Recently we discussed some great examples of using counters for splitting numbers apart to see that 4+0=4 and 3+1=4.  
Now let's turn our attention to groups of 10 for adding and subtracting.  The goal is to introduce concretely how we group and ungroup tens for adding and subtracting in base-10.  Notice that I did not say carry and borrow.  
Use fewer than 20 counters and start to break up numbers the same way.  

1. Create numbers greater than 10 using counters of your choice.  Be sure to group 10 and push them to the left leaving the remainders to the right.  (15 would be a group of 10 and 5 single pieces.)
2. Take the next step and put two groups of counters together that are each less than 10.  For example a group of 8 and a group of 7 counters are added separately and then mushed together.  Make the group of 10 with the 5 remainder.
3. With subtraction, let's subtract 7 from 12.  To subtract take away counters. The counters are arranged in a pile of a 10 and 2 singles.  To subtract the 7, first ungroup the 10 pile and bring it back to the singles pile.  Then subtract the counters to find the five remaining.

As always, work your way through with verbal discussion, playing with the counters, writing the symbols to match what is happening. 

What I just wrote sounds a bit clinical.  How have you or can you do this process organically or by "living math out loud"?

Here are some on-line virtual counters for the discussion above.  Try it out yourself first and see what comes to you.

Operations and Algebraic Thinking
Understand addition as putting together and adding to and understand subtraction as taking apart and taking from.

Basic introductions to addition and subtraction start with taking a set number of items and breaking them up in multiple forms.  This is an important skill that will be used through math and will be reused often in Algebra.

Start with fingers.  How many fingers do you have?  Here you have the ability to show that 1 + 4 = 2 + 3 = 3 + 2 = 4 + 1 = 5.  Then 1 + 4 = 1 + 2 + 2 = 1 + 3 + 1 = 5 and so on.   Not only are children learning addition and subtraction but they are starting to get to understand, without the words, the concepts of commutative and associative property.  Commutative is moving around numbers doesn't change the answer and Associative means that the numbers stay in the same order but are grouped differently.  Decomposing 1 + 4 into 1 + (3 + 1) is important to building these skills.

In time, expand the number using different numbers of objects and have your child experience a solid sense of numbers.  Some teachers will make up a number triangle 1 - 2 - 3 for kids to show that 1 + 2 = 3 or 3 - 2 = 1 or 3 - 1 = 2.  Do this in real life with physical objects that can engage multiple senses.  Be sure to have fun relaxed conversations around the different possibilities.  Finally, translate the words, experiences, and concepts into number sentence and equations.  We are always building up from the concrete to the symbolic to the abstract in as many senses as possible.

The most powerful tool that we have in math is our Base-10 number system.  We really take it for granted and many people don't know what it is even though they use it all day long.  As far as I am concerned, public education spends too little time on the number system.  I find that children and adults who have problem with arithmetic start from having no to poor understanding of Base-10.  

Learning Base-10 is perfect for organic learning using items in the world, bundling them in groups of 10, and bundling those bundles in groups of 10.  Children leaving kindergarten just need to start to understand base 10 for two columns.

The key here is two things:

1. For children to experience a collection of items in groups of 10 with a remainder of single sticks giving them a two digit number
2. Learning that having 3 groups of 10 and no single sticks requires writing in both columns to get 30.
1. 3 in the 10s column and 0 in the 1s column.

It is key that the child begins to create a simple two column table and to understand that both columns need to be written in.

Here is a very good virtual manipulative for you to play with.  Can you create something in real life that mimics this process?  Take the time and play with it yourself and explore by putting in more than 10 in the single cube column and then grouping bunches of 10 and move them into the 10s column. Clicking on the picture of the cube puts them down below.  
http://nlvm.usu.edu/en/nav/frames_asid_152_g_1_t_1.html?from=category_g_1_t_1.html

Any kind of teacher needs to ask a basic question before they start to teach: what is it that I have to teach and how do I make it happen. 

In this post, I would like for you to get a chance to see how very finite the standards are for a child to complete kindergarten math in public school. As homeschool families, we don’t need to follow curriculum standards like teachers do on a rigid timeline and test their students to grade them.  However, these standards are a good guideline in following the progression of how students learn math at an average level at a particular age.

One of the problems with the standards is that they tend to pigeonhole and/or stigmatize a child when being compare to the average child.  So, please don’t get caught up on the grade level as a way to judge yourself or your child.  This grade leveling can create a great source of anxiety for you and in turn for your child.

Here is the link for the common core standards that a majority of the states have been adopting as their standards.  It is a pdf file. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Below is the overview for the kindergarten standards.  In following threads we will break down each section and begin to share ideas on how to teach each topic, finding out what works and what doesn’t. 

 Grade K overview
1.     Counting and Cardinality
 1.1.  Know number names and the count sequence.
 1.2.  Count to tell the number of objects.
 1.3.  Compare numbers.
2.     Operations and algebraic thinking
 2.1.  Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
3.     Number and operations in Base ten
 3.1.  Work with numbers 11–19 to gain foundations for place value.
4.     Measurement and data
 4.1.  Describe and compare measurable attributes.
 4.2.  Classify objects and count the number of objects in categories.
5.     Geometry
 5.1.  Identify and describe shapes.
 5.2.  Analyze, compare, create, and compose shapes.

I teach a course every year at a local college for elementary school teachers and the most important issue that comes up in the first class is math anxiety.  Think about this, about half of the candidates coming in are deathly afraid of math and they look at me like the dentist or the grim reaper. 

Candidly, I go probably where you go as homeschool parents, which one of their teachers made it so painful as to drive them to phobia?  Sadly enough, it is a self-perpetuating problem handed down from teacher to student, parent to child, friend to friend, and as a broad accepted punch-line in our society.   There are some great articles that I will share with you later on the subject.

Here is my first question: what are your experience with/around math anxiety and how has it affected you or those around you?

Here is a blog post that I wrote on Math Anxiety:  http://www.mrgelston.com/1/post/2012/06/can-we-really-overcome-math-anxiety.html