math, Number Sense & Numeration

Update: Assessment

I’m interviewing everyone in my class to make sure my report cards are up to date and accurate. It’s been very telling!

I often get one-on-one time with students, but they are usually at different places in their work. During the interview, I’m asking the same 6-8 questions, and talking about the strategies the kids use from beginning to answer. One question in particular is standing out because so far my friends fall into 3 categories.

The question is: I have 7 crackers, you have 9 crackers. How many do we have altogether?

One child said, without pause, “16.” This child was confident, and didn’t falter at all when I asked how he’d gotten the answer so quickly. “I just know things like that.” When I asked other questions he was equally confident and had very efficient strategies.

Another child, same question: “…mumble…mumble…it’s…16?!” I asked for an explanation. “Well, I know 9 is almost 10, so make it 10, then 10+6…yeah…16.” Earlier in the year this child told me he solved problems by reading my mind until he found the answer. I’d say he’s made excellent progress in his meta cognitive and communication skills!

Another child, same question: “….2?” I repeat the question. “7!!!!” I repeat the question. “9!” I take a handful of counters out of the nearby basket & make a pile of 7, and a pile of 9. Then I say, “These are mine. I have 7. These are yours. You have 9. How many altogether?” Response: “If I take away 2, then we’re even!” And “Is it almost time to eat?”

So I put the counters away and write on a piece of paper “7+9” and the child says 16. Rote memorization for the win!

There are three things going on here, and if I made each of these three the team captain I’d have no trouble finding people in the class with similar thinking to fill their teams. Each of the other questions I’m asking further shows the thinking behind the answers I’m getting from the class, including showing me the preferred strategies each child has. It’s so much more interesting than just getting a worksheet filled with answers.

Data Management, Geometry, Measurement, Number Sense & Numeration, Patterning & Algebra

Another One About Reporting

As the end of Winter Break approaches, it’s time for me to sit down and do some planning for the coming weeks.  Reports cards are due at the end of the month and I need to get all of my assessments up to date and my comments organized.  The report card should reflect what the child is capable of at that time, not what they were doing 2 or 3 months ago. I last formally reported on everyone in November. I know there has been growth for everyone, some big and some small.

For math assessment, I am going to re-do the interview I used in September.  I know that for some children I can start in a different place because they have shown mastery in areas I previously assessed.  I will have to go beyond where I left off with them because they have shown growth toward the end of year goals. I also need to add in some geometry and data management questions so I can report accurately on that as well.  I have a lot of anecdotal notes to draw from, but I want to be really sure of what they can do now.

As I have been reflecting on this, I am struck once again with how hard it is to divide math into 5 strands.  I suppose it is easy in the Primary grades to do that with Geometry, Data Managment/Probability and Measurement.  But even at this point they are all starting to blend together. Everything we learn in Number Sense is related to everything we learn in Patterning and Algebra.  I can hardly decide how to mark everyone sometimes because I’m not always sure if the things they need to build understanding about exist in one strand of the curriculum document or another.  I have to consult it every time because in my mind it’s all mashed together into “math”. Everything we do in Number Sense is related to what we do in Measurement too, but it’s a little easier to seperate out the skills that will be reported on.  Same for Geometry and Data Management/Probablity.

Here is one example of this from the Grade 2 curriculum document (2005):

  • identify and describe, through investigation, growing patterns and shrinking patterns generated by the repeated addition or subtraction of 1’s, 2’s, 5’s, 10’s, and 25’s on a number line and on a hundreds chart (e.g., the numbers 90, 80, 70, 60, 50, 40, 30, 20, 10 are
  • count forward by 1’s, 2’s, 5’s, 10’s, and 25’s to 200, using number lines and hundreds charts, starting from multiples of 1, 2, 5, and 10 (e.g., count by 5’s from 15; count by 25’s from 125);
  • count backwards by 1’s from 50 and any number less than 50, and count backwards by 10’s from 100 and any number less than 100, using number lines and hundreds charts (Sample problem: Count backwards from 87 on a hundreds carpet, and describe any patterns you see.);

Two of those are from the Number Sense strand and one is from P/A.  But I teach them simultaneously. And if a child is having trouble with skip counting is it because s/he isn’t understanding the patterns associated with the skip counting, or is having trouble memorizing the order, and if they seem to not be having any trouble is there some rote counting, or is the child processing the numbers and thinking about the patterns?  It’s tricky to assess sometimes. And sometimes it isn’t. For instance, if a child can say, “2, 4, 6, 8, 10” but then stops and can’t figure out what comes next, I know the first 5 terms are acutally just counted by rote. Or if a child can count by 2’s even further, but then isn’t able to do this when there are actual things to be counted, I know there has been some memorizing. And if a child gets to ten, then pauses to work it out in his head, comes up with 12, then slowly with 14, and so on, I know there is some understanding.  It’s tricky to boil all of that down to a letter grade.

Someday when I open my own school and can make my own rules, I am not going to assign letter grades to Primary kids ever. The report cards at my school will be all about the comments.  And I will definitely not divide math up to strands!  But for now, I’ll sit down and go through my assessment and the curriculum documents, then I’ll sit down with everyone in the next 2 weeks or so and ask them the questions I’m wondering about.  And then I’ll sit down and give them all a grade that reflects what they can do.  Easy, right?

 

Guided Math, Number Sense & Numeration, Number Strings

Addition of double digit numbers

There were about 50 of these on our whiteboard at different times over the last few weeks. We’ve gotten pretty good at adding the tens using mental math strategies. 20+20 -> 2+2=4, so 20+20=40…no problem! But it’s was time to move on!

I really wanted everyone to learn how to effectively use a number line. We’ve been working our way through a Context for Learning kit called “Measuring for the Art Show”. I demonstrated it about 1 billion times. First we used cubes to make a line, then we annotated this on some cash register tape, and then we moved to the whiteboard. Finally, I gave everyone some problems (from the kit) and some paper they could use for drawing the number lines.

As I walked around I could see lots of kids with lots of right answers but no number lines. “How are they doing this??” I wondered. So I asked. And I was amazed! So many of them were using the mental math strategy of splitting. They thought about how many ones there were in each number, and how many tens were in each number, then they found a total.

But the number line isn’t the go-to strategy yet. So I’m annotating the problems two ways now. The way lots of them are doing it, and the way some of them are doing it. Both are ways I hope all of them can do these problems eventually.

Our next step…my next step…is to organize into small groups (I know…back to some guided math. It keeps coming up!) I need to help the kids who are splitting learn the number line, and help the number line kids do some splitting and help the “I have no idea what to do kids” get some ideas.

I’m ALWAYS happy to reach Winter Break, but it always comes about a week before I’m ready. I don’t want to interrupt our math learning. But I’m confident the stuff we’re doing now will stick. I won’t have to start from scratch on January 7.

Here is more work from today:

math, Number Sense & Numeration

Understanding vs. Memorizing

When I was in elementary school my teachers regularly asked us to complete Math Mad Minutes.  These were sheets of math problems, usually just 1-digit numbers, and we had to complete as many as we could in just one minute.  Some years we did addition and subtraction, some years multiplication and division.  Sometimes we even had to do a Mad Minute that had a variety of operations on it.  When I first started learning how to become a teacher, my mentor teacher used these.  Children started with a sheet that had 20 problems, and if they could do all of those in a minute they upgraded to a sheet that had 30 problems!  The super fast kids got a sheet with 50 problems.

I hated doing these.

I remember having only one strategy:  I went through the Mad Minute, week after week, and did all the problems that had 0, 1, or 2 for an addend, subtrahend, or factor.  If I saw a number along the way that was a “double” I would do it (3+3, 6×6).  Basically, I memorized the location of the problems for which I knew an answer.   I have a clear picture of myself sitting in Mr. Goodrow’s 6th grade class and reciting to myself the answers to the top two rows of problems.  I was certainly memorizing a bunch of stuff but I wasn’t actually memorizing anything useful beyond the Mad Minute.

True confession:  In my first classroom as a teacher, I finally “memorized” the times tables for good. Nobody gave me a sticker when I could recite them all, but I did it anyway.  I was teaching math on a rotary to 90 fifth grade students every day.  I have a clear picture of myself standing at the whiteboard writing answers to multiplication problems and realizing there was a pattern to the answers.  I was 27. I was university-educated.  I feel quite confident nobody had every told me about these patterns.  It opened a door for me.

What if I had understood this sooner?  Sticking with the multiplication example (though I could also talk about how understanding addition and subtraction is equally important!) if I had understood these connections and patterns I’m sure division, fractions, decimals, algebra and statistics would have all come much easier for me.

I’m listening right now to a Ministry of Education “Town Hall” call.  People are advocating for spending the Primary grades memorizing facts. The thing is, nobody ever says, “In the Primary grades kids should just memorize words.  We’ll teach them to understand words, read sentences, and write sentences once they get to the junior grades.”  Sounds ridiculous, right?

So if you are at home at night and want to work on helping children memorize math facts, then go for it.  But in class, I have some really important foundations of understanding to build. I have concepts to connect, I have patterns to point out, and I have number sense to build. You will not find any Mad Minutes.  Do I want them to have facts memorized?  Absolutely!  Are we actively working toward that?  FOR SURE! But I’m not going to focus on this at the expense of spending time on building understanding.

math, Math Workshop, Number Sense & Numeration

Games

This past week I had a chance to think about, and talk to colleagues about, how my family did math at home when I was a kid. My answer: we played games. I don’t recall having much homework until high school. I always got off the bus (5th grade and up) by myself and had about an hour before my mom came home. I typically did my homework while watching Days of Our Lives.

After dinner, however, it wasn’t uncommon for us to play Rummy, Gin Rummy, or Uno. We also enjoyed Yahtzee and Monopoly. If nobody was into it, I’d play Solitaire by myself (yes, I was a kid before it was possible to start a game online with someone.) When I was a teenager my dad taught me to play Black Jack. (It’s much less stressful with my dad than it is in Atlantic City for sure!)

Now I like to play games, in class and at home with my own children. Some of our favourites are War, Addition (or multiplication) War, and Tens Go Fish. You can play any of these with a standard deck of cards. Remove the Face cards if your children aren’t ready for adding, or multiplying, 11-13!

Today in class we did 100 chart puzzles. I copied 100 charts on card stock and cut them apart. Since I made these I have been using a 120 chart in class, but I couldn’t let the game go to waste. I suppose I should pass it off to the grade 1 class and make a new set. It was easy, and cheap. I’ve got nothing to lose. You can see here that some of my puzzles are more challenging than others.

Besides practicing math, games are a great way to practice taking turns, and losing with grace. These are important skills for kids to learn too!

Data Management, math, Number Sense & Numeration, Patterning & Algebra

Which Way Do I Go?

The beginning of the year is hard for me in math. There are so many things that need to be done!  This is especially true for those of us who are teaching split grade classes.  Some things are the same: number sense, for example.  I can figure out where everyone is and take them to where they need to be.  But my grade 2s are supposed to learn about some things that the grade 3s are supposed to already know (which sometimes they do and sometimes they don’t) and the grade 3s are supposed to do things that the grade 2s are not (which sometimes they are ready for and sometimes they are not!)  And I know I can still do the things, and it won’t hurt anyone to learn about something a year early, but it all takes time. And even though it’s only the 29th of October, I feel like time is slipping away and I need to GET ON IT!

So this week, I was feeling like it was time to move on from adding the tens and the ones.  I gathered the balances so we could talk about balancing equations.  I started planning in my mind where we’d go next.  But by Friday, I realized that I might be moving on a bit to fast.

Remember when I wrote about how we were having trouble communicating our math thinking? Well, that hasn’t gone away yet.  Now that we are adding, and even subtracting those double-digit numbers, I thought, wouldn’t it make sense to stop there and do some problem solving?  Wouldn’t it make sense, I asked myself, to take this thing we are pretty good at doing and use it to practice the communication piece?

So this is what we are doing.

  1. Trip over the balances that are shoved out of sight behind my desk. It was a pain to get them into the room so I’m just going to live with them for a while.
  2. Monday’s problem:  (Two versions because I am differentiating!)There are 14 red apples, 15 green apples, and 8 yellow apples.  Can each child in the class have one apple? 

     There are 4 red apples, 5 green apples, and 8 yellow apples.  Can each child in the class have one apple?

     

  3. Tuesday’s Problem: I bought some Halloween candy this weekend!  I have 15 suckers, 23 Smarties, and 30 Kit Kat.  Do I have enough for every child in our class to have 3 pieces of candy?   

    I bought some Halloween candy this weekend!  I have 10 suckers, 12 Smarties, and 4 Kit Kat.  Do I have enough for every child in our class to have 1 piece of candy? (The Smarties are stressing me right now because I mean 23 of those little boxes of Smarties, but there are 10 actual Smarties in each.  There’s a unitizing thing in there.  I think I’ll just have to verbally clarify with the class before moving on.  I’d just take out the Smarties all together, but I’m sort of feeling committed to them now because it’s going to give us something good to talk about.)

  4. Wednesday: Give in to the evil of Hallowe’en and graph some candy.  (I try to do random survey’s and graphing instead of a data management unit.  I’m going call it spiralling, like all the cool #iteachmath teachers.)  Then they’ll work on these alone, not with their Learning Partners:Make a list of 10 ways you can add two numbers and get the answer 37 every time. 

    Make a list of 5 ways you can add two numbers and get 10 every time. 

  5.  Thursday and Friday: Depends on how the other days are going.  I really want to make sure that I am not rushing through.  I want to take the time to congress the solutions properly, and to talk about what makes a good visual representation of the groups thinking.  We are starting up with November Learning Partners (a few days early because we were all just DONE with the October groupings!) I have a fun nrichmaths activity that we will do if things are going well.  And I have some 100 chart puzzles we can do, which will help reinforce the work we’ve been doing about noticing patterns in the 10’s and ones that help us take leaps of 10 and 1.  We are on to Measuring for the Art show next, and this is an important understanding for that unit.
  6. Then it’s Monday again, and we can balance some equations.  Probably.  Most likely. “It is highly likely that the class will work on balancing equations next week.” to put it in data management and probability  language.  And then we should move on to some geometry because that is something I have a hard time integrating on it’s own at this particular grade level.

Even though I am feeling compelled to get moving, what I really want to do is make sure everyone understands what we are doing now.  These adding and subtracting and patterning and data management skills are so important and there’s no sense in moving on until everyone is ready, not just me.

 

math, Math Workshop, Number Sense & Numeration

Counting

I never used to worry too much about teaching students to count. I mean, the year I taught kindergarten we did a lot of counting, but in grade 2…or 3…or 4? Nope.

In the last few years I’ve become aware of how important counting is, and the layers of skills that are involved.

After interviewing my people, I discovered most can start at 30 and get to 100 without difficulty. Some had trouble not starting at 1 – and they also had trouble assembling a 100 chart. This is not, I think, a coincidence.

One of the classes figured out we need 12,000 laps around our track to equal Terry Fox’s journey across Canada.  We’re keeping track of our contributions with tally marks!

Some students were very organized with their counting…

And some students were not organized with their counting.

After we congressed these photos, I sent everyone off the do more counting. I asked them to count out 17, or 24, or 52, or 65 of the math tool they wanted to work with. Everyone tried an organization strategy of some sort! Even when they sorted into rows or groups of 5, many were counting by tens.

I loved this one because this child said, “10, 11, 12, 13, 14, 15, 16, 17!” It was great to see her counting on!

I think when we look at these images and congress them today, I need to make sure I talk about how the organizing helped. One of the things I know I need to do better is point out these things that seem so obvious to me.  Some students will have already realized the advantage of using groups, but some will not have.  I need to help them with that.

I used to always start the year with addition. This made more sense I guess when I was teaching grade 3/4 classes. But starting with counting and making sure everyone has a strong foundation of number sense to build on has truly made my addition and subtraction units go more quickly. Everyone seems more prepared for the addition and subtraction work once I know they are solid with counting skills.