“Fingers, fingers, 1-2-3…how many fingers do you see?”  

We are playing one of our favorite finger games. I hide one hand behind my back. When I bring it forward, I hold up some fingers and the children shout out the number of fingers that they see.

“Three!” shout the friends playing the game.

Finger games can be played anywhere at any time because our fingers are always, well…handy! Besides, there’s a lot of math to be learned in those little fingers. Fostering a love of math in children begins with building a basic understanding of numbers.

I watch as two-year-old Jade repeatedly looks at his fingers and then back at mine as he attempts to duplicate my patterns. Children learn through their senses, and Jade is visually and physically working his way through an early math skill. He is also engaging in a sensory-motor experience that helps build abstract thinking skills.

When children engage in finger play, sing counting songs and play counting games, they are building a strong number sense. Number sense is a person’s ability to understand key math concepts such as quantities and the numbers that represent those quantities, as well as concepts such as more or less. Children with good number sense can think flexibly and fluently about numbers.

While using his fingers, Jade can feel and see the difference between the numbers 2 and 4. This developmentally appropriate math game is helping Jade connect a quantity to its numeric name—and his vocabulary is growing as he chants along with the rhyme.

Compelling new studies are also revealing how hands literally “help the brain think.” According to the website Science Translated—which educates students and the public about ongoing scientific research in a simple, jargon-free way—”Children clearly ‘think’ with their hands while learning to count.”

Neuroscientists and educators agree: Children who learn to use their fingers as a mathematical tool in the early years experience more success in math than those who don’t.

When children use their fingers to count, they are strengthening their number knowledge and their ability to visualize numbers in their minds. Counting is more complex than simply memorizing and reciting number words. Children need to understand the counting sequence, as well as one-to-one correspondence, cardinality and subitizing.

• Counting sequence: Counting involves using the same sequence each time, starting with one.
• One-to-One Correspondence: Exactly one number from the counting sequence is assigned to each object in the collection.
• Cardinality: The last number assigned to an object when counting the collection indicates the total quantity of objects in the collection.
• Subitizing: The ability to recognize a small group of objects without counting.

Watching and listening to children’s counting will help you see what they know and what they still need to learn. Once the children have a strong understanding of the numbers up to five, try adding your other hand to the game. For example, I show two fingers on my right hand and three fingers on my left hand. The children have to add the two sets of numbers to give me a total number.

“1- 2-3, let me see…the number two!”

We also use our fingers to play with shadows. Using the sun as a light source, I call out a number. The children then hold up the appropriate number of fingers to represent that number, casting “finger shadows” on a wall or on the sidewalk. 

This is a great way to help children build their number sense. It allows the children to work on:

• Finger-isolation activities such as pointing with the index finger, counting out the fingers on their hands or wiggling all of their fingers individually
• Thumb-opposition activities such as touching the thumb to each finger to build strength and dexterity for pencil-holding and cutting with scissors

These are all good reasons to add some finger play to your days! Keep it fun, keep it spontaneous and keep it simple. What looks like child’s play will help build a strong foundation for later math learning. You can count on that!

A pair of four-year-olds in my classroom are happily playing “dice wars,” a simple but fast-paced game. To play, each child rolls a die and the player who rolls the highest number wins. No one is keeping score today—and nobody seems to care who wins each round. 

I watch as Juan shakes the die between his palms, rolls it, counts each pip (dot) on the side that’s facing up and announces, “1, 2, 3!” I rolled a three!”

“My turn!” exclaims Maria as she shakes the die between her hands and rolls it. “SIX! I win! Six is more than three!” Maria doesn’t need to count the pips. She recognizes the pattern immediately and her number sense tells her the value of the pips on the die.

Rolling a die is fun in its own right, but these friends are working on the math skill known as subitizing. When children begin to recognize the pattern on the die and associate it with the number of pips (dots) without counting each pip, that is subitizing!

Children develop subitizing skills in much the same way that they learn to read sight words.

In a previous Early Math Counts blog post, Jen Asimow, M.Ed, explained it this way: “Remember when you learned about ‘sight words’ and how children learn them? According to one school of thought, children memorize sight words by taking a mental snapshot of the entire word. By using context clues, they learn the word as a ‘whole’ rather than as a series of letters. Consider how children learn the words EXIT or STOP.  Both of these words appear in the same way—on signs above doors or on red octagonal street signs—and nearly always in the same colors and typefaces. All of these clues help children form a mental picture of these two words, and they often learn the pictures before they learn the individual letters that make up the words.”

Maria and Juan are playing with a die, so they are only working with numbers ranging from one to six. As they play successive rounds of the game, they are beginning to recognize the patterns on the die without counting the individual pips.

With every roll of the die, Juan’s pattern-recognition skills are growing stronger. Before long, he can recognize the total number of pips on each throw without counting.

According to child development experts, the ability to subitize quantities up to and including four by the age of five represents a significant developmental milestone. 

Subitizing is a fundamental math skill, and dice games are a good way to help foster the development of this skill.

“Hey! Do you want to play that block-building dice game?” asks Pierre as he grabs a die from the jar and joins the group. 

Roll and Build is another dice game that we’ve played in our classroom for years. One child rolls a die and the other children add that number of blocks to their towers. Children learn to recognize four dots on a die, which helps them understand the cardinal value (how many four represents), which they can then link to the symbol (4) and the word (four).

Games like this provide repeated opportunities to interpret the dot images. As children figure out how many pieces to collect or add or how many spaces to move on a game board, they develop their number sense and other early math skills such as counting, number identification, the correlation between numbers and the items being counted and concepts such as greater than or less than.

Keep a jar full of dice within easy reach to give the children plenty of opportunities to practice and make up their own games.

Begin by subitizing quantities of 1, 2 and 3. In a math workshop that I attended, the trainer had blacked out the pips representing 4, 5 and 6 for the younger children.

If a child is having difficulty subitizing, reduce the quantity of dots. 

Dice games help young children develop math and social-emotional skills in a fun and engaging way.  So grab some dice and introduce your gang to subitizing fun! 

Check out our Early Math Counts lessons page for dice game ideas. Be sure to click on the Connect With Families button in the left-hand column of each lesson to download a Parent Letter that you can customize to share the day’s learning activity with parents and other family members.

A couple of weeks ago, I observed one of my student teachers read The Very Hungry Caterpillar aloud to a group of 18 three-year-olds.  She did  a great job reading; using all of the skills and techniques of dialogic reading that we teach our growing teachers. While watching her, I realized that there are so many great math concepts in that book.  There is sequencing, number sense, number recognition, one-to-one correspondence, and predictability.  Using an engaging book to explore these concepts is so much more interesting than so many more typical didactic exercises that tend to be less developmentally appropriate and definitely more boring to young children.

I found this cool website that is filled with ideas about how to use The Very Hungry Caterpillar in a variety of ways,  Check it out here.

It is pretty big just like the other one, but this one only goes from 0-10.  It offers a different set of mathematical possibilities because of its design differences.  You can see that this one has large numerals and a corresponding number of objects next to each.

There are aspects of this number line that might make more sense to young children.  It only goes up to 10, but the inclusion of “0” complicated matters.  Children generally learn to count beginning with “1” not “0”.  It is also hard to illustrate nothing as the absence of objects is supposed to indicate. You and I can understand that the empty space around the number “0” indicates nothing, but I don’t think young children will.

If I were to make a jumbo number line for my classroom, I would take aspects of this one and aspects of the other, put those together to create the perfect (and appropriate) prototype.

Keep the large, easy-to-read numbers but begin with 1.  Make the number line 6 units long and then add-on units as children develop stronger number sense.  Provide other context clues, ideally removable, so the line can change as needed.  It might be nice to have objects, like in the example above, but it might also be nice to have “pips” like dots on a die, to indicate the number.  It would be cool if the units could be arranged to create a circle or to go around a corner.  That way, if space was an issue, the line could be used on an area rug.

What would you do to create the perfect prototype?

It is big enough that children can stand or sit on it, move around on it, or line up on it. The numbers are large and easy-to-read.  Odd numbers are red and even numbers are blue, giving additional context clues to support the children’s concepts.

Begin introducing the mat when children are transitioning from one activity to another. Have each child pull a number out of a basket and go stand on that number.  Make sure the numerals you draw on the cards look exactly like the numbers on the line, so if needed, children can match their numbers easily.

You can also use the number line as a place to play a game like Simon Says.

Simon says, “Everyone find a blue number.”

Simon says, “Move to a red number.”

Simon says, “Change places with one of your friends.”

“Stand on one foot.”

Use the number line like hopscotch. Have the children line up at the zero spot and ask them to jump on each number and call it out as they go.  Switch it up by asking the children to hop on one foot, or to only step on the even (or blue) numbers.

If you have older children in your program, the number line can be used for counting on or taking away.  These skills are much more difficult and I would not recommend introducing them until the children have a strong and developed number sense.

Adding gross motor movement to any activity is more inclusive of learning styles and definitely enhances learning.  Leave this mat out during free choice time and see how the children choose to play with it.  Let us know what you find out.

That is why I thought I should mention the trouble with the English counting words above ten and below twenty.  In many languages, each of those numbers’ values is described by the words themselves.  Eleven should really be “ten plus one,” and twelve should be “ten plus two,” and so on.  This would make so much more sense as children begin to associate the counting words with their values.

Perhaps it is wise to explain to your children as they begin counting above ten, that eleven really means ten plus one, and fourteen really means ten plus four.

Remember, many children can count because they have memorized the counting words (rote counting).  This is not an indication of number sense as much as it is a growing competence in language and memory. Meaningful counting begins when children connect the number words with objects correctly as quantity.

Knowing “how many” is not an innate skill like crawling.  It takes frequent opportunities and exposures to see quantity.  This begins with a visual perception of relative amounts.  One of the best ways to encourage this kind of thinking is by providing opportunities for children to determine “More” and “Less”.  We do this through regular, everyday interactions like eating, by asking “Do you want more applesauce?” and even earlier, “Do you want more milk?”

When you observe interactions between caregivers and infants, do you see the adults put bottles back in the babies’ mouths and jiggle the bottle about, rather than asking, “Do you want more milk?”  It is irrelevant if the infant has oral language.  This is an opportunity to expose the child to the notion of quantity.  We don’t expect the infant to respond to the question. We use this language in order to work on conversational skills, to allow the infant some control over her feeding and to encourage the development of the concept of quantity. It is up to her to decide if she wants more.  Eventually, she will seal her lips together and shake her head so the caregiver cannot put the nipple back in.  She has decided that she doesn’t want more; that she is done.

Don’t begin working on counting before the young child is ready.  Keep exposing her to quantity in general and observe how she become better able to determine more and less. Think of ways to engage her in number activities that don’t involve counting but do involve number.  Keep modeling counting. Let the development of number sense unfold naturally.  It will.  I promise.

In Kindergarten, instructional time should focus on two critical areas: (1) representing and comparing whole numbers, initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.

• 1. Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away.
• 2. Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.

So, although there are clear expectations for each area, teachers are expected to spend more time focusing on number than on any other area. This expectation reinforces the idea that number sense is the foundational skill on which all other mathematical skills are built.  That means that we too, should focus on number.

Next week I am going to explore the first area of the mathematics Core – Counting and Cardinality.  Can’t wait!

The first goal in the Mathematics section is

“Goal 6 – Demonstrate and apply a knowledge and sense of  numbers, including numeration and operations”

 

The associated learning standard is

“Learning Standard A – Demonstrate beginning understanding of number, number names and numerals”

and the benchmarks are

6.A.ECa – Count with understanding and recognize “how many” in small sets

6.A.ECb – Use subitizing (the rapid and accurate judgment of how many items there are without counting) to identify the number of objects without counting in sets of four or less

6.A.ECc – Recognize and describe the concept of zero

6.A.ECd – Connect numbers to quantities they represent using physical models and representations.

6.A.ECe – Differentiate numerals from letters and recognize some written numerals

6.A.ECf – Verbally recite numbers from 0 – 10

 

Without looking at the example performance descriptors, I think we could come up with a thousand and one ways to look for examples of the children meeting these benchmarks.  Using a variety of math manipulatives, regularly as a part of your everyday program, children will begin to know how many pips are on a die without counting them (6.A.ECb), count the number of Unifix cubes there are in a set, (6.A.ECa, 6.A.ECf) identify numbers in a matching game and name them (6.A.ECe), and so on.

It is important to note that although the authors of this document provide performance descriptors, that we as practitioners, do not get caught up in “teaching to the test.”  It would be easy to use these examples as specific ways that we look for successful achievement for children, but it is much more developmentally appropriate to expect that there are a variety of ways that children can show us what they know.

This goal is about number- recognizing a written numeral saying its name and differentiating those symbols from letter symbols. It is about understanding the concept of “nothingness” and that “nothing” can be represented by the symbol “0”.  It says that, just by looking, children should be able to tell how many of something are in a set of 4 or less and that they should be able to count individual items in a set accurately.  Children should be able to answer the question, “how many?” and make representations of that number by creating a set using physical numbers and representations of that number.

In April, I plan on exploring perhaps the most important book ever to be written about young children and number….it is aptly entitled “Number” and hopefully, this discussion will continue to shed let on how children achieve these goals and meet these benchmarks.