Tag: counting

  • Why Learning to Subitize Is Important

    Why Learning to Subitize Is Important

    Previously, I gave a definition and some resources for subitizing – assessing quantity without counting. The questions I posed in the previous article were

    • If you do it, how did you learn to do it?
    • How do we teach children to do it?

    Having thought a great deal about it, I wondered if it was even relevant to teach it. And Husband asked the question, “Don’t they need to count and know their numbers before learning to subitize?”

    You don’t have to learn to count to know “how many.”

    Although evidence indicates that subitizing and counting happen in the same part of the brain, you don’t have to know how to count to subitize.

    Subitizing up to 4 or 5 using recognition bypasses the “counting” of numbers.

    Daughter is currently subitizing 2 – without counting. I haven’t taught her how to count objects at home and they don’t do it at school yet. So she has no concept of one. But “two bows” (one for my hair and one for hers) is very important to her. Likewise “two bowls” (one with goldfish and one with Rice Chex) is an amazing thing.

    This article by Ernst von Glasersfeld explains how number words can be associated with the quantity without ever counting. Children give the appearance of counting because they’ve memorized the number words and they use the cadence to tap their finger on objects. But they’re usually doing this to be rewarded with praises and cheers. They don’t really understand the concept of counting. This is what I saw the little girl doing at the playground.

    It IS important for kids to subitize.

    What’s the point, anyhow? If I can quickly see 8 things  – does it matter? And is it important to tell the difference between 8 things and 9 things? If there are 8 hungry tigers heading for me and I subitize that there are 9, will it make a difference in how fast I run from them?

    I researched more and found out that  subitizing does matter to the understanding of numerical concepts.

    Subitizing 2 items and putting it with a subitzed 3 items can yeild the perceived 5 without ever knowing what addition is. And having the group of 4 items and seeing within it a group of 3 or 2, leads quickly to subtraction and division.

    Thus subitizing “teaches” various mathematical concepts without ever getting into the language or construct that grownups have put on it.

    So how did we learn it?

    My initial guess was that we’ve learned to do it the same way we’ve learned to read without sounding out the words. Over the years we’ve seen and counted many hundreds of bundles of 3 things that when we see another bundle of 3 things, we know there’s 3 of them.

    Upon further research I’m now of the opinion that I learned it from Ma saying to me “there’s three of them” while I looked at a collection of three objects. I bypassed the counting and went straight to learning the cardinality – with the help of grownups.

    The cliffhanger…

    So I’ve yet to answer the question, “How do we teach subitizing?” Alas, it’s coming. (UPDATE: it’s here, complete with downloadables!)

    In the meantime, where do you observe subitizing and does it seem important?

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  • What is Subitizing?

    What is Subitizing?

    Suzanne at Guided Math Study Group emailed me about subitizing the other day.

    I used perceptual subitizing to “know” that there were three green hair curlers here.

    Alas, I had never heard this term. So I did some research and thought I’d share what I’d learned.

    Subitizing is a way of instantly counting. In fancy math terms it would be getting to the cardinal number of a set (how many) without without going through the ordinals (counting each one’s position).

    There are two types, perceptual and conceptual. Perceptual is perceiving the number of objects immediately. Conceptual is putting a little effort into it.

    For instance, I glanced at the hot rollers Daughter had scattered on my side of the bed. I perceived there were three on the nightstand. But I had to do a little conceptual subitizing to arrive at the number of curlers on the floor (see the pictures).

    I had to mentally “chunk” these hair curlers into smaller perceivable bundles to conceptually subitize the number here.

    Here is a list of resources I found about it:

    In the next article, I’ll attempt to answer the two questions I see associated with subitizing:

    • If you do it, how did you learn to do it?
    • How do we teach children to do it?

    What do you think of these questions? Do you do it? Do you teach it?

    (By the way, it is pronounced with ooo – like “Ooo! Subitizing is neat!”)

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  • Counting Isn’t an Inherent Concept

    Counting Isn’t an Inherent Concept

    We spent our vacation a couple of weeks ago in central Texas. We had lots of fun – including lots of counting.

    One to one association of fingers to Discovery Toys’s Giant Pegboard pegs

    I noticed that some of the basic counting principles grownups know, aren’t inherently known to kids.

    I was observing Daughter as I was counting. She continued to look in deep concentration as I counted from one direction and then recounted from the other. Then I would rearrange the items and count them again. She was still enthralled.

    We teach our young children to count to 10, but never realize they don’t know what that means.

    Counting is a way to associate how much with a group of objects.

    If there are three things in a bundle, we associate the word “three” and the symbol 3 with that bundle. This number is called the cardinality of the set/bundle and means the number of things in it.

    My lone Calculus classmate in high school decided to create her own number system when we were freshmen. Everyone else laughed at her, but I got it. She was noticing that there was no rhyme or reason that we called a set of two objects “two.”

    We’ve decided to say out loud “two” and label it two and 2 and that means this many things: X X

    Counting is a way to order things.

    A bunch of things can be lined up and counted – even if they aren’t technically in a line. Each subsequent number is associated with another object. These numbers are the ordinal numbers. The final number that you count ends up being the cardinality of the set (from above). In this way you use ordinals to determine the cardinality.

    Counting is a way to compare one group of objects with another.

    Take two groups of objects. Pair one object from one group to an object from the other group – set them up in a one-to-one fashion. This shows that the two groups have the same number of objects. It doesn’t determine how many there are, but very young children don’t have to know the numbers to grasp the concept of “the same.”

    This eventually leads to the concepts of equality as well as less than and greater than.

    Counting isn’t dependent on which object you start with.

    This was the craziest concept for me. I noticed this when reading Brown Bear, Brown Bear one night.

    To mix it up (to keep my sanity) I would count the children in the book in differnet directions. After 3,000 nights of reading the same book over and over, something occurred to me. There is no reason for a 2 year old to know that counting in one direction will yeild the same number as counting in another direction.

    This is taught – not directly, but through experience. After counting a bazillion times, we eventually figure out that no matter which way you count things, you’ll get the same number.

    Well, unless Little Brother starts eating those things.

    Counting can be stopped and picked up where you left off.

    This is another concept that grownups “just know.” If you can mark your place (and Little Brother isn’t involved), stopping and coming back won’t change the result. This is the forerunner to addition, too.

    Counting is the foundation of all mathematics.

    This is the kicker. Counting is the beginning of it all.

    If you can get your kiddo to count, the rest is cake. And not just saying, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10,” but really getting him or her to understand the totality of the concepts.

    • How much is there?
    • Is there an order?
    • Does one group have more, less or the same as another group?
    • Did the number of objects change when you counted differently?

    Grownups get it, but we aren’t born with it. Imagine that you don’t know these things inherently. How does that change the way you observe the world?

    This article was originally shared on Homeschool Creations.

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  • Teaching Math with Counting

    Teaching Math with Counting

    Counting is a big deal in our household these days. Daughter’s 21 months old so teaching advanced multivariable calculus takes way too long for her attention span.

    Walking Steps and Counting Them

    But we count everything. Things in books, steps, bobby pins, people… you name it.

    We count in negatives, too.

    If you’re following my tweets, you might know that I count to Daughter in the bathtub – from negative 30 to positive 30.

    My neighbor suggested that early introduction to negatives might have helped her friend’s 14 year old son. He now has all sorts of difficulties with math. Not the least of which is arithmetic among positives and negatives.

    This makes me even more excited about counting in negatives to Daughter.

    Counting in negatives shows order.

    You may not need 61 seconds of anything. But you can count from negative 5 to positive 5. The point in the counting of negatives is to introduce the order of the numbers – since negatives seem to go “backwards” when you list them in order.

    And you can introduce distances with counting. You can show how distance is different than the number of points. Counting from -5 to +5 is actually 11 numbers. In this video you can see how I “rediscovered” this and then explained it:

    Parents often focus on the alphabet and reading in the very early years. There’s some neglect of math things outside of counting to 10 (starting at 1). You can do so much with counting – counting negatives, counting distances, skip counting (2, 4, 6, 8, etc.), finger counting, counting backwards.

    Counting is the foundation of all of math. And there’s tons of fun ways to use it in the early years.

    Share your ideas about counting in the comments below!

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  • Practicing Math Skills Early in Life Is a Brain Changing Event

    Practicing Math Skills Early in Life Is a Brain Changing Event

    Thanks to Will Summer who inspired this post with a retweet to me.

    Seems researchers have previously studied the differences in math skills among children vs. adolescents and grownups. Dr. Vinod Menon with has done new research on early math training that focused on kiddos who where merely one year apart.

    Turns out that in one year of math, the brain changes quite significantly!

    You can integrate early math learning into just about anything.

    Daughter is heavy into Brown Bear, Brown Bear by Bill Martin, Jr. and Eric Carle. This is not a math book. Nor does it pretend to be. But I’ve found a way to use it to develop math skills.

    When we arrive at the page with the children, we practice counting the kids. First in one direction, then in the other – as direction matters to kids. It isn’t inherent that if you count one way you’ll get the same number as if you count the other way.

    Sometimes we’ll count the top row of children and add it to the number of the bottom row. We can practice the commutative property by adding 5 + 4 = 9 and the adding 4 + 5 = 9.

    As an alternative twist, we’ll count and add the kids on the left page to the kids on the right page. This one gets fun, because that little blond girl is half and half!

    So we can now add fractions and practice the commutative property at the same time!

    What else?

    With Dr. Menon’s research, we now know how important it is to get started early with math learning. What other things can you do to start teaching math skills early?

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  • Counting with Ordinal Numbers

    Counting with Ordinal Numbers

    In the last post I discussed cardinal and ordinal numbers. I gave definitions but not much in the way of examples.

    Here’s an example of using ordinal numbers to count up to arrive at the final cardinal number. Oh – and I’m using the fun little critters from Discovery Toys!

    Whatcha think? Does it make you want to count some bugs?

    What other ways can you use to teach counting?

  • An Observation of Cardinals and Ordinals at the Playground

    An Observation of Cardinals and Ordinals at the Playground

    We were hanging at the playground on Mother’s day. Daughter had taken up with a cute little girl – a good playmate for the teeter-totter. I watched Daughter’s new friend do something interesting. She was “counting” some gumballs that had fallen off the sweetgum tree.

    Here’s what she said: ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT

    I looked and there were ten gumballs. I watched her more and saw that while she was reciting the terms and pointing at the gumball pile, she wasn’t associating each number word with the next gumball in the pile.

    Developmentally, I’m not sure if this is okay or not. I wonder.

    Mathematically, I know that there’s still a disconnect with this girl and what numbers mean to her.

    There are two types of numbers.

    In the realm of counting numbers, there are two types. Cardinals and ordinals. A cardinal number tells us how many we have. Like this:

    And ordinal number tells us the order:

    When we count, we use both types of numbers.

    The group of gumballs has the cardinal number of 10. But to arrive at that cardinal number, to determine how many there are, the friend had to “order” them. She was trying to point to each one and assign it a position. “YOU, I label 1st, you are 2nd … and since you are 10th, I know I have 10 gumballs.”

    We grownups take advantage of this procedure. Indeed most people don’t know what cardinals and ordinals are.

    But knowing this when you teach counting is quite helpful. At home you can do something similar to the potatoes in the pictures above. At school, you can buy some great posters (I found one at Teacher’s Heaven last night) that demonstrate this.

    Will you change the way you think and talk about numbers?