discrete math Articles

Economics is a fragile and often overlooked subject matter. It effects every one of our lives, whether we want it to or not. Will we have a house to sleep in, can we afford college, what are we going to eat tonight, do I get that 42 or that 56 inch TV?

Sadly, even though every person's life is influenced by our economic state, a surprisingly small amount of people understand it or even care about it. I feel obliged to enlighten those who want to understand and compelled to spark an interest for those who have no care.

I like to explain economic theory by utilizing an analogy. Economic laws follow the same principles as a poker match. If everyone joins into a poker game with a certain amount of money (chips), each person is motivated to play with hopes of achieving a large stack. Of course for some to be high stacked, the chips had to come from someone's wallet other than their own. It is interesting to study how the amount of chips influences a person's decisions in what they will do and whether they will leave or stay. From observing people's actions, we can formulate theories of how chip magnitudes and how others' chip amounts can influence the state of minds of the individual players, not only influencing them but many times governing the decisions that a player makes.

I enjoy teaching and studying economics as well as many other subjects. I enjoy explaining complex ideas such as economics, calculus, philosophy, and many sciences in clear, precise, and of course FUN ways. After all why should one waste his precious time with acts that illicit boredom, when he could be having fun.

My name is Andrew and I am a Professional Engineer, Professional Tutor, and tennis player extraordinaire. Take a look at my tutor profile for more interesting facts about me. If you liked the way I explained this, if you enjoyed an idea that this blog elicited, or if you found something interesting on my profile, don't be shy! Email me and help me help you help you.

Ken B., known as "The Best Little Tutor In Texas", has just surpassed the 400 hour tutoring mark in Houston, Texas! What makes Ken so good and popular in Houston? It is because of his diverse background and of being able to do the following: mathematics, statistics, chemistry, physics, computers, and computer programming. He can help a student in many many different areas. Ken does both high school and college and does regular, honors, IB, PAP, AP, etc... All that is quite a talent. Ken says that the subject most tutored in the past several months is statistics, and the reason for that is that most teachers use the 'dump' method...they 'dump' a copious quantity of power point files onto the student but the teachers do not really teach how to 'do' the problems...he has seen the same trend with other subject areas, and this is most unfortunate for students taking the classes...so, if you need to get on top of your mathematics and science courses (except of biology), then Ken in Houston is the person to contact.

One of the hardest things for students to do is to keep re-writing steps they already know how to do. So often, we get used to doing something, so we shortcut and skip steps, because *we know* what those steps are. But someone else, reading through your page, doesn't understand how you went from point A to point D without seeing point B and C, too. So one of the easiest steps I make my kids (in my classroom, and those I tutor) do is to write down their process - sometimes even making them write it in words.

For example, when working an algebra problem of 3(x-2)=18, I'd make them do all the work:
3*x - 3*2=18, distribution to get rid of the parentheses
3x-6=18, multiplication
+6 to both sides, to get the x term by itself
so 3x = 24
divide both sides by 3, to solve for x instead of 3x
x = 24/3
x = 8

Now, I wouldn't make them write it down with words all the time, but on a test or quiz that might be a 10-point question. And the first few times they learn a new process, I would *definitely* make them do it. Review the above equation. Can you see how much easier it would be to do the next problem, say, 5(x+3)=52, if you had the one above to look at and get the process from? The words help SO MUCH!

Word problems are considered the most difficult Math problems to solve for the following reasons:

1) Reading Comprehension: Before you even figure out how to solve a problem, you have to really understand its description, the facts, and what you are supposed to solve for. It is not always easy since sentences can be complicated and not all problems are well written. In fact, in my experience, many problems are not clear and have ambiguous information that can be understood in different ways. There are also many students who do not read well; reading comprehension is known to be a big issue. Furthermore, to really understand a problem you need to have a mental model of it, which is not always easy to do EVEN if you understood the parts.

a. You will learn how to break complicated descriptions into small parts, how to understand those parts, and how to see the relationships between the parts. You will also learn to use a set of tools that will help you understand and create Mental Models of the problem.

2) Translation: After you read and understood the problem, which does not mean you know how to solve it, you have to translate it to Mathematics. Mathematics is a language, and while it is not fundamentally difficult to translate human speech to Mathematics, the principles of translation are not really taught in school and students are expected to figure it out by themselves. While some students figure it out, most students only do so partially, and therefore get stuck at this stage. Even when the principles are taught (mostly not explicitly even then), they are taught with unintuitive symbols and methods (all those x and y).

a. You will learn how to systematically translate Word Problems to Mathematics in a simple way.

3) Problem Solving Strategy: This is the least understood, and the least taught (if at all) element of problem solving. It is also the most critical one, because even if you have mastered steps 1 and 2, you can get stuck without a good strategy. The few text books that do talk about “strategy” do not provide a strategy but are actually discussing the tools I mentioned in (1). Strategy is an overall approach that applies to most, if not all, problems. A tool is a way to solve a specific problem, or a one type of problem, but does not fit other types. It is a very important difference that underlies the problems in how Word Problems are taught. Note that you never study “word problems” as a topic, but only as part of a specific subject.

a. With this book you will learn a strategy that applies to ALL problems, not just word problems. It is a general strategy to solve any type of problem, not just in Math but also in life. Use this strategy for all the problems you encounter and you will see how powerful it is.

4) Mental Block: Due to all these problems mentioned above (and the one mentioned in 5), people (not just kids) often get stuck when they try to solve problems, either math or non-math ones. It is a normal and natural human reaction – freezing in the face of complexity and ambiguity. No one teaches how to overcome such mental blocks, except maybe say “have confidence”, which is not very helpful when you are not confident and cannot see what is going on. What is more important is for you to have a way to solve problems EVEN if you do not fully understand them, and you do not have “confidence”.

a. You will learn to solve problems even if you don’t have confidence, and as a result – develop confidence!

5) Knowledge: If you do not have the “technical” knowledge required for a given problem, you cannot solve it. That is a requirement that cannot be circumvented by having a good strategy, or solution tools.

There was a time when I struggled with an advanced mathematics class in High School when the teacher presented math theory from an abstract perspective. This bothered me, but I was determined to allow the subject to seep into my mind, as well as I possibly could, while listening to my teacher lecture. When he, the teacher, decided to give us one problem for homework, I secretly thought that I got off easy for a 1 problem math homework night. Little did I know that the question was posed in such an abstract, new language, manner that I really did not know how to approach solving the problem. I really did not completely understand the question.

First of all, it is difficult for any of us to admit weakness. This is just not built into our DNA. We want to feel capable, and when we do not, we panic. This is a major issue for many students who want to more easily learn mathematics. When I stumbled upon the same issue with trying to understand the one question given to me, I was petrified. One question my teacher gave me and I did not get it. I remember thinking that the new language he used during his lecture went over my head. I did not record his sentences but I did take notes. I struggled desperately to understand what he said as he jotted notations on the board. But, no, I was at a stand-still. Mentally, I just could not wrap my mind around the new concepts presented.

What was I to do? I did not want to continuously struggle, for this was wearing me out. In my stupor, I decided to go to the school library, hoping that the studious atmosphere might rub off onto me and somehow the answer would pop into my head. So, I quietly entered the library and found a quiet remote table to sit at. I put the paper on the table directly in front of me and stared at it. "I can't believe this," I said to myself, "I just don't get it. It's like a foreign language to me." Sitting there frustrated, I again became much stressed, and then the tension simply wore me out. "I need to not think about it so much," I decidedly told myself. "Hmm, let me lay my head down on the table and take a little rest."

I simply followed this little voice inside of me. I put my head down and went into a semi-meditative resting state. All the stress left my body, and my mind was free to roam with new information that left my conscious mind and seeped into my subconscious. Suddenly, I visualized while resting with my eyes closed, the ideas clearly needed to answer the question. Suddenly, the teacher's presentation of newly formed ideas came together perfectly. When I opened my eyes, I hurriedly wrote down what I discovered. Then I reviewed it again with such clarity, that it was hard to understand why my conscious mind could not figure it out. I needed my subconscious to communicate to my waking mind and it did so; calmly and without hesitation once I allowed my resting mind to reach a peaceful state pulling all the new information together in a very clear, logical manner.

I was lucky to learn this technique prior to college. Many times, especially when studying difficult math theory, I used this technique to ease the completion of homework while either dreaming at night, or going into this meditative state during the day. I hope to help students overcome unproductive tensions while attempting to learn mathematics and teach them this stress-free technique. My students will be amazed of their minds' innate capabilities of learning mathematics in this peaceful manner.