Let’s sort this out!

posted by Dr. Bilge Cerezci

sorting rocksAt all ages, children classify intuitively to make sense of their world that seems largely out of their control. By 2 weeks of age, infants distinguish between objects they suck and those they do not. By 2 years, toddlers form sets with objects that are similar. In preschool, children begin to sort objects according to a given attribute and form categories. Many parents have likely walked into a room to see their four-year old putting their blocks or other toys in piles based on color or type. So why sorting is important you may ask. By sorting the objects around them, children start using their analytical thinking skills that is the lifeblood of mathematics. Studies have even been shown that by comparing objects to one another and understanding the relationship between set of objects, children engage in transitive thinking: A blue block is bigger than a red block and smaller than a yellow block. So, blue blocks need to go into a medium-sized block pile. Practicing sorting skills also provide children with models for organizing things in the real world, such as putting toys into the right toys boxes or putting the socks in a sock drawer and underwear in the underwear drawer.

Sorting Ideas

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 sorting skills. Here are some of the sorting ideas you can implement in our home:

* Collect real-life objects such as rocks, marker caps, marbles, and buttons. Ask your children to guess which objects will together and which items will not. Ask the children to sort them according to different attributes such as; color, texture, type and etc.

* When it’s clean up time, ask your child to sort toys by attributes. For example, ask your child “Can you pick up all the toys that are the same color as this?”

* Encourage your children to name groups of things or activities. For example, at the dinner table, talk about attributes. You might say “2 people at this table wear glasses, 4 don’t.” or “3 have curly hair, 3 have straight.”

While you are doing these activities, use words such as “same,” “different,” “math,” “group,” “collection” and “set” as they apply and encourage your child them to use when they are describing their groups and comparing the groups they have created to one another. You may also ask your children questions such as, “Can you figure out what goes together?” “Can you sort these a different way?” “Why do these go together?” “Why do these not go together?” These kinds of open-ended questions will allow you to better understand your child thinking and push your child to be more precise in explaining their mathematical thinking processes.

Different children, different decisions

Children at different development stages are equipped with different mathematical abilities. A younger child will likely require less categories (sorting by two attributes) while an older child often can handle three, four or more. What you use for sorting also depends upon the age and ability of the child, as well as their interests. Some materials may be more challenging to sort for younger children (e.g., visually ambiguous materials) while others too simple and even boring for an older child (e.g., colored unifix cubes). Using real-life objects and situations to provide sorting experiences is always beneficial for all-around learning for all age groups. The bottom line is to know your child’s abilities, interests and to meet them where they are at, so you can just give them the right amount of challenge without underwhelming or overwhelming them.

Focus on Attributes!

posted by Dr. Bilge Cerezci

As she sits on the floor, a three-year old starts stacking blocks with various shapes and sizes. After some experimentation, she realizes that it is hard to build a tower if a block lays on its curvy side.


What does this 3-year-old discover about shapes?

From an early age, young children notice different shapes have different characteristics, even if they don’t know their names yet. They realize that some shapes have points while others have none. They also discover some shapes have flat sides while others don’t. Traditionally, we teach children the names of basic two-dimensional shapes: circle, square, triangle and rectangle and assume that being able to name these shapes indicates a higher level of geometrical understanding. Unfortunately, this can be any further from the truth. In reality, young children need your help to focus on attributes of shapes rather than overall appearance. For example, as you build a block tower together, encourage your child to pay attention to defining attributes of the each shape you are using. You might say, “I see you are stacking up the blocks that have flat sides. Look, all of its sides are flat. How is this one (i.e., cube) different that this one (i.e, half circle block)?” As you continue with the activity, encourage your child to use her fingers to trace and feel the shape. Give them a plenty of time to feel the shapes, count the sides and even ask them to find an item in your home to that resembles that shape.

As children manipulate various three-dimensional shapes, they will eventually build deeper understanding geometrical shapes such as flat faces of solid (three-dimensional) shapes are two-dimensional shapes.

There are many ways to encourage and help your child to learn about shapes. Here are some of the games you might play with your children at home:

* Drawing shapes in sand or foamshapes in shaving cream

* Walking around shapes drawn or taped on ground

shapes on the floor

* Making shapes with bodiesbodies making shapes

Shapes are all around us and it is easy to play games like these at home, outside and elsewhere. Most importantly, make sure to have fun while doing it.

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 Classification

posted by Dr. Jeanne White

When older children and adults perform chores such as doing laundry or putting away dishes, they may not even realize they are making decisions about how to classify objects into categories—washing the white clothes together and putting the plates together on a shelf.  Young children can begin to see how objects can be sorted into categories with the help of several children’s literature books.

Young children naturally sort objects by color.  Have a variety of books available that introduce colors and show objects of a particular color so children can make associations with the object and the color.  A fun book that can be used to explore colors is The Color Box (Dodds, 1992), in which a monkey named Alexander crawls into a box to discover a world where everything is black, then another where everything is white, then yellow, and so on until he ends up back in his world where there are objects of every different color.  Then provide groups of objects that can only be sorted by color such as Unifix cubes or square tiles.  Once children have sorted these objects by color, they can count the number of cubes or tiles in each group and compare them—which group has the most?  Are there more red cubes or more orange cubes? Then make several types of objects available for children to sort by color such as beads, tiles, cubes and blocks.unifix cubes scatteredunifix cubes and other shapes

After children have had several experiences sorting objects by color, they can begin to explore other attributes.  In the book, The Button Box (Reid, 1990), a boy visits his grandma and finds a box with hundreds of buttons inside.  He sorts them into categories such as buttons covered with cloth, sparkly buttons, metal buttons and buttons from uniforms.  He also lines them up based on size and color.  At the end of the story, the boy and his grandma close their eyes and each choose a button from the box.  Then they look at them and talk about all of the ways the two buttons are alike and the ways they are different.  Follow up the story with a game similar to the one in the story, by comparing two buttons or other toys that have similar attributes such as two dolls or two cars.

Another book that can be used for classification of objects is How Many Snails? A Counting Book (Giganti, 1988).  On each page there are illustrations of one type of object but they vary by size, color or design.  On the first page there are eight clouds for children to count.  Then there are more questions to answer, “How many clouds were big and fluffy?  How many clouds were big and fluffy and gray?”  Children can use their toys to count, sort and answer questions such as, “How many frogs?  How many frogs are yellow?  How many frogs are yellow with a green stripe?”  Incorporate science by introducing various types of insects and asking children to tell you how they can sort them—by putting all of the insects that are the same color together or by putting the insects with wings in a group. four frogs

When children sort and classify objects into groups, they are building a foundation for graphing and data collection.  grasshoppersThey can create unique ways of sorting objects into groups and once they know how to put objects into categories, they can help with all of those chores!

What is Math?

posted by Lisa Ginet

When you hear or see the word “math,” what do you think of? Your high school algebra class? Balancing your checkbook? A geeky engineer with pocket protectors? When you add “early childhood” to “math,” what do you think of then? A little one learning to say, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10”? A bright poster with a circle, triangle and rectangle neatly labeled? All of these are common ideas about what math is and how math starts, but none of them are what I mean when I say “foundational math.” Before I tell you what I do mean, I want you to try something.

Look at this image:
shapes-pictureConsider this question:

Which of the figures are the same?

Often when I ask this, a person says, “They are all different from each other.” Another says, “They are all the same; they are all shapes.” Both of these answers make sense, but I often ask people to keep looking to see if anyone can come up with another answer. Usually, people then generate these six answers:

  • top two shapes are both orange
  • bottom two shapes are both green
  • left two shapes are both striped
  • right two shapes are both solid
  • top left and bottom right are both circles
  • top right and bottom left are both triangles

In fact, although none of the two shapes are identical to each other, any two of them are “the same” in some way. Figuring this out involves logical thinking about the attributes of the shapes.

This shape activity demonstrates one definition of mathematics – a logical way of thinking that allows for increasing precision. We can use math to make sense of the world. We can use math to solve problems. To use math in these ways, though, we cannot just memorize facts. We must build our own understanding, so that we can think flexibly in different situations. Without a strong foundation, a tall building would not stand for long. Likewise, without a strong foundation in mathematical concepts, children can struggle to understand the more complex mathematical thinking they need later in life.

At the Early Math Collaborative, we have developed a set of 26 “Big Ideas” – key mathematical concepts that lay the foundation for life-long mathematical learning and thinking. While these concepts can be explored at any early age, they are powerful enough that children can and should engage with them for years to come. As you engaged in the shape activity earlier, you were using two of the Big Ideas:

  • Attributes can be used to sort collections into sets.
  • The same collection can be sorted in different ways.

Most likely, you were not thinking about these ideas consciously; rather, you were looking at the shapes and thinking about them. You were using math to make sense of the puzzle I posed and to come up with a solution. This type of math may not match your prior notion of math as quickly-recalled facts and properly executed procedures. You may need to set aside some of those notions in order to develop a deep understanding of foundational math that will help you have fun doing math with children.


Tools that Support Mathematical Understandings in a Preschool Room

Over the years, I have blogged about 100+ kinds of manipulatives, from the homemade variety to the expensive stuff, from the kinds designed for toddlers to tools for early school-aged children.  I really like interesting, diverse, multidimensional tools that can be used in a variety of ways and for many developmental levels.

Today, let’s take a look at how tools are used to support mathematical understandings and how tools can be misused so they are not really effective in terms of meeting early math outcomes.   IMG_0103 FullSizeRender-10

Here you can see a few sorting trays designed so that children can create, extend, and copy patterns (green tray), sort and organize items by attribute (yellow tray), and graph (blue tray).  IMG_0105

This red one is divided into quadrants so that more or larger items can be grouped or sorted in other ways (I.e., animals that fly, animals that swim, animals that walk on 4 legs, animals that walk on 2 legs).

These all come in a set and are beautifully made, large enough that children can use smallish items with them but not so smallish that they present choking hazards.

So, if you had these in your room, how would you use them?

The day I saw these in a Head Start classroom, they were sitting out on a table with several bins of small manipulatives nearby.  A few children were playing with them and from my observations, it was clear that they did not know or understand the purpose of the trays.  They were using each tray much the same way they might use a plastic plate, as a receptacle for putting the toys, storing the toys, and moving the toys around.  The teacher never came over and neither modeled how to use the trays nor explained how to use them.

Here you can see how one child was putting dinosaurs into the spaces of the tray in a disorganized fashion.  At no time did a teacher support his play by suggesting alternatives, or sitting next to him and encouraging more purposeful use of the materials.IMG_0104

It reminds us that our role as “set designer” and “provocateur”  demands that we do not put materials out in a haphazard fashion but systematically think about our curricula, materials and environment in thoughtful ways.  We  must observe the children to know where they are developmentally and what they need from us so we can support their growth.

These tools are not meant to be put out on a table at the same time without direction.  They are meant to be used to support developing mathematical competencies.  That is NOT to say that young children should not use them to play.  Of course they should.  In fact, before I used them for their designed purpose, I would allow  the children to explore the trays, one at a time, over a period of time, to see how they explore them organically, without direction.  Eventually, I would take one over to group time and show the children how they can make patterns in the green tray or sort items in the yellow tray.  I would show examples and sit at the math table and work on patterns with the children, creating them and asking the children to extend them.  I might ask children to create their own patterns and see if I can extend them. I would set out very specific manipulative that have 2 or 4 attributes so they can be easily distributed in the sorting trays.

I would set the scene and facilitate the play.  I would use math language throughout these interactions and make it fun.  I would show excitement about the challenges these trays present and pose thoughtful and stimulating questions that the children could answer.  I would encourage the children to “try” and get excited at their efforts.

In short, I would do my job by being the best, most thoughtful teacher I could be.

Think before you set up your materials and arrange your space.  How are you going to be the best teacher you can be?

Mittens, Hats, and Scarves – Oh My!

The multicolored layers of puffy winter wear that are required to keep a small child warm during February in Illinois provide endless opportunities for exhausting trips outdoors.  It often takes an equal amount of time to get everyone dressed as  it does to actually be outside.  Snow pants, zippers, scarves, gloves, hats, and coats require dexterity and patience to get assembled and on, two qualities that are only beginning to emerge in the preschool child.

That said, all of these fun winter clothes are a great resource for sorting and graphing as a large group.

OLYMPUS DIGITAL CAMERAThis picture shows a large graphing floor mat that is so perfect for preschool children.  They can put their own hats or mittens in the squares (rather than having a teacher do so on a piece of tag board).  This simple physical involvement will make the activity so much more interesting for the children.  Once they have put their items in the separate squares, they are easy to count and provide appropriate visual cues so the children can “see” which has “more” and which has “less”.

I would start with Mittens vs. Gloves and see where it goes!

People to People

I have no idea if there is a real game called People to People or if I learned of it at one of my previous positions, but this is a great one that you should know too.

People to People is a gross motor game that is fun and engaging and children love it.  It is noncompetitive and is played for as long a time or as short a time as interest dictates.  It requires no materials, just a leader and (ideally) a large space.

Gather the children together and explain the rules.  The rules are simple:

1.  There is a leader.

2.  Everyone else is a player or an observer (children can choose to opt out and watch).

3.  Players have to find partners while they play.

4.  Players should find a new partner every time the leader calls out, “People to people.” Players should never pick the same partner again.

The leader calls out two body parts and the children find a partner and connect the body parts.  For example, the leader says, “Elbow to elbow,” and the children find a partner and put their elbows together.  The leader then calls out another set of body parts, i.e., head to head, or foot to foot.  This continues until the leader calls out, “People to people,” and the children have to find a new partner and it begins again.  With very young children keeping it simple is best.  Choose body parts that they know and are easy for them to identify on themselves and on others.  As the children get older, you can mix it up and get tricky, with body parts like, hips, calves, shins, and pinkies.  You can also make it even trickier by having the children connect two parts that are not the same, i.e., finger to foot, or head to belly.  The children not only have to identify each body part but they also have to negotiate who is the head and who is the belly.  This requires a lot of social navigation.

People to People supports matching skills and asks that the children sort and resort themselves out while finding partners.  They have to find new partners and use the concept of “elimination” so they don’t choose the same partner more than once.  Not only are they practicing following directions, this game focuses on prosocial behaviors that encourage turn-taking, compromise, and social generosity.

Give it a try and let us know how it goes.

Data Analysis and the Young Child – Week 2

Last week we took a first look at how teachers of young children approach data analysis.  I thought that we could look at the way the NCTM lays out this standard.
According to the NCTM website:

Programs should enable all children to:

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them

This can be demonstrated by the children in the following ways:

  • pose questions and gather data about themselves and their surroundings;
  • sort and classify objects according to their attributes and organize data about the objects;
  • represent data using concrete objects, pictures, and graphs.

And…programs should enable all children to:

Select and use appropriate statistical methods to analyze data…

This can be demonstrated by the children in the following ways:

  • describe parts of the data and the set of data as a whole to determine what the data show.

Sorting and classifying is something we do all the time with children.  We can have children sort nearly anything that we have many of.  It is easier for children to sort by one attribute at a time (for children under 4, only sort by one attribute at a time).  When putting the cutlery out for lunch, put it out in a pile, all mixed up with spoons, forks, and knives.  Have the children sort them before setting the table. There is a great lesson about this very idea in the Math at Home site.  Click here to see it.

There are a couple other lessons available in the Math at Home site about sorting and classifying.  Click here or here to check them out.

If you try any of them out, let us know how they go.

Sorting Books

Both of my boys have left for college and I am in a state of shock.  Looking for ways to fill my heretofore parenting time, I decided to clean and tidy my house.  I bought the book the life-changing magic of tidying up, by Marie Kondo as I thought this would be the motivation needed to get going on this monstrous project.

It got me thinking about this time of year; going back to school, the beginning of fall, change in the air, kids growing up, and opportunities to do things a new way. Now is the time to make small changes in your classroom that can impact the children in meaningful ways.

My very good friend Alison has an artist’s eye and a great sense of style.  When I last visited her out in California, I fell in love with the way she organized her children’s books.  Rather than sorting them by author or title as most of us usually do, she organizes them by color.



You can see how the books create this rainbow effect on the shelves but more than that, this visual style of sorting is a wonderful way for young children – pre readers- to participate in the upkeep of the classroom library.

Most children are highly visual and remember books by their covers, colors, and pictures.  Most know their colors before they read, and are able to choose favorite books by their look rather than by title or author.  Imagine putting away books at clean up time, and rather than having the books stuffed haphazardly on the shelves, children are able to sort them on their own into their color family.  You can see that it is not an exact science.  The oranges flow into the reds and yellows, but it is not perfect and does not need to be.  In addition, this is a sorting system that supports young children’s early math development by asking them to consider where the books go based on a visual attribute and then sorting them just so.  I love this idea.