Math

Measurement-Data and Common Core Standards

In addition to working with numbers, kindergarten students should also be introduced to measurement and data. As students work to meet three standards of Measurement and Data, they note measurable attributes such as length and weight, make comparisons, as well as sort and categorize information. Through exploration in these areas students gather, organize, analyze, and interpret information about the world around them.

There are some great books to get you and your students going. with my affiliate links.


STEM Center

If you haven't tried a full-blown science or in this case STEM center before, try setting up a center on measurement. Start simple with some rice and measuring cups, then build from there. Over the years I have accumulated materials from Lakeshore and others, but regular every-day measuring items for the center are easy to find.

The cornerstone for learning in kindergarten is formed by hands-on investigations. This is especially true in STEM. Touching, moving, and manipulating objects helps students gain a better understanding of mathematical concepts. 

Take measurement for example. Most workbooks adequately address length, weight, and volume, but students gain a deeper and more complete understanding of these measurable attributes when given an opportunity to measure, weigh and examine volume with measuring tools. 

Set up your area with a variety of standard and non-standard tools. Rulers, measuring tapes, craft sticks, and/or linking cubes are all effective means for measuring length. A balance and a tub of classroom objects allows students to compare objects by weight. A tub of colored macaroni and a set of measuring cups is a great way to learn more about volume.

Hands-on activities are engaging and interesting to students. “Discovering” the answer is exciting and satisfying to young learners. Whether the hands-on exploration is guided or independent, you will find that it enhances student learning.


Supporting Curriculum

Awesome resource! Thank you!
Using these this year. Thank you!
Good ideas, I haven’t tried them out yet.

The lessons vary in style and format. Many lessons can be used with small groups while others can be completed with a large group. All lessons can be adapted to support struggling students or to challenge high-achieving students. It can also be found on TPT.


For some great cross-curricular thematic fun, enjoy the measuring lessons found in our comprehensive thematic unit, Giants, Trolls and the Big Bad Wolf.


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Or, find it here at TPT.

Used it last year in May, and my students (kindergartners) loved every activity in this unit! Very engaging and fun! Planning on using it again this year. Thanks!
This was an absolute lifesaver! I used multiple products for my formal observation and was given great compliments! Thank you!!!
This unit is AWESOME!!!!! Thank you so much for including CC standards with these fantastic activities! Great, great purchase!!!!!

Teaching Young Children About Time

Small children are always asking when certain things are going to happen:

"When are we going to eat lunch?"
"When is recess?"
"When will Mom get home?"
 "When can I go and play?"

Of course, small children also don't have an accurate grasp on the concept of increments of time. Days, months, hours, and minutes are all very fuzzy concepts to them. So to help kids start to understand how to measure time, I borrowed an idea from mathematics guru John Van de Walle. His brilliant idea is to take the minute hand off of a clock so that children can focus on the hand that gives them the most important information: the hour hand. 


I had to break the plastic covering to get to the minute hand, but after that, it was easy to snip off the minute hand with a pair of scissors. Now that only the hour hand shows, the children can begin to conceptualize how long an hour is, and when certain things will happen. If I tell them, "Lunch, will be at 11". They know to watch for when the hour hand is pointing exactly at the 11. And, during that time, can watch the speed at which the hour hand moves from number to number to gain an idea of how long an hour is exactly. We can also use the hour hand to begin to use time vocabulary in a way that makes sense. For example:


"It's almost 9 o'clock" 
"It's just past 12"
"It's half past 1"
"It's exactly 10 o'clock"


I've found that using an "hour hand only" clock has given my kids a lot of independence. They can now check it on their own to find out if it's time for certain things to happen that are always at the same time. They can find out if it's lunch time or play time on their own, which gives them a sense of control over the day as well as an introduction to how time is measured.

Enhance Learning with Hands-on Activities

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The cornerstone for learning in kindergarten is formed by hands-on investigations. This is especially true in math. Touching, moving, and manipulating objects helps students gain a better understanding of mathematical concepts. 

Take measurement for example. Most workbooks adequately address length, weight, and volume, but students gain a deeper and more complete understanding of these measurable attributes when given an opportunity to measure, weigh and examine volume with measuring tools. 

Set up an area with a variety of standard and non-standard tools. Rulers, measuring tapes, craft sticks, and/or linking cubes are all effective means for measuring length. A balance and a tub of classroom objects allows students to compare objects by weight. A tub of colored macaroni and a set of measuring cups is a great way to learn more about volume.

Hands-on activities are engaging and interesting to students. “Discovering” the answer is exciting and satisfying to young learners. Whether the hands-on exploration is guided or independent, you will find that it enhances student learning.

For a great hands-on measurement activity, enjoy this free guided reader and these other great "Measurement-Centered" Products.

 

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Using Ten Frames

A ten-frame is a great math tool. You can use it to work on skills such as 1-1 correspondence, number and quantity matching, addition, and subtraction. A ten-frame allows students to manipulate objects to solve a problem and/or see the solution for a problem.

 

With a second ten-frame you can help your students reach a better understanding of teen numbers. Choose a teen number and have your students identify its numerals. Next fill one ten-frame completely. Continue adding objects to the second ten-frame while counting to your chosen number. Finally, analyze what you have together, writing the problem next to the ten-frame.

 

A large magnetic ten-frame from a school supply store such as Lakeshore provides a means to practice math skills with a large group. You can “think aloud” as you compose and decompose teen numbers. By presenting one teen number each day with the ten-frames, your students will quickly grasp the concept of a teen number being “ten ones and some further ones.” Vary your approach from day to day - at times present the number first, some of the time have the counters in place, and at other times start with the addition problem. In all cases talk about what you are doing and/or have your students explain the process.

 

Ten-frames can be beneficial in presenting and demonstrating many math skills. They are also helpful for guided and independent practice activities. For some hands-on practice with ten-frames, check out lessons in our Number and Operations in Base Ten unit as well as the following units:

January Interactive Math Worksheets


These are not your typical worksheets, they are made for adult/child interaction and teaching to complete. These higher-level thinking pages will challenge students and deepen mathematical understanding.



Contents Include:
Winter Wonderland: Properties of Addition and Subtraction
Candy Jar: Equations that equal 6
Winter Birds: Equations that equals 7
Measuring Mittens: Comparing Lengths
Favorite Winter Activities: Gathering and comparing data
Favorite Winter Animals: Gathering and comparing data
Hibernating Bears: Finding Missing Addends
Counting Snowflakes: Grouping in 10's plus remainders
Snowball Fight: Ordering numbers starting with a given number 1-6
Penguin Party: Finding missing addends to make a given number

Subtraction Worksheets: Winter

Tomorrow, my students will begin the chapter on subtraction, and yes, I already know they will need lots of practice to become fluent subtractors with numbers 0-5.

Subtraction is a crucial math skill. It is important that students understand that subtraction involves taking away or pulling sets apart. As they grasp the underlying concept of subtraction, students can solve problems and begin to learn the subtraction facts.

These worksheets will allow independent practice in counting objects, taking away, and identifying differences.




To further meet objectives for Operations and Algebraic Thinking, check out:

http://www.teacherspayteachers.com/Product/Operations-Algebraic-Thinking-Kindergarten-Common-Core-Essentials-726817 .

https://www.teacherspayteachers.com/Product/Addition-and-Subtraction-Fluency-Worksheets-1870731

https://www.teacherspayteachers.com/Product/Kindergarten-Subtraction-Worksheets-1118326

https://www.teacherspayteachers.com/Product/Subtraction-Worksheets-for-Kindergarten-238626

Numbers All Around: Building a Solid Foundation 0-10


Numbers surround us; they are part of our lives every day. Before coming to school, students see numbers on clocks, remote controls, license plates, etc. We need to continue to surround our students with numbers. Have the numbers permanently displayed, post numbers with that number of objects, and put up charts for number chants (1 Potato, 2 Potato) and nursery rhymes (1,2, Buckle My Shoe). 

Although we will expose our students to many numbers during the kindergarten year, we need to develop a sound foundation with the numbers 0-10. Instead of rushing on to the teen numbers when your students recognize the numbers to 10, continue working with those numbers. Students need to be able to recognize the numbers 0-10, write them, count out the correct number of objects again and again, compare numbers, match number and quantity, and (eventually) add and subtract numbers within 10. If you develop a solid foundation, then using numbers beyond 10 becomes much easier for students.

To help your students recognize and use numbers, check out the following materials:










Building Number Sense

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. 


My students are loving these Multi-Approach NO-PREP worksheets, and are developing a focused grasped on number sense along the way! This product currently features the numbers 1-10, but will be updated to include the numbers 0 and 11-20. 
These pages can be ran double sided to optimize use of paper




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

Kindergarten And Preschool Math and Language Arts Common Core Assessments

Are you looking for assessments that are authentic, strategic, and linked to Common Core Standards?  Are you looking for assessments that will lead you to be organized, effective, and powerful in your teaching.

These assessments gather information about the language arts and mathematic strengths and weaknesses of your students. You will be able to immediately use important information to guide your classroom instruction, strategically target intervention groups, and inform parents of individual student progress.

Additionally consider using ESGI, a secure assessment platform to digitally conduct your one-on-one tests, score, and report your assessments.  The digital and paper format are a great way for you to visualize your assessments and keep yourself organized!

You will love the ease that ESGI gives this assessment product. I have done the work for you; all of these tests are there: click, click, done. Simply click on the test explorer tab (within ESGI) and look for my name: Kathy Crane to find the correct tests. (Important: At the very end of the testing documents are screen shots of how to organize your tests on ESGI to correlate with the paper copy).

Even if you are not sure whether paperless testing and data retrieval is for you, you will want to take advantage of their free 60 day trial! Simply go to  https://www.esgisoftware.com/ESGI/ click on the orange button!

Please remember to enter promo code B7227 to receive $40.00 off the already low price.

             




                                      



    












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)

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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:

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.

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).

 

(This book by Walle and Lovin is one of my very favorites for teaching mathematics. You can get it through this affiliate link)

 

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 what it 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)

 

(Cathy Fosnot is an excellent resource for mathematics teaching. Here is her book in this affiliate link.)

 

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. 

In our next post we will discuss the development of mathematical representation. Be sure to check it out!

Number Sense and the Common Core: Part-Whole Relationships

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

kindergarten-number-sense

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!


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Clocks, Calendar, and New Year Fun

Although learning to tell time is not a math skill, and is not included as a Common Core Standard in kindergarten, the teaching of clocks as a tool for mathematical thinking is crucial. 

A clock is made of the numbers 1-12 laid out in numerical order. This alone makes the clock a handy aid when teaching counting and cardinality. Giving a student opportunity to work with clocks help develop number recognition, sequencing, and numerical order.

I think I have introduced clocks the first day back from winter break for the entire 23 years of my career (thank you Ruth Hepworth). Waiting for Monday morning, right at my carpet/calendar area, I have my little student size Judy Clocks ready to go and my copy of Hap Palmer’s, Paper Clocks ready for the play button to be pushed. After we have manipulated the clocks to the song a couple of times, I love to have the students construct their own clocks to take home and show off their new skill of telling time to the hour. 


You may wish to check out our Happy New Year Unit that focuses on ways to measure time: Clocks and Calendars.

Why is Place Value Important?

When I was a child learning mathematics, the emphasis was on memorizing facts and algorithms. When I was taught place value the lesson went something like this: a number in this location is in the tens place, a number in this location is in the ones place, remember that, there will be a test.

I never understood the purpose of place value, and it was only when I became a teacher myself that I came to understand the invaluable tool that it can be when solving mathematical problems and in understanding our number system!

It is not an understanding of the labels “tens place” and “ones place” that are important when teaching place value, it is the concept of exchange that they represent. In our number system there are only 9 numerals, 1, 2, 3, 4, 5, 6, 7, 8, 9. In order to represent any quantity larger than these numerals we “reset” one column and start over in the next, creating the number “10”. Our number system is a base ten system because that is the amount at which we run out of numerals and must start in another column.

Young children often confuse large numbers like 19 and 91; this is because they do not have an understanding of the exchange that has taken place to create the numerals. 19 is really 10 + 9 and 91 is really 90 + 1. Once a child understands what the 1 and the 9 mean, depending on where they are located, they will have no trouble distinguishing between large numbers. Understanding what a numeral means in context will also serve children well when they add and subtract digits with more than one numeral. For example, if a child needs to add 31 and 42 and knows that they are really adding 30 + 40 + 1 + 2, the solution becomes more apparent. If a child needs to add 48+32= and they understand that they are adding 40+8 and 30+2 they can "borrow" the 2 from 32 and add it to 48, making the problem 50+30, a much simpler problem to solve!

These place value lessons are designed to increase a childʼs understanding of the number system and how context determines the meaning of a numeral.

In order for children to be successful in these lessons they must first understand the concept of skip counting and be able to count by tens. The lessons are organized in order of difficulty, progressing from the earliest concepts of place value to those that develop later. Ideas have been included at the end of some lessons of ways in which you can extend a lesson if your students need more practice on a particular concept. I hope they will be helpful in your classroom and will help your children not only be able to improve
in mathematics, but to think more like mathematicians.

                  ----Submitted by Lyndsey

Math Timed Test: Are They Valid?


If you are not a member of youcubed.org, you should be. It's a great resource for teachers and parents about the most current research into how to teach math. The site was set up by Jo Boaler, professor of Mathematics Education at the Stanford Graduate School of Education, who is a mathematics hero.

One of Boaler's points for which I have strong personal feelings, is that timed tests in mathematics are absolute garbage (absolute garbage are my words, not hers, but that's because of the personal connection). Timed tests were something I dreaded as a child. I knew that I could never pass them with the required speed. All I could hope for was to get them over with without too much embarrassment. They absolutely convinced me that I was no good at mathematics. I wish I'd have known then what I know now, which is:

Mathematics is not about speed. It is about depth of thought. If you want an example of this, you can read about Chinese mathematician Shing-Tung Yau.

Children with exceptional math skills are just as likely to preform poorly on timed tests as children with poor math skills.

Timed tests create a situation of anxiety, which actually shuts down the brain's ability to think.

Perhaps the worst thing about timed tests is that they don't teach anything. Children with poor math strategies try to use those poor strategies quickly and children with excellent math strategies try to use their excellent strategies quickly. At the end of the test, nothing has changed about the way the children are approaching mathematics. This is a major problem, because the thing that really separates children's mathematical ability, is the way they approach problems. Children who struggle with math use clumsy and inefficient strategies. Children who excel at math use elegant and efficient strategies. For example, imagine you gave a kindergartener two piles of objects, and ask them to count the piles separately. One pile has 6 objects and one pile has 4 objects. You then ask him how many there are all together. A student whose strategies are underdeveloped starts counting all of the objects again--even though they previously counted both piles. A student whose strategies are more efficient might start at 6 and count on. A student with excellent strategies might move one of the objects in the pile of 6 to the pile of 4, creating a problem of 5+5=10. It is the method that students use that leads to efficiency, not the time in which they do the task. Timing our inefficient mathematician is only going to result in him trying to count faster.

When I was a student, I was deeply embarrassed of the fact that I couldn't remember all of my multiplication tables. I was terrified that my math teacher might discover that I could never remember what 8x6 was and always had to start with 7x6=42+6=48. Now I've discovered that this kind of flexibility with numbers is exactly what mathematicians need. In fact, my computer programmer husband and I have had many discussions about the math classes he took in college and how they were about teaching him this kind of flexibility on a higher level, how they taught him to problem solve, and about how accuracy was always more valuable than speed.

So, if we want to improve our student's ability to do math, please, please, please, never pull out a timer, because we never want our children to get the idea that math is about speed. Just like good writers are concerned with how well something is written, and not how fast they can type, and good scientists are concerned with how accurate their results are, and not how quickly they came up with the results, good mathematicians should be concerned with the processes of mathematics, not the speed of their sums.
                                                   --- Contributed by Lyndsey