# Kindergarten

### Important notice to Think Math! kindergarten teachers and coaches

Think Math! provides all you need. But you won't need all that Think Math! provides. The demands of kindergarten are diverse. You get to choose what makes sense for your class. To give the greatest flexibility to teachers, Think Math! provides broad options for kindergarten. The key to successful use of Think Math! kindergarten is to choose. To help you make the best choices, this page explains the components and options.

Do not try to do everything. Trust yourself as a teacher. The first edition of the teacher guides do not make it clear enough that teachers have choices. This page is to supplement and simplify the teacher guide.

See Getting Started with Kindergarten: Core and overview, and structuring centers

### The thinking behind the design of kindergarten

#### Diverse demands of kindergarten

While the range of skill levels may increase in elementary school, no grade is as diverse in developmental level as kindergarten. Some read and write; some don’t even know the letters. Some add and subtract (and there are the odd few that even multiply a bit); some can’t count; and some are still unsure how many fingers they have and need to recount them every time. Some are very "school-ready"; some still have little notion of how to be with other children.

The great skill of a kindergarten teacher is the ability to meet such a challenging situation and, over the course of the year, knit together a group that functions socially and intellectually.

For curriculum developers, the developmental diversity of kindergarten creates a different challenge. We must serve the teacher -- providing a structure and a diverse-enough repertoire of activities to meet the needs of a diverse group of children -- but we must not even imagine that we can dictate what "should" go on in the classroom. Only the teacher, and the school mathematics coaches in those schools that are fortunate enough to have them, can make these decisions.

In addition to the diversity of the students, the teachers of kindergarten whom we consulted during the development of Think Math! also had more diverse expectations than at any other grade. Split about half and half, some very strongly preferred little or no written component, feeling that at the PK/K age, children should be playing and experiencing and talking, not working on paper. This was, in fact, our original design. Others equally strongly felt that Kindergarten is when children are first encountering letters, learning to hold a pencil or crayon, and developing fine motor skills. While they certainly did not want a "paper-driven" curriculum, they did want every math experience to include at least some option for recording the activity.

Think Math! provides for both styles and, appropriately, leaves considerable room for teachers to use their own judgment. Of course, try some new things; some will reveal surprising abilities you might never have guessed the children would have. But also remain alert and flexible. Not everything will work with every child, and it doesn't need to. You can't do everything that is suggested, and are not expected to do everything.

#### Kindergarten children as natural mathematicians

Four and five year olds -- even the ones who can't count -- are, in many ways, thinking like little mathematicians.

**They classify and make categories.**This is how they learn language. Often, they'll hear a new word, and then use that word all over the place, and a bit more broadly than adults might, inventing a "category" of things that they think it applies to. (And they are astonishing language learners, acquiring new words so fast that many have almost half of what will be their adult vocabulary by the time they are five!)**They compare and quantify,**sometimes even when we wish they wouldn't. ("She got more than me!") They may not be accurate, but they are engaged in the mathematical idea of comparing quantity.**They sort.**Given a pile of buttons of, say, two sizes and two colors, children often sort them into categories or arrange them in some way. They might sort by color or size, as an adult is likely to do; and the rare child might sort by both color and size. Or they might sort in idiosyncratic ways, perhaps putting two big buttons and a little one together as "mommy, daddy, and me." But sorting by some kind of "meaning" is natural at this age.**They abstract.**A child's drawing is not what the child sees, but what the child knows. The preK child's drawing of a person shows the belly button, and might place arms and legs around the head. It is a "diagram" more than a photograph. And it's not just that the child' "can't draw well." That could explain poor proportions, but does not explain the use of extra details! The drawing is abstract -- everything he or she knows or cares about -- and not the concrete representation of what the child sees.**And they're fascinated with counting**even if they can't actually do it correctly!

All of this gives kindergarten teachers wonderful opportunities -- already naturally built into the child's way of playing -- to help these young children notice what clever little mathematicians they already are, and then to refine and extend their ideas naturally and still playfully, building their power as mathematical thinkers. Just by virtue of being adults, let alone college educated, teachers' logic and knowledge are way beyond what these children can learn, even about the topics (like counting) that are age appropriate. The children are little mathematicians, but in their own way, and not as consistent as adults are. They are also intrepid and ready to learn because they have not yet had the experiences that, unfortunately, leave some adults forgetting how mathematically intelligent they once were (and really still are, if they let themselves play again).

### Morning time: the core of the program

#### Make sure you do the morning time activities! They are essential!

The very beginning of each chapter describes that chapter's "morning time" activity. (Because it appears only at the beginning of the chapter, and is not repeated with each lesson, it is sometimes not noticed.)

If you do the morning time activity faithfully and engagingly every day, and nothing else, your children will be ready for first grade. Of course, there is much, much more that your children can do, and that is what the other components of Think Math!'s kindergarten program are designed to develop.

### Why Kindergarten begins with number

**FAQs** Many kindergarten programs start with shapes and then go to numbers. Can you tell me why numbers are first in the kindergarten curriculum?

Part of the answer is that, though we start with number, there is a lot of attention to shapes from the beginning as well. Here's a quick overview. In Lesson 1, we have children select from all the pattern blocks -- square, triangle, diamond, trapezoid, hexagon -- and describe one pattern block. In Lesson 1, children also make pattern block designs and explore connecting cube towers. Throughout the chapter we have a heavy emphasis on studying pattern blocks and their attributes, as well as making designs with them. Quantity cards are used frequently to practice number, but they also picture numbers using a variety of shapes. Children continue to explore cube patterns. Headline stories feature shapes, such as stars in Lesson 3, and circles and triangles in Lesson 14. Lessons 12 and 13 have activities with Cuisenaire rods. Lesson 14 has an activity where different shapes are used to make a real-object graph. Chapter 2 explores many of the same topics, but with somewhat more direct focus on shapes and somewhat less on quantity. Both are certainly important in both chapters.

One reason that kindergarten programs often start with shapes is because shapes are interesting and accessible to young children and because they are important for the study of mathematical ideas in a way that kindergarten children can understand and express. Describing the shapes (typically with colors and sizes) is also seen as part of the language development of young children. We do this too. However we also introduce number early for two reasons. For virtually all children, the numbers 1 and 2 (not the reading or writing of them, but the meaning of those quantities, the ability to recognize whether they have 1 or 2 books, for example) are well established; the vast majority are curious and interested in other numbers, too, even if they don’t quite know what they all are, or can’t count them; and most children (still a pretty large majority) can count through many numbers up to about 10. Only some can read or write those numbers (despite being able to use several numbers correctly), just as only some know their letters (despite having a huge vocabulary of words that they speak and use correctly). The reading and writing of numbers—just like all of the written work in Kindergarten—is provided for use at the discretion of the teacher. Some teachers prefer to emphasize reading and writing; some prefer to de-emphasize it; because we needed to support both judgments, we had to provide the options, but from a mathematical learning perspective, either approach is fine. But it is also important to honor and feed children’s natural curiosity about number, the several numbers they already know, their eagerness to describe the world not only qualitatively (color, shape, comparative size) but also quantitatively (with counting and, much later, with measuring). So we start with both.

Much of the number-related work, especially early in the year, can be seen as an exploratory activity so that children experiment with the concept of grouping and counting objects as well as looking at their attributes. For the children who are developmentally ready to think about number and can use 1-1 correspondence, the number part of this chapter (and the organization that the number line provides) supplies a fun challenge. For children who are not there yet, they learn the number part of the chapter more like one learns a song or poetry. Both developmental groups find the activities fun. No single child will do all of the activities of any lesson. The large range of activities gives all developmental groups activities that challenge them, but do not overwhelm them.

Number, like shape, is revisited throughout the curriculum.

### Structuring Math Time and Centers

Each lesson in Think Math! includes a whole group session (with 1-3 activities) as well as three center activities. Structuring all of these activities into one 60-minute block of math may feel overwhelming, especially at the beginning of the year. Below are five models that teachers have developed to make this work in their classrooms. Find a structure that works best for you. You may choose to adapt as the year progresses, or even each day, depending on the amount of support you have in your room.

#### Groupings for centers

When you plan your groupings, there are a variety of questions to consider, such as:

- Will the groups be the same for a week? Month? Year?
- How will they move? Together? Individually?
- Who will be responsible for the materials?
- Will students have individual folders?
- How will you use LAB pages? Whole class? Small group? Tear out of book? Not at all?

Everyone agrees that once a game or an exploratory center has been introduced to and practiced with the whole class, it is easier for children to successfully work at an independent center.

#### Models for Structuring Math Time and Centers

**Whole Group Lesson - Whole Group Activity**

One of the simplest ways to structure a Think Math! lesson is to begin with the whole group lesson. Once the whole group lesson is complete, move to tables and do one activity/game from the “centers” located in the Teacher’s Guide. You choose the activity/game that is best for your class. All students will work on the same activity, but the activity may be differentiated to meet the variety of needs in the classroom.

**A-B-C Model**

In this model, you choose the three activities that you consider the most important from the lesson. Part A, the one you decide is most important, can be a whole-group lesson taught by the teacher and followed by a whole group activity. When a child finishes his/her work, the child gathers the materials for part B, a choice that has been pre-introduced to the group; the child works independently or with a partner. If a child completes the activity in part B, the child takes the materials for the activity for part C. Not all children need to get to this activity. In this model, the teacher is able to support children needing extra help during Part A, while others go on to B and C. Prepare and introduce materials for B and C ahead of time. Students work at their own pace.

**“Children Move”**

The teacher begins with the whole group lesson. Then, students go to designated tables to work at one of the centers. The Teacher Guide usually gives ideas for three centers, but for a large class (over 18), you may want to create a fourth center to reduce the number of children in one center at one time (4-5 children is ideal). Each table has the one center’s materials prepared and ready. One center is “teacher led”; the others are games, activities, or explorations that have been introduced in the whole-group lesson or in a previous lesson. The teacher will direct students to switch tables after a designated amount of time (generally 10 minutes). Ideally, all students visit each center in one lesson.

Alternative: Some teachers choose to keep the same 3-4 centers available all week, so that the students have multiple opportunities to work at a center. This model also allows students to visit a center for longer; they can visit other centers the next day.

**“Children Stay – Materials Move”**

This model begins with a whole group lesson. Then the students go to designated tables to work at one of the centers. (Again, the teacher may introduce a fourth center to maintain smaller working groups.) Each table has the one center’s materials prepared and ready on a tray so that it can be moved. Once the students complete the center activity, the teacher initiates a transition. Instead of the students moving, the teacher rotates the materials among the tables. The teacher may want to support students at the Center by traveling from table to table. Ideally, all students have the opportunity to work with all different materials in each lesson.

Alternative: Some teachers choose to keep the same 3-4 centers available all week, so that the students have multiple opportunities to work at a center. This model also allows students to visit a center for longer; they can visit other centers the next day.

**Materials for All Three Centers at Tables**

This model begins with a whole group lesson. When completed, students choose which center they’d like to work at. Each table has the center’s materials prepared and ready. Once the students complete that center activity, the child initiates a transition to another center, where space is available. Since the students initiate the move, you may want to have an extra space at each center to ease the transition.

### Frequently asked questions

#### Why coins? Our state standards don't call for that in K

"My state's standards only have 1 Performance Indicator about Money and Time, but I see a lot of references to coins and the clock in Think Math! Could you tell me why?"

We could have used plastic counters, teddy bears, or blocks, but children love the coins, especially when they are real. They like teddy bears, too, but the real-world grown-up feel of coins, even if they have no notion of their value, feels "important" to them, and is fun. Though they don't need to learn the values, many become curious, and so we include activities that satisfy that curiosity, available to any interested child, but not demanded of any.

Even at later grade levels, when children do need to learn the values of coins, coins were always intended to be used as manipulatives in Think Math!, not as the focus of the lesson. In later grades -- not in K -- children use dimes and pennies to model place value, nickels and pennies (along with hand and fingers) to build numbers under ten. The same is true of clocks. Think Math! uses clocks to support counting by fives, to vary children's experiences with half and quarter (hours), and to build other arithmetic ideas. And, of course, to learn to tell time!

**Teacher comment:** In second grade, some of the lessons on time that had originally been spread out over a few chapters were eventually aggregated into a single chapter, responding to teachers' expectations. As a teacher myself, I'd be inclined to spread them out again, teaching a different topic and periodically returning to build a new clock skill.

In K, none of that is explicit, but getting to handle and distinguish coins, and even learning the value (much as they learn the name, just as a thing to know) is interesting to them. "Understanding" is another matter. Nobody expects the K child to be able to count a pile of change, but children feel proud and smart to know what the coins are called (even if they slip often!), and to say how much they are worth (even if they aren't reliable about that, and cannot, in any event, say what two nickels are worth together). Children who do have rich and varied experiences early on -- playing "store" in class (even if they have no clue about who pays whom the money, let alone how to make change) -- do better in later years! And some will surprise us and even learn to count change.

#### Children at this age will not become experts at money, and don't need to be

-- for tests, for life, for state standards, for anything. But they like the coins, partly because they are "real" things, and partly just because they are money. Coins are excellent counters, even if children don't know their value, and they are a great opportunity for differentiated instruction. Some children will be able to see a nickel and two pennies as a chance to "count on"; others will just see "three coins." In later grades, the value helps teach the arithmetic, and so it is a real investment in the future to give K kids a chance to see coins, ask what they are, and perhaps even remember.

#### What is required? What is optional?

Are all parts of the lesson required? What would you recommend I focus on when time is short?

The core of the program is in morning time. The richness of the program is in the lessons and centers. If you have very little time for mathematics, your children will (of course) get less of a mathematical beginning than will those with more time, but if you take the morning time seriously and use the centers as guides for the various ``choice stations that most kindergarten classes have, your children will be well enough prepared for first grade. Before children at this age can work/play independently enough at their choice stations, you will, of course, have to introduce them to the activities. Some suggestions for those introductions can be found in the Math Time sections of the Teacher Guide.

#### Are all the activities developmentally appropriate?

There seems to be a lot of sorting, then recording or transferring of information in K. Is this developmentally appropriate?

**A two-part answer:**

- The developmental spread in Kindergarten is so great that virtually nothing -- except, perhaps, the spirit of playfulness -- could be developmentally "appropriate" for all children of this age. A perfect challenge for one may be boringly easy for another and frustratingly out of reach for a third, even though all three children may be perfectly matched learners a year or two later.
- Every part of the Think Math! Kindergarten program is developmentally appropriate for a significant number of children who can be found in every kindergarten class. Only you can decide which of the activities fit with which of your children.

#### Is sorting, classifying, naming, and describing categories developmentally appropriate?

Children of this age acquire new words (naturally) at a phenomenal rate. The language learning is (mostly) a matter of forming a mental category (recognizing a certain kind of animal, for example) and assigning a symbol (a sound) to it. Kindergarten-age children are constantly and actively engaged in language acquisition and making meaning of (spoken) symbols and simple visual symbols (pictures or letters or numbers). This is a good age for children to be developing their language, including their mathematical language -- seeing things and putting them into words (including words about shape and placement and quantity and comparison...). Think Math! takes advantage of this natural aptitude of young children.

Think Math! also asks for a lot of "translation" from one representation to another: children translate among the languages of pictures, spoken words, and numbers constantly, in all directions. Not all children at this age are developmentally ready for written symbols, but the translation part (not including the written form of it) is very age-appropriate and developmentally sound for virtually all children of this age.

#### Is written mathematics developmentally appropriate?

**Teacher discretion advised!** For some, yes; for others, no. Of the various representations children might use for their mathematical ideas -- spoken words, actions, manipulatives, pictures, and writing -- the appropriateness of writing as a record-keeping method is the most variable. At this age, children are just beginning to learn to record their language. For those who cannot yet draw or write, those ways of recording are not yet appropriate. (Our own first child was not reading or writing letters or numbers in Kindergarten.) At the discretion of the teacher, children might use stickers, or objects, or leave no permanent record at all. Other children are ready and eager to write a number to show they know "how many" or to draw a picture, or make a sticker-graph, or even write a few words of description. Some feel pleased to "have homework" like an older sibling. Only the teacher can know. A curriculum-writer’s job is to support the teacher by providing a sensible path through the mathematics and good activities that span a reasonable range of developmental levels. The teacher's job -- difficult at all ages and especially subtle in kindergarten -- is to understand each child well enough to know how to choose those activities that help that child along that mathematical path.

### Pacing

#### The pace seems ambitious? Is it practical?

Pace at K, as at any grade, must be adjusted so that it is lively and varied enough to remain interesting without being frustrating, and "calm" enough to be enjoyable without being boring. In all grades, we are more ambitious than many curricula. Children can learn much more than they typically do, and part of everyone's concern about mathematics learning in the U.S. is that children are not learning enough. What turned children off most in the classes we researched before and during this project was the sluggish pace. They were bored; not enough was happening.Pace needs to stay lively, but not pressured. The judgment must be made by the teacher.

"This looks too hard for my kids!"

I have some very advanced children.

Planting seeds and tending the seedlings while waiting for fruit