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.

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.

Geometry, math, Math Workshop, Number Talks

Real Geoboards vs. Virtual Geoboards

This past week we’ve been doing some geometry work in class.  The grade 2 curriculum expectations for geometry are fairly simple:  name, sort and make 2D and 3D shapes.  In general, children arrive in grade 2 already knowing most of these.  The more common the shapes are in the natural environment, the more likely this is true.  Octagon and hexagon usually give everyone a hard time with their tricky names, but by the end of grade 2 few children can’t recall these names.  In grade 3, we have to do a few more things. The vocabulary is increased (quadrilateral, angles) but again other than folding and unfolding nets of 3D solids, it’s nothing too complex.  Of course, I say that from this point of view – some kids do find it a bit tricky.  In all, however, it’s about 1 week’s worth of expectations.  I like to teach them early on because there are a lot of problem solving opportunities that can involve geometry and once we have the vocabulary learned the problem solving comes more easily.

This week I had rubber bands on hand.  That’s not actually something that happens all the time.  Since we had them, I pulled out the old geoboards.  Lack of rubber bands is actually one of the main reasons I don’t always pull them out.  The virtual geoboards, available here, here and here, are so much more reliable.  And nobody can shoot a virtual geoband across the room at somebody.

In the activity shown below, our Friday lesson, students were asked to make some shapes according to a rule.  Then their classmates had to figure out the rule.  Was the rule: shapes that have 3 sides?  Shapes with 4 corners?  Shapes I enjoy making because they create cool patterns? Here are some of our results:

A few years ago I remember reading an article about how important it is for students to have a real experience with a manipulative before they move to the virtual version.  I tried to find that this morning and couldn’t.  My brain doesn’t remember the source!  So I put it out to my virtual PLN (Professional Learning Network) on Twitter, and found a lot of teachers agreeing with my thinking. Reflecting on our week today, I was so glad that I had used the real geoboards.  There was some really interesting stuff that happened.

First, students were making shapes of different sizes over and over in different ways.  On the apps, they tend to get busy playing with other tools – like changing the colour of their geobands, or colouring the shapes in with a variety of colours.  They get focused on the wrong things and come up with rules like:  Shapes that are orange.  And I’m sorry, but orange is not a geometric attribute.

Second, I noticed that some children were struggling to stretch the bands across the pegs.  Some of the rubber bands are smaller than others, so this became a problem solving challenge.  I feel like they were motivated by the task to work through the challenge and find a new rubber band or change the perimeter of their shape.  This simply doesn’t present as a challenge to be worked through in the virtual environment. I had forgotten about this part!  Developing learning skills needs to be embedded in every part of our day, and I’m glad they got this chance to work on some problem solving skills.

Finally, there were some social things that we could work on.  Sharing is always an issue for 6 and 7-year-olds. They had to cooperate and collaborate to share the rubber bands on their table, and to decide who would get which geoboard.  I tried to make sure that every board at least matched the other boards in a group, but I didn’t always make that happen.  Again, students had to talk through this because everyone wants the one that is different, and therefore special.

Now that we have spent some time with the geoboards,  they can become one of the activities students can do during a Math Workshop session.   I can put them on a table with some task cards, or the students can request them to help solve a problem.  When we move on to perimeter and area (after we spend some time working with number lines for the next 3 weeks!) I can incorporate questions about shapes and feel confident that everyone knows the shape, and can work with the shape.  And I can add some

 

math

Games

Last week I wrote about using the 100 board in class. There are many games that can be played with a 100 board, and all of them help strengthen a child’s number sense.

I recently bought some 100 board games from Arnold Tutoring, and we’ve been having a great time with them at home. We like to play all sorts of games, so it was easy to convince my children to play these!

Both games were definitely worth the money! I can download a 100 board for free, but by the time I copy, laminate, buy all the game supplies and a nice container to keep it all in, it will have cost me a bit. Plus: I have no time! So the games were definitely worth the $— I spent. And I didn’t have to pay for shipping. And I love internet shopping. Really, this was more like an investment in my quality of life. (Seriously!)

“What’s my number?” is like one of our favourite games, which is called “Guess Who?” Except this is played with numbers instead of people. “Add to 10” is a bit tricky for us right now, but we easily changed it to suit us (my son is good with numbers, but he’s only in grade 1, so not quite ready for double digit addition.). We rolled the very nice 0-9 die, and then talked about our strategy for moving forward on the board. If nothing else the game would have paid for itself just using it this way. But I know we’ll get to more involved math before the end of the school year. This game set will grow with us.

These are games meant to replace math worksheets. They are for families that want to strengthen their child’s math skills, and have a good time doing it.

math

The 100 Chart

It used to really bother me when my students filled out a hundred chart in random order. I wanted them to start at and finish at 100, demonstrating that they can count in order, and that they could print all the numbers. But I’ve changed my thinking on this.

Here is some work we did this week:

In each of these examples, the child is following a pattern. It’s not always the standard counting pattern usually it’s the patterns in the one’s place.

This tells me more about my growing mathematicians than whether or not they can count to 100.

Next week we are going to fill in the numbers from 101-200. I can’t wait to see what patterns they continue to use to fill out the chart! And I really can’t wait for the conversations we’ll have around this!

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.