One of the hardest questions for many math teachers to answer in a way that is relevant to students is: “why do I need to know this?” “For the next course you take”, the easiest answer in many cases, does not answer the question that was usually being asked. My answers to this question obviously depend on the topic being studied at moment, and I don’t have “good” answers for all topics… but here is my list of key life skills I learned directly or indirectly from math class, with some examples of situations where I find them indispensable.
Sums and differences
How much will all three of these items cost? How much more would I have to spend to get that one instead of this one?
Integer products and quotients
How much would three of this one item cost? Which is cheaper per unit: the 10oz or the 16oz size (when cost per unit is not displayed)? If our four person band will receive $160 for this gig, how much will my share be?
Decimal products (percentages, multiplication by reciprocals)
What dollar difference will a 3% raise in my weekly paycheck represent? How much am I saving if this item is discounted by 20%? How much should I tip the server at the restaurant?
Mental Math Skills
The above calculations usually arise at times when, or places where, I do not wish to take the time to pull out a calculating device, or let the world know I cannot figure things out without help.
Facility with the rules of algebra allows me to re-arrange problems to make them easier to solve… particularly when I am trying to work them without a calculator or something to write on. 17 x 12 is much easier to calculate in my head if I think of it as
(17)(10 + 2) = (17)(10) + (17)(2) = 170 + 34 = 204
You could also break it up this way:
(10 + 7)(12) = (10)(12) + (7)(12) = 120 + 84 = 204
Algebra has also taught me to be comfortable tackling problems that will take many steps to solve, by first breaking the problem down into smaller tasks or goals, then solving each in turn (in the same way I might write a 20 page research paper). Or, if that approach does not work, to try working backwards from the desired solution… or perhaps even starting “in the middle”, and working from there to both the start and the end. These problem solving approaches are useful in many walks of life, even non-quantitative ones.
Word problems often present information in a less structured, more challenging to wrap your thoughts around way. I have to think a bit to figure out what information is relevant to the question being asked, along with how best to use it. They force me to determine what mathematical tools might be relevant to the situation, more so than problems which are already expressed either entirely in numerical form or as equations. In other words, they help me learn to apply what I know about math to the “messier” problems I might encounter in the world around me.
Deductive reasoning is very broadly useful (ask any lawyer), and influences all of my attempts at communication greatly. It was easiest for me to grasp as a concept in the context of developing geometric proofs, which provide a visual aid to the deductive part of the reasoning. “Problem solving” involves tackling problems that are new to me, which I have the tools to solve, but for which I do not know “where to start” from prior experience. Almost every student struggles with this process as they learn it, and I see geometric proofs as an easier place to learn this challenging and valuable skill than others.
A bank balance earning compound interest is an exponential function. If you wish to understand the key concept behind accumulating enough savings over your lifetime to retire in some degree of comfort, you must understand how to work with exponential functions. If you are interested in working with biology, ecology, or economics – all fields that can involve populations or systems that are growing or decaying exponentially, this is an important topic to have mastered. To work with or solve exponential functions you need algebra, the laws of exponents, and inverse functions including logarithms (which are also needed to understand the Richter scale for measuring earthquakes, or decibel levels used in measuring sound volume).
Sequences and Series
The time value of money is just as critical a concept in managing personal finances as it is in corporate finance. Series are needed to calculate the effect of inflation on your savings and investments over many years, or to decide which investment (in your education, a solar system, a car) will provide you with the greatest return over the years of its life, etc.
Descriptive and Predictive Statistics
Many news reports and advertisements use statistics, but seldom provide enough information for the viewer or reader to make their own evaluation of the data. A strong background in descriptive statistics helps you to understand how easily incomplete statistical information, or poorly designed polls, can mislead at election time, in advertisements, or in making organizational decisions. Recent research results on brain development also seem to lean towards describing our brain as a marvelous statistical engine that allows us to make reasonable inferences in situations that have a history of varied results.
Probability and Expected Values
If you are ever tempted to play the lottery or gamble in hopes of winning more money than you lose (as opposed to purely for entertainment purposes), probability and expected values are something that will benefit you greatly. Investors seek to achieve a good return based on the amount of risk taken. How do you decide what level of return is needed to compensate you for the level risk taken?
While I seldom resort to the procedures and techniques taught in calculus in my daily activities, the concept that a problem can be broken down into a series of pieces which sum to the whole is a critical principle in many walks of life. Every “impossibly big” project I have completed has relied heavily on this approach. Calculus concepts helped me immeasurably when studying college Physics, and therefore in understanding many of the principles people have discovered that govern our universe. In particular, some topics in electricity and magnetism did not make sense to me as explained by the professor, but the integral describing the situation made things clear. Continuous probability distributions can certainly be studied without calculus, but understanding what you are looking at is easier when you are familiar with integral calculus. Have you ever wondered how Boeing can ensure that the fuel tanks in the wing of a new plane will be able to carry enough fuel to provide the plane with the desired range (without building a prototype first)? How can you possibly calculate the exact interior volume of a complex shape like an airplane wing section? Calculus! But before you can really tackle the processes of integral calculus, you need to first master differential calculus.
Being able to easily determine the rate of change of a function, either on an average or instantaneous basis, makes it easier to solve a number of science, economics, and even practical every-day problems (like the remaining cooking time needed for something on the grill based on several internal temperatures you have measured). An understanding of differential calculus often changes the way you think and communicate about quantitative topics that involve rates of change, particularly if you are attempting to model real-world situations. Rates of change are a central part of our daily lives: How fast is that car going? Will I have enough gas to get to my destination? How much longer will the turkey or roast take to cook fully? How much longer before the snow is gone? Rates of changes should receive greater focus in math courses throughout high school.
Using scrap paper to do work on topics not yet mastered, and copying work over before handing it in. It is amazing how much can be learned from copying work over:
– checking for mistakes
– correcting post-mistake work efficiently
– finding a new way to approach a problem
– finding a better or more effective way to present your answer
Keeping course materials organized in a way that makes them easy to review regularly, and find material when needed. Organization skills and habits of mind can help improve both your efficiency and your results on most any task or assignment.
Self Awareness and Reflection
How do you feel about a task as you complete it? Are you certain you are doing your best work? Are you certain this will meet expectations? Are you working with confidence and speed, or do doubts and hesitation creep in from time to time? Pay attention to these feelings, and let them guide where you spend your time.
Doubt, hesitation, or anxiety are often alarm bells telling you that you do not understand something well enough yet… therefore you need to do something about it, now!
What helps you do you best work? Does the setting, or the time of day, or a looming deadline, or the presence (or absence) of friends matter? Think about this, and plan your work to use such insights to your advantage.
Things often seem easy to do when you watch someone else do them (such as when a teacher is presenting at the board). Ideas often sound straightforward when you hear someone else talk about them. Yet most people don’t really understand something well until they have done it completely on their own (no help from friends, no looking at notes)… preferably after some time away from it. If you really want to ensure you understand something, teach it to someone else who is having difficulty with it!
When working in groups, are you helping to move the group forward, or are you along for the ride? You can’t fool the other members of the group for long. Always strive to be both constructive and productive. Are you working for the group’s benefit as much as your own? Both are important. The more skills you master, the greater your value to a group. The better the group works together, the greater its value to all.
Individual achievement is marvelous, fun, wonderful, exciting, and great for self-esteem. Group achievements are different, and can be even better. There is something magical about meeting challenges as a group, particularly challenges that none of of the group believes they could have met individually. This is just as true in Music as it is in Engineering, Business, or a class project.
Communicating My Thought Process
Most math teachers will require their students to “show their work” in order to receive full credit. This is no different than an English teacher requiring a conclusion to be justified by preceding persuasive paragraphs. Writing an “essay” that consisted only of a concluding sentence, even if it is a very good conclusion, would not result in a passing grade. Neither will most work in math and science if all you do is show the result. We always, always need to either convince the reader that our result makes sense, and/or help the reader verify that we have not made any mistakes. In English, we use words to do so; in Math we use both words and the more concise notations developed in each Math course.
The more challenging a task, the more creative the solution approaches need to be. Can a problem be made easier to understand by summarizing or doodling? Can it be broken down into smaller pieces that are easier to tackle? Can you start at the end and work backwards? Can you start somewhere in the middle, then work from there to both the beginning and the end? Can the task be described or illustrated in a different way, one that might bring completely different approaches to mind?
Try, Try Again
If an approach does not seem to work, try a different one. If no other approaches come to mind, do something else for a while then try again. Are you sure there are no other approaches you could use with this problem? What if I re-drew my diagram? Could I have taken an alternative approach at any step of my earlier work? If a first attempt at a task leaves you feeling as though what you produced is not quite good enough, re-do it. You usually do not have to start from the absolute beginning, as your presentation of your ideas or work is often the culprit. Find someone to help review or edit your work. Rethinking your work can help find errors, but can also help build confidence and comfort with concepts. Re-reading and re-editing will often greatly improve the presentation of ideas.
Sometimes, Things Will Get Worse Before They Begin To Get Better
A problem may look nice and concise as stated, but the process of solving it can sometimes turn it into a page full of terms, parentheses, fractions… a mess. But then you start to spot some like terms which can be combined, or common factors that cancel out, a substitution that reduces the number of terms further, and from there it can be all downhill to the solution. Or it might get worse again. Having the patience to take such events in stride, and the perseverance to continue looking for a path to your goal, are invaluable in many situations.
Great Satisfaction Often Arises From Initial Frustration
Many, if not most, new topics are likely to generate anxiety and frustration long before you experience success, mastery, pride, and satisfaction. Get used to it, both in school and in life. Use your recollections of successfully overcoming past frustrations to help you persevere with the ones you face now. It is normal to get frustrated – it happens to everyone, including those who appear to master things with ease. Frustration, followed by work that leads to mastery, can help you remember things better. Think of how many times you have been frustrated in the past, then mastered the topic that was frustrating you. How great a feeling was that? You will probably remember those experiences and the related ideas better than most others. This is part of the reason you will often hear people say “That was a really hard course, but it was also one of the BEST I have taken!”.