Guest Post: Spooky Math

Today’s guest post comes from Madison Hodson, an instructor at AcerPlacer who studies mathematics at Weber State University.

What do you get when you divide the circumference of a pumpkin by it’s diameter..? Pumpkin Pie! It’s Maddy here, and since it’s Halloween, I thought I would open up this little article with a Halloween math joke – I’ll try to keep the theme of connecting Halloween and math going throughout the article. Thanks to the internet, I was able to do a little research and find out some weird, creepy, and spooky facts about some numbers!

Vampire Numbers

A number v = xy with an even number of n digits formed by multiplying a pair of  n/2 digit numbers (where the digits are taken from the original number in any order) x and y together. If v is a vampire number then x and y are called its “fangs.”

• 21 × 60 = 1260
• 41 × 35 = 1435
• 15 × 93 = 1395
• 30 × 51 = 1530

 

Tombstone

Also known as the halmos symbol, the tombstone ▮ indicates the end of a proof.

 

Napier’s Bones

An abacus created by John Napier used to calculate the product and quotients of numbers.

By The Article Genius – Own work, CC BY-SA 4.0, Link

 

Devil’s Staircase

The Devil’s Staircase, also known as the Cantor Function, is an example of a function that is continuous, but not absolutely continuous.

Source Link

 

Witch of Agnesi

A curve studied by Maria Agnesi in 1748 in her book Instituzioni analitiche ad uo della gioventù italiana (the first surviving mathematical work written by a woman). The Cartesian equation is 

Source Link

Guest Post: Why Some People Actually Enjoy Math and How You Can Too

Jaden Steele is a recent recruit to the AcerPlacer crew. He is studying education at Weber State University.

Throughout my life, I’ve always been intrigued by mathematics. This has caused people to question my normality. How could I enjoy something that basically everyone else hated? To answer this, we must consider what many people enjoy, and question what makes them enjoy those things. I will list some reasons that I believe people find pleasure in certain things using examples of enjoyable activities. Then I will use those same examples comparing them to why I, and many other mathematicians, enjoy math.

1. People like feeling that they are good at something.

Many people are good at a lot of things, including sports, playing music, or any number of unique talents. Because they are good at those things, they tend to enjoy doing them. It makes them feel good knowing that they have talent, and it builds their confidence and self worth. Most likely, they will continue to build on those talents for the rest of their lives.

One of the most famous researchers of the learning process is Edward Thorndike. To paraphrase, he states that any behavior that is followed by positive reinforcement will increase in likelihood. On the flip side, any behavior that is followed by some kind of punishment will decrease is likelihood. This basically means that when we do a good job at something, and someone lets us know that, we feel good about ourselves and continue to perform exceptionally well. Also, if we don’t do well at something, and others make us feel like we are poor at it, we’ll likely never want to try that thing again.

For me, my talent was always math. From the time I was two years old, I learned to count, and I would literally count myself to sleep every night. I could count as high as any normal adult could count and would normally fall asleep by the time I got into the three or four hundreds. My parents noticed my weird and unique talent and they praised me for it. They helped me to have a higher understanding of numbers that lead to me develop a love for math. As I went through school and brought home tests and quizzes all with a 100% and a smiley face drawn on the top of the page, my parents always let me know just how proud they were of me. Their encouragement only lead me to thrive in all things math.

Unfortunately, not everyone was born a math wizard, and even worse, most people believe they are terrible at math. This belief not only comes from impatient parents, teachers, and tutors who make them feel inferior, but people telling themselves that they are horrible at math, when in reality they’re not. This punishment causes people to not find any sort of joy while doing math. If you wish to start enjoying math, the first step is to stop believing that you have no math skills. As your skills improve, math will become much more enjoyable.

 

2. People like what other people like.

A word that I always use as a joke with my family is “sheeple.” We’ll use that word when someone does something just because they saw somebody else doing it. They act like sheep, following the actions of the shepherd. This isn’t a bad thing, it’s just how most people’s brains work. Before they do something, they like to feel that others are also doing what they are doing. It feels good to fit in, especially if what you’re doing is what the “cool kids” are doing.

Many people develop interests from the interests of other people. If one of your role models listens to a certain type of music, then you might decide that you like that music too. If your dad likes a certain sports team, you’ll probably end up liking that team as well. We like what other people also like. This principle has been coined as “joining the bandwagon.”

Math isn’t something that people generally like. It definitely doesn’t make you cool to like math. In fact, it is cool to be bad at math, which is why some people pretend like they can’t do math. The world has made fun of people who like math. They call mathematicians nerds, and people don’t want to be associated with math geeks. If you wish to enjoy math, you’re going to have to care less about being viewed as “cool” and about being part of the bandwagon.

 

3. People like what they believe to be important, good, or useful.

People value things in life very differently. What people value high, they also enjoy. Almost everyone cares deeply about their family, and they enjoy being with them. A great example that shows how people enjoy more valuable things would be a dinner date. If a couple went to a super nice restaurant with a fancy environment and expensive food, they would enjoy it much more than going to Burger King and ordering cheap, two dollar burgers.

Sometimes people can be misled into believing that one thing is more valuable than something nearly identical to it. The price of an object can sometimes make people believe that the more expensive thing must be more valuable – like soda, for example. A two liter thing of Coke from a plastic bottle is the same exact product as a smaller coke from a glass bottle, but the glass bottle is more expensive, and you look cooler drinking from a glass bottle, so people consider it more valuable.

Math teachers all over have always heard students ask, “When am I ever going to use this?” Although they may be right that they probably won’t use math on a daily basis, this attitude really causes the student to dislike math. The reality is that they probably won’t ever need to remember historical facts, or the chemical makeup of molecules, or how to construct Shakespearean literature on the daily either, but you hardly hear students complain about the importance of those subjects.

If you wish to begin enjoying math, stop telling yourself that it isn’t important, and maybe start thinking of ways that math skills can help your life. Practicing problem solving will help your brain to figure out ways to solve real world problems that you’ll encounter later in life.

 

4. People like what they understand.

Perhaps the biggest factor on why people enjoy or don’t enjoy something is the level of understanding they have of that particular thing. This is true in sports, music, art, or any aspect of culture. An athlete might look at someone who enjoys comics and question why in the world they enjoy such a weird thing, and the comic nerd might ask the same question about the athlete. They understand what they enjoy, which is why they enjoy it so much. They don’t however understand other people’s interests.

The more understanding I gain of mathematics, the more I enjoy doing it. I promise as you begin to understand math, you’ll start to enjoy it as well.

 

These points will not only help you enjoy math, but they can help you to enjoy any new thing that you didn’t previously enjoy. Try to understand the topic, find reasons why it is important or good, this will help you relate to others who enjoy that topic, and finally, once you’ve mastered that new topic, you’ll feel really good about it.

Guest Post: Math With Purpose

Chris Allen is a new addition to the AcerPlacer family. He is working towards a degree in engineering at Weber State University.

Often times the study of math is derided as a secondary subject, essentially a means to another subject such as physics or engineering. We sometimes see our math classes as merely an object to overcome to get to another goal. It’s somewhat rare for a student to stop and really consider, “Why even all the fuss about these?” In an age that we can simply take 15 seconds and look up a formula for anything we could ever want then magically we get an output that’s presumably correct, why bother with learning these archaic methods that can be thousands of years old (read: out of date)? Sure, the people who program the black boxes that feed us the right answers should/need to know this stuff, but I don’t.

The answer to that lies in part simply to build an intuition for when the black box might be feeding us a helping of bovine excrement. Take the story of an engineer who was running analysis on a trailer. This was a completely enclosed trailer and no part stuck outside of the trailer’s base. After putting a model of the trailer into the computer, it gave the center of mass about 3 feet outside of the trailer, which is physically impossible. However, this engineer trusted the computer all the way to the next project meeting, much to the engineer’s embarrassment.

Building this intuition for how the math outputs should look like is only part of what we learn as we learn different methods and approaches in math. The real gold in a strong mathematical education is not expecting 2+2=4, but by teaching us a diverse way of thinking about the world around us. How we can use a few basic rules and a bit of thinking out side of the box to solve nearly any puzzle. It’s a field that, despite appearances, creativity in mathematics is the most rewarded attribute.

We sometimes are told it’s important to learn math because it teaches us how to think, or some might say that that it teaches us different ways to think. But looking at the vast diversity in the mathematical fields of thought, I think the real reason why we should all study mathematics is that it teaches and reminds us that we can think.

Guest Post: The Most Common Math Mistakes and How to Fix Them

Jodie Larsen has a BS in applied mathematics from BYU-Idaho with a minor in biology. In addition to teaching at AcerPlacer, she tutors students on several topics in math.

After many years of teaching / tutoring, I have helped and observed an enormous amount of students and have come to recognize the types of mistakes students most often make which has led me to realize what the most common ones are, and it’s not generally as simple as not knowing how to do a problem. Here are the most common mistakes I’ve observed and some suggested remedies.

1. Trying to memorize versus really understanding

I feel this is the most critical problem to correct in order to be accurate in all things math. Math isn’t a subject of memorization. It won’t be mastered simply by knowing verbiage and formulas. Many people have eidetic memories – a trait I always wish I possessed – however, that isn’t enough. Math is beautiful in so many ways, but in large part (in my opinion) because concepts can be combined in new, and endless, ways.

The key to fixing this problem is to ASK QUESTIONS. Students need to be voracious in wanting to know the WHYS in everything they are doing and in understanding the whys, students can then apply them in new situations with much more ease and accuracy. In order for this to be effective, though, the students need good teachers who are willing to teach them the reasons and meanings behind what is being done rather than just talking at them.

2. Not understanding the fundamentals before moving on

For the same reasons you shouldn’t build a house on sand, a person should make sure to learn the fundamental rules (learn and UNDERSTAND them) before trying to use said rules in more complicated applications. If students struggle with exponents, for example, but then try to do intense factoring, the factoring will be much more difficult. This mistake goes along with the mistake listed above because when learning the processes and rules which will continue to be used and applicable, a student must make sure they have a firm hold on the rules and will be able to apply them whenever needed.

The remedy to this problem is to be very vocal with your instructor / tutor and let them know that you’d like more practice and clarification until you feel you’ve mastered each rule or concept.

3. Poor handwriting and disorganization

This one seems perhaps a bit obvious, but it’s actually amazing how often this can throw people off and will cause major frustration when, for example, x’s look like y’s and 5’s look like s’s. Students will often say such things as, ‘I could get it right if I could read my own writing!’ and though they know it and can laugh about it, they don’t often work on it. This may be the hardest mistake to fix, but it just takes care and really slowing down.

Along this same topic, students sometimes aren’t properly taught how to organize their work, or they may just not take heed when they are. I often see cramming in tight spaces, overwriting to avoid rewriting, etc., and thus it becomes a sort of ‘Where’s Waldo?’ situation whenever I need to help students error-check a problem.

One of my biggest recommendations is to get a notebook of graph paper to do work in. There are built-in columns which make it very visually easy to place one number in each box and have everything nicely spaced out. As an added bonus, this can drastically help students who may suffer any form of dyscalculia – I witnessed this with a student years ago and it helped her (and her daughter) tremendously.

4. Using pen

I know, I know, some students are die-hard pen enthusiasts and though I understand the reasons why, it often really hinders them in math. If any mistakes are made, the student often tends to scribble them out or write over them instead of just doing the problem over.

Mistakes are made much more often with these types of actions and so I always recommend working in pencil. If students don’t like the scratching sound of a pencil, then I recommend a pen such as a FriXion pen which ‘erases’ with friction instead of an eraser (we all know how well those don’t work!) which can help avoid the mess.

5. Skipping steps (head math)

Though head math can be impressive, it’s also very error prone. I understand that students want to go as quickly as possible in order to get homework done sooner, but if mistakes are made, then the problem needs to be redone anyway, rendering the step-skipping rather useless.

I firmly believe that writing more steps out yields more correct answers and higher retention overall. A phrase I often use is, “When in doubt… write it out!” If you are working on a problem and you get it wrong, I will recommend you restart the problem and SHOW ME the steps while talking them out. Oftentimes, you will find your own mistake, and that’s empowering!

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If I can summarize in one sentence, it would be this: Always ask WHY and practice extra material until concepts are mastered, use a pencil to write out ALL steps neatly, and graph paper is your friend. It’s as simple as that and will greatly increase your chance of success in math (and life) overall!

Guest Post: Strength in Numbers

Ryan Brown is an instructor at AcerPlacer and is working on a BS in mathematics with a minor in secondary education. He is also an avid rock climber.

After teaching various types of mathematics, I have been asked countless times by students, “ When are we ever going to use this?” This question is asked when trying to isolate specific variables, simplify rational expressions, or evaluate complex numbers. Some answers that are given are very arduous and stretching quite far. Some ideas and concepts may not have a proper answer to the student, but in the complex world that we live in all that we have was designed using mathematics. Everything from the car we drive to the cell phone in our pockets – someone performed innumerable amount of math equations to see the safest design of a car, or perform tests to see how to make the battery in our phones last longer. All around us is magnificent architecture that was not just imagined and then someone starting digging the hole. There was intense preparation that went into each structure before we ever see the physical building and foundation being built.

Taking another stance on the exercise that mathematics provides the human brain is extremely powerful! Learning mathematics helps your learning skills on many levels. It helps you learn deeply and focus on tasks at hand. It helps to keep you organized and learn complex concepts without giving up easily. The human brain’s frontal lobe is not done developing until about age 25; the more we learn by that time, the more we can retain and remember. Mathematics is one of the best ways to train our brain how to remember the most information in the shortest amount of time. Our brains retain about 90% of the information that is input when we teach someone or try ourselves to perform a given task. Why mathematics helps us learn this form of learning so quickly is because we are given instant results and answers back in math. The second we make a mistake, we often cannot proceed with the equation, or our answer is not provided as an option. We don’t like to make mistakes; that is why we often take the easy way out and try to do a form of studying where we cannot make mistakes, like reading and listening to audios, which has a 5% retention for our brain.

To finish up the point of this article is that we use math everyday. We see the benefits of math everywhere. Our brain needs complex problem solving to stay young and active. We can mimic the study plan with math with all other courses to help us save time and energy. I hope this article was worth your time! Thank you for reading!

Guest Post: Why Study Math?

Jodie Larsen has a BS in applied mathematics from BYU-Idaho with a minor in biology. In addition to teaching at AcerPlacer, she tutors students on several topics in math.

“Why did you decide to study math?,” and, “Were you always good at math?,” are two questions which we, as instructors, hear on a near-daily basis, and they may seem to have simple answers but I, when I really think about it, realize that that is not the case.  I will use myself as an example on how to answer those questions.

I would not consider myself one of those Good Will Hunting type of people where I come up with complex equations and proofs off the top of my head which I write on windows to confuse people.  Though I find those types of things completely fascinating… I am more of a learner and user of mathematics rather than a discoverer and prover. Granted, I learn it more quickly than many, but my brain just likes puzzles, and that’s what I consider math to be.  Giant, elegant, beautiful puzzles.

Why did I personally decide to study math? Many people can’t fathom that I would want to focus my attention there.  There are many answers to that question, actually. I would say the biggest reason is that I had incredible teachers throughout my years who taught me well and, therefore, inspired me to love the subject and want to do more of it.  When you have great instructors, the subject matter will be more enjoyable no matter what you are studying. I know many students say they hate math and when I dig into why they do, oftentimes it comes down to the fact that they had poor teachers or were pushed through a flawed system for various reasons.  Having good instructors is paramount, I feel, and therefore it makes me want to be the best instructor I can be. I want to instill a love for math in as many as I can. It hurts me a little every time someone says, “I hate math.”. It shouldn’t be that way and I try, every day, to change that mindset for students and when the students’ mindsets start to change, you can see it in their demeanor and in their eyes.  They want to learn. They want to understand and enjoy it.

Another reason I went into math is simply that I was GOOD at it.  I excelled and felt confident and smart when doing it, so naturally one would gravitate towards things which make them feel that way.  To be very honest, when I was about to start college, I wasn’t sure what my major should be as I felt I needed to know what I wanted to BE when I grew up and then tailor my major to that career.  I had no idea what I wanted to be, actually, which was a little scary. I declared math to be my major as I loved it, and I knew it was applicable in so many fields so I ended up studying applied mathematics for my major and biology as my minor and the subjects are so beautifully harmonious together that I thoroughly enjoyed all my classes.

Was I always good at math?  Well, that’s a complex question.  I’ve already addressed it a bit above, but really, it was simply that I caught on quickly and had fun doing so; however, I did need to be taught… by excellent teachers. Students sometimes think that we came out of the womb being math geniuses, but I had to be taught like everyone else. The difference is, I thoroughly enjoyed it and chose to make it my field of study.  I hope that, through our course, students will discover they enjoy math as well when they understand it, and THAT is the ultimate key.  “I like it when I get it!!!”

What is the moral to this passage?  Other than just giving my personal history, I think it can be wrapped up into the following:  I have a profound love of math because of excellent teachers and my |mindset| (positive mindset for those not familiar with absolute value notation).  I, and my colleagues, aspire to create a love, or at least an appreciation, for math and to assist students in seeing how it can be applied and in what instances.  The ultimate compliment is when we hear a student exclaim that they like math, even though they may look around guiltily like it’s some sort of taboo thing to admit.  🙂

Don’t be scared of math – embrace it in it’s beauty and complexity and know you accomplished something great by mastering it.

Guest Post: The Math Stigma

This week’s post comes from Andrew Petersen, who graduated with a BS in theoretical physics from Weber State University. He recently accepted a post at a company doing data analysis, making this his last guest post as an AcerPlacer instructor.

Struggling with the attention needed to do well in math in my elementary school, my friend nudges me, “Uuugh – I suck at math, want to go ride bikes?” I agree, knowing that I will never get anywhere with math, and riding bikes sounds immensely more entertaining. As children we develop into a social construct already in place, slowly built upon by thousands of generations of humans in our cultures. This social boundary subconsciously forces us to think and do certain things to fall within the norm. One of those things is that math, as a subject learned by a lot of people, is hard.

From birth and within our social boundaries, we are told that math is hard by our peers, mentors, and often times our parents. Those we look up to have labelled mathematics as the F-chord of our academics, and we probably aren’t good at it. After being told this, when attempting to learn math, we expect it to be difficult – we know it’s challenging – it seems like an impassible hurdle. This thought process and structure, I think, is entirely ironic. The only reason some people are inherently good at math, or anything for that matter, is due to their development and environment when growing up. This tells me that how we think of and structure things (math, baseball, reading…) is completely moldable. Then I also think – “Isn’t that how we learn…everything?”

Often I thought being good at math was due to a higher intelligence and a crafty creativity that I simply didn’t have. After years of studying and graduating university, I finally have the realization that an inherent intelligence was not the key – persistence and good habits were. A lack of this realization manifests into students often misdirecting their blame and anger. Every once in awhile I will have a student who struggles through the whole class even while working hard, retakes the class, then repeats. That student then begins to blame the institution – “My instructor was at fault, the math at my school sucks.” Yet this is misdirected anger, because those students don’t lack the intelligence, they lack the habits when writing and learning mathematics. They don’t work problems top to bottom, they skip steps, and they just get lazy. It’s understandable, all of us are lazy at some point, but this is the skill that needs to change.

We all learn differently, and many times while sitting in class I have struggled to keep up with my notes. For a long while I would finish writing out my thoughts, then move on with what the instructor was at next – but I am behind at that point, and continue that progression for the rest of the class. In most classes, no matter the subject, the instructor must get through a set amount of material. Due to time constraints, that often means they must teach faster than the students are comfortable with. As a student, I would blame the professor, until I realized this was misguided. It took years to discover that I need to listen, regardless of what I get written down. It is far more important to absorb what the instructor is saying through my senses, then fill in the rest later.

To break this stigma, we need to instill a social construct around our children that math is like any other subject, we just must learn how to learn. First though, we must do this to ourselves, and redefine how we think of mathematics in the first place – it does not take a especially smart person to learn math, it takes persistence and good habits.

Editor’s Note: An F-chord is a particularly difficult chord to play on a guitar.

Don’t Take My Word For It

Although it may be hard to believe, I am not the sole person who sees value in studying math. Before we begin with my thoughts, it might be helpful to see what others have to say.

5 Reasons Why You Should Study Mathematics – This article from The Complete University Guide talks about why you should pursue mathematics as your main focus of study. While that isn’t the main focus of this blog, it brings up some good points for how mathematics can make you more marketable.

10 Reasons You Should Study Math – In this six minute video, Danny Doucette argues for why math should be studied and touches on things such as the transferability of skills, the nature of developing industries, and the responsibilities of citizens, among others.

The importance of maths in everyday life – The author of this article from the Times of India is a math teacher. Not only do they give great reasons to study math, they also give great advice in how to teach it, especially for younger students.

Why Do We Learn Math – While the entire series of articles this is a part of is fantastic, this most directly addresses our topic at hand by taking the statement, “Math makes you think,” and explaining one way it actually does that.

6 Everyday Examples of Math in the Real World – Are you curious how math is used at home? This article focuses on math outside of a potential workplace to how it might benefit you more generally.

Examining How Mathematics is Used in the Workplace – This article seeks to examine a few studies on how math is used in workplaces from automobile production to nursing. Published by the Mathematical Association of America, it is a dense but interesting read.

Table of Examples – Tired of articles and just want to look up an example for your chosen career? Then this page is for you. Published by the Math Department at British Columbia Institute of Technology, it provides examples of problems that might be seen in various fields.