Preventing Students From Saying, "I'm Not Good At Math."

  

                                         Photo courtesy of: www.motherpedia.com




   Here I am writing as the summer semester winds down and I find myself again thinking of my future students.  When you pursue an Elementary Ed. Degree, you know that your license will be a K-6 license, which means that I could one day have a classroom filled with five and six-year-olds or a classroom full of twelve and thirteen-year-olds, or any age in between.  Whatever the age, as their teacher, how will I help my students to engage in math?  to feel confident and competent in it?  to be interested in it?  to, dare I say, enjoy it?
   There are many articles out there about math and kids, most echo what we already know as students and parents of students.  What I know is that I will need to be proactive about both easing my students' math anxieties and encouraging them to embrace the math concepts that I am trying to teach them.
   We all have at least one memory of being called on in class, or worse yet, called up to the chalkboard and not having the correct answer to a math problem.  Those types of scenarios do not help students to feel comfortable with math.  Public embarrassment is not a good way to instill confidence in a child.  Confidence is key when it comes to math.  When a child believes that he or she is capable of coming up with the correct answer, or is at least comfortable asking questions that lead to that end, math becomes enjoyable.
   Not only do teachers need to provide many opportunities for students to be successful and to gain confidence in their own abilities, but we also need to make sure that our lessons are relevant to the real lives of our students.  How many times have I heard my teenagers ask disgustedly," When am I ever going to need to know this?"  We need to show our students that math is useful in everyday life.  If we don't, we lose their attention.  Lessons need to be linked to things that matter to them so that they care about what they are learning and see its purpose.  Annie Murphy Paul echoes these sentiments in her blog post entitled, "How Do You Spark A Love Of Math In Kids."
   Math is all about knowing the formulas, recognizing the patterns, making note of previous learning and knowing which strategy to "pull out of the toolbox."  It will be my responsibility as a teacher to help my students to do just that.  I hope to help them to love math in the process.

Teaching Children About Fractions

   The neat part about being in math classes right now as I pursue my Elementary Ed. degree is thinking about how I will apply what I'm learning to my future classroom one day.  How do I take this information and prepare to present it to my students in ways that they can understand and actually use in their real lives?  How will I break these large mathematical concepts down into smaller bite-sized chunks for them?  For example, how do we teach young children about fractions?
   Even children of very young ages can understand the idea of "half."  "Give half of those animal crackers to your brother..."  "Here, we'll split the pile of Legos in half so that you can both build with them..."  Half is pretty understandable, but what about more complicated fractions?  For example, which is bigger, 1/8 or 1/4?
   I guess I have a bit of a theme here lately with references to food.  I didn't intentionally do that, but here I go again.  In the classrooms that I work in as a Para Educator, I find that food can be very helpful when discussing fractions.  A Kit-Kat bar works nicely to show fourths since it is conveniently packaged as one chunk of chocolate that can be broken apart quickly and easily into four separate but equal parts.

  We can show our students that one of the four smaller bars is one fourth of the whole candy bar.  Simple enough.  The fun part is to lay another Kit-Kat bar under the overhead projector so that the children can watch each "fourth" being cut in half.  Now the candy bar is divided into eighths.  This makes it easy to see that you would rather have one fourth of the candy bar than one eighth.  Children tend to think that the bigger denominator must be a bigger number.  This visual helps them to understand what fractions mean.  Would they rather have 4/8 of a candy bar or 4/4?
   Another idea that makes learning about fractions easier is the use of pizza.  We can cut one pizza into eight equal pieces and another into sixteen equal pieces.  Then let the fun begin.  Which is more? 4/16 or 2/8?  There are some surprises for children when working with fractions.  Even those children who are not typically visual learners benefit from visuals when it comes to fractions.
  I found an interesting article from the U of M titled, "Teaching About Fractions: What, When, and How."  The article states that we need to let our students use manipulatives and give them many opportunities to explore fractions in hands-on ways.  It doesn't always have to be food.  It can be blocks or coins or paper cut into equal parts.  (Kids do get excited about getting to eat an 1/8 of a Kit-Kat during math, though.)
   Here is one example of a neat manipulative for learning about fractions:

Following the Recipe

      I like to bake.  Actually, I really really like to bake.  It's one of my only hobbies that finds me indoors now that I think about it.  I'm not one of those bakers, though, who can just whip up something delicious without a recipe in front of me.  I like recipes.  I know several women who can rattle off the recipes for their favorite treats off the top of their head with 100% accuracy.  I have always marveled at that ability because I do not have it.  I've been using the same chocolate chip cookie recipe since I first learned to bake about thirty years ago and I am not even close to having it memorized.  I wouldn't even try.  I get that little recipe card out and follow it each and every time.  Maybe the recipe card is my security blanket.  I don't know.
   What I've realized this summer while taking these math classes is that math is a lot about knowing the recipe.  In the language of mathematics, the "recipe" is called a "formula," but it's really the same thing.  If you don't happen to have the formula memorized, you simply need to remember that there is in fact a formula and then use it.
   There have been many times over the course of these past several weeks that I have read and then reread a homework problem, wondering how in the world to even begin solving it.  I've started scratching out some preliminary guesses as to how to plug in the information that's given and then realized, "Hey, there's a formula for this!"  It's amazing how even the most difficult of problems can be solved with the formula that was designed specifically for that purpose.
   If you want to figure out the volume of a pyramid or cone, you'll want to know that V=1/3 bh.  If you want to be able to figure out the value of each interior angle of a polygon, you'll want to know that the formula (n-2) x 180/n is the one that you need.  The "n" is where you plug in the number of sides of the polygon that you are working with.   For example, if you are figuring out the interior angles of a square, you'd plug the number 4 for the four sides into the spot in the formula where the "n" is.  (4-2) x 180/4 will give you the answer of 90 degrees.  Pretty neat.  Try it for a hexagon, an octagon, or even a 22-gon and you'll soon have your answers.  Try it without the formula and you won't get very far very fast.
   Do you remember hearing the formula for slope: "rise over run?"  That's an easy one to remember.  How about that area of a rectangle is "length times width?"  Could you memorize the formula for the equation of a line from two points, though?  What about the distance formula show below?

   Not only do we need to know these formulas in order to "know" math, but we also need to understand the order of operations.  We again need to follow the recipe.  When solving an algebraic equation, which part do we start on?  Here's a handy little tool courtesy of coolmath.com for that:

   If you don't know to do the part of the problem that's in parentheses first and to do all multiplication and division before you do any addition or subtraction, you'll just make a big mess of the problem and end up with the wrong answer.  Like I said, it's a lot like baking.
   I have found that knowing the formulas is empowering.  Knowing that there is a formula to plug the numbers into is empowering.  Solving the problem with a correct answer at the end is satisfying much like biting into that warm perfectly baked chocolate chip cookie is satisfying.  If I'm being honest, I enjoy the cookie a little bit more than the math problem, but you get my drift.

 photo courtesy of www.meals.com



    

A Slice of Pi

  
   I find myself thinking about math more these days than ever before.  I have to admit that these past four weeks have been intense.  My brain has been inundated with math concepts to the point that I even ponder ideas having to do with math when I could be thinking of other things.
   After revisiting many formulas, I began wondering about the concept of Pi.  Where did it come from?  How long has the idea been around?  Who came up with it?  Was his last name "Pi?"  I'll answer that last question right away.  No, it wasn't.
   After doing some investigating, it seems that a mathematician named William Jones was the first to use the symbol for Pi, although he did not invent the concept of Pi.  According to an article on Math.com, "Ancient civilizations knew that there was a fixed ratio of circumference to diameter that was approximately equal to three.  The Greeks refined the process and Archimedes is credited with the first theoretical calculation of Pi." 
   I have been finding out that there are many people who have devoted their entire lives to the study of mathematics.  I wish I could have met them and been in the presence of such intelligence.  I also wish that one of them could sit next to me as I do my homework, but I digress. 

                                                                 William Jones
                                                   Photo courtesy of: en.wikipedia.org




   I found this very interesting article on some of those math-minded geniuses titled "5 brilliant mathematicians and their impact on the modern world."  It is very clear that the contributions of these five men; Isaac Newton, Carl Gauss, John Von Neumann, Alan Turing, Benoit Mandelbrot, helped to make our world what it is today.  We would not have the everyday lives that we currently have without them and without what their minds helped to develop.   Many of the gadgets that we have come to rely so heavily on; computers and cell phones to name just two, would not be possibilities without the math concepts that are foundational in their operations.

                                                                  Isaac Newton
                                                 Photo courtesy of: www.brighthub.com

   So back to this notion of Pi.  The number represented by the symbol Pi, or π, is 3.1415926... ...
and goes on...and on...and on...  Here is a list of the first 500 digits of Pi as taken from an article titled,"Memorize the Number Pi to 500 Places:" 


500 digits of pi, written as 50 digits in each row:

3.1415926535897932384626433832795028841971693993751
05820974944592307816406286208998628034825342117067
98214808651328230664709384460955058223172535940812
84811174502841027019385211055596446229489549303819
64428810975665933446128475648233786783165271201909
14564856692346034861045432664821339360726024914127
37245870066063155881748815209209628292540917153643
67892590360011330530548820466521384146951941511609
43305727036575959195309218611738193261179310511854
80744623799627495673518857527248912279381830119491


Pretty amazing, isn't it?  Believe it or not, there have been many many people who have spent more time than we can imagine memorizing thousands and thousands of digits of Pi.  There's even a world record category for it.
   For those who have no desire to try to memorize the endless list of numbers in Pi, we can at least appreciate the old "Area equals Pi r squared" from elementary school.  The next time you're trying to figure out the area of a circle, you'll be serving up a slice of Pi with that formula.  If I were you I'd stop at 3.1415.  A little Pi goes a long way.

 
 
 

A Whole New World

   Here I am on this adventure of taking two college math courses during the summer session and I find myself stunned at the amount of information that is coming my way.  I've already admitted that math is not my strongest subject, but what I've been studying this past week has really given me a lot to think about. 
   I've been studying and working problems on topics with names such as:  "Connecting Euler's Formulas for Planar Networks and for Polyhedra," "Working With Angles in Regular n-Gons," and "Determining the Interior Points of a Simple Closed Curve in the Plane," to name a few.  Those are topics from my Mathematics For Elementary Teachers II class.  This blog is actually designed as part of my studies for Mathematics For Elementary Teachers I class.  I mention both classes simply to say that while summer is normally the time for outdoor fun and swimming in the sunshine, the only thing swimming around here right now is my mind.  It's a good thing.  Don't get me wrong, but it is a new thing.  New things always take a little getting used to.
   I did give myself a break from the studies the other night and went outside to play on the trampoline in the dark with my daughter.  We laughed and jumped and sang songs like a couple of dorks.  After a long time we laid down on our backs and looked up.  I laid there looking at the stars and feeling the breeze of a cool summer night.  As I noticed the hexagonal shape of the net that surrounds the trampoline, I thought to myself, "I bet I could figure out the angles of each adjacent side of the net."  That's when I realized, this math stuff gets in your head.  I suppose that's the whole point.
   After spending several hours this past week learning about numeration systems of other cultures from both the past and the present, I have a new found appreciation for all things math but especially for our numbering system, which now seems so much "easier" than others.  It's only my second blog post and I am already referring to math as "easier?"  Huh.
  In all seriousness, we can be glad to have the system that we do for counting and combining numbers.  I learned about the Egyptian, Babylonian, Mayan, Indo-Arabic, and Roman numeration systems.  I even learned something called "The Russian Peasant Algorithm" for Multiplying."  It consists of making two columns of numbers beginning with the first number to be multiplied placed at the top of a column on the left and the second number to be multiplied placed at the top of a column to the right.
    From there, the numbers in the left column are halved and the numbers in the right column are doubled until the last remaining pair of numbers begins with a 1.  The pairs of numbers are set up in a table.  Rows of numbers beginning with an even number are crossed off of the list.  At the end, the sum of all numbers remaining in the right column are added together to find the answer to the original multiplication problem.  It is time-consuming, but really rather amazing.  You'll want to try it.  Go ahead.  Just do it.

Multiplying Using the Russian Peasant Algorithm

 

 

 

 http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml


   Each of these systems is equally impressive.  The Mayan and the Babylonian systems were pretty incredible and left me scratching my head.  It's one of those things that at first is like a puzzle that you have to try to wrap your mind around and then once you get it, you get it and you can appreciate it.  If you've ever mastered one of those metal "brain teaser" types of puzzles, you'll know exactly what I mean.  You know, the ones where you have to get one part of this metal puzzle separated from the other and it seems impossible...until you do it...and after that you can't not know how to do it?
   Mathematics is a language of its own, but it is one that has been spoken in every culture across all times.  The wording and the symbols may be different, but the ideas are the same.  As for me, I all of a sudden really really like the way that we "do" math here in America.  And with that, I have some math to do... ...

Here is a chart of the Egyptian symbols for the powers of 10:

https://www.google.com/search?q=Egyptian+Symbols+for+powers+of+10&client=firefox-a&hs=FJV&rls=org.mozilla:en-US:official&channel=fflb&source=lnms&tbm=isch&sa=X&ei=dXCoU4XgNtevyASshoD4Bg&ved=0CAgQ_AUoAQ&biw=1218&bih=638

Getting Started

   I have this opportunity to create a blog for a Math Class that I am taking this Summer.  I can't say that I'm an expert on blogs, but just like so many other aspects of being back in college, I'm willing to give it a try.  
   It's been many years since I took my last Math Class, and I find myself thinking back to High School.  I can picture the classrooms where I took Algebra and Geometry.  I can remember where many of my friends and classmates sat too.  (In a graduating class of 39 kids, it's really not that difficult to do.)
   I also remember the feeling of fear; fear of being called on to come up to the board and show an answer, fear of not having the correct answer, fear of being laughed at.  My dad used to try to help me with my Math and if any subject made me cry, it was Math.  Well, that or Phy Ed., but that story is for another time.  I can still feel that feeling of sitting at the kitchen table trying not to cry over Math problems that I could not wrap my mind around.
   Some things in life change and some stay the same.  This week we've been learning about sets and operations on sets.  Again, I find myself back at the kitchen table trying to wrap my mind around a Math topic and, well, trying not to cry.  I'm finding that there are symbols that are very new to me.  I realize that their meanings simply need to be memorized, much the same as the multiplication sign, the minus sign, the plus sign and others.
   Here is a list that I found that is helpful for understanding the meaning of these symbols:
  
Table courtesy of: http://enpub.fulton.asu.edu/powerzone/fuzzylogic/chapter%202/chapter2.html
   
   Sets are basically collections; numbers, objects, anything that can be put into a category.  Georg Cantor was a German Mathematician who came up with the idea of sets in Mathematics.