Tuesday, June 25, 2013

Mental Math and the Decline of it due to Computers




Blog 3: Mental Math

            In today’s fast paced world small calculations are done for us instaniously wherever we go, whether it be shopping online or at your job. Thanks to computers small calculations are done away with in lieu of time, but have you ever been to a place where their tills are not working, or had a delivery boy who could not do mental math very well? Since we are so use to computers and calculators doing all our math for us many freeze when it comes down to having to do math in our heads, and the customers get impatient and mad, and they start to demand the person to go faster which makes them more flustered and it goes on and on until the customers leaves or the employee gets the math done and hands over the change. Now think about this situation, is it really the employees fault? No, how about the customer? No, it seems like this is a problem where we can place no blame, but we can. To find who is to blame we only need to look at two sources, ourselves and our schools.
            You may ask yourself why we, the people of this earth, are to blame for the failing of employees to do mental math. Well simply put, we made it this way! We created computers to help us, but we have become reliant upon them for everything from medical records to entertainment, and yes, even to math. With our reliance upon computers so high there are few people who can do mental math, or mental computation as others put it.  I think this is best explained through a personal story of mine. I deliver pizzas Friday through Sunday evening, it’s not a great job but it’s a paycheck and that’s all that matters. I remember on my first day of work I was being trained in and I had to deliver with one of the other deliverymen, and for the first couple of hours all the extra money that we were given by the customers were tips so we didn’t have to do much math, but then we got an order where we were told to bring change for a 100 dollar bill, not so bad but the order was $12.86. Now I myself wasn’t so deterred by it and we went to the house and rang the bell, when the man handed my trainer the 100 my trainer had to pull out his phone and use the calculator! I interjected as soon as he pulled it out and said, “The change is 87.14.” and my trainer looked at me with wide eyes and said “How did you do that without a calculator?” and I simply replied “in my head”. On the way back to the store he asked me how it was possible to do that in my head and I explained it to him and he was absolutely stunned. Now this is just one story and this story has a pretty easy calculation but I’ve seen even easier ones completely overwhelm employees at places like McDonalds when their tills have given up the ghost. So we, the human race, are partially to blame because we created computers and have relied on them far too much for far too long. But there is another culprit and I’m sad to say it is our schools.
            Now many think that American schools are among the best, and whilst we are not number 1 we are definitely in the top 10. Being that we have such a great school system, one would think that our students are getting the best education possible in everything from reading to math, well that is where we are incorrect. Now don’t get me wrong I love America just as much as the next guy, and our kids do get a fantastic education, but ask yourself this, how many times have you heard a teacher say “Use a calculator” or seen a test that says calculators are allowed? From first grade children are taught to use calculators on the simplest of problems, calculators are great tools for graphing and for typing in long string of numbers to save precious time, but as a high school student I would use them for simple things like 14+8! (I now use more mental math but, this is how I once was) And I’m not the only one who did this, many other students did it for this and sometimes even on their times tables! There was a time without calculators where students used their fingers, and minds to figure out questions like these, and we should still teach ­­­that, to foster mental math, but even our schools have fallen prey to the “computer reliance” that has our very society in a stranglehold.
            In order to fix this problem there are two things that we, as a society, must do. First, we must break our dependence upon computers and calculators, and second (and this one leads into the first mind you) we must teach our students how to use mental math. If we do not start to do something now it may be too late and our students will keep having the same problems class after class. You may ask, well how can we help? Well I have two very simple changes that we can make, first don’t allow calculators unless they are needed (by needed I mean things like parabolas and graphing etc.) and also, even in high school, ask some questions that they must use mental math for and the first to answer them gets a piece of candy or extra points or some sort of reward. Now, rewards are sometimes thought to reduce the child’s want to do the work unless there is a reward, but if you think about it this is one where it does not matter because it puts the mental math in their brain and they will use it in other situations without even thinking twice.
            Mental math is truly an important skill, and it is one that many people do not possess due to our dependence upon technology, and it is a skill that we must foster and grow again, lest it become a lost technique and our children become entirely reliant on computers for our mathematical computations.  Finally, below I have added a video for you all to take a look at, one that shows how important mental math is.


Problem Solving More Than One Way to Skin a Cat



Blog Post 2:  Problem Solving
When one is doing any mathematical problem they use one or more problem solving strategies. Now since it is obvious that we, as adults, use problem solving strategies every day, it would only follow that we must teach our students the same problem solving strategies. The only thing is that there are many different ways to solve a problem so how do we choose which strategies to teach? Also, many may ask why it is so important to teach different strategies.
When it comes to the many ways that a person may solve a problem it is more of a question of what works best for that particular individual, sure it would be great if you could teach every person a specific strategy and have it work but everyone’s brain is different and as such different people must use different strategies. Some children learn best by seeing a graphical representation and as such it may be best to have them make a table or graphical representation, on the other hand there are children who are more logical and methodical and as such it may be best to teach them to make an equation, and then there are even more children who are more “hands on” learners and as such it may be best to act out problems with things like blocks or cubes. Thankfully all of these are possible for every problem because there is always more than one way to solve a problem. For instance say you have a child who is more of a visual learner and they get a problem like:
 Jan has 5 apples Ike has 7 and Mike has 3. How many apples do they have combined?
Well it would be best to have them make a table:
Name               Apples
Jan                   5
Ike                   7
Mike                3
---------------------
Total                15
This way they had a graphical representation and so they saw that they could just add up the three straight up and down, now this is a very simplified problem but it make a great example Now let’s move on to a child who is more of a methodical learner, they like stability and rules when it comes to work so it may be best to teach them to make an equation. So let’s take the same problem and make an equation
Let J=Jan’s apples I=Ike’s apples and M= Mike’s apples T= total apples
J+I+M=T
5+7+3=T
T= 15
Again a simple question but a great example of how a child can make an equation out of a question to help them further understand that the question is merely asking them to add up the three people’s amount of apples. Finally, let’s say you have a kid who is more “hands on” it may be best to supply them with some block or small cubes and they can then set up groups of 7, 5, and 3 and then count them all up. This will help give them a physical object that will help them to better visualize the problem.
            So why is it so important to teach several different ways to solve problems, simply put it’s a classic case of “different strokes for different folks” and so you can’t make a cookie cutter way of solving a problem and expect every child to get it. I think the best way to think about it would be in our own terms, by that I mean “adult” problems. Say you have to pay your gas bill, water bill and phone bill how would you write out the problem, well I like simple equations so I’d make an equation and add up the three amounts, some may make a list (like a table) then add them up, and some may take the cash they have and put it into three piles individually. So if we, as adults, do things like this differently why do we constantly try to teach children that there is a “right” way to do math and a “wrong” way?
Let me regale you with a story to drive this point even farther home. There was a time when I was in 3rd grade and we started to learn multiplication and I had trouble with it, so I would do my homework by doing my multiplication by doing addition. By that I mean I would take 5x2 and I’d just do 5+5, this may seem odd but it eventually made much more sense to me and I still do this today at times. Now this was apparently not okay and was threatened by my teacher to be held back just because I refused to do it the way she wanted us to…now that seems a bit odd because if it worked for me  and I was gradually learning my times tables that way why stop it, well it’s because she wanted every child to do their math in a specific way, but not every child is the same, just like every adult is different and we need to realize this and teach several different problem solving strategies to children to give them ways to choose from that make the most sense to them.
           

Thursday, June 13, 2013

Ancient Number Systems



Blog Post 1:  Ancient Numeration

            Whole numbers are “counting” numbers like -1,0,1,2,3,etc.; and we use them every day. We use them to communicate everything from speed limits to page numbers and every other kind of data one can think of. This week I had the pleasure to have homework that dealt with an entire chapter on the subject these numbers. Contrary to the thought many people would have, it wasn’t as simple as one would think, but there was one specific subject that was specifically hard, but also fun,  ancient numeration.
            The numbering number that we use currently is the Hindu-Arabic system which is numbers like 1,2,10,700 etc. and it is so widely used today that we hardly think of numbers as anything else but these, but what if I told you that  meant one million to the Egyptians? Well it is true, and there are many other types of numeration systems that have been used throughout time. This week I learned about the Egyptian, Babylonian, Roman, and Mayan numeration systems, and this writer was a bit overwhelmed at first but now is very comfortable with them and has many thoughts on how they can be used in a modern classroom. Now, I could describe the systems but at the end you’ll find a couple links to check out the symbols and their meaning for different ancient numeration systems.
            Now many may ask, “How does mean 330?” well it is simple really. The Egyptians were a very advanced civilization and they created a very simple number system, each symbol represents one number of the corresponding place value and the symbol was used for the numbers one through nine. So to make 36 in the Egyptian system all you have to do is   because is ten so three of them makes 30 and then you add the six mean 330?” well it is simple really. The Egyptians were a very advanced civilization and they created a very simple number system, each symbol represents one number of the corresponding place value and the ’s and you get 36, simple isn’t it? Well, I thought it was and so I’ve come up with a fun game for these symbols. If you wanted to teach your students about history and math at the same time you have the perfect way to do so here. Take the Egyptian number and create a basic code like  5=A and 6000=z or something of the sort and then create an “encrypted” message with them ( like a famous Egyptian quote or name) and then have them decode it. Not only will it teach them of this ancient number system but also tell them some information about history.
            Now I gave a list of links earlier for you to see the various number systems that I learned this week, but I’m going to explain a bit more about the harder one, the Mayan system. This number system is extremely overwhelming at first glance, but once you get the hold of it, it all comes pretty easily. Now, a fair warning, this system includes a bit more math than the previously explained Egyptian system, so you’ll be doing a lot more thinking so just bear with me one this one. Let’s say you see something like this:
  You may think, “That’s a nice picture but what does it mean?” Well, simply put, each “tier” has a formula you use to figure out its value then you add them all up, and bam you got yourself a spicy Mayan number sandwich, so let’s figure this one out together
Is the top most symbol and it is on the third tier (now it can go all the up to five tiers but let’s keep it simple shall we?) and as such a Mayan symbol on the third tier is figured out by taking is value (16) and multiplying it by 360 so:
16x360= 5760 
Then you then move on to the second tier which is  which  is 12, and on the second tier you multiply that by 20:
12x20=240
Then the final symbol we have is which stands for 13 and you multiply anything on the first tier by one:
13x1=13
Then you just add them all up
5760+240+13= 6013
So the number above is 6013, seems like a lot of work doesn’t it? Well it is, but I think of it more as a puzzle to be figured out and I enjoy a good brainteaser every once in a while.

So there you have it, ancient numeration. It’s a very challenging yet interesting subject and I enjoyed it more than I have ever enjoyed any subject in math. Now, I provided you with a couple links to learn about Roman numerals and the Babylonian system, so go ahead and take a look at them they are very interesting and you definitely will give your brain a workout.

Also, as promised, here a couple sites to check out the several different number systems mentioned above.


Sources:
1.)Babylonian numeration system. (n.d.). Basic mathematics. Retrieved June 12, 2013, from http://www.basic-mathematics.com/babylonian-numeration-system.html
2.)Edkins, J. (n.d.). Mayan Numbers. Not the Home Page!. Retrieved June 12, 2013, from http://gwydir.demon.co.uk/jo/numbers/maya/index.html
 3.) Roman numeration system. (n.d.). Basic mathematics. Retrieved June 12, 2013, from http://www.basic-mathematics.com/roman-numeration-system.html
4.)Egyptian mathematics and Numbers Hieroglyphs. Egypt Pyramids                                                                                    Pharaohs Hieroglyphs - Mark Millmore's Ancient Egypt. Retrieved June 12, 2013, from http://www.discoveringegypt.com/egyptian-mathematics-numbers-hieroglyphs.htm