MATABOO

Mataboo is a game which is derived from taboo. It is played like taboo, but with mathematical terms.

When I was a mathematics teacher in Istanbul/Turkey, I developed this game for K-4 and K-5 students to play in the Living Mathematics Student Club. They enjoyed the game very much.

Playing Mataboo helps the students to improve their body-math language and verbal communication skills. It gives feedback about how they construct the mathematical terms and concepts. And since the game is played with groups, it improves their social interaction and communication skills.

How do we play Mataboo?
First you prepare a game board with a cardboard and 3 different colored papers (I used post-it). You build a road on the cardboard with these papers. Also with the same colors you prepare the game cards. Number of the colors is preferable in fact; you can use only one color to play classical taboo (telling without using the taboo words written on the card). I used 2 more colors: pink, for telling the words with puppet and green for telling 3 words at the same time by using only 15 words (using connotations).

You need at least 4 people (2 teams, each consisting of 2 persons) to play the game. But you can play with as many people as you want. You choose a starting point. Teams put their pawns on the first block of the path which they have to go on to finish the game (I made the pawns with cardboards but it can be a small stone, a small toy or whatever the teams choose to represent themselves). Teams, in turn, draw a card from the same color of the cards with the path they are on. And they try to get their teammates say the word within a preplanned time (we used 1 minute sandglass). Now let's look what will happen when teams are on the

Orange Road

Stepping on to an orange road, one of the team member draws a card from the orange cards. Then s/he tries to get her team find out the word written on the card. S/he may only use speech (but of course not allowed to say the taboo words) to prompt his or her teammates. For example describing the 'trapezoid', s/he cannot use the words polygon, quadrilateral or skew.

Pink Road

A pink card is drawn and told with using a puppet. Words like circle, ray, angle, diagonal can be written on these cards.

Green Road

Similarly a green card is drawn and the team member tries to describe the 3 words by using maximum of 15 words. Words like 'Meter-Origami-Denominator', 'Gram-Compass-Degree', 'Number-Cube-Addition', 'Second-Dividend-Area' can be written on green cards.

So what are you waiting for? Make your game and play now. I wish you a lot of fun!

PRIME NUMBERS

MODULUS

What time is it? What is the date today? The day is 24 hours, why did you say 2:25 pm or why did you say the month and day but probably not 14:25 or 734142^nd day after Christ?

Does time exist in the natural world or is it an invention of the human beings? For example, is the motion of the world around the sun determine time? Does the motion of the moon around the earth, or the world around itself? Actually, our calendar is organized upon the motion of the world around the sun and we name it as year, the world itself as day. And similarly there is a lunar calendar which was being used by Chinese and Arabians.

What are the devices used to measure the time? Sundial, sand glass, Quartz minerals, Cesium atom, else? All of the devices used are based on a motion in one way or another.

Hence, in a circular statement we can say that “we use time to measure motion and we use motion to measure time.”

But which motion are we interested in? Can we consider the motion of a comet for measuring time? What is common for the motions mentioned above? If I say these motions are ‘periodic’ or simply ‘circular*’, does it make sense? And in all of the above cases, we always calculate the measurement of the smallest period. (Say ‘year’ for one turn of the world around the sun)

That is where the modulus comes from: it means ‘small measure’ in Latin. Gauss used this term in 1801 to introduce modular arithmetic; mod meaning ‘according to modulus’ and arithmetic ‘the subbranch of mathematics, dealing with operations on numerals’.

That is why we say that today is January 18^th, 2008 in place of 734142^nd day. We use small measures like day, month, year, etc.

Number Circle instead of Number Line

So, we have a motion which is circular and we are interested in the smallest measure. For example, let’s consider the usual clock based on 12 hours. The modulus (small measure) is 12.

To show 13 in normal arithmetic we use number line:

To show 13 in modular arithmetic we use number circle such as:

That is why modular arithmetic is also referred as clock arithmetic.
Modular/clock arithmetic is used in many fields. Besides calendars and clocks it is also the arithmetic of the computers, it is used in music, cryptology, banks and in lots of other places for security reasons. And it will be kept using as long as the humankind loves both simplifying and complicating the things.

*The term ‘circular’ is used to indicate that the motion’s starting point and ending point is the same. According to this definition, the motion of an oscillating arc is considered also circular.

Secret Meeting of Letters and Numbers: Cryptology

Cryptology is the study of “secret writing”. It deals with ciphering information: takes the letters, matches them with numbers, then these numbers are changed to letters again by a secret way. Designing this secret meeting, modular arithmetic helps a lot.

Due to increasing security problems, our need for this secret meetings has grown. Cryptology is used in many areas such as the security of banks, computer and network passwords, between messaging of persons and government foundations, etc.

Wkh Klvwrvb ri Fubswrorjb

One of the earliest and simplest ciphers is the ‘substitution cipher’ which was used by Julius Caesar during his military campaigns in 100-50 B.C in Egypt. Caesar, Roman military and political leader, replaced the letters by letters three positions further down the alphabet. For example, with a shift of 3, A would be replaced by D, B would become E, and so on. We may write C (x) = x + 3 (mod 26). So, according to this ciphering method we can read the subtitle of this section: The History of Cryptology.

It is easy to break the Caesar cipher. Because, the frequency of letters changes in every language. For example in the English language, the letters E and T are usually most frequent, but in Turkish the letters A and E. By using the frequency analysis and computers we can break even the long ciphers in few seconds.

Terminology

Cryptology is the study of cryptography meaning “secret writing” (from the Greek kryptós, “hidden”, and gráphein, “to write”). Cryptology deals with both ciphering and deciphering information. While the roots of ciphering information extend to Ancient Greek, cryptology is founded by Arabians in 800s, long after that.

Cipher in English, chiffer in French, şifre in Turkish are all coming from sıfır (sat-fe-rı: bending, changing into, turning to) in Arabic language.

The meeting of letters and numbers which is known as Ebced in Arabians was used in many areas besides ciphering information. In Ebced numeric values were given to every letter in the Arabic alphabet. And the name of this method comes from the letters elif, be, cim and dal which were matched with 1, 2, 3 and 4.

First Ciphering Devices

First ciphering devices in the history of cryptology started to be used in 16^th century. But the most important one was Enigma which had determined the destiny of the Second World War. Enigma, made in Germany, was used in the military submarines and caused the Entente Powers to lose their power. However, in the last period of the war, German military used Enigma very inattentively and used uncreative ciphering letters like AAA or ABC. That is why Britain army broke the ciphered messages easily and gained power.

In fact, we can say that the ciphering systems which were used in Second World War in order to send strategic messages, the algorithms used to break the codes, the invention of ciphering and deciphering devices has resulted in the rise of computer science.

Who is using cryptology today?
Today, not only the intelligence departments and the military need cryptology for ciphering messages but also we use cryptology in our daily lives. We use it when we withdraw cash from ATMs or shop from the internet, in our computers or when we use online banking, etc.
Day by day, the mathematicians develop new encryption methods. Nowadays, AES (advanced encryption standard) is being used by U.S. government. AES has a fixed block size of 128 bits and a key size of 128, 192, or 256 bits; which is sufficient to protect classified information up to the SECRET level. For the TOP SECRET level 192 or 256 key lengths are used. And of course attacks are unavoidable where there are secret meetings of letters and numbers. But, in the conclusion all these attacks –whether successful or not- serves as an incentive to improve the encryption methods and cryptology.

A Mathematical World

Below is a ciphered text which was encrypted with a method which I derived from the substitution cipher and developed a bit. Our key to break the code is ‘A Mathematical World’. That is, the first word of the text is encrypted with A (first letter of our key), the second word with M (second letter of the key), third word with A (third letter of the key), … Here is the ciphered text:

J JVFU ZPV U TQNM KZQQ BS BSJUINFUJD: BYUFNB JMMNM, SDLQ TVCUSBDUFE, XAHQ JRIQFMIFBA PCS XJAWFVKZAH ZQHQD HMZMHIH.

Hint: For example, first word of our text is encrypted with the letter ‘A’ and the numeric value of A is 1.

A	B	C	D	E	F	G	H	I	J	K	.	.	.
1	2	3	4	5	6	7	8	9	10	11	.	.	.

You should think: “To which letter’s numeric value 1 was added to get a ‘J’ (numeric value: 10)? i.e. x + 1 = 10 (mod 26) then x = 9 which matches with I.

So, the first letter of our text is 'I'. To see the decrypted text click here.

CONSTRUCTIVIST VIEWS OF LEARNING

How do we learn? For example how did we learn that ‘if it is 8 am now and it will be 2 pm six hours later.’? In order to make calculations in the clock arithmetic, is it necessary to know some knowledge of modular arithmetic? The example might seem exaggerated but that’s the underlying mentality of the classical educational system! Most of us learn the clock arithmetic in the childhood without any prior knowledge of modular arithmetic. We learn this through our experiences. We learn it when we adjust our sleeping time or decide when to return home.

Constructivism views learning as a process in which the learner actively constructs or builds new ideas or concepts based upon current and past knowledge. In other words, “learning involves constructing one's own knowledge from one's own experiences” (Ormrod, J. E., Educational Psychology: Developing Learners, Fourth Edition. 2003, p. 227). But what our educational system does is to assume that the students have no current or past knowledge; rather, they are seen as empty pages. Especially in mathematics education, students are expected to memorize. They learn by memorizing that 2+2=4; which is not always correct.

2 + 2 is not always equal to 4

At a first glance, it seems as if it is quite difficult to apply the constructivist paradigm to mathematics education. In a literature class, each student can have his/her own understanding of Shakespeare for example; but in mathematics 2 + 2 = 4, and how can we change this basic fact in another way?

Here is a list of questions, a constructivist mathematics teacher might have asked:

• What is the mathematical setting we are working with?

• How many elements does our space have? In particular, does it include 4?

And the following ideas might have helped him/her:

• If we use the daily arithmetic then the answer to 2+2 is 4;

• But if our space only includes the numbers: 0, 1, 2 and 3 then our answer cannot be 4 since it is not in our space.

• The numbers that are needed to count the days of the week are 1, 2, 3, 4, 5, 6 and 7. If today is Saturday, it is the 6^th day of the week and 2 days later it will be the 1^st day of the next week. Then in weekly based arithmetic 6+2=1 (modulo 7). And if a week were to consist of 3 days then 2+2 would be 1 in this system.

Constructivism suggests that knowledge of mathematics results from forming models in response to the questions and challenges that come from actively engaging with the problems and environments - not from simply taking the given information, nor as merely the blossoming of an innate gift (Mathematics Ed.: Math Forum @ Drexel). So emerging from the interaction of people and environmental problems, the knowledge should be interactively learned.

Teacher's Role in a Constructivist Classroom

Contrary to the criticisms by some educators, constructivism does not dismiss the active role of the teacher or the value of expert knowledge. Constructivism modifies that role, so that teachers help students to construct knowledge rather than to reproduce a series of facts (Educational Broadcasting Corporation: http://www.thirteen.org). A constructivist teacher should always make extra researches about the subject being taught. S/he should know the relevancy of the subject with daily life; also should create or find activities for the students. The teacher should make the classroom atmosphere participatory with questions, which helps to keep the students motivated.

In my opinion the desire of the students for participation and knowledge is innate. If you observe the children before and after they begin the school, you will see that the current education usually hinders their curiosity. They usually ask more questions before beginning the school. Thus the problem is to decide what is more valuable for us? “If we want students to value inquiry, we, as educators, must also value it.” (Brooks, 1993; p.110). If we ask questions with only one and unique answer, such as what is the general term of Fibonacci sequence, then we cannot expect the students to calculate the population growth of bees, or at least have the idea that Fibonacci created this sequence to calculate the population growth of rabbits. Because if all the teacher wants is some general formula, then the students will not try to explore the rest.

As a result, teachers should not forget that we are living in a changing world; even 2+2 is not always 4.

DO WE NEED DISCIPLINE?

Do We Need Democracy? “The most effective way to restrict democracy is to transfer decision-making from the public arena to unaccountable institutions: kings and princes, priestly castes, military juntas, party dictatorships, or modern corporations.” says Noam Chomsky. In Turkey’s educational system, discipline is generally seen as vital which must be and can only be applied by teachers and school government. Yes! Discipline is really vital but cannot be applied exteriorly if it is not agreed by students. What must be done is to explicate the concept of discipline as an inner power which is necessary to improve strength, self control and socialization. Therefore, if we need democracy we should not transfer decision-making, establishment of discipline rules to institutions like school governments or teachers.

“Be kind to others” and “Do our best work”

In a school, second graders decided they needed only these two rules in order to do their work well (Eight Models of Discipline: The Glasser Model, p.120). Great! All we need is just two rules. Then we can construct a classroom with a positive discipline atmosphere. There may be other rules of course; but the important thing is, there should be fewer rules and and these rules should not be so specific; rather they should be for more general cases –like in democratic constitutions. Also, students will not forget easily when there are only two rules they should obey.

When Rules Are Broken

Of course, it would be folly to expect students to obey the rules every time. Moreover, some people think that ‘Rules are set up for to be broken!’. First of all, participation of students in this decision making process is very important. But before setting up the rules with students, a discussion of the necessity and importance of rules should be held. And if still the rules are broken what should be done, has to do nothing with punishment. According to Dreikurs, punishment teaches what not to do, but fails to teach what to do (Eight Models of Discipline: The Dreikurs Model, p. 63). So when setting up rules with students, the consequences should also be established. And applying these consequences, students shall know that they make their own choices about how they behave.