Over the past decade or two, you may have heard some chatter about the declining levels of interest and achievement in mathematics among American students.

According to a survey about math relevance to U.S. middle school students, more than half of the students surveyed said they would rather eat broccoli than solve math problems. A whopping 44% would prefer to take out the trash.

Another assessment compared the math performance of students in developed countries across the world. The USA ranked a pitiful 36 out of 64 nations that participated.

Way too many students in America hate mathematics, but all U.S. citizens must study the subject regardless. Compulsory education laws require children to attend a public or state-accredited private school for a specified period. Mathematics is one subject that all students must learn and pass examinations about to graduate and progress in life.

As with any subject that we are forced to spend years of our lives learning, mastering, and performing on tests, it is natural for students and parents alike to start chiming in with, “can I use this math knowledge in my everyday life?” “…and, how?”

The short answer? Absolutely.

The longer answer? Infrequently, so perhaps we need to find better ways to justify teaching mathematics to all students. Read on to find out how. (And be sure to leave a reply with your thoughts!)

**What is Mathematics, and How is it Useful in Real Life?**

To answer the question, let’s first take a step back and consider what the whole project of mathematics is all about. Then we’ll compare this authentic experience of math to the standard math curriculum taught in American schools today.

According to W. W. Sawyer’s book, *Prelude to Mathematics*, “Mathematics is the classification and study of all possible patterns.” We can take the word ‘pattern’ in a broad sense to mean almost any kind of regularity that we can observe and recognize.

For Sawyer, “Life is only possible because there are certain regularities in the world. A bird recognizes the black and yellow bands of a wasp; man recognizes that the growth of a plant follows the sowing of a seed. In each case, the mind is aware of the pattern.”

Mathematicians view mathematics holistically, often describing the field as “the study of patterns.” In contrast, the average student or adult usually thinks of mathematics as a set of skills, concepts, and formulas to be remembered and performed within highly specific applications. Why do most people have such a warped view of mathematics, which causes them to see it as unpleasant, irrelevant, and futile?

Math is useful to you and me because it solves problems of everyday life. We use basic arithmetic every time we make a purchasing decision or perform a simple calculation. In early times, humans needed math for trading, managing supplies, distributing properties, describing the motion of the stars and planets, and building calendar systems to know when to plant and harvest crops and when to conduct religious activities.

Mathematics has been developed over the course of at least 4,000 years by humans and utilized because it solves many everyday problems. It is both a language and a toolset. It changes and evolves with our increasing human understanding, yet it deals with fundamental and eternal truths of space, time, motion, and matter. It has developed into the most significant cultural achievement in human history.

Over the millennia, mathematics has evolved to the point where it can now be used to solve many kinds of problems across all fields of human knowledge. It is not only useful for measurements and statistics. It also is needed mainly to formulate and investigate the laws of nature. Mathematical models provide us with valuable and vital information about climate change, economic trends and predictions, and the workings of the human body, to name a few.

It has played a significant role in many technical developments, more recently including space exploration, CD players, mobile phones, Internet technologies, and GPS systems for navigation.

Cryptography is a branch of the mathematics underlying the distributed public ledger technology known as the blockchain. The mathematics of secret codes enabled Satoshi Nakamoto to design the groundwork for the Bitcoin protocol, which he published in 2009 in his now famous white paper. New careers are cropping up right before our eyes as some of the top tech talents are pouring their energy into building systems to enable digital assets, secure communication, and tamper-resistant voting systems. None of these applications of blockchain technology would be possible without the mathematics of cryptography.

Higher mathematics such as linear algebra, geometry, statistics, calculus, and differential equations often deal with topics that are so particular that they have no use for the average person. At the same time, having a firm grasp of all these subjects is crucial for any professional scientist, engineer, or physicist.

**Plunge Towards Infinity in a Psychedelic Shore**

Theoretical mathematics deals with many interesting features of mathematics, some of which don’t have any useful purpose at all. Take for example the Mandelbrot set, a set of complex numbers for which the recursive function f* _{c}(z) = z^{2} + c* does not diverge when iterated from

*z = 0*.

The Mandelbrot set exists in the world of imaginary numbers, forever untouched, yet unseen in a visual format until 1978 when Robert W. Brooks and Peter Matelski published the first graphical representation of it. The first picture was rather low-resolution. It looked something like this:

Later in the 1980’s, mathematicians and computer scientists decided to run the Mandelbrot set algorithm through CPU’s and graph the results after thousands and millions of iterations. High-powered computers are capable of performing many more calculations than a human could do manually. They can also use the full-color spectrum of visible light to illustrate the rate at which the series diverges at each point on the graph.

In the four decades since 1978, computer scientists have re-run the algorithm to display graphs of the Mandelbrot set at extremely high resolutions, revealing precise fractal geometry and recursive self-similarity at many different levels of zoom. No human being designed these patterns, and no artist drew them. The images genuinely are embedded within the immutable properties of numbers.

If you have a few extra minutes to contemplate the grand universe of numbers and mathematical relationships, watch this video:

It’s hard for me to wrap my tiny little head around this infinitely complex structure. In what sense does the Mandelbrot set exist as a real object, anyway? Was it created or discovered when a human decided to graph it with a computer and did it exist before that point in time? This is truly mind-blowing stuff.

With all of the countless ways in which mathematics helps billions of people every day while offering us a chance to marvel at the nature of reality itself, it’s a puzzle why not everybody gets it. Why would somebody not want to learn math, a powerful tool that applies and relates to every other field of knowledge?

**Why Are There So Many Math Haters?**

Many students will tell you that they hate math because it’s pointless. They are not lying to you. They genuinely want to see the purpose, but they don’t. And who can blame them?

We’ve all had those experiences in a stuffy math classroom when the topic of the lecture seems like it couldn’t possibly be relevant to anything in real life, other than passing the exams to be done with school finally, so we don’t have ever to do it again! The result? Negative attitudes are widespread not only about math but also about our competency with regards to learning how to use math.

Make no mistake; I don’t think it’s correct to blame teachers for the problem. The disengagement in question usually happens in spite, not because of teachers’ best efforts. And the source of the problem isn’t even necessarily in the schools.

The majority of American adults dislike math and believe that they are incapable of doing mathematics at a high level. These adults become parents who, unconvinced of the importance of math, don’t instill positive attitudes about math in their children. The cycle continues across generations, resulting in declining interest and achievement in mathematics.

We desperately need to find ways to make math interesting, relevant, and worthwhile for today’s students. If we don’t, American youth will continue to lag behind their international peers in math competency to the detriment of our future medical, scientific, and technological advancement.

We need to break the cycle of seeing math as pointless and annoying and start connecting math with the next technological breakthroughs that are unfolding all around us at a rapid clip. Math is a powerful gateway to solving real problems in society. We need to figure out how to encourage everybody to want to learn it for personally meaningful reasons.

The real-world context of mathematical problems can be invaluable for somebody trying to appreciate math. If we vividly understand that there is a specific application that some individual human on earth uses to do their job, then we can understand the utility in solving an individual problem. If we think the question is relevant to our lives, then we naturally want to know how to answer it.

Explaining the specific real-world applications of each math skill might be an excellent way to get more people interested in learning math. Context is indispensable for demonstrating the relevance of particular math skills.

I’ve talked with teachers who are comfortable explaining to students why they need to learn the math skills and concepts they teach in class but struggle to explain precisely how they will use them in the future.

To demonstrate, when was the last time you used the Pythagorean theorem? You know, the equation that describes a fundamental relationship between the three sides of a right triangle.

*A*^{2} + B^{2} = C^{2}

^{2}+ B

^{2}= C

^{2}

It’s likely that you haven’t used it since you graduated high school.

If you needed to come up with a tangible example of the Pythagorean theorem in action, you could draw up some plausible situation in which a man is trying to climb a wall using a ladder. He knows how long the ladder is, and he knows how far away he is from the wall, but he needs to figure out the exact height of the wall. Pythagoras saves the day!

See, math is useful, I told you! You may need to measure the length of a brick wall, using a ladder, someday. You know, when you grow up. Like everyone does… right? Or maybe not.

**Okay, But I Still Don’t Really Care…**

Given enough time and creativity, there are usually ways to for a knowledgeable person to come up with plausible applications for any math concept. However, I see two problems long-term with this approach to persuading young people to study mathematics.

The first problem is that teachers, parents, tutors, and other adult figures may not always have an extensive enough “mental encyclopedia” of these colorful illustrations to be able to provide context for every last concept on the syllabus.

How can we possibly be prepared to explain how math can be applied if we ourselves haven’t applied even 10% of what we learned in school?

For example, what would you do if your kid came home from school complaining about the pointlessness of math class, asking when they would ever need to factor a trinomial or find a vertical asymptote on a rational function graph?

Good luck with answering that one.

If the assumption is that we’re trying to convince students that everything they learn is going to be applied in their real life somewhere, it becomes a losing battle. We don’t want to fight that battle. We run out of ammo before too long, and we ultimately lose.

The second problem is that students may not honestly care about the real-world application even if they are fully aware of it. They don’t even want to know when they would use imaginary numbers in real life (or that complex numbers are needed for AC circuit analysis in physics and electrical engineering).

They ask the question because of a different motivation – usually, a frustration with mathematics. “What else am I taking away from this?” That is what millions of students want to understand.

The majority of people dismiss math as pointless and futile. This pattern is not surprising. It’s part of a little defense mechanism we all use to justify to ourselves why we don’t need to put in the effort to learn something complicated. It takes hard work to practice and master mathematics.

Unless you care deeply enough about your report card and will do whatever is necessary to earn those precious A’s and B’s, you may not have a sufficient reason to put in the work required to do well in class. Sadly, even those students who earn A’s and B’s rarely crack open any math books later in life.

Sometimes even knowing the useful purpose of math concepts is not enough to inspire interest in learning them. Students often have a general idea of which careers they would have zero interest in pursuing in their future, making it easy for them to rule out particular reasons why they should learn what is being taught in class that day.

We might do a disservice to students when we tell them they will use what they learn in math class every day. The truth is that they probably will not. You or I might use these things depending on which career we go into, but then again, we may not ever use them after graduation if we go into many successful careers.

**Everybody Needs Arithmetic (and maybe Pre-Algebra, too)**

The way the curriculum misrepresents math in school has caused people to hate mathematics and vow to never go anywhere near it after they graduate. Something needs to change so that people can rediscover the joy and happiness that mathematical understanding can bring!

Somehow, the math that millions of American students experience in school is an excruciatingly dry, annoying, irrelevant, mere shadow of the subject that has little to do with real life or professional work or even the mathematics in which mathematicians engage.

Perhaps math class would be more engaging if it dealt with topics that are important to the everyday adult, such as money management. Handling money is an essential part of adult life, and you wouldn’t be able to manage your spending budget, credit cards, loans, savings, or investments without a solid background in arithmetic!

One example of a mathematical tool for managing your money is known as the “Rule of 72.”

The Rule of 72 is a real-world application of the compound interest formula that uses arithmetic to determine how long it will take for an investment to double. It simplifies complicated math and helps you to approximate the answer in your head.

This is just one example of many math skills that a typical person could appreciate as useful in their own lives. Because there are plenty of ways that you and I use elementary and middle school math in our lives, it’s easier to justify mastering arithmetic and pre-algebra skills and concepts.

The difficulty is how to defend why you or I would need to learn advanced math.

A philosopher and mathematician named Reuben Hersh wrote a book called *What is Mathematics, Really?* in which he explores the true nature of mathematics. Hersh makes an important point: math pedagogy often misrepresents “real” mathematics. People end up disliking math because they feel that it lacks something essential.

Maybe you are wondering why you ever had to learn trigonometry or pre-calculus when you never planned to go into specific technical fields that would require that background knowledge. The math learned in grades K-8 is vital for most people’s daily lives, but after Algebra 1 (usually studied in the 9^{th} grade) the everyday context of the actual math content starts to become less and less apparent. As the grade levels progress into high school and beyond, parents often don’t know how to tell their kids exactly why the math they’re learning is important and relevant.

Everyone can agree that arithmetic is essential. But I’m guessing it takes a more involved approach to inspire people to study algebra and beyond rigorously. Maybe students shouldn’t be required to enroll in so many years of math classes, but should still have the option, since the skills are potentially useful to some people but not the majority of people. Most people could probably get by in their lives with only a 5th grade level of mastery. They’d know how to add, subtract, multiply, divide, and use fractions.

Some pre-algebra skills up to the 8th grade involve topics like ratios and proportions, basic geometry, rational numbers, graphs and modeling, equations and variables, basic statistics and probability. These concepts are helpful for almost everyone and especially those with an interest in higher education. However, why is it necessary for everybody to learn algebra, geometry, or calculus?** **

**But What About Algebra and Beyond? **

Why should I? Why do I have to? What’s the benefit of learning and acquiring competence in mathematics? Many students, parents, teachers, and other adults alike have this question.

Are there ways to find intrinsic or intangible value in learning and practicing mathematics? Perhaps these values could include acquiring social-emotional development (a.k.a. Soft Skills), abstract reasoning and problem-solving abilities, perseverance in the face of difficulty, precise thinking, clear verbal expression, or even fun and enjoyment as in recreational Sudoku puzzles or geometrically inspired artwork.

Math in grades K-8 is a natural and essential part of everyone’s general education. A strong background in these fundamental skills and concepts sets people up for success when entering any other field containing any math (pretty much all of them). Maybe we need to think differently about how to justify learning K-8 math vs. high school, college, and advanced math.

Regardless of your life background, I would love to hear your thoughts in the comments section below!

Feel free to respond however you’d like, or answer one of the following:

- Did you ever hate math in school? How do you feel about it now?
- How have you found mathematics to be useful in your everyday life?
- What are some good “intrinsic” reasons to study mathematics?
- How would you encourage students that math is worth the effort?
- What are some math skills that you would like to improve?

Jeb SturmerThink about your length. If you want people to read your blog I’ll give you 700-800 words. This is too long. The graphics are very nice.

DawsonThanks for the tip, Jeb! I’ll aim for a shorter post next time 🙂

KelWow! Great first blog post and a great first subject! When you started mentioning “patterns” the philosopher in me immediately went back to the Pythagoreans. To them EVERYTHING was numbers. All of reality itself. Physics clearly shows this and also points to the likelihood of an intelligent mind behind all of nature. “The heavens declare the glory of God.”

As a Myers-Briggs INTJ I use pattern recognition every day too. I even use it to guide people into better ways to configure their love life at my website Relationship Options (www.relationshipoptions.ca – I hope the not so subtle plug is allowed? 😉

Lastly I thought about how language and logic themselves are all numbers and how every statement we say can be reduced to a mathematical formula.

Everything really IS math! Very cool!

Thanks for the great start to a really great blog! 🙂

DawsonGreat insight! Yes, the Pythagoreans were a committed bunch of mathematicians.

I am intrigued by how Pythagoras had a devoted inner circle of followers called the mathematikoi who lived ascetic lifestyles in order to study and practice mathematics with him so they could understand reality.

Thank you, Kel.

W Brad SwiftHi Dawson,

I have some good news! Your blog post is too long. Why is this good news? Because you can break such a post into multiple posts over a longer period of time which will make it easier for you to maintain the blog while also helping your readers to stay with the subject and retain more.

With that being said, I found several points you explored of interest. It seems to me that educators need to better differentiate the various levels of math and not group it all under one four-letter word. Also, after basic arithmetic is introduced, why not invest time into the practical use of those principles like how to balance a checkbook, how to use math to plan a budget, the ins, and outs of paying bills, etc.

BTW, the video was awesome!

DawsonHey Brad, thanks for the suggestion! I’ll try to make the weekly posts shorter and more sustainable in the future.

I’m glad you liked the video and I agree that it’d be helpful if people could know what the different levels of math are.

MeredithGreat post!!! The earlier comment noting that even language is mathematical rings especially true to me– as a student of both Linguistics and Computer Science, I work with the intersection of language and math every day! It’s amazing how many applications there are of advanced math, even in fields that you wouldn’t necessarily expect.

My view is more pragmatic– if kids don’t get exposure to advanced curriculum in each subject in high school, how will they be absolutely sure of what they love and want to study later in life? My TRUE love for math came in advanced Calculus my senior year of high school, my love for science came in 10th grade Chemistry, and in 8th grade, I thought I wanted to study International Relations. My curiosities fluctuated greatly throughout high school, and I’m thankful for the advanced classes in ALL subjects (government, physics, history, math) that shaped my educational interests and world view today.

If high schoolers got the option to never take a math class again (or any other subject) after 8th grade, I think many students would miss out on exploring their passions fully. Let’s keep exposing students to advanced curriculum in each subject, including math, to prevent missed opportunities for them to become well-rounded and duly self-aware of their greatest strengths and interests.

DawsonHow is math-as-a-language different/similar to English, Spanish, Mandarin, French, etc.? As a linguist would you say math is a universal language?

That’s a good point about getting broad exposure to the advanced courses! Without any experience, there’s no way someone can say for sure what they’d like to continue studying in more depth.

Maybe it’s a good idea to take as many courses as possible just to explore those interests, which might develop into long-term passions.

Ben BrissetteHey Dawson,

I really enjoyed this blog post!

I found myself asking the same questions that you posed at the end of your post: how do we convince students that this study of mathematics is worthwhile, especially considering we ourselves probably don’t have the requisite encyclopedic knowledge to find an interesting application for every topic? I was reminded of something I was told/learned only partway through my undergraduate education: what we learned there, we were learning not for the subject matter, really, even if it was in our “chosen field of study” but in order to “learn how to learn.” Perhaps this is relevant, since although interests can still fluctuate drastically in college (as a previous commenter mentioned regarding high school), many people are already decided regarding their future career paths and feel that unrelated courses (especially in liberal arts institutions) are a “waste of their time.” Do you think this argument of “learning how to learn” is a helpful one for students studying mathematics beyond the eminently practical levels you described?

I apologize for my comment getting a little long!

DawsonThanks, Ben! I’m glad we resonated with some of the same questions.

Yeah, it’s interesting that you bring up the argument that there’s value to learning anything because it teaches you “how to learn.” I would agree that a good learning experience can establish strong habits when it comes to learning. For me, taking Computer Science 201: Data Structures and Algorithms was one of my favorite experiences in college. It taught me so much about how to think about problems and, at the time, it was one of the most challenging learning experiences I’d had. I loved it!

I’m a proponent of taking courses unrelated to your career path if only to get a new perspective on your chosen field 🙂

MichaelGreat post Dawson — looking forward to the next one!

DawsonThanks, Michael!