Wednesday, January 27, 2010



Monday Jan 25, 2:25 PM.  My cell phone rang.  I looked at the clock.  Dismissal time.  I figured my oldest daughter was calling to ask me to give her a ride home from school.  I grabbed my phone, but the incoming number said "unavailable".  I don't normally pick up incoming calls that say "unavailable".  But since it was dismissal time I picked it up figuring that perhaps my daughter had lost her cell phone or something, and she was using another phone to call.  When I opened the phone and put it to my ear, I heard that short second of silence that lets you know that you are receiving an automated call.  Annoyed, I almost closed it again.  However, the voice started, "This is ... Middle School.  It is Monday Jan 25.  We are calling to inform you that the school is in Lockdown".

My stomach flipped.  This was already the second time this month that the school had to go into lockdown.

The call continued, "No one may enter or exit the building.  Due to the timing of the incident, we will not be dismissing students at this time.  There will be a late dismissal.  Please press 1 to confirm that you received this message."

I pressed 1 and hung up.

Since I am both a parent and a teacher, perhaps I got a double dose of panic.  But images of both Columbine and Virginia Tech planted themselves firmly in the front of my mind.

"Take it easy" I said to myself, "Statistics are on your side."

True, it's a fairly good size school, lots of classrooms and hallways.  The likelihood of my child being in a problem spot was very small.

Nevertheless, this was already the second time in a couple of weeks.  The first time there had been a weapons threat.  This time I didn't know what to expect.

Furthermore, the middle school that my oldest attends shares a campus with the elementary school.  Therefore, all three of my kids are on the same campus.  Any lockdown at the middle school also sends the elementary school into lockdown. Any threat to the middle school potentially poses a threat to the elementary school also.

I held my breath for half an hour waiting to see if they would allow the elementary students to leave at 3:00.  No phone call at 3:00.  Good.  That meant that the elementary kids should be leaving their building and getting on their buses.  3:20 PM my two elementary school kids got off the bus at the house.  Big sigh of relief!  However, still no word from my oldest.  I kept an eye on the clock.  Every minute felt like an hour.  3:39 PM my cell phone rang.  I grabbed it and looked at the incoming number.  Relief!  It was my daughter.

My kids, of course, know nothing of Columbine.  And the words "Virginia Tech" mean nothing to them.  We have never talked about those incidents in front of the kids.  However, we did have quite a bit of conversation about the events at the middle school that evening after everyone was home.

To my two younger kids it's just an inconvenience.  While the elementary school kids were not allowed to leave their building, they were free to move about the building.

However, in the middle school, since that is where the threat was made, as soon as lockdown is initiated everything comes to a halt.  Wherever you are, you sit down and don't move again until lockdown is lifted.

But this time it was more.  The first time the middle school kids were allowed to continue to use the computers.  Not the second time.  The first time the premises were thoroughly searched.  The second time was more extensive.  In addition to searching all lockers and all rooms, this time all students were patted down, shoes off, and every inch of every bag, purse and pencil boxes were searched.  And absolutely no computers in use the entire time.


Tuesday afternoon 2:25 PM.  My cell phone rang.  I picked it up and looked at the incoming number:  "unavailable".  My heart stopped.  I opened the phone and put it to my ear.

"This is ... Middle School calling.  It is Tuesday Jan 26.  The school is in lockdown.  No one may enter or exit the building."

Any semblance of logic that I have disappeared.  Severe panic took over.

".... late dismissal....."

That's it?? No 'press 1 to confirm that you have received this call'??

What is going on??  The third time this month, and the second time in two days!!  And this time the automated message didn't even take the time to tell me to confirm that I had heard the message??

Ok, now I'm scared.  Really scared.

3:00 PM.  No phone call from the elementary school.  Good.  Hopefully they are allowing students to leave the building and get on buses.

3:17 PM.  My two younger kids get off the bus at the house.  Big sigh of relief!

Still no word from my oldest.

3:34 PM.  My cell phone rings.  The incoming number is my daughter.  Thank God!

More conversation that night with everyone, especially my oldest.

What do I say?  How do I refrain from communicating my own sense of panic?  How do I tell my oldest child that her peers are capable of such serious and deadly crimes?  How do I acknowledge that, yes, middle school can be a dangerous place.  How do I say that without making her live in unnecessary fear?


Wednesday Jan 27.  I resisted the urge to keep my oldest home from school.  8:00 AM.  I dropped her off in front of the middle school.  1:25 PM.  Early dismissal time.  My cell phone rings.  The incoming number is my daughter.  She wants a ride home.  I breathe a sigh of relief.

"How did it go today?" I ask as she gets in the car.

"We can't leave our classrooms unless we sign out and get a pass.  Only one bathroom in the building is open now.  A teacher guards the bathroom.  There is a table set up in front of the bathroom and you have to sign in when you go into the bathroom and you have to sign out when you come out of the bathroom.  Then we have to sign back into our classroom when we come back in the room.  We need a separate pass to go get a drink of water."

I sighed.

"Yeah" she said.

She got it.  She's pretty smart sometimes when it comes to understanding what doesn't get said.

Tuesday, January 19, 2010

Welcome to Wide Reading (Just Don't Bring Your Kindle or Your Laptop!)


At the middle school my daughter attends, there is a program called "Wide Reading".  Four times a week, for 35 minutes at a designated time during the school day, all students have to read.  If you are in grades 7 or 8, you may choose to attend a school club instead during that time period (6th graders don't have that option).  Otherwise you are required to read.

I think this is great.  I've always loved reading.  My husband loves reading.  All three of my kids love reading.  We encourage the kids to read as much as they can.

So far, so good.

Here's the catch: No computers allowed (every student in the jr high has a laptop), no online research allowed during this time, no Kindles (or similar item) allowed.  NONE!  Absolutely not!  Not even open for discussion.

Is it just me, or is there something wrong with that kind of thinking in the year 2010??

Now I personally do a lot of reading.  On some days the truth may be closer to "skimming", but nevertheless, I find myself doing a lot of reading.  Most of the reading I do these days is in the form of links that have been sent to me from all of the wonderful educators I follow on Twitter.  I also follow @COSMOSmagazine and @MATH_SCIENCE along with others which provide me with interesting articles about things happening in the world of science and mathematics.  I follow @timeshighered and @chronicle which supply me with a good amount of articles related to higher education.  I follow @themainemag which keeps me up on things happening around Maine.  I follow @DayTripNation and @visitmaine which gives me ideas for places to take day trips and where to vacation.  These are just a few examples of reading material that comes to me via twitter.  I also read blogs.  I read different types of blogs:  some educators, some just for fun.  But whatever the case may be, I read a lot of interesting things, I gain a lot of insights, I learn a lot of information.  Sometimes I comment, often I don't.  But even if I don't comment, I usually think for a while about what I read and whether or not I agree, whether or not it helped me learn something or if it changed my perspective about anything.  Many times I will read through the comments on the blogs.  That allows me to hear other people's thoughts and responses, and it makes me think further about what I just read.  Sometimes I read blogs that are funny or entertaining, and it improves my mood or makes me laugh.

Sometimes I need to find out some information about a topic that is completely new to me.  So I head over to my computer.  I may send a tweet asking for suggested reading material.  But more likely, I just start searching online.  I try to sort through the information responsibly, and make sure that I'm reading accurate information and not some whacked-out ideas.  But usually I have pretty good luck finding information.  After that I sometimes will email a friend or colleague who knows something about the topic in question, and send them the links I just read and ask them what they think.  I may get back an email with more links or more information.

Yes, I definitely do a lot of reading.

Sometimes I read a book; a good old-fashioned print-on-paper book.  I love those.  But I don't read them as much any more.  I would have to say that these days the majority of my reading is online.

I don't yet own a Kindle, or similar item.  I have not yet downloaded an entire book and read it on a computer screen.  I still like the idea of sitting down with a print-on-paper book and reading through it a little at a time, going back over some of the pages, and, yes, sometimes falling asleep with my nose in the book.  If I am reading an online article that is extremely long, I sometimes print it out so that I can read it a little at a time, underline things, make notes in the margin, and write questions to myself about things I want to research further.  However, it does not have to be print-on-paper in order to qualify as reading!

Yes, I definitely do a lot of reading - mostly online.

So what to think of a "Wide Reading" program that only allows you to read print-on-paper books?

Now I understand that jr high school students might try to get on facebook or other social networking sites and waste time.  They might try to get to material they shouldn't be reading.  They might not really be "researching" when they should be.  Yes I know you have to corral them in sometimes and peek over their shoulder.  I know that because I'm a parent.  But really now, that is NO reason to define reading as print-on-paper books only!  No reason at all!!

"Can't you use your laptop to do research during that time?" I asked my daughter.  "After all, that's reading."


"Can't you read online articles related to something you're studying in one of your classes?"


"What if the student owns a Kindle", I asked.


"So it's only considered reading if you are holding a paper book in your hand?"


"What if a student doesn't have a book with them?"

"The teachers always have extras."

I roll my eyes....

It's 2010. We must define "reading" as more than just print-on-paper books! 

Monday, January 18, 2010

Memorization vs Information Mining In the Digital Age


I am currently reading The Dumbest Generation by Mark Bauerlein. (I know, I know, the book is already a couple of years old and I'm first getting around to reading it.  Oh well.)  I'm only in chapter 1, and I already don't like it.  However, I will keep reading.  Perhaps I'll blog again when I'm finished.

In the meantime here is a question for you:
If your life depended on it, would you rather rely on your memory or your ability to mine information?

In other words, let's assume that for some bizarre reason, your life was at stake, and the only way you would be able to save it is if you could successfully name off all the countries in South America and their corresponding capital cities.  You then would be given two choices: 1.  Start listing them off from memory or 2. Find them on the computer.  You must choose one or the other of those methods and you only get one shot.

Obviously, it's not even up for discussion.  Only a fool would choose their memory.  And he would soon be a dead fool at that.

Ok, so every capital city of every South American country may seem a little unnecessary to commit to memory.

How about Reconstruction?  Shall I expect every American I ever meet to be able to define Reconstruction, give me a brief, one page synopsis, and then explain some of the factors that led to the North abandoning its commitment to Reconstruction?

Oh, I can hear it now, especially from History majors and Civil War buffs.  "Yes! Yes!", you scream.  "Every American should know that!!  It should be committed to memory!  You should, on a moments notice, be able to spout off quotes from every author who ever wrote a book on the subject.  You should understand the impact of Reconstruction on American society from the mid 1860's up through the mid 20th century and beyond!"

Really now, should I know all that off the top of my head?

Ok, how about something a little more current?  Should I be able to name all of the Supreme Court Justices?  How about the President of Russia?

Oh I can hear it now.  "Yes. Yes.  Everyone should know that".

Really now?  Should they?

"Of course they should!  How will they ever be productive, responsible American citizens without that basic knowledge?"

Really now?

So some more questions for you:  Do you know what Congressional District you live in?  Can you name the Mayor of your city?  Do you know where the phrase "separation of church and state" originated?  Can you recite the 4th Amendment?

Excellent!  Good for you.  Here's your gold star.  You may go to the head of the class.

What if I were someone who could not answer all (or any) of those questions off the top of my head.  But what if I know how and where to find accurate and reliable information online?  What if I could access more information online in five minutes than any university full of professors and college students combined could hold in their heads?  Is that ok?  I think so.

However, for those of you who still believe that Memory trumps Mining, here are some more questions for you:
1.  Do you know the percent of high school students in your school district that are active users of illegal drugs?
2.  Do you know how many babies are born each year in your city to mothers who are actively using?
3.  Do you know what percentage of the people in your city live below the poverty level?  In your State?
4.  Do you know the estimated percent of young girls (elementary school age) in America today who are being forced to have sex with drug dealers as an arrangement by a parent who doesn't have cash to pay for heroin?
5.  Do you know the estimated number of people world wide who are subject to human trafficking and slave trade each year?
6.  Do you know what's been happening in the Darfur region lately? (More info here.) Rwanda? The Congo?

Just wondering, that's all.

Here is what I have observed about people who place a very high value on memorizing facts.  You memorize what is important to you.  And then, unfortunately, you try to pretend that any information you don't have memorized isn't worth knowing.  Guess what, you're wrong.  And our young people today can see right through you.  What they understand is that memorizing doesn't make you intelligent, and it never has.  It just makes you a good memorizer.  Memorizing facts does not in any way ensure that you can do anything worthwhile with that information.  Memorizing historical facts does not in any way mean that you care about the present or want to change the future for the better.

Furthermore, it's the internet age.  How about we have a little respect for people who are good at Information Mining, even if they aren't so good at memorizing.

So next time I want to know something about my world, either close to home or abroad, I think I'll do some Information Mining.  If I can't find it, I think I'll go ask one of my students.  Chances are, they'll be able to help me search.

Saturday, January 16, 2010

Orthogonality vs Current Reality (A Mid-Life Math Crisis Part 3)

Problem 1:  This is Orthogonality: (general formulation and web site here)

Problem 2:  This is Current Reality:  (additional  video and pictures here)

Which of these two problems would you like to be able to say you can solve?

So what do orthogonality and current reality have to do with one another?

Nothing.  Correct.  That's what I said.  Absolutely nothing.

So what's with the orthogonality? 

Yesterday I was reading this article because the link had been sent via twitter from someone I follow.  Apparently the Supreme Court Justices don't know the meaning of the word "orthogonal".

Wow, that's a real shame!

Exactly.  Not a single person reading this thought that, did you?

No, you didn't.

Why is that?

I'll tell you why.  Because while it's a pretty little word, knowing the definition of "orthogonal" doesn't mean you're making a bit of difference in your world.

Head back over to the "orthogonal" article for a minute.  Be sure to take a look through the comments!!  Notice especially the response from Stas Peterson:  apparently this person thinks some Lawyers are morons because they don't know the definition of "orthogonal".  Mmmmmmm.....  Really now, people, come on.  While I enjoy the negative stereotypes and jokes about lawyers as much as the next person, let's be honest here.  For whatever their faults may be, not knowing the definition of the word "orthogonal" isn't one of them!  I noticed that the non-lawyer population got a few good jabs for being morons also.  Mmmmm.  After all, who doesn't use the word "orthogonal" every day?  Why I was just saying to my neighbor this morning, "I failed to eschew an ice patch yesterday resulting in my compromising my orthogonal position in relation to the terra firma".  I mean doesn't everyone go around talking like that?

Math professors, and others of great academic and intellectual accomplishments and extensive (albeit generally unnecessary) vocabulary, have you nothing better to do for your world than take pride in the fact that you know the meaning of the word orthogonal???

You are making me embarrassed to be a math teacher!!  People are dead and dying in Haiti, as well as so many other tragedies going on in the world today, and you're puffing out your chest because you know the definition of "orthogonal"??? 

Get on your knees and confess your sin of arrogance!  Then while you're there, start praying for the things that really matter in your world.

A Mid-Life Math Crisis Part 1: click here
A Mid-Life Math Crisis Part 2: click here 

Wednesday, January 13, 2010

Math: The Class No One Wants To Take (A Mid-Life Math Crisis Part 2)

If you would like to read Part 1, click here.

So I'm a college math teacher.  Yep, the class no one wants to take.  That would be the one.  Wow, I'm feeling good about myself already.  Now granted, there are a few people out there (for instance, myself, 25 years ago) who willingly sign up for college math classes.  But let's face it, those people are a small minority.  The majority of the students who find themselves in a college math class are there because it's a requirement for graduation.  Or, in the case of developmental level math classes, the student didn't score high enough on the Accuplacer or the SATs to place into a 100 level (or higher) math class, so s/he has to take a math class for which s/he is not even getting any credits toward graduation.  Now in this latter scenario, although I fail to see how this could possibly be my fault, I often notice that the student comes into the class resenting me, the teacher.  Why is that?

I've also noticed that many students come into college math classes with the attitude that this is the least important thing they will ever do.  Even decent students have that attitude.  I don't understand it.  I've had students in my math classes that I know are capable individuals and catch on pretty quickly.  They don't have a problem understanding the material.  And with a little bit of effort, they could likely pull off an A in the class.  However, they are perfectly happy to do just enough to get out of math class with a C.  In any other class they would be appalled and/or embarrassed to get a C for a grade.  But they are perfectly fine with it - no actually they are quite happy with it - if it is a C in math class.  Why is that?  It seems to me that somewhere along the line in their educational experience there was an unspoken arrangement that math was "hard" and if you could get a C in a math class then you were doing really well!  And why argue with that?  I mean seriously, if no one is going to question a C, why not do the bare minimum and just get by with a C?  So why does it bother me that my potential A students are happy with a C?  Shouldn't it bother the student?  Actually, what bothers me is that it doesn't bother the student.  In fact, I have students that are happy to do just barely enough to pass with a D.  They figure that if it isn't a pre-requisite for anything, and they don't have to worry about transferring the credits, then a 63% will do the trick - after all it's passing.  What?????  Where is this attitude coming from??

Now don't misunderstand me.  I have legitimate learners in my classes, and I have students that put out a lot of effort and I have students that strive to get the best grade they can get.  But they are a minority.

Somewhere along the line in the 12 or 13 years of education before these students get to me, they have acquired this attitude.  Does it have anything to do with the emphasis placed on reading in schools but hardly anything is said about math?  I don't know.  I have three kids in elementary and middle school.  All I've heard about for the past 7 years is "DEAR", "wide-Read", etc.  I haven't heard a thing about a program called "Drop Everything And Math".  I haven't heard a thing about "wide-Math".  In fact, I don't think I've heard the word math at all when it comes to focus areas and special programs.  Interesting.  I do know that 2 out of my 3 kids have scored very high on the Raven test, for whatever that's worth.  The school administers these tests twice during the elementary school years, once during jr high school and once during high school.  Apparently the school now considers two of my kids "gifted and talented", and they go to something during the week that emphasizes math.  Wow, so about a dozen kids a week are considered "gifted and talented" enough to be part of a program where they spend extra time focusing on math?  Ridiculous!  I mean I think it's great that my kids got high scores on the Ravens and all that jazz, but really now, I know for a fact that my kids and a handful of others are not the only students out there that are capable of doing math!

Incidentally, I feel compelled to be honest and let you know that I took the Raven test a few years ago.  At the time I was working part time in the "Gifted and Talented" program so I decided to take the Raven just for laughs.  And sure enough, what a laugh it was.  In order to qualify for the "gifted and talented" program, your score is compared to others in your age group.  Since we were testing only up through grade 10, I scored myself against the 16 year old age range.  For the record, I didn't score high enough to be considered "gifted and talented".  (And my undergraduate degree is in Mathematics. And I graduated with a 3.6 GPA back in the day when I did my undergraduate degree.)  In other words, had I been a student, I would have lost out on the opportunity to participate in extra time during the week that focused on additional, higher than grade level math activities intended to further enhance a student's math skills.  Interesting.  Oh, and I must also admit to the following: my husband, who has a BA and an MA in History, and half of an M.Ed, outscored me by a noticeable margin.  He would have been more than qualified for the "gifted and talented" program and would have been encouraged to pursue additional math classes and educational opportunities in the area of mathematics.  Needless to say, I am not convinced that the Raven alone should be determining who has mathematical potential.  As my husband frequently likes to say, "I have a History degree, I work in IT and program computers for a living, and the most useful course I ever took in college was my horticulture class".  Well that about sums it up nicely, doesn't it?

But surely I digress....

The point here, I guess, is that I do believe that many students have the potential to do quite well in math classes that I teach.  Currently I am teaching developmental classes and College Algebra classes.  Those are classes that are well within the capabilities of every student who finds himself there.  Not everyone is an A student in math.  But I do believe that many more are A students than the grades would indicate.  I believe it with all my heart.  And my style of teaching and my willingness to help my students reflects what I believe.  The students still refuse to believe it themselves.  Or perhaps they are afraid that I might expect something of them if they believe it themselves.  It is hard to say.  But I do know that, sadly, by the time they get to me, they are 12 or 13 years into some seriously low expectations of their capabilities in a math class.

If you would like to read A Mid-Life Math Crisis Part 3: click here

Tuesday, January 12, 2010

A Mid-Life Math Crisis Part 1

Yes, there really is such a thing.  I know because I'm having one.  Actually, I'm not sure the "crisis" is that new, nor is it happening due to my now "mid-life" status.  But it's one that's really bugging me at the moment.

So imagine...  You're at a family get together.  You know, those things you attend once a year or so when you get to see all the extended family together in one place, talk about what you've been doing over the past year, who has gotten married, divorced, had a kid, graduated college, etc.  The conversation tends to go something like this:

Carolyn:  "So what have you been doing for work lately?" 

Relative 1: "Oh, I'm still in my own practice.  I'm starting the fourth year now.  But I'm investigating Doctors Without Borders and similar organizations.  I'm wondering what I can do to be involved in organizations like that."

Carolyn:  "Oh good to hear."

Relative 2:  "I finally finished up the last of the classes I needed.  I passed the Patent Bar exam and everything.  So now I'm working on intellectual property and patent law in the bio-medical research field."

Carolyn:  "Wow, what types of things do you specifically deal with?"

Relative 2: "Patents on various medications.  One of the most recent cases involved a medication for Rheumatoid arthritis"

Carolyn:  "Sounds like an interesting job"

Relative 3:  "I just got back from a year in Haiti.  We were working with an orphanage there and at the same time we were also involved in a project to get clean drinking water to a village.  It's incredible when you see some of the devastating conditions the rest of the world lives in.  It is so wonderful when you have an opportunity to affect the daily lives of people in such a positive way."

Carolyn:  "That sounds wonderful"

Relative 4:  "So, how about you Carolyn, what are you doing these days?"

Carolyn:  "I still work part time teaching college math classes at a couple of different local colleges."

Dead silence.

Carolyn:  "Yeah, it's really neat.  I just finished teaching my students how to graph Logarithmic Functions!"

Dead silence.

Carolyn:  "Yeah, did you know that every exponential function has a horizontal asymptote?  It's so interesting.  In fact, the function f(x) = a ^ x for 0 < a < 1 has a y-intercept of (0,1)?  Furthermore, any transformation of an exponential function can be called an exponential function also!"

Dead silence.

Relative 3 (in an attempt to recover the conversation back to something worthwhile):  "I've been really burdened lately about the reality and horror of the extent of human trafficking and sex slave trade that currently exists in the world today.  Now that the clean drinking water project in Haiti is completed, I'm going to focus my efforts on getting connected to an organization that actively is involved in stopping human trafficking around the globe.  I think I'd like to work with something like that for a while to better the life of all humanity."

Ok, so maybe it's not really that bad.  First of all, if I'm asked, I just usually say that I teach math classes part time at a couple of colleges, and I leave it at that.  If I do talk about it further, I talk about amusing things that have happened in the classroom or interesting, or sometimes infamous, students that I have met.

And no, I don't have an entire extended family full of people who are all trotting around the globe involved in Doctors without Borders, Ending Human Trafficking, or Clean Drinking Water in Third World Countries projects.  But I do know some of those people.  And I know that they didn't need a math degree to make a difference.  By their own admission they likely could not pass a College Algebra final if I handed one to them.  Do these people ever worry about the equation of a hyperbola that is not in standard form? I doubt it.  Do they know that not every function has an inverse function?  Probably not.  But they are making a difference.  They are changing lives.  When they leave a village in Haiti knowing that for the first time ever the people there will have access to clean drinking water, is that a good feeling?  You bet!  When one of my students can correctly find the domain for f(x) = (x-3) / (x - 1) is that a good feeling?  Not so much.

To be continued...

Click here for Part 2

Monday, January 4, 2010

Creating God In Our Image

(or not all functions have an inverse that is also a function)

No, this isn't going to be a math lesson.  But for the sake of the point I'm going to make here, think about it for a minute. Not all functions have an inverse that is also a function.  On a very simplistic level, (and math teachers will have to forgive the extremely abridged explanation here - but it's for the sake of argument) you can think of an inverse as "reversing the roles of x and y".  In math, when you do that, the interesting thing is that you will get an image, but you won't always get a function.  Or you could say that the new image may not be something that will pass the "vertical line test"; or perhaps you could say (although you wouldn't necessarily put it this way in a math class), "it just doesn't function".

And so is the case when we attempt to "inverse" the "function" of creation.

Genesis 1:27  "So God created man in His own image... male and female He created them."

No, this isn't going to be a theological dissertation. And I am not here to argue whether or not the Genesis account is literal or allegorical. There are thousands of books written over millennia that have done that already. Refer to one of those for an in-depth study on that topic.  However, so that you do understand my position in this blog post, I do believe that man was created in the image of God. I do believe that I have a Creator and I do believe that I was created to reflect the image of that Creator.

This function of man created in the image of God is a function that only works in one direction.  It was only meant to function when going from God to us.  But in our arrogance, we attempt to inverse the function.  We are always determined to create God in our image.  We take our limited ideas and understanding, our flawed and prejudice thinking, our own ideas about right and wrong and justice and our cultural influences, and we attempt to create God.   And in so doing, it is true that we do get an image.  But the resulting image that we get does not pass the "vertical line test".  See the problem?

In my mind, this problem has only one solution.  Look at the original equation.  Study the original function only.  Graph the original function if you want to know what it looks like.  Understand the beauty in how it works. See for yourself how it passes the "vertical line test".

Now enter another problem:  I also believe in sin.  And I believe that whatever original beauty the image once had, it has now been marred.

The way that I think about it in my mind is similar to looking at a photograph of someone I know.  If you took a black marker and drew an X across the face, I would likely still recognize the person.  Likely I would still know whose image it was.  I would still be able to see some features.  But the image would be, to one degree or another, ruined.

That is how I look at humans.  I do believe that there is the image of God stamped on us.  I believe that it is something about our "essence".  I know that sounds mystical, and perhaps it is.  But I believe all humans carry that stamp.  And when we truly see one another's "essence", we see something of God.  Unfortunately, I also believe that we all have a big X drawn across us.  I believe it is sin.  And it prevents us from reflecting the image of God in a completely accurate manner.  You might even argue that it is very difficult to see God in us at all anymore.  Perhaps that's true.  But it is still there.

Exodus 20:4 "You shall not make for yourself an idol in the form of anything in heaven above or on the earth beneath or in the waters below."

Now, go back to the example of the ruined photograph for a moment.  Let's say that you handed me that ruined photograph and told me that from that picture, you would like me to create a picture of what the original looked like.  Ahhh, we now have a problem.  And depending on how ruined the photograph is, we may have a very big problem.

So let's say I give it a try...  No need to expound on that further, we all see what the problem would be.  But we do it anyway.  And in so doing, I believe we break the second commandment.  No, we may not have physical representations that we worship.  But yes, we do create an "image" or of God, or an "idol".  And often we do a very bad job of it.

The solution?  First, get rid of the idol.  I know we all hold our image that we've created very dear.  It is our nature to do so.  But it has to go.  Then look at the nature of the original.  Study it.  Consider it.  See what the Original has to say about Himself.  Appreciate that we are the reflection and not the original.  And don't try to reverse the roles.