math

## I’ve done it again…

…I planned to blog every week about math in my class, and I have not followed through.

In my defence, I have been really, really tired. Exhausted, in fact. And after spending time planning and organizing my day I didn’t have it in me to properly reflect on how things have been going. December is ROUGH.

Alas, a two week break has been really good for me. I have a few days until our return to school, and have started to think about what that will look like. January is the end of term, so I need to do some math assessments for everyone. I have blogged before about doing math interviews. I do love them, but have decided to create my own set of questions for this mid-year assessment. I know what we did, I know where the curriculum says we need to be in June, and I want to figure out exactly where we go next. I can do that best if I create my own assessment.

Up until yesterday I was pretty sure we would be doing some online school in January, so when I was thinking about the problems I wanted to pose I made them into slides. I’ll still use these when we are working in person.

More than any other year, my students have such a wide range of skills. They are definitely not divided into “grade 2” and “grade 3”. We’ve had to spend quite a bit of time on counting, especially skip counting. I decided to use Toy Theater to make some images. I anticipate that some of my people will still count by ones. But I think most people will at least count by 5s and 10s, and at least one will count by 20s. (These are just 3 of the 8 images I am including.)

I’m also asking some problems that ask them to work with single digit numbers and double digit numbers, and I have a multiplication question. That is where I know we need to go soon, so I want to see what happens. I predict a few will count by ones, a few will skip count by 4s and three will say, “4×4=16.”

We’ve done some geometry and money as well, so I will have coins and shapes in front of us that everyone can manipulate.

Two summers ago I updated my spiralled math map to reflect the new curriculum. I am pretty much on track with that, although we are a bit off track. By the end of first term in previous years my grade 3s would have been working with three-digit numbers and we haven’t done that yet. All my 2s would be working with double digit numbers, and we aren’t quite there yet. But we will have half the year to go so I am not worried about it.

I am also really interested to see where everyone is in their communication skills. That was definitely a “need to work on” area for lots of and I think we’ve done a pretty good job so far.

Finally, I want to know how they all feel about math.

The interviews will take a while. I predict I’ll be spending at least 10 minutes with each child, and I have 20 in my class. But the information I gather will help guide us in the right direction for the next few months so I know it will be time well spent. I have a total of 18 slides. Some students won’t need all of them at the end, and some can start in the middle. I’ll try them out on my son before school starts. They’ll be too easy for him, but it will help me work out any trouble spots.

## Baking

Are you still baking bread? At the beginning of the world-wide shutdown, everyone was baking bread and cookies and their own pizza. We are still doing this – we were doing this before. We’ve slowed down a bit because it’s been too hot to turn the oven on. But yesterday I made some bread and today my daughter is baking cookies.

She wants to do this all by herself. I have a recipe that’s meant to be easy for children to follow, and she has a lot of kitchen experience for a 9-year-old. She’s only needed help so far with the “1 slash 2 cup” of butter. We’ve talked about 1/2 dozens of times but it still eludes her. I showed her how to use the markings on the butter to figure out 1/2 of a cup. She then needed help with 1/2 cup of honey. Partway through pouring she realized she had the “1 slash 4” cup and thought it was going to be too much because she only needs “1 slash 2”. We’ve talked about this a bunch too, but in the moment she was confused again.  I coach her through it.  “1/4 is 1/2 of a 1/2.  So if you have 1/4, you have 1/2 of what you need.  So what do we need to do?”  She figures it out, I pour her another 1/4 cup of honey, and she’s back to working on her own.

I’ve been thinking a lot about baking and the learning that goes along with it. Little of it actually shows up in our curriculum. There’s problem solving, some collaboration (her brother doesn’t like chocolate so this is always part of our conversation when making cookies), communication, following instructions and, of course, measurement and fractions. At home this type of learning is very important to me. I want my children to head off to university with the ability to cook more than Kraft dinner, grilled cheese and scrambled eggs. The curriculum connections are a bonus. At school I cook or bake with my class maybe twice in a school year (less if there are students with allergies or special dietary needs in the class.)  I learned to cook at my grandma’s house, at my parent’s house, in Home Economics classes, and in neighbourhood 4H clubs.

If I was in charge of the curriculum, I’m not sure what I would eliminate in order to make space for cooking in the elementary grades.  I’m not sure there is anything we should exchange for cooking time. Secondary students still have the option of doing catering courses so they can learn to cook. Mostly I feel like learning to cook is part of how we pass our family and cultural traditions and values down to our own children.  I know some parents don’t teach their children to cook, but maybe that needs to be up to them.  Maybe this is one thing the schools have let go of for good reason. If I cook with my class it’s because we have a connection to something we’ve read about, or it’s a snow day and we need to fill the time with a worthwhile but fun activity.  I don’t feel compelled to teach them life skills like cooking or sewing.

math

## Summer Math: Vacation

I handed my 8 year old this packet off magnets and, “Half for you, half for your brother.”

“Wait…so we get 4?” I wasn’t sure how she’d decided on 4 as half of the package. As I tried to formulate my probing question, she said, “Wait. (Long pause). 1, 2, 3, ….we each get 5. Wait…5 is half of 10? Yeah. We get 5 each.” She wasn’t in the mood to explain her thinking though, so now we’ll never know what was going on in her head! One interesting thing I’m noticing often is that she says an answer quickly, then continues to work on the problem in her head nearly always catching her own errors and correcting then unprompted. I’m going to have to talk to her about why she gives the answer so quickly, before she’s sure she has the right one. And I need to get her to do some stuff in writing to see if this is only happening with mental math calculations.

This past week we spent a few days in Toronto. We had an Air BNB in the building where my brother in law lives. We found it when were visiting the CN Tower.

Tall, right? We were on the 6th floor. To get to his condo, we had to ride the elevator to the 15th floor, exit, wait for the other elevator, then go up to the 34th floor. Here are the things we practiced: counting backward, figuring out how many floors until we would reach our destination, how long we’d been waiting for the elevator, counting forward, paying attention to the pattern of button pushing so we’d know whose turn it was to push buttons. These questions have come up in other elevators, but there was renewed interest since my children have never been in such a tall building. The CN Tower elevator doesn’t have numbered buttons or a countdown display, so that was a bummer. The elevator was pretty slow most of the time, but slow enough for us to have time to work out “how many floors until…” Maybe I should ask the kids to figure that our now!  I feel like these conversations were all about developing number sense. Thirty-four isn’t a very big number, but it really high up in the air when one is in a skyscraper.  Fifteen also isn’t very big, but looking over the pool railing from the fifteenth floor makes 15 seem really high!

We also had lots of conversation about the size of the pool and how it compared to other pools we’ve used this summer. It’s deeper than 2 of them, shallower than 1. It is longer than 2, but shorter than 1. It is a rectangle (clear in this picture but not as clear to kids standing right beside it) but we were in 2 square pools earlier in the summer.

## Summer Math: Math Before Bed

We love doing “Math Before Bed” as part of our “read at bedtime” routine.  We get out of the habit sometimes though because we also love to play card games (UNO, Go Fish, Old Maid, Memory) before bed. Last night I pulled up this picture:

I quickly counted them:  10 per column, 4 in each row. 40.

My 8 year old started counting by ones.  She said, “I think there are 38.”  Knowing she was not correct, I asked, “Besides counting by ones, how else could you find the answer?”  At the same time, I started counting the far right column. Only 9.  Hmmm. I counted again.  Yup.  I had assumed all 4 columns had 10.  We chatted about this.  We talked about how we could use my original answer of 40 to find the answer.  “There are 2 missing from the last row!  How can we use that?”  She had a bit of trouble figuring this out.  She kept saying, “10, 20, 30, 40.” over and over. I said, “Well, 40 but two are missing.  Maybe someone ate them!”  She counted backward to 38 and we were done.

Then she asked, “Can I make my own picture like this tomorrow?”  So that is what we have just finished.  She decided to use plasticine.  I was recruited to mix colours together and help her make tiny balls.  She decided she needed 60 of them. She also decided she wanted to do rows of three because 2’s and 5’s are too easy and she likes a challenge. (HOORAY!!!)  After counting over and over by 3’s, making a few mistakes along the way, I prompted her to notice that there were 10 in a column.  “10, 20, 30.  Oh.  Halfway there.”  🙂

In the end, we had more than we needed. She put those into groups of 5 (and one group of 4) to figure out how many were left. “5, 10, 14,” she said.  It’s so interesting to me that she can skip count, but often counts by ones.  She says this is because “ones is more easier.” She only switches to larger numbers and skip counting when she has a lot of things to count. I suppose this makes sense.

## Summer Math: Maps

There’s some mapping skills in the Geometry and Spatial Sense section of our curriculum, so that’s the connection I’m writing about today. One of the Grade 1 big ideas is: “describe the relative locations of objects using positional language.”  For Grade 2 students, one of the specific expectations is: “describe the relative locations (e.g., beside, two steps to the right of ) and the movements of objects on a map (e.g.,“The path shows that he walked around the desk, down the aisle, and over to the window.”)

The girls at our church go on a camping trip every summer for a few days.  The girls have to be between the ages of 11 and 18 to go, but of course there are adult leaders.  I’m been about a million times and I love it!  We go to a place on Highway 144, north of Sudbury.  It’s in the town of Levack, and just past Onaping Falls. This year I only needed to go for one night, so I had my two children with me.  It was such a beautiful day that I decided we’d leave a bit earlier than necessary and stop at the Onaping Falls Lookout.  A.Y. Jackson painted a painting called “Spring on the Onaping River” here.

Thinking I remembered the way, we set off on a trail.  We got to here, but it was a dead end.

Back to the map we went!

Turns out we had followed the trail to the Handicap Lookout Area (it was wheelchair accessible.)  We used the triangle to orient ourselves, then re-parked the car in the “picnic and parking area” closer to the trail-head.  I wish I had a picture of the rocks we had to climb to get down into the river valley!  It was a lot of work and I didn’t have time to take photos on account of trying not to fall and break my neck – or allow my children to do the same.  At the bottom we enjoyed some time by the river.

After looking at the map, both children wanted to walk all the way to the lookout bridge, which we could easily see in the distance.  However, after this short hike, we all agreed that the bridge would need to wait for another time.  I think the trail would have been much easier after our descent, but I was already thinking about going back up the hill.

Car trips…or van trips…are a great time to practice lots of practical math skills.  For a while we played a “game” of finding numbers higher or lower than 50 on the road. The speed limit was 90Kp/h, we had to go 400 m to the next turn, there were 17 km until we got to Sudbury, etc. We then challenged ourselves to figure out how far away from 50 each number would be.  We mainly did this with the single- and double-digit numbers.  I feel like this is all part of gaining spatial sense.  By the end of the trip they were saying, “500 meters isn’t that far, right?” or “250 KM!  That will take forever!”

We’re headed off on another road trip today – this time going south. Both of my children are weirdly obsessed with taking surveys.  I’m going to challenge them to come up with some data they can gather while we are driving.

Finally, how beautiful is this:

math

## Summer math: money and even& odd

What does 50 + boat + beaver = ?

If you are Canadian, you may have known that it equals 65 cents. “The one with the boat” is what my 6 year old often calls a dime, and of course the nickel is the beaver coin. Not sure why the names of these coins elude him. He has no trouble remembering the value and I suppose that’s what matters now.

This conversation came up because we saw yard sale signs. Last summer we started letting the kids do their own yard sale shopping. It really helped them start to understand the value of money. I don’t mean the actual value of the coins and bills, but the whole concept of working hard to earn (or find!) the cash and then having to decide if the desired item was worth that amount. Of course they have to count it themselves, and they are both getting pretty good at it. I’m getting pretty good at making them think it was their idea to not buy the junkiest item on the table.

Money came up again today at Canadian Tire. I received 40 cents in Canadian Tire money after my transaction (I haven’t embraced the electronic version of this.) The self-checkout (which I only used because the boy was not getting a new bike helmet like his sister and needed a job to distract him from the injustice of it all) gave us eight 5 cent CT dollars. These had to be equally shared. There was a “some for you, some for me exchange”, some negotiation and finally each was convinced they had an equal amount. It fit nicely into an ongoing conversation we are having about even and odd numbers as well.

Speaking of even and odd, did you know that 13 is odd? 6+6=12, so if you have one more than that it’s not even because there is one extra. (Explanation courtesy of the 8 year old!)

## Summer Math:Counting and Subitizing

I’ve been in a “blogging about math” funk for a few months.  One of my summer goals is to write more, so I thought I’d start a series of blog posts about the math that I am doing with my children at home this summer.  To be clear, this is not a series that is planned.  Instead, I am going to try to be very mindful of the times we do math together formally or informally.  My children, who just finished grade 1 and 2, are probably involved in informal math conversations about the same as many children of teachers. Both of them are pretty good mathematicians, and by that I mean they use flexible strategies to do mental math calculations, they notice math in the world around them, and come up with strategies for solving math problems that naturally occur around them.  I’m a firm believer that this happened because I am intentional about helping them mathematize their world, just as I am intentional about making sure they learned to read by reading to them at least every night before bed (and usually more often!)

Earlier in the week I received an e-mail about the latest Mathies resources.  This morning we finally had time to sit down and explore a bit.  I picked a game for my 6 year old called “Representation Match” and if it wasn’t for his Minecraft addiction I think we’d still be playing it (he only gets to play on Saturdays and when he sneaks my phone into his closet unnoticed so it’s tough competition!)

I chose the numbers 0-20 for him, and I chose all the representations of those numbers.  He had to find matches – two ways to make 14, or 19, or 17, or any number between 0-20. These are all the choices available.

He had to work at this!  He was not able to subitize all the numbers so we had a few conversations about how to figure out the number represented.  For example,  there were 3 dice, two showing 6 and one showing 5. I prompted him to think about 6+6 which he knows is 12.  Then I pointed to the 5. He counted on by 1’s to get to 17, then chose the numeral 17 as it’s match. Sometimes he had to match two picture representations.

When he played again, he chose 2 or 3 representations for himself, always a different combination.  My daughter did the same.  She played with the 0-20 cards, even though she is in grade 2.  She likes to get answers fast, so this appealed to her. She was also playing with the cards hidden, more like a traditional memory game and said she had a lot to keep track of in her mind if she was playing with the higher numbers.

We use Dreambox a lot at school.  I love it!  But I also like to have students doing some targeted math activities that keep them immersed in a specific skill for a while.  Dream box allows them to pause a game and move on to something else, which is fine, but also lets them give up too easily sometimes.  I think this Mathies game would make a great supplemental activity for us during the first month or more of school when we are talking about counting strategies, as well as for practice throughout the year.  Did I tell you I’m scheduled to teach a grade 1, 2, 3 split next year? I haven’t taught grade 1 before so I am anticipating how that will look.  The “Representation Match” game will let me set them up to match numbers 0-5, 0-10, 0-20 and 20-50, I think it will be good for the whole class.

## Math at Home

My son, who is in grade 1, has really good number sense.  He has a lot of mental math strategies that he uses efficiently and flexibly.  He adds on, he counts back, he finds landmark numbers, he even splits numbers!  And no, this is not because we spend a bunch of time every day drilling math.  It’s because we play lots of games and have math conversations that pop up throughout our day.

As I watched him play “Sorry” I was surprised that he was having some counting trouble.  He has been able to count in sequential order with one-to-one tagging for quite some time. He can count a variety of object by ones, more than 100, and when he makes a mistake he notices it on his own and fixes it.  He subitizes, and I feel like this what he is doing  while he counts and that his how he notices his own mistakes.  But that’s a tangent I won’t go on right now.

What surprised me as we were playing “Sorry” this week was the trouble he was having  moving his pawn the correct number of spaces on the board.  He recognizes every number in this game, and connects the number symbol with the amount. He’s done this with other games many times, such as when we play other games and he has to compare which of two numbers is larger. (I had a hard time writing that sentence because I kept thinking about how we haven’t played War in a long time!)  When he drew 5, for example, I know he knows that is 1, 2, 3, 4, 5.

When he would draw a number he would count to that number as he bounced his pawn around the board, but invariably any time he had a number higher than 3 he would bounce a different number of spaces.  Sometimes he would go fewer than he was allowed, and sometimes he would go farther than he was allowed.  If you draw a 4 in this game, you have to go backward, and he did OK with that but he would count slower than usual, so I built that into my intervention. I told him about the problem.  “Just like when you are counting things, your pawn has to touch each square when you count it.” I started by putting his hand in mine, and making sure that every bounce had his pawn landing in just one box without skipping any boxes.  After several rounds of this, he started doing it on his own.  He would slow down his counting and he’d land in the right spot.

The next day we played again, and the problem resurfaced.  This time I explained the problem to  him, then instead of holding his hand I put a finger on the square as he counted.  If he got ahead of me, or skipped a square, he would recognize this on his own and correct himself (and sometimes his big sister had to butt in and point out his mistake, but that’s a different post altogether!)

The third time we played the game, he needed a verbal reminder, but that was it.  And the fourth time he needed the verbal reminder.  And if we have time to play it again tomorrow, which I hope we will, I expect he’ll need the reminder again, but I’ll wait and see.

This whole thing has surprised me some, mainly because as I said before he knows how to count with one-to-one tagging and has for a while.  So why was he having trouble? This is what I think: there was a little pressure on him this time that isn’t normally there. First, he loves to win and he knew that winning in this game requires getting around the board quickly.  That was a distraction and a stressor when he was trying to count. Second, besides just counting, there was some other thinking that had to happen.  If you land on a square with a triangle you get to slide, and if you land on a square that already has a pawn on it then you say “Sorry!” and bump that pawn back to start, and sometimes I could see that he was making a move with one pawn while also thinking about how maybe he should actually be moving a different pawn to get a better outcome. He’d be in the middle of a move, suddenly stop, put the pawn back where it was and move a different one instead.  Third, …I don’t actually have a third.  I think those two things are enough to explain why he was having some trouble. I did double check to make sure he was wearing his glasses the first time I noticed it, and he was, so we can’t blame the vision.  And his coordination is such that moving a pawn around the board is not a physical difficulty for him.

Counting is such an interesting thing, isn’t it? I feel like I have some new insight into him as a mathematician.  I have since noticed that he also needs reminders to slow down when he is doing calculations.  He also does a better job when it is just me and him and he doesn’t have to worry about his sister butting in with answers. (Are you noticing a theme here?  It’s hard to be the little brother!) Finally, he does a much better job and enjoys the whole thing more when he can do single step problems. I feel like that last part is developmental and will work itself out over time.

My diagnosis is that there is an executive functioning thing going on.  He is using his working memory to do multiple tasks each time he takes a turn, not the least of which is to manage his emotions around the fact that his big sister is always butting in.

I am, of course, thinking about how to help my son with this particular thing.  But what does this look like in a classroom?  I’m thinking it would be useful to sit down with a few of my students and play a round of “Sorry” or “Trouble” or even “Snakes and Ladders” and really play with them.  They do these sort of things sometimes during indoor recess, but if I were to set this as an activity during class it would be so a group of children would be busy while I work on the real math with other kids.

Time to rethink that practice.

## Finding Connections

A few weeks ago, I sort of made my husband famous when I wrote about how he and I had each solved a problem about a good deal.  I had used an elegant solution, to quote Cathy Fosnot, and he had used the long division algorithm, which was just fine, also to quote Cathy Fosnot.  (You can hear the whole thing here.)

I had occasion to ask my husband to solve some math problems again this week, and I thought I should make sure that everyone knows he is my go-to double-checker. His methods may be old-fashioned, but he gets the job done.

I have applied for a Teacher Learning Leadership Project grant (TLLP) and am pretty close to getting it approved (I HOPE!!)  At the beginning of the month, I received a request for some clarifications about the project, which is apparently what they do.  Makes sense.  I am asking for just shy of the maximum allowed amount of money, so I’m actually glad to know they are making sure people are being fiscally responsible with this money.

One thing  I was asked to do was to make sure my budget aligns with the project goals.  I went over the entire thing with a fine-toothed comb, making sure I had the right number of days, and had figured it all out properly.  I have to account for the number of days each member of the project will be out of the classroom, and how much it will cost to provide coverage for the class.  My principal is joining, so I have also had to account for the extra money paid to a “teacher in charge”.  One member of our group doesn’t actually teach at my school, so I have factored in some mileage for her.  Even though I’ve been over it a few times, I needed someone to double-check it all for me.

I suppose you could say we worked as a team here.  I gathered the information we needed, and organized it into problems.  My husband, new to using an iPhone took one look at the work I needed him to do, pointed at his phone and asked, “Does this thing have a calculator on it?”  I showed him, and he proceeded to answer each question  on it’s own.

Again, the interesting thing here is that he didn’t see that he could solve one and use it to solve the others.  The 4 day option is double the 2 day option, but he didn’t use this. I think I am paying extra close attention to this right now because in my class we have been talking about splitting.  If we know that 40 + 50 = 90, then we don’t have to start over to solve 44+50.  We know it is 4 more than 90!  For some people, this might not be a huge revelation.  But for me, when I first learned to do math without using algorithms, these important connections between problems were completely missing. The only time I used anything like this was when I figured out 3 x 7 = 21 (for example) and then found all the 3 x 7 or 7 x 3 on a Mad Minute and wrote 21.  But I wouldn’t, for example, notice that if I knew 3 x 7 = 21 I could use that to help me with 3 x 8.

Connecting is one of the 7 mathematical practices in the Ontario Mathematics Curriculum: Grades 1-8 (2005).   On page 16, it says:

Experiences that allow students to make connections – to see, for example, how concepts and skills from one strand of mathematics are related to those from another – will help them to grasp general mathematical principles. As they continue to make such connections, students begin to see that mathematics is more than a series of isolated skills and concepts and that they can use their learning in one area of mathematics to understand another. Seeing the relationships among procedures and concepts also helps develop mathematical understanding. The more connections students make, the deeper their understanding. In addition, making connections between the mathematics they learn at school and its applications in their everyday lives not only helps students understand mathematics but also allows them to see how useful and relevant it is in the world beyond the classroom.

Since I have started to focus on teaching students to see the connections in math, I have noticed an increase in their over all number sense.  For many, as soon as they see a connection, it’s like a switch was flipped and they “get it”.

## Is it a good deal?

Perhaps the only thing worse than being a teacher’s kid, is being a teacher’s spouse. My husband is both.  His mom and dad were both teachers when he was growing up – and  his dad was teaching at his high schools!  The worst possible fate, right?  His dad was also a principal, but luckily after my husband graduated, and in another town.  So, yes, it could have been worse for him.

He doesn’t remember this happening to him, but I am always experimenting on my children.  I try out new games, and practice my mini-lesson and conferring language, and I read them the books I am previewing for lessons planning.  This week, I decided to give them a break and experiment on my husband.

The next conversation with Cathy Fosnot and Stephen Hurley on VoiceEd.ca takes place tonight.  We were challenged to solve a problem in preparation for the learning we are doing.  I teach grades 2 and 3, and thought this was a problem that would not be appropriate for them, but I wanted to try it out myself.  Here is the problem:

A neighbourhood store is selling 12 cans of cat food for \$15.  Another store is selling 20 cans for \$23.  Which is the better deal?

You can watch a teacher introducing this to some students here.

My first thought was, “Division algorithm”, but I am pushing myself to think in new ways, so I did it like this:

I found halves of each side, until halves no longer made sense.  Then I found 3rds for the first store, and 5ths for the second store, all in order to find out the price per can of food at each.  I feel confident I am right. (Which I only mention because I don’t always feel confident when I don’t check my division with the standard algorithm.  I’m getting better! I teach primary students, and haven’t taught anyone older than grade 4 for a really long time, so division isn’t something I’ve had to think about as a teacher in a really long time.  I have taught basic, beginning division of course, but mostly I’m focused on adding and subtracting – the Number Sense, Addition and Subtraction Landscape skills.)

On a whim, I decided to ask my husband to do the same. He is not a teacher, so hasn’t done any learning about math since he was a kid, unless you count the stats class he barely survived when he was working on his master’s degree.  Here is what he did:

I conferred with him after, and you can see how well that went.  🙂

Here is the interesting part for me:  He has always been good at math.  He plays cribbage too quickly for me to learn from him, and has done so forever.  He calculates in his head really quickly, and can seriously look at numbers and just know (as he indicated in our conference!) He is a social worker and has to do math all the time, but all of it is calculating, and all of it the same week after week.  He calculates mileage, works with money, figures out time sheets for the people he supervises.  He’s had this job for 15+ years though, so he doesn’t ever have to problem solve with math.  Or at least he rarely has to. Mostly, he figured out how to do this 14.75 years ago, and now he just does the same thing week to week.  And he does it every day/week/month – so lots of practice.

The problem really came for him when I asked him to explain.  He was so irritated with me for pushing him to talk about his strategy, how he knew what he needed to figure out, why he needed to divide. He had trouble putting his thoughts into words, and that is not typically a problem for him at all.

This is one of my favourite things about teaching math now:  I’m not just training kids to do math tricks.  I’m teaching them to be math thinkers, and math communicators, and math users.

(Update:  Click here and you can listen to Cathy Fosnot and Stephen Hurley talk about the math in this problem, including my solution.)