Compare numbers.

• CCSS.Math.Content.K.CC.C.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.¹
• CCSS.Math.Content.K.CC.C.7 Compare two numbers between 1 and 10 presented as written numerals.

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When small groups of objects are presented to young children, unless they are very different (1 toy car next to 10 toy cars), they may not be able to see the difference in quantity right off the bat.  Remember, they are confused by appearances so if you put 3 large cars next to 5 small cars, the child may believe that the 3 large cars are “more” than the 5 small cars, because they look like more i.e., they take up more room so in the child’s mind that is “more”.

There is a developmental process that needs to shift for the child to conserve quantity.  This often does not happen until the kindergarten year, and sometimes even after that.  However, that doesn’t mean that we shouldn’t provide children with as much experience as possible in comparing quantity.

When distributing items to children, ask them to compare who got more, who got the same and who got less.  Using their counting skills and one-to-one correspondence, they may be able to count the items and determine the answers.  If they are 3 and under, they may simply guess.  If they are able to count, you can use a number line to help them see which number is bigger or great and therefore, “more”.  The best way to reinforce the above concepts is to compare numbers frequently throughout the day, so you can maximize children’s exposure to the concepts, use as many visual cues as you can so they can access different ways of knowing, and give them credit for trying.

Make sure you provide lots of written numbers to be “read” around the room.  If your program is filled with written words, try and match the words that children are exposed with the same quantity of written numerals.  This will make a huge difference in gaining familiarity with written numerals.

Count to tell the number of objects.

• CCSS.Math.Content.K.CC.B.4 Understand the relationship between numbers and quantities; connect counting to cardinality.
  • CCSS.Math.Content.K.CC.B.4a When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
  • CCSS.Math.Content.K.CC.B.4b Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
  • CCSS.Math.Content.K.CC.B.4c Understand that each successive number name refers to a quantity that is one larger.
• CCSS.Math.Content.K.CC.B.5 Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.

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This entire domain is something I think most early childhood teachers focus on.  We count with children all of the time looking for opportunities throughout the day for children to develop this skill.  There are some interesting specific expectations here that I think are worth exploring- things you might not be thinking about, but should be.

The first part of this standard refers to “one-to-one correspondence” even thought the authors are not calling it this.  When a child counts one object to one number and understands that each object represents “1” and only 1 number name and quantity.  So when children set the table we ask that they put one napkin at each chair, and one plate, and so forth.  You may periodically have the children count aloud when putting each item in its place, to reinforce the concept of one item = one number.  The last number they arrive at is the total number of items they distributed.  So, if there are 5 chairs at a table and the child distributes 5 napkins and 5 plates, there are a total of 5 places for children to sit and eat.  That number is the same no matter what chair they started from.  Sometimes, you can be explicit about this by asking, “How many kids can sit at this table?” or “How many napkins did you pass out?” Follow this up with a prompting question, i.e., “If you put 5 napkins out, how many plates did you put out?  Why is it the same number?”  Explaining this, or putting it into words is going to be a work in progress, but the questions will encourage mathematical thought and exploration.

The next part sub standard is a much, much harder concept for most children.  Many children 3 and up, can tell you that 5 is more than 3, or 10 is bigger than 2, but understanding that there is an actual algorithm that 6 is one unit bigger than 5… that is highly complex.

So, let’s break this down.  If I can count, “1,2,3,4,5… and so on, I have memorized a specific set of words in a specific order that describes a mathematic algorithm of

1+1+1+1+1+1…. and so on.

That means that I have to possess an understanding of the concept of “1” or in other words, the “oneness” of something so that I can then make the leap to the larger principle that counting by ones is based on the addition of “1” to the previous number.  This is also true when we ask children to count by 2s or 5s or 10s.

As adults, it is hard to break down a concept that is seemingly very simply and something that we have had known for what seems like our entire lives, into this highly complex process.  Many teachers think that if children can count then they understand number.  That is patently untrue and is a disservice to them.

Simple addition will support this concept and by simple I mean adding 1 to other small numbers.  Put two fingers up on one hand and put one finger up on the other.  Have the children add the fingers together.  This clear and consistent approach to adding “1” will reinforce the eventual understanding that 1 more, means the next number in the sequence of numbers.

Keep those kids counting….whenever you can as often as you can.

Know number names and the count sequence.

• CCSS.Math.Content.K.CC.A.1 Count to 100 by ones and by tens.
• CCSS.Math.Content.K.CC.A.2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
• CCSS.Math.Content.K.CC.A.3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

Although I believe that this first standard may start out as  a language and memory skill, I do agree that eventually children should be able to count.  Even if it begins with children memorizing the sequence (1,2,3,4,…) and not really understanding what that sequence means, it is a good beginning.  The problem is when teachers and parents confuse the child’s ability to recite a list of numbers (10, 20, 30, 40…) and his understanding of number.  These are two very different skills and competencies.

We often hear children counting from a number other than 1, but usually it is in the context of completing a sequence, for example, when they join in when others are already counting.  In that case, they have heard the sequence beginning with 1 and are completing the phrasing.  It is harder to find opportunities for children to count beginning with a number other than 1, but finding ways for children to count starting at 4, or 5, or 8 will help develop this kindergarten skill. Try using visual cues, i.e., fold down three fingers and have children count the rest of the fingers on your hand.  Remember, it is OK for them to need to start with 1 and point to one of your folded fingers.  They do not need to master this skill during their preschool years, they only need to have ample experiences so they will eventually master it.

Writing the numbers is also important, but children may know the numbers even f their fine motor skills are not yet able to create them.  Again, confusing two different areas of development is problematic.  Remember, allowing children to have multiple ways of showing what they know is the ultimate form of assessing their progress.  Whenever an opportunity arises for children to write numbers, try to encourage it.  My kids liked to write numbers on the bottom of pages of their art as well as date numbers.  So when you ask them to write their names on their work, ask if they want to write the date as well.

In Kindergarten, instructional time should focus on two critical areas: (1) representing and comparing whole numbers, initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.

• 1. Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away.
• 2. Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.

So, although there are clear expectations for each area, teachers are expected to spend more time focusing on number than on any other area. This expectation reinforces the idea that number sense is the foundational skill on which all other mathematical skills are built.  That means that we too, should focus on number.

Next week I am going to explore the first area of the mathematics Core – Counting and Cardinality.  Can’t wait!