A brief history of the affects of Technology on Mathematics Collaboration

Mathematics and Technology

In the world we live in today, technology is constantly changing the way we do things. It has evolved the way we are able to teach and to learn.

In past centuries, collaboration with other professionals in your field was limited to carrying stone tablets, and sending hand written letters by way of messengers. I don’t even want to imagine discussing complex proofs by way of messenger pigeon.  

Beginnings of modern collaboration

The Plimpton Library tablet from around 1700 BCE, written in cuneiform, contains the most influential and profound mathematical insights that are still used today. It was written on wet clay, essentially by making impressions using wedged instruments. 
As a mathematician, this made collaboration with others a long process. You would have to mix your clay, engrave your theories , and if you did not make any typographical errors allow your clay to harden  .After your clay had hardened, you would then have to carry a big clay tablet with you across the dessert to collaborate with mathematicians in other towns.

However, this beat the alternatives. If they had simply written their theories in the sand and hoped it stayed until someone came to look at them, the famous Pythagorean Theorem, and Euclidian geometry might not have existed today.

As a result of the ineffective means of sharing knowledge, one can only wonder what great mathematical discoveries were completely lost to us. For all we know the tablet with the method for finding all of the nth primes with one simple equation is buried in the sand, or was shattered by someone falling while carrying it.

The Renaissance Era

Now skip ahead to the 1400’s. The hundred year’s war is over. The Byzantine Empire has fallen, and most importantly the invention of the printing press. Now mathematicians could share their work with a much wider audience. With all of the new innovations of this era, mathematicians still had the time lapse of sending their proofs by way of messenger. Collaboration was still more efficient, and produced more mathematical advancements than previous centuries. Number theory, calculus, and Newtonian physics all came from this era, along with many more.

The computer age

The computer age gives us a whole new meaning of collaboration, innovation, and the ability to calculate very large numbers quickly. Mathematicians all over the world began to use these amazing new machines to assist in making new discoveries.

Branching from this invention, come the intranet/internet allowing collaboration on a scale never before seen. No longer were we bound by waiting for the mail to deliver our work to other mathematicians to check our proofs. We could digitize and zap our theorems to all parts of the globe in minutes.

Social Media

As internet speeds improved, we were able to communicate in ways we never thought possible.  Community websites allow us to find and connect with others in our field of interest globally. We no longer are bound with traveling the globe to find a particular expert who is working on a similar project. Now we can just “tweet” our way into new partnerships, learning teams and even employment. 

Math Poetry: The pi poem

        This poem is a free verse poem I wrote to remember the digits in pi. How it works is it uses each digit in pi as the number of letters in each word for instance the first few words are " Now I view a world"
'Now' has 3 letters ' I ' is 1 letter ' View ' is 4 letters ' a ' is 1 letter and ' world ' is 5 letters.  or 3.1415

                 Now I view a world            

                 Imaginary in itself                              

                 Solid for those unstable believers                  

              Naively compliant 

                   Set in the ignorant seal

                        What can the thinkers say 

                      to clarify intricate codes

              is realness realness. 

                live a quizzical reality 

                                                            3.141592653589323844338326952884197
                                                                 

A sad story about Math

 

         Your change will be $3.63 mam...

        My wife worked as a cashier at Mcdonalds, and one day a lady came in and placed her order. Her total came up to $16.37. The lady paid with a $20 , and when my wife handed her back $3.63 the lady got furious.
She swore that she was owed $4.00 because her total was $16.37. My wife, being an excellent judge of character, could tell that the lady was being totally serious. She truly thought that her change was supposed to be $4.00. My wife tried to explain to her what her change was every way she could. The manager and other employees tried to explain it to her, and then finally another customer waiting behind her had to break out his cellphone calculator to show the lady what her change should have been. After looking at the number on the calculator, she apologized and grabbed her food and left extremely embarrassed.

                       The sad truth

       Unfortunately, this was not the first time this has happened. The number of people who are unable to even properly calculate change is staggering. It was an occurrence that happened a couple of times each week. Levels of math literacy are steadily declining, and it should not be ignored. There are many programs and alternate learning centers that offering after school and even weekend classes to those struggling with math. However, they are not everywhere.Small towns like the one I am from do not have those resources at hand, and any of the alternatives are more expensive than the average household can afford.  

.  

Converting a number from base 10 to base 6

steps to convert to base 6 from base 10


Ill illustrate by example and explain each step as I go.


First you will set up your problem.  lets say we want to convert 200 to base 6

the problem will be set up as follows  200 =      x  6  +        in the first  blank,     , we divide 200 by 6 and drop the remainder which is 33  which makes our problem look like this 200 = 33 x 6 +    
now for the second blank we place the remainder,  which we dropped in the first step  which gives us this:
 200 = 33 x 6 + 2

The next step is you take the 33 and repeat what you did in the first step and placing it under the first part of or problem   like this        

200 = 33 x 6 + 2
33   =  5  x 6 +3


Then you take the 5 and do the same thing. and you end up with  200 = 33 x 6 + 2
                                                                                                   33   =  5  x 6 +3
                                                                                                    5  =   0 x 6  + 5
You then will take the remainders and from bottom to top and you get 532


This method works when converting to other bases as well


                                                       

Many ways to multiply: Russian Multiplication

           Russian Multiplication 

Here we are going to discuss Russian multiplication. This method is also known as binary multiplication. The simplicity of this type of multiplication stems from the fact that you are only halving and doubling numbers then adding. I will demonstrate by example.       

                                                             Multiply 17 X 18 = ?

1.)                  17 | 18                         first we place the two numbers side by side and separate them into 2                                          

                                                          columns

                                                                                                                                                                           

2.)                   17 | 18                        in this step we started with the column on the right and cut 18 in half 

                            |  9                         to get 9 then we cut the 9 in half and dropped the remainder to get 4

                            |  4                         then we cut the 4 in half to get 2 and the 2 was cut to get 1. we stopped

                            |  2                         here since we got to one this is as far as we need to go for this method

                            |  1                          to work

                                                                                                                                                                              

3.)                  17 | 18                         Here we took the left column and instead of halving we doubled the

                      34 |  9                          17 and got 34 and doubled the 34 and got 68 doubled the 68 and got

                      68 |  4                            136 and doubled the 136 to get 272. and we stopped here because

                    136 |  2                              the 272 is in the same row as the 1

                    272 |  1

                                                                                                                                                                       

4.)                  17 | 18                      

                      34 |  9                          What we did here was we looked in the right column and found all of the 

                      68 |  4                            even numbers that are in the right column and crossed out the 

                    136 |  2                             entire row that had an even number in the right column 

                    272 |  1

                                                                                                                                                                       

  5.)                17 | 18                                 

                      34 |  9            1             in this final step we took the numbers that were not crossed out in the

                      68 |  4            272           left column and added them together, and as you see we have the 

                    136 |  2            +34                so 17 X 18 =  306

                    272 |  1            306

Once again I am going to explain that although I use the words, " easy, quick, and simple" , everyone should pick the types and methods that work best for them. I feel it is important to understand that there are multiple ways of doing things in mathematics and when we stop searching for new approaches in math we stop the heart of mathematics and destroy what its was originally created to do. Which is explore, explain, examine, and expand our understanding of the world around us.

Many ways to multiply long multiplication in reverse

In our public education system we are unfortunately only taught that there are only two ways to multiply.
One is "long multiplication", and the other is a calculator. Long multiplication is the process of starting with the ones column and multiplying and if the number is greater then 9  you carry the value over to the tens column then you multiply and add and so on... It is a very lengthy process depending on the numbers you are multiplying. How ever there are many ways to multiply numbers together and there are many of them that do not test your ability to memorize a multiplication table. Which , in most public school systems that is all we are truly learning, is how to memorize and regurgitate facts. In the real world, there is much more to life than memorizing. Don't get me wrong a  strong memory can be a beneficial talent to posses. However, the truth is that everyone is different. Some people have strong memories and some do not. So to address the fact that everyone is different and not every one has strong memories I will go over some alternate forms of multiplication. I will skip over long multiplication as most people are already familiar with it.
 Lets start with the alternate long multiplication. So rather than starting on the right hand side and carrying over to the other columns we start on the right. I will demonstrate by example.

WE will multiply 47 X 32 = ?

                  47                    we started with the left hand side which is 4 X 3 which is 12. but if you notice I
                x32                a zero after the 12. The left most column is actually the tens column, which means
              1200                     that we are not multiplying 4 and 3 we are actually multiplying 40 and 30 so we
                                           need put 1200 in here.

                                                                                                                                                                   
               47                       in this step we cross multiply the 4 and the 2  which is 8, and once again we see                                    
            x 32                          that we are actually multiplying 40 and 2 which is 80. 
            1200
               80
                                                                                                                                                                   
               47                         here we multiply the3 and the 7 which s 21, and again we note that it is actually 
             x32                          30 and 7 we are multiplying.so we add the zero to the end to get 210
           1200
               80
             210
                                                                                                                                                                              
              47                            In this step we multiply the two numbers in the ones column 7 and 2 and we 
            x32                            get 14, and in this step there is no need to add a zero because we are
           1200                            multiplying only 7 and 2.
              80
            210
              14
                                                                                                                                                                   
             47                                     in this final step we simply add up the rows at the bottom and we 
           x32                                        get the final answer which is 1504
         1200
             80 
           210
             14
         1504          




This method may not be the best for everyone. I personally prefer this method.  However,  the point I am trying to make is that everyone learns and understands things differently. So it is best to evaluate all the options and choose what works best for you.

Multiplying by 9, 99 ,999 etc

Here is an interesting trick for multiplying numbers by 99, 999, 9999, and so on. This trick works with any number of 9's you have. To multiply numbers by 99  you subtract 1 from the number to be multiplied and you get the first digit in your answer. then you take the number you just got and subtract that from 99 and you get the second part of your answer. This method works for all numbers less than 99, 999 , 9999 depending on which one you are multiplying by. So if you were multiplying by 99 it would work for all numbers up to 98



 Example:     99 X 3 = ?



1.)subtract 1 from 3:    3-1=2 First part of answer
2.)the subtract the result from 99:   99-2=97    second part of answer

and combine the two answer and you get 297

as you can see all you are doing is taking the answer you get form the first step and making it the first digit(s) in your answer, and the result from the second step is the rest of the digits.


Example 2:            99 X 12 = ?

1.)subtract 1 from 12,   12-1=11
2.)subtract 11 from 99,  99-11=88

and you combine the two and you have 1188
                                           
Example 3:    999 X 998 =?

1.) subtract 1 from 998,    998-1= 997
2.)subtract 997 from 999,  999-997=2

combine the two answers and you have 9972... but wait that's not right. So what happened? In this example, the second part of the answer only has 1 digit yet 999 has 3 which means that the second part of the answer must have 3 digits. so how do we show the number 2 as 3 digits. Its simple we just add 2 zeros at the beginning turning 2 into 002 so when we go over our example again:

1.) subtract 1 from 998,         998-1=997
2.) subtract 997 from 999,     999-997= 002

combining the two we get  997002

This is the same for the 2 digit examples the second half of our answer must have the same number of digits as the 9's we are multiplying by
 Example: 99 X 98= ?

1.) subtract 1 from 98,     98-1= 97
2.) subtract 97 from 99,   99-97= 02 (here we added the 0 at the beginning to make the number of digits the same)

combining the two we get 9702

Summary:
 To multiply numbers by 99  you subtract 1 from the number to be multiplied and you get the first digit in your answer. then you take the number you just got and subtract that from 99 and you get the second part of your answer. This method works for all numbers less than 99, 999 , 9999 depending on which one you are multiplying by. So if you were multiplying by 99 it would work for all numbers up to 98

The second half of our answer must have the same number of digits as the number of 9's you are multiplying by.

Suggestions and comments

If anyone has any suggestions, comments, or requests I will gladly  talk about or explain anything math or mathematics education related. So please leave your comments, I will be screening so please only serious or relevant comments. Thank you and hope you enjoy

Knowledge versus Intelligence

    



       Math is something people use everyday, although it can be difficult at times to use effectively. Mathematics education has been in need of a major overhaul for some time now. Too often, mathematics educators have fallen into a habit of teaching only one way to do things. In some cases, educators are unaware of multiple ways of teaching mathematics.

     There is too much emphasis on memorization, and not emphasis on logic. In classrooms many days are often spent memorizing one task or method of calculation, or one formula. or in some cases many formulas.This task can make math quite overwhelming. The goal here should be to ensure that students understand the concept, rather than just to be sure that they can memorize a bunch of formulas. Unfortunately, this is not the case.

      Logic is the foundation by which mathematics is built. All of the theorems, formulas, and rules that we use today are the result of the mathematicians starting with the very basics of numbers and building upon them by using logic. If we were to learn how to logically arrive at new theorems and formulas in mathematics, we would possess a greater understanding of mathematics as a whole. This means that educators should encourage students with an interest in finding new ways to approach a problem. After all, we would not have any of the great discoveries of math processes today, if we did not first look for an alternate way of using formulas and numbers. However, students are graded on their ability to memorize and regurgitate facts and formulas, and not on their ability to think and understand.

           Rote memorization is simply a method that will increase knowledge. It is more important to increase a person’s intelligence. Often people confuse knowledge and intelligence as the same thing, however, they are two very different things. Merriam Webster defines knowledge as the sum of what is known: the body of truth, information. They also provide the definition of intelligence as the ability to learn or understand or to deal with new or trying situations: reason; also: the skilled use of reason, the ability to apply knowledge to manipulate one's environment or to think abstractly.

      First, we will examine the definition of knowledge. The definition of knowledge shows us that knowledge is simply a quantitative measure of what we have memorized or experienced. From the definition of intelligence, we can conclude that intelligence is a qualitative measure of our ability to understand and to gain knowledge.

        Ole Roemer, the first person to calculate the speed of light, accomplished this through use of his intelligence not his knowledge. His knowledge did play a hand in his ability to make the calculations. However, without his ability to reason or to think abstractly he would not have made this exciting and profound discovery. As you can see knowledge and intelligence are both extremely important, but knowledge can be obtained by anyone.

        Just because a person doesn't know the definition of a word doesn't mean they're not intelligent, it just simply means that particular word is something they have not been exposed to. However, an intelligent person knows they can simply open a dictionary and find the definition of the word they are unfamiliar with. Also, an intelligent person will not only understand that definition but deduce the proper way to use the word.

About Math anxiety

     

Introduction

         Math anxiety is a common problem that effects a large number of people in the world today. It is important to discuss some of the misconceptions about this problem. The first and most important myth is that people who suffer from math anxiety have poor grades. This is simply not the case. People of all intelligence and education levels have experienced some level of math anxiety.   Second myth about math anxiety is people who suffer from math anxiety fail math classes in school. Even people who have math anxiety can receive a passing grade in math classes. The nervous and uncomfortable feelings people get when looking at math problems is not completely debilitating in all cases, and it can be overcome with proper guidance and awareness of the issue. The key is to prevent it by finding its root. Some doctors describe math anxiety by its symptoms rather that its root causes. There are many factors that contribute to this problem, and every person has their own experience in how it effects them. Even if we eliminate the common problems those who suffer from this epidemic share, we must still look into the individual experiences that perpetuate the symptoms.
   
 

Barriers to Understanding Math Anxiety

     One important fact to consider when you are fighting math anxiety, it that most people who understand math do not fully understand why other people have a fear of mathematics. It has been an obstacle for me.
 My wife suffers from math anxiety, and I am a math enthusiast. Which at times caused a conversational paradox. I would begin talking about  something "really exciting" I found about math, and slowly realize after a few moments of blank stares that I completely "lost her". She like most people who suffer form math anxiety, began a seemingly natural shutting out process at the sound of anything mathematical.So to alleviate this problem I began to try and find ways to relate the areas of math I am interested into something she is good at.
       She is really good at writing and enjoys books. So I took the area of mathematics that I am most interested in and tried to explain it in terms of composition.  Prime numbers is something I have spent a vast majority of my time studying. While my wife understood what a prime number was she was unable to grasp why they were an important research topic for me. I explained to her that prime numbers are used heavily in encryption and security. The "really big" number used to encrypt data is basically the Title of the book,or data, and to find your way into the chapters of the book you want to read  you use the table of contents. which is the prime factorization of that really big number. The prime factorization of the really big number is as unique to the really big number as the table of contents is to that specific book.

What is the root cause of Math Anxiety?

   While the explanation I gave her, did help her understand that one concept, it did absolutely nothing to help with her math anxiety problems. It was at that point I realized her problem with math went even deeper than having nothing to relate it to, or even her understanding of the subject as a whole. She went to high school in a small town with a very small educational budget. As a result of their limited budget, the schools were forced to assign faculty, who were not truly qualified to teach math. She had many bad experiences while attending these math classes taught by unqualified teachers. The instructors she had were set in the "there is only one way  to do things" mentality. This is due to them not having the proper education themselves to realize that there are more than one of doing things, and to not penalize students who only understand the concepts being taught  in their own way.  The reality of mathematics is that there are many ways to do everything, and that finding these new ways has been at the heart of mathematics's evolution over the centuries.

Are Teachers to Blame for Math Anxiety?

No, teachers are not the sole cause of math anxiety. Teachers who are improperly prepared to notice and help relieve math anxiety can perpetuate the problem, but it is not the fault of all teachers. There are many educators out there that are fully qualified to teach math and to help students overcome math anxiety. Unfortunately, there are not enough of them. Class sizes continually get bigger and budgets get smaller.

Are there causes other than education?

Yes, other factors can contribute to math anxiety. In some cases, there can be underlying psychological problems affecting a persons ability to understand math. Brain injuries or development issues can also be factors, but this is not the focus of our discussions here. We will instead be focusing on the educational aspects of math anxiety.

How Do We Cure Math Anxiety?

    There are a number of ways to fix this problem, just as there are many ways to solve all problems in mathematics.  We first need to get to the root of the problem and make some changes. In the example of my wife's high school she had teachers who were not truly qualified to teach math. However in my high school we did have similar problems, only I got lucky and had the head of the math department who also taught university level mathematics. The combination of the individual attention she gave to her students and her fluency in teaching the subject made an already present drive to learn more flourish within my mind. So what this tells us is that one solution to the problem starts with the school boards and the policies that govern the criteria for being a mathematics teacher. I feel the teachers themselves should be required to have classes on logic, history and applications of number theory, and more in depth critical thinking courses. These classes can easily be blended into the core requirements for math teachers. Another problem is that the teachers who are qualified have their hands tied by what the individual state and federal standards are as to what and how they teach. Some of these restrictions should be lifted and allow the qualified educators to actually educate. Instead of guiding the students on effective memorization and regurgitation techniques.