Mathematics
For the Young and Young At Heart

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Studying Math

  • Math books are meant to be read... but not like novels.  Work though the book very slowly and carefully with pencil and paper in hand.
  • Do not expect to comprehend each new concept with equal ease.  What is easy for 1 person is hard for another.
  • Math is a cumulative subject.  Missing one topic impairs the ability to master new material.
  • Math is similar to a sport.  It cannot be learned by just watching or listening.  It requires practice!

Why Study Math?
by Bill Breitsprecher

For many, math classes represent our greatest “challenge” at school. It may seem abstract and far removed from our lives. Some that may have had difficulty with some math concepts when they were younger and are now afraid. Others may feel overwhelmed by the fact that there is just one right answer – there is no room to “bluff” or “fudge” the numbers.

Math is all about patterns – there are patterns in numbers. Because numbers can be used to describe and predict the world around us, math reveals hidden patterns in our lives. Yes, in math class we do calculations, but math is much more than that. It involves looking for patterns, testing our observations, and estimating results.

Many enjoy math because it is about solving puzzles and bringing order to chaos. Do you enjoy puzzles? Is it fun to create or discover a different way to look at things? For some, math is fun because it stimulates curiosity and creativity. This is referred to as “pure mathematics.” It is about the satisfaction of seeing something from a mathematical point of view. Many enjoy math for math’s sake.

For others, math is a tool to solve practical problems. This is called “applied math.” Of course, the type of math problem that one person finds interesting or useful may be very different from the type of problem another finds interesting or useful. Perhaps even more important, solving practical problems requires a set of skills that need to be developed. In order to learn to apply math to practical problems, we need to develop a set of tools and strategies.

Perhaps that is the most important point – most people would agree that solving practical problems with math is valuable. We might disagree on what represents a practical problem. We might also disagree on how to learn the basic skills that are needed to learn “applied math.”

Math is a cumulative subject, one that builds on previous learning. Perhaps the “basics” that will allow us to solve more practical problems do not look useful, but without them, applied mathematics is unmanageable. Math is taught to develop the tools needed to apply more powerful problem solving strategies later.

The world we live in is more complex than most assume. We are interested in things like weather forecasts or how much money we might earn and save this year. These may sound like simple questions, but the answers vary with many variables, each constantly changing. To understand where these numbers come from, one needs a good understanding of linear algebra and multi-variable calculus.

Solving practical problems in a meaningful way requires a variety of techniques and strategies. In math class, students learn “story problems” that are designed to teach applied mathematics – many students dislike these types of problems. The assumptions and math that underlies them is not as readily apparent as a mathematical expression or equation. They require students to understand a problem, determine “useful” information to find the solution, translate that information into an equation, solve that equation, and then propose a meaningful solution that answers the original question.

Unfortunately, in some math classes, these problems sound phony and unimportant. Who is really interested in finding the time it takes for a car to travel some distance when we know that a truck travels another distance 6 mph faster in the same time.

Remember, we live in a complex world. In order to teach meaningful techniques and strategies to solve problems, we have to start with simpler examples that have fewer variables. Getting ready to solve the types of problems we are interested in will require preparation. Perhaps those of us that prefer to work with “applied math” and practical applications need to be patient – we learn to walk before we run.

Think of a baby playing with blocks. The child grasps at them and moves them around the playpen. If we watch that baby, through trial and error, the child will succeed in moving them about, perhaps even stacking stack some so that they will not fall. What is the benefit of this play? Is the child learning?

Yes, as any parent can see. The child is learning eye-hand coordination, motor skills, and spatial reasoning. The blocks provide an opportunity for growth – in and of themselves, they represent little meaningful at all.

When we study math, the real skill we are learning is to think carefully. Often, this means learning to see things differently. Many times, a problem seems hard not because the solution is difficult; it just requires a different point of view.

Some scientists believe the human mind was not built for careful analysis; it was designed for survival in a dangerous world. Because we often “act and react,” the human mind can be tricked and can mislead us. We have all seen optical illusions that illustrate this. Even when we understand that we are looking at an optical illusion, our mind still “tricks” us.

Reasoning does not come naturally. It is something we build. It takes effort. Math is the only thing humans do where the outcome is an absolute truth. Either 6 does or does not equal 9. The truth becomes obviously clear.

Math underlies all sciences because the process of objective truth and reason form the basis of the scientific method. Knowing the difference between valid and invalid reasoning is the key to problem solving. It is also the heart of a democracy.

This is why we study math – its really not about the “x’s” and “y’s” we learn in algebra class. It is about preparing the mind to seek more important truths. The reasoning skills are more important than the theorems and definitions.

Math represents the intellectual power of some of the greatest minds in human history. Concepts taught today are hundreds, even thousands of years old. Those that came before us worked hard to understand and develop these skills. They have passed them along to us. Learning math may not always come easily, but then math is not natural to our environment. All things of value take effort to learn. When we learn math, we are nurturing and developing our minds.

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Math is Distinctly Different
by Bill Breitsprecher

Math is very different than other subjects in school. It is all about applying concepts, processes and ideas. It is not possible to “memorize” all the answers, we need to learn and use the steps to finding the answer. It takes practice.

In many ways, it is more like learning a musical instrument than it is like English, History, or Science classes. If we wanted to learn to play guitar, simply watching all or favorite bands would not do it – we need to practice. Like learning to play guitar, math is learned by doing. Talking notes in class, actively reading your text, and completing all homework is how we practice math.

Another way that a math class differs is that it is a linear learning process; it is sequential. Math courses and textbooks carefully present ideas in an order that builds on previous learning. The skills used in the beginning of the class form the foundation for learning new skills later. In a History class, I can learn about the early 1900’s without a full understanding of the 1600’s. In a math class, skipping a chapter will make later chapters difficult or impossible to master.

Don’t skip homework in a math class, even when it looks “easy.” In fact, that is the time to do the homework, when it looks fully manageable and understandable. Students that consistently do ALL homework will likely find that the class never becomes unmanageable. Students that keep fully up-to-date on assignments and that ask for help when it is need often have to do little studying before an examination because the understand the process and are ready to apply this understanding to the problems on the test.

In many ways, math is also like learning a new language. It will require a vocabulary of new words. Learning to speak fluently in another language takes study and practice. Just learning new words is not enough – they have to be used to create meaning. Math also requires a new “vocabulary” of terms and ideas. Like the words in a foreign language, he vocabulary in a math class means little without being able to apply them solve problems.

Math classes are designed to challenge students and “stretch” their understanding. Do not feel embarrassed if you need help or feel lost – that is common. For many of us, it is part of the learning process. Many students find that math classes are sometimes frustrating; it is not unusual to struggle with math. Do not give up – use this as a learning experience. For some of us, this is just part of the learning process. Ask for help, work with a student, or sometimes just take a short break and try to study when you feel “fresh” again.

Before class, be sure you have read any assigned reading. Even if you have not been assigned a reading, look over the next unit. Being prepared will make note taking easier; it will also allow you to ask better questions and follow your teacher.

Participate in class. Take good notes and ask questions. When asked, don’t be afraid to propose an answer and don’t worry if you are wrong. Seeing why an answer is incorrect can be a valuable learning experience. Usually, when you have a question or misunderstanding, someone else does too.

Do all homework and get help with it as soon as you need it. Typically, students that do not get help early fall farther behind. This will likely make math class a frustrating experience, certainly not something anyone needs. Do what you can to keep math as positive an experience as possible. Failing to keep up and asking for help will not be fun for anyone.

Above all, remember that math is a process. Saying I understand it is not enough, I have to be able to do it with confidence and accuracy. For some, the key will be to relax, especially when taking quizzes and tests. Students that find math stressful will benefit from learning ways to deal with stress.

One good way to learn confidence and accuracy is to carefully write out all problems in a neat, organized fashion. Write out all steps, even the ones that seem easy. Remember, we want to build confidence in our skills. Writing out things we feel confident with will surely help. When possible, check all your answers by substituting them back into the original equations. This will verify your work and build confidence in accuracy in the skills needed to perform the check.

Math is distinctly different than other subjects, but the skills you learn will help you in many direct and indirect ways. Different subjects demand different things from students. Accepting that math requires a unique set of skills allows us to work towards mastering mathematics. Along the way, we will also learn a great deal about ourselves and how we learn.

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Reading a Math Book
by Bill Breitsprecher

Do you like to read? Do you like to read math books? Getting the most out of a math textbook means reading it carefully – math books are very different than other types of books. They do no “flow” like a novel – typically, math books alternate between text and examples of math problems.

Do not “speed read” a math book – unlike other types of writing, authors of math books use as few words as possible. They do not want to “overshadow” their work – the mathematical examples and solutions represent the key concepts. The writing is meant to supplement the examples and solutions.

Read a math book slowly, carefully reading each word. Understand each part of each sentence. Technical writers believe that extra words and repetition distracts from meaning. Assume that each word is there for a reason. Read each sentence for understanding before going on to the next one. Expect to re-read or review previous sections.

Don’t get discouraged if something looks difficult. Most of us can remember any number of things that seemed hard before we finally “got it.” Think of other challenges you have overcome. Most of us have read a math book before and felt confused, but we worked through that confusion and mastered the math. There is great satisfaction in learning something mastering concepts that seemed complex.

A math book requires active thinking at all times – don’t read them when you are tired or not alert. A math book requires asking yourself about the author’s purpose. What were you suppose to learn from each paragraph? Remember, math books never have sentences that are just there to “fill-up” a page. Continually ask yourself what the important concepts in each sentence are.

Math books are never collections of unrelated facts. They are carefully crafted presentations that link ideas and processes. Try to think of how information is related and “chunk” the material so that learning is sequential – one step naturally leads to the next. Usually, when something looks overwhelming, it is because we did not undersand or we forgot something from a previous unit. Be ready to go back and review as needed.

Don’t try to memorize each unit – few of us could possibly remember each and every example in a math book. Math is a process – its about doing, you are learning tools and methods to solve problems. Read a math book with a pencil and paper in hand.  Few learn math by watching someone else do it, nor will we learn it by simply seeing someone else’s work.

Get involved while you read. Take notes about definitions, key concepts, or theorems. Try to state them in your own words. If you do not understand an idea, make yourself a note to ask someone about it later – assume that everything in a math book will be important.

Math books are full of examples – write them out too. Don’t just copy the author’s work. Use it as a guide to help you solve the same problem. A math book is meant to allow you to work out each line by yourself with the author. Write out each step, including any that the author chose to omit. If you can, try to finish the problem without looking at the example and check your solution with the book. If you need help solving the problem, be sure to carefully follow the example and not just rewrite it.

If you needed to closely follow the author’s work, work the problem through a second time, this time covering up the book and without looking at your guided practice. Don’t try to memorize the solution; try to learn the process of creating the solution. If you can work your way through a challenging problem without consulting the author’s example again, you are really learning!

Once you have developed an understanding, confidence, and accuracy with a given type of problem; look for similar problems. Math books illustrate the skills that will be needed to solve that section’s “homework.” Use the examples in the book to help you recognize the steps needed to solve related problems. Math books never expect students to “invent” new problem-solving techniques. They are written to review and reinforce the procedures of solving similar and apply understanding.

A math book is meant to teach 2 types of understanding. At first, students learn how to follow, reconstruct, and apply a given concept or procedure to a particular type of problem. Next, students “create connections” between new learning and previous learning so that a fuller understanding of the process results. We don’t memorize complex procedures like math; we build a working body of knowledge that allows us to go forward.

When reading a math book, remember the “say and do” principle: we remember only 10% of what we read, 20% of what we see, but 90% of what we say and do. Math is learned by practice – using a math book to “guide” that practice is a powerful tool.

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