What is Mathematical Modeling?

modelMathematical models are a depiction of a problem, process, or technology in the form of mathematics. Models are built to learn about a system or process, study the affects, and predict outcomes. They explain why the system or process works the way it does, and why the results occurred. They allow us to study and predict results and phenomena that cannot be seen or cannot be measured. Mathematical models are created and used by those in natural science and engineering, social sciences, economists, business, and statisticians.

The process required for creating a mathematical model is the same steps used for the scientific method. This includes observing, creating a model, and then predicting the outcome. The observation stage includes observing and measuring what is occurring in the real world. This can consist of gathering empirical evidence or quantitative measurements. The modeling phase includes creating the necessary equations, and analyzing the observations noted thus far. The prediction part of the scientific method uses the model created to predict what will happen in a yet-to-be-conducted experiment or in an anticipated set of events in the real world.

Most scientists and engineers use the experimental approach rather than the modeling approach. The experimentalist designs a study, conducts the experiment, and then records and analyzes the results. In contrast, the modeler converts some of the aspects of the real-world problem into a mathematical system.

Engineers in particular uses models of varying complexity to predict process, device, or technology results in order to properly design devices and processes. For example, every vehicle, airplane or building represents a model-based prediction that the vehicle will operate correctly, the building will stand, or the airplane will fly without unanticipated consequences. Prediction in engineering design assumes that resources of time, imagination, and money can be invested with confidence because the predicted outcome will be a good one. Depending upon the design and the model, it may also eliminate months or years of experimental time. In some cases, experiments only adequately measure and analyze a certain percent of the variables. Also, when investigating a new technology, the importance of certain variables may not be known or are unable to be measured. In order for a mathematical model to be adequate, it must include and account for all of the important variables.

Steps for Creating a Mathematical Model

Usually when you begin thinking about creating a mathematical model, you have some observations about the real world. You should be able to easily identify the need for the model. When you are ready to being creating a mathematical model, some questions and/or thoughts that may be helpful are listed below:

1) What are we looking for? How should we look at this model?

Identify the governing principles. You usually begin with some observations about the real world, and gather all the information that you currently have that is relevant for the mathematical model.

2) What do we want to know?

A list should be created regarding the questions that need to be answered. After you have decided on the initial scope of the problem, all available relevant data should be identified. The question “what do we know?” or “what information is available to help solve the question of what we are looking to answer?

3) What do we already know from experiments and/or literature?

If you have not done so already, conduct a thorough literature search. It may be possible that someone already created a mathematical model of the process or problem that you are trying to solve.

4) How should we look at his model?

Create as many diagrams of what is actually happening with the process that you are trying to model. This will help to clarify your inputs and outputs, and will begin to clarify the define the scope around the problem that you are trying to solve.

5) What assumptions can we make to eliminate some of the variables?

Create a list of all of the assumptions that you will use to narrow and clarify the scope of the model.

6) What will our model predict? Start with a simple model, and then add complexity as needed. Identify and/or construct the equations that will be used, and the resulting answers. If you are building an empirical model, then create equations from the data that you have obtained.

7) What are the input & output variables?

Create a list of all of your input and output variables. Define each constant, and determine which variables that you need to solve for, and others that you do not have constants for.

8) Are the results valid?

Validate your model with new experimental data or data that you have not used to create the model. Identify tests that can validate the model.

9) Constantly test your model and update your equations based upon new data and information.

If there is good agreement between what is observed and what the model predicts, then there is some reason to believe that the mathematical system does indeed capture correctly important aspects of the real-world situation. However, some of the predictions of a mathematical model may agree closely with observed events, while others may not.  When this occurs, the model need to be modified to improve its accuracy. The incorrect predictions may suggest ways of rethinking the assumptions of the mathematical system. The incorrect inferences of the revised model will lead, in turn, to yet another version, more sophisticated more accurate than the previous one. The goal is not to make the most precise model of your problem or process, but that model includes all the essential variables in order to accurately predict an outcome.

This may sound intimidating for the beginner in mathematical modeling, however, if you are new to creating mathematical models – I recommend starting as simply as possible. This may include using equations from textbooks or previous classes that you have taken, and create an answer that seems reasonable to you. This makes it easy to start creating a model, and complexity can be built into a model as needed. Experienced modelers may begin with more complexity because they understand how to code it properly, and to fit it in with the rest of their model.

Importance Of Mathematics

41196162 - smart female high school student writing various high school math and science formula
41196162 – smart female high school student writing various high school math and science formula

When I was a high school student twenty years ago, I was not a big advocate of math. In fact, I thought that it was boring, and I often did not understand the real world application. However, I ended up with a PhD in engineering!

Now after being in industry for 16 years, I understand how math has helped us to develop as a society, and how it is involved in almost everything that we touch. Some of the items that have been designed using mathematics includes televisions, computers, cell phones, cars, houses, traffic lights, satellites, medicine, and roads. Actually every item that is made through modern manufacturing techniques uses mathematics. We also take for granted that we constantly do simple math in our heads; we calculate the time between activities or meetings, and how much we spend from our recent paycheck or business transaction. The importance and contribution of math in our daily lives is very high. However, we all realize how mathematics is commonly perceived. The most common misconception is that mathematics can only be comprehended by people with a high IQ or that math consists of complicated formulas and equations. In short, mathematics is seldom considered to be for the average person. This misconception is due to several factors:

• There is a lack of knowledge of the subject itself. It involves understanding a structure or pattern of symbols that represents actual item counts.

• Most people have been taught math incorrectly. This in-stills an early “fear” of mathematics that is difficult to reverse in later years.

• In order to reduce fear of math, practical application and “hands-on” activities help students to associate symbols with real-life problems.

Introducing math at a very early age is the best way to solve this issue. The sooner the child becomes familiar with the world of numbers, the easier it will become for him or her to understand the subject and start loving it. Before they begin learning math in a classroom, they can begin to learn the basics through shapes, counting, objects, and activities around them.

Young children can be taught basic mathematics and counting through cooking in the kitchen or grocery shopping. Cups can be counted while creating the child’s favorite recipe or items can be counted as they are added to the cart. The child can be involved with comparing product prices in the grocery store, or adding up the costs so that they know how much the total bill will cost. There are many other ways that a young person can be taught to understand the importance of math in their daily lives. Helping them to explore the world of math around them will make it much more than just a subject that needs to be learned at school for grades. These simple ideas can have significant impact on both your life and a child’s life in the long run.

As we get older, mathematics helps to provide the basic knowledge and understanding about how the universe works. It helps us to improve our knowledge in science and technology, and it opens a world of opportunity and career options. It is not a secret that most of the four-year colleges require three to four years of high school math and science for admission. Recent job market statistics show that close to 90% of the jobs require the job applicant to have mathematical knowledge at a certain level.

It is also interesting to note how recent research shows that people who learn mathematics at a very early stage of their life have comparatively high reasoning powers and problem solving skills. Practicing mathematical concepts provides clarity to the thought process by gifting it with the capability to imagine different scenarios or probabilities for a particular situation and come out with a relevant solutions. In this day and age, we have to make many choices in a short span of time, and math can definitely come to our rescue in helping us to make logical and rational decisions quickly.

Should You Go to College or Grad School?

48788794 - portrait of positive asian male in glasses with laptop in cafe
48788794 – portrait of positive asian male in glasses with laptop in cafe

During my undergraduate years, my goal was simply to finish my bachelor’s degree. It was not an easy degree (chemical engineering), and working full time made it even more challenging. I had a good friend that I went to college with, and we both worked graveyard shifts in the medical field. Around exam time, we slept between classes in our cars! Somehow, we made it through, and my friend vowed never to go back to school again. I could not blame him.

Although this “initiation” into the engineering field was tough, I still began to contemplate going to graduate school in the year 2000. Neither of my parents went to college (although most of my dad’s side of the family did), and they always felt like they were limited in their job options due to their lack of degree. Learning from them, I did not want to go through the same struggles. Therefore, I applied for graduate school during the year 2000 and began my Masters in Chemical Engineering in 2001. There are many reasons for and against college and graduate school, and this depends upon your individual situation. I list some of these reasons in the following paragraphs.

Why go to college or graduate school?

There are many reasons to go to college or grad school, and some of these include:

1. The credentials that come with completing the degree.

2. It may open doors to certain positions within a company, or provide a competitive “edge” compared with someone that does not have a degree.

3. Depending upon your field and company, it may provide a salary increase.

4. It will help to develop your professional skills.

5. You will be in a program with other smart and interesting people who will end up being your colleagues.

A college or graduate degree often makes it much easier to advance within a company. It may open up a plethora of professional opportunities that might not be easily obtainable otherwise. However, do not just simply believe what I am saying — do some research into the careers and/or companies that you are interested in. Investigate the opportunities and salaries that are available to you with and without a college or a graduate degree. If you are interested in academia, then going to graduate school is a “no brainer”. If you are interested in industry, you should do a significant amount of research to make sure that this is the right decision for you. You should also remember that the number of jobs that you can apply to will decrease with the number of degrees that you have (especially without the required job experience). In many companies and industries, job experience is at least as important as a graduate degree — if not more important. Regardless of whether you want to be in academia or industry, the market is very competitive, and you have to realize that obtaining a degree alone does not automatically entitle you to a great job and career.

Why you should NOT go to college or grad school

The worst reason to go to college or graduate school is because you do not know what to do. Some of the reasons not to go to college or graduate school include:

1. You cannot find a job.

2. You do not know what to do.

3. There are high costs associated with college & graduate school.

4. For many programs, the rate of success may be uncertain.

5. The work can be difficult.

6. The social environment in college may not be what you have envisioned.

7. Especially for PhD, medical, or law programs, the work takes years, and there may be an effect on your family and perhaps other areas of your life.

8. Depending upon how much you take on, you may feel isolated due to the solitary nature of your work. It is not uncommon to lose friendships due to the limited amount of time that you have available for socializing.

9. Every year spent in college may mean another year that you are without job experience or savings.

10. You may have to put off having children until you are finished.

I am personally a huge advocate of going to college. No matter what your path will be in life, it will most likely open doors that would not otherwise open. With that being said, if you are interested in a skilled trade, this is a great option because it can also lead to a steady career with many business opportunities. We have many friends who are successful tradesmen and some who also have built successful businesses around their trades.

Before going to a graduate program, I recommend working in your field of choice to make sure that it will be worth the additional school, time, and money investment. I enjoyed working in my field, and I realized that additional schooling could open up additional opportunities. Therefore, I decide to go back to school. However, graduate school is not for everyone. Depending upon the program, it may not lead to additional opportunities or pay increases. In PhD programs, approximately 50% of students drop out, even after investing many years in their program. In addition, you may have to consider the reality that your friends may have houses and children many years before you are able to and/or may be more financially secure if you go to college for an extended amount of time.

The other topic that I have not touched upon is entrepreneurship. I believe entrepreneurship has its own learning curve that cannot be learned in school, and it has many of the same challenges and obstacles. Keep in mind that anything worth achieving will require significant time and focus. Find your passion, logically plan your decisions, work hard, and be persistent!

How to Write a Math Research Paper

14216981 - concept of  a businessman that solves problems
14216981 – concept of a businessman that solves problems

Researchers read journal publications when they are looking for new ideas about the research that they are conducting. Since journal articles communicate new ideas in a particular field, they must be clear, concise, and written as simply as possible. Even if your ideas or research are ground-breaking, the importance and validity of your work can go unnoticed if you do not communicate those ideas clearly. The best scientific or mathematical writing provides clear meaning in the fewest words.

Evaluate Other Journal Articles

Before you consider submitting your paper to a journal, research as many journal articles in your field as possible. Look at the content and structure of each paper, and then pick a few papers to use as a model. Write down the answers to the following questions while studying these papers:

1)   How many pages is each article?

2)   What sections do each article include?

3)   How long is each section?

4)   How many tables and figures are in each paper?

5)   Do you want the same section headings in your journal article?

After you have examined several journal articles, make sure that you understand the formatting, section, and bibliography requirements for the journal that you will be submitting your article to. The next step is to create a plan or outline of what your paper will entail.

Create a Plan

After reviewing several articles, the next step in writing a math research paper is to create a plan. Now that you have reviewed several journal articles related to the topic that you would like to write about, take a few minutes to answer the following questions:

1)   Why do you want to write the journal article?

2)   How many pages will your article have?

3)   What sections will your article have?

4)   What will the title be?

5)   Will there be any co-authors?

A few common reasons why individuals want to write a journal article is to (1) build their resume, (2) meet the requirements of a masters or doctorate program, (3) become known or have an impact in a particular field, (4) or make a difference in a particular area of research. The length of an research article usually ranges from 5 – 20 pages. There is not a steadfast rule for the number of pages to submit to most journals. When considering the title of your journal article, make sure that it is more specific than most papers that you have written previously. Most journal titles are a very precise and unambiguous statement of the research that the article will contain.

Roadblocks

The largest roadblock in writing a journal article is “starting” due to fear of criticism. Many high-achieving students are not used to getting criticized; they are only used to receiving praise. No matter how talented and brilliant that you are, criticism is more common than praise in both academia and industry. Excelling in any field will cause you to push boundaries and often result in failure. Therefore, it is important not to fear criticism; criticism that you receive will only help you.

Start Writing

It does not matter how good or bad your writing is — it is important to just start writing. Start with the section that you find easiest, and draft the sections in an order that makes the most sense to you. Individuals that have a difficult time starting to write something are often trying to obtain perfection from the beginning or are trying to write the paper sequentially.

Revise Your Writing

It is common to revise your writing 5 – 10 times before it is in good shape. Good writing is about good revising. Revise your work as many times as necessary to make sure that all of the points are clearly stated, and your work can easily be read and understood. Ask yourself the following questions when you are revising your research paper:

1)   Does the text include all relevant information?

2)   Is there additional information that needs to be included to make the arguments more comprehensive?

3)   Is the text clear and concise?

4)   Have you adequately reviewed previous work conducted in this field?

5)   Are all of the figures and tables labeled properly?

6)   Have you followed the instructions for that particular journal?

Finishing Your Paper

After you have finished writing your paper, give your paper to an interested friend, teacher, or parent to review. The reviewer will preferably be able to understand the content, and provide valuable feedback to make sure that you have not missed anything from a technical, mathematical, or grammatical point of view. One last important point to mention is that no paper is perfect. If you have reviewed your paper several times, and it seems to be in a finished state, then you have finished your paper! Congratulate yourself on finishing the research paper! The final step is to submit your paper to the journal for review.

Feedback on your submission

After you have submitted your journal article, you will obtain feedback on your submission. It is common for journals to ask for revisions prior to accepting the paper. When you resubmit your article, make sure that you submit a separate document explaining how the reviewer’s feedback was addressed. Most feedback will improve your paper and your journal writing. If you receive an acceptance, or an acceptance upon revision, celebrate that your paper was accepted!