Is Math Creative?

Is math creative?

As a math major in college, I was required to take a computer programming class. In retrospect, the reasoning made perfect sense: successful programming follows a natural logic, very much the same way math does. But at the time, I was resentful, and a little scared.

Sure enough, I was lost by week two. I enlisted in some tutoring from a dear friend in my section. And she demonstrated to me a completely different way of structuring the code. Her process made much more sense than the methods taught by our instructor, so I adopted it. Three days later, I sat in shock, as the prof announced that some of our assignments looked suspiciously similar.

Let me be clear: I had not copied my friend’s coding. I had identified with her way of thinking and modeled my code after her approach. But it was such out-of-the-box thinking, I understood why the prof thought we were cheating. And sadly, instead of talking to him about it, I simply reverted back to his methods. Yeah, I didn’t get much out of that class.

My friend demonstrated some amazing creativity in her approach to coding. She did this in all of her math classes as well — for which she was greatly rewarded. I learned from her that thinking creatively is critical for succeeding in math of any kind. And I mean any kind — from proving Fermat’s Last Theorem to finding out how many gallons of Symphony in Blue you need to paint your living room.

Too often, math is described in black-and-white terms. There’s a right and a wrong answer. There’s a step-by-step process to follow. If you think of math this way, it’s no wonder. Most of us were taught that math is about a right answer.

But those teachers were wrong. Sure, the right answer is important, but just like those inspirational posters say, it’s all about the journey. How you get to your answer is just as important as the right answer.

And that’s where creativity comes in. Because we all access this information in different ways. Some of us are visual. Some of us need time to think. Some of us like to talk things out. Those of us with true numeracy use creative methods for solving ordinary problems. Take 23 x 6, for example.

Most of the world would stack these numbers up, multiply 6 by 3 and then 6 by 2, add (remembering to align the numbers properly) and get 138. But there are many other ways. I like this one:

23 x 6 = (20 + 3) x 6

                       = (20 x 6) + (3 x 6)

         = 120 + 18

          = 138

With that method, I can do the problem in my head!

But you don’t need to solve the problem that way. Come up with your own process. Be bold! Set off on your own! Be creative!

So in answer to the question, Is math creative? YES! You’ve just got to access your own out of the box thinking.

Photo Credit: Yuri Yu. Samoilov via Compfight cc

Do you agree that math is creative? Why or why not? What examples of creativity (or lack thereof) can you share?

3 Website Statistics Tools Reveal Your Visitors’ Secrets (in a good way)

Web site analytics

We writers have all heard about platform, and so many of us have blogs — both to share what we’re thinking and to reach audiences that might also be interested in reading our books. But how can you tell if your website or blog is really working for you? Stats, of course. Since this can be overwhelming, I’ve asked Donna K. Fitch — an expert on WordPress and a fellow writer — to share her favorite tools for gathering and analyzing website stats. She’s broken it all down, making the process easier to understand and to follow. Plus, she’s offering you a free glossary of analytics terms. (I’m snatching that bad boy right up!)

Who’s visiting your website? Is anyone reading your blog posts? Site statistics provide the keys to these questions. Many options exist for providing these statistics, depending on what site platform you use or how much depth of analysis you desire.

Your hosting provider will often show you the rough numbers in your dashboard or cPanel, such as “x number of visitors this month,” but that’s not helpful. That number is the equivalent of the counters people were enamored of a few years back. Seeing “302,455 site visitors since 1997” is not that impressive if you think about it. Site visitors is a raw number that may include the site owner herself, and tells nothing about how many of those visited multiple times.

I design and maintain WordPress sites, so I’ll be focusing on that platform, but you may find many of the same tools available on your site. If you’re not familiar with the terms used, I’ve provided a glossary of analytics terms as an exclusive free PDF download for you.

Jetpack

A suite of free features available on WordPress.com hosted sites, Jetpack offers website statistics. If you use self-hosted WordPress, you can link your account to a WordPress.com account and take advantage of Jetpack as well. At a glance, Jetpack shows you a graph of site visits, referrers, search engine terms, subscriptions, top posts and pages, and clicks. As you dig into the information, you will find an “About the math” section, where their logic is explained.

When should you use Jetpack? When you’re mainly interested in how many site visits occurred when. This number is useful if you want to determine the optimum day to post on your blog. If more people read it on Friday than on Monday, you can adjust the day you post accordingly.

Google Analytics

Available for free to anyone with a Google email address, Google Analytics can be quite daunting, but the wealth of information provided is astounding. Hooking your site up to the analytics tracking requires signing up at http://google.com/analytics. You must use a Gmail account for the requested email address. Google provides you with an ID code which you then insert into either all the pages you wish to track, or your site template. After that, wait about 24 hours to see the results.

When you log into the Google dashboard, you immediately see a graph of visits, with information on sessions, users, pageviews, pages per session, average session duration, bounce rate, and percentage of new sessions, plus a pie chart showing the percentage of new and returning visitors. Below that you see three sections: Demographics, including the language of the top ten visitors, their country or territory, and city from which they visited; System, showing what browser they used, what operating system (Windows, Macintosh, Android, iOS, Linux), and what service provider (Charter Communications, Time Warner, etc.); and Mobile, listing operating system, service provider and screen resolution.

While you could spend hours on each section alone, skip down to the Behavior section to see interesting facts about how people interacted with your site. (The Acquisition section is primarily useful to those purchase ads on Google AdWords.) You can learn what pages were visited the most, how fast pages were accessed, and the flow of the user’s behavior through the site. You can even click on Real Time and see who is visiting your site at that very moment.

When should you use Google Analytics? When you’re interested in knowing in detail who’s visiting your site, from where, how often. When you want to tweak your site’s content or architecture in answer to visitor patterns, to enhance the user experience. When you just love seeing all the numbers generated by your website.

WP SlimStat

This is a highly-rated free web analytics plugin for WordPress. I have only recently installed it, but am amazed at the range of information it provides, some of which isn’t provided by Google. A real-time log shows you who’s viewing your site at any given moment. The Audience section uses the term “daily human visits,” separating out people from visits by bot. Site analysis gives you at a glance how many content items your site has, as well as the number of comments. The Map overlay provides a fun way to see if you’re getting visits from around the world.

When to use WP SlimStat? When you’re a WordPress self-hosted user who wants a rich, broad array of statistics in an easy to use format.

I’d love to hear from you about your experiences with these tools. Please use the comments below to share!

Donna K FitchDonna K. Fitch, MLS (Master of Library Science), MCert (Master’s Certificate in Web Design and Development), is the founder and CEO of Maximum Author Impact, creating beautiful WordPress websites, training webinars and other resources for indie authors. She is the independent author of Second Death, The Source of Lightning, and The Color of Darkness and Other Stories, newsletter web editor and member of the Horror Writers Association, and a member of the Alexandria Publishing Group, aimed at raising the level of professionalism among indie authors. In her day job, she is the digital communication specialist in the office of marketing and communication at Samford University in Birmingham, Alabama, USA.

Top photo credit: trekkyandy via Compfight cc.

Bottom photo credit: Donna K. Fitch.

Math at Work Monday: Joe the Platform Consultant

 

In the IT field, there are many machines and programs that are really confusing and difficult to understand. Not only do we have to trust and depend on these machines, but also the people who service them. Joe Thompson is one of the good guys. He provides assistance to the users and companies when they need it most. From consulting to maintenance, Joe and his colleagues are there for us when our technology isn’t working quite right. (Joe is also one of my former geometry students. It’s been great to reconnect with him and see how accomplished he is now!)

Can you explain what you do for a living?

Red Hat’s consultants help customers get our products working when they have specific needs that go beyond the usual tech support.  We are essentially advanced computer system administrators on whatever our customers need us to be to get Red Hat’s products to work for them.  Common consulting gigs are setting up Red Hat Satellite to manage the customer’s servers, or doing performance tuning to make things run faster or a “health check” to verify things are running as efficiently as possible.

We just put out a marketing video about our consulting for public-sector clients, actually:

https://www.youtube.com/watch?v=eMzANG3Yhlk

(We do more than just public sector and cloud, of course.)

When do you use basic math in your job?

The most common is when tuning a system to perform well, or configuring various things.  Unit conversions and base conversions are especially important.

IT has a long-running math issue actually: does “kilo” mean “1000” (a round number in base 10), or “1024” (a round number, 10000000000, in base 2)?  There are various ways people try to indicate which is intended, like using a capital K vs. a lowercase k, or using KiB vs. KB.  This matters in a lot of cases because when you get up into large data sizes, the difference between round numbers in base 10 and base 2 gets pretty big.  A 1-TB hard drive (a typical size today, maybe even a little small) is a trillion bytes — 1000 to the fourth power, not 1024 to the fourth power.  The difference is about 10% of the actual size of the drive, so knowing which base you’re dealing with is important.

Then there are units that have to be converted.  A common adjustment for better performance is tweaking how much data is held in memory at a time to be transmitted over the network, which is done by measuring the delay between two systems that have to communicate.  Then you multiply the delay (so many milliseconds) by the transmission speed (so many megabits or gigabits per second) and that gives the buffer size, which you have to set in bytes (1 byte = 8 bits) or sometimes other specified units.

Sometimes software writers like to make you do math so they can write their code easier.  If a program has options that can either be on or off, sometimes a programmer will use a “bitfield” — a string of binary digits that represent all the options in a single number, which is often set in base 10.  So if you have a six-digit bitfield and want to turn off everything but options 1 and 6, you would use the number 33: 33 = 100001 in binary.

Do you use any technology (like calculators or computers) to help with this math? Why or why not?

I’ve always done a lot of arithmetic in my head and I can at least estimate a lot of the conversions without resorting to a calculator.  I’ll break out the calculator if the math is long and tedious though, like averaging a long column of numbers, or if I need a precise answer quickly on something like how many bytes are in 1.25 base-10 gigabits — I can do the billion divided by 8 and come out with 125 million bytes per base-10 gigabit, and then multiplying by 1.25 I know I’m going to be in the neighborhood of 150 million bytes, but I need the calculator to quickly get the exact answer of 156250000 bytes.  If I’m on a conference call about that kind of thing I’ll use the calculator more than otherwise.

Google introduced a new feature a couple of years ago that will do basic math and unit conversions for you, so if I’m deep into things or just feeling lazy I can also just pull up a web browser and type “1.25 gigabits in bytes” in the search bar, and Google does it all for me.  But recently I noticed I was reaching for the calculator more, and arithmetic in my head was getting harder, so I’ve been making a conscious effort to do more head-math lately.

How do you think math helps you do your job better?

Without math, I couldn’t do my job at all :)  Even so little a thing as figuring out how long a file will take to transfer takes a good head for numbers.  As soon as you dig under the surface of the operating system, it’s math everywhere.

How comfortable with math do you feel? Does this math feel different to you ?

I’m pretty comfortable with math.  A lot of my off-time hobbies touch on computers too so it’s a lot of the same math as work even when I’m not working.

What kind of math did you take in high school? Did you like it/feel like you were good at it?

I took the standard track for an Advanced Studies diploma from grades 8-11 (Algebra I, Geometry, Algebra II, Advanced Math), plus AP Calculus my senior year, and always did well. I didn’t expect to like Geometry going in because it’s not one-right-answer like a lot of math, but I ended up enjoying the logical rigor of proofs.  (Though I do recall giving my Geometry teacher fits on occasion when my proofs took a non-standard tack…)

Did you have to learn new skills in order to do the math you use in your job? Or was it something that you could pickup using the skills you learned in school?

Most of it was learned in school, although base conversion isn’t something we spent a lot of time on.  I got good at it through long, frequent practice as you might guess…

Do you have a question for Joe? Send me your question and I will forward it to him.

Photo Credit: Dan Hamp via Compfight cc

The Brain Ordered: A review of The Organized Mind

The Organized Mind

These days I’m devouring just about any writing I can find that features the cross section between neurology, sociology and psychology. Daniel Pink’s Drive completely changed my mind and confirmed my suspicions about how motivation actually works. And now The Organized Mind by behavioral neurologist Daniel Levitin has helped me better understand how the brain helps us organize our time, thoughts and things — and how our brains can get in the way.

It’s a big book. And parts of it are very dense, including sections that explain the anatomy of the brain and almost an entire chapter devoted to the probabilities of medical outcomes. But the rest of the book is quite narrative, with funny and relatable examples. This science and geek loved it all.

For me, the takeaways were in productivity and learning. It’s not fair to boil it all down to two categories, but I will. At the same time, I’ll point out how all of this relates to math, with a few quotes from Levitin‘s book.

What the Brain Does Well

Categorization

Turns out the brain is perfectly designed for identifying similarities and differences.

In the last few years, we’ve learned that the formation and maintenance of categories have their roots in known biological processes in the brain. … Theoretically, you should be able to represent uniquely in your brain every known particle in the universe, and have excess capacity left over to organize those particles into finite categories. Your brain is just the tool for the information age.

Where’s the math in that? Everywhere. It could be argued that math is the study of categories. Start with our number system. Positive numbers that are not fractions and decimals fall in the category of whole numbers. Add negative numbers to that group, and you’ve got integers. (And so on.) Or you can group numbers as prime and not prime or even and odd. Graphs of equations can be lines or curves — and some curves are parabolas, while others are circles. See where I’m going with this?

This is all good news. Because the brain is so excellent at forming and maintaining categories, your brain was made for math.

Discovery

But how can we make sure we remember all of these categories?

The last two decades of research on the science of learning have shown conclusively that we remember things better, and longer, if we discover them ourselves rather than being told them explicitly.

This idea has huge implications for math education. For the most part, approaches to teaching math fall in one of two categories (see what I did there?): telling and discovering. Most of us who grew up in the 70s and 80s learned math through the “telling” method. The teacher gave a lecture, demonstrating how to perform a skill, and asking students to practice the steps shown in the lesson. Discovery turns this process on its head, giving students the opportunity to figure things out on their own, even finding new ways to solve problems. When they can discover ideas on their own, students have a much better shot at remembering what they’ve learned.

Of course discovery is messy and difficult, which brings us to ways that our brain gets in the way.

What the Brain Doesn’t Do Well

Frustration

This idea from Levitin blew my mind. Apparently it’s a proven fact that people don’t manage frustration well. It’s why we procrastinate, and that feeling of frustration is rooted in our brains.

The low tolerance for frustration has neural underpinnings. Our limbic system and the parts of the brain that are seeking immediate rewards come into conflict with our prefrontal cortex, which all too well understand the consequences of falling behind. Both regions run on dopamine, but the dopamine has different actions in each. Dopamine in the prefrontal cortex causes us to focus and stay on task; dopamine in the limbic system, along with the brain’s own edogenous opiods, causes us to feel pleasure.

Then we play into this automatic system with two “faulty beliefs: first, that life should be easy and second, that our self-worth is dependent on our success.” So, when the going gets tough, we quit — shoot for an easier option.

Unfortunately, this is just something we need to fight against. And Levitin has some great strategies to offer. At the same time, I felt very validated in my instinct to choose low-hanging fruit, rather than reaching for loftier goals. That also goes for the math student who is immediately frustrated by assignments he can’t understand, and the grownup who always lets someone else split the restaurant tab.

Probability

For years I’ve struggled with my inability to internalize the concepts of probability, so I was really relieved to learn that my brain is wired this way.

Cognitive science has taught us that relying on our gut or intuition often leads to bad decisions, particularly in cases where statistical information is available. Our guts and our brains didn’t evolve to deal with probabilistic thinking.

No wonder I have to work so hard to understand the probability I’ll suffer from a medication’s side effects or even the chance I’ll win in Roulette. Unlike categorizing, my brain isn’t set up to have an intuition about probability. (This isn’t to say that others can’t find calculate probabilities quickly, of course.)

Of course much depends on our understanding of probability, including life-and-death situations, like choosing the right medical treatment. It’s important to think about these things in a clear and focused way. That’s one reason Levitin spends so many pages on something called FourFold tables. (More on those in a later post.)

I encourage you to pick up a copy of The Organized Mind. (No, you can’t borrow mine; I’ve been referring to it over and over since I finished reading!) It’s a great look at how we can maximize the things our brains do well and work against the tricks our brains play on us — to be better organized and productive, while learning and using math.

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Have you read Daniel Levitin’s book? If so, what did you think? Share your comments and questions.

 

Tough Equations: Aging parents and housing

Tough Equations: Aging parents and housing

I’ve hit the age when many of my friends and colleagues are managing the realities of having aging parents. Luckily, I’m not there yet — my mom is still very active, both physically and mentally. But many of us in our 40s or 50s are probably at least thinking about how we might manage our parent(s) affairs if/when they are unable to handle things on their own.

My friend and fellow writer, Beth, faced this problem last year, when she, her husband and her mother moved to another state. Beth’s mother needed a little more supervision, and so Beth and her husband arranged for her to live with them. That brought up some emotional and practical questions, which Beth shared in an online writing’s group that we both belong to. She gave me permission to share them here:

Mom lived independently until we combined households. She wants to pay us a monthly fee that covers “room and board.” The question is: How to figure a fair and reasonable amount.

It’s been a long time since [my husband] and I had a roommate. In those days, we simply divided the big stuff by three (rent, utilities, cable), and each person was responsible for his/her own food. That doesn’t seem fair in the current situation for a variety of reasons (not the least of which we’re talking about my MOM, not some friend).

I feel I’m making this unnecessarily complicated. Can anyone help me sort this out? I bring it up because Mom talks about it constantly. She seems to feel the amount she’s paying is too low, and I keep putting the brakes on changing the dollar figure until we have better data about our expenses.

Naturally, I think math can help us find some simple solutions to emotional problems. So I offered this:

I have a really easy and non-biased way to look at this. Calculate your total household costs — mortgage, utilities, food, etc. Then divide this by three. Each of these is a share.

Next, you can decide how many shares each person should have. For example, your mom may have only a half-share, based on what you think she can afford or how much she eats, etc. Take half of a share, and that’s her monthly rent.

Naturally, I like taking a mathematical approach, because it can help reduce the emotions. And if any of the variables go up or down — utilities, for example — you can adjust the rent really easily.

And that seemed to do the trick for Beth. In fact, she took things even farther, considering fair market value, as suggested by another group member:

Here’s how we solved the problem in the end:

1. I drew up Mom’s current monthly budget.

2. I drew up a list of household expenses that apply to her (including the mortgage payment). I didn’t include things like pet expenses or [my husband's] fuel for commuting, obviously, because those are our sole expenses.

3. I used Laura’s methodology to divvy up the total household expenses into three full shares. Then I calculated partial shares: 3/4, 2/3, and 1/2.

4. I used [another member's] data about the fair market value of a studio apartment in [my county] for comparison purposes.

5. Then I sat down with Mom and first explained her current budget. Next, I went over the household expenses.

6. I told her about the fair market value of a studio apartment and explained how that related to our attempt to determine what was a fair amount for her to pay us each month.

7. I showed her the share information.

8. I showed her how each share amount would affect her net income. Even at a “full share,” she still retains about 45% of her net income for “mad money,” and that’s without touching any investments. (I didn’t point that out to her, in terms of trying to steer her. I think what I wrote kind of reads that way. I just used a calculator to show her what each share amount would leave her, in terms of disposable income.)

9. I had written all these figures down on paper, so I stepped away to giver her time to peruse the numbers for awhile and consider what SHE wanted to do.

10. After a few minutes, she called me back and said she’d decided to pay a full share. She’s the type of person who likes to “pay her own way,” and she’ll still have plenty of mad money left over. She also was very happy she wouldn’t need to dip into any investments.

It’s important to note that this cut-and-dry approach didn’t erase all of the feelings in Beth’s situation. She was very nervous talking to her mother, and her mother felt responsible for paying a full share. See? Feelings.

Another interesting aspect is how flexible this process can be. With some simple parameters — the value of a full share vs. a half-share, for example — Beth’s family can alter the process depending on where everyone is financially. And if her mother needs more resources or Medicare helps to pay for things, the entire formula can be changed.

Just a bit of math helped Beth gain some perspective and offer her mother tremendous autonomy. The process also set them up to avoid conflict later on. Nice work, math!

Photo Credit: VinothChandar via Compfight cc

I’m currently reading The Organized Mind, by Daniel Levitin, and I can’t wait to share a review with you when I finish. He offers some really terrific math to help when medical decisions are tough. Four-square decision tables anyone?

What do you think of the process Beth worked out? (I also offer this approach as a way to divvy up the cost of a beach house among several family members.) Have you used math to help you come to a difficult or emotional decision? Do you think this approach would work for a young adult who hasn’t flown the nest? Share your stories in the comment section.