The explorations below are for parents of child ages 2-6 (although teachers may find them useful and adapt them for their purposes). These explorations are organized according to a list of big Ideas from The Big Ideas of Early Mathematics: What Teachers of Young Children Need to Know (referred to as Big Ideas below) by Erikson Institute’s Early Math Collaborative. In this book, 26 big ideas are organized under 9 topics. You can find out more about this book here on my blog www.mathbookmagic.com. We wrote about the first topic, Sets, and its three related big ideas here. This post continues with the second topic, Number sense, and its three related big ideas.
When you think of early math, I’m sure counting comes to mind. I hate to admit it, but I am guilty of comparing my children with others when it comes to counting. Your 1 and 1/2 year old can count to 20?! Oh no, my 3 year old can barely count to 7! Ahhh!! Then thankfully I come to my senses and remember counting is not a competition! And if I make it a competition, I’ll miss the magic that comes with observing and wondering with children as they develop facility with this integral skill. Learning the counting sequence is an amazing feat. But it is not all there is. We can’t assume that the child who can correctly count to 20 or higher understands their numerical meaning. And for the child who can’t count to 20 yet, we must remember children have a wealth of informal knowledge about quantity and developing a robust number sense takes time.
Number sense is a key building block for arithmetic, which makes up the majority of a K-8 math content. Erikson Institute’s Early Math Collaborative defines number sense as” the ability to understand the quantity of a set and the name associated with that quantity.” (Big Ideas, p. 30). Here are three big ideas associated with Topic 2: Number Sense.
Number Sense: Developing a Meaningful Sense of Quantity
Big Idea 2a*. Numbers are used in many ways, some more mathematical than others. Ever stop to think about all the ways we use numbers? Some numbers are nominal or categorical (e.g., sports jersey numbers). Some are referential (e.g., time and temperature). Other numbers are ordinal (1st, 2nd, 3rd). For these numbers, order matters. Other numbers are cardinal (5 cupcakes is more than 1 cupcake) and provide answers to the question, How many? There is a huge psychological difference between ordinal and nominal numbers for young children when it comes to which is more. For example, it you are #1 (1st) in the race (ordinal), that is good. But if you only have 1 cupcake (cardinal) and your sister has 3 cupcakes, not so good. Further, ordinal and cardinal numbers are inherently mathematical. However, typically, categorical numbers don’t indicate quantity, rank or any other measurement and while we can rank and compare referential numbers, we usually don’t perform mathematical operations with them.
Big Idea 2b. Quantity is an attribute of a set of objects and we use numbers to name specific quantities. This big idea centers on developing a concept of “fiveness”. “The idea of 5 must be constructed in your head, since it is a quality that we best in in relationships between sets.” (Big Ideas, p. 33) When we count the oranges in the bowl, we say “1, 2, 3, 4, 5” and the last number we count is the answer to how many oranges in the bowl. There are 5 oranges in the bowl, even though we said all the numbers 1 through 5. It is important to remember that numbers are abstract. The abstract nature of numbers is why Erikson talks about not using “naked numbers” with young children and highlights the importance of attaching a concrete object to numbers (saying “5 oranges” versus “5”) .
Big Idea 2c. The quantity of a small collection can be intuitively perceived without counting. There are times when a child will tell you how many without counting one by one. This is called subitizing and it part of a visual number sense. Two types of subitizing are perceptual and conceptual. With perceptual subitizing, a child can instantly tell you how many are in a small set (5 of less). They sometimes can do this even before they can count up to 5. Conceptual subitizing is recognizing smaller groups within a larger set and adding those small groups together, such as two dots plus three dots equals five dots. More about subitizing here.
*The Erikson does not include these letter labels (a, b, c) in Big Ideas. I do this for clarity. They are not meant to denote any sort of learning progression. For more on learning progressions, you might find Graham Fletcher’s summaries helpful.
These three big ideas provide helpful anchors for exploring counting and number sense in early math. The games, questions, books, resources, and activities are possible ways to engage with big ideas 2a, 2b, 2c in informal ways throughout your day. Be sure to listen closely for the magical ideas of your brilliant children.
Number scavenger hunts around your neighborhood or home are great way to observe the different ways numerals show up (e.g., license plates, house numbers, mailboxes) and link number names to objects (e.g., 1 fire hydrant on our block, 4 stops signs at the corner, 5 petals on a flower, 2 robins on the lawn). Here is a number game printable from Math on a Stick booth at the Minnesota State fair and here’s a lovely description of using this number game from Kent Haines’s website. This number game focuses on finding hunting for a quantity (3 of something) as opposed to the numeral 3. [Big idea 2a, 2b]
We’ve also picked a card from a standard deck or Tiny Polka Dots deck (see below) and go on a hunt for that number. Here is an image of some 3s around our house. [Big idea 2a, 2b]
Fingers are a convenient counting tool (image is of my oldest child, my grandmother, my mother and me). And it is important that children are encouraged to use their fingers to support understanding. Here are a few other counting tools.
This Montessori inspired puzzle is a great one. It connects concrete (rings), pictorial (dots), and abstract (numerals) representations for number. [Note: The pieces from puzzle linked above are a choking hazard. I wouldn’t get that one if you have children under 4 in the house. Here is an alternative with larger pieces. [2b, 2c]
A Rekenrek is a nice tool to have on hand. We have this one, however there are larger ones as well with 100 beads. You can make your own as too. Here is a great blog post by the Brown Bag Teacher about the rekenrek with directions for how to make your own. [2b, 2c]
Five frames and/or ten frames are another useful tool. Here is are some printables (5 frame, 10 frame). Children can use coins and counters on them or dry erase markers if you decide to laminate them. Here is an electronic version. [2b,2c]
Card games are great for familiarizing children with numbers as they include both numerals and a pictorial representation. Play a version of a Memory/Concentration game, Go Fish or a game of War, which we call MORE.
In addition to playing with a standard deck of cards, you could play with the Tiny Polka Dot card deck. This awesome deck was created by educators Dan Finkel and Katherine Cook. Play card games mentioned above or they have a bunch of other games to choose from. These thoughtfully-designed cards present quantities in unique ways by varying dot size, arrangement and representation. [2b, 2c]
Bears in a cave is a fun game to play. We learned about it from this awesome math wisdom book Math Fact Fluency by Jennifer Bay-Williams and Gina Kling. This book has over 60 games and tools to support math learning. Here are the directions for Bears in a cave (or Sleeping Bears as the game is called in this book). Each player needs a five frame, a plastic cup/bowl, and 5 bears (e.g., counting bear manipulatives (ages 4+) or our favorite Teddy grahams). One person places some bears in the “cave” to sleep. The other person looks at the bears “awake”/unhidden and tries to figure out how many are sleeping/hidden. The five frame may be used to place the unhidden bears to help them determine how many sleeping/hidden bears. [2b, 2c]
Any board game with a die/number spinner is a great way to develop early number sense (e.g., Count Your Chickens, Trouble, Chutes and Ladders). For more game ideas, check out math educator Kent Haines’s www.gamesforyoungminds.com, a website full of magical games to inspire wonder and joy. [2b, 2c]
MATH ART AND CRAFTS
Make some art inspired by classic picture book 10 dots by Donald Crews. 10 dots provides a springboard of examples of things that can be made with 1-10 dots. One dot (the sun). Two dots (a foxes eyes). There is a lot you can do with 10 dots and some imagination. I cut up a bunch of black dots and keep them with our art supplies. Here are a few pieces we did in our house (a tent, flowers, and a smily face with a mustache). Check out this blog post by art teacher Jenny Knappenberger. [2b, 2c]
And lastly one of our favorite ways to share math is through picture books. Here are some great ones Number sense ones. Absolutely One Thing: Featuring Charlie and Lola by Lauren Child and I Know Numbers by Taro Gomi are great for exploring how numbers are used in different ways (2a). For comparing quantities, check out our post about Lia & Luís: Who Has More? by Anna Crespo and Giovana Medeiros on www.mathbookmagic.com [2b]. CRASH! BOOM! A Math Tale is great one from the Mathical list. Children can build their own block towers like the main character [2b]. Counting Kisses by Karen Katz is a sweet bedtime read [2b].A classic that you can never can go wrong with is Anno’s Counting Book by Mitsumasa Anno [2a, 2b, 2c]. Tana Hoban’s book are always great for noticing and wondering about numbers. In particular, check out Count and See. [2b, 2c].