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

Cooking with Preschoolers

posted by Alison Balis Hirsch

Cooking with kids offers a wonderful array of learning opportunities for young children. It provides practice in language arts (vocabulary and “reading” a recipe), science (chemistry and exploring the senses), and developing social skills (cooperation and turn-taking). The kitchen also provides a range of math practice such as counting, measuring, and understanding order.

IMG_5879In my son’s pre-k class, the teachers and children cooked together almost weekly; the recipes coincided with their Letter Of The Week. So for B week they made banana bread and for O week they made omelets. The recipes were simple enough for the teachers and children (ages 4 to 5-years-old) to manage, each having a minimum number of ingredients. My son LOVED the rice pudding so much that I asked his teacher for the recipe. When I saw how simple it was, I suggested we collect ALL of her recipes and create a cookbook to share with other parents, whom I imagined were equally excited to cook with their kids at home. After all, these were recipes already vetted by our experienced and talented teacher.


As a teacher in the classroom, we sought out parent/caregiver volunteers to assist with cooking projects. Having the child’s special grown-up allowed those participants to engage in the school life of their child, providing them with an opportunity to better know the other children and teachers and also observe their own child in the context of the classroom. It also allowed us to maintain good adult/child ratios while working with small groups of children (typically 4 – 5), in the kitchen. The children who cooked or prepared snack (sometimes it was simply designing bagel faces with cut fruit, vegetables and sprouts) usually delighted in their food and seemed proud to share their creations with their classmates. The learning continued through the service portion of the meal, since the child chefs, with the help of the grown-ups, needed to figure out how to divide what they made into equal portions for their friends. Something like bagel faces required counting and one-to-one correspondence; baking a quiche required cutting it into enough equal-sized pieces to serve everyone.

For recipes that were made frequently, we made recipe booklets that were much more readable for children. Play dough was something we made with children on a weekly basis and for that we created cards, bound by binder rings that had visual instructions and described quantities with pictures.january photos 070

Cooking presents children with plenty of opportunities to learn and is also a great way to teach principles of good nutrition and encourage an adventurous palate: in my experience kids are much more likely to try foods they’ve grown or prepared themselves.


Classroom Jobs: The Snack Helper

posted by Alison Hirsch Balis

There are many reasons to provide “jobs” for children in the preschool setting. A job shows the importance of the child’s contribution to the group and his/her affect on the social fabric that is the class community; it provides practice in children’s developing social skills, such as speaking in front of the group to give a weather report or choosing a song to sing together with the whole class; it allows each child to have a turn in a “leadership” role by being in charge of a task; and then a particular job can present a pre-academic lesson in manner which is meaningful to the children in the group.

Setting the table for snack is one example of using math in a relevant way while practicing one-to-one correspondence, a foundational math skill needed in counting. In his/her role, the Snack Helper assists the teacher to prepare the tables for group snack. This responsibility ensures that each table has the appropriate number of snacks and also that each place where a child will sit has a snack… because it doesn’t really help to have all 18 snacks placed on one table when the 18 kids sit at 4 separate tables.

My school was a 5-day program and our class had the same 18 children each day, minus any absences. Because we occasionally had fewer than 18 children, and also because there may not have been a child to match every chair at each table (we had two tables that sat 6 kids and two tables that sat 4 kids, for a total of 20 seats), we designed four individual number cards showing the numeral, the number word and the number quantity (for practice subitizing) of either 4, 5 or 6 (four, five or six) – the possible number of kids at a table depending on attendance that day. Just as a reminder, the definition of subitize is to perceive the number of (a group of items) at a glance and without counting. It’s more or less the math equivalent of sight words.


The teachers placed the laminated number cards on each of the four tables, indicating to the Snack Helpers (there were two) how many snacks to set out. The Snack Helpers needed to “read” the cards and set the corresponding number of snack bowls and drinking cups for their classmates. It was not at all uncommon to see the two children working together to correctly set out snack for their friends. So in addition, we were supporting and encouraging collaboration and vocabulary while providing a great benefit to the teachers – help setting the table!

As an aside, we began the school year pouring milk or water FOR the kids, but after a couple of months, we provided small pitchers so that once seated, the children could practice this skill on their own. A pitcher like this one (almost 17 ounces /.5 liter) works really well for children since the lid helps minimize over-pouring.

water jug

Baby Brains and Math- Chanting and Singing

Did you know that babies respond more to the “rhythm” of speech than the words themselves?  It is a natural impulse to speak to babies with a higher-pitched-than-normal voice, a sing-song lilting quality to the words, and a repetition that is particular to these interactions.  Throughout my career, I have had students and new parents ask me about speaking this way.  There are many people who don’t like it and feel like there is something wrong with it.  I have heard that this type of speech is discouraged by their families and even ridiculed, calling it “baby talk.”

It isn’t baby talk.  Baby talk is when you speak like a baby to other grown-ups, or older children.  The above-described speech was traditionally called “motherese” but is now called “infant-directed speech” or “parentese.”  Babies’ brains respond positively to this type of speech and their whole bodies respond to chanting and rhythm.  If you put an infant on your lap and bounce her to the rhythm of music or a chanted tune, the infant will nod her head, bounce her body up and down, and kick her legs and arms to the beat.

We can use this as a means of supporting mathematical concepts with infants.  Through repetitive chanting of songs, infants will begin to internalize number words and concepts.  If you sing “The Wheels on the Bus” with an infant in your lap, you can encourage the notion of “round and round” (spatial thinking) and “up and down” (more spatial thinking).  If you chant, “This Little Piggy” touching each of baby’s toes one at a time, singing,  “this little piggy went to market…” you will encourage “one-to-one correspondence.”

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)!

Observing One-to-One Correspondence

Last week I was visiting a student at her center so I could observe an activity that she planned.  While observing my student at one table with several children, a 3 or 4-year-old girl was very busy at the table behind me, organizing and sorting, and lining up all sorts of small math counters.  It was great to watch my student, but I was equally mesmerized by this other child.

1 to 1 1


She had gathered up all of the math counters and put them out on her table.  She then began lining them up.  She began with 6 trucks and then lined up 6 bunny rabbits next to them.  She told me that each bunny needed their own truck.

If you look carefully at the photo above, you can see that there are mor bunnies than trucks.  Why, you might ask.  Well, the bunnies take up less space than the trucks, so even though there are more, if counted, the space they occupy is the same as that of the truck’s.

Her intent was to have each bunny with their own truck.  In her mind, she accomplished this. Next, she added a car for each bunny.  The cars and the bunnies are almost exactly the same size, so her ability to arrange them in a one-to-one pattern was more successful, because she didn’t have to consider the size differences.

1 to 1 2

Eventually, her friend came over and had many ideas about the activity.  She suggested they add more figures and was pretty geared up about having dinosaurs join the party. The first girl was determined to make more lines of figures and wasn’t quite ready for the dinosaurs.  So after the cars, she lined up the boats. Here, you can see that the boats are a little bit wider, so she only needs 7 boats to match the space requirements of 8 cars. Next came the train engines1 to 1 3 and finally the ducks.  She ran out of room on the table, and simply walked away.  She was done when the space was done.  Fascinating.

Are You Odd or Even?

Last Friday I had the pleasure of visiting my friend B’s kindergarten class in San Francisco, California.  I met her teachers and her friends.  I saw the class toys and books.  B introduced me to the class frogs, birds, and fish.  I watched her dance with her friends and then get ready to go on a field trip to the pumpkin patch.  It was so lovely and reminded me of how much I loved being a kindergarten teacher.  The room was alive with activity and the children were electric with excitement about the day’s activities.

Outside the room, B showed me this poster.

photo (29)

B explained how they learned “odd” and “even”.  The teacher asked them to count using their fingers.  For “1” you put up your pointer finger on one hand, and for “2” you put up your pointer finger from the other hand. If both fingers are up, then they are partners and you put them together.  Every number that has a partner (2, 4, 6) is “even” and every number that is alone is odd (1, 3, 5). You continue that way, middle fingers, ring fingers, and pinkies.  In the picture below, the number 5 is represented by the ring finger that does not have a partner.  Therefore, it is odd.

photo (28)I know that these children are a bit older than many of yours and most of them already have a strong number sense, but I still like the idea of this even for younger children. First, I appreciate that the age board is a departure from the typical Birthday Boards that I usually see and second, the counting system is interesting as it reinforces one-to-one correspondence, is easy to do and can be expanded as children get older.

B explained to me that she was “odd” because she was 5.  I asked her what would happen next week when she celebrates her 6th birthday.  She looked at me like I was a simpleton and said, “I will be six and move to the other side of the board.”



Graphing – Caps For Sale


I had an opportunity to observe a wonderful teacher use the children’s classic Caps For Sale as the foundation for a graphing activity.

She first read the story with the large group.  The children knew the book well and read along with her.  They acted out the monkey parts and tried on all sorts of different caps.  They had a blast.

She then told them that during free choice they could come over to the table and vote for their favorite colored cap. She created this wonderful board so they could vote.

graph caps for sale

Once the votes started rolling in, some of the children stayed at the table so they could watch the results.  There were a couple of children who wanted “blue” to get the most votes, and a couple of children who wanted “red” to win. This became very exciting as the votes for blue and red were neck and neck for a time.  The children had loads of opportunities to talk about which color had more and which had less.  The graphing exercise itself became a vehicle for a lot of conversation about more and less, favorites, counting, counting on, and one-to-one correspondence.

Later, once all of the votes were in, the teacher brought the graph to the rug so the group could revisit their data.  You can see how this all played out in the video below.


After watching the video, do you have any suggestions for improving the activity?  Tell us what you think.