Other Kinds of Counting (#2 young learners series) March 30, 2020

In the previous blog post I talked about the most fundamental ideas about counting and how we can help children develop cardinality. Until children have grasped this understanding, there is nothing else to be learned in mathematics. Nothing. They need to know the names of numbers, the order they come in, and that the last number names the quantity. However, we don’t want a child’s mathematical development with numbers to get stuck in counting. It is important in the beginning but we want them to develop a rich understanding of numbers apart from counting. I will share more about that later.

 

Here are some other kinds of counting that you can do with your children as their knowledge of numbers develops.

 

Counting Up and Back

Counting up to and than immediately back down from a target number can be a rich exercise. It can be done with movement (such as clapping, jumping, air fist-pumping, etc.) in a rhythmic manner. Or you might move a set of objects from one line to another as you chant, “ 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 3, …”.

 

Another version is to count up and back between two numbers such as 8 and 13. As before, keep a rhythm.

 

What about counting between numbers such as 28 and 37? That raises the difficulty level again.

 

Counting On

Counting on is the practice of starting the counting sequence from a number other than 1.

 

WITH COUNTERS: Give your child a collection of small counters and have them line them up. Then have them count a few of them, hide them under their left hand and count from that number. For instance, if they have 10 counters, you might have them hide 4. Point to the hand and ask how many are there under the hand. (Four.) Then say, “Let’s count like this: four (pointing to the hand), five, six, …”.

 

Instead of covering the counters with a hand, you can use a piece of paper, a small bowl turned upside down, or something else that is handy. The important idea is to know how many are hidden (because they were counted first) and continue “counting on” from that number.

 

Older children can practice counting by 1’s from other larger numbers such as 37, 142, etc.

 

Other ideas for counting:

Most kindergarten-aged students can also practice counting by 5’s and 10’s. I have found it very useful to use hand motions for these. For instance, when counting by 10’s you can flash all 10 fingers every time the next multiple of 10 is said, thus illustrating the increase by 10 each time.

 

Similarly, when counting by 5’s, flash the fingers of one hand each time a number is said. Students may or may not have full understanding of the concept of “tens and ones” here, but still practicing the counting sequence of words is useful.

 

Counting by 2’s and 3’s are both useful exercises. These can be done using counters and/or by looking at numbers on a 100 chart.

 

IMPORTANT: Count past 100!

When you are counting up to 100 by 1’s, 2’s, 5’s or 10’s do not stop counting (as may be your instinct) at 100. I have asked dozens of children what comes after 100, and their answer is almost always “200” no matter what they were counting by. It is very important to continue the counting sequence past 100:

…97, 98, 99, 100, 101, 102, 102, …

…96, 98, 100, 102, 104, 106, …

…85, 90, 95, 100, 105, 110, …

…70, 80, 90, 100, 110, 120, …

 

Happy counting!

Mathematically yours,

Carollee

Counting: the Beginning of all Mathematics (#1 young learners series) March 29, 2020

I am inspired to begin a series of posts about math for younger children. For each topic we will discuss it on several levels, so wherever your child is on the math journey, hopefully there will be some ideas for you to engage with him or her.

 

It is important here to recognize that mathematics is reasoning and not memorization. While it is useful to commit to memory certain facts and procedures, it is essential that these facts and procedures develop with understanding. We want children to build networks of ideas that allow them to cope with novelty and solve problems they have not previously considered or encountered. Just memorizing facts and procedures can actually be debilitating, as it bypasses the important activity of building inter-connected mathematical ideas.

 

We begin this series with counting which is, indeed, the first foundation in math.

 

Counting involves at least two separate skills: saying the numbers in order and connecting this sequence in a one-to-one manner with the items being counted.

 

First, the child must learn the counting words in order, so say them over and over and over with the child. Make if fun, of course; a game, if you will, but it is in the repetition that the child will learn them (as is true for so much of her early learning).

 

Often we start with counting to 10, but don’t hesitate to count farther. (Of course this may vary with the age of your child. One may want to count a shorter sequence with a two year old than with a four year old.) In English there are no patterns at all in the first 12 numbers, and even in the teens the pattern is difficult for many children to recognize it. The obvious pattern of numbers does not really become apparent until we get to the twenties. Children must just memorize the counting sequence of numbers (yes, one of the places memorizing comes in handy!).

 

Saying the order is important but equally important is that the child is able to count one item for each number. Most of us have seen a child “counting” some items like blocks and randomly touching the blocks either more slowly or more quickly than actually saying the numbers. They seem to know touching is part of the process but have not understood the “one touch, one count” idea, otherwise known as one-to-one correspondence. How do they develop this? By doing it over and over. Experience and guidance play major roles in the development of counting skills, so counting often with your child benefits him greatly.

 

Matching the spoken counting words one by one to objects is generally easier if the objects are such that can be moved compared to those that cannot be moved. Movable objects allow the child to actually slide the object away from the ones yet to be counted. If objects to be counted are in a picture then a set that is ordered in some way is easier to count than a randomly displayed set. In the case of the latter a child often either misses items or tries to count them again.

 

There is one more important aspect here to consider, and that is that there must be meaning attached to the counting process. There is a difference between being able to count and knowing what the counting conveys. When we count a set of items, the last number word we say is used to represent the magnitude or the cardinality of the set.

 

When children understand that the last counting word said names the quantity of that set, they are said to have developed the cardinality principle. How can you tell if your child has developed the cardinality rule? Give them a set of objects and ask them, “How many are there?” If they emphasize the last number they likely have developed this. For example, they might say, “One, two, three, four, five. There are five.” I have asked children to tell me how many items there are and had them count the items, look at me, and when I repeat the question, “How many are there?” have them begin to count again. In that case the child seemed to interpret the question as a command to count rather than a request for the quantity in the set.

 

The only way to develop cardinality is by counting! Two activities are helpful here.

• Have the child count several sets of items that all have the same number of things but the items themselves are very different in size, e.g. six apples verses six grapes. Discuss how they are alike and how they are different.
• Have the child count a set of items, then rearrange the items and ask, “How many now?” If the child sees no need to count them again they likely have connected the cardinality to the set regardless of the arrangement of the items. If they want to count again, discuss why they think the answer is the same.

More on counting next time!

In the meantime, count often! Count everything!

Mathematically yours,

Carollee

 

Early Counting: the Foundation of Math May 12, 2014

The meaning attached to counting is the most important idea on which all other number concepts are developed.

Counting Involves at Least Two Separate Skills:

• A child must be able to produce the standard list of counting words in order: “one, two, three, etc.” This must be learned by rote memory.
• The child must be able to connect this sequence in a one-to-one manner with the items in the set being counted. In other words, each item must get one and only one count. This important understanding is called one-to-one correspondence.

Meaning Attached to Counting:

There is a difference between being able to count as explained above and knowing what the counting means. When we count a set, the last number word used represents the magnitude or the cardinality of the set. When children understand that the last count word names the quantity of the set, they are said to have the cardinality principle.

Give a child a set of objects and ask, “How many”? After counting, if the child does not name how many are there (as, “There are 7 of them,”), then ask again, “How many?” If a child can answer without recounting, it is clear he or she is using the cardinal meaning of the counting word. Recounting the entire set again usually means that the child interprets the question “How many?” as a command to count.

Almost any counting activity will help children develop cardinality.

• Have the child count several sets where the number of objects is the same but the objects are very different in size. Ask the child to talk about this.
• Have the child count a set of objects, and them rearrange the objects. Ask, “How many now?” (If the child sees no reason to count again, likely the child has a good sense of number and has developed cardinality.)

Happy counting!

Mathematically yours,

Carollee

Kindergarten Collaboration: Synergy at Work January 29, 2014

Many thanks to the three K teachers at Bert Ambrose Elementary School who invited me to participate in their math collaboration afternoon. It was wonderful to see what two hours of “math chat” did to energize and inspire them. One of the teachers admitted right at the beginning that when he saw me, he felt he ought to do a “woohoo!” (since that comes out of me so often regarding math) but that it just wasn’t in him. By the end of the collaborative session the “woohoo!” was back and he and the two others were excited to go back to their classrooms and integrate more mathematics into both “centres” and the “free play” parts of the school day.

One of the things we talked about was the need for kindergarten (and pre-K) children to count. Saying the numbers in order is important, as is counting in a one-to-one correspondence (one count for each item).  It is also important for children to realize that the last number spoken names the number of items in the set (a principle know as “cardinality”).

GRAB A HANDFUL:

Here is one way to give students the opportunity to practice counting. Place at the centre several containers of things which can be counted. These can be blocks, large beads, erasers, plastic animals, etc. Students are to take a handful from one of the containers, count the items, and write the number of items on one of the hands on the recording sheet provided. Alternately you can provide only a single kind of counter and have students vary the amount grabbed each time (i.e., grab a large handful or a small one).

Download the recording sheet for Grab a Handful here.

I hope you will give the activity a try!

Mathematically yours,

Carollee

12 Days of Christmas: Canadian Style December 11, 2013

If you are looking for a great Christmas-themed book for the young (or young at heart), you will want to check out A Porcupine in a Pine Tree: A Canadian 12 Days of Christmas (written by Helaine Becker, illustrated by Werner Zimmerman). Of course, the math connection is in the counting and in the adding up of all the items being given during the 12 days – all totaled it adds up to quite a tidy sum.

Certainly, any book with numbers tends to make me smile, but this one more than most. It is seriously delightful! It offers Mounties frolicking, squirrels curling, moose calling, hockey players a-leaping, and more.

The hardcover version is available from Amazon.ca (not Amazon.com) for the bargain price of $12.26 (price valid as of date of posting). Actually, for just a bit more, you can get the gift set version with the hardcover book plus an adorable little plush porcupine (sounds like an oxymoron doesn’t it!)

I am not affiliated with the author or publisher – I was given a copy of this book last year for Chrismas and I love it! I know you will, too.

May your days be merry and bright!

Mathematically yours,

Carollee