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The Little Red Hen

One of my favorite tales to tell.. The Little Red Hen
Here is a link to some writing center activities.





Making The Red Hen: Collage Portrait

Writing the Hen's Biography

Finished Product!


Acting out the Story at the Drama Center




                    
Learn The Process

Grinding the Wheat


                 









Constructing The Elements as a Team.
            
Constructing The Elements: Building Teamwork

 

Adding Finishing Touches 

Our Story Element Mural



The Development of Mathematical Representation

According to the research of David Sousa, children progress through three stages of mathematical understanding as they develop an understanding of concepts. The stages are Concrete, Representational (Pictorial), and Abstract. It will be important to remember that each of our children will be in a different stage of development for each concept that we are teaching, and, therefore, it is important to differentiate the method by which the children are allowed to work with problems. Differentiating in this way is sometimes known as the CRA (or CPA) approach. First, let's define the different stages of development:

Concrete
All children must start here when learning mathematical concepts. Concrete models tie mathematics to the real world and include anything that the child can use physically to represent a problem.

Representational/Pictorial
The representational stage provides the mental scaffolding for children to move their mathematical understanding from concrete to abstract. In this stage, children are able to use visual or pictorial representations to represent concrete examples. Teachers deliberately help children see how pictorial representations tie to concrete examples.

Abstract
The abstract level of thinking represents mathematical thinking symbolically. It is important to realize that this is the final level of understanding for children, and that we must help each child through the first two stages before they will be able to grapple with abstract representations. "Numerals were developed to signify the meaning of counting. Operational symbols like + and - were constructed to represent the actions of combining and comparing. While these symbols were initially developed to represent mathematical ideas, they become tools, mental images, to think with. To speak of mathematics as at mathematizing demands that we address mathematical models and their developments. To mathematize, one sees, organizes, and interprets the world through and with mathematical models. Like language, these models often begin as simply representations of situations, or problems, by learners... These models of situations eventually become generalized as learners explore connections between and across them" (Fosnot and Dolk, 2001).

Many lessons in kindergarten can be specifically designed to move children from one stage of understanding to another. For example, one lesson could ask groups of children to count out objects from a bag and then draw a picture of the objects they found (Concrete to Pictorial). The teacher could then write the number each group found in their bags on the board (Pictorial to Abstract). Lessons can also be designed where each stage of development can be used to answer the question.

Here is such a lesson: The teacher places a container in view of the children and tells them their are five bears inside. Some of the bears are red and some are blue. How many of each color could be inside? Children can use their own sets of bears to answer the question. They could draw a picture. They could use numbers and equations. The important part of the lesson is that each child is developing an understanding of the part/whole relationships of the number 5, and allowing each child to work within his own stage of understanding will better strengthen his mathematical knowledge.


Number Sense and the Common Core: Compensation, Unitizing, and the Landscape of Learning

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 hierarchical inclusion, magnitude, and subitizing. In Part 3 I discussed part/whole relationships (a concept which will probably fill the entire kindergarten year). The whole of the math Common Core (except for geometry and measurement) fall under the umbrella of number sense, and today we will discuss some of the concepts of number sense that are on the horizon for kindergartners.

Compensation

Compensation is the ability to play with numbers. It is the understanding that if 5+5=10 then 6+5 must be 11 because 6 is one greater than 5 and so the sum must be one greater than 10. Or that if 5+5=10 then 6+4 must also equal 10 because 4 is one smaller than 5 and 6 is one larger than 5. This is a complex skill that some kindergartners will not be ready for, but some children may begin to use compensation, and teachers should feel free to conduct Number Talks introducing compensation.

The following video is an example of a 2nd grade Number Talk involving compensation. Number Talks involving compensation in kindergarten would obviously be less complex.








Compensation strategies can be used in the following Common Core standards:

Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

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).

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.

Fluently add and subtract within 5.

Unitizing

Unitizing is a child's ability to see numbers in groups. It is an ability they use to see that there is simultaneously 1 chair with 4 legs, to count by 2's with understanding, to hold one number in their head when counting on, or to begin grouping numbers into tens.

Grouping numbers into tens is especially significant, because "A set of ten should play a major role in children's initial understanding of numbers between 10 and 20. When children see a set of six with a set of ten, they should know without counting that the total is 16. However, the numbers between 10 and 20 are not an appropriate place to discuss place-value concepts. That is, prior to a much more complete development of place-value concepts (appropriate for  second grade and beyond), children should not be asked to explain the 1 in 16 as representing "one ten". The concept of a single ten is just too strange for a kindergarten or an early first grade child to grasp. Some would say that it is not appropriate for grade 1 at all. The inappropriateness of discussing "one ten and six ones" (what's a one?) does not mean that a set of ten should not figure prominently in the discussion of the teen numbers" (Walle and Lovin 2006).

In kindergarten, the goal is not to formalize unitizing, but to begin to help students see numbers in groupings. The following standard requires unitizing:

CCSS.MATH.CONTENT.K.NBT.A.1
Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.


Let's dissect this standard to figure out exactly what it is and is not asking you to evaluate. According to the standard, children should compose and decompose a number in the teens into a group of tens and some ones. Nowhere in the standard is it required for the teacher to use place value language (as Walle and Lovin discourage) but to make representations of the teen numbers using objects and drawings. Although the language of the standard includes the terms "ten ones and one, two, three, four, five, six, seven, eight, or nine one" this vocabulary is there for the teacher, the standard itself specifically requires "understanding" from the student.

Watch the following videos, if kindergartners can see the images and identify the teen number that is represented, then they have met the conditions of the Common Core Standard.






The Landscape of Learning

We should keep in mind that these concepts of number sense do not describe a linear progression of understanding. "Historically, curriculum designers... analyzed the structure of mathematics and delineated teaching and learning objectives along a line... [f]ocusing only on the structure of mathematics leads to a more traditional way of teaching--one in which the teacher pushes the children toward procedures or mathematical concepts because these are the goals. In a framework like this, learning is understood to move along a line. Each lesson, each day, is geared to a different objective, a different "it." All children are expected to understand the same "it," in the same way, at the end of the lesson. They are assumed to move along the same path; if there are individual differences it is just that some children move along the path more slowly--hence, some need more time, or remediation. As the reform mandated by the National Council for Teachers of Mathematics has taken hold, curriculum designers and educators have tried to develop other frameworks. Most of these approaches are based on a better understanding of children's learning and of the development of tasks that will challenge them." (Catherine Twomey Fosnot and Maarten Dolk 2001)

According to Cathy Fosnot, a child's development of number sense looks less like a line and more like the following chart, developing in a way that she describes as the "Landscape of learning".

https://jessicapartridge.files.wordpress.com/2014/02/fosnot1-e1393552630905.jpg

"The paths to these landmarks and horizons are not necessarily linear. Nor is there only one. As in real landscape, the paths twist and turn; they cross each other, are often indirect. Children do not construct each of these ideas and strategies in an ordered sequence. They go off in many directions as they explore, struggle to understand, and make sense of their world mathematically... Ultimately, what is important is how children function in a mathematical environment (Cobb 1997)--how they mathematize." (Fosnot and Dolk 2001)

Because number sense development is nonlinear, the best activities we can use in our classrooms will weave together different components of number sense and engage children on multiple planes of development. Many of the links in these posts will lead you to some excellent books with activities in them, that do exactly that. 

But we must be honest, many of the textbooks that have been adopted are more concerned with checking off the Common Core standards than developing the understanding behind them, much less teaching in a non-linear fashion. Therefore, we, as teachers need to be more discerning lesson planners using textbooks and workbooks only as a resource to teach the Core in the way we know best, instead of letting textbook companies dictate to us how the Core should be taught. 

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)




video

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 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.1
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.
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 and the Common Core: 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!

Kindergarten Retention: What is the Right Call?

Spring is approaching, and with it, comes the reality that some students are not on target to master a majority of academic materials assigned to their grade level. Will retention help these struggling students reach grade-level proficiency?

The decision to retain is often a contentious topic. Some say that retaining a student will simply set him/her up for social and/or behavioral challenges in the repeated grade. According to Goos,M., Van Damme, J., Onghena, P., & Petry,K. in their work, First-grade retention: Effects on children’s actual and perceived performance through elementary school (2010) “Although retained students started their repeated grade with an advantage in math and reading relative to their grade-mates, this advantage was lost by the end of elementary school.”

Conversely, others may say that retaining a student will give him/her the gift of time. The time to review all academic concepts previously presented with further opportunities for strategic intervention. The time to develop, mature, and hopefully thrive.

When considering retention, there are many factors to determine. 

Why is the student’s academic performance poor?
Is the performance related to the child’s developmental level?
Is the child from a low income household?
Is the child a English Language Learner?
Does the child have a summer birthday?
Is the child a boy?
What is the child’s physical size?
Does the child exhibit immature behaviors?
Do the parents help with homework, participate in school activities and/or volunteer?
Did the child have excessive absences?
Is the child transient?
Is the child motivated to complete tasks?
Is the child of typical intelligence?


The decision to retain is not to be taken lightly. It will effect the entire life of the child in question.  My opinion? After years of experience I have to agree with this article that was published in September 2014 in the Journal of Social Forces. However, I will say that August birthdays are a completely different subject. Kudos to the states who have moved the kindergarten entrance to July 31st.

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