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Common Core Who-Dun-It Mystery

As a member of Teachers Pay Teachers, I often read and share on their Seller's Forum.  As the Common Core State Standards (CCSS) become more "common", many teachers are asking about things being omitted or totally left out.  Let's start this discussion with what the Common Core supposedly is.  CCSS is a state-led effort coordinated by the National Governors Association and the Council of Chief State School Officers. The standards establish common goals for reading, writing and math skills that students should develop from grades K-12.  Although classroom curriculum is left to the states (which actually had no input into the process), the standards emphasize critical thinking and problem solving and encourage thinking in-depth about fewer topics.

With that said, this is the way I perceive these standards.  When I started teaching, (I have been at it for 30+ years) the curriculum was a nice, juicy apple. Included were subjects like spelling, geography, history, and cursive writing.  In addition, areas such as effort and behavior were evaluated. I can't remember ever giving a state or national test, but I did have to teach art, music and P.E. The majority of the children went home for lunch where some adult was waiting for them.  Later, the arts were added to the curriculum and qualified teachers were hired to teach art, music and P.E. (Thank goodness!)

Then came the slicing of the apple.  One test was introduced and given each year.  (We gave the ITBS.)  Objectives were written that were different from the textbook, and more children were staying at school for lunch.  As time progressed, additional slices of the apple were removed as "before and after" school programs became necessary for children and free lunches became common place.  In addition, more than one test was required because now the district and the state wanted data.  History and geography became social studies, and phonics and spelling were replaced with the whole language approach.  I was directly a part of our district's benchmark test writing project where much money and time were devoted to the test that would reveal all, make teachers better, and students smarter.  Of course, none of those things occurred, and the money wasted could have been better spent on teachers who really make the difference in the classroom.  (By the way, all of those assessments are now gone.)

Many advocate that the CCSS will become the tool that can successfully turn around education.  Let's remember that these standards are merely the minimum each grade level is to master.  Think of the CCSS as the core of an apple; there is no "meat" on the core, just the left over part of the apple.  The basics are there; but teachers need to add the meat, but will they, especially since the high stake tests that are imminent will most likely only test the core?  My question is: How much testing will be required; how often, and at what expense in money and time?  And who will pay the price?

I did some of my own reading of the CCSS, particularly those for the "key" grades of K-3.  Yes, the CCSS requires the multiplication tables be taught through 10, but does that mean a teacher shouldn't go to 12?  I personally want all of my algebra students to know the doubles through 25 because it makes finding the square root so much easier.  Since the multiplication fact is in the student's head, no calculator is required!  I also observed that money is not mentioned anywhere in the common core for grades K-3 except in second grade.  Covering that standard is going to be a daunting task for 2nd grade teachers if students have never seen it before.  (I did find the money standard in grades 4th, 6th, and 7th, and after that, money was considered Consumer Science.)  In addition, there is no standard for patterns in kindergarten which I find quite disturbing since all math is based on patterns.  Time and introductory place value have also been deleted.  If students do not get these basic concepts in kindergarten, it is obvious they cannot grasp the more complex ones in later grades.

As I read the many different responses on the TPT Forum as well as various articles about the Common Core, I realized that many teachers are viewing them as the all-in-all.  If that is all that will be taught, education is in BIG trouble.  I would suggest reading a rather thought provoking and eye opening article by Carol Burris, principal of South Side High School in New York. She was named the 2010 New York State Outstanding Educator by the School Administrators Association of New York State, and she is co-author of the book Opening the Common Core.

My primary concern is that the Common Core will become so focused and fixated on a limited number of standards that little will be left of well-rounded education except a very inadequate and flimsy core. If a teacher only follows the common core and nothing more, students will miss important building blocks in between. I believe education is an exciting and engaging lifetime journey, not a final destination or a binding contract with any government. I trust and hope parents (families) are the constant in this equation (a math word!) while schools and teachers are the variable. (both will change over time) How can any test adequately measure that?

Here is a ten minute video which explains the Common Core so that
anyone can understand it. Check it out at: Common Core

Much Ado About Nothing

I have decided to post (this is an updated previous post from 2011) some questions about zero that my college students have asked me in class.  I will say this, "Zero can surely give you a severe headache unless one knows its properties."  

Question #1 - Do you know why zero is an even number?    All mathematics is based on patterns.  Because of this, I know that an even plus an even number will always give me an even answer; an odd number added to an odd also gives me an even answer, and an odd number plus an even gives me an odd answer. In other words:    E + E = E     O + O = E     O + E = O

The numbers 4 and -4 are both even numbers. If we add them together, their sum is zero.  Based on the math pattern of  E + E = E,  then zero has to be even as well.  If we substitute zero in other problems such as 1 + 0 = 1, it fits the O + E = O  rule just as 2 + 0 = 2 fits the E + E = E  rule.

In Algebra, even numbers can be written as 2 x n where n is an integer.  Odd numbers can be written in the form of  2 x n + 1.  If we have n represent 0, then  2 x n = 0 (even) and  2 x n + 1 = 1. (odd)

I say all of this to relate an actual incident that occurred in my classroom.  I wrote the number 934 on the white board, and commented that since it was even it was divisible by 2.  One of my students was perplexed because he did not understand how 934 could be even when it contained two odd numbers and only one even number.  He actually thought that all the digits of a number had to be even for the number to be even.  Funny?  Not really!  Amazingly, he had made it through 12 years of school without understanding Place value as it relates to even numbers. Unfortunately, I had assumed that everyone (especially my college students) knew what an even number was.  I no longer make assumptions about students and their math knowledge!

Question #2 - Is zero positive or negative? The definition for positive numbers is all numbers greater than zero, and the definition for negative numbers is all numbers less than zero. Therefore zero can be neither positive or negative.

Question #3 - Is zero a prime or composite number? To be a prime number, a number must have only two positive divisors, itself and one. Zero has an infinite number of divisors so it is not prime. A composite number can be written as a product of two factors, neither of which is itself. Since zero cannot be written as a product of two factors without including itself, zero, it is not composite.

Question #4 - Why can't you divide by zero? I love this question. Back in the dark ages when I asked it, I was always told, "Because I said so." Being an inquisitive student was not a blessing when I was growing up. Math teachers who knew all did not want to be questioned!!!! Anyway, I don't mind the question, and here is my practical answer.

First, we must understand division. Division means putting or separating a number of items into a number of specific groups or sets. When you divide, such as in the problem 12 divided by 2, you are really putting 12 things into two groups or two sets. Therefore, if you have the problem 8 divided by 0, it is impossible to put eight things into no groups. You cannot put something into nothing!

Hopefully, this clears up a few things about zero.  I leave you with this math cheer.  (I always wanted to be a cheerleader!)

                            Zero, two, four, six, eight,

Who do we appreciate?

Even numbers! Even Numbers! Even Numbers!


Can't Memorize Those Dreaded Math Facts!

Many of my college students come to me without knowing their math facts. Some do, but most do not. Since we use calculators in the class, it really isn't an issue.  It just takes those students longer to do a test or their homework. One day, the students in my Basic Algebra Concepts class (a remedial math class) were playing a math game to practice adding and subtracting positive and negative numbers. We were using double die (see picture) where a small dice is located inside a larger dice. (I have to keep an eye on these because they tend to "disappear". The students love them!)  I noticed one of my students continually counting the dots on the die. He was unable to see the group of dots and know how many were in the set.  It was then that I realized he could not subitize sets. (to perceive at a glance the number of items presented)

Subitizing sets means that a person can look at a grouping or a set and identify how many there are without individually counting them.  (i.e. three fingers that are held up)  When a child is unable to do this, they cannot memorize math facts since memorizing is associating an abstract number with a concrete set.  Many teachers as well as parents fail to recognize the root cause of this memorization problem.  AND no amount of practicing, bribing, yelling, or pulling out your hair will change the situation.  So what can you do?

First of all, the problem must be identified.  Use a dice and see if the child must count each dot on each face. Try holding up fingers or laying out sets of candy (M&M's - yummy!) or using dominoes. Put five beans in a container, and ask the child how many are in the box. (They may count them the first few times.)  Take them out, and put them back in.  Ask the child again how many there are.  If, after several times, s/he is unable to recognize the set as a whole, then s/he cannot subitize sets.

How do you help such a child?  If you have small children at home, begin the subitization of sets by holding up various combinations of fingers.  My granddaughter just turned four; so, we worked on holding up two fingers on one hand and two fingers on the other; then one and three fingers, and of course, four fingers. I also like to use dominoes. They already have set groupings which can be identified, added, subtracted, and even multiplied. A dice is great because the child thinks you are playing a game, not doing math.  Roll one dice, and ask the child to identify the set of dots. Try the bean idea, but continue to change the number of beans in the box.  My granddaughters love the candy idea because they are allowed to eat them when we are done.  (All children need a little sugar now and then even though their parents try to control the intake.  I love being a Grandma!)

Gregory Tang has written two wonderful books for older children, The Grapes of Math and Math for All Seasons, which emphasize subitizing sets. At times, I even use them in my college classes!  I was fortunate to attend two of his workshops presented by Creative Mathematics. He not only has a sense of humor, but his books can be read again and again without a child becoming bored. Check them out!