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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!


First and Ten

I was scrolling through Facebook last night, procrastinating going to bed, when I was reminded that this is a good opportunity to talk about math and football. Knowing the partners that add up to 10 is a huge concept in kindergarten math that very much supports first grade and second grade mathematics in terms of addition and subtraction.

How does that apply to football?

Image courtesy of antpkr at

Image courtesy of antpkr at

Every time the quarterback passes or hands the ball off he gets a portion of 10 yards completed and the commentators always talk about how many more yards to a first down. If you are watching the Super Bowl this weekend with your kindergarten, first, or second grade child consider throwing a little math into the mix. Ask, “How many more yards to 10?” or “How many more yards to the first down?”

This is the perfect chance to bring out the math in our everyday lives!

Building a better shapes book

I am so excited to repost this blog. Christopher Danielson, who writes Talking Math with Your Kids,  has created a super shape book that is accessible for all ages. What’s great about the book is that there are no right or wrong answers. This book is all about explaining and justifying your thinking. I can’t wait to share this resource with my kindergarten teachers who are just about to begin their geometry units!

When I first read the post I was sitting next to my own kindergartener (the one who I take home every night) and I thought, “Hey, I’ll try this out on her.” We scrolled through each page and had a great conversation around why we chose each shape. It was interesting, when I disagreed with her, choosing a different shape for a different reason, she was pretty willing to go along with my idea. I asked her, “Who’s right?” and she quickly said, “You must be, I guess!” It took a few pages of convincing her that we could both be right and by the end it was a bit of a game to see just how different our thinking could be. I especially appreciate the developmentally specific prompts given in the post so families and teachers can use the book with varying age groups. I’ve included some additional supplemental pages on my Downloads page if you want to add to the book.

When I printed the PDF the pages came out, with a border, to be a 7 3/4 square so I tried to size my supplemental pages to fit the originals.

Please take the time to read more posts from Talking Math with Your Kids. So many goodies!

Talking Math with Your Kids

There are many shapes books available for reading with children. Most of them are very bad. I have complained about this for years.

Now I have done something about it.

Most shapes books—whether board books for babies and toddlers, or more sophisticated books for school-aged children—are full of misinformation and missed opportunities. As an example, there is nearly always one page for squares and a separate one for rectangles. There is almost never a square on the rectangles page. That’s a missed opportunity. Often, the text says that a rectangle has two short sides and two long sides. That’s misinformation. A square is a special rectangle, just as a child is a special person.

After years of contemplation, I had a kernel of an idea the other night. The kids are back in school before I am, so I had some flex time available. One thing led to another and…

View original post 405 more words

Checking Your Work

Homework is a tricky subject in our house.  One family member is an elementary teacher, one family member is in kindergarten and one family member has chaperoned a field trip, helped on the weekends and attended elementary school many, many years ago.  You can imagine that a certain someone – ok, it’s the dad in our house – tends to get told what to do all to often when trying to help out with homework.

This week, as I was fixing dinner, Adam sat down to help out with the weekly math homework.  Things are pretty simple at this stage of the game.  We’ve seen lots of tracing numbers 1 through 5 and some matching numerals with dots or cubes.  This week’s homework asked the kids to extend a bug pattern.  Lots of cutting and pasting – right up kindergarten alley.  Since patterning didn’t seem to require too much nuanced discussion, I happily let Adam take over.

Math Practice 6

After assembling all of the materials and lots of cutting, they set about figuring out what bug came next in the pattern.  Then, the magic happened.  I heard Adam say, “Check it to make sure it sounds right.”  Ah, be-still my heart.  Without even knowing it, he was instilling in our barely 5 year old daughter, an overarching habit of mind – oh so important to mathematical proficiency.

Outlined in the Common Core State Standards of Mathematics are content standards (what the kids should know) and the practice standards (how kids should “behave” with math).  The content standards are what many think about and, unfortunately, the practice standards are often brushed aside.  This might be be due, in part, to their location in the standards document but it is also because the wording tends to be a bit complex.  Many teachers have been working on trying to understand the eight math practice standards and apply them at the level they teach.  While the ideas are big, and extremely important, at the earliest grades, these practice standards can look quite simple.

Math Practice 6, Attend to precision, emphasizes precise use of math language and vocabulary as well as accuracy.  This obviously looks different at different levels but with kindergarten, “Checking your work” can elicit this standard.  Every time you ask your child to check his/her work or praise him/her for doing it independently, you are reinforcing the idea that review supports precision.  If you think about it, you can apply this thinking to a variety of subject areas.  How many times did your teacher ask you to reread your writing looking for errors and opportunities to improve?

Just like everything you do with your very young children, establishing routines early can lead to habit.  Maybe I’ll let the novice take over more of the responsibilities with math homework!

You can learn more about the standards for mathematical practice from Dreambox.  If you want more in depth information you can get it from Think Math and Illustrative Mathematics.

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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