Math Mondays: Ready, Sets, Count!
Preschool children are often good at counting when it comes to identifying “how many” objects are in a set, but they have difficulties putting their counting skills to work when it comes to comparing sets. Why?
The issue is more complicated than you would think, and in fact, science is still trying to figure it out. Nevertheless, recognizing numerical equivalency between two sets – despite any differences in other dimensions (such as size, density, or color) — is a fundamental numerical competence, and hence, it is the topic of our second week of Math Mondays.
This little game may help explain why such a seemingly easy topic may be complicated for children… and for adults too!
1. Without counting, guess which set has more candies?
2. Now try this: without counting, guess which set has more candies?
It was much harder to guess,wasn't it?
3. Now how about this:
I bet it was even harder to guess which set has more... Most likely, you were tempted to count.
4. One more- now compare the same two sets with a different spatial arrangement:
Easier, wasn't it?
Recognizing numerical equivalency between two sets, despite any differences in other dimensions, is a funfamental numerical competence (Mix, 1999, Ginsburg & Opper, 1987). s adults, we engage in different strategies when comparing things; sometimes we rely on the visual properties of things (like in Example 1 above) and sometimes the best way to compare is simply to counts (as you experienced in Example 3).
As children, we could also easily know that the set on the right is more than the set on the left in the first example. However, children may not know the language involved. What do we mean when we say, “more,” or “less,” or “the same”? Examples 2 and 3 are much harder to solve at a glance, for children and for adults too!
Children need to learn what strategies they can use to solve the problems — strategies such as counting and matching. Possible strategies for comparing two sets of items:
- Visual inspection. This is what you did in Example 1, and this strategy is most accurate if the relative size of the two sets is more than 2:1. The bowl on the left contained 3 M&Ms, and the bowl on the right contained 7, so it was pretty easy to guess that the bowl on the right contained more.
- Matching items from one set to the other. It’s a good strategy for comparing smaller sets — in Example 2, you could align one chocolate chip with one M&M until you have two leftover chocolate chips, indicating that the bowl on the left had more. But it’s a hideous strategy to compare a set with 100 candies versus one with 98 candies.
- Visual and spatial patterns. This is seen in Example 4. It’s especially useful when you can arrange items, like we did with the chocolate chips in rows of 4.
- Counting. The most accurate tool in your toolbox, but it also has many possible sources of errors — especially for young children who haven’t mastered their counting skills yet.
And, as always, we make mistakes! You might have made one or two of these common mistakes yourself in the game above:
- Being distracted by irrelevant properties of sets; we usually perceive bigger as more (see Example 2).
- Counting errors (skipping items, double counting, using incorrect numbers, etc.).
- Memory. Some children may count the two sets correctly but forget their counting results by the time they want to judge which one has more.
- Not knowing order of numbers — is 8 bigger or 9, or the other way around?
So how can we help children master these concepts?
- Contextualize the concepts of more, fewer, and the same in everyday activities. Compare the number of trains with the number of cars on the coffee table. Compare food: “Do you have more goldfish or grapes?” Arrange stickers based on their sizes. Give each stuffed animal a teacup. Have fun with it!
- Language, language, language. Read stories to your kids; encourage them to use the relative words (“more,” “less,” “same,” etc.) and to compare things around them. Talk with your child about most, more, fewer, fewest, or same number; biggest, bigger, smaller, or same size; tallest, taller, shorter, or same height; and widest, wider, or same width.
- Model different strategies. Compare sets by matching, by arranging them in patterns, by counting them, and so forth.
- Play the M&Ms game outlined above. Encourage them to check their work by counting. And don’t forget to let them win the opportunity to eat the game pieces after you’re through!