math, Number Sense & Numeration, Number Strings, Number Talks

Subtract

I know it’s only Tuesday, but I’m in a “celebrate the little things” frame of mind.  It’s helping me cope, which is pretty important for teachers under stress.  And other humans of course.

This week, only 2 days old, I am teaching area and perimeter.  But I like to continue with Number Talks even when we’ve moved away from the computation portion of our work. Yesterday I started with one from Sherry Parish’s “Number Talks: Whole Number Computation” book. I chose one from the grade 2 section:  20-19, 20-14, 20-12.  We talked about how we could use all of our strategies from the math wall, but that counting up seems to work better than counting down.  We used the math racks for this activity, and I celebrated my own ability to say “subtract” every time instead of reading the problem as 20 TAKE AWAY 14.

Then today, we started with 20-19 (because one thing I have definitely learned from Cathy Fosnot and her Number String work is that starting with a helper problem is very important – this is not part of the Number Talks book work.)  Everyone remembered or quickly figured out that the answer is 1.  Then I wrote 30-19.  They weren’t so sure at first.  It took us a while to get to 11.  Some kids wanted to count all the way down (their go-to strategy).  Some wanted a math rack so I pointed out it doesn’t have 30 beads so what would they do, they didn’t have a plan. Some made random guesses AS ONE DOES!  We started to talk about our potential answers.  I purposely called on a student that I knew had the right answer AND that I knew could explain the thinking to share.  This child did a great job of noting that the helper problem was helpful.  If we think about how 30 is just 10 more than 20, but we still subtracted 19 then the answer should be 11, or 10 more than 1. I pulled out the math racks. I showed them 30-19 using a 20 rack and a 10 rack.  This way I could show how I added 10 more in but didn’t remove any more than I had the first time, leaving 10 more remaining after I “took away” or “removed” the 19 beads on the rack. After we did 40-19, someone noticed a pattern.  This is honestly the thing that makes me happiest sometimes.  Some groups of kids will notice patterns right away and then stop puzzling over the math because they have found the short cut.  But this class always takes a bit longer to see the pattern.  It’s fine with me, because we have a lot of good conversations along the way.  But I’m also glad when the pattern becomes part of our conversation too.  Even after this child thoroughly explained the pattern, some were skeptical.

Along the way I was drawing the number lines.  I decided to draw a new one each time because I wanted them to clearly see how each time the 19 didn’t change, but our starting and ending point did.  I also wanted to highlight the iterated units.

We talked about each of our strategies:  does it make sense to count up?  to count down?  to try and “take away” something?  to think of an addition problem that would help?

Two weeks ago I blogged about a moment when everyone was working independently.  It has rarely happened since. But it is happening, throughout our day.  Slowly but surely we are edging forward, and today’s math was a reminder to me that we are indeed doing good work. Not every day is easy.  In fact, today wasn’t that easy.  But this was my shining moment.

We’ll continue on with this tomorrow, but I will use some other number besides 19.  I am also going to write out today’s work on a chart paper because I think it will help us going forward and, therefore, deserves a spot on the math wall.

math, Math Workshop, Number Sense & Numeration, Number Strings

Subtraction

What does “-” mean in math? As in 5-2=

We had an excellent conversation about this during a Number String this week. I went off script after discovering that some of my friends didn’t know how (as in “no idea” how) to subtract 23 T-Shirts from their inventory in our math lesson.

This is what we decided:

The next day we talked more about “take away” using the math rack. Then we talked about how we can count back on a number line. Then we talked about how we can actually count up on a number line in order to find the space, or difference, between 2 numbers. One friend was very keen to keep explaining how adding and subtracting are opposites, and during one explanation solidified his own understanding of how he uses what he knows about adding to subtract.

I had my strings planned out for the week, but on Wednesday I realized we actually needed to do something different than planned. Literally everyone in grade 3 is doing well adding double digit numbers. They need more practice for sure, but the Strings were not moving us forward. At the same time a weaker skill (from way back in grade 1) popped up and I felt it was a good time to address it.

I’m thinking more and more and more about how math learning happens on a developmental continuum. Everyone travels along the continuum at a different pace. Hopefully nobody dawdles in one place for too long, and hopefully they remember what they’ve learned. I’m confident everyone has past experience with subtraction. However, for some reason, it didn’t stick. It’s for this very reason I had my grade 2’s dawdling in their own spot on the Landscape of Learning all week. They played games that had them practicing addition facts, and started creating their own flash cards to take home and practice. Next week they’ll be doing the same with subtraction facts.

math, Number Sense & Numeration, Number Strings, Number Talks

Number Strings/Number Talks

Math was fine this week. We started doing more place value work while working on “The T-Shirt Factory” Context for Learning unit by Cathy Fosnot. It’s always an interesting one, but I actually didn’t do it with my class last year.  We weren’t ready for it until much later than this and when we were ready for it…I forget what we did instead.

This week was Halloween. That means an interrupted day on Thursday because of the Halloween Parade.  I anticipated a day of difficulty on Friday as well, and while we’re at it, let’s just admit that Wednesday wasn’t going to be easy either.  See how hard it is to stay on schedule?  That’s why we didn’t exactly stay on schedule with the unit.  However, I didn’t skip math any of those days – even the snow day on Friday!

I did a fun mapping activity with a Halloween theme one day when I was pulled out for a meeting, and we did a lot of work with the base ten blocks.  But every single day I made sure that we were doing a Number Talk.

During a Fosnot unit, there will be a lot of Number Strings.  But when I am not teaching a specific skill and want to review things that I hope everyone already knows or that I know they need to practice, I go back to Number Talks.  This week we used some from the Grade 2 section of the book “Number Talks” by Sherry Parish.  We started with single digit numbers and I found out on day one that most everyone understands commutative property.  I repeated a talk that would reinforce this with double digit numbers on the second day.  On Wednesday, Thursday and Friday we talked about the “doubles plus 1/doubles minus 1” strategy.  These strategies are now displayed on our math wall so we can refer to them often.

This week, I am doing Number Strings to support the learning in the unit.  Because I can barely remember what day it is on most days I have to write my numbers on a Post-it note.  These will sit on my lap top all week or I will lose them.  I find it also has me thinking many times during the day about what we are doing in math, which is good for my brain.

Some of these are from the unit and some are from my head.  I know my learners well enough at this point that I’m sure we will need to do practice the skills that are in these strings multiple times.  They’ll be practicing them while doing the work in the unit as well.

I’m going to finish off my planning today by making my anecdotal record sheet for this unit.  (Just double checked and I already made one a few years ago! WOOHOO!

math, Number Strings

Math Workshop Thoughts

I’ve been learning from Cathy Fosnot for many years.  I first started using her Context for Learning Math Units about 15 years ago.  I’ve read her books, even the newest conferring book.  I’ve attended in-person workshops with her.  I listened to every episode, sometimes more than once, of her podcasts on VoicEd Radio. (Go here if you want to listen!)

You might think I didn’t need to go to a 2 day workshop to learn some more from her, but HOLY COW I learned so much in the last days.

Both days we focused on using Number Strings to promote the development of numeracy.  I wrote in my notes:  We do STRINGS to promote a development of NUMERACY – a deep understanding of number and operation. We want to eliminate as much working memory stress as possible. We want SO MANY things to become automatic and known so they (students) don’t have to work every piece out.  

After spending a few days working through Number Strings with Cathy and many of my colleagues, I have come up with a few things for me to work on adding to or refining in my teaching practice.

  1.  Cathy had all the problems in the string listed on the side of the board.  She added her models and numeric representations on the side.  Some of these were erased as they got messy or as she ran out of room, but the equations in the string stayed up during the whole conversation.  This will help students have the answers from previous questions, and helps them see the patterns in the questions, which will hopefully help them see the patterns in the solutions that can help them learn how to solve equations.
  2. Cathy talked a lot about using multiple representations for a single strategy.  This helps children who understand one begin to understand another.  This is also because if they don’t understand the first they have a chance of understanding the second. Overtime they can develop some fluency and flexibility and choose for themselves which strategy or representation makes the most sense. In my notes I wrote: “DIFFERENT REPRESENTATIONS DEVELOP DIFFERENT THINKING! Choose the model carefully. Go back and forth between representations so that they can constantly see the connections.” This is also something Monica Neagoy really stressed in “Unpacking Fractions” (Summer book club, which I just realized I never blogged about!)  Showing things with many different models helps kids understand the math instead of just understanding the model.
  3. I have been modelling decomposing using carrots, and Cathy was modelling them using parentheses/brackets.  She admitted this is new thinking for her.  I learned about the carrots from her!  I’m going to use the brackets from now on.  The carrots do make it messier. I also think that going straight to the brackets, using associative property and commutative property from the start, even in primary grades, is going to remove an attitudinal barrier that exists for students who think they are starting to learn algebra for the first time in grade 5 or 6.  I think this makes it so obvious that algebra is taught all along!

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Those are 3 pretty good goals, I think.

I am excited and mostly ready to go back to school. Workshops like this, at the end of August, help me get even more excited about starting with a group of children and helping them grow as mathematicians (and readers and writers, and humans.)  I’m also looking forward to more podcasts with Cathy Fosnot on VoicEd starting at the end of September.  I feel like I STILL have a lot to learn from her.

math, Math Workshop, Number Sense & Numeration, Number Strings, Number Talks

Early Algebra

We started the “Trades, Jumps and Stops” Context for Learning unit quite some time ago. A variety of inclement weather days has interrupted us a lot, so we are behind where I thought we’d be at this point. That might be part of the issue I’m having with this unit.  I’ve not taught it before, so that has to take a bit of the blame as well.  And finally, I’m thinking I might have misjudged our general readiness for this unit.  But I talked to my down-the-hall neighbour who is also doing the unit and she concurred:  It seems that some kids are easily getting it, and some are really, really having to work hard to get it.  There’s not a lot of in-between here.

Day 4 of the unit begins with a mini-lesson that we needed quite a bit of time with.  I was to fill 2 separate bags with a certain amount of coins in each, plus a “mystery” coin.  We have had a bit of trouble adding up coins, so I decided to stretch this out a bit instead of trying to get through it in 15 minutes.  Instead of putting the bags out of reach, I gave the coins to kids.  I started by explaining, “X and Y have some coins.  They have an equal amount of money, but they have different coins.  I want to see if you can figure out which coins they have.” On the first day only one child had a “mystery” coin (a poker chip!), but on the second day both kids had one.  The children holding the coins were very excited to be given this job. It gave them practice identifying the coin by name and value.  Have you ever thought about how we interchangeable use “dime” & “10 cents” when talking about coins? Some of the kids are still calling this “The Boat”.

We unpacked the bags a bit at a time.  I don’t have pictures of the whole process, but this is how it looked in the end.  Obviously our mystery coin was worth 10 cents this time.  This is the beginning of the children learning about a variable, and I think they did OK!

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On the second day, we talked more about the signs < > and =.  We added up the coins in chunks, as suggested in the lesson, and then decided if we needed to have a < or a > or if we finally needed the =.  This time, instead of adding them up as a group, I had the students work with partners.  We knew that X had 40 cents, and Y had 5o cents, but how much would they have once they each got another quarter?  This partnership was trying to make a number line across the bottom end of this photo, without actually making the number line.  It’s a step in the right direction for them!

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They wanted to have the numbers in a long line, but couldn’t hold all those totals in their head. Writing them above helped them work on the math and compensated for the stress load on their working memory.

We had lots of people able to do this:

Finally we made it to the mystery coin.  We knew how much money everyone had in actual coins, but what was that mystery coin worth? This led to one of those really cool moments when I felt like (most) everyone was excited about the math.  You can see here where I recorded some of the different responses to the value of the mystery coin.

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I was even able to use a double number line to show two of the answers, and since that is the goal of this unit (developing an understanding of the double number line is in fact the very next lesson we will do!) I felt very good about that.

The next day I needed a Number Talk to reinforce the understanding of the variables.  I didn’t really NEED to do it, since this doesn’t come into the curriculum until a later grade, but they were excited about it, and it certainly can’t hurt them so we did it anyway. I found some images on Math Before Bed and use them for our Number Talk.  I feel like they really help to reinforce the student’s number sense because there is more than one way to make 10, or 12, or any number really.

The weather here is terrible today (Sunday) but I’m hopeful there will be plenty of actual school days this week when we can move forward with this unit.  I don’t want to lose our momentum!  Next time I do this unit, however, I am going to maybe wait a bit.  Of course, the thing that keeps getting me through is that I have to trust Cathy Fosnot! She says this will work, and she has seen it work with many, many children, so I am going to go ahead and finish the unit.  The students are mostly getting the ideas behind the math. Some of them are actually in need of more practice with adding up money.  I am going to make sure we have a day this week when we have some activities that require counting money and we’ll rotate through those to give everyone lots of practice. Often these blogs help me think through what has happened and what needs to happen next!  Time to stop writing and plan my money counting activities.

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!

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: