This is common core state standards support video for mathematics; the standard is K.CC.4abc, so actually we're dealing with three different standards. The standard reads: understand the relationship between numbers and quantities connect counting to cardinality. a) When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. Part b states: understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. And then part c states: understand that each successive number name refers to a quantity that is one larger.
Let's start with part a. It states again, when counting objects, say the number names in the standard order, pairing each object with one and only one number name, and each number name with one and only one object. Now, this statement links the idea of one-to-one correspondence, which can be defined as the ability to match one object to one corresponding number. And that's basically what we will be dealing with here. Now, another perspective of one-to-one correspondence is that it deals with the ability to match each member of one set of items to a member of an equivalent set of those same type of items. But here we won't be dealing with two different sets. Again, it's just the idea of matching the object to the corresponding number.
Now, part b is two statements; understand that the last number name said tells the number of objects counted, and then the second statement is the number of objects is the same regardless of their arrangement or the order in which they are counted. Now, these statements are connected to the concepts of cardinality and conservation of number. For the first, cardinality refers to the number of items in a set. So, in turn, cardinal numbers are whole numbers that are used to specify the number of items in a set or group. So, cardinal numbers...we're really talking about quantities. We're not dealing yet with number in terms of being part of, like the real numbers, or being a position on a number line. We are dealing with numbers as they refer to quantities.
The second statement is more about conservation of number. Now, this refers to the idea that if we have a group of objects and we rearrange them, the number of objects is still going to remain the same. Then the last part, part c, deals with both the concepts of cardinality and cardinal numbers, the idea that as you state successive numbers, the quantity becomes one larger.
Let's look at part a. The key here is that when students are counting, they need to match the physical object with the number that they're saying. So, here, when they say the number one, they associate that number with this object. Likewise, when they say the number two, they associate the number two with this object. We go with three, they associate three with this manipulative, and then the same thing with four. What you'll want students to do is mentally start having this type of picture, where again, this particular manipulative is associated with the number one, this one. And now we have a quantity of two, so this object here is associated with the number two and so forth.
Let's go to part b. Now here, for the first statement, students need to understand that the last number name said tells the number of objects counted. So we start off with one, and then two. So we said the two, so that's how many we have. Then we say three, and so we have a total of three here. Likewise, here, we say four, so we have a quantity of four. Now, what might happen with kindergarten students is that they might have just finished counting off these four, and then you ask them "well, how many balls, or how many objects or whatever do we have here?” And the students might turn right around and start counting over, you know, one, two, three, four, instead of realizing that, over here, when they counted, four, that that's the last item. So that's how many we have.
What might happen here is that there's no comparison. So, this is an idea maybe to try. Okay, we have a quantity of one. Then we have a quantity of two, but notice that we're doing it as separate sets instead of just changing the one set that we start off with. Likewise, here we have three, but we put it as a different set, so that again, the students can see that the last number that we count is in fact the total quantity that we have. So now we have four, then five, and then six. So, again, students can look and make comparisons and again realize that the last number that they counted is in fact the total quantity that's in that particular set or group.
Now let's look at the second sentence in part b. The number of objects is the same regardless of their arrangement or the order in which we count them. So if we start off with say, six items, and we count them, you know, left to right...one, two, three, four, five, six. And then, let's say we still have the same arrangement where we count them differently. We start here, and we count this way...one, two, three, and then four, five, six. Then what we can do is start putting them in different arrangements, like here, and we count this way, counterclockwise...one, two, three, four, five, six, or rearrange them this way and count differently, sort of haphazard. Rearrange them, still in yet another different way, and another different way, and another different way of counting. But, the point is that in all of these instances, we still have the same quantity of six.
The last part of the standard states: understand that each successive number name refers to a quantity that's one larger. So let's say we start off with three, and we're going to compare it to four, so now we add one, so that's one larger. And then the same thing with five...we add one more. That's a five, so we have, again, it's one larger than the previous four. But just like the previous examples, students still may not see this because all we're doing is taking the same set and just adding one more, and they have nothing to compare to. So, again, something like this might work where we're going to compare three to four. So we start off with another set of three, and we add one more to it. Now students can easily see than this set here is in fact one more than the previous set. So, four is one more than three.
Let's say we're comparing four with five. Again, we start off with four. Here's another set of four, but we add one to it, and again, now students can see that we do in fact have one more here than we did over here. And one more example, we're comparing five to six. So we start off with five, so we have another set with five, add the six to it. And now again, they can see that this set over here is in fact one larger than that one over there, that when we have a six, we do have one more than five. And so doing something like this, again, students can visually compare and be able to tell that, yes, I do in fact have one more here than I had over here with the three. We have one more here with the five than we did with the four, and one more with six than we did with five.