Emerging Math, part 1: Introduction to Counting

So cogsci-toddler’s big thing these days is counting. And growing up with English, Korean, and Japanese in a primarily Spanish speaking neighborhood, he can count to 10 (sort of) in 4 languages. But the next blog series will be devoted to why this is a cognitively unimpressive feat! After all, any toddler parent can tell you, knowing number words does not mean number knowledge!

I busted out the camera a little late so the video captures Mo in the midst of counting…

Here he is counting in Korean… note his Korean pride as he rejects his dad’s Japanese counting in the middle… and then says, “Gracias!” at the end. What a linguistic bi-bim-bap (Korean mixed rice dish)!

Mo’s current knowledge of counting is really just about pointing and saying a series of words. This behavior is paramount to pointing at people and saying “duck-duck-goose” or “eeny-meeny-miny-mo.” It’s not really about numbers at all. But to make that claim, I have to grapple with two questions: (1) What does it really mean to understand counting? (2) How would you know whether a child “truly” understands counting?

(1) What does it really mean to understand counting? Rochel Gelman and C. Randy Gallistel (both formerly of UCLA, my alma mater!) theorize that there are (at least) 5 basic principles to counting:

    • Stable Order: The words have to occur in the same order. You don’t necessarily have to point to the objects in the same order but the words always have to go one-two-three-four (you can’t say, three-one-two-four).
    • Order Irrelevance: Even though the words have to have the same order, the objects can be counted in any which way!
    • One-to-one correspondence: Each object can only be tagged by a word once. This one is violated all the time by cogsci-toddler1; he’ll count all the objects and then go back and keep saying number words until he reaches his limit (10 in English/Spanish; 12 in Japanese, 14 in Korean).
    • Cardinality: This just means that the last number is significant because it marks the number of items present in the set. The end result of counting is figuring out “how many.” If you ever ask Mo “how many ___?” he is most likely to say, “Two!”
    • Abstraction: Anything can be counted! Ducks, shirts, cups! Even amounts!

Mo probably sort of has some understanding of stable order and order irrelevance and maybe even abstraction but he definitely does not know one-to-one correspondence and cardinality.

(2) How would you know whether a child “truly” understands counting? The strategy in research is to trust consistent performance across a variety of tasks. Each number task is limited and can be “gamed” in some way. But converging evidence is a decent indicator of solid counting! Here are a few of the tasks you can do with your toddlers!

  • Equivalent sets: Shown a set of items (e.g., 3 marbles), child can produce the set (give you 3 marbles).
  • Match equivalent sets: Shown a set of items (e.g., picture of 3 dogs), child can choose a set that has the same number (e.g., 3 bones vs 5 bones). This task is like equivalent sets except it is multiple choice.
  • “Give me x”: When asked for a number (e.g., “give me 3!”), child give you that number of objects correctly.
  • Answer “how many?”: When shown a set and asked “how many,” child says correct number. (Need this one to master cardinality.)
  • Count high: Instruct child to count as high as he/she can.
  • Identifying number symbols: When shown a number symbol (“3”), they can say it and produce the amount.

Here’s another task that I use with Amos that I’ll talk about more in a future post. I give him the option of x vs. y objects. When I do this with treats (in the video below, I offer capers… Amos really likes capers), Amos usually is pretty accurate with numbers <= 5. But when I do the task with random objects (like sticks), Amos totally ignores me or just says “two” no matter what. But when I tried to get a video of him being random, he is intensely accurate. Bah.

I thought he might be tripped up by 2 vs. 4 because he really prefers to say 2… Next time I have to try and mix up the order and give him the choice between 4 and 2. Perhaps he is just using the heuristic of choosing the last number. More number posts to come!



2 thoughts on “Emerging Math, part 1: Introduction to Counting

  1. teachingninja says:

    I love it!! I love numbers! And I love Amos!! Soooo cute!

    I also love how Dr. Cogsci’s Flickr name is jithepee.

  2. treevalley says:

    hahah! korean pride eh? he should embrace all of his identity (including his japanese one)! granted he is getting a lot more people he loves speaking korean to him. thanks for the most through break down of counting I have ever encountered!

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