Prime Factorization, GCF and LCM

   I was relieved to know we were moving on to this topic in chapter five.  I didn't enjoy our first chapter, sets and shading etc.  I absolutely hated that chapter and it even showed on my test.  Therefore, I am thankful we have test corrections in our class.  I am not going to lie, I was surprised when I realized what we were going to be doing.  I haven't heard of prime numbers, GCF and LCM in a very very long time which means I haven't done it. 
  I am expecting to do much better on this upcoming test.  I feel that I know and understand the material much better than last unit.  I don't mind doing the factor trees and figuring out problems.  I hope to do much better on the true or false section on this test. 
  What I really like about this unit is that there is a choice of methods to use to figure out the GCF or LCM.  I mentioned above that I don't mind the factor trees, which is one of the theorem's to choose from.  I am not a fan of the set intersection method for finding GCF.  It it too time consuming and there is always the possibility of missing a factor.  It can also be difficult for someone who doesn't know all the factors of different numbers, I am one of those people. 
  Another theorem I consider using to find the GCF is the subtracting method.  This method is good to use especially when the numbers aren't too large.  To start out it'd look like this GCF (350-250,250) and one would continue that until there are two similar numbers in the parentheses.  There is another theorem that deals with long division.  I personally haven't done long division without a calculator since elementary school.  I don't really bother with this method which is why I'm glad there's a choice. 
   It's always important to know that GCF, with prime factorization, needs the lowest exponent with only COMMON numbers.  Where as LCM needs the highest exponent with ALL numbers. 
  The prime factor test is the main thing I struggle with in this unit.  I confuse some of the rules to determine whether or not a number is prime.  I know how to figure the square root in that first step, which is why it's important to know all the numbers up to 100.  I know I have to test primes up to the first number.  From there I just seem to get confused.  I will just have to practice more on that part of the unit.  This time around I am not worried about this test and seem pretty confident.  I just hope my mathophobia doesn't kick in once I receive the test.   
    
  

Piaget

    The following article "Applying Piaget's Theory of Cognitive Development to Mathematics Instruction" by Bobby Ojose, discusses concrete stages of cognitive development in children.  Jean Piaget is a primary focus in children's psychology and has contributed a great deal of information in that field.
    All together, there are four primary stages of development.  They are sensorimotor, preoperational, concrete operational, and formal operational.  Some of this information is a summary to me because I have taken two psychology courses here at Dutchess Community College.  Each course discussed Piaget and the stages that were mentioned above. 
    The sensorimotor stage is “characterized by the progressive acquisition of object permanence in which the child becomes able to find objects after they have been displaced, even if the objects have been taken out of his field of vision”.  This stage is also important for a child to link numbers to objects. It’s interesting to know children at such a young age are able to have this concept of numbers and counting.  Mathematics starts early and it is just the beginning.
    The next stage is the preoperational stage.  Now, children engage in problem-solving tasks. Children can use materials such as blocks for building and sand and water for measuring. 
    The third stage is the concrete operations stage.  Cognitive growth in this stage is remarkable.  Hands-on activities are great because they have mathematical ideas and concepts which is useful for problem-solving.  Manipulative materials include: pattern blocks, Cuisenaire rods, algebra tiles, algebra cubes, geoboards, tangrams, counters, dice, and spinners.  These are all great items to contribute to growth in mathematics. 
    The last stage is the formal operations stage.  Children develop abstract thought patterns where reasoning comes into play.  Such reasoning skills include clarification, inference, evaluation, and application. 
    With these stages, teachers have the opportunity to develop lesson plans based on where a child stands.  Teachers are able to know what children know math wise and can provide proper materials that children will know how to use.  At certain stages, I would only focus on what type of growth is occurring.  For example, in the sensorimotor stage, one can read books to the children that show objects linked to numbers.  There could be a picture of three dogs with the number three right next to them.  Children could easily begin this concept and make the necessary connections. 
    I always wonder where things would stand if people like Jean Piaget never discovered stages of development in humans.  Piaget made it possible for teacher's to teach math at a proper level to children. This was an interesting article and it really refreshed my memory. 

Visual Learner

     Everyone prefers a specific learning style.  A person can choose from over three learning styles when dealing with a subject.  These styles include visual learning, auditory learning, and kinesthetic/tactile learning. In math, I consider myself a visual learner.  I need to see examples and pictures drawn.  This way, I am able to make more connections and have a better chance of remembering the material.  Being a college student, I take many notes from lectures and I prefer those notes to be written by the professor on a blackboard.  I feel more confident by this because when test time comes around, I know I have material ready to be reviewed.   

     All teachers are different and I never know what to expect on the first day of class.  Some teachers prefer to talk the full hour and fifteen minutes while students jot down facts/statements that have been said.  Others make use of power points and blackboards for students to copy from.  In math, visuals are a must.  It's easier to make graphs and draw diagrams to figure out a problem. 

     The following article "Learning Styles" is a good way for someone to figure out which type of style is strongest.  I pretty much answered yes to all of those questions in the visual learner section.  I like to see things written in order to read it as many times as I need to.  Which is why I prefer written instructions as opposed to oral instructions.  I know where to look and don't have to bother asking over and over.  When I study, I rely heavily on notes that I've taken in math class.  I write down everything that's written on the board.  It could be a number of examples and terms, as soon as I see them, I write them down.  Later on, I know exactly where to look and can remember what we did in class. 

     During math lectures, I don't doodle or have my mind elsewhere.  Knowing that math is my weak subject, I focus and am too busy writing as well as looking at the examples.  I give a lot of credit to those who are strong in auditory learning.  In one lecture alone, a lot of material is covered and a ton of information is said.  Those who can listen a great length of time and be able to remember the information, are not simple tasks.  That is just my opinion, like I've said above, I need to write notes in order to remember information.  In a sense, I can see how auditory learning could be helpful in some aspects.  One being, saying words or terms aloud over and over.  That is something I do when I am really struggling to remember something.  It sinks in more this way, but overall I am a visual learner and I will continue with writing notes, using pictures and looking at my math book.