The focus of the first three years of formal schooling for children Kindergarten through second grade is to provide experiences that help them develop:
- Neuro-psycho-physiological maturation
- Socio-linguistic maturation
- Quantitative Reasoning
- Spatial orientation/space organization.
All experiences from psychomotoric/physical (games, toys, activities on water and sand tables, kitchen, playing with pieces of wood, bead-work, etc.) to social (games, toys, story time, sharing, etc.) to emotional (games,toys, community building, making friends, sharing, etc.), and cognitive (reading, writing, language acquisition, number work, building/taking apart, organizing, classifying, making and observing patterns, designing, etc.) develop the above areas. These experiences should support each other and in order to have maximum impact, they should be integrative. The objective of these experiences is to move children from their egocentric, centered, and perception bound perspective to observe and appreciate others’ perspectives, focus on more than one idea, and take initiative.
Kindergarten through second grade is the most important period in children’s lives. Children make more neural connections, acquire a large number of brand new words, begin to understand and use the structure of language in communication and socialization. They learn—recognize, extend, create and apply patterns, gather and use information, and begin to form and test social relations in and out of school. The learning habits—personal and social, they form in this period are the bedrock of their future studentship. One can predict what their achievements will look like later in life based on what happens to them in these years. For example, the most important skills that can predict achievement, with a high degree of predictability, at high school and beyond are:
- phonemic awareness, a “good” vocabulary in the native language, and the ability to read and willingness to apply basic reading skills,
- decomposition/recomposition of numbers up to ten and the related sight facts to show the foundations of quantitative reasoning, and mastery of additive reasoning—the concept of addition, addition and subtraction facts, procedures of addition and subtraction, and, most importantly, the understanding that addition and subtraction are inverse operations, and
- spatial awareness of objects around him/her (to my right, left, above me, below me, next to me, near me, far away from me, etc.) to understand space organization/spatial orientation (by the end of second grade children should be able to identify objects not only from their perspective but also from the opposite perspective).
Quantitative reasoning and spatial orientation/space organization form the basis of mathematics and mathematical way of thinking. In the next few posts I focus on how we can transform classroom routines to enrich mathematics activities, mathematical thinking, and mathematics content—language, concepts, and procedures during these years. One of those routines during the Kindergarten through second grade is the calendar activity. There are definite goals to be realized from this activity. We want to focus on the mathematics component of this activity.
A. Calendar Activity—Introduction to the School Day
All over the United States, teachers from Kindergarten through second grade open their day by gathering children around in a circle. Circle time is a social activity—a content rich process of community building.It has the potential of providing an opportunity for every child to become a contributing member of this learning community. Effective teachers are able to set the tone for the day through this activity. Here the rules and responsibilities of the membership to classroom learning community are acquired and are the harbinger of being a productive member of the future world they will inhabit. Circle time also serves as a venue not only for social learning but also for exploration and testing of one’s potential.
This circle activity is based on the principle that all learning is socially constructed while we individualize it for personal competence. During this socialization period several things happen: New children are welcomed to the class, special events in individual lives (e.g., birthdays) are acknowledged, children share their accomplishments, and they learn about the day—the day of theweek, the date, the temperature, the weather, the number of school days passed and remaining, important historical events, etc. It is also an opportunity for the development of socio-linguistic, emotional, and quantitative reasoning. It is planned to integrate cognitive, affective, and psychomotoric development. Effective teachers make use of this time for important learning in all of the domain related to children’s development. In this post, I want to focus on the quantitative reasoning component.
1. The Setting and Activities
It is another day in Mrs. Hills’ first-grade classroom. Nineteen children are sitting around her in a circle. Each one occupies one letter on the rug. The rug has all the letters of the alphabet woven into it.
Mrs. Hills has begun her class just like each day. The routine is predictable, and the children know it well. They knowtheir place on the rug. She takes the same seat.
On the surface, the day appears just like another day—things appear to go almost the same way: she takes attendance, the lunch count, assigns jobs to children and reminds them of the old and regular assignments and selects one of the children as the person of the day.
She looks out the window. As she looks out, children’s eyes follow her eyes. They begin their comments about the weather—the physical aspects, their feelings about it, and wishes. They talk about the leaves turning color. They mention their mothers talking about the weather and winter clothes. One of the students, David, almost as if reacting to a pat on his back goes to the window to observe the weather outside and tries to read the temperature. David is having difficulty reading the number/numeral. There is a little line before the numeral. Mrs. Hills asks him to tell her what is creating the difficulty in reading the temperature. “There is a line just before the number,” David announces. “Yes, this time of the year, we will see this line quite often. Does anyone know about this line?” Mrs Hills asks the class. Several children raise their hand to help him read the temperature. Jonathan is always there to help, but Mrs. Hills sees the raised hands and asks Roland to help David read. Roland helps David read: “− 2 degrees.” Mrs. Hills now asks Jonathan to explain to the class what the line before the number means. Jonathan is pleased to explain the reason. Mrs. Hills talks about the relationship of the weather and the temperature—she talks about different seasons, temperatures, and surroundings outside the classroom. After several questions and comments from the children, she steers the class discussion to their daily opening activity—the calendar activity.
Mrs. Hills is a veteran teacher of thirteen years. She used to teach Kindergarten before she was moved to first grade five years ago. She also used to begin her teaching day in the Kindergarten class by the calendar activity with her children.
Mrs. Hills: Susan go to the calendar and point to today on the calendar.
Susan stands near the calendar on the easel and touches the square of the day and moves her finger above the day and points to Thursday.
Susan: Today is Thursday andit is the 29th of October.
Mrs. Hills: Look at the number line. Can someone point to the number that tells us today’s date?
Several students raise their hands. Two children try to point to the date. Finally, Mrs. Hills asks one of the students who is looking for 29 in the nineties. He points to the number 92. Mrs. Hills asks him to point to 20 and then asks him to count sequentially till he reaches 29. She asks children to look at the number the child is pointing to.
Mrs. Hills: Read the number.
The child reads the number. Mrs. Hills asks another child to read the number on the calendar.
Mrs. Hills: Michal you go to the calendar and put your finger on today’s date.
Michael points to the location where 29 is written on the calendar. Mrs. Hills asks the whole class to give Michael a hand.
Mrs. Hills: Does any one knowhow to make 29 using Cuisenaire rods?
Only a few years ago, Mrs. Hills used to use Unifix cubes, blocks, and other counting objects (coffee stirrers, straws, buttons, etc.) to make the number representing the date and the number of school days. It used to take a long time, and as a result the only mathematical skill the children would learn was one-to-one counting. For example, using straws, children will make bundles of ten straws to represent ten or they will fasten ten unifix cubes to makes groups of ten. Then, she started using Base Ten blocks. That cut down the time as the “longs” in the Base Ten blocks represented 10s and the “flats” represents 100s. Even with these materials, children counted the units when the one’s place was a number bigger than 5 and some even counts the ten marks on the 10-rod. Now she uses Base Ten blocks for hundreds and tens and Cuisenaire rods for ten’s and the one’s places. Now children, in her class, routinely make numbers using Cuisenaire rods and Base Ten blocks together. For example for displaying the number 124, they would use a one hundred block, two orange rods, and the purple Cuisenaire rod. They have become quick and fluent in making numbers, place value, and number relationships. Their numbersense is so much better. She is able to cover the curriculum in allotted time with almost all children demonstrating mastery. Even her children know the definition of mastery (efficient strategies, fluency, and applicability).
Early in the academic year Mrs. Hills defines what she considers mastery of a math idea: One understands the mathematical idea, can derive the answer using efficient strategy, can do it in more than one way, has fluency (where fluency is needed, e.g. arithmetic facts, key formulas, etc.), and can apply the idea in solving problems. Every time she introduces a new language, concept, or a procedure she reiterates the definition of mastery.
Contrary to her earlier fears that children will take long to learn how to use Cuisenaire rods, she found that it took only a few days for them to learn their Cuisenaire rods—the relationship between numbers and colored rods. First, she helped them discover the number names of each rod and then memorize them by using them and discovering patterns and number relationships. She kept a graphic of the Cuisenaire rods (stair case of rods from smallest to largest) for a few weeks and then removed it when the children knew the rods well.
Mrs. Hills has realized that the earlier her students know the rods, the sooner they will learn, master, and apply number relationships—facts and place value. Today also, before the children make today’s numbers, she does a brief exercise: She says a number and children in turn show the corresponding rod (if the number is less than or equal to 10) or make their number (if the number is larger than 10) using Cuisenaire rods. The children have already mastered the number names of Cuisenaire rods that match the colors (Sharma, 1988). They have been using these rods to make numbers and add and subtract numbers.
Similarly, it took Mrs. Hills some time to accept the definition of mastery. She always believed that children can have either conceptual understanding or fluency. She thought if children could arrive at answers by counting objects, on fingers, on number line, or hash marks on paper, they knew the fact. She thought fluency of facts was not necessary and it was counter productive to mathematical thinking. But she now realizes that the language, conceptual understanding, procedural fluency, and applications are complementary and support each other. She is sure of the idea that it is better to achieve mastery in the current concept before going on to the next concept. In the beginning of the year, it takes longer to master concepts, facts, or procedures, but later because of the mastery of earlier concepts the new concepts become easier to master and applications are much easier. In fact, she and her classes are able to do more meaningful mathematics, efficiently andeffectively in less time. She really understands what effective teaching is all about. Now, she routinely practices the following concepts almost every day:
- Counting forward, backward, from a given number beginning the academic year by 1 and then progressing to 2, 5, and 10. Towards the end of the year, her children are able to count by 100 from any given number.
- Number names of the rods till children are fluent.
- One more and one less than a given number as a preparation for introducing strategies for developing arithmetic facts.
- She picks up a rod and asks what number will make it ten?
- What two numbers make a particular teen’s number?
- She practices Sight Facts of a particular number using Visual Cluster cards till they master all of the 45 sight facts.
- She uses Cuisenaire rods and Base Ten blocks for making the numbers during calendar time (the date, number of school days, number of the day).
For example, Mrs. Hills asks children to make 29 (today’s date) using Cuisenaire rods. Children make 29 using the Cuisenaire rods. They display 29 as 2 ten-rods and a nine rods.
She displays the number by using the magnet
ic cuisenaire rods on the board and writes 29 below the rods (2 below the orange rods and 9 under the blue-rod.)
Mrs. Hills: What two numbers make 29?
Children: 20 and 9. 29 = 20 + 9 or 9 + 20!
Mrs. Hills: Great! What two digits make the number 29?
Children: 2 and 9.
Mrs. Hills: Very good! What is the value of digit 2?
Children: Two tens or 20!
Mrs. Hills: Very Good! That is true! Yes, 2 is in the tens’ place. What is the value of digit 9?
Children: 9 ones or 9!
Mrs. Hills: Great! Can anyone tell me what two other numbers make 29 as you saw in the Cuisenaire arrangement of 29? What two numbers, other than 20 and 9, make 29?
Child 1: 10 and 19, 10 + 19 = 29.
Child 2: Or, 19 + 10 = 29.
Mrs. Hills: That is very good. Can you show me this by the rods?
Child 1: Yes! See.
The child shows 10 and 19 (as seen in the figure below).
Mrs. Hills: That is very good! Please give her a big hand.
Child 2: What about 0 and 29?
Mrs. Hills: That is also right. Give him a big hand too!
Mrs. Hills, then children to write the combination of two numbers that make 29 as seen earlier on their white-boards. Children write:
0 + 29 = 29; 29 + 0 = 29;
10 + 19 = 29; 19 + 10 = 29.
Every child holds their white-boards and she checks them from her seat. If there any corrections to be made, she solicits children’s input. In case of a child having difficulty, she asks the child to make the number using the Cuisenaire rods or points to the board, where these number combinations are displayed using the Cuisenaire rods. After this she continues the calendar activity.
Mrs. Hills: Could someone tell me what will be the date tomorrow?
Child 3: That will be 1 more than today. Just add one to 29. It will be 30.
Mrs. Hills: That is right! Can you show us?
Child 3: Yes, I will take 30 Unifix Cubes.
The child first counts 29 Unifix Cubes and then adds one and declares: “Here are 30 cubes. These show tomorrow’s date.”
Mrs. Hills: That is correct. Can someone show us another way?
Children show 30 using several counting materials. Some children make 30 using Cuisenaire rods. Mrs. Hills observes their progress.
Mrs. Hills: Can someone show how to make tomorrow’s date more efficiently?
Child 4: I can do it more efficiently. Let me show it.
The child shows 29 and 1. He places 1 above the 9 in the number 29. (as seen in the diagram below). And then replaces 9 + 1 by 10 (an orange rod). He also writes the equation for the operation,
Mrs. Hills: That is great! He deserves a long hand.
Children applaud the child with several claps.
(The rule in Mrs. Hills’ class is “big hand” means two claps and “long hand” is several claps or till Mrs. Hills stops clapping. Children yearn for Mrs. Hills’ “big hand” and “long hand.” When they get the long hand, that is a big day for the child. Children keep score of the big and long hands earned. Generally, they only get “great” or a “great job.”)
Mrs. Hills: Great! What two digits make 29?
Children: 2 and 9.
Mrs. Hills: Great! What is the value of digit 2?
Mrs. Hills: Great! What is the value of digit 9?
Children: Nine ones.
She asks children to write number 29 in the expanded form on their white-boards. 29 = 20 + 9.
She writes the expressions: Number in standard form (29) and the number in expanded form (29 = 20 + 9)
Mrs. Hills writes few more numbers on the board to assess that all of her students have understood the number concept and its decomposition/recomposition and asks questions from each of her students and makes sure that each one of them had a chance to answer few questions.
B. Number of School Days
Another important and interesting activity related to calendar time is the number of the school day.
Today is the 67th day of school. The number pouch next to Mrs. Hills displays the school day from yesterday in symbols 66 and the six orange rods in the pocket marked tens and one dark green rod in the pocket marked ones. The pocket marked with 100s is empty. Children are eagerly waiting for the hundred pocket to have something in it.
She takes the number 66 from the pouch and displays the number on the white board next to her. She also displays the number with the help of rods as they are magnetic. Then she asks children to read the number displayed on the board.
Mrs. Hills: What number day was yesterday?
Almost all children have their hands up. Mrs. Hills picks David—a shy little blonde whose hand is half way up.
Mrs. Hills: Who is going to tell me how many tens are in sixty-six?
Mrs. Hills picks Marina.
Marina: I think six.
Mrs. Hills: Touch the six tens.
Marina touches the six orange rods.
Mrs. Hills: What is the number of the school days today?
Children shout out 1 more than 66.
Mrs. Hills: Cameron, what is 1 more than 66?
Mrs. Hills: Great! How will you make 67 from 66?
Cameron puts the 1-rod on top of the 6-rod (dark green) and then replaces the 6-rod by the 7-rod (black Cuisenaire rod). Mrs. Hills all children to make their own 67.
Each child makes 67 using 6 Orange Cuisenaire rods and a black rod. Each child has a small white board to write the numbers on. They place their rods making the number 67 in front of them in the same way as they will write the number on paper. Below this arrangement they write the numbers ‘67’—6 below the six orange rods and 7 below the black rod.
Mrs. Hills, then, asks all the same questions she asked in the case of “29” to make sure that children knew how to make the number and decompose it as:
67 = 60 + 7 = 50 + 17 = 40 + 27 = 30 + 37 = 20 + 47 = 10 + 47 = 0 + 67, concretely, orally, and then in writing.
By the end of the activity, each child has been asked questions related to these numbers. The time devoted on these activities varies, depending on what other pressing demands of the day are. However, this period is used for “tool building” for her main concept lesson little later in the day. She generally teaches reading and mathematics in the morning. The formal mathematics period involves a three-part lesson: (a) Tool Building, (b) Main Concept, and (c) Supervised individual, small-group, and large-group practice to achieve mastery. She conducts formative assessment during all three segments. The formative assessment is to collect information about her teaching and children’s learning. It informs her immediate teaching activity and her work with children. The formative assessment information also helps children to assess themselves as learners. The information from the first segment informs her how to shape the main concept teaching, and the formative assessment information from the main concept teaching how to design/redesign children’s practice activity (the quantity and quality).
C. The Hundredth School Day Necklace
Many Kindergarten and first grade teachers celebrate the hundredth school day by children making a necklace of hundred fruit-loops. The completed necklace for their mothers. Children count colorful fruit-loops and then string them. It is a very interesting and engaging activity for children. However, there are two problems: (a) first, it is a very time consuming activity, the mathematics payoff is very little. Of course, there are payoffs—social, fine motor coordination exercise, and emotional satisfaction; (b) by the time the child presents this necklace to his/her mother many of the fruit-loops have fallen down and the child still thinks there are 100 fruit loops on the necklace. There are better alternatives for making these necklaces.
1. Necklace One
Children can make a 100-necklace by taping (or stapling) ten strips (cut from orange oak-tag paper/heavy stock and each the size of orange colored Cuisenaire rod) and writing 10 on each strip. The following shows the partial necklace (with 4 tens).
This will help children to learn, very easily, that ten 10-rods (or 10 groups of 10) make 100 (without counting).
2. Necklace Two
Children can also make a hundred necklace by taping (or stapling) ten strips (cut from oak-tag paper/heavy-stock and each the size of an orange colored Cuisenaire rod) and writing 10 on each strip. Each strip is equal to ten. The following shows the partial necklace.
Whereas making the first necklace teaches children that 10-tens make 100, the second teaches children all the sight facts of 10 and that 10 can be made in several ways. Making ten is fundamental to learning addition and subtraction facts as most efficient strategies for deriving addition facts are dependent on making ten. When children know the sight facts of 10, they can easily arrive at all the other arithmetic facts. With the mastery of arithmetic facts and place value prepares them for arithmetic operations.
She knows children have mastered place value when they can answer the following questions, correctly, consistently, and fluently.
- What digits make this number?
- What is this place (pointing on a digit in the number)?
- What is the value of this digit?
- What digit is in the ___ place?
- What place is the digit __in?
- Can you write this number in the expanded form?
- Can you write this expanded form in the standard form?
- What numbers make this number?
Making the number representing the date and the number of school days so far help children to learn to answer the questions posed above. Mrs. Hills knows that if children can answer these questions correctly and fluently for three-digit numbers, they can easily extend this knowledge to any digit whole number.