math, Number Sense & Numeration

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.

math, Number Strings

Minus/Subtraction/Take-Away

Minus…subtraction…take-away.  Do these all mean the same thing?  They are certainly all represented by the same symbol.

Last week the math coordinator was in  my class for a few days.  (Here name is Melissa and she blogs here!)  After watching me do a number string related to subtraction, she encouraged me to always say “subtract” when I am reading the problem to the class, rather than “take-away.”  Some kids will actually do some adding to solve these types of problems, and by always saying “take-away” I would be restricting their thinking and maybe even imply that they need to use a certain strategy, namely that they need to remove.

I know that there are different ways to solve a subtraction problem:  add on, count back, think of it as a part of a fact family and figure out the addition problem.  But I hadn’t really been intentional about my language when discussing subtraction with the class.  I was more focused on the answer!  (I’m hanging my head in shame!) (not really…but you know what I mean!)

On Thursday and Friday we had bus cancellations, so I didn’t really get a chance to try this out until today.  We were working a Number String from Cathy Fosnot’s mini-lesson book.  We talked about 14+1, then 14-1 (Did you read that 14 subtract 1? or 14 minus 1?)   Then I gave them 14-13.  You can’t see them in this picture, but I had the maths.ca relational rods going in the background, and had build 14+1, and 14-1, and those were still visible to the students. I saw lots of kids with their fingers out counting back.  It’s an okay way to get  correct answer, but very inefficient.  However, I then asked my favourite student (my daughter!!)  to tell me how she solved it.  I’d seen her working away on those fingers, and I know that if she spent a tiny bit of time thinking before she started that, the answer would have been obvious to her.  Knowing this, I had to ask her about her strategy.  “Well, I thought about having 14 cookies, and then I ate 13 of them, so yeah…one is left.”  This is not totally unreasonable for her (don’t judge my parenting!) especially if they are Viva Puffs!   I annotated her thinking like this:

I pointed out to everyone that CC was thinking of subtraction as “taking away” something.  And then asked others what they thought about when they saw a subtraction sign.  Someone else said, “Well, I knew you would only need one more to get from 13 to 14, so I knew it would be 1.”  I talked about how that child was thinking about the difference between 14 and 13, which was different from CC’s but they both still got the same answer.  Then we did 2 more problems from the string, and talked about the “take-away” strategy and the “find the difference” strategy.

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Someone even mentioned that they thought about 9+2=11, which is a great connection to some work we did a week or so ago, so that was awesome too.

It’s funny how being intentional about how I was reading that symbol to the class changed the strategies they used.  This wasn’t truly the goal of the Number String, but “m delighted by the results.  I am hoping the forecasted 25-35 cm of snow (and 80 Km/h winds!) hold off until late on Tuesday so we can work on this again tomorrow.  I feel like we are developing a really big understanding about subtraction!

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.