## Mathematical Milestone: Wowed by the Power of Twos

##### August 30, 2018

Last night my daughter was working on an assignment for school. She had to write her name in big letters and decorate it. The teacher is apparently going to put the names up on the wall, to mark off an area for displayed artwork in the classroom. In an effort to make hers more interesting, my daughter started adding little math equations around the border. I was kind of half listening as she counted out loud to herself: "Two plus two equals four, four plus four equal eight", etc. As she got up to "1024 + 1024 = 2048" I started to pay more attention.

When she got to the point where she couldn't add them in her head anymore I helped her by writing down the numbers so that she could add them:

8192
+ 8192
_______

?????

Old school math. Carry the one, etc.

Quite soon she started marveling at how large the numbers were getting. She ran out of room at 131,072, which (I had to check this) is 2 to the 17th power. I told her that this what she was doing (multiplying by two again and again), and that yes, you end up with very big numbers that way. We looked up what power gets you to a million. And she said, unprompted: "I am wowed by the power of twos." She was a bit giddy with the excitement of the whole thing, to the point of disturbing my husband, who was trying to work himself. But I loved it. And he loved it, too, once he realized what was going on.

She ended up deciding to keep that version of her name and do a different one for the classroom wall, but she took the power of two page in to school this morning to show to her teacher and classmates. This is joy of learning, my friends. When a kid just starts noodling around and discovers something that she thinks is cool. [Hey, if you just keep doubling numbers, you get to really big numbers quite quickly. Cool!] You never know where such opportunities will arise, but you can certainly notice and encourage them when they do.

Thanks for reading! I hope that some of you will find this little anecdote enjoyable.

© 2018 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## Quick Tip for Including Math Practice with Reading

##### November 28, 2016

I 'm always keeping an eye out for opportunities to give my daughter a bit of practice with math and show her that math is useful. This morning we finished the first chapter of a book (West Meadows Detectives: The Case of Maker Mischief). She looked at the number at the bottom of the page and remarked: "We read 20 pages." I said: "Well, no, because the story doesn't usually start on page one." So we looked, and sure enough, the text of the first chapter of this book started on page seven. She was quick to tell me that we had read 13 pages. (Technically, we read 14 pages, because we read pages seven and 20, but I didn't see the need to get into that right way. That will be a topic for another day.)

We attempted a similar calculation when we started reading a storybook collection with a table of contents (Biscuit's Christmas Storybook Collection), but the numbers weren't as easy (page five to page twenty-one), and so we dropped the effort for now. But I intend to try this again.

Page numbers and book chapters provide a natural opportunity for practicing subtraction. Obviously, I wouldn't want to turn every reading session into some sort of drill concerning number of pages. But if you have a child who is achievement-focused ("How many pages did we read today, Mommy?"), I don't think that there's any harm in using page numbers for a bit of extra math practice from time to time.

What do you think?

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## Mathematical Milestone: Doing Division, aka Math for Breakfast

##### October 03, 2016

The other morning, right after I woke my daughter for school, she announced, "Mommy, I know that six divided by three is two. AND I know that six divided by two is three. I just figured it out last night, and Daddy said it was right." I agreed that this was indeed correct. She claimed that this was the only division that she knew, and I proved her wrong there by asking her what ten divided by two would be. She counted by twos on her fingers and came up with five. We also discussed the fact that since anything divided by one is itself, she actually knows things like 1,900,017 / 1.

Asked after this what she wanted for breakfast she said "More math!" So, I set out twelve pieces of cereal next to her plate and I shot division and multiplication problems at her while she ate her bagel. She understands intuitively that if twelve divided by four is three, then twelve divided by three must be four. She's starting to understand that in that case, four times three must equal twelve. We went a bit beyond the 12 (twenty-four divided by six, etc.), but it started to be less fun, and we eventually moved on to reading picture books.

What I'm finding interesting, something that I hadn't expected, is seeing HOW she solves problems. She doesn't always get to the answer the way that I would, or even in a way that makes immediate sense to me. But she gets there. For example, she knew that 12 times two was 24 because she had memorized that 12 + 13 = 25, and calculated it would be one less than that. Whatever works, I say.

With Common Core at school, there is considerable emphasis on showing how you come up with a solution, so I'm encouraging her to share her thinking with me. She's still working in school on addition, with numbers that add up to 10 or less. This is actually ok, because she's learning to memorize those sums, instead of having to calculate them each time. In the meantime, I figure it can't hurt for her to practice multiplying and dividing at home, if that's something she finds interesting.

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## The Power of Spending Your Own Money

##### May 23, 2016

My daughter and I had an entertaining day recently, with an experience that I thought was both educational and empowering for her. It started out when she was looking through her Girl Scout Daisy handbook. [Have I mentioned that she ADORES Girl Scout Daisies? It is true.] She found an exercise that her troop had not gotten to this year, in which participants are supposed to list something that they want, and figure out how long it will take to save for this item.

Her first item was a fish, with tank, which is going to require some time for saving. But her second item was three packets of seeds (flower or vegetable). I told her that this would likely only cost \$7 to \$10. She ran upstairs to check her "spend" box (which still contained some leftover birthday money), and came down brandishing a \$20 bill. She wanted to know if we could go seed-shopping immediately. We were somewhat at loose ends, with my husband away, so I said "Sure."

Here's how she spent her \$20. First she spent just under \$10 for three packets of seeds. Then she bought herself an ice cream cone from the nearby Baskin Robbins, at a cost of ~\$3.25. Then she decided that she wanted to buy small gifts for the friends she was going to see later in the day. She picked out three items from the dollar bins near the front of the nearby Target (one was for herself), at a cost of \$5.46 with tax. She had about \$1.50 left over.

Leaving the hardware store, she remarked: "This was already a good day, but now it's a GREAT day."

At all three stores, she paid with her own money, with much pride (though I did hold on to the change in between). As made sense, I encouraged her to do the required math. The three Target items cost \$1, \$1, and \$3, so getting to \$5 was easy, though she's not quite ready to compute San Jose's 8.75% sales tax in her head. When she handed over \$5.50 at Target, I had her figure out what her change would be. I rounded the prices of the three seed packets and had add those numbers together. But I didn't push it too hard. I wanted our time together to be fun.

But this whole experience highlighted to me why it's important for kids, once they are old enough, to have some small amount of money of their own. My daughter was empowered by the whole process of deciding what she wanted to buy, figuring out  how much things were going to cost and what she could afford, and physically being the one to pay the sales clerks. The day would not have had nearly the same feel had I just been buying her things. In fact, at one point, I offered to buy some cookies to take over to her friends' house. She said: "Mom, they're MY friends. I should buy the presents, not you." What parent could argue with that?

We are currently deferring her allowance for the next few months, to save up for that fish tank. I'll keep you all posted.

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## Mathematical Milestone: Counting Redeemable Cans Backwards from 100

##### May 16, 2016

The other day my daughter brought home Earth Day -- Hooray!! by Stuart J. Murphy and Renee Andriani as her book report book. It's about three kids who want to plant flowers in a local park that they are cleaning up for Earth Day. They decide to redeem cans to make money to buy the flowers, which their teacher tells them will take 5000 cans. They end up launching a can drive at their school and around their neighborhood, and of course succeed in time for Earth Day. There's a fair bit of math involved as they group the cans by 10s, 100s, and eventually 1000s, and add each day's haul to the total.

This type of overtly lesson-driven book would not normally be my personal cup of tea, but my daughter enjoyed it. And there are some cute details (e.g. squirrels helping to collect the cans). But I knew what was coming next. I read Earth Day -- Hooray!! to my daughter at breakfast. As soon as she got home from school she started digging through our kitchen recycle box, looking for cans. She is now on a mission, though she does seem to understand that 5000 cans would be too many to target on her own. She stood at my side while I drank my lunchtime Fresca, waiting impatiently for me to finish, so that she could have the can.

She decided that she wanted to collect 100 cans, and then have them taken in for redemption. She seems to be motivated by a combination of environmentalism and an interest in the money. Where the math milestone for her comes in is that she is counting backwards from 100 as she adds cans to her bag. A friend kindly contributed a bag of cans (which my daughter simply had to go pick up within the hour). She counted them up, and excitedly came to me to say "We're in the 70s now." No, we don't have 70 cans, but we are in the 70s if we are counting backwards from 100.

So, redeeming aluminum cans turns out to be another unexpected way to incorporate math into the life of a six-year-old. Setting and counting down from targets, estimating how many cans can fit into one garbage bag, and, eventually, figuring out how much money she'll be due. Math is everywhere!

Thanks for reading! I hope that some of you will find this useful.

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## Learning Math from the Scholastic Reading Club Flyer

##### April 13, 2016

My daughter has been a fan of the Scholastic Reading Club flyers since preschool. She's now in kindergarten, and in addition to using the flyers for selecting books, I've started using them to get her a bit of practice with additional and division and general money sense. This has been a gradual process. Our iteration has gone something like this:

1. I just chose for her. (Preschool)
2. I let her go through and circle everything that caught her eye, and then I still chose which ones to order. (Start of kindergarten)
3. I asked her to put stars on her top 3. I ordered mainly those, with a couple of extras that I would pick. (A couple of months into kindergarten)
4. I started pushing back when she would pick things that were expensive (like a fossil excavation kit). She would then offer to pay, out of her allowance, for things that I wasn't willing to pay for.

This month she picked her three items, and I pushed back a bit because two of them were relatively pricy (e.g. Dig it Up: Lots of Rocks for \$10). She suggested that she pay for half of the order and I pay for half. I said ok, and then she went through and added up the total cost of the items that she wanted, and then (with a little help) divided the total in two. Then she counted out the money from her "spend" box. She ended up selecting four items costing a total of \$30, for which she paid me \$15 (there was some birthday money involved). I then quietly added two paperbacks that I thought were both a good deal - I don't think that she will notice by the time the order arrives.

Bottom line is that my child is very interested in what books (and other things) she's going to be able to get from the Scholastic flyer. This makes her eager to do the math, if that's what it takes to get to what she wants. So here I am showing her that math is useful, and giving her a bit of light-hearted practice. It's just a matter of keeping one's eye open for these types of opportunities.

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## Mathematical Milestones: Counting How Many Days

##### March 31, 2016

I noticed recently that my daughter has acquired a new math skill. She can (with some help regarding the number of days in each month), figure out how many days it will be until some future date. This is an important skill for kids, because they often urgently want to know things like:

• How many days until my birthday?
• How many days until April Fool's day?
• How many days until summer vacation?
• How many days until Christmas?

Of course counting the days is fairly easy if your target date is within the same calendar month as today. It still requires either counting forward or using subtraction, so even this simplest case is useful. My daughter used to do this by going to the wall calendar and counting the days, but she has grasped the abstraction of the numbers at this point, and usually doesn't need to.

When the target date is in a future month, the calculations are more complex. You have to be able to count how many days remain in this month, and then add the relevant days from the next month (plus other months, as applicable). We went through this exchange on March 20th:

Daughter: "How many days until April Fool's Day?"

Me: "Let's figure that out."

Daughter: "OK. How many days in this month?"

Me: "31."

Daughter: "OK." Pause for thinking. "Ten, eleven, twelve. Twelve days until April Fool's Day!"

I was very proud. She counted by 10s to get to the 30th, then added in the 31st and the 1st. The next step will be counting the days until some event that's more than a month away. But there's no rush.

I think a key to the fact that she can do this calculation stems from my asking: "Let's figure it out." I ask that question ALL the time. If you train your kids to figure out the answers to numeric questions, instead of just telling them, they get used to it. They learn how to do it. They'll eventually do it on their own. Especially when the question is of vital importance, like knowing how long they'll have to wait for April Fool's Day.

Thanks for reading! I hope that some of you will find this useful. Wishing you all a fun April Fool's Day tomorrow!

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## Keeping Math Fun: Cooking with Kids

##### March 16, 2016

I've blogged previously about ways to keep math fun for kids (following sports, building a lemonade stand). Today I'd like to share an idea that was suggested by Lara Ivey on Facebook. In response to one of my many Facebook posts on this topic, Lara suggested that cooking is a great activity for showing kids that math can be both fun and useful. I probably wouldn't have thought of this myself because, in truth, I'm not much of a cook. [The bible in our house is How to Cook Everything by Mark Bittman, but we don't break it out very often.] But even the simplest of recipes offers the opportunity to practice math skills.

For example, my daughter got it into her head the other day that she wanted to make a Tang popsicle. She chose a plastic cup that she wanted to make it in, and jettisoned a long-expired sorbet popsicle to gain access to a popsicle stick. Then we got out the Tang canister and looked at the directions. One eight oz. serving required half a scoop.

• Easy math question: how many servings can you make with a full scoop?
• Harder: if a quart takes two scoops, how many servings is that?
• Bonus question: if two friends come over, how many scoops do we need for everyone to have one serving? What if we want everyone to have two servings?

This is useful stuff!

We always make a double batch when we make brownies. Plenty of math practice there, doubling all of the ingredients. Then we can decide how many rows and columns to cut if we want there to be a particular number of brownies to share. The opportunities are endless!

Do you use cooking time to help your kids build math skills? What other suggestions do you have for demonstrating that math is both essential and fun?

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## Mathematical Milestones: Estimating #PictureBooks

##### February 01, 2016

In this new blog series, I am documenting some of my daughter's milestones on her path to numeracy. She will be six in about 2 months and is in kindergarten. The first entry in the series is here

The other morning my daughter, somewhat out of the blue, demonstrated her understanding of estimation. She was looking through a stack of picture books for something to read during breakfast. She called out to: "Mom, what's three eights and one five?"

I had to go look to figure out what she was talking about. She had counted the bottom eight books and measured the height of that part of the stack with her fingers. Then she moved her fingers up to find two other same-size sets, and then counted the remaining books at the top of the stack. Then she counted the actual books, to see how close she was. It was pretty good - the estimate was 29 and the actual number of books was 31.

I think that she picked up on the idea of estimation from two different places. First, there's an early reader that we enjoy called Gumballs: A Mr. and Mrs. Green Adventure (link goes to full review). In this title, an alligator uses estimation to win a contest (guessing the number of gumballs in a jar). Second, her kindergarten class did a math project a couple of weeks ago involving little bags of M&Ms. The teacher asked the kids to estimate the number that they would find in their bags. So she had the idea of estimating in her head, though we hadn't discussed it recently at home.

Estimation is a useful skill, so I was pleased to see my daughter using it. Thanks for reading! I hope that some of you will find this of interest.

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## Encouraging Your Child to Like Math: Why and How

##### January 27, 2016

In today's quantitative world, it's important that kids grow up with a positive attitude concerning math. When we raise our children to believe that they "hate math" or are "bad at math", we do them a grave disservice in terms of both life skills and the job market.

"Taking math courses matters. Research studies have established that the more math classes students take, the higher their earnings ten years later, with advanced math courses predicting an increase in salary as high as 19.5% ten years after high school (Rose & Betts, 2004). Research has also found that students who take advanced math classes learn ways of working and thinking--especially learning to reason and be logical--that make them more productive in their jobs." (Jo Boaler in Mathematical Mindsets, Introduction)

This is especially true for those of us who are raising girls. I have a kindergarten-age daughter, and this is something I work on every day. To have the full spectrum of options available to her in the future, she needs to not be afraid of math now.

Here are some tips for making math a positive experience for young children:

1. Stop saying that you yourself are bad at math, or hate math. Don't grumble when you have to balance the checkbook. Kids absorb these messages, even if you aren't directly talking with them. (See Mathematical Mindsets for more on the impact of both parents and teachers giving out negative math messages.)

2. Find ways in your day-to-day life to use math, and to point out occasions when math and numbers are useful. For small children, this could mean pointing out speed limit signs when you are driving, or counting out the Cheerios that you put on their plates. As kids get older, you can incorporate ever more complex examples. Does your child want to know when dinner will be ready? Instead of saying "in an hour" try this: "It's going to take me about 10 minutes to finish preparing, and then it will have to cook for 40 minutes, and then cool for 10 minutes. How long do you think it will be?" Which brings me to my next piece of advice...

3. Whenever you can, turn your child's questions into word problems. When your child asks a question involving time, dates, or anything else with numbers, this is an opportunity. Whatever the child is asking about is something that (at least for the moment) she cares about. You can have her practice doing a bit of math in her head, and see why she would want to understand math. I do this constantly with my daughter. If she says: "How many minutes until my Girl Scout Daisy Meeting?" I tell her, "It starts in two and a half hours. An hour is 60 minutes, which means that a half hour is 30 minutes. So how many minutes is two and a half hours?" Of course I help her - I don't want these little word problems to become onerous.

4. Surround your child with toys that build her spatial skills. Your child can never have enough blocks. In my house, we have a mix of plain wooden blocks, castle-themed wooden blocks, and Magna-Tiles. I also recommend Legos, once your child is old enough for them. It's better to get the big tub of classic Lego pieces rather than having a lot of specific sets with step-by-step instructions. Tinker toys and Lincoln Logs are still available, too. Also good for building math skills are toys that can be sorted and categorized. We have a set of 100 little plastic teddy bears in six colors which my daughter has played with for years now.

5. Include books that have positive depictions of math in your child's library. When your child is small, buy or borrow plenty of different counting books. Try to steer towards books that are FUN, rather than anything that seems dry. And if your child rejects a book, do NOT press him. Find a different book. Turning the reading of counting books into a chore runs the risk of turning your child off from reading AND math.

Some math-themed picture books and early readers that my daughter and I like are:

6. Most importantly, push back when other people erode your child's enjoyment of math, and guard against this in yourself. Many schools are unfortunately full of dry math worksheets and repetitive homework problems. Your job, as your child's advocate, is to push back on this if you can. Talk to your child's teachers. Refuse to work on the "optional" homework problems or apps. If your child uses a tablet, replace dry, quiz-based apps with apps that your child enjoys. Do not get sucked into those informal parental competitions about how high your child can count, or how many worksheets he did yesterday.

Your goal is for your child to ENJOY math. The rest will follow. Do anything you can to make math more, not less, enjoyable. A positive attitude about math will keep doors open for your child that might well close otherwise. This is worth some effort.

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook

## Two Apps Making Math Learning Fun for My Daughter

##### January 26, 2016

I don't usually do app reviews on this blog, but I wanted to mention a couple of apps that my daughter LOVES that are helping to keep math fun for her. You should consider these more recommendations than formal reviews.

First, some background. My 5 year old daughter enjoys playing games and watching shows on her Kindle Fire. This is something she is allowed to do only within defined time limits. Her teacher recommended / suggested that the kids work at home on math using an app called ScootPad. And I tried to do this, despite the fact that my overall goal is to limit the amount of time that my daughter spends using her device.

It turns out that my daughter, who really enjoys math, HATES ScootPad. As we've experienced it, the app consists of a series of quizzes. They are repetitive and too easy for my daughter's experience level (she went to a fairly academic preschool). I tried to get her to get through them, because I thought that if we could get through the early ones, more challenging material would follow. But we never made it, because she was literally crying with frustration, and I eventually stopped having her use it. I'm not saying that ScootPad doesn't have value, and wouldn't be a good fit for some kids. But for my daughter, it was taking away her joy of learning

Fortunately, there are two apps that my daughter really enjoys playing with that are helping her to build her math skills. The first one is from Scholastic, and it's called Sushi Monster. It is only available on iPad and iPhone, so I have to let her use my iPad to play it. Scholastic's website says:

"The game, which meets Common Core State Standards, offers students practice, reinforcement, and an extension of math fact fluency in a completely engaging and challenging way. Students will strengthen reasoning strategies for whole number addition and multiplication by helping monsters make a target sum or product. Students earn points, stars, trophies, and personal bests to challenge themselves and unlock new levels of play.

The addition mode gives kids a solution, and asks them to select which numbers add up to that solution, from a displayed set of numbers. Then you get another solution, and you pick from the remaining numbers to try for that number sentence. If you choose the wrong numbers for one sentence, you probably won't be able to complete successive sentences.

Each round increases the number of choices, so the game gets harder and harder as you go. If you successfully get through one level, the next level has higher numbers to work with. (E.g. at first you might be choosing which two numbers add up to 10, eventually you are selecting which numbers add up to 120 or 900 or whatever).

There are fun sounds, and it's very fast-paced and entertaining. Then you get virtual points and stars and such, which my daughter does find motivating. There's also a multiplication mode, which we've dabbled with, but she's not quite ready for it. But my general view with this app is that she's excited about it and asking to play, while practicing doing addition and even multiplication in her head. She basically learned that 20 plus 20 is the same as 2 plus 2, with an extra 0 at the end, by playing around with this app.

More recently I found several recommendations for math-based apps in Jo Boaler's book Mathematical Mindsets. We tried a few of them out, and found one that was a hit: Hungry Fish from Motion Math. This one is available on iOS and Google platforms, including the Kindle tablets (which is definitely a bonus for us). Here's the Motion Math description:

"Your fish is hungry – hungry for numbers! This fun addition and subtraction game for iPad, iPhone, and iPod touch has instant addition: touch two numbers together to instantly add. Most addition games teach in the form 3 + 4 = __; Hungry Fish challenges players to find different ways to make a 7 (1+6, 2+5, 3+2+2, etc.). There are 18 levels of challenge (for 4-year-olds to adults) and bonuses to customize your fish with new colors and fins."

So, it's similar to Sushi Monster, but with an underwater theme. The fish has a number on it, and then bubbles float out with different numbers on them. Kids have to combine the bubbles to make the number displayed on the fish, and then the fish eats them. As the fish eats, it grows bigger, and the kid is getting points. At certain intervals, you earn a new color with which to decorate your fish (which my daughter again finds motivating). She's experimented with the addition, subtraction, and negative number levels. In the latter, the target is a negative number, and you have to combine positive and negative numbers to get to the total. The first few times my daughter played this app she was positively giddy with how fun she found it.

I've found it interesting to watch her with this app. There are lots of different ways to combine the numbers to get to the target numbers, in some cases. She doesn't just have the answers in her head all of the time (particularly for the two-digit numbers), but she uses strategies, like combining smaller numbers at random until she gets something that she recognizes will lead to a solution. Sometimes she sets the difficulty level too high and gets frustrated, but it's easy for her to slide it back to an easier level. She is sometimes learning through trial and error, which is how a lot of learning happens. I am happy to see her using this.

We don't get "credit" at school for using Sushi Monster or Hungry Fish. And I only let my daughter use them as part of her device limit, not on top of that limit. But if she's going to be on the device, I can't tell you how happy it makes me to see her practicing math effortlessly, enjoying herself, while continuing to build her knowledge and intuition.

Do you all have any other math app recommendations for us?

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook. This site is an Amazon affiliate, and purchases made through affiliate links (including linked book covers) may result in my receiving a small commission (at no additional cost to you).

## Mathematical Milestones: From Recognizing Number Order to Playing Shop

##### January 15, 2016

Today I'm kicking off a new series. I've been sharing my daughter's milestones on her path to literacy. But as I've expanded the focus of my blog, I realized that I've been remiss in not also sharing her milestones along the path to numeracy. So I'm going to recap a few here, and then share them going forward.

I actually remember the first milestone quite clearly. It must have been about two years ago. We were in a parking lot in Monterey. There were numbers painted on the parking spaces. She pointed and said: "Look! They're backwards." I assumed that she was commenting on the fact that from the way we were standing, the numbers were upside down. But no! She proceeded to say: "Sixty-seven, sixty-six, sixty-five. Backwards!" I thought that this was pretty good for her age at the time, and the moment stayed with me.

A couple of other more recent milestones stand out. We have a household rule that if she is ready for bed by a certain time, she can have 20 minutes of playtime. She came up to me a few weeks back holding up five fingers on one hand and four fingers on the other. I said: "9?". She said: "No, Mommy. Four fives. Twenty minutes." Since then, whenever she has to indicate 10 she holds up five and two fingers. Twenty is always four and five fingers. She is positively gleeful about her use of multiplication. Shortly after this, she came to me, very excited. "Mommy! Five twos is the same as two fives!" So it seems she has figured out The Commutative Property. To me it's fascinating to watch these concepts that I've understood for so many years dawn on her, new and fresh.

Last night (and this is more of a game than a milestone) my husband and I were eating dinner. She was finished, so without any help from us, she decided to set up a store. She piled up a bunch of oranges, apples, and bananas, together with a couple of packages of crackers. Then she wrote numbers on some pieces of construction paper and taped one to each type of object. Then she started hitting us up to buy things, using napkins as currency. Her prices were a bit steep (\$23 for a box of crackers! \$9 for an orange), but we went along.

It was a great exercise for practicing math. I asked for two boxes of crackers, and what that would cost. She first thought \$43, but we got her to the right total. Then she had to tell me my change from \$50. The bananas didn't fare so well from being played with (and dropped at least once), but it was otherwise a fine game. I liked that she set it all up herself, and we didn't need any fancy props.

One of the reasons I've expanded the focus of my blog is that while my daughter enjoys books, she also seems to like math. I would like this to continue, and I think it's high time I started sharing her progress in that area. Along the way, I'll try to include tips for other parents. My biggest tip at this point is to just turn everything that is remotely related to numbers into a word problem when you talk to your kids.

For example, my husband recently had to go to a meeting in San Francisco. My daughter wanted to know how long he would be gone. I said: "Well, the drive up and back will each take about an hour, and the meeting will take at least two hours. How long do you think he'll be gone?" She first thought "three hours", and I had to tell her to include the ride up and the ride back, but she got there. Then she illustrated her understanding of variability (in Daddy's arrival times) by suggesting that the meeting might run over, and that it might end up being closer to six hours (which was pretty accurate).

Do you all do this? Turn life into a series of math problems for your kids. Or is that just me? Of course I'm exaggerating a bit with this last question - I want it to stay fun for her, and I back off right away if I think that she's feeling pressured or impatient. But I do think that, just as surrounding kids with books and reading aloud to them are key to developing literacy, talking with kids about numbers and playing mathematical games with them are key to developing numeracy

Thanks for reading! I hope that some of you will find this useful.

© 2016 by Jennifer Robinson of Jen Robinson's Book Page. All rights reserved. You can also follow me @JensBookPage or at my Growing Bookworms page on Facebook