Counting & Numerosity

We find great joy when we see our children counting things. We applaud them when they say, "Look, there are 1...2...3 balls!" Or when they count the days on a calendar, touching each numeral with a fingertip. But do they really understand what they are counting? And what does it mean to count? A child who can count, as in the examples above, does not necessarily understand the numerosity, or "how many-ness" of what she is counting.

To help her develop a strong "number sense" we must provide her with a variety of opportunities to explore numbers through hands-on explorations with objects, such as mathematics manipulatives and things used in her everyday world. She must be given tasks that encourage her to think, reflect, and consider, so that she may construct her own understanding of mathematical ideas, as true learning occurs only when we create it for ourselves.

Think about growing a tomato plant....surely, you can read about how to do it and explain it to others based on what you have read, but when you actually plant the seeds, transfer the sprouts into the soil, determine how to steak the drooping vines, keep the bugs off the leaves and fruit...it is then, and only then, that you construct an understanding of how to grow a tomato. Prior to doing it yourself, it was just knowledge.

In the WRC Educational Playroom, children participate in a wide variety of activities, designed to help them develop, over time, the following (just to name a few):

• the notion of relative magnitudes - a child can discern one set as more or less than another without counting the objects in each set
• the one-to-one principle of counting - each item to be counted is counted once and only once
• the stable order principle - number words must be recited in the same order
• the cardinal principle - the last word counted represents the "how many-ness" of a set
• the principle of increasing magnitudes - the later number words refer to greater quantities
• the one-to-one principle of numerosity - two sets are equal if the items in each set can be matched one-to-one with no items remaining

We often talk about learning in terms such as "building blocks," "stepping stones," or even "a good foundation," which is really quite inaccurate. Cognitive development occurs in more of a give-and-take way where children move forward and backward in a recursive manner as concepts and ideas begin to take shape in their minds, over a long period of time--usually years.

And while it is true that we want our children to have a "strong foundation" in math, their development will not happen in a linear, static way; rather, it will be much like an Impressionist painting--when you stand very close to the painting, you see lots of dots that appear to simply be dots, and the dots may seem rather disconnected, which is similar to the seemingly random bits of concepts, skills, and knowledge we may see in our children. But, when you stand back a bit farther, the painting begins to take on a more recognizable shape, and the relationship between all the dots becomes evident, just as we start to see our children making connections as they construct their own understanding of mathematical ideas.

So what do we do in the playroom to help children develop their own mathematical ideas?

• We visualize quantities using dot patterns independent of and within ten-frames for quantities 1-10. The patterns are ordered and predictable, to help children develop strong mental images.
• We count using base-10 language, in which the words we say actually mean the quantity. So your child may say "10-two" for "12" or "2-ten-7" for 27. Have you ever considered how confusing it is for a child to say "ten... eleven...twelve....thirteen..."? The words after ten have no relationship to the quantities they represent. In contrast, ask a child to count to "2-ten" (20) and he will be able to do so, with relative ease, AND will be able to match objects to the count.
• We solve problems that are difficult and messy, but situated in their own world. How do we share markers equally? In what order do we place the numerals on the calendar? Are there more purple kinds of flowers in the garden or yellow ones?

So how can you support your child's learnings in the playroom? Play games that require your child to build and decompose sets, count objects, assign quantities, and solve problems. Situate them always with number patterns, and help her see those patterns in her everyday world. And have fun while doing it together!

And while you're mathematizing with your child, check out the National Council of Teachers of Mathematics website activities that use the same ten-frames and counters we do in the playroom! It's a great way to weave developmentally appropriate technology activities into their play, too.