The Five Senses of Math

posted by Diann Gano

We spend a lot of time outdoors. Playing. That play involves math in such natural ways that it is easy to overlook how often math comes into our lives. Research has found that early math proficiency is a better predictor of future academic success, high school graduation and college attendance than any other childhood skill. (National Governors’ Association, Unlocking Young Children’s Potential: Governors’ Role in Strengthening Early Mathematics Learning

So, here at Under the Ginkgo Tree, we encourage that investigation and provide materials that support children’s development of math concepts. Outdoors, the process for mastering the fundamentals of math is truly enhanced in a holistic and inviting environment. Come see how we create these opportunities using all five senses for all types of learners.

SIGHTexamining a tree trunk

Regardless of the season, we often spend time seeing how tall or small our friends are using resources that are available to us. Sometimes we use tape measures, but often we have other ideas.

Adding magnifying glasses, kaleidoscopes, and binoculars will slow your children down. They will look closer, longer, and think harder. Remember, we are giving them lots of time to investigate and explore and enjoy their childhood. Add these to your play space!

measuring with apples





mesauring with carrots








magnifying glass







Right! Three! Did you count? No, you subitized! We love to subitize. Subitzing is seeing a small amount of objects and knowing the number without counting. Playing with dice, you roll a six and without counting the 6 pips, you know it is six. We often play this game with fingers. We place both hands behind our back and then bring forward one hand with a few fingers showing. As they get better at this, we make it faster, or change the finger arrangement, or add both hands. We also work with subitizing outdoors with any given object. Kids love this game and I encourage you to play it often. Restaurants, long car rides, or waiting rooms are all great places to subitize! If you have fingers, and a bored child, subitize!


cutting vegetables cutting vegetables 1 snacks

Well, this one is almost too easy. If you put six cookies in front of three children and tell them to figure it out…they will. F A I R is a favorite math word that we hear around here. Everyone has to have EQUAL amounts of everything! Every meal we serve has some sort of math discussion available.

I invited the four-year-olds to help set up lunch. When they asked if they could cut the cucumbers, we discovered lots of investigation on size, direction, more, less, and equal.

When it came time to put out the fruits and vegetables, they created sets and counted out tomatoes, cut sandwiches in half. We had a math buzz going on!

Cooking is a great opportunity to bring in math vocabulary and concepts. When cooking with children, we always try to have enough bowls, utensils and, of course eggs, for each child to make their own portion. Think about it, if you watch your friend whip up her award-winning recipe, it’s just not the same as doing it side by side with your own version. This also offers great opportunities for mentoring and scaffolding with their peers on tricks for measuring and cracking eggs. It may not seem like math, but it is. If your body and brain don’t have the energy to deal with the egg mess on any given day, we have been known to cheat with the infants and cracked the egg into the measuring cup. That counts as an egg turn for anyone not old enough to roll her eyes at the thought of it. (Usually your four- year old.)snacks

SMELLsmell dandelion holding an iris

Those little noses often lead us to food. We routinely have

discussions about how many raisins or chocolate chips are in our cookies. We also like to smell in gardens and parks. Did you know that there is a math pattern in nature called Fibonacci? Some refer to Fibonacci as nature’s number system. From the pattern of florets on a flower to the bracts of a pine cone or the leaf arrangements in plants, the same number pattern appears over and over. The basic pattern is 1,1, 2, 3, 5, 8, 13, 21, 34… The next number is found by adding up the two numbers before it. It forms a spiral. This is a bit complex for most of the young learners in our program, but we talk about it and the spirals that we see in pineapples, and pinecones and flowers.

bunch of flowers

We often count the number of petals on our flowers. Last summer we planted perennials to our play space and included a number of flowers that include this Fibonacci pattern. Cala lilies have one pedal; euphorbia have two; triilium and some iris have 3 petals; buttercups and columbine have 5; bloodroot have 8, black-eyed Susan’s have 13; shasta daisies have 21 whereas other daisies can be found with 34, 55 or even 89 petals. Isn’t that crazy fun?

spring flowers


baby at fence baby ringing the bellSOUND

We live a block from a college campus and the bell tower. So every hour on the hour we have some counting sound opportunity in our own backyard. Music is an easy way to add math to your outdoor play space.

We have bells placed throughout our play area. We often have obstacle courses that include ringing a bell or drumming on a drum. Music is a big part of our lives. Singing songs, counting rhymes and fingerplays combine music and math. Really is there any better place for those loud drums, bells, tambourines and maracas? What were we thinking? These are outdoor toys!




bells on deck
baby with tamborine children with musical instruments




TOUCHacorns child holding an acorn

When you give children real materials to touch and smell and feel, the learning is deeper and more authentic. There is a major difference between touching real apples and moving them from basket to basket, than counting apples on a worksheet. Looking at stripes and circles on a page is not the same as touching and understanding that each rock has stripes or circle textures or has six shades of green. It makes sense to them. They control it. They aren’t rushed to move on to the next box on the worksheet. Give them real materials.

We talk about blocks and rocks often in our program. Those two things create an unbelievable amount of building. Building is math. We spend hours and hours building things that we may not ever even play in or with. We have spent days making forts. Sometimes, we don’t ever actually play IN them. We just build them. We build zoos, and fairy homes and squirrel traps. It is the building that is the fun, the creativity, and the play. Give your children time to touch, and think and process, and arrange and rearrange. It’s all good.stack of rocks

We also play games where one child plays a rhythm and the other needs to repeat it. There’s very often pattern play going on with this. For many children that rhythm and counting go hand in hand. When we follow singing commands to go in or out, or up and down, around and through, those build spatial awareness and reasoning skills that are important skills for geometry. Everyone is happy when we have music in our lives. Make your own or fire up Pandora!

Have fun this week noticing how often math is in your child’s life. It will make you smile and give you peace to know that you are doing just fine as a parent and an educator. Keep creating math environments and playing with your kids. It’s all quite simple. Put the worksheets away. It will come when their brain development is ready and it is relevant to them. Until then, just watch, listen and smell the learning coming their way!





Number Sense: Make it Real!

numbers opaqueposted by Dr. Bilge Cerezci

Young children are motivated to explore mathematical concepts they encounter in their everyday interactions with the world. Through these interactions, they develop a range of informal understanding of numbers including ideas of more or less and one-to-one correspondence. For example, a child as young as two knows if she gets more or less crackers than her friend next to her. She also exhibits her basic understanding of one-to-one correspondence when she insists on getting a cookie because her brother had one and she had none. Such intuitive understandings of number sense may help lay the groundwork for later understandings of numerical equivalence and operations, such as addition and subtraction. While serving as important building blocks, such understanding does not necessarily help young children explicitly examine and interpret their experiences in mathematical forms. So, how do we help young children make connections from these informal knowledge around numbers to a deeper, more concrete understanding of numbers?

Helping children recognize math in the real world and finding everyday math activities at home is a great way for parents to reinforce young children’s developing number sense. For example, when you are setting your table for breakfast, ask your child to join you. You can ask them how many plates do you need to set the table or whether you have enough eggs for everyone or not. While they are taking the plates from the cabinet, encourage them to count. When young children practice counting, they’re also learning one-to-one correspondence. A child that understands one-to-one correspondence knows that 4 plates equals 4 or that 5 eggs equals 5. To help them practice this concept, give your children large groups of objects to count. For example, you are making a strawberry cake for dessert and you only need 10 strawberries. You may ask your child to help you figure out whether you have enough strawberries or not. As they are practicing this skill, children may count some of the strawberries twice and/or skip counting some of them. Therefore, it is important to closely observe your child as she is counting. When she is double-counting some of the strawberries, does she realize what she has done? Does she self-correct? In such instances, resist the temptation of correcting them. Instead, ask her to double-check her answer and give them enough time to check their work and self-correct their mistakes. If she is struggling, provide them with some strategies she can use(e.g., moving strawberries to a different pile as she counts).

Taking this kind of approach not only allows children to see math as fun, but also helps them see numbers as useful tools that they can use to make sense of the world around them. While doing these kinds of activities, the most important thing you can do is to help your child see math is something that makes sense and it is practical and enjoyable. This will help your youngsters to build a strong understanding of math and develop a love of learning math that will last a lifetime.

square blocks

The Early Math Experience Matters

posted by Dr. Bilge Cerezci

Traditionally, mathematics education has not been considered developmentally appropriate for young children (Battista, 1999). Math is abstract while young children are deemed to be concrete thinkers, and some cognitive developmental work done in the mid-twentieth century has been used to suggest that young children’s mathematical ideas develop on their own timetable, independent of environmental factors like teaching (Piaget, 1969). Over the past two decades, however, a growing body of literature has indicated that many mathematical competencies, such as sensitivity to set, size, pattern, and quantity are present very early in life (National Research Council [NRC], 2009). magnetic numbers plus symbolsYoung children have more mathematical knowledge, such as an understanding of number and spatial sense, than was previously believed. For example, research suggests that young children have a basic understanding of one-to-one correspondence even before they can count verbally (e.g., pointing to items in a collection and labeling each with a number) (Mix, 2001). Further, young children also enjoy exploring spatial positions and attributes of geometric shapes by building towers with blocks and cubes and by manipulating various materials, such as puzzles and two- and three-dimensional shapes (Clements, 1999; Clements & Sarama, 2008). They also demonstrate emerging awareness of measurement, when they begin to notice and verbalize similarities and differences in the size, height, weight and length of various objects and materials (Clements & Sarama, 2008). In addition, research also suggests that 3 and 4 years-old children engage in analytical thinking as they collect and sort materials by various attributes (e.g., color, size, and shape) and in algebraic thinking as they copy the patterns they observe in their surroundings and create their own patterns by using pattern blocks and other materials (Epstein, 2003; 2006). In fact, as research points out, most children enter school with a wealth of knowledge in early mathematics and cognitive skills that provide a strong foundation for mathematical learning (Clements & Sarama, 2009; Ginsburg, Lee, & Boyd, 2008; Mix, 2001).

There is also new evidence that achievement in early mathematics has a profound impact on later success. For example, Duncan and Magnuson (2009) examined the mathematics achievement of children who consistently exhibited persistent problems in understanding mathematics in elementary school and analyzed it in comparison to children who had stronger early math abilities. The results of the study revealed that 13% of the children with persistent problems are less likely to graduate from high school and 29% of them are less likely to attend college than those who had stronger early mathematics abilities. In other words, the initial differences in mathematics skills in early years may lead children to remain behind their more knowledgeable peers not only in primary grades but throughout their formal schooling (Geary, Hoard, & Hamson, 1999).

Studies also showed the predictive power of early math skills compared to other academic skills, such as reading. Lerkkanen, Rasku-Puttonen, Aunola and Nurmi (2005) investigated the relationship between mathematical performance and reading comprehension among 114 seven-year-old Finnish-speaking children during the first and second years of primary school. The results suggested that the level of mathematical knowledge children have before schooling is very important because these skills are predictive of their subsequent reading comprehension. In other words, early mathematics skills predict not only later achievement in mathematics but also later reading achievement. Similarly, Duncan and colleagues (2007) conducted a meta-analysis of 6 large-scale longitudinal data sets to examine the relationship between early learning and later school achievement. Of them, two were nationally representative of U.S. children, two were gathered from multi-site studies of U.S. children, and last two focused on children either from Great Britain or Canada. The researchers focused on the relationship between school-entry skills (i.e., reading achievement, math achievement, attention, internalizing behavior problems, social skills, and anti-social behavior) and later math and reading achievement while controlling for children’s preschool cognitive ability, behavior, and other important background characteristics such as, socioeconomic status, mother’s education, family structure and child health. Their meta-analysis revealed that only three of the six sets of school entry skills and behavior are predictive of school achievement: math, reading, and attention. Further, early math skills were consistently a stronger predictor of later achievement compared to reading and attention (Duncan, et. al., 2007). Consistent with the educational attainment analyses (Duncan & Magnuson, 2009), early math achievement was found as the most powerful predictor of later school achievement (Duncan, et. al., 2007).

Even though young children are natural mathematicians (NRC 2009) and capable of developing some complex mathematical ideas (e.g., addition) and strategies (e.g., sorting by multiple attributes to analyze data), it is also true that they do not become skilled in mathematics without adult guided rich and intentional interactions with those foundational math concepts. This month, we are going to focus on three of these foundational math concepts (e.g., number sense, sorting and geometry) and how you can provide your youngsters with rich and engaging math experiences that offer for opportunities and structures for the development of deeper math understandings.

Silly Putty Recipe Card

Using Children’s Literature to Reinforce Counting and Cardinality

posted by Dr. Jeanne White

Young children love it when an adult sits down and reads a book to them, carefully studying the illustrations before the adult can turn the page.  Why not seize these opportunities as a way to introduce or reinforce mathematical concepts?  There are four reasons why I like to use children’s literature as a mathematical resource:

  1. Literature can offer examples of real-life problem solving.

When I read The 3 Little Pigs to a child, we discuss how many pigs there are and how each one has a way to solve the problem of how to prevent the wolf from blowing down their house.  Even though pigs can’t really talk or build a house, the young child begins to understand the idea of a problem and solution as well as the lesson that sometimes we have to go back and try a new solution.

  1. Children can discuss and demonstrate how characters use math.

In the book, Pete the Cat and His Four Groovy Buttons (Litwin, 2012), Pete the Cat sings a song about his four buttons.  But as he loses one button at a time, he alters his song to include three buttons, then two, then one.  Young children can see, and hear, how Pete the Cat uses math in his everyday life by counting the remaining buttons each time he loses one.

  1. The text can provide common language and context for problem solving situations.

When I would read the book, The Doorbell Rang (Hutchins, 1986), to my primary students, we used little chocolate chip “cookies” cut out of brown tagboard and small paper plates to act out the story.  On the first page, Mom makes 12 cookies for Victoria and Sam to share.  This provides an opportunity for children to talk about how to distribute the 12 cookies on the two plates and then how to make sure the same number of cookies is on each plate.  Throughout the story, more children come to the house to share the 12 cookies, which are continuously distributed evenly among the growing number of children.

  1. Children can apply mathematical concepts with literature.

In the 12 years I have been teaching math methods for the pre-service teachers at Elmhurst College, the teacher candidates are always amazed at the number of children’s literature books used to teach mathematical concepts, from PreK through eighth grade.  I have also conducted workshops for preschool and kindergarten teachers, as well as for families of young children, who are also surprised at how easy it is to use literature to teach and reinforce mathematical concepts including counting, patterns, geometry and sorting.

Using Children’s Literature for Counting and Cardinality

red dots in a row

Young children learn to count to ten with meaning—they should not only be able to rote count from 1 to 10 but be able to count up to ten objects.  Adults and older children can become role models for counting and demonstrate the concept of cardinality, the awareness that the last number said is the total amount. When first introducing the concept of counting, use a book with the same objects on each page such as Ten Black Dots (Crews, 1986).  As you read each page and model how to count the number of black dots on each page, the child only has to pay attention to quantity since each object is the same size, color and shape.  Then practice counting up to ten circle counters so the child is only paying attention to the dots in groups of 5


Once children are able to count objects that are the same size, color and shape, read a book such as Math Fables (Tang, 2004), in which animals are shown in different configurations along with a rhyming fable, from one spider up to ten beavers.  This book can be read to children again when they are ready to break down numbers and group them into more manageable and familiar amounts.

colored frogs


An activity that can follow this book can be counting familiar objects such as toys or food.  Allow the child to touch each object while counting, whether counting illustrations in the book or toys on a table.  The child should also be able to repeat the total number of objects.  For example, “One, two, three.  Three frogs.”

yellow dots


Once children can count up to ten objects, they can begin to learn the complements of ten (one and nine, two and eight, etc.) with the book Ten Flashing Fireflies (Sturges, 1995).  In this book, two children are outside collecting fireflies in a jar.  First there is one firefly in their jar and nine fireflies in the night sky.  Then they catch another firefly and two can be seen in the jar and eight in the night sky.  After repeated readings of this book, try playing a game in which some “fireflies” are in the jar and some are in the sky.  Show the number of fireflies in the sky and have children figure out how many are in the jar.  They can have ten counters of their own to help them figure out the math problem.


I hope you have as much fun as I do, reading children’s literature and creating related math activities to introduce and reinforce these counting and cardinality concepts.

Thunder and Lightning

thunder and lightningThis one requires a really big storm one with thunder and lightning but it is a nice way for the children to think about “how far” the storm is.

If you are inside and the skies get very dark and the rain is coming down hard, bring the children to the window to look for lightning.  You can explain that lightning is a loud noise caused by electricity in the clouds.  Once the children see the lightning, have them count slowly until they hear the thunder.  It takes about 5 seconds for the sound of thunder to travel one mile so the higher the number the farther away the storm is.  Repeat this each time you see lightning. They can then figure out if the storm is coming closer or if it is moving away.

This is a great way for the children to think about storms, to work through their fears about the loud noises thunder makes, and to use their counting skills for something that connects to their lives.

The Abacus

An abacus is an ancient counting tool that has been used all over the world, for centuries, primarily in Asia.  The frame is traditionally made of wood with wire or small wooden rods running through it.  On each of the rods there are beads that move from one side to the other. Children being using an abacus by moving the beads from one side to the other and counting them as they go.

abacusThe abacus pictured above is designed for an early childhood environment.  This one has 10 rows of 10 beads.  An abacus designed for older children or adults provides “decks” or separate areas to represent place value.

Since children take in information through their sense of touch (as well as their other senses), the abacus makes good sense.  It reinforces one-to-one correspondence and number sense.

Did you know that blind children all around the world learn mathematics with an abacus?

A Great Estimation Activity

FullSizeRender-11I had the great good fortune to observe a very interesting Estimation Activity the other day at a local child care center.  Before the observation took place, my student and I discussed how estimation can be a pretty engaging activity for young children because it feels like a game – a guessing game. We talked about the counting skills of the children in her group and she felt very confident that they all had a pretty secure sense of number, at least up to 10, and were all able to count reliably.

We discussed ways of creating the jars so the children’s number and counting skills would be challenged appropriately; enough to be stimulating but not too much to be frustrating. My student decided to stick with small items that fit easily into empty baby food jars and chose items that at first glance, seemed easy enough to count.

The children came over to the table at their leisure during free choice and my student explained the game to them.  She defined estimation and explained what they should do.  Each child estimated how many of each item were in each jar.  They then wrote the numbers on small pieces of paper and stuck them to a graph next to their names and under the items.


What I found fascinating was how the careful choice of the items created a challenging math exercise for the children.  The pom pons were large and nearly filled the jar but because two of them were red they looked almost like one, making it hard to see where one began and the other ended.  Many children counted the five pom pons as four as they were “tricked” by the red ones.

The beads were straightforward; seven beads in seven colors, easily discernible and easy to count, as evidenced by the chart above.  The marbles were harder as they rolled around the jar a lot and it was hard for the children to know which marbles they had counted and which ones needed counting.  The really challenging jar was filled with Cheerios.  First, there were 8 Cheerios in the jar, which was the biggest number they had to count to.  Second, all of the Cheerios looked the same, so it was nearly impossible for the children to know if they had counted each one once, or if they had recounted some.

These small challenges are important to consider when setting up an activity.  For children with a secure sense of number and solid counting skills, the jars did not allow the children to point to each individual item or to line them up or to separate them for counting.  Many children still use these strategies to ensure they are counting correctly and following the counting rules.  One-to-one correspondence provides a framework for counting so that children know that each separate item has a one number attached to it, no more and no less.  One bead = one and the next bead = two, and so on.  The Order Irrelevant Rule  says that as long as each item is only counted once, it doesn’t matter what order the items are counted in.  This activity challenged both of these rules which is what made it really engaging and interesting both for the teacher, the children, and the observer (me)!

Counting the Pips Vs. Subitizing


While observing a kindergarten classroom the other day, I observed a child counting the pips on a die each time she rolled it.  The lesson asked that children roll one die, read the number of pips, and complete a word chart determined by the number.  Interestingly, this child could both read simple sight words and write them, but could not read the number 3 on a die.

This got me thinking about the activity itself.  It was clearly set up as a literacy lesson, asking that children practice their reading and writing, but the addition of the die asks that children also practice their subitizing skills.  Remember, subitizing is the ability to look at a group of objects, in this case the pips on the die, and know “how many” there are without counting them.  This ability to take a mental snapshot works much like reading sight words does but we work on sight word reading all of the time with children and focus on skills like subitizing much less often.  Even while observing this activity, the teacher worked diligently with the child on sounding out the words, reading them, and writing them down.  However, when the child tossed the die, she was left to figure out the number without the same sort of support.  Why is that?

If I had been supporting this activity, I would have focused on the die as much as on the words.  The math is as important as the reading and writing and deserves equal time.

There are several ways to support children as they begin to subitize.  One thing you can try is to have the child toss the die and then before she counts the pips, pick up the die and see if she can tell you “how many” were on it.  If she can’t, show her the side again and ask her “how many.”  Let her count and tell you how many there are.  The next time, try it again until she becomes more familiar with the patterns of the pips.  With ongoing support a child of 4 and 5 should easily be able to subitize up to 6.

Indoor Gross Motor Games That Support Early Math Competencies – Simon Says

As the weather turns colder and outdoor time becomes shorter, indoor gross motor time becomes a much more important part of the early childhood curriculum.  If you are lucky enough to have a large indoor space that allows for running and climbing, jumping and riding, then count yourself among the lucky. Children (all children, not just young children) need sufficient time and space to move their entire bodies.  Physical development, just like the other domains of development, is encouraged through positive opportunities for practice in supporting and engaging environments.

Over the next several Thursdays, I am going to write about an indoor gross motor activity that supports the physical development of the young child while providing additional opportunities to encourage mathematical thinking.

Simon Says

Simon Says is one of those games that grows as the child grows.  Even two-year olds can play a simplified version of Simon Says, one without consequences, right or wrong, or a set of rigid rules.  This game is exactly like Follow the Leader, but Simon gives verbal directions rather than modeling the action. The rules are simple.  At first, the teacher is Simon.  The teacher gives directions that begin with the words Simon Says and follows with what the children should do.  Once in a while, the teacher gives a direction without preceding it with Simon Says, and the players are supposed to ignore the directions because Simon didn’t say. When children are older, Simon Says becomes a game of elimination, but I wouldn’t play that way during the early years. (Sitting out because you failed at preschool is simply not an option for me.)

When playing with preschoolers, try to include numbers in your directions. “Simon says, jump 3 times.” or “Simon says, stand on one foot.” You can include notions of spatial awareness. “Simon says, walk to the edge of the rug.” or “Simon says, spin in a circle.” You can also include some simple counting. “Simon says, count to 5.” or “Simon says, count our friends.”  The whole game doesn’t have to include math, but some of it can.  It is a natural fit and works well.

Once the children in your group become familiar with the game, be sure to encourage the children to take turns being Simon. The opportunity to give the directions, rather than following them, is quite powerful, so let them try.


New Early Math Videos

If you are not connected to the Zero to Three organization, you really should be.  According to their website:

ZERO TO THREE is a national nonprofit organization that provides parents, professionals and policymakers the knowledge and the know-how to nurture early development.

Neuroscientists have documented that our earliest days, weeks and months of life are a period of unparalleled growth when trillions of brain cell connections are made. Research and clinical experience also demonstrate that health and development are directly influenced by the quality of care and experiences a child has with his parents and other adults.

That is why at ZERO TO THREE our mission is to ensure that all babies and toddlers have a strong start in life.

We know that as babies, the way we are held, talked to and cared for teaches us about who we are and how we are valued. This profoundly shapes who we will become.

Early experiences set a course for a lifelong process of discovery about ourselves and the world around us. Simply put, early experiences matter. We encourage you to learn more about very young children, early development and the work of ZERO TO THREE by exploring our site.

Recently, I received an announcement that they have unveiled three new videos from a series called “Let’s Talk About Math” that focus on math and children ages 0-3.  There will be six videos in all, but the first three that are available are: Shape Awareness, Spatial Awareness, and Counting.

If you work with infants and toddlers, you must see these.  They are very well-done, provide insight into early math experiences, and show real-life children in real-life scenarios that involve math.

Check them out here.