Author: admin

Where does the term “abacus” come from?

The term “abacus” has a meaning not only in mathematics, but also in architecture. There it stands for a column top plate, a board-like upper member of the capital. In other words, a “square plate at the head of columns. But it also has the meaning ” of a “command staff at temples” (in: Wiktionary). However, all this is not meant in the following. Here it concerns the “abacus”, which can be called probably rightly the oldest calculating machine of mankind. How to use an abacus?

According to Duden, the word “abacus” comes from the Greek, from “abákion”, and stands for arithmetic or game board of antiquity. If we go back to Latin, it comes from the word “abacus”.

Of course, the word abacus also exists in German sign language. To see it, please click here:

If it goes after the Duden, then as plural of the word “the abacus” “the abacus” is to be used (Duden, 1999). This sounds strange at first, but it is so. I stick to the plural formation. Since the Duden is no longer binding for the German language, the plural formation 1 in the Wiktionary should also be mentioned here. Here the plural 1 is namely “the Abaki” and obviously derived from Latin, as well as the word “abakus”. This makes sense in my opinion. The plural 2 is therefore “the abacuses” (in: Wiktionary).

The abacus has practically no meaning as a calculating device in Europe today. Electronic pocket calculators have replaced it. However, there are various abacuses for learning basic arithmetic operations for children, especially in pre-school and primary school age. In Asian, but also Arabic countries, arithmetic with the abacus is still learned in special arithmetic schools. It has been proven that it promotes the ability for quick and secure mental arithmetic even with very large numbers.

The historical roots of the abacus

The historical roots of the abacus probably lie in Asia, because “already around 1100 B.C. this ‘calculating machine’ was known in China” (Microsoft ENCARTA Encyclopedia PLUS 99). Originally, in this ancient advanced civilization, number rods were placed on printed tablets. Later, pierced counting stones were slid back and forth on sticks. This was already the basic form of an abacus. These stones now had a clear, always constant order, a place value system. They could be moved according to one’s own idea, but better according to a uniform calculation rule. Other sources assume that the abacus was developed by the Babylonians (National Geographic, November 2010, page 32): https://argoprep.com/blog/k8/abacus/.

The Chinese abacus came to Japan and was further developed into the soroban. The Indians also used similar things for arithmetic, namely so-called “dust tables”. The Babylonians already had tables for calculating with the multiplication tables around 1350 BC. By the way, why shouldn’t mankind in their advanced civilizations of that time make parallel and independent inventions of abacus or calculating boards? By the way, also the Maya knew the abacus. They called it Nepohualtzintzin. Whether it made the leap across the Atlantic, or was invented in parallel, certainly still needs clarification.

The cultural basis for the abacus used today in Europe is probably to be found in ancient Greece. The unique find of the “Salaminic Arithmetic Table” (National Museum in Athens), represents such a mathematical primitive device from the Greek cultural sphere. It was made of white marble and its age is estimated to the third century BC. On this historical abacus a division into fields is achieved by horizontal and vertical lines. Further one finds on it Greek number signs. On this arithmetic board it was possible to move loose arithmetic blocks back and forth. Here was a clear mathematical order, which could be used according to fixed rules. A miscalculation became improbable. There must have been something similar to the “Salaminic arithmetic table” also as an arithmetic table. The Greeks called their version of the abacus “abákion”.

“In ancient times … the Romans used a sand-covered wax tablet, a marked plate, or a notched tablet or plate” for arithmetic (Microsoft ENCARTA Encyclopedia PLUS 99). It is known from finds that they used hand abacuses made of clay for their arithmetical calculations. The Roman name “abacus” was used for this, the eponym of all modern abacuses.

The illustration of an ancient Roman hand abacus shows seven rows of four beads at the bottom and one at the top. Between the two rows of slits in the vertical you can find Roman numerals. It was probably used in a similar way to a Japanese abacus (soroban). With this Roman abacus, even fractions were possible. The beads of the two additional right rows stand for whole, half, quarter and third ounces. The similar construction of the Roman abacus with the Greek and Chinese abacus of that time is obvious. But there is especially an astonishing similarity with the Japanese soroban. This is probably an ancient technology transfer from the Asian to the Southeast European area, perhaps due to wars, trade relations or travelers.

With these ancient abacuses, one could safely handle all calculations in the commercial or statesmanlike field at that time. It should be remembered that it was not only a matter of actual arithmetic. The known world at that time was still far away from a uniform currency or measurement system. Just such conversions, therefore, were common in all places. In the old Rome, with their Roman numbers, one knew no “zero” and thus also no units, tens or hundreds etc.. The balls or calculating stones, Latin “calculi”, represented a certain number only by their position. Consequently, a place value system is used as a basis, which, however, is to be understood differently than our today’s decimal system.

“In medieval England, a simplified version of the abacus was used. It consisted of a board divided into squares representing the positions of the tiles. To calculate, coins, buttons, stones or other small objects were moved. The checkered tablecloth, from which the name exchequer comes, was originally a device for arithmetic like the divided table” (Microsoft ENCARTA Encyclopedia PLUS 99).

The famous German arithmetic master, Adam Riese, described the use of arithmetic boards in his work “Rechenbuch auff Linien und Ziphren in allerlei Handthierung / geschäfften und Kaufmanschafft”. His merit lies in particular in having popularized arithmetic in the decimal system commonly used today in German-speaking countries. It originated in the Arabic world and was perfected here. E.g. the digit “zero”, originated in the Indian area, was cleverly used for arithmetic. The Arabic origin was exactly the reason for the Catholic Church to dismiss it as pagan. Accordingly, the church at that time worked very much against the introduction of this form of arithmetic. However, especially the merchants of Europe quickly recognized the enormous advantage of this Arabic way of calculating, because until then they had been calculating in the Roman system. This was very cumbersome, even if one used an abacus or an abacus. Methodically, Adam Riese used a new type of abacus for his Arabic calculation. Perhaps the basis of all European abacus forms?

The abacus still allows complicated arithmetic operations today. Actually, it is still very simple in construction. It consists of a frame with balls or stones, called “beads”, which are threaded on rods or wires or guided in slots. It was used extensively in our country until the 16th century. Today it is only used in Asia, especially in China, rather in small stores. In addition to the four basic arithmetic operations, it is also possible, among other things, to draw roots with the help of an abacus. Addition and subtraction of large numbers are probably the absolute strengths of an abacus, even compared to a modern pocket calculator.

A definition of the term is important to me. An abacus is not a slide rule! With an abacus you calculate in a place value system. Today this is usually the decimal system (10s system). The mathematical basis of a slide rule is the calculation with logarithms.

The entrance test for the city’s elite specialized high schools might be given online due to the coronavirus, Mayor Bill de Blasio said Friday. SHSAT test online School practice  are hand crafted by SHSAT experts.

A caller to Brian Lehrer’s weekly WNYC spot with Hizzoner quizzed him asked about the fate of the exam this year.

“It will be administered,” he said, without offering any timetable. “The timing we’re working on.”

Given the ongoing coronavirus surge, de Blasio said kids are not likely to take the test at a desk in front of a proctor.

“Historically many of the standardized tests are in-person and that’s not a scenario we are likely to do, especially with what we’re talking about right now,” he said. “So we’re going to work out that time and we’re going to work out the methodology. We’ll have an announcement on that and certainly in the next few weeks.”

Parents have been demanding clarity on the administration of the test that determines admission to some of the nation’s top public high schools — including Brooklyn Tech, Stuyvesant, and Bronx Science.

Roughly 30,000 kids take the Specialized High School Admissions Test each year.

Mounting confusion heightened this week when the Department of Education abruptly canceled a series of information sessions on high school admissions a day after announcing them.

Those meetings were supposed to clarify admissions processes for all competitive high schools across the city — including the specialized campuses.

While he signaled the survival of the SHSAT Friday, de Blasio suggested that parents might have to extend their wait for specifics still further.

“We’ve got to first deal with this immediate challenge right now that we’re talking about,” he said. Use an shsat test online practice to help prepare for this exam. 

The SHSAT has become the subject of intense debate in recent years.

Opponents have called for its abolition and assert that it’s a narrow measure of student talent that gives the upper hand to kids with better preparatory resources.

Backers argue that it has helped to forge famed academic bastions that consistently feature heavy immigrant and low-income populations.

It is more Dra reading levels complicated than that: it is a non-state educational center at the Faculty of Philology of St. Petersburg State University. It is an institution with dual management through partnership with foreign universities. In other words, it is an independent educational center located between the two spaces of Russian state education and foreign private education.

So, getting back to the question of the correspondence between the diploma and the graduate’s situation on the labor market in France… Today, unlike the situation 30-40 years ago, the diploma does not grant any privileges on the labor market. In a situation of mass education, it only serves as a necessary and minimum condition of access to qualified employment: according to the level of the diploma, the graduate has access to a starting job. That is, for those who receive a bachelor’s degree, the set of available professional positions is much narrower than for the holder of a master’s degree. At least in the normative model of the labor market.

As for the doctoral diploma (the European equivalent of postgraduate studies), there is a somewhat different story, on the whole, close to Russia. Successful completion of doctoral studies is not directly related to bonuses in the labor market: in science departments, it is the first necessary step in a research or university career. Although doctoral studies in a for-profit school are seen as analogous to MBAs, if not formally as such. And as the value of the scientific doctorate in the current situation is getting lower and more and more students are getting degrees from the first two levels, there are reasons for the third educational level to become a kind of market accessory some time later.

In France in the last ten years there have been several scandalous cases of misuse of doctoral studies: the successful defense of a dissertation by a newspaper astrologer or a far-right politician. However, these cases are rare and fall at the center of professional and public scandal. It should be recognized that the credo of most university professors in France is professional integrity and republican concern for education, if only because they are much more massively and thoroughly involved in the management of universities than in Russia, and in their professional competition the criteria of intellectual competence play an essential role.

In Russia, the value of the PhD degree has also been devaluing rapidly over the last 15 years, admittedly much faster than in France. As a result of the deinstitutionalization of teachers’ personal involvement in the functioning of higher education institutions and the unscramble my word denial of responsibility for their results, the commercial overreaching of higher education qualifications is much more pronounced and dramatic. PhDs and doctorates are becoming a distinction that big politicians or wealthy market players, rather than academics, are beginning to look for in order to add another mark, a kind of optional but prestigious vignette on their business card.

The introduction of Cogat test prep inclusive education will indeed be useful, given the nature of violations among special children and their needs. The existence of mild and severe forms of impairment, combined impairments, should lead to the understanding that not all children with disabilities can attend a regular preschool. The manifestation of different forms of violations – behavior, well-being of a child, and the impact of the environment on this – can degrade the health status of both the child and other children and teachers. For example, if we talk about a child with attention deficit and a child with cerebral palsy, mental retardation and speech disorders, we should compare the type of assistance and correction and the need for the right specialists.

A mild form of impairment requires much less conditions for its correction than a severe form of impairment. If, in the first case, a small number of methods and stimulating materials are needed to diagnose and identify this disorder, a program consisting of corrective and developmental classes, a teacher, a psychologist, and a tutor will select methods to work with the child according to specialist recommendations. In the second case, there must be much more conditions and more work with the special child. Installation of special equipment (ramps and supports), which is not available in a regular kindergarten, availability of specialists (a teacher – psychologist, orthopedist, teacher – defectologist, LFK teacher), constant attention and control by a tutor (assistance in moving around, during hygienic activities – individual approach), when there are about twenty other children in the group. In remote places of residence there are no such specialists, additional rates are not provided in general educational pre-school institutions, and for the pre-school institution itself it is not reasonable to hire such a specialist, which will be needed for several years, and on the date of release of such a child from the pre-school institution there is no guarantee that in a sparsely populated place a child with a similar rare and severe disorder will enter the pre-school institution. It turns out that a child with disabilities will acquire knowledge and skills within the framework of general education standard in the same period of time as normally developing children only if the level of psychophysical development corresponds to or is cogat exam close to their age.

The caregiver to whom this child will enter may experience fear: whether he or she will cope, whether he or she will be able to help, whether he or she interacts with the child correctly. This fear will contribute to the teacher’s failure psychologically and morally – increased responsibility, uncertainty, non-acceptance. Ignorance of how the failure manifests itself, what can harm it, and failure to use special and recommended techniques and methods of working with a special child will primarily harm the child himself and create a tense environment around him.

Failure to accept a special child may result from a teacher’s incompetence – the educator does not know and cannot interact with this category of children. This is done by specially trained people who are specialized in particular violations (a teacher – speech therapist with speech disorders, a teacher – psychologist – escorting all categories of children, a teacher – defectologist can also work with such violations as mental retardation), who guide and advise teachers and the administration on special children. Whether a special child will be provided with the right specialist and whether the specialist will be professional and competent in his/her activities is difficult to control.

By the end of this decade, four out of every Kinesthetic learner characteristics professionals in the world will be from only two countries – China and India. Economic growth in Asian countries will also change the balance of higher education graduates. In the next eight years, China and India are expected to lead the way in training students.

Studies by the Organization for Economic Cooperation and Development (OECD) show far-reaching changes in the balance of number of graduates, and growing Asian economies should bypass the United States and Western Europe.

Projections of the kind that the “global talent bank” will have in 2020 show that China will continue to increase the number of its graduates – up to 29% of the world’s 25-34 year olds. The U.S. will lose the most – decline to 11% – the country for the first time will take third place and will be after India.

The U.S. and European Union countries together are expected to have slightly more than a quarter of the world’s young graduates. Russia will also reduce its indicators – its share of graduates in the world will almost halve since the beginning of the century. Indonesia, according to forecasts of the OECD, will rise to fifth place (on the photo – students from Indonesia).

Is this the end of the empire? Higher education has gre vocabulary list become a mirror and a magnifying glass of economic activity. In the period after World War II, universities in the United States, Western Europe, Japan and Russia dominated. The USA, in particular, was a superstate for universities – in understanding their wealth, influence, and, until recently, the number of statistical indicators. Until 2000, the U.S. had a particle of young professionals relative to China. Japan also had many young specialists, as did India.

Today, China and India are the biggest players in the industry. The increase in the number of graduates reflects a change in the ambitions of these countries – now they want to compete with developed economies for high-income jobs. Instead of developing low-cost industries, they aim to attract high-tech jobs that the Western middle class had previously reserved for them.

Fivefold growth
As OECD data show, this is not only the case of countries such as China, whose economy is expanding while others are standing still. In all industrialized countries, the number of graduates is growing – just not as fast as in China, where their number has increased five times in a decade. OECD notes that by 2020, the number of young graduates from China will be approximately equal to the total population of the U.S. population aged 25 to 64 years. On this altered world map, we will see that Brazil will have more graduates than Germany, Turkey more than Spain, Indonesia three times more than France. In the UK, the opposite trend is observed, the country is projected to increase its particle size from 3% in 2010 to 4% in 2020. This drive to increase the number of graduates has clear economic goals, according to the OECD study.

Will there be enough jobs?
The transition from “mass production to the knowledge economy” means better employment and income – so there are “strong incentives” for countries to expand higher education. But will there be enough jobs for graduates with higher education? The OECD has tried to analyze the situation by looking at one side of the labor market: the number of jobs in science and technology. The number of such jobs has grown rapidly – and the report says that this is an example of how expanding higher education can create new types of employment.

These S&T jobs – for specialists and technicians – account for about every four out of 10 jobs in some Scandinavian and Nordic countries. By contrast, and this is an indicator of the older economy, these technological jobs represent only a small proportion of the workforce in China and India.

The OECD believes that there are significant economic benefits to investing in higher education – this creates new jobs for the more educated, while unskilled jobs disappear.

Today, no one is surprised by distance learning; most educational institutions of different levels in Russia and abroad use remote technologies to some extent in the teaching process. Many students today consider this form of education as an alternative to the usual one not only in the process of obtaining a second degree, but also choose this method for their first higher education. Just 10 years ago the number of students in the world who receive education in distance technologies exceeded the number of full-time students [4]. There is no escape from modern trends, and many universities understand this and try to develop in this direction. By the way, it is correct to say “distance learning”, not “distance education” as it is just another way of forming knowledge and skills of a student, and the content of education remains the same.

In the works of many modern authors the problems of distance learning development in Russia are considered, but probably each of them, before starting to study the problems, studied the question of expediency and effectiveness of this form of education. Let us make a short review, and also give our thoughts on this issue.

So, the arguments are in favor:

1. most researchers call the first “plus” of distance learning the opportunity for students to receive in-service education [1, 2]. This is indeed a very important argument for choosing this form of education, especially for those who have decided to pursue higher education as a mature student;
2. it is not necessary to travel to the institution, at least not frequently. This is especially true for students from the periphery: it reduces financial costs and provides the opportunity to obtain a degree from the capital city or from abroad;
3. Those who are physically unable to attend classrooms because of a disability also have the opportunity to receive an education. Our university and personally the author have experience of working with such students in distance;
4. There have always been and are people in our society who are able to “educate themselves,” i.e., self-educate, but for one reason or another have no officially recognized degree. The remote form of education allows to reduce to some extent the dependence of a person who wants and is able to obtain knowledge independently, from educational institutions and have the opportunity to obtain a diploma of higher education on an equal basis with others;
5. possibility for students to participate in the organization of the educational process: to choose time and a place for work with an educational material, to define speed of studying of a material corresponding to features of the thinking;

Of course, distance learning has its own “disadvantages”

1. the student does not have the opportunity to consult personally with the instructor;
2. there is no opportunity to study “live” to build relationships in a team (with teachers, classmates, administration of the university), to speak to the audience;
3. not every profession can be learned remotely (a doctor, veterinarian);
4. not every student is able to maintain his or her motivation for independent work. In addition, the lack of such an effective motivator of academic activity as permanent control by the instructor is also affected. Those teachers who work with first-year students in universities know how important it is for freshmen, especially at first, to check their homework and regularly organize checkups and tests. The percentage of students who do not need such supervision is very low, and they already realize that independent work on a subject is necessary;
5. The student does not have the opportunity to compare the intermediate results of his or her studies with other students, and does not have the opportunity to compare “live”: when working at the whiteboard, speaking at conferences, etc. read from here https://argoprep.com/blog/k8/benefits-of-homeschooling/
6. the absence of a person who submits material with emotional coloring, which affects the degree of his understanding (remember a fragment of the movie “The Great Change”, in which Nestor Petrovich is giving a lecture about the Emperor Nero, who ordered the burning of Rome, which would have inspired him to write poems about the fire – none of the listeners remained indifferent).
7. It is important for the teacher in the classroom to feel the students’ understanding of the material (according to their views, the questions asked, the answers to their questions) and to quickly adjust the teaching process: to repeat difficult moments once again, to give additional explanations on some questions, to change the pace of the presentation. In distance learning, this connection is lost;
8. the student has the temptation and opportunity for “off-campus” study and the instructor has no way of controlling such costs of distance learning;
9. For an institution of higher education, the introduction of distance learning is associated with high material costs: technical equipment, software and hardware, training of special personnel, etc.

At the same time, however, it should be noted that to some extent this compensates for some of these shortcomings: it is not possible to get a face-to-face consultation – it is possible to consult with the help of messages or Skype; there are no “live” relationships – but for some students this provides a more comfortable psychological environment for studying, etc. In addition, the traditional form of study has not been canceled, and students always have a choice between full-time, part-time, and distance learning.

Our university has so far little experience in the use of full-fledged distance learning, and most teachers use distance learning technologies in addition to traditional ones. For example, the author has developed an e-learning course “Probability Theory” based on which is successfully used by several streams of full-time and part-time students. We regularly conduct a survey of students in order to identify problems they have when working with e-courses and to learn their point of view on the effectiveness of using remote technologies in the educational process [3]. Students believe that the combination of conventional classroom activities and independent work with e-learning courses is quite effective (table), as evidenced by the results of the analysis of student performance.

Thus, distance learning, like any other form of knowledge acquisition, has many advantages and disadvantages. A significant disadvantage is the lack of a centralized system of certification and accreditation of e-learning courses, resulting in a multitude of “handicrafts”, loudly called e-learning courses and e-textbooks, which are in fact ordinary word files. An important factor that prevents more intensive introduction of distance technologies into the educational process is the lack of motivation of university teachers to work in this direction [5]. Perhaps the reason for this is the high labor intensiveness associated with the creation of methodological materials for distance learning, but it is important to realize that in the future the time and effort involved should be compensated for by reducing the time spent on some of the usual types of educational workload of the teacher.

At this stage of development of distance learning, our task is to organize the learning process in such a way that new forms of learning produce at least the same quality results as traditional ones. In addition, there are many questions about how to measure the effectiveness of distance learning. But anyway, the huge “plus” of distance learning technologies is that they allow anyone to learn continuously – all their life.

Multi-sensory mathematics teaching methods

The multi-sensor approach helps children learn.

Using vision, kinestics, hearing and movement can make it easier for children to understand what numbers and symbols represent.

Method 1 of 7. Visualization

Using balls, dried beans or even M&M dragee as an object of manipulation is a great way for children to imagine the essence of mathematical operations.

For example, children can solve a problem by adding more beads. Or they can find out how much is left after subtracting (physically disappearing) some. Children can also group different numbers of items to multiply and divide.

Method 2 of 7. Using Lego Details

Using Lego elements to create shapes allows children to get acquainted with the theme of part and whole.

It also gives them a concrete idea of the properties of the shapes they create. Tiles and cubes also work perfectly when teaching methods and operations.

For example, the teacher can stack items in groups of 2, 4, 6, and 8 of the same pattern and then ask the student to do the next three stacks in the template (by adding twice each time). The teacher then helps the student to connect the items to the numbers they represent.

Method 3 out of 7. Drawing

Drawing mathematics is the next step after working with practical materials such as beads or coloured lego dice. It’s a way of showing children how to think – and it takes them one step closer to writing numerical sentences with numbers and symbols.

For example, a teacher may ask students to solve a 4×6 multiplication (4 take 6 times) by typing 6 groups of 4 asterisks (apples, balls).

Method 4 out of 7. Adding music to math

There are many ways to connect math and music. For example, children can use songs to memorize algorithms or mathematical rules. And playing music notes can help children learn how to group or split.

For example, a teacher can play one note on the keyboard and hold it for a short time. It’s a “whole note.” Then students repeat it. Then the teacher may ask, “How many quarters of notes make up a whole note?”

After some discussion, the teacher or student can play four short notes that are the same length as the whole note.

Method 5 out of 7. Integration of motion into mathematics

Multi-sensory Learning: Techniques for Math learning is an attractive way to help students maintain what they have learned. There are many ways to do this. For example, children can show angles by turning their bodies while they are in a hoop.

There is an interesting game.

A teacher writes numbers on the outside of the ball (big ball). (It can be integers, fractional numbers, or decimal places). The ball passes through the room, and when the pupil catches it, he must perform a mathematical operation with two numbers on which his hands land.

Method 6 out of 7. Graphic Organizers

You can create a table of hundreds. A chart of hundreds can help children with mathematical problems see numerical relationships.

For example, a child can get a black and white grid of 100 squares (a large square divided into 100 units of a smaller square). It can be asked to paint 1/4 of its total volume.

When he or she has to find the number of square units to be colored (25). The child will determine for himself that ¼ is the same as 25 of 100, or 25 percent.

Method 7 out of 7. Use TRIZ in mathematics teaching

TRIZ is an acronym that stands for “the theory of solving inventive problems”. Like any other theory, it has its structure, functions and algorithm.

The theory of inventive problem solving is one of the most unique methods for development of creative abilities of a child, development of sense of mathematics in children.

As G.S. Altshuller, the founder of this method, said: “TRIZ is a controlled process of creating a new one, combining accurate calculation, logic, intuition”.