Monday, October 31, 2011

Math: What is it all About?

            Math intimidates a lot of people.  Public schools are frequently, almost constantly, looking for Math teachers for middle and high schools.  Homeschool moms who have felt confident in teaching most subjects suddenly start looking for coops and tutors when it comes to “higher level” math.  Statistics show that most adults do not feel confident about their math abilities, and most say they never used this math once they left formal schooling.  So, if most adults don’t use math, is it worth the time and energy to teach?

            To answer this question, first we should look at the relevance of math in the real world.  In the book “A Thomas Jefferson Education,” the author suggests reading the classics in all subjects, including math.  What is there to read about math?  Math is simply a list of problems, right?  Wrong.  That is how it is taught, and it is why most adults think math does not affect their daily life.  However, just like history is about people, math is about patterns which people found in nature.  Reading about the “masters” who discovered these patterns helps us to realize exactly how math affects our lives, and we are much more apt to learn if we know something will be needed in the future.

            In that same book, van de Mille suggests certain objectives which should be learned from any math program.  Here are the thirteen “focuses” which he lists:

1.     seek and recognize patterns

2.     explore the relationship between things

3.     see similarities and distinctions

4.     analyze logically, but with a deep sense that there is a right answer and a set ideal worth detecting

5.     compare and contrast

6.     see things in black and white

7.     see infinite shades of grey and therefore avoid jumping to conclusions

8.     seek evidence for conclusions

9.     check opinion with first-hand research

10.  put his own pen to paper before accepting what society tells him

11.  seek for absolutes

12.  remain open to surprising new information which makes past conclusions limited, though perhaps still accurate

            Is he talking about math or science?  Actually, once you start reading some of the classic books by master mathematicians, you discover that there really is not much difference between the two subjects.  Most of what we consider higher level math was designed to help predict natural events or to know what would happen if something else were to happen, and so it would be possible to prevent disasters (like the classic tale of 2 trains colliding at a certain speed) or to predict success (like knowing how much you will have at retirement if you put $2,000 a year for 8 years into a mutual fund that earns 18%, then leave it alone and let the interest roll over into principal for 20 years – compound interest is fascinating!).  Math can help us with most of our problems, if we simply understand why a particular study of math is necessary.

            How can you teach required math by reading biographies, even if they were mathematic geniuses?  Here is an example.  If you want to study geometry, begin reading Euclid (I also think he is great to read when you are studying algebra).  Have your child (and you) make up numbers to check the absolute conclusions that he came up with.  See if they are accurate.  Keep a journal.  Some of his propositions, you will be able to check in a couple of minutes, they are such obvious conclusions.  At the time, though, they needed the absolutes presented to understand the more difficult concepts.  Others, however, you may have to attempt over and over, possibly even skip and come back to, before it makes sense.  However, the act of struggling through, trying to discover the truth behind the theory, will cement that knowledge into your head and you will find yourself (and your child) using what you learn in daily life.

            If you are like me, not quite sure where to start, find the “mathematics” section of the library and pick a book that looks like it is about one of the classic mathematicians.  Don’t pick a current person initially – choose someone who is regarded highly even after centuries of other mathematicians checking their data.  You will discover how the mathematician thought, which is invaluable if you are going to use the math you and your child will learn.  Keep a journal, writing down concepts and people you would like to know more about, and work out any mathematical concepts which are brought up in the text, even if it is a very rudimentary idea.

            Math is valuable in daily life.  Algebra teaches logic.   Geometry teaches methods to recognize patterns in the natural world.  Calculus teaches methods to solve problems connected with objects in motion.  There is a reason that every level of math is considered important – learning about the “master mathematicians” help us learn how we can apply math is our everyday lives.

Tuesday, October 25, 2011

Protests and Ignorance

            Even though I have pretty strong political views, I try very hard not to mention them in my blog.  The purpose for this blog is to provide helpful smidgeons of advice when I stumble across them, and politics does not usually fit into that category (there’s nothing little in that topic!).  However, I have recently been bothered by something in the news which I feel that I need to bring up in the blog, as it deals with education about politics (or the lack thereof).   I apologize in advance for this lengthy post, but I could not rightly cut any of it and I felt it was necessary to post.

            The Occupy Wall Street protests have been in the news a lot recently.  Along with you, I have heard commentators and news reporters all try to analyze who is in charge of this seemingly mismatched mob which has encamped itself around what is considered the center of business in the United States, with occasional bands sprouting up across the nation. The biggest question seems to actually be two questions:  Who are they? and What do they want? As basic as these two questions are, they have not been easy to answer.

            I heard a very interesting radio program recently which helped shed some light on this “movement” for me, and I want to share a little of what I learned with you.  I do not normally listen to Dave Ramsey (I should, I know I should, which is why I get his e-mail updates and why I am “friends” with his program on Facebook, but I just normally don’t take the time to listen).  However, when his October 19 program advertised that it was about the “OWS” (as he called the Occupy Wall Street Protesters), I decided to listen.  I have heard other radio hosts take callers from this group, but I like how Dave talks to callers he does not agree with, and I knew he would not agree with this group.  I listened to the entire 3 hours (going to bed at 12:30 am – a great feat since I normally wake up with my husband at 5:00am). What I heard really opened my eyes.

            Dave had only planned on using one of his three hours for these calls, and he began by saying he only wanted to hear from those who support the OWS protesters.  Each person who called, he asked what they supported about the movement.  Caller after caller listed their grievances – some Dave agreed with (i.e., against the bank bail outs) and some that were so far out there that I wondered how Dave could keep a straight face (like the guy who said plastics were poisoning the world).  Caller after caller was certain that the reason everyone was protesting was because of their own “pet” complaint, yet, caller after caller, Dave explained to them that their call was not one of the reasons listed on the OWS official website (I think two calls could actually have been loosely connected with these official reasons for the protest).  They were there for one reason, and they were certain that their own protest was a valid purpose for the movement.  Dave struggled through the first hour to enlighten them that the movement was NOT there to support their claims.       

            Each protestor knew why he was there, but every caller had no idea why the protest had been organized.  Ignorance, it seemed, was the only common thread between the calls.  Ignorance, the lack of knowledge – not necessarily a lack of intelligence (though a few calls fit into that category, as well). 

            Some people called saying that they were against Wall Street.  Dave asked them if they realized that Wall Street was simply a street in New York City.  Many callers had not made that connection. On Wall Street, as Dave explained, was a building called the New York Stock Exchange, a place where investors exchange stocks.  He called it the flea market for stocks.  That’s all they do, and it’s only a limited number of businesses that participate in that.  None of the callers were against honest businesses, and they all lost some of their steam as Dave talked to them.  Also, Dave pointed out to several of them that what they were upset about was not on the list of 15 items which the website listed, so they were adding their energies to a movement that was not supporting them. 

            Other callers said they were against the Bush tax cuts, yet when Dave asked them what that meant, none of these people could explain.  Again, ignorance of their main issue ruled these particular calls.  Dave asked if they knew that putting $2,000 aside for their child’s college education tax free was one of the tax cuts, and asked why that was a problem.  None of them knew that. So, Dave would ask them exactly WHICH tax cut they were against, and none had an answer. One caller replied, “Well, I’m not against that. I’m just against the phrase, ‘Bush Tax Cuts.’”  Once again, ignorance ruled these calls.

            Some claimed to be against the inequality of wealth, though when Dave pointed out that 8 out of 10 millionaires had started out in either poor or middle class families and had worked their tail off to make their money, listing specific examples by name, none of these callers wanted to take their hard earned money.  Dave did point out that this would be stealing.  Everyone who called in to complain about the lack of jobs actually HAD a job themselves, they were just grieved that people out there “somewhere” didn’t have a job.  Ignorance of business and the way money works were the common thread in these types of calls.

            In the middle of his second hour, a Twitter message went out from the OWS “leadership” saying that Dave was only taking the “stupid” callers, which he immediately addressed on-air, challenging anyone to call.  They were not turning away supporters of this protest if they could halfway put two sentences together and if they had a phone that had signal strong enough to be understood.  Toward the end of this second hour, a man called in who called himself a “Trainer” for the OWS protesters.  Dave asked what he taught these people, and the guy simply said that he taught them whatever he felt they needed to know.  Some things were about life as a protester, while other things he said he taught was about life in general, but he could not even name one item he taught.  His own personal protest, his reason for joining this group he was training, was also not even on the official list.

            So why were these people there?  No one, not even those who were standing on Wall Street, seemed to know.  Then came the last caller of hour three – one of the original founders of the Occupy Wall Street movement.  His call, in my opinion, was the most enlightening.  After hearing three hours of supporters of this movement, which was not actually moving anywhere, hearing one of the “founders” helped bring the entire protest into perspective, but it only helped after hearing all the other callers.

            This man who said he was one of the originators of protest was in his early thirties, like most of the callers (most of whom were born in the 80’s).  He was thrilled that the movement had gotten as big as it was.   Dave asked him why the callers for the past three hours, many of whom had actually stood on the street to protest, all had different reasons for being there and not even the trainer knew the “demands” made on the official website.  The young man reiterated that he was very excited that what had started as a small group of people had now extended across the United States.  He said it didn’t matter that they were there for different reasons – he was just glad that they had a venue to express themselves.  The leader of the movement had NO INTENTION OF LEADING THIS GROUP!! 

            Why is this a problem? Why not just allow these people to express themselves?  As Dave told this young man and other callers, if you have a legitimate protest, come up with a solution and start your own protest, but THERE NEEDS TO BE SOME WAY TO RESOLVE THE ISSUES, and the way things stand right now, there is no way for this to happen in the OWS protest.  Each person is there for a different reason.  Dave told this young man that they need to communicate with those who are protesting, so they know why they are there and when it’s time to go home.  So far, almost a week after the show broadcast, this has not yet happened.

            Another problem is that, with this “movement” not having a leadership or any organization, there is room for other, well-organized groups to step in and take control.  This is already happening.  Communist and Marxist organizations have started setting up platforms, talking to those who are protesting in various cities across the US.  Labor Unions have tried organizing the people in these crowds.  All have had a small amount of success, though with everyone there for a different reason, none of these groups have control over the protest.  Criminals have already found that there is no leader and rapes, muggings, and general unlawful activity are ruling many of these protests.  Yet, we still have people blindly, ignorantly getting angry at the local law enforcement who are trying to keep the peace during these unorganized protests. 

            Ignorance is ruling the protests.  There is a website, for anyone interested in finding it, that lists fifteen specific reasons for the protest, though very few have actually bothered to look this up.  For others, the ignorance is about the actual issues that they claim to be protesting.  Still other people seem to be ignorant of the fact that they have a legitimate protest that they should start on their own, organized protest. 

            Simply protesting to say you are mad is not a true protest.  There is no way to resolve the fact that someone is mad.  A peaceable protest is one of the great rights we have in this country, however, an unruly mob has no place in a civilized society. 

            We need to educate ourselves about the facts. Don’t just blindly join a group if you do not know why they are there.  If you have an issue you want to protest, you need to clearly communicate that issue and protest it in a lawful manner.  "I disapprove of what you say, but I will defend to the death your right to say it" was written by Evelyn Beatrice Hall, attributing it to Voltaire.  People have a right to protest, even the guy who believed plastics are poisoning the world, but they need to do it in a peaceable manner with some method of resolving (or dissolving) the protest in mind.

Wednesday, October 19, 2011

The Clockwork Universe, by Edward Dolnick

The Clockwork Universe

By Edward Dolnick

            I have started looking for “classics” for my daughter and I to read when she gets into middle and high school years.   If you have not yet read my blog about A Thomas Edison Education, then I highly recommend for you to read it so you understand my purpose for looking for educational classics (of course, I recommend you read the BOOK, too, not just my blog!

          In the Thomas Edison book, I was surprised to discover that there were “classic” books in every field of study, not only in literature, and so I started a quest to discover books written by the “great minds” of history.  The first which I found was a book written by Euclid.  Since that book was written by him, with notes added by another, more modern mathematician, I had no idea who he was or why I wanted to read about what he said about the subject of Math – it was simply the first name on a list which I made of names to look for.  While I learned a fabulously easy way to find the Greatest Common Factor, the rest was difficult for me to follow. I would not recommend reading Euclid to begin reading about the classics (although I discovered that the MOST boring parts of the book were not actually Euclid’s writing at all, but instead were the writings of the person who chose to edit his works!).  I will probably eventually go back to him, but I’m not ready yet (and my daughter certainly isn’t!).

             The book I am reviewing today, The Clockwork Universe by Edward Dolnick, was my second attempt to discover a Mathematical Classic.  I think I struck gold! 

            It is not that this book is an especially deep thinking book (thankfully! – I don’t think I was ready for another of those so soon!).  What struck me as exciting about using this book for our homeschooling instruction in middle school years is that this book tells me about several of the mathematical geniuses.  I have heard of Sir Isaac Newton, Aristotle, Kepler, and Galileo.  Descartes’s name came up in some of my reading, though I couldn’t remember EVER having seen his name before my recent readings.  I knew that between the 1600’s and the 1900’s there had been a lot of changes in the way Science and Mathematics were viewed, but I quickly discovered that I had never really read or been taught about any of that in any depth. 

            TheClockwork Universe, I believe, is a good book to start with in educating your children about Mathematical and Scientific classics.  It gives a brief biography of these early geniuses, and allows us to see a brief glimpse of how their thought processes worked.  It shows how in their day, science and math were not separate subjects of study.  They were also not designed to be taught separate and apart from the real world, but instead, the real world demonstrates daily what math and science prove.  Most importantly, to me at least, it does not take today’s view of ignoring a person’s religion altogether when writing about what a person did in his life.   Whether or not he agrees that the God of the Bible is the one and only true God, Edward Dolnick acknowledges throughout the book that these great scientists believed that, and a good many of their experiments and discoveries were designed to prove what they already believed. They saw beauty in the orderly system of nature, and fully believed that only a divine being who was much greater than we are could have created so much precision and beauty. 

            The Clockwork Universe tells me who these people called geniuses actually were, why they did the experiments they did, what serious mistakes they made, what serious misconceptions they maintained through part or all of their lives, and who their contemporaries were.  It tells how the bubonic plague affected many of these men, either directly or indirectly, and gives some insight as to each person’s personality.  I can use this information as a springboard to dive into other studies in the future, this time understanding a little better who I will be reading about and knowing why I care what they have to say.

            Whether you’re homeschooling, helping a child who attends public or private school, or just interested in learning more on your own, The Clockwork Universe gives interesting insight into the men who helped advance our mathematics and sciences into the technological advances which we have today.

Wednesday, October 12, 2011

How Deep is Your Ditch?

              Ditches are long narrow passages made in the ground by digging.  People dig ditches for various reasons, at varying depths and varying purposes.  Someone once compared the educational system to men digging a ditch.  How deep is your ditch?

              If you think about it, ditches have different lengths and depths, depending on their proposed purpose.  Some ditches are for laying pipeline, necessary for water or sewage.  These have to be dug at a certain depth so that they avoid problems with weather, but not so deep that the diggers have to dig through rock and other obstacles that are unnecessary for the purpose.  Some ditches are short but deep, like a trench used by soldiers on a battlefield or a trap used to catch animals.  Other ditches are designed for directing run-off water, and the depth of those depends on the amount of water expected at any one time.  The purpose of the ditch decides the depth and length which it will be dug.

              So how does this compare with education? The depth of a person’s education shows how much that person understands a particular skill.  The length of that education shows how many individual skills in one particular subject that person learns.  If someone spends a great deal of time and energy understanding a particular topic, then we can say that person has a deep understanding.  If that same person knows many things about a special subject, then we can say that person has a long “ditch.”  People only have a limited amount of time in school, whether homeschool, public school, or private school, so how “deep” and how “long” should a person’s educational “ditch” be? 

              This person who compared today’s schools to a ditch (sorry, I cannot remember who it was) said that if America’s Educational System were a ditch, it would be 2 inches deep and 2 miles long. Wow!  That is a really long ditch!  There are many purposes which could be satisfied by a ditch that long! But, only 2 inches deep?  Really?  What use is that to anyone?  Good question.  And that’s where you should be examining your own educational program.

              How deep are you teaching the students in your life?  Whether you teach one child or thirty, teaching them a lot of things on a checklist with no depth to each item is not really valuable, is it?  Just like a 2-inch deep ditch, there is no purpose for it.  Instead of teaching a little of a lot of things, teach a few things with a lot of understanding.  Like a person digging a deep ditch, you may spend a lot of time in the same place, doing the same thing, but you will end up with a product that can be of value later, once this “ditch” is finished. 

              What does a “deep understanding” look like?  Let’s take something many teachers teach a lot of but frequently don’t teach with depth – addition.  Typically, children will learn to count their numbers, then they will learn how to use either a number line or their fingers to add single digits.  From there, they go on to double digit addition, and then harder problems.  (Yes, I am skipping subtraction skills, which are usually taught with addition, but subtraction is not my focus right now).  Many children fully understand that when you have five apples and your friend brings you 2 apples, then you will end up with 7 apples.  However, most math programs may only have one or two questions that are written that way.  Usually, the child sees 5+2=___.  What does that mean?  Some children see it as the same type of problem, but other children do not.  Are these children behind?  NO!  These children are exactly where they should be developmentally – they understand things concretely but not abstractly (see my other article “Not Just a Little Adult” to understand more about concrete and abstract). 

              If a child is given a problem of 5+2=____, the teacher should use the concrete, pictorial, abstract method for this problem.  First, for the concrete part, the teacher should get 5 items and 2 related items (these could be 5 red grapes and 2 green grapes, 5 stuffed animal dogs and 2 stuffed animal cats, 5 red cars and 2 brown cars, or whatever you have available).  Have the child take them out and put them in 2 separate piles, then push them together and count the entire group.  Write the answer.  Then get another set of 5 items & 2 items, so they can see that you can apply this type of problem to many things, not just what you took out. Do this 2 – 3 times.  Then, go to the pictorial method.  Have your child draw 5 circles in one box, then 2 circles in another box.  Have them put the plus sign between the boxes and put an equals symbol after them.  In a box after the equal symbol, have your child draw 7 circles, crossing off one circle from each of the other boxes as they put them in the equal box.  Then count what is in the box and write the numeral 7.  Do this again, only this time, cross off the squares (or whatever you have them draw) and count aloud. This time they do not need to draw them in a box on the other side of the equal sign – only write the answer in numeral form.  Finally, after a couple of different problems, have the child write “5+2=____.”  Then they can answer this problem.

              That seems like an awful lot of work to solve one problem!  It would take FOREVER to have the child do an entire worksheet full of those addition problems, especially when it is so much easier to teach them to count on a number line or their fingers.  If this was your thought, then you are in the habit of digging 2 inch, 2 mile ditches.  Yes, it would take too much time to do an entire worksheet full of these problems (doing 3 concrete examples, 2 – 3 pictorial examples, and 1 abstract notation for each problem), so don’t! Only do 3 – 5 problems like this or possibly only one problem if the child becomes very frustrated.  However, where only a few students understand what they are doing when they count by number lines and fingers (an abstract concept), almost EVERY student would understand what they are doing with the depth method (which begins concretely).  THAT is your goal, isn’t it?  To make sure they understand what you are doing.  As a 4th & 5th grade teacher, it was very frustrating to find students (very intelligent students, by the way) who did not really understand what they were doing when they were adding.  Since multiplication is simply repeated addition, it meant even less understanding of the multiplication tables.  These students were on a downhill slide of education which is harder to stop the longer they are allowed to continue.  If they don’t know what they are doing when they are multiplying (just writing down something they memorized), then how can they understand Algebra?  Geometry?  Trig?  If they can understand the basics, then the “harder” work will be much more readily understood.

              Math is not the only subject this is true for.  If children have a chance to play with water, then they can be ready to understand the three forms of matter (solid, liquid, gas).  If they take field trips to a nearby lake or cave or mountain, they are more able to understand landforms and varying bodies of water.  Education should be useful, not just philosophical.

              How deep is your student’s ditch?  Spend time in the same place, doing the same thing over and over.  When you feel they understand one part very well, go to the next step (instead of 5+2, learn 7+6).  Just like when you dig deeply in one place it makes it easier to dig deeper in the next step, when your student deeply understands one thing, the next step is much more easily understood, and probably won’t take so much time.  Dig ditches that will be the foundation for a valuable education – your student deserves it!