# Building Number Sense

*Number Sense*should be one of the primary focuses of any kindergarten program.

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Read more about the importance of number sense: http://kindergartenkiosk.blogspot.com/search?q=number+sense#ixzz3l7AQGLMu

# Number Sense and the Common Core: Part-Whole Relationships

In Part 1 of this series I discussed how important number sense is to a child's development along with early number sense skills and how they relate to the Common Core. In Part 2 I discussed the importance of hierarchal inclusion, magnitude, and subitizing. The whole of Common Core math (except for geometry and measurement) fall under the umbrella of number sense, and today we will discuss the aspect of number sense that should be the major focus of any kindergarten math program.

**Part/Whole Relationships**

"Count out a set of eight counters... [a]ny child who has learned how to count meaningfully can count out eight objects as you just did. What is significant about the experience is what it did *not *cause you to think about. Nothing in counting a set of eight objects will cause a child to focus on the fact that it could be made of two parts. For example, separate the counters you just set out into two piles and reflect on the combination. It might be 2 and 6 or 7 and 1 or 4 and 4. Make a change in your two piles of counters and say the new combination to yourself. Focusing on a quantity in terms of its parts has important implications for developing number sense. The ability to think about a number in terms of parts is a major milestone in the development of number" (Walle and Lovin 2006).

All of the following Common Core standards involve an understanding of part/whole relationships. Notice that the word *equation *is mentioned in only three of these standards, and in those standards it is only one option that students can use to represent addition and subtraction. In actuality, using an equation may not be developmentally appropriate for most kindergartners. What they should be doing in order to meet the standard, is show addition and subtraction in terms of the part/whole relationships of numbers.

For example, if you ask a child to show combinations of 10 with colored plates, as shown in the previous video, and he/she can tell you all of the different combinations that make ten, **they have just solved a problem involving addition to 10 or subtraction from 10 using a drawing. This meets Core Standard K.OA.A.3 without solving any abstract equations. **

"To really understand addition and subtraction, we must understand how they are connected... By modeling addition and subtraction situations and then generalizing across these situations, children are able to understand and represent the operations of addition and subtraction... Children who commit the facts to memory easily are able to do so because they have constructed relationships among them and between addition and subtraction in general, and they use these relationships as shortcuts. When relationships are the focus, there are far fewer facts to remember, and big ideas like compensation, hierarchical inclusion, and part/whole relationships come into play. Also, if a child forgets an answer, she has a quick way to come up with it" (Fosnot and Dolk 2001)

CCSS.MATH.CONTENT.K.OA.A.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

CCSS.MATH.CONTENT.K.OA.A.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

CCSS.MATH.CONTENT.K.OA.A.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

CCSS.MATH.CONTENT.K.OA.A.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

CCSS.MATH.CONTENT.K.OA.A.5 Fluently add and subtract within 5.

It would be difficult to overstate how important part/whole relationships are to a kindergartener's mathematical development. I highly recommend looking into the books that I have used as sources in these discussions if you would like more information as well as some great lessons on teaching part/whole relationships. Our Interactive Math Worksheets for March and Interactive Math Worksheets for January also include some activities designed to develop this skill. Tomorrow we will discuss some of the number sense skills that are on the horizon for kindergartners.

# Number Sense and the Common Core: Hierarchical Inclusion, Magnitude and Subitizing

In Part 1 of this series I discussed how important number sense is to a child's development. In fact, number sense could be considered the most important part of the kindergarten year. I also discussed early number sense skills and how they relate to the Common Core. **The whole of the math Common Core (except for geometry and measurement) fall under the umbrella of number sense.** Here are more of the components of the Core and how they relate to an understanding of numbers.

**Hierarchal Inclusion**

Hierarchical Inclusion, as explained in the following video, is the concept that a number contains all of the previous numbers. Imagine a number as a Russian nesting doll, if working with the number 4, imagine the largest doll is 4 and inside that doll are the smaller dolls, 3, 2, 1. Without this understanding a child will think that, when counting, the number he points to and names "3" is "3" in of itself without the other objects and when you ask him for "3" he will give you only that object.

Hierarchical inclusion also helps students understand other number sense concepts, such as part/whole relationships and compensation because a child cannot understand that 4 can be broken up into the parts 3 and 1 without also understanding that the number "4" *contains *3 and 1.

In order to complete the following common core standard, a child must understand hierarchical inclusion:

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).

### Magnitude

Magnitude is a child's ability to compare groups. Even children who cannot count have the ability to judge the relative size of groups of objects, but as a child's number sense develops, so should the sophistication of her understanding of magnitude. The following Common Core activities depend on a child's understanding of magnitude:

CCSS.MATH.CONTENT.K.CC.B.4.C Understand that each successive number name refers to a quantity that is one larger.

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.

CCSS.MATH.CONTENT.K.CC.C.7 Compare two numbers between 1 and 10 presented as written numerals.

### Subitizing

Watch the following video. In it, groups of objects quickly flash on the screen. Can you tell how many objects there are in each grouping?

As explained in the following video, subitizing is the ability of a child to quickly recognize a number visually. Children do this by mentally grouping the objects they see (and you probably did this too, for example, seeing a group of 3 and 3 and 3 and knowing that there were 9 objects).

"Looking at a quantity for a short time and then being able to tell how many are in the group(s) without counting each object in the group begins to develop from small sets of two, three, four, and five objects, to parts of sets of six up to twenty. Generally, this development begins between ages 2 and 6. Later, the subitizer sees objects as groups of 10s and 1s and, combined with an understanding of place value, is able to see the numerosity of a large group of numbers quickly." (Copley 2010).

Subitizing strengthens a child's understanding of what numbers mean, and how they relate to one another. In fact, when children are taught to subtize, and their attention is drawn to the groups and patterns they see, it becomes a visual representation of addition and subtraction, as seen here

When strengthening a child's subitizing skills, we are teaching the following Core standards:

CCSS.MATH.CONTENT.K.OA.A.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

CCSS.MATH.CONTENT.K.OA.A.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

CCSS.MATH.CONTENT.K.OA.A.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

CCSS.MATH.CONTENT.K.OA.A.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

CCSS.MATH.CONTENT.K.OA.A.5 Fluently add and subtract within 5.

It is important to note that the word *equation *is present in only three of these Core standards, and in those standards it is only one *option* for representing addition and subtraction. In actuality, representing addition and subtraction with equations in kindergarten will not be appropriate for most of our students, but if we have the children participate in a subitizing activity where they are shown a variety of images with a quantity of 5 arranged in different ways, and a child can state that each group contains 5 because they saw a group of 1 and 4 or 2 and 3, they are **fluently adding and subtracting within 5**, and in a way that is more appropriate than asking them to solve 2+3=__, because instead of working from the end result backwards, we are building their foundational knowledge. In fact, testing a kindergartener's understanding of addition and subtraction by asking her to solve an equation, is like testing her phonemic awareness by asking her to read a story!

Tomorrow I will discuss the rest of the components of number sense, including the concept that the majority of your kindergarten math lessons should be focusing on. See you tomorrow!

# Number Sense & Rudimentary Math Skills

**Number Sense is King**

A child's development of number sense is of utmost importance. Not only does it predict a student's future success in mathematics, it may also predict future success in literacy. Because of it's importance, early *Number Sense *should be one of the primary focuses of any kindergarten program. "Unfortunately, too many traditional programs move directly from [the rudimentary concepts of math] to addition and subtraction, leaving students with a very limited collection of ideas about number to bring to these new topics. The result is often that children continue to count by ones to solve simple story problems and have difficulty mastering basic facts. Early number sense development should demand significantly more attention than it is given in most traditional K-2 programs" (Walle and Lovin 2006).

One benefit of the Common Core, is that we, as teachers, do not (and should not) have to depend on textbook companies to interpret the Common Core for us, we can use the Core itself as the basis for our teaching, and all of the concepts listed on the Core (except for measurement and geometry) fit squarely into one or more of the areas that constitute Number Sense.

**Rudimentary Math Skills**

**Counting**

Counting is not a component of number sense, but I mention it here because it is one of the rudimentary concepts that a child needs to develop before they begin to work with numbers. I am referring here to counting by rote, or memorizing the number names and their sequence. The following songs teach rote counting.

The following Common Core standard refers to rote counting:

CCSS.MATH.CONTENT.K.CC.A.1 Count to 100 by ones and by tens.

**One-to-One Correspondence**

A rudimentary concept, one-to-one correspondence (as explained in the following video) is a child's ability to match their rote counting sequence to one and only one object that they are counting. Some children arrive in kindergarten with this ability, but many do not.

This Common Core standard is referring to as one-to-one correspondence:

CCSS.MATH.CONTENT.K.CC.B.4.A 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.

This Common Core standard involves one-to-one correspondence, however, it is important to remember that standards like this include **writing skills**, and, therefore, are not wholly mathematical. A child's fine motor development should be taken into account, and the methodology of teaching writing skills should be used:

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).

**Conservation of Number**

Another foundational skill is an understanding that the organization of a group of objects does not change the *amount* of objects. The following child is struggling with conservation of number:

"What children *see* plays an important part in their understanding of the world... When adults watch a child count out eight objects and then say that there are more than eight when the objects are spread out, it is often difficult to understand how the child is thinking. However, imagine some situations in which we adults are also fooled by our perceptions. Thirty adults in a room may seem, even to us, like more people than if we saw thirty children in that same room. If we don't actually count, our estimate of the number of people might reflect that general impression. Our experiences over long periods of time have taught us to check our perceptions and trust our logic when perception and logic contradict each other. Children, however, are still tied strongly to their perception. They need many different experiences, along with maturation, before they understand what we describe as *conservation of number" *(Richardson 1999).

When we involve children in activities that develop number conservation, we are working the following Common Core standards:

CCSS.MATH.CONTENT.K.CC.B.4.B 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.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.

In the next post, we will discuss the components of number sense and how they relate to the rest of the Common Core. Stay tuned!