I'm Bon Crowder and the photos above are both of me - in 1989 and today. I'm a Generation X mom of Generation Z kids.

I began peer tutoring in high school in 1984. MathFour.com is the 2015 version of me helping peers be comfortable in math.

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Tag Archives: sets

How to Teach Subitizing: 1 – 4

The comparison of numeracy to literacy is curious.

Learning math is the opposite of learning to read. When you read, usually simultaneous to learning a language, you sound out words and then put meaning to them. When you learn to count and do math, you know the meaning inherently and then put a language to it.

At some point we learn to recognize words without sounding them out. And at some point we learn to recognize quantities without counting them out. This is called subitizing.

The Your Baby Can Read program uses the concept of subitizing to teach reading – you show your baby the word alongside the object. So the shape of the word car is as recognizable as a car itself.

The children using Your Baby Can Read don’t learn to sound out words. They don’t understand the concept of letters any more than babies not using the program. But they instantly recognize the shapes of the words – giving them an (assumed) advantage.

Aside: We didn’t use the “Your Baby Can Read” program, not because it was gimmicky (I love anything that looks gimmicky), but because there is a huge DVD element to it. We decided not to put Daughter in front of the TV for her first 2 years. A decision we stuck with, but sometimes was a struggle!

This article contains a “your baby can count” type program. (And it’s a free download!)

How did we learn subitizing?

I don’t recall having been taught it directly. Although I could be wrong. The research on it has been happening since the early 1900s, so it might have been taught without being labeled “subitzing.”

In a previous article about why learning to subitize is importantChristine Guest commented that she learned it out of frustration for counting with chanting.

I wonder how many of us do that. Are grownups so adept at subitizing that they forget that’s how we assess quantity? Maybe we’re taught to chant-count because that’s the way we think counting is.

But it isn’t!

How do you teach subitizing?

Images are accompanied by the written numeral as well as the number spoken aloud. The images would be printed on cards, done via video or “live” with 3D objects.

I’m still working on the numbers 5-10 and up, but for the numbers 1-4, the following 8 styles of image sets would be done twice. Once using the same objects for each image set, and once using different objects for each image set.

  1. Organized in a row vertically.
  2. Organized in a row horizontally.
  3. Organized in a row diagonally.
  4. Organized in a row other way diagonally.
  5. Organized in a regular shape (triangle, square).
  6. Organized in a differently oriented regular shape.
  7. Organized in an irregular shape.
  8. Organized in a different irregular shape. (There will be more of these for 4 than 3, etc.)

The objects could be blocks, cars, little dolls, just about anything. I created the set below from blocks I found left in Daughter’s block set.

Each zip file contains a few .jpg files with 4″ x 6″ pictures. You can print them at home or ship them to Walmart, Target, CVS, etc. for printing. I left off the MathFour.com logo so the kiddos wouldn’t get distracted. Please share them along with links back here.

What do you think? Can you use these? Did you?

This article is also linked in the Preschool Corner on Homeschool Creations.

4 Responses to How to Teach Subitizing: 1 – 4

  1. Hi Bon,
    I got into a discussion at school yesterday about subitizing and remembered reading about it on your blog. I searched and read all of the posts and just printed/laminated your cards. Thanks!

  2. I have a daughter who cannot subitize. She is dyscalculic. She has no basic number sense. She cannot memorize any basic math fact. Related to this, she has no sense of the passage of time and no sense of past and future and no real sense of money, even though she can add money together using her fingers or paper and pencil. The total she added means nothing to her.

    In doing extensive research in this area, I have read many research papers on dyscalculia and subitizing. The scientific community, based on extensive brain scans and other studies, seems to feel now that subitizing is something most human beings (and many animal species) do automatically, probably from birth, and that this skill develops further over time. Research is showing that explicit teaching of subitizing can help anyone and quite a bit, but most already have this ability at a basic level and tend to advance in this skill without explicit instruction.

    There is a small percentage of the population that does not have the ability to subitize. These children end up struggling tremendously in school and frequently as adults because this basic function is what allows us to do simple arithmetic without counting on our fingers, “memorize” times tables, process elapsed time, understand calendars and the passage of time, handle finances etc. There is a book called My Thirteenth Winter, written by a girl who has dyscalculia. She cannot subitize. If you have a student or a child who, no matter how hard they try, cannot memorize addition, multiplication or subtraction facts, or seems to struggle with time, or the concept of money, read this book to understand how debilitating this issue really can be.

    We are just starting to do systematic subitizing exercises with my 7th grade daughter, after years of tears and frustration because of an inability to function in simple math efficiently, yet the cognitive ability to understand many more advanced math concepts (that function using a different part of the brain than the area that subitizes).

    Your info here is also helpful. Thank you.

    Wish us luck!

    • Wow, JC! Thanks for the info. What power you, and she, now have knowing what’s going on.

      Please keep us informed – and indeed, GOOD LUCK!

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5 Responses to Union, Intersection and the Empty Set

  1. In the paragraph “What happens with the empty set?”. I like the idea that two cups, one empty and the other with some objects in it, have nothing in common, hence the empty set as intersection!

    What puzzled me a bit is what if we try to find the intersection of two cups with all different objects in them? The result is again an empty set. Moreover, if the question arises, what are the ALL subsets of a given set, it will be {T, S, R}, {T, S}, {T, R}, {R, S}, {T}, {R}, {S}, O (empty set). Here I used your example Square, Triangle, Rectangle as a shorthand. We can see that power of set (set of all subsets) is exactly 2^3 = 8 i.e. 2^(# of elements) and this includes empty set too!

    In other words it can also be said that empty set is common element for empty set and any other set, which looks like a different statement that empty set has nothing in common with a set of objects. Actually, empty set is a subset of any set. If we empty the cup we will see that there is an empty set right there as well and that it is common with the other (empty or emptied) cup, i.e. other empty set, on the left side of intersection sign.

    • Thanks, Nash! I love the power set – just can’t get enough of it (maybe because I’m so jealous, I wanna be the Power Bon).

      You write, “it can also be said that empty set is common element for empty set and any other set.” I think you might have mistyped. The empty set is a common subset of both the empty set and any other set, but not a common element. This was the point of the previous article on empty sets. There is not an empty set inside the empty cup – the empty cup IS the empty set.

      I think you might mean that the empty set is a common element of the power sets of the empty set and any other set.

      Power set of empty set = { {} } (the set containing the empty set)
      Power set of {T, S, R} = { {T, S, R}, {T, S}, {T, R}, {R, S}, {T}, {R}, {S}, {} } (what you wrote out)

      In which case the empty sets are elements of the bigger sets – and are the intersection of the two.

  2. Fantastic examples and graphics! A person could literally scroll through and learn about unions, intersections, and the empty set just through the pictures and reading the captions. Super brilliant.

    My suggestion? “Write” a math picture book just with labels, captions and titles. It will be like summarizing novels in 160 character tweets! (or is it 140?)

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4 Responses to Empty Sets – When to Use Curly Brackets

  1. I think that the analogy might be misleading, because you can have many empty containers, but there is only one empty set. A set is like a container in some ways; but unlike a container, a set is determined by its elements. Two sets that contain the same elements are identical, but two distinct containers could have the same contents, or no contents.

    I do agree that the analogy can be useful, but one should avoid phrases such as “two empty sets” which can cause misconceptions.

    • Wow, Dave. You’re totally right.

      Which makes me really think about the possibility of having different empty sets. Since the empty set is an element of the power set of any set, might the empty set as an element of the power set of integers be technically different than the empty set as an element of the power set of the set of polynomials (for instance)? I guess, no. But something feels like it might be yes.

      If we say, “There’s nothing in it,” it might be very different if we know what the possible things are that MIGHT be in it.


  2. Hmm. The axiom of extensionality states that two sets are equal if and only if they have the same elements. This is a standard axiom of set theory, and it implies that the empty set is unique.

    But it does seem to me that there are instances where we would want to allow different sets to have the same elements. For example, suppose that a high school has a math club and a chess club. It might happen that the two clubs have the same membership list, but we would still consider the clubs to be different. Maybe there is a non-standard set theory that would model this situation.

  3. Don’t confuse the name of something with the thing itself. Two clubs can have the same set of members, just like you can define sets S=\emptyset and T=\emptyset. Four containers gives you four copies of the empty set, which we lazily call four empty sets. (Except that some students will correctly think that the containers are not empty, and have different sets of air molecules.)

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2 Responses to Why Learning to Subitize Is Important

  1. I remember subitizing as a little girl because I got mixed up chanting to count, or lost my place. But I could see triangles, quadrilaterals, and groups of doubles, so I’d make shapes, then add them up.

    I think it just happened during block play when my Mom needed me out of her hair while she made dinner.

    Why I felt I needed to count my blocks I’m not sure though; maybe I was imitating counting books or Seasame street?

    • That’s really neat Christine. I wonder if there’s research on shape seeing – there’s most definitely some better, fancier word! I’ve never been good at “getting” spacial stuff but I know others (my dad) who are very good at it. Are you good with spacial stuff – like being able to “see” things in 3D when they are just drawn or you only see one part of them?

      And I’m sure Sesame street had a lot to do with all of our desires to count. What a great show!

      Thanks for your comments, Christine!

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