# 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 foam

* Walking around shapes drawn or taped on ground

* Making shapes with bodies

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.

# Using Children’s Literature to Reinforce Geometry

posted by Dr. Jeanne White

As young children are formally introduced to the names of shapes, they begin to notice these shapes in their surroundings.  They see their plate as a circle and their napkin as a square when they eat dinner.  They look at the windows and doors in a room and recognize them as rectangles.  Tana Hoban’s book Shapes, Shapes, Shapes (1986) uses photographs of familiar objects such as pots and pans, and scenes such as construction sites, to present various shapes. Children will find more shapes on each page as they look at the photos again and again, and as they learn to name more shapes such as trapezoids and ovals.

An activity that can follow the introduction of this book can be allowing children along with family members to take photos of shapes in their home, their neighborhood or school.  They can display and compare the photos and name the shapes in each other’s photo.

In addition to two-dimensional, flat shapes, young children should be introduced to three-dimensional, fat shapes.  Reading the book, Changes, Changes (Hutchins, 1987), can open a child’s mind to the endless possibilities of how to arrange 3D blocks to build structures.  In this wordless picture book, a wooden couple builds a house but it catches on fire, so they must build a fire engine, then a boat to deal with all of the water, and so on.  Encourage children to find 3D objects in their environment such as food containers that represent cubes, cylinders, and rectangular prisms.  They can build their own structure with these containers and name them as they build.

Once children are familiar with the names of shapes, they can expand their vocabulary to include attributes of shapes.  The book, If You Were a Triangle (Aboff, 2010), includes illustrations of triangles that are slices of watermelon, Yield signs, faces of pyramids, designs on wallpaper, and more.  The text repeats the phrase, “If you were a triangle…” and lists attributes such as “three sides,” or “three corners” and introduces the terms polygon and angle.  At the end of the book, specific triangles are shown—equilateral, right and isosceles—along with examples of these triangles put together to form a new, composite, shape such as a rectangle or rhombus.  Children can look for triangles in their environment as well as practice putting the triangle Pattern Blocks together to form new shapes.

Another concept children learn in early geometry is relative position.  Young children are gradually exposed to words used to describe the position of an object or person relative to other objects or people such as above, below, beside, in front of, behind, and next to.  Young children are also starting to distinguish between their right and left and are learning to move, and count, forward and backward.  A book that is fun for children to use to learn these concepts is Bug Dance (Murphy, 2002).  The bugs in this book go to school together and in gym class they learn a dance that teaches them to take steps to the right and to the left, then hop forward and backward.  Young children can perform the dance as the book is being read over and over.

After children have practiced their dance moves they can practice the terms in the book, as well as other position words, to describe the position of Pattern Blocks.  For example, children might say: the square is below the hexagon; the triangle is on the right of the square; the trapezoid is on the left of the square; the triangle is next to the square.

There are many children’s books that are written to introduce shapes, however many use the word “diamond” instead of rhombus.  I try to avoid these books or let children know a diamond shape is called a rhombus when we are learning math.

# 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:
Consider 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.

# Geometry

Geometry is so much more than learning the names of shapes for young children.  When we think of Geometry, we might harken back to that high school class where we had to memorize loads of formulas to determine circumference, area, diameters, and volume. This is NOT what we do with young children.

For young children, geometry is really housed in a larger concept that we call “spatial thinking.”  This includes mathematical skills such as categorizing shapes and objects, measurement, perspective, mental transformation of shapes (being able to turn a shape upside down), scaling, proportion,and location. This list is in no way complete, as there are many more ways that spatial thinking can be taught and learned in the early years.

An examination of the physical environment is one sure-fire way to get kids talking about geometry.  Using examples from the area around you and them, try to look for shapes, edges, lengths, and areas.

Copying shapes using manipulatives such as tangrams is another way to explore geometry.  We are going to look broadly at tangrams another day, but for now, take a look at these two sets of tangrams and consider how children can explore them.

# Symmetry and Snowflakes

Do you remember the day you were told that each and every snowflake in the entire world is unique and that no two snowflakes are alike?  The idea of infinite possibilities still rattles my brain.  How can each of the billions and billions of snowflakes be unique?

Spatial awareness or concepts about space and shape, are pretty interesting to young children.  Snowflakes are one way to explore shape in an engaging and meaningful way, especially if you live in a part of the world that is filled with the cold, white stuff a good part of the year.  Also, as children begin working on their cutting-with-scissors skills, creating snowflakes is great way to practice.  Some kinds of paper are harder to cut than others but the easiest paper to cut is also the most likely to tear.  I like simple copy paper for snowflakes as it is sturdy enough to withstand some three-year old torture, but light enough to cut easily with children’s scissors.

Begin by folding the paper in half and then in half again.  The difficulty in cutting increases with the number of folds, so fold the paper to meet each child’s individual developmental needs.  One fold reveals the least interesting patterns and more folds reveals more complicated designs.  Ask the children about the shapes they have created.  Show them that they can fold the paper back up and continue cutting, if they so choose.  The snowflakes will just become more interesting.

If you want them to look more like snowflakes, and you think the children may enjoy it, you can begin with circular paper.  Again, fold at will, but the thicker the folded paper is, the more difficult it is to cut.

If you are working with older children, snowflakes are also a great place to discuss symmetry.  They provide concrete examples of “mirror images” that are easily (maybe not easily) seen. Notice how unique and distinct each of the children’s snowflakes are. No two are alike and that is what makes them special.

# Tessellations

At the Museum of Math there is an exhibit about tessellations.  Do you know what a tessellation is?

A tessellation is created when a plane is covered by repeating shapes that leave no gaps or overlays. Tile floors are tessellations as are honeycombs.  You can tessellate with very simple shapes (rectangles and triangles) as well as extremely complicated ones.  The artist M. C. Escher is known for using extremely complex shapes to create tessellations that are also optical illusions. What do you see in the pictures below.  Do you see white birds and white horses?  What happens when you look for black birds and black horses?  Do you see those?

Notice how the shapes cover the entire surface of the plane but do not leave any spaces.  At the Museum of Math, there are buckets of shapes that fit together to create tessellations.  Tessellations would make really interesting puzzles for children who enjoy complicated spatial activities.  Do you have any examples of tessellations in your classroom?

# Shape Scavenger Hunt

Turning an ordinary activity like “Find a Shape” into a game like “Shape Scavenger Hunt” is pretty easy to do, and much more fun for children to engage in.  Frequently, while visiting centers, I hear teachers ask the children, “Who can find something in our room that is a circle?” or “Who can find something red?”  The children look around, everyone raises their hands, one child gets called on, and then he tries to find something that fits the criteria.  Usually, everyone raises their hands because they want a turn, more than they know the answer.  This is the usual course of events.

It is far more interesting to create a very simple chart for each child, clip the charts to clipboards (I LOVE clipboards, so each child can work on his/her own), tie a good pen or felt tip marker to the clipboard and then have the children look for items from their charts.  It could look like this:

Each child could take his/her clipboard and find something in the room that is either this shape, or with this shape on it.  They could draw what they find on their charts and then tell the group what they found. If adding color confuses them, then be sure to leave it out.  If you want to add the dimension of color so that each item has 2 attributes; i.e.; red squares, yellow circles, etc. then be sure to explain that as well.

There are very clear messages on this chart.  The first message says, “Find 2 rectangles.”  The second says, ” Find 3 circles,” and the thirds says, “Find 1 triangle.”  Before you have the children set off on their scavenger hunts, be sure that they can “read” their charts.  Explain that there is a space for each item they find and they can draw their items in the chart when they find them.

Use these to determine how the children in your group are identifying their shapes, recognizing numerals, and following directions.  Have fun with scavenger hunts.  The more the children get used to this format, the more they will enjoy it.

# Early Drawings

Yesterday, while walking my pugs I saw these chalk drawings.  I love how you can see what the adult has drawn and what the child has drawn.  The bottom picture is a “mandala” – the first symbolic representation children draw.  They are often circular with lines intersecting the edges, looking like rudimentary suns.  Sometimes, children’s mandalas are triangular, sometimes crosses.   For children, these are the elements that provide practice for drawing more sophisticated representations later. They are also the foundation for creating numbers and letters.  If you ask children about their drawings they will often report that they represents “people”-circles representing heads or torsos, and lines representing legs and arms.  As these become more sophisticated, children may add more circles and lines, to represent the mouth, they eyes, the nose and the ears.  I love these early drawings and the early ability to create and identify shapes.

# The Common Core – Geometry Pt. II

## Analyze, compare, create, and compose shapes.

• CCSS.Math.Content.K.G.B.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).
• CCSS.Math.Content.K.G.B.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
• CCSS.Math.Content.K.G.B.6 Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”

_____________________________________________________________________

This half of the Geometry Standard is quite complex and subsumes many aspects of the previous standards.  Take for instance the phrase, “number of sides and vertices/corners”.  This requires that the child can count the sides or vertices, using one-to-one correspondence, understands the attributes and the vocabulary of “sides” and “vertices”, and then is able to compare all of those aspects of a shape to another.

Of course, if we are talking about triangles or squares, this isn’t very complex.  But when we are talking about solid shapes (3 dimensional) and then moving them in space to present different orientations, the ability to meet this standard is much more difficult.

Breaking this standard down into smaller parts will make the most sense for teaching.  First, children need to have exposure to 2 and 3 dimensional shapes and solids.  Next they need repeated opportunities to use the associated vocabulary to describe their attributes.  Then they need to see several examples of shapes and solids using manipulatives and real-world objects.

Another way to introduce these concepts is by using the second substandard above to support the first substandard above.   When children are afforded opportunities to build and create shapes and solids in many sizes, using a variety of materials, they will experience them on a sensory level as well.

The third substandard above may come easily to some children and may be much more difficult for others.  Children who are naturally drawn to puzzles and tangrams and who can easily manipulate shapes so that an “upside down triangle” is still a triangle will probably be able to put 2 triangles together to create a square.  Other children’s spatial skills may not be as developed, so working with these manipulatives will be important, but may be frustrating. Take a look at this post about Tangrams to see how this type of manipulative can provide a foundation for shape building.

# Shape Books- See the Difference

Here are two examples of bookmaking – one that I consider to be developmentally appropriate, interesting, engaging, and best practice and another that I consider to be boring, and – for lack of a better term – lazy.

I frequently hear early childhood teachers introducing shapes as a curricular concept to children.  This is good – understanding that shapes have attributes and names is important. But shape goes far beyond circles. squares, rectangles and triangles.  This example is the kind of thing I often see in the field.

These are the first several pages of a mini book that was available for children to work on during free play.  You might argue that it is OK.  It is a book about “Shapes” and children can color in the shapes to explore their attributes.  As you can clearly see, Louie couldn’t be bothered to even finish coloring in the pages (there were 2 more after these, also left blank).  As a coloring activity (supporting fine motor skills) it is OK although I know there are better ways for children to use their developing fine motor skills.  Mostly, I think activities like this are a waste of paper.

Next, take a look at a “Shapes” book that Noah made when he was 4.  I am not going to tell you what each page says (you have to figure it out by practicing reading his invented spelling attempts).  Notice how there isn’t a circle, square, or rectangle in the pages- but they certainly are “Shapes”.  Why do you think I believe this to be a much more meaningful exercise for children?