mathwizards10

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Math in the Philippines: Issues and Trends

by: Alvin M. Cariño

Mathematics, as what we have learned in the previous modules, is not only enclosed on the typical academic setting in which only revolves on extensive mathematical figures and formulas but also covers its practical application on the different fields in everyday living. We can find math on arts and humanities; social sciences; life science and operation research. By these, we can say that math is definitely everywhere and if we say everywhere it also covers its own issues and trends that might affect societal framework.

 People often look at mathematics as an unrealistic practice since not all of its subject matters can be applied to our daily basis; we only need to know the basic arithmetic in order to survive. How are we able to apply the rules square roots in buying basic commodities? How are we able to use the x and y n paying fares? These points at issue are already inculcated in the minds of people since then; they often perceive that why do we need to study its complexity if in the real world we commonly use the basic addition, subtraction, multiplication and division. The question now is: Why do we really need to study all these complicated things? Though many people are looking at it from this point of view, still, learning mathematics is beneficial to all. The main value of its complexity is the fact that it exercises our analytical skills that helps us to think critically. As we develop this skill, we will be able to find structure and strategies not only for solving mathematical equations but also for solving life and societal problems.

However, the issue of mathematics education arises on how are we going to address problems if many students tend to fail on this discipline? It is said that happened not because of their negative presumptions about math but due to some aspects such as gender, socio-economic background, race and culture that have been shown to impact mathematical learning. Thus, the important and current issues around mathematics should be discussed deeper for everyone to realize the significance of valuing this discipline in understanding the critical role mathematics plays in building a just and humane society. This would be further realized if we narrowing our scope on what the current state of mathematics is in the Philippine context.

TRENDS IN MATHEMATICS

Mathematics, as a human concept is a discipline that has background story, is dynamic and evolves constantly. Some trends that we should be considered are the issues of math regarding the culture called ethnomathematics—is a field of study that looks at the interaction between mathematics and culture and the increasing ubiquity of technology in mathematics. These trends might affect the resolution of mathematics in providing its substance to the people.

On the case of technology as a trend in mathematics, it is becoming a vital tool that aids not only the instructors but also the students in widening the area of their learnings and information. These technological devices are also important in reinforcing mathematics in society. At the present time, learners prefers visual representation and interactive approaches in order for them to better comprehend information. As a result, certain fields develop a tool that would represent mathematical concepts clearly that could deepen the conceptual understanding of the learners, such as computers, graphics calculators, interactive smart boards and other various apps are increasingly useful to aid in mathematical learning.

However, there is an arising issue concerning the use of technology which is known as digital divide in which tackles about the access of this certain tool. Not all schools, not all teachers, and not all students have the capacity to acquire these technological tools to aid them in teaching and research because of several factors like socio-economic, locations, and the readiness in applying such technologies like computers and graphics calculators in mathematics. The growing use of digital technology is also influencing how students are accessing mathematical knowledge. These technologies help everyone to broaden their perspective on math but having a lack of access on these devices might also deprive the people from widening their knowledge on this field.

In the Philippines most especially rural areas, they do not have enough access in these devices especially when it deals with internet connections. Learners on rural areas often focus only on textbook given to them and not able to explore more on the topic because they lack resources. The distribution is learning is very distinctive if we compare those people who have access and those people who do not. In this developing country, poverty is still evident that is why perhaps, learners who have economic hardships do not have enough financial resources in availing the learning material. Lastly, unlike other countries, we are not yet equipped yet in incorporating advanced mathematical learning technologies in classroom instruction. Therefore, we can see that the use of these technologies have a big influence in framing our learnings.

MATHEMATICS AND DIFFERENCE

 The learning of mathematics among individual can also be compared through their differences. These differences can help us picture out on what is the extent of mathematical understanding in a given situation. There are two issues that tackles on mathematics and difference: the first issue has to do with socioeconomic status as an important factor impinging on success in mathematics learning; the second pertains to Indigenous peoples and their access to quality and culturally relevant mathematics education.

 Some research shows that students from low socioeconomic status households and communities are slower to develop academic skills than those groups with high socioeconomic status perhaps due to the fact they might not have enough learning materials to be used and maybe they might have not given a chance to go to school. According to figures from the Department of Education and the National Statistical Coordination Board in the Philippines, 1 in 6 Filipino kids will not attend school. Further, only 7 out of 10 kids will complete elementary school. Of those 7 kids, only 4 will complete high school, and of those 4, 1 will proceed onto university. The main reason for this is identified— Poverty. The given scenario does not only focus on the deprivation mathematical learning but also for education as a whole, This relationship between SES and academic achievement shows that financial state is a big factor that affects not only on understanding mathematics but also on education itself.

Furthermore, education at the present time needs for more inclusive approach to give not only on particular individuals but also for the growth of culturally relevant education for Indigenous people worldwide. Education tends to be selective at times, it focusses only on the mainstream learners and disregarding those learners from different cultures. This is the reason why indigenous people receive a different level of comprehension on mathematics because of cultural differences. As a result, the government and the education department should develop culturally relevant mathematics education to help indigenous students succeed in mathematics. In the Philippines, we cannot evade the fact that these group of people are sometimes discriminated especially on field on learning since the lesson are commonly centered on conventional approaches that is why The Department of Education is pursuing the localization and contextualization of the basic education curriculum as part of its K-12 program. The implementation of IP education follows this approach. In particular, mathematics teachers are expected to develop mathematics lessons that are localized and contextualized to indigenous culture in order to make mathematics accessible to students.

MATHEMATICS EDUCATION

The practice of teaching and learning mathematics also impacts the reception of this discipline. Mathematics education is a field that problematizes how mathematics is taught effectively, and how mathematics learning takes place. The problem on the way of teaching of mathematics also makes a problem on mathematics education. Educators on the Philippines are capable in providing a quality education but still they still need to undergo further studies to be able to ride on the global education trend especially on sciences and mathematics. According to the article of Business Mirror, teachers have roles to play in the advancement of their students. It’s important for teachers to understand that math disabilities can arise at any stage of a child’s educational development. Many factors may stand in the way of a student’s mastery of the subject, even when her classmates claim that math is fun.

 With the implementation of the K to 12 program, math teachers have opportunities to further their own content knowledge for teaching. Opportunities for ongoing professional development that connect research in education to implementation in the classroom are now within their reach. Big universities in Metro Manila are currently offering advanced courses for teachers who want to further their professional development, math teachers included.

These issues block our understanding on the subject matter that is why it must be addressed as soon as possible. Mathematics would be accessible and inclusive if all these concerns will be taken into account. These discipline should never be disregarded since it helps everyone to develop critical skills that would also help solve problems around us.

Photo Credits

www.worldvision.org%2Fblog%2Fchilds-daily-journey-school-philippines&psig=AOvVaw2frXeyJMM3XVbR9YAqQVu9&ust=1576035787681743

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IN DEPTH: Voting Systems Problems and Fairness Criteria

by: Alvin M. Cariño

Problems with Different Voting Systems

Every one of us is given a chance to exercise their right to choose. Given that when choosing we are always handed more than one option, we are tasked not only to simply pick one of those options but also sort them out according to our preferences in order to derive to our final and best choice. However, coming up to our final pick is not that easy, we need to undergo a certain process that is why we have voting methods in which gives a systematic ways to bring out our votes.

There are various kinds of voting system that are used by different countries namely: Plurality Method, Borda Count, Pairwise Comparison, and Plurality with Elimination Method. These methods has different procedures but all of them has one purpose, to accumulate everyone’s choices. Though, these voting systems help the process of election to be more efficient, each one of them still has identified potential problems that would question its overall reliability.

In the case of Plurality Method which the commonly used voting system which suggests that the candidate with most first-place votes should win, it has presented points that makes it disadvantageous for the public. This method focuses solely on the first choice of an individual and this property makes this Plurality method problematic since it does not take into account the preferences other than first. It is also susceptible to insincere or strategic voting for the fact it neglects the other preference of the voter. As one of the problems of Plurality Method stated by Aaron Hamlin (2015), this is inexpressive. Plurality voting is among the least expressive voting methods there is. A plurality ballot puts a slate of candidates in front of us and forces us to choose only one. No more. It rejects the fact that a person might have his/her second choice and this would change the game. We likely have opinions about all those candidates. And yet, we only get a say about one. Different voting methods allow you to express yourself in all kinds of ways such as choosing as many as you want, ranking, and scoring. But plurality lets you do none of that.

For instance, suppose in an area with a population of 50, 20 citizens voted candidate A as their first choice; 18 voted for candidate B; the 12 voted for candidate C. In the rules of plurality, candidate A wins but it discards the possibility that in 50 people, 20 of might chose as their first choice but there is a chance that the remaining 30 people voted for the other candidate might place candidate A on their last choice. As a result, there is a contradicting point of view in the public that is why in this kind of voting system public satisfaction is hard to attain.

The ranking system shown on the example is the other kind of voting method which is called Borda Count. This is a method in which places on a ballot are assigned points (perhaps in an election with five (5) candidates, first choice will have five point, second will have 4, and so on). Generally, the candidate with the highest accumulated points wins the election. Borda Count unlike Plurality acknowledges the other preference of people aside from their first choice and it is believed by many as a solution for every country to have a best leader that everyone is looking for.

 However, even if it is consider as the most appropriate method to satisfy everyone, it still hides a possible problem. This rank-based voting system produces a winner that is a compromise candidate (that may or may not be a bad thing). Aside from that, maybe one of the reason why many country do not practice this method according to Breana Noble (2015), is the fact that it’s confusing and many cities may do not have the proper equipment to count the ballots. The elections for multiple positions will become more complex as well. While it is possible to perform ranked-choice voting elections when more than one position is up for grabs, it involves setting a threshold for candidates to obtain, complicating the process.

Another kind of voting method is called Pairwise Comparison where the candidate winning the most "matchups" (pairwise comparisons) wins the election. This method can also be considered as head-to-head fight (e.g. Candidate A vs B; A vs. C; A vs. D; B vs C; B vs. D; and C vs. D). The main problem of this voting method is the possibility that it may produce an election in which everyone wins. (Provide an Example)

In relation to Plurality method, there is also a voting method with almost the same principle in which it eliminates the least favorite candidate, based on the number of first-place votes, until there is a winner— this is called Plurality with Elimination. The same with Plurality, the problem of this method is its susceptibility to insincere or strategic voting. This strategic voting occurs in elections with more than two candidates, when a voter supports another candidate more strongly than their sincere preference in order to prevent an undesirable outcome. For example, in a simple plurality with elimination election, a voter might gain a "better" outcome by voting for a less preferred but more generally popular candidate.

Though, these voting methods has their own disadvantages, still the important means to be used is the method that could possibly satisfy the public. Moreover, aside from these voting we also have this Fairness Criteria that would give us an idea on what are the voting methods that satisfies and violates certain principle of fairness.

Fairness Principles

        There are different principles for fairness, these are Majority Criterion, Condorcet Criterion, Monotonicity Criterion, Independence of Irrelevant Alternative Criterion, Unanimity, and Non dictatorship.

           Majority Criterion. This principle states that if candidate A has a majority of the first-place votes, then candidate A should be the winner of the election. This criterion is satisfied by the Plurality Method, the Plurality with Elimination Method, and Pairwise Comparison Method since these voting method relies on first-place vote to identify the winner. On the Other hand, the Borda Count Method does not satisfy the majority criterion. This means that the Borda Count Method does not always select the candidate with the majority of first place rankings.

           Condorcet Criterion. This principle suggests that if candidate A is preferred by the voters over each of the other candidates in a head-to-head comparison, then candidate A should be the winner of the election. It is always satisfied by the Method of Pairwise Comparison since this is a voting system in which undergoes with head-to-head fight to derive the winner. The Borda Count Method, the Plurality with Elimination Method, and the Plurality Method might select a Condorcet candidate by comparing two or more candidates, but they can also fail to honor the criterion.

           Monotonicity Criterion. This states that if candidate A is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor A (and only A), then A should still be the winner. The Plurality Method satisfies the monotonicity criterion. Conversely, the Borda Count Method and the Pairwise Comparison Method can fail the monotonicity criterion because while a rearranging voter is required to move winning Candidate X favorably in his or her ranking, how that voter shifts the other candidates is not restricted. Also, the Plurality with Elimination Method can violate this criterion.

           Independence of Irrelevant Alternative Criterion. This principle says that if candidate X is a winner of an election and in a recount one of the non-winning candidates withdraws or is disqualified, then X should still be the winner. The Plurality Method, the Borda Count Method, the Pairwise Comparison Method, and the Plurality with Elimination Method can fail to satisfy the Irrelevant Alternative criterion.

           Unanimity.  This principle says that if everyone prefers A to B, the voting system should rank A above B. Plurality method is believed to satisfy this unanimity criterion. A ranking method also satisfies the unanimity criterion if it guarantees that A≻B if A is preferred to B by every voter.

           Non dictatorship.  This criterion suggests that the property of non-dictatorship is satisfied if there is no single voter i with the individual preference order P, such that P is the societal ("winning") preference order, unless all voters have the same P. Thus, as long as there are voters in the society that have different preference orderings, the preferences of individual i should not always prevail.

References

Hamlin, A. (2015, March 30). Top 5 Ways Plurality Voting Fails. Retrieved from Election Science: https://www.electionscience.org/voting-methods/spoiler-effect-top-5-ways-plurality-voting-fails/

Mathematics of Social Choice. (n.d.). Retrieved from Society for Industrial and Applied Mathematics: https://epubs.siam.org/doi/10.1137/1.9780898717624.ch12

Noble, B. (2015, July 5). Pros and Cons of Ranked-Choice Voting. Retrieved from Newsmax: https://www.newsmax.com/fastfeatures/ranked-choice-voting-pros-and-cons/2015/07/03/id/653400/

Simonson, M. (2016, November 21). Mathematical Democracy: Mission Impossible? Maybe not…. Retrieved from American Mathematical Society: https://blogs.ams.org/mathgradblog/2016/11/21/mathematical-democracy-mission-impossible-not/

What does the non-dictatorship principle of the Arrow theorem mean exactly? (n.d.). Retrieved from Politics Beta: https://politics.stackexchange.com/questions/32503/what-does-the-non-dictatorship-principle-of-the-arrow-theorem-mean-exactly

Photo Credits

https://www.google.com/url?sa=i&source=images&cd=&ved=2ahUKEwjJ1tbXk6rmAhXC63MBHdmvDtsQjRx6BAgBEAQ&url=https%3A%2F%2Fwww.cityandstateny.com%2Farticles%2Fpolitics%2Fnew-york-city%2Fnew-york-city-charter-revision-commission-voters-guide.html&psig=AOvVaw1wH9nVxjeOfXBNQoX9jCCx&ust=1576035424352476

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Mathematics in Medicine, Life Science, and Operations Research

by: Alec Gabrielle Gonzales and Andrea Trisha Buenaobra

It is clearly noticeable that mathematics is present around us - it is everywhere and in everything! It is not just present, but we are applying it in real life scenarios. Mathematics played and is continually a major contributor in discoveries and inventions. To make it more simple, science and mathematics are interconnected.

         Math is significantly helpful in the field of Medicine and Life Science in terms of the complex problems and complicated circumstances. Actual problems can be formed into mathematical forms for more understandable process in acquiring its solutions. It can also be used to identify population change that is contributing to conservation biologists.

         Operations Research on the other hand is a strategy that can help make better choices in decision making circumstances. It elaborates procedures and narrow down possible steps for more thorough solving of the problem.

          I hope that we can all see that Mathematics is vital in addressing health-related concerns, explaining biological phenomena, and informing decision-makers about using an optimal choice.

Photo Credits

https://www.wklaw.com/practice-areas/unauthorized-practice-medicine-california-penal-code-2052/

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Look and See – Math is Everywhere

by: Alec Gabrielle Gonzales and Andrea Trisha Buenaobra

Mathematics is present in everything in this world, and when I say everything there is no exemption. Even in breathing and sleeping, Mathematics is there. We might feel nauseous and problematic upon seeing mathematical formulas and equations, and deny the fact that math is in there.

Before I can fully convince you that Mathematics is present everywhere, let us define it first. According to Merriam-Webster Dictionary, Mathematics is the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations; it is way beyond its branches namely algebra, arithmetic, calculus, geometry, and trigonometry. Basing in its definition, its scope is very vague and can touch all that exists.

Since it is visible and anchored in every field, it is very recognizable in the discipline of Humanities and Art. It has certain mathematical elements, concepts and equations present in each artwork. We may be second guessing in assessing this knowledge, but we must open our minds to the reality that Mathematics has a lot of contribution in all endeavors.

Let us point out how mathematics affects the music industry. Long ago, when classical music dominated the industry, it used much more complex notes and measurements compared to the pop culture we have today.

Fractions are used in music to indicate lengths of notes, and it also signifies the beat of a certain piece. In a musical piece, the time signature tells the musician information about the rhythm of the piece. There are also well-known mathematical relationships between the pitch of various notes on the musical keyboard. An octave is separated by a factor of two; a fifth interval (say C to G) by the ratio 3/2, and two adjacent notes on the keyboard are separated by the twelfth root of two = 1.059463.

A lot of great musicians are mathematicians. Albert Einstein is the greatest example; he can play violin and piano. According to his second wife, Elsa, he may be in a deep concentration at first, then playing the instrument later. The great Baroque composer, Johann Sebastian Bach whose specialty was canon, can be analyzed with the use of mobius strip. His compositions can be analyzed and can sound the same when played reversed or as the original. Bach’s work, Frère Jacques and Canon 3, have the same structure topologically: after the introductory measures, the canon settles into a cylindrical steady state. All of Bach's canons are organized this way.  

Mobius strip is a spiritually significant symbol of balance and union. It is a 2-dimensional surface; has length and width but no thickness. It doesn’t only play a significant part in analyzing music, it is also present in the arts. In real life, it is bet portrayed in conveyor belts and bicycle chains, and it will last longer since all the surface areas get the same amount of rasp.

Math and dancing are also intricately linked. Sequence and power of set (or cardinal number) is observed in this activity to create a choreography. Sequence is an ordered list of objects, whereas the same elements can appear multiple times at different positions in the sequence; power of set (or cardinal number) is a finite set is a natural number which is the number of elements in the set. These elements are present in dancing since choreographers use sequential and repetitive steps to create a whole dance routine.

Geometry – with its shapes, patterns, angles and symmetry – is perhaps the most apparent field of mathematics present in dance. Looking at a solo dancer frozen in one position, we can see the lines of the body, their angles and directions in relation to each other and to the room. In a moving group of dancers, we notice the lines and shapes created by the ensemble, their change with the music, and the patterns of beats that cause those changes.

Mathematical concepts can be used to understand dance at a more profound level and to create better choreographies, and at the same time dance analogies can make math lessons more vivid and accessible for the students.

The connections of mathematics and visual arts are so many. For example, angles and perspective can be used to create dimensions inside the artwork; points, lines and shapes are used to produce an appealing artwork; and measurements are also applied to achieve the accurate size desired by the artist. It may not always be visible when we’re not looking closer and not open for the possibilities, but there is much symmetry, geometry, and measurement involved in creating beautiful art.

Its prominent connection started when the sculptor Plykleitos wrote his Canon, prescribing proportions based on the ratio 1:Ö2 for the ideal male nude back in the 4th BC. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. This ratio is approximately 1.618 and can be achieved when you divide a line so that the longer part divided by the smaller part is equal to the actual whole line divided by the longer part. It is also represented in represented by the Greek letter phi.

Artists also apply different techniques in doing their artworks; these techniques serve as their “signature” or as a trademark. Mathematics is still present in the picture. For example, the famous Pablo Picasso pioneered the cubism, that is characterized by the use of geometric planes and shapes.

A perfect example is the well-known work of Leonardo da Vinci, Mona Lisa. When looked closely, it has numbers put inside the painting itself. The golden ratio (1.618) is also noticeable when we measure the appendages of the subject. It is considered as a perfect artwork.

Another example is tessellations, the work of the Dutch graphic artist M. C. Escher and mathematician Sir Roger Penrose who brought attention to the concept. A tessellation is a pattern of shapes that fit together perfectly and repeat in patterns. It may not seem like it has anything to do with math, but it is all about math as they can be tiled to form a continuous pattern on an infinite plane.

Personally, I also think that the price of an artwork is another participation of Mathematics in visual arts. Since, money involves numbers and currency.

To sum up everything, Mathematics is also a study of patterns, and we can study everything in arts and humanities from different mathematical perspectives, including geometry, number theory, trigonometry, differential calculus, and signal processing. As we can also observe, arts in all forms observe series of patterns for it to come up with an aesthetically pleasing output.

Mathematics is evidently related to Architecture. The concept can even be considered null without it. Architecture is simply a whole other world of Math. This connection simply plays with numbers, measurements, shapes, proportions, perspectives, and everything in between. Architects use geometry to create outputs that are according to different principles such as aesthetic and mathematical.

Let’s go down to history lane. In ancient Egypt, Greece, India, and the Islamic world, the different buildings during that time, such as pyramids, tenokes, and palaces were made with specified proportions due to religious beliefs.In regards to architecture in the Islamic word, Math is used in geometric tiling patterns and shapes for the internal and external structure.

On the other hand, Renaissance architecture used symmetry and proportion as emphasized by known architects namely Andrea Palladio, Leon Battista Alberti, and Sebastian Serlio. They were influenced by De architectura of Vitruvius from ancient Rome. In addition, they were also influenced by Pythagoreans’ arithmetic.

Mathematics in weaving, or fiber arts is deeply inspired by a variety of mathematical concepts. Such creation includes quilts, knitting, cross-stitching, crochet, embroidery, and weaving. The mathematical concepts covered in this relation includes graph theory, number theory, topology, and algebra.

While this connection includes numerous techniques, some are naturally geometrical like the counted-thread embroidery. They also exemplify mathematical concepts through physical expression or visual utterance.

Media Arts and Mathematics go well with each other. Especially using media as an academe for the subject. The collaboration is very useful as it innovates and develops method of learning and teaching. Math can seem easier immersed with Media Arts as various means can be used. Aside from the helpful collaboration of both concepts in learning, you can also see Math in different forms of arts in media. For example, movies. The duration, special effects, and animations also involves Mathematics. It may not be salient in expression but it is evident especially in the process.

The combination of Math and Literature is significantly evident as it also gives us the reason why Math is important. Aside from that, there are also several reasons as to why they work well combined. Literature introduces the subject in a lighter approach and it also has specifications that triggers people’s interest with matter.

In addition, literature makes Math more interactive and less boring and traditional. This is very helpful especially with complicated problems and struggling audience. Examples would be books, they are written in a more understandable way in order to give a better learning process for the people especially complicated equations and problems.

References:

Bailey, D & Borwein, J. (2017, December 6). Why Are So Many Mathematicians Also Musicians? Retrieved in https://www.huffpost.com/entry/why-are-so-many-mathemati_b_9814796 on December 2019.

Phillips, T. (2016, November 25). Bach and the Musical Möbius Strip. Retrieved in https://plus.maths.org/content/topology-music-m-bius-strip on December 2019.

Wasilewska, K. (2012). Mathematics in the world of dance. Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture, 453-456.

https://en.wikipedia.org/wiki/Mathematics_and_architecture

https://ourjourneywestward.com/math-and-literature-connection/

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Evolution of Numbers: The Glow-up We Need

by: Alec Gabrielle Gonzales

Can you imagine yourself using sticks and symbols to count and identify numbers? Gladly, our numeral system undergone the glow-up we need for us to count in our own preference. The numbers didn’t look and function as what we use today.

         Even though our numeral system undergone the glow-up, there are some that looks and functions as it is until today. For example, the tally mark system is still used up to this day. In the academe, before we are exposed in adding complex numbers, we are taught to use “sticks” as a basis of adding and multiplying numbers. We just simply draw the number of lines asked, count it, and we are already good to go. In sports, tally mark system is also exercised to count the scores of two opposing teams. Long ago, it was used to count the flock of sheep and other animals.

          As life has become more complicated, civilizations need a better method of counting to use it sufficiently and to make their life comfortable. They create ways to count bigger and higher amount of numbers. The numeral systems created by Greek, Hebrew and Egyptians are only an extension of tally mark system; new symbols are just added to represent different values. This method of counting is called non-positional number system, which means that symbols could be written randomly (in no particular order) and it still portrays the same value.

          . If the systems invented by the different early civilizations used repeated symbols to count numbers and can be placed randomly to attain the number that is to achieve, the late civilizations used positional number system that reuses symbols and it assigns value based on their position in the sequence. It means one in which a digit’s value depends on its placement occurs in the representation. It is observed by the Babylonians, Chinese and Aztecs. The positioning of the symbols in this system indicates the different power of tens, starting from the right and increasing as we move to the left. It is much easier to use this number system because a certain symbol is not needed to be repeated, but the placement dictates its value. In fact, our decimal system is a positional system.

The Roman numeral system used letters to represent numbers and it also has different method in counting numbers; instead of just adding, we can also subtract by putting the smaller value symbol before the higher number. Interestingly, it is a combination of positional and nonpositional number system. The values of each letter are constant; hence, it will change based on its placement.

According to most scholars, the numeral system used today emerged in the North Maghreb Region of the Arab Empire. It was the base-ten system – only using 10 unique glyphs. It was then spread by the Indians and Arab merchants. This gave birth to the Hindu-Arabic numeral system that we’ve been using up to this day that was discovered in the 15th century. Now, it is the most common and widely used number system in the world.

Just a fun fact: did you know that the number 0 (zero) was invented late by the Mayans in ancient Mesopotamia? Zero is absolutely important to measure nothingness. It gave way to the discovery of hundreds, thousands and millions. The invention of zero immensely simplified computations, freeing mathematicians to develop vital mathematical disciplines such as algebra and calculus, and eventually the basis for computers.

          We commonly use base-ten or decimal system because of the major anthropogenic reasons – we only have 10 fingers. I usually use this most of the time, whenever. It is also the reason why the Aztecs and the Mayans use base-twenty or vigesimal system. Even though we are widely exposed in using base-10, there are different base systems used in counting. For instance, the Babylonians were sexagesimal or they used base-60. There is also a base called duodecimal system which is based-12. 60 and 12 are highly composite numbers that could be divisible by 2, 3, 4 and 6 – which is common and better in representing fractions. It is used in packaging eggs since 12 pieces of egg in a dozen. I use duodecimal in reading and telling the time. There is also what we call binary numbers that is base-2 [could also be base-8 and -16], mostly used in programming. No matter what, we should acknowledge these base systems since we use these up to these days consciously and unconsciously.

          For me, it is much better to use decimal system than duodecimal. I can’t deny the fact that both are functional and useful in their own special ways. Other than we are already used in exercising the decimal number system, it is also much easier to count it since it is a whole number ending in zero. We also only have 10 fingers that we can use in counting effortlessly with the decimal number system.

“With just these ten symbols, we can write any rational numbers imaginable,” said Alessandra King. The sky's the limit in producing numbers of any value. We must thank all the civilizations that invented these numeral systems. We must not take these numbers for granted since it has undergone a lot of processes to make our life better.

References:

https://www.quora.com/Why-do-humans-count-in-a-ten-base-system-Have-humans-always-counted-in-a-ten-base-system-Are-there-reasons-other-than-anthropologic-our-10-fingers-why-we-use-a-ten-base-system-How-different-would-the-world-be-if-we-used-a-base-like-8-or-12

·         TED-Ed. (2017, January 19). A brief history of numerical systems - Alessandra King. Retrieved from Youtube: https://www.youtube.com/watch?v=cZH0YnFpjwU on December 2019.

[mathwizards10]

Axiomatic Game: Crazy Words

by: Alvin M. Cariño

Crazy Words (or CRODS) is a word game that is inspired by a children play in early 2000s. This game will test the vocabulary of the players and their capability in finding a distinctive term that would associate on a certain category. It is essential in this game to be “Crazy on Words” which means the players must think of a word as fast as much as possible since it is time-bounded. The special about CRODS is that it can be played in various contexts; as a matter of fact you can use English, Filipino, or Vernacular as a medium in this game. It will not require you to move from one place to another but it will make your brain go round for sure. Invite your friends to Play, Learn and Have Fun with the Crazy Words.

PRE-GAME GUIDELINES

●      This game should be played with a minimum of two (2) players and a maximum of eight (8).

●      The players should prepare a Letter Bucket. It can be a fish bowl, a small empty box, or anything that can serve as (small) container.

●      Inside the Letter Bucket are letters from A to Z that is written in small pieces of paper and then rolled afterwards. Thus, there should be a total 26 rolled papers that corresponds each letter.

●      Every Player must also have a Crazy Sheet. It can be printed or can be written by the player him/herself; any kind of paper will do.

●      In the paper (Crazy Sheet) , the player should make five (6) columns (Each column has a relating classification; written on the first column is “Picked Letter”;  “NAME” in the second; then “PLACE”, “ANIMAL”, and “THING” on the following columns. On the sixth column, the player must write “Total Score”.

●      Further, It should also be divided into 28 rows (Written on the first row are the classifications; the next 26 rows are allotted to be the player’s answer space; and on the last row in the sixth column is where the final score will be written).

●      Before having the game proper, the player must decide unanimously on what medium will they be using (English, Filipino, or Vernacular).

 GAME PROPER GUIDELINES

         After preparing the needed materials and deciding the medium that will be used, here are the rules or guidelines during the game proper:

The players will decide     on who will be the first one to get a rolled paper in the Letter Bucket     (it can be voluntary or systemic). NOTE: Everyone will be given a chance     to pick a letter.

Once the first player     picked a letter, everyone will be given 10 seconds to fill in each column     (expect the sixth one) with what the column is asking. The player should     write the letter on the “Picked Letter” column; and write on the column     NAME, PLACE, ANIMALS, OR THINGS the word that are asked that start with     the picked letter. NOTE: One word only per category.

After 10 seconds, each     player will check each answer (the proper scoring and checking will be     found in a separate section named SCORING GUIDELINES).

Put the accumulated     score (each row) on the last column.

Right after checking,     the picked letter should be set aside and the next player will picked     another one. Then repeat on what the players did in the 2nd and     3rd rules. THEN JUST DO AGAIN RULES 4, 2, 3 until the players     finished getting all letters.

 SCORING GUIDELINES

●      Every column (except the sixth) has a corresponding points. The first column has 5 points; however, if you fail to write or you wrote a wrong letter, automatically you will not have a score.

●      The next four (4) columns has 10 points but it may vary because of various circumstances:

a.)  The player will only receive five (5) points if he/she has written the same word with the other player.

b.)  If the player misspelled the word he/she will receive no points. (Note it is only applicable in column PLACE, ANIMAL and THING)

c.)   If the player wrote a word outside the chosen medium, he/she will receive 3 points. (Note it is only applicable in column ANIMAL and THING)

d.)  If the player fails to write a word, automatically he/she has no point.

●      If the player is still writing after the 10 seconds limit, he/she will automatically receive no point for that certain round.

         The player who has the highest total score will be the Crazy Word Victor.

Photo Credits

https://www.kidsdiscover.com/teacherresources/6-writing-prompts-to-jumpstart-your-science-class/

[mathwizards10]

Math Statements and Connectives

by: Andrea Trisha Buenaobra

Mathematics is a big world of different components. It can be seen in almost everything and everywhere. One of which is in Logic or also known as Propositional Logic. What is logic? It is the study of the principles of correct reasoning. It is the process of coming up to a certain conclusion or idea through different contexts. It usually bases on statements and different propositions. Mathematical Logic is the application of mathematics to formal logic. It is concerned with the foundations of mathematics and metamathematics which is the study field of structures and other formal properties of the subject.

Mathematical statements can be considered as an element of Mathematical Logic. Simple statements are logical statements that entails a certain piece of information. They are the foundation blocks of logical arguments. Declarative assertions that can either be true or false, but never both. An example would be “The sum of 5 and 4 is 10.” That is an example of a mathematical statement because there is a definite answer, however it’s false because the real sum of 5 and 4 is 9.

Compound statements on the other hand don’t only contain a single statement or information. They are formed by combining simple statements together. These are operations on statements. Forming a new statement from other statements. They are constructed and connected with each other through the use of logical connectors. There are specific connectors for each type of compound statement.

The five basic connectives of compound statements are Conjunction, Disjunction, Conditional, Biconditional, and Disjunction.

Conjunction statements are simple statements connected using AND. Both statements are supposed to be true in order for the truth value to be true, otherwise false.

Ex: All dogs are brown AND dogs are direct descendants of wolves.

The truth value of the statement above is false. It is because the first statement is not a fact. Not all dogs are brown. However, the second statement is true that dogs really are direct descendants of wolves. Unfortunately, the truth value for conjunction can only be true if both statements are facts. Therefore, it is false.

Disjunction statements are combination of statements through the connector OR. The truth value can be identified true if at least one of the statements is true. If both are false, then the truth value is false.

Ex: Cats are carnivores OR dogs are omnivores.

The truth value is TRUE because indeed cats and dogs are carnivores and omnivores respectively. The rule for the statement to be true in disjunction is that at least one of the two is true. Regardless of one statement being false, as long as the other is a fact then the truth value will be true.

Conditional statements are statements in the form of “If p, then q”. The truth value can only be declared false if P is true and Q is false, otherwise it is true.

Ex: If this year is 2019, then next year is 2020.

The truth value of the conditional statement is TRUE because both p and q are true.

The Biconditional statements are statements in the form of “p if and only if q”. The truth value can be considered true if both statements have the same truth value. Regardless if it’s true or false, as long as they are the same then it will be true.

Ex: Pets are small if and only if they are dogs.

           The truth value of the statement is false. It is because dogs vary in size and there are also other small pets that are not dogs, like kittens.

Negation is the NOT of a simple statement. The truth value can be identified as true if p, or the statement itself, is false, and vice versa.

Ex: Dog pounds are not for dogs.

The truth value of the statement is true if it is presented as a negation statement. It is because if you negate the simple statement, it explicates that dog pounds are really for dogs.

The possible operations on statements was discussed above, which concluded outcomes’ truth values. However, they don’t necessarily make sense or relate with each other. We have two identities to know the relation of statements. They are implication and equivalence.

Implication is the incrimination of connection. The involvement of something in a certain environment. An example would be dogs’ behavior. They are known to be polygamous. This implies that if a pet or an animal is a dog, then it is polygamous. This can be stated as “If the animal is a dog, then it is polygamous.” It uses “→“ as its connective. This relation implicates that if something is a dog, then it is automatically true that it is polygamous. P as the statement that the animal is a dog, implies Q as it being polygamous.

Equivalence rather are equivalent and definition statements. They are usually compound statements that can be restated. An example would be “A person can be pregnant if and only if it is a woman”. If we separate the context of the whole statement, it would be p as “A person can be pregnant.” and q as “Only women get pregnant.”

Photo Credits

www.toppr.com%2Fguides%2Fmaths%2Fmathematical-reasoning%2Falgebra-of-statements%2F&psig=AOvVaw1m_n7dMI2JKHH4DMzIIZG5&ust=1576032812445169

[mathwizards10]

Number Power

by: Alvin M. Cariño

Many people often perceive numbers as the antagonist of every story. They usually see it as a fatal chemical that wants to disturb everyone’s minds to bring them into madness. Numbers are also remarked as the enemies of every student since it is a threat of for them in attaining high grades. Thus, this is considered as the main stressor of the general public. However, if we try to change the way we look at these things, we might able to see its lighter side.

 Numbers are not always cruel. The reason why people hate number is the fact that they are already occupied with their presumptions. If people just try to appreciate its value, no one will be having a hard time in understanding it. Numbers is our companion in our everyday lives. We meet them everyday; we use them in our daily basis. They exist not to make our lives complicated but to make it even easier. Therefore, numbers are everywhere, we just need to find its beauty.

Everyone might think that numbers are just plainly numbers but it is not. Numbers hide a fascinating aspect that might change our perception on them. Numbers are categorized according to its characteristics. There are five main kinds of numbers identified by Alex Neill (2016) namely: Natural Numbers, Whole Numbers, Integers, Rational Numbers, and Real Numbers.

●      Natural Numbers. It is also called positive integers, counting numbers, or natural numbers. They are the numbers where students generally start with (these are the numbers: 1, 2, 3, 4, 5 …).

●      Whole Numbers. This is the set of natural numbers where 0 is also introduced, i.e., (0, 1, 2, 3, 4, 5 …).

●      Integers. This is the set of all whole numbers plus all the negatives (or opposites) of the natural numbers, i.e., (…, ⁻2, ⁻1, 0, 1, 2 …).

●      Rational numbers. This is all the fractions where the top and bottom numbers are integers; e.g., 1/2, 3/4, 7/2, ⁻4/3, 4/1 [Note: The denominator cannot be 0, but the numerator can be].

●      Real numbers. This also called measuring numbers or measurement numbers. This includes all numbers that can be written as a decimal. This includes fractions written in decimal form e.g., 0.5, 0.75 2.35, ⁻0.073, 0.3333, or 2.142857. It also includes all the irrational numbers such as π, √2 etc. Every real number corresponds to a point on the number line.

These kinds of numbers can be divided into two which are called discrete and continuous numbers. Natural Numbers, Whole Numbers, Integers, and Rational numbers fall under discrete which means that they are separate and distinct entities. In fact each of these sets is countable unlike the Real Numbers that are continuous that means that between two real numbers, however close they may be, there are infinitely more real numbers. By looking at this, we can say that numbers are not simply 1, 2, and 3. Numbers can be an interesting venue if we just know how to switch our point of view in its opposite angle.

The existence of numbers can be considered as the greatest discovery/invention of mankind since this provides a way for us to share thoughts and ideas beyond words. Numbers became the new language of different fields like business, social science, arts and humanities. It provided a way to uncover new concepts and to solve societal concerns. Also, through numbers, everything can be possibly explained undoubtedly by numerical representation.

Without these numbers, there will be no change, solutions, development, and there will be no new things that will be discovered. Communication will be harder and many things will not exist like technology. Thus, we will be stocked in the small box forever and we are not be able to experience these things that we are experiencing right now.

Numbers might often perceive as the big bad but if everyone just learn to appreciate it, we will find out how fascinating and powerful it is. It might get tricky as time goes by, we just need to calm down; they are not our enemy at all, they are our companion to survive every single day. Numbers are not just numbers— everything revolves around them.

Photo Credits

www.teleradioamerica.com%2F2017%2F05%2Fsabias-que-contar-con-los-dedos-te-hace-mas- inteligente%2F&psig=AOvVaw3Lw0uFaUZ78Og8vHO6K5mU&ust=1576032153151897

[mathwizards10]

How Could We Not Appreciate the Beauty of Mathematics?

by: Alec Gabrielle Gonzales

“Beauty is in the eye of the beholder” – it may be cliché, but it is highly applicable in the reality of the society. As humans, we have different preferences.

It is clearly undeniable that we all have our own definition of beauty. For instance, something aesthetically pleasing for me is not lovely for someone. If somebody will say, Mathematics is an art, it is indeed, but we all have our own interpretation and level of understanding.

The playing of shapes, repetitive patterns, and contrasting choice of color schemes in artworks can undeniably trigger our brain the sense of beauty that leads us to appreciate things. Mathematics is even present in music, dance, photography, visual arts and everything that involves art. Artworks are exhibited to be admired by audiences and to recognize the artists.

May I ask, how do you feel upon seeing complex mathematical equations? Is it the same when appreciating works of art?

Believe me or not [because I can’t even believe it myself], there was a research conducted in the journal Frontiers in Human Neuroscience whereas 60 formulas were rated by 15 mathematicians.

This study found out that the more beautiful they rated the formula, the greater the surge in activity detected during the fMRI (functional magnetic resonance imaging) scans. Professor Semir Zeki, one of the researchers said that a large number of areas of the brain are involved when viewing equations, but when one looks at a formula rated as beautiful it activates the emotional brain - the medial orbito-frontal cortex - like looking at a great painting or listening to a piece of music.

Even I really can’t believe that there’s a new study that says appreciating mathematics is like adoring art.

Personally, I really despise mathematics, because I already have the presumption that it is really complicated to the highest level. Why is it always present in everything? I am already traumatized upon seeing large sum of numbers, especially when letters and fractions will already enter the picture. I know that I am not the only one who turns hysterical upon seeing such complex formulas.

Shopping is one of the activities I love the most. I highly find beauty and enjoyment in buying new things. Is math present in it? Yes, of course! As someone who doesn’t have a lot of money, I need to budget my money properly, and to categorize what I need from what I want. Adding all the sum of all my shopped items is Mathematics, obviously addition. Even though I really don’t like solving, but whenever an item says “20% off” I am eager to find the new discounted price, just to make sure that I can save a lot of money. I also need to master subtraction to check if the change is right.

Since we are living in a new era, online shopping is a new trend, replacing the traditional one. I also assess critically whether it is much more practical to pay the shipping fee or to add some items to lessen the over-all total of my shopped items. I even started selling things; math is obviously present because I need to calculate the capital and the interest for me to make sure that I will garner enough money to support my funds.

Also, when shopping, I need to count the quantity of my shopped items; and compare its quality to other brands. When shopping for clothes, I double check that the measurements fit me well.

For me to escape my problems and as a better alternative than breaking down, I sleep. Undeniably, math is still present in this activity. I bet I’m not the only one who counts the number of hours I slept to assure that I have enough rest. Amazingly, there is Mathematical professor named Mark Holmes that used Mathematics to develop a new computer model that can be easily manipulated by other scientists and doctors to predict how different environmental, medical, or physical changes to a person’s body will affect their sleep. Their model will also provide clues to the most basic dynamics of the sleep-wake cycle.

We can’t deny the fact that the power of Mathematics is present in everything. The patterns of leaves, the paws of a dog, the meter in a musical sheet, the time of the day, even in the activities we enjoy and love… name it all - everything!

“Neuroscience can’t tell you what beauty is, but if you find it beautiful the medial orbito-frontal cortex is likely to be involved, you can find beauty in anything,” said Professor Semir Zeki as a support to their research. Beauty may be in the eye of the beholder indeed, for we can control what we find beautiful.

We just need to have a shift of mindset due to the bad implication math related lesson had already inculcated in our minds.

We need to widen our imagination and think freely of the wonders Mathematics can influence our lives.

Photo Credits

https://image.businessinsider.com/5277c97d6bb3f79a71022cdd?width=1100&format=jpeg&auto=webp

[mathwizards10]

Math in Different Angles

by: Andrea Trisha Buenaobra

Mathematics. Just the thought of it gives me a headache already. What is math really all about? Do we really need Math? Do we like it? Do we hate it?

Mathematics is a very broad subject. It is different in so many ways but it still somehow finds a way to connect with each other.

I personally loved Math in my early years in school. The subject itself challenges me which triggers my excitement and persistence. I loooooved Math. Especially when it was just numbers. The times when the alphabet was still unknown to the subject. Basic Math was my asylum in the world of numbers. The satisfaction I get after solving equations knowing that I easily aced it, what a great memory.

However, Algebra happened. I liked it at first. Actually it was a pretty cool lesson to tackle. The increase in the level of difficulty from just adding and subtracting challenged me and gave me more reasons to analyze and get my brain to work with me. Life was okay with Algebra, but then Calculus entered my life. And there it was, the end of me. The complication was just too much for my soft and weak persistence. I don’t even have any words to express how much I dislike the subject anymore.

Overall, Math is significant. Math is everywhere. Despite the problems we encounter with learning it, I can’t imagine life without Math. But then again, I like the challenge, but I prefer the comfort. If I were to choose a specific lesson in the world of Math, you already know what it is.

While we’re on the subject, I actually surveyed some of my relatives and friends on what they like and dislike about math. Let’s take a look at the table below to see what I have gathered.

LIKED

Algebra – 2

Fibonacci Series – 2

Business Math –1

DISLIKED

Calculus – 5

(No. of Respondents: 5)

General Perception and Attitude towards Math

For starters, it is evident how Algebra and Fibonacci Series were the most liked areas in mathematics. As I asked them the reason for it, they told me that both were fun and easy to understand. They also stated that they were uncomplicated because the basic requirement was just common sense which is a very simple demand. Algebra was also chosen because it was identified as, if not the main, one of the basic fundamentals of Mathematics. In addition, Business Math was also mentioned. The person told me that he likes the difficulty it provides as it makes him more determined to find out the answer and he is quite fond of analyzing things. Personally, it would be surprising for me as it is one of the subjects I truly despise. But as we all know, we all have different skills and learning preferences.

Calculus turned out as the most disliked field. The reasons stated were that it was very complicated, a lot of principles to memorize were needed, and there were plenty of rules. Calculus is more than just what you see. It goes way back and each process should be precise and done properly. Failure to do so would get you in a more difficult stage which will make you do the process all over again.

Therefore, we all have different preferences, learning techniques, and interests. But overall, Math is fun and helpful. You just have to know what lesson fits you and try to use it as a start to see the subject in a different angle.

Photo Credits

www.thoughtco.com%2Fdefinition-of-binomial-2312369&psig=AOvVaw0W8ZFbpsJefv8P7iIjfGow&ust=1576029097907223