the added value that the class brought to theAssessmentvalue-added students.

There are many papers that experimentally compare effectiveness of different teaching techniques. MostAssessmentof teaching techniques of these papers use traditional statistical approach to process the experimental results. The traditional statistical approach is well suited to numerical data but often, what we are processing is either intervals (e.g., A means anything from 90 to 100) or fuzzy-type perceptions, words from the natural language like “understood well” or “understood reasonably well”.

When students select a university, one of the important parameters is the average class sizeAssessmentof universitiesClass sizeaverage. This average is usually estimated as an arithmetic average of all the class sizes. However, it has been recently shown that to more adequately describe students’ perception of a class size, it makes more sense to average not over classes, but over all students – which leads to a different characteristics of the average class size. In this chapter, we analyze which characteristic is most adequate from the viewpoint of efficient learning.

Education is a very important process. We are all aiming to teach better. And in the 21st century, this means using computers to automate (and improve) as many aspects of teaching as possible.