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.

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.

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.

The Language of Multiplication

Recently, my kindergartener, Lilly,  came home and declared, “I’m bad at times.”  Let me clarify a couple of things.  First, when she said that, she didn’t mean that she was occasionally evil.  She meant she was bad at multiplication.  Second, we don’t do, “I’m bad at ___.” in our house.  We work pretty hard to send the message that we’re all capable of most things if we only apply ourselves.  Lilly knows this, but as the youngest kid in her class, and likely her entire grade district-wide, she’s constantly comparing herself to classmates who are much older and wiser.  She’ll tell you all day long, “You just have to practice, practice, practice.”  And at times, she actually believes this.  Over spring break she was determined to skip a bar on the monkey bars and by jove, she did it.  Other times, she needs some convincing.

So your first thought at a kindergartener declaring her deficiency in multiplication might be, “Why are they doing multiplication in kindergarten?”  I assure you, they’re not (at least not formally).  This statement came from a classmate, whose older brother just went through third grade.  Third grade is the year students are supposed to know from memory (and by the end of the year) the product of all single digit numbers.  Needless to say, her classmate spent a year listening to his brother complete math homework and picked up a little – perhaps a lot – here and there.

My first thought was, “You’re not bad at times.  You’re using the wrong language.”  Most of us learned our multiplication facts from drill and practice.  We learned tricks to remember the nines and were always stumbling over the sevens and eights.  No one bothered to explain the meaning behind multiplication until after we’d memorized our facts.  Turns out, if we do the reverse, understand before we practice, memorization comes a whole lot faster.  And better.  Part of understanding is knowing what “times” means.  I sat my daughter down and explained that “times” just meant “groups of”.  We practiced this way:

Me: “2 times 3 just means 2 groups of 3.”

Lilly: “Oh, I know that!  That’s just 3+3.  It’s 6!”

Me: “Let’s try another one.  4 times 2.”

Lilly: “4 times 2 . . . so that’s four groups of 2.  So 2, 4, 6, 8!”

When multiplication language is understood, especially the language of operations, competency and fluency increases.  Learning multiplication requires an understanding of the language (times, factor, product, etc.) and an ability to think flexibly about numbers.  Most kids are comfortable multiplying by 1, 2, 5, and 10.  That’s because they all have had practice skip counting in these number patterns.  If they’ve had good instruction, they understand that counting by 2s is just adding 2 each time.  This connects to student understanding of multiplication as repeated addition.  The other facts anchor around knowing these facts and are based on students being able to decompose (break) numbers into parts.  There are classroom appropriate posters that are useful for parents to use as references for helping their children at home too.

Practice with multiplication strategies and work with word problems allows third graders to develop an understanding of multiplication, which in turn supports recall of “basic facts”.  Because kids have had so much practice developing a foundation of multiplication in K-2 in looking for patterns, working with repeated addition and arrays, they are able to more quickly develop fluency for multiplication.

If your little one isn’t yet learning multiplication, don’t hesitate to pose questions that support multiplicative thinking.  As I was writing this post, I took a break to put my daughter to bed.  We finished reading The Lion the Witch and the Wardrobe earlier in the day.  At bedtime, Lilly asked how many pages we’d read.  I wasn’t sure, and since I was in the middle of writing this post, I couldn’t help but respond, “I’m not sure how many pages but, if we read 4 chapters and each chapter was about 9 pages, how many pages do you think?”  She said, “So 9+9+9+9 . . .”  She understood the situation and was thinking multiplicatively.  I’m assuming, with more conversations like this, she’ll be ready to go when she gets some formal instruction in multiplication in three years!

Count the World

I 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 FreeDigitalPhotos.net

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

Just the Basics – Quick Reference for Math Support at Home

So it’s true that there’s a lot to learn in math each year.  But there are some major themes of each grade level, especially concepts that parents can help with at home.  This is a nice resource for parents and teachers of students in grades K through 3.  Think of it as a tip sheet, or book mark, whatever you like.  If you’re not sure how to help your child with math homework or what to say, this tool can help!

Click on any image below to access the tip sheet you’re looking for.

Common Core Math Resources for Parents & Families

It’s amazing how controversial learning mathematics can be.  It seems like everyone either loves, hates or isn’t interested in the Common Core State Standards.  As someone who spends an excessive amount of time exploring the math standards, I happen to fall on the side of love but am perplexed about how best to educate parents and families about what these standards are all about.  Because of all the politicizing, it’s become difficult to address what is most important – what do these standards say kids should be able to do and how do we help them achieve this?

Regardless of your opinion of the standards, it makes sense to spend some time learning about what they say.  Council of the Great City Schools has created a fabulous resource to help parents and families navigate the terrain of the Common Core math standards.  Fittingly, they’ve named them Parent Roadmaps.

I strongly encourage you to spend a few minutes learning about the math your child is learning in school.  It turns out these standards aren’t nearly as juicy as the news would have you believe!

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.

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.

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.

 

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.

 

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.

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.

Bedtime Math

My principal has started sharing all the math related articles he finds with me.  On the one hand, it means I’ve got more reading to do.  On the other, I find out about awesome resources like . . . Bedtime Math!

So the idea is this: everyone knows about the importance of reading a bedtime story.  Why not some bedtime math problems?  This nonprofit, working in partnership with the Overdeck Foundation, has developed bedtime math problems associated with a short story.  Each story has 3 levels of problems based on age and skill.

 

They’ve even got a book for sale that builds on the same ideas you find on the website.

Upload the app for bedtime math on the go!

Perhaps my favorite at bedtimemath.org is the Crazy 8s Club.  This club is designed to meet after school for about 8 weeks for varying grade levels/ages.  Anyone can lead the club and the materials are free.  What an amazing resource!