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Count the World

Counting the WorldI know church is supposed to be a reflective place.  Can I help it if I happen to reflect on math?  I hope not!  As we were leaving one Sunday morning, there was a significant back up on the stairs.  This is frequently caused by the very dainty, and somewhat fragile church goers who live in the retirement home next door.  On this occasion however, it was a different kind of dainty parishioner – a toddler.  As she carefully graced each step with both feet, her dad peered behind his shoulder and gave us all the, “I’m so sorry but I know you understand.” look.  He then returned his attention to his daughter and began counting.

The act of counting steps, or anything tangible, is a vital component to supporting young children while they mathematize their world.  Mathematizing is really just about bringing out the math that is inherent in the world and space around us.  That things (anything really) can be counted is mathematizing.  Kids mathematize when, given the option of a portion of cake, choose what they perceive as the larger portion.  They learn about volume and dimensions when trying to build towers with varying sized sets of blocks.  They mathematize when they’ve calculated that there is only one “fun” swing on the playground and the likelihood of loosing it is good so giving it up to play on something else is not an option!

Early counting and grouping is of particular importance.  The action of that father counting steps with his toddler supports her development of cardinality.  Cardinality is the idea that number and quantity are related.  Each number represents a set of that many things.  While this is obvious to you and me, it is not clear to the youngest mathematicians of our world.  Watch a very little child, 2 or 3 years old, try to count a set of objects.  He may understand the idea that he is supposed to say the count sequence while pointing to objects but he may not yet know that each number he says has to correspond to one of the items.  And not only that, each number has to correspond to a different item.  He doesn’t know you can’t count it twice!  This one-to-one correspondence develops over time and through repeated opportunities to practice with guidance.  As the toddler on the steps felt each count underfoot, she was developing one-to-one correspondence and cardinality.

These ideas of cardinality and one-to-one correspondence are components of the kindergarten Common Core State Standards for Mathematics.  They are really quite basic but so vital for mathematical proficiency.  While they “live” in the world of kindergarten, they are skills that can and should be developed much earlier.  At home and at daycare, adults can help children mathematize their world by subtly applying the count sequence to objects.  Imagine all the times you could say, “Let’s count them!”  Rocks, legos, beads, toys, shoes, birds, swings, diapers, grapes, forks, blocks, friends, etc.  Adults can easily teach children to touch and count each object and to help them distinguish between the counted and uncounted by demonstrating pushing the counted collection aside, one by one.  While one-to-one correspondence takes time and fine motor skills to develop (so don’t fret if it takes awhile), modeling of this behavior is invaluable.

One more thought on cardinality.  When children finish counting a set of objects and are done saying the count sequence, we assume that they understand that the last number said represents the total in the group.  This is not necessarily the case.  If you follow up a counting sequence by asking, “So how many?” you may notice that your child repeats the count sequence.  This is a good indication that he does not yet understand this component of cardinality.  You can’t force him understanding, that will take time and experience,  but you can model it yourself.  When you model you can say, “1, 2, 3, 4, 5.  Oh, there are 5 markers!”  This indicates to your child which of the numbers represents the total of the group.

Happy counting!

Restaurant Math

I was never amazing at packing enticing activities for my little one while waiting at a restaurant.  I’d always remember, right as we handed our waitress the menus, that prepared moms packed a goodie bag to keep their children occupied.  Luckily, I learned how to make due with what we had.  As my daughter hit toddlerhood, we began playing games with the sugar and jelly packets often found at breakfast diners.  From this, restaurant math was born.

The idea of this activity is to support a child’s ability to subitize (seeing a quantity without having to count it).  This is a vital skill and kids are able to subitize small quantities from a very early age.

So the idea is this . . .

Select a few different sugar or jelly packets.  Quickly scatter them in groups for your child to see.  Initially, put out one or two items and ask, “How many?”  Your child will likely be able tell you without having to count.  Here’s the most important part.  After your child tells you how many, always follow up with, “How do you see it?”  This will seem silly with just 1 or 2 packets because it’s pretty obvious, even if you’re 3!  Even so, it’s ok for your child to learn to say, “I can just see it!”  When you start to increase the quantity, it helps your child develop a  verbal pathway to explain his/her thinking about the connection between quantity and adding.  So if you put out 5 packets, your child might say, “I see 3 and 2.” or “I see 4 and 1.”  The colors you use and the arrangement you use will impact what your child sees.  When you first start increasing the quantity, separate the items with a lot of space and by color.  This will help your child “see” the groups.  As your child gets stronger at subitizing, you can put the items in one group.  Think about dice patterns.  Here are some photos of what this might look like early on.

 

"How many?" "5!" "How did you see it?" "I see 2 purple and 2 red.  That's 4 and 1 orange is 5!"

“How many?”
“5!”
“How did you see it?”
“I see 2 purple and 2 red. That’s 4 and 1 orange is 5!”

 

"How many?" "Um, 6!" "How did you see it?" "I saw 2 blue and 1 pink.  That's 3.  And 3 yellow are 6!"

“How many?”
“Um, 6!”
“How did you see it?”
“I saw 2 blue and 1 pink. That’s 3. And 3 yellow are 6!”

Even though your child may be able to see the groups, don’t expect him/her to know the total without counting.  Early on, children will need to count to determine the total.  In the picture above, your child might say, “I see 2 blue and 1 pink, that’s 3.  There are 3 yellow!  So 1, 2, 3, 4 ,5, 6!”  As children become more familiar with the different groupings, he/she will begin to recognize how smaller groups are combined to create a larger total.

 

For more information on early subitizing, read the article, Beyond Counting by Ones, by Deann Huinker.  This article is full of activities you can do with your child.

What activities have you invented to keep your kiddo occupied?  I’d love to hear about it!

Counting by Twos

Counting by twos supports a variety of other mathematical tasks.  Counting by twos allows students to count more quickly.  More importantly counting by twos supports doubling, pairs and grouping for multiplication.  Here are some activities you can do to help your child learn to count by two numbers and understand what it means to count by two.
 
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  • Take turns counting.  You say “1”, your child says “2”, you say “3”, your child says “4”, etc. through 20.  Do the counting again but this time you whisper (your child still says the number out loud).  The third time you just mouth the number but your child still says the number out loud.  This will reinforce the counting by twos skills.
  • Think of a toy that is both small and interesting, that already belongs to your child.  Legos, blocks, cars, etc.  Play a game where you set out a pile and ask your child to figure out how many.  Your child should touch two and count two as he/she pushes them away from the pile.  Also, when he/she is done counting, ask if every item had a partner.  Ask if the group was even (each item had a partner) or odd (one item was left without a partner).  Take turns selecting and then counting piles.

Penny Stacking

Penny StackingI still remember the giant tootsie roll piggy bank that held all of my pennies for college.  I’m pretty sure that those pennies didn’t pay for more than a semester’s worth of books but they did create some very fond memories of my childhood.  When I was a little girl, my dad was the best playmate I ever had.  Sometimes I think he had more fun playing than I did.  One thing we did was play with pennies.  Nope, I didn’t lack toys.  It was just something different to do.  Somehow he knew, without formal training, ways to incorporate math into play.  Here are a few penny ideas (with lots of credit to my dad):

  • Count them . . . So this is obvious!  But, since the value of a penny is 1 cent, it’s a perfect item to count.  Starting a penny collection is both interesting to kids and financially responsible!

Counting Pennies

  • Stack them in towers of 5 or 10 . . . This reinforces skip counting which is a vital skill to counting mixed sets of coins (down the road).  It also shows the difference (in height) between 5 and 10 and lends itself to the idea of 5 being half of 10.  Build two 5 towers and one 10 tower, then let your child explore what happens when the five towers are combined.  Compare with the height of the 10 tower.

Penny Towers

  • Build structures with them . . . Pyramids, mazes, designs, anything!  This activity helps kids see what can be done with circular objects and consider structure.  After building, count the pennies to see how many it took to build each structure.

Penny Building

  • Tell stories . . . Young children can solve simple word problems, especially about food and other things at their interest level.  Money is a bit trickier because it doesn’t mean to kids what it means to us.  If you use pennies to solve word problems, focus on the coin, not the value.  For example, say, “Lilly had 5 pennies.  Then she got 2 more.  How many pennies does she have now?” instead of, “Lilly has 5 cents.  Then she earned 2 more cents.  How much money does she have now?”  At early ages, coins are more about collecting the object than the financial value they hold for adults.
  • Pretend to “buy” things around the house . . . You can help your child develop the idea of using money to buy something.  They see you buy things all the time but what are you usually using to pay?  Even if it isn’t your credit/debit card, you’re probably not paying in coins!  But kids idea of money and what they will learn about money early in school revolves around coins.  Set up a store with things around the house.  You can even label items with their value (5 cents, 2 cents, etc.).  This would help them with recognizing numbers and connect the numeral to a quantity (the fancy word for that is cardinality).
  • Build a penny staircase . . . So this is similar to the idea of using pennies to build different structures.  The important thing about the staircase is the increase by 1 nature.  This visual shows kids clearly how our counting sequence refers to quantity (cardinality again) and how when we count forward, we are actually adding and when we count backwards, we are subtracting.

Penny Staircase

Can you think of other fabulous things to do with pennies?  Please leave a comment.  I’d love to hear your ideas!

Understanding Teen Numbers – Kindergarten/First Grade

This video shows a quick activity, families (or teachers) can do at home (or school) to support understanding of teen numbers.  I used what I had – buttons, paper, marker and a four year old.  This activity is more appropriate for kindergarten age students or first grade students still developing their sense of teen numbers.  It gets at the idea of ten being not only a group of ten ones, but that it could also be considered one ten, an important concept for children to grasp to understand place value. It moves from using concrete objects, to representing them on a ten frame and then using the number symbol for teen numbers. It also helps children see the “hidden” 10 in teen numbers, something not obvious since we say “teen” not “ten”.

Credit for the idea of this activity goes to Melissa Hedges and Beth Schefelker who facilitate the Numbers and Operations in Base Ten, K-2 module for the Brookhill Institute of Mathematics.  They are amazing teachers and mathematics leaders!

Here are a couple of items to note:

  • This activity requires that your child/student know how to count and read numbers to 20.
  • After counting, it is important to ask, “So how many?”  This indicates that your child/student understands an aspect of cardinality that the last number you say represents to total collection.  If your child needs to recount the collection each time, this task might be too advanced.  Working on basic counting and one-to-one correspondence would be more appropriate.
  • If your child is an older kindergartener, a good question to ask about the group of 10 ones would be, “What else could we call this group?”  The idea is to help your child think flexibly about 10 as a set or group.  Ten can be considered 10 ones (that can be broken apart when regrouping in double digit subtraction) or ten can be 1 ten that can be thought of as a clump or group (so that 2 can represent 20 in the numeral 25).
  • The steps of this activity are important.  Children/students move from working with concrete objects (i.e. buttons) to representations (ten frames – adding a step where students match a ten frame to their button ten frame would be good) and then to the symbolic (using the numerals to represent the teen numbers).  For children/students who have worked with or have understanding of number bonds, number trees or equations, ask the child to represent the activity with one of these abstract concepts as a concluding step.

The handmade ten frames are available for free as (nicer) PDFs on  Teachers Pay Teachers.  The place value cards are available there too but I had list them at a cost (too many pages).  I’ve added a new page for downloads so you can find the place value cards right here for free!

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