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Math Talks – Episode One

Math Talks – Episode One

For the first episode of Math Talks I used my daughter, Lilly, as my student.  She tends to be my educational guinea pig for many, many things.  Lilly is entering first grade but thinking about numbers more like a second grader.

Here is a breakdown of the discussion:

The talk begins with a dot pattern image flash.  This is an activity that helps kids to subitize (recognize quantities without counting).  I started with this activity for two reasons.  First, as a warm up and a way to get going.  More importantly, I did it to reinforce the question I posed later.  I wanted her to think about the ways to make 7 so my dots were different patterns of 7 (six and one, three and four, five and two).  Later I was going to ask her to try to think about 7 as a group of two and five more so I knew this activity would get her thinking in this way.

Math Talks 1 Dot Image 2Math Talks Dot Image 3Math Talks 1 Dot Image 1

When I explain the activity to Lilly, she immediately asks how fast I’ll flash the dot cards.  You might notice my misspeak as I tell her that it won’t be so fast that she can’t see them and then I meant to say, “. . . but not slow enough that you can count them.”  It is important that kids understand the idea isn’t to count the dots.  They idea is to either immediately know the quantity (sets up to five) or to visualize the picture of known quantities and combine them (i.e. see two and two and two and one and that’s six and one more and that’s seven).

An important part of this section of our math talk is the follow up questions I asked.  By asking her how she saw the dots and how many, it confirms and cements the idea that the total can be thought of as two parts combined.

Math Talks 1 Question

As we get into the meat of the talk, I made sure to write my math story in a context that was interesting.  Lilly’s show of choice this summer was Wild Kratts so I used that as a hook to get her interested and engaged.  The question I posed was an add to, change unknown problem.  That’s teacher speak for an addition word problem in which one of the addends (the quantity you are asked to add to the first quantity) is unknown but you know the first addend and the total.  This problem is trickier than when you known both addends because you have to count up or add up to the total and attend to how many it took to get there.  This is a problem first graders should have skill with by the end of the year.

What you see in the video is a visual of her thinking process while solving.  I used the Show Me app to record her steps and did so after our discussion.  It’s important to note that what you see is an exact match of what she wrote on her paper.  After our talk, I spoke with Lilly about what she saw in her head when she was solving problems like this.  She shared that she is thinking in numbers rather than in pictures.  While that is good, and where she should ultimately be thinking, it is important that she have the ability to work with a picture model as well.  In future talks, I’ll work on modeling her thinking with a picture and relate it to the number models with which she is comfortable.

We wrapped our talk with a discussion about challenge.  This summer I’ve spent a lot of time learning about growth mindset.  I really want to instill in my child, and others for that matter, the importance of attempting challenge, using mistakes as opportunities to learn and the value of effort.  I’m doing that by emphasizing those qualities in the praise and feedback I give.

Here is the podcast and the youtube video is below.  Enjoy!

For more Math Talks, visit the Math Talks page on the menu bar.

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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!

The simplest math tool you never knew existed . . .

How are you feeling about your child’s knowledge of addition and subtraction facts?  Are you worried that he hasn’t memorized them yet. Worried that his teacher isn’t focusing on facts enough?  Well, if you’re older than 18, you probably learned math very differently than your child is learning math today.  Here are a couple of tidbits that differ greatly from math instruction in “our day” and math instruction today:

1. The Common Core State Standards for Mathematics specify that children should learn their addition and subtraction facts through strategy work, not from drill and practice.

2. The standards outline that, by the end of kindergarten, kids should “add and subtract fluently within 5” which means that they should be able to add or subtract any numbers up to 5 and get the correct answer with relative speed.  By the end of first grade kids should be able to do this within 10 and that not until the end of second grade should they “know all sums within 20 from memory.”

So how can you help your child with math homework if you don’t know how they’re supposed to learn their facts?  I’m just going to lay it out on the table – talk to your child’s teacher first if you’re not sure what to do or how to help.  Also, remember that your job is to help your child “figure it out” not to do the thinking for him.  Beyond all that, here’s an amazing tool you can create and use to help your child at home.

Ready for it?  It’s called a . . . Rekenrek.  If it’s easier, call it a number rack.  Literally translated it means arithmetic rack and it was created in the Netherlands.  It was invented a good time ago to help children understand the number system, develop number sense and improve computational fluency (learn addition and subtraction facts).

Rekenreks are easy to build.  All you need is 2 black pipe cleaners, 10 red pony beads, 10 white pony beads and a piece of foam board (a piece about 4 inches by 8 inches is plenty big).  Cut 2 slits on the short sides of the board, evenly spaced between each other and the edges of the board.  String 5 red and 5 white beads on each pipe cleaner and pull through the slits on the foam board.  It is important to make sure that the bead color lines up on both pipe cleaners.  When completed, all beads should be pushed to the right and the beads on the left should be red and the beads on the right should be white.  See the images below.

The simplest math tool you never knew existed . . .The simplest math tool you never knew existed . . .The simplest math tool you never knew existed . . .

 

These are easy to build and there is value in having your child build it (with guidance).  On a Rekenrek, each bead represents 1 so by building it himself, your child will see the embedded structure of 5 red and 5 white to make 10 and the overall structure of 10 and 10 to make a total of 20.

So here’s how the Rekenrek can help your child.  When working on facts, typically your child only sees the numeral (the symbol for the number).  Using the Rekenrek allows your child to see the quantity attached to each number.  Through practice your child will eventually come to just know these facts.  What’s more, your child is likely learning strategies for adding and subtracting.  For example, many children learn doubles  (4+4, 5+5) automatically.  Children can use this knowledge to solve near doubles.  Children recognize that in the problem 6+7, 7 can be split into a 6 and 1 to make the problem 6+6+1.  The problem with this strategy is that because children easily memorize doubles they don’t “think” about what the fact means.  So asking them to use the double to solve a close fact isn’t always intuitive.  Demonstrating this on the Rekenrek lets your child literally see the double plus 1.

 

The simplest math tool you never knew existed . . .

Work with Rekenreks support much more basic understanding of number as well.  Rekenreks help children subitize (recognize small quantities without counting) and work with composing and decomposing numbers.

The simplest math tool you never knew existed . . .

Below are links to 2 activity guides (meant for the classroom) you could use with your child to support understanding of numbers and operations.  They are free!

Using the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics

From the Math Learning Center – an in-depth series of lessons that builds from most basic number sense skills to addition and subtraction activities.

Mathrack Activities and Directions

From Mathematically Minded – a brief description of activities as well as assembly instructions.

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!

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