The Other Side of Math for Infants and Toddlers by Dr. Jill Uhlenberg

 

Many adults focus on numerals, counting objects, and helping young children learn the order of numbers. While these are important basics for mathematical learning, there is another side to math that is just as important; that is, learning about spatial relationships. Spatial understanding is a foundation for a substantial amount of math that we use as adults. This includes map-reading, measuring, classifying, recognizing and developing patterns, and more.

Infants and toddlers can develop spatial understanding through exploration of simple household items like plastic storage containers with lids, nesting cups and spoons, and muffin tins, along with objects to place inside these containers. Given plenty of time and adults who are willing to observe rather than direct the children’s experiences, young children can construct basic knowledge of spatial relationships. We collected 50 hours of video in order to examine how 12- to 36-month-olds used contents and containers and what they learned in the process. Adults can support this learning in the following ways:

    

Provide a variety of containers and objects for the children to explore. If possible, include a container that is large enough for the child to climb into. You will observe them placing containers on their heads, feet, and hands. Placing everything in a tub or drawer that is accessible will allow the children to engage freely, learning about the materials, then combining the materials in different ways to see what happens.

    

Spend time observing these explorations. You may want to name the materials and actions using spatial terms: you put the scarves in the bowl; I see you put the red ball beside the yellow ball; oh, the canister is on your head! You may be surprised at how the children use the materials. Over time, you will see growth in the children’s understanding of space and shapes. We also noticed that the children spent long amounts of time exploring, busting the myth of short attention spans typically attributed to toddlers.

  

 Allow the children to decide whether a problem has been solved rather than the adults expecting perfection. That is, we suggest accepting approximation. If a child is satisfied with the result of some experience, note this as a step in their development of spatial understanding, and watch for refinement in future explorations.

    

The favorite materials in our observations were plastic balls (ball pit type) of different sizes and colors. The children combined these balls with muffin tins (we had three sizes from mini-muffin to large muffin sizes). Using these materials, very young children began by testing whether one ball would fit in each of the 12 cups of the muffin tin. As adults we know these sizes are constant, but at 12 months of age, this is new knowledge. Other spatial understandings/math learning constructed with muffin tins and balls included these examples:

• One-to-one correspondence—one ball fits in one cup;

• Estimating space—using the medium ball in the medium-sized muffin tin; noticing which balls fit and which did not; pouring a bowl full of balls into a too-small canister; trying to place a lid on a too-full container;

• Classification—sorting balls by color or size; older preschoolers may use two characteristics, but infants and toddlers can generally only sort on one attribute;

• Patterns—placing balls in the muffin cups in a particular order (color, size, selected cup, etc.) that is repeated;

• Distribution—performing the same act on the same materials repeatedly, like placing the same ball in each of the muffin cups one by one; this mathematical process shows up later when adults are calculating equations.

    

The more experiences, the more advanced the children’s use of materials became. Eventually they began
combining materials in creative ways by constructing towers of bowls, canisters, nesting containers, and other objects. These explorations lead logically into the use of blocks, another strong component of spatial understanding. When you recognize and celebrate infants’ and toddlers’ bent for objects and containers, you are supporting their beginning spatial understanding. You will be surprised at the wealth of knowledge they construct over their first three years.

Addition without Worksheets or Flashcards: POINT

If you have a child at home practicing addition, Point is a great game to print off and play with them. It is much more engaging than flashcards or worksheets, and involves much more mathematical reasoning.

This game has two types of cards - numbered dot cards and tile cards. The dot cards are dealt out to all of the players and the tile cards are placed face down in the middle with the first one turned up. The tile card shown has 9 gold bars. Players have the opportunity to use 2 or more numbers to total 9. The player must prove to the other players their cards add up to the tile card number before getting rid of their cards. The winner is the player who gets rid of all their cards first.

Tile Card

As you play with your children, watch to learn how they count and add. Ask them to explain how their combination adds up to the tile card's number by asking, "How do you know that adds up to 9?" Do they always count one at a time? Do they count on from a larger number (Example - the tile card on the right has 9 gold bars. Do they start with 5 and count on 6, 7, 8, 9? Or do they start with 1 and count each bar up to 9? Be prepared to explain your combination cards as well!

As you play more games, you may find them figuring out the strategy of saving their small cards (1s and 2s) to make larger numbers to be able to get rid of their cards more quickly.

Keep a tally of who wins each hand and add again to figure out  who won the most.

Click here for a PDF to print all the game cards. We include a backing for the dot cards and tile cards, but you can eliminate those to save ink. Important: When you are ready to cut apart the tile cards, cut on the red lines.

Memory (Home-Made Version)

The previous post, Memory, explains the mathematical benefits of playing the game. This post shows how one can make a home-made version. Making a home-made version can be a rich, integrative experience with your children as it involves reading, writing, and fine motor skills.

Materials needed: images, scissors, glue or tape, material for making cards

Step One:

Finding Images: search your recycling or mail for catalogs, magazines, or advertising flyers with photos. Challenge your child to find pairs of items that can match such as two lawn mowers, bicycles, etc. Hand them a pair of scissors and ask them to cut them out.











After your child cuts the images out, trim the excess around them. As you trim, talk with your child about the name of each item. What an item is called may be different depending upon where you live. For example, are they a pair of pants or a pair of slacks?

Step Two:

Making the cards: Next look for a material to use to make the cards. If you have an old deck of cards with some missing cards, this will be perfect! Block out the existing image with masking tape to provide a space for your images. Other materials that can be used are old Christmas cards, envelopes, or cereal boxes. For the rest of this post, we will be using a cereal box. 







Regardless of what you use to make the cards, work to make them uniform in size. This is a good opportunity to model using a ruler and standard units of measurement. Think aloud, modeling what you are thinking as you use the ruler. I want to make them 2 1/2 inches wide and 3 1/2 inches wide...

Step Three: 

Attaching Images to Cards: Use tape or glue to put the images on the cards. If you are lucky to have clear shipping tape in the house, this is perfect. 

You can easily address literacy learning by asking your child to help you label the pictures using a sharpie. 

What sound do we make at the beginning when we say shirt or shoes? Help me remember what two letters make the /sh/ sound. 

Step Four:

Checking the Pairs: Ask your child to help you check to make sure you have a pair of each object. 

If you have labeled your cards with a sharpie, point to the words and do a bit of reading as you check. 

Remind me what this says. Hat? What sound do you hear at the end of hat? What letter makes that sound?

Step Five: 

Begin to Play: Mix up the cards and lay them out in an array. If you are working with an older child, you can model talking about rows and columns if you want. 

I can make two rows with six cards in each row. But that leaves me with only four at the bottom. I wonder what would happen if we put seven cards in a row? Eight? 

If you have a very young child, start by only putting 4 pairs out at first. As they gain confidence, add more pairs to the deck to challenge them. 

Begin to play as directed in the previous post. You may think that your child will recognize the pairs by the print of the cereal box. If they do, they have an excellent memory! 

Tomorrow's post: Tapatan!

Memory

This is a very simple game that most preschoolers can play successfully. Kindergarten, first, and second graders enjoy it as well. The mathematical concepts that can be addressed: the concept of pairs, counting, counting in twos, more and less.

This game can be played with any matching cards. Many different sets of matching cards are available commercially with various pictures, such as farm animals, insects, birds, household items, paintings by famous artists, etc. Standard playing cards or number cards can also be used. The number of pairs used to play the game should remain low (around 6 pairs) for the youngest players. With older children, more pairs can be used.

When children first play this game, they tend to choose cards at random; they often do not replace cards in the same place from which the cards were taken; and many times they do not pay attention to which cards other players turn over. As they become more experienced at playing this game, they begin to figure out that watching the other players turn over cards will give them useful information.

An interesting way that young children often play Memory is that when children see a player turn over a card and they think they know where its match is, when it is their turn they will often turn over the card that was just turned over first, and then try to find its match. Adults recognize that if they are not positive about where a card is placed, they should turn over the card that they are not sure about first, so that if they are wrong, they have not wasted a turn.

Young children often play Memory cooperatively. That is, when a player turns over a card and another player knows where its match is, that child will often tell the player which card to turn over in order to make a match. Here is a video of 4-year-olds playing the game: https://regentsctr.uni.edu/sites/default/files/videos/original/memory_0.mp4

The versions of Memory sold in most stores are made by Hasbro. You may already have this game. If not, you can download and print these files for use at home: 

Memory cards: https://regentsctr.uni.edu/sites/default/files/games/Games/Memory/Memory_Cards.pdf

Memory game board: https://regentsctr.uni.edu/sites/default/files/games/Games/Memory/Memory_Grid.pdf

Memory rules: https://regentsctr.uni.edu/sites/default/files/games/Games/Memory/Memory_Rules.pdf

If you do not have a printer or enough ink at home, check the next post for how to make a home-made memory game.