math, Number Sense & Numeration

Equal Playgrounds?

Believe it or not, playground equity is something my students have talked a lot about.  At my school, we have a “Primary” playground and a “Junior” playground.  The Primary students think the junior side is better, and the junior students agree, but sometimes wish they could go back to the Primary side to play. I mention this because the context of a community playground and having equal play space is something I know they could relate to.  This week’s problem can be found in detail here.

Trying to solve this problem reminded me how much I still struggle to see and draw a model of my thinking.  I really didn’t have any trouble with those rectangular subs!  But 3/4 of 2/5 and 2/5 of 3/4 sort of messed with my mind.  I wanted to work it out with some manipulatives at school today.  I was going to use colour tiles. Unfortunately, I have this thing called a “full time job” that tends to interfere with my blogging life and I didn’t get to it.  Add to that the fact that we are right at the beginning of the “Measuring for the Art Show” unit and any spare time I have to work things out for my own understanding is being devoted to that right now.  (I’m going to blog about that unit at the end of the week! Or I’m going to work on report cards.  Or maybe both.)

I was able to work this one out with the numbers though. This is my first attempt at drawing a model:


I was confusing myself, so I went back to numbers. After, tried to model it again (on the top sheet.)

img_9149Working this problem took me right back to school.  I was forever answering the question I thought was being asked instead of making sure I understood the problem.  Time and again I would have a test returned to, read a problem to see if I could figure out what I had done incorrectly, and discover I simply hadn’t answered the right question. With age comes wisdom and this time I went back and listened to the teacher set up the problem a second time before I tried to solve it.  The first time I thought I was to find 3/4 of the whole and 2/5 of the whole and compare to see if the playground and pavement were equal.  Now I know what I have to ask myself if the question makes sense, and this one did not.  After listening  the second time I knew I was to find 2/5 of 3/4, and 3/4 of 2/5 and compare which is a different question!

Without going into too much detail, I’ll tell you that this turns out to be a common concern amongst my colleagues. Students answer the question (in reading, in writing, in math, on the provincial assessment) they think was asked, not the question that was asked.  I love the way putting the problem in an understandable context, and presenting it in a way that encourages students to really discuss the problem until they understand what they need to figure out, helps us make sure we are really assessing what the students know about math, not about following directions. And I also love that the conferring process can focus on straightening out misconceptions so the student can go forward with a set of skills that help them make sense of problems, not just find answers.


And I love that if I was actually going to teach this unit (I won’t because I don’t teach the grades these units are recommended for) there are some models modelled in the books and I could learn how to draw them before I teach the unit to my class.

math, Number Sense & Numeration, Uncategorized

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 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: img_8827.jpg

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.)

Guided Math, math

Sometimes I get into deep Twitter conversations that leave me thinking for days.

A few days ago, I got involved in one of those Twitter conversations that reminds me why I love having a professional learning network of strangers.  I love being pushed to think about my practice and how I might change it, or why I might keep it the same.

I have settled, I think, into a Math Workshop teaching structure rather than a Guided Math structure this year.  And by settled, I mean we do what suits us when it suits us.  When we are deep into a Context for Learning unit (a.k.a. The Fosnot Units) the workshop hums along.  When we are working on some of the other problems and activities, I feel like it still moves along, but not quite as smoothly. I’m cutting myself some slack for that.  But I am definitely not running a centre-based program with a set schedule that we follow (today is game day, tomorrow is iPad day, and blah blah -whatever else people do in centres.)

One day last week Mark Chubb posted about targeted instruction, and one method for organizing groups so those with similar ability levels, based on their answer to 1 question.  In summary, he wasn’t sure if students should be organized into ability groups.   It got me thinking about how I organize my students so I thought I’d write about that this week.  (You can read the post here.  It’s about a lot more things that what I have written today!)

I agree with what he says about students who are struggling being paired into a “low group”, and that this will only really serve to teach them basic skills and not push them into higher level thinking, thus widening the gap between them and peers who “get it” quicker. In his exact example, it would keep kids in a vocabulary learning group while their peers are moving forward to problem solving. I wholeheartedly agree that a child can learn lots of things about geometry while still, occasionally or always, calling a hexagon a “pentalellogram”.  

There are lots of ways groups and partnerships are organized in math class – or really any class.

  • Random partner assignments:  pulling sticks with names on them, or slips of paper, in order to create partnerships.
  • On the fly, sort of organized partner assignments:  the teacher calls out pairs, figuring out who to put where in the moment and trying his/her best not to leave two for the end who shouldn’t work together.
  • Planned in advance partnerships:  the teacher makes intentional learning partnerships in advance of the activity.

I have been using a method I love, and will continue to use, for a while now.  I plan out the learning partnerships I want in my class, and then I have those students work together, throughout the day, for an entire month.  Here is why I like this way of doing it:

  • I figure out who will work well together because of their learning skills, not just their subject specific skills.  I think this helps children see each other in a variety of contexts, and helps them polish their collaboration skills.  I like to pair two really quiet people together, for example, so they are “forced” to start talking and collaborating.  I like to pair two children who naturally fall into leadership roles in a partnership together so they learn to negotiate in their communication – hopefully learn to listen to each other and not just talk.  I like to pair three kids when one is perpetually absent – with a group of three it’s OK if one is away.  And on, and on, and on.
  • Working together for a long stretch of time means they get a lot of the social stuff out of the way early on, and spend the rest of the month more focused on learning. You know those kids who will spend a whole work period moving around the room trying to find the perfect spot?  Or the whole work period arguing over who is going to be the scribe (this mostly, but not only, happens when markers are involved!)  Well, when kids are working on the 2nd or 3rd task together, they don’t even talk about where to sit, or who will write.  They just go to the spot they picked the time before, check their memory to see whose turn it is to scribe, and get to work.  
  • I can make sure that students don’t have the same partner two months in a row.  Inevitably they will work with each other again and again.  This is especially true since I find myself using triads quite often and there are only 20 of us…sounds like a math problem!  10 months, 20 kids how many different combinations can I make??

So I think this all fits into my ongoing thinking about Guided Math vs. Math Workshop.  I think I am grouping based a bit on ability (I don’t want a strong grade 3 with a grade 2 who is still trying to make sense of it all…for both their sakes!)  but mostly I am thinking about how a group of 2 or 3 kids can push each other across the day – in math, but also in their communication skills (written and oral).  I get to shuttled around the room offering guiding questions and moving kids along.  I also know that if I don’t get to someone on Monday, there’s always another day or another time during the day when we can connect and talk about what is happening with the group.  I like that kids work with everyone in the class eventually.  And I like that I can spent about 20 minutes at the beginning of a month organizing groups, and then not worry about it in the end.

I do ask students for their input, and ask them to do some self reflecting.  Who do they want to work with and why? Do they think they are being a good partner?  What could they do better?  That sort of thing.  They like having the input, and they do get quite good at thinking reflectively about themselves as learners.


Non-standard units

After some interesting discussion about thermometers, we measured the temperature of hot water, snow and the classroom air. We had time to jump into some measuring of length, etc.

We’ve been measuring the length of things with rulers all year. But this was the first time I asked the class to use non-standard units. I put different kinds of math manipulatives out, and said, “You’ve been sitting here a while. Why don’t you go use the tools I put out to measure your tables.”

Some groups decided this meant they were to measure the perimeter of the table. Others measured the distance across.

I was surprised when these students used different units (blocks) but didn’t think it was a problem. (they are in grade 3.)And these grade 3s saw nothing wrong with spreading the blocks out. And the kids with the snap cubes counted every single cube by 1s (181 cubes total.)

This was a good assessment in that it showed me where my students are in their understanding of measurement. They all know how to use a ruler, but clearly they still have things to learn about measuring.

Geometry, math, Measurement, Number Sense & Numeration

I’m making a plan!

This is the point in the Winter Break when I have started to think about sitting down to do some lesson plans. Instead, I just spent 30 minutes on social media reading garbage, and am now writing.  My lesson plans will wait until Sunday night, right?

I have started to work on report cards this week though.  This always gets me in a reflective mode.  Mainly I am asking this:  Have I covered something from at least 4 strands of the math curriculum in a way that will allow me to write a good report card?  I still have 4 weeks to go, so if the answer to this is “no” then I have time to make up for that.  I know we have done plenty of Number Sense and Numeration, and lots of Patterning and Algebra.  I think we have done enough Geometry, if I spend another day or 5 (or 6) on that, and I am going to comment on Measurement this term too, but will need to spend a few days doing some of those activities.  We did a lot of measurement in science, but I haven’t asked them do anything lately.  I want to make sure I have done something recently that I can comment on.

That leaves Data Management.  We have done quite a few things that are part of Data Management, but I don’t feel like I have done enough to comment on this strand yet. One of the things I have been working on this year is integrating math into other parts of our day.  We did lots of measurement in science, for example.  I wanted to do more data management in science as well, but we got side tracked.  I have not taught a single measurement lesson during math though, so I feel good about that.

Number Sense and Numeration, as well as Patterning and Algebra, are the areas I have always felt I needed to spend a lot of time on during math.  As a result, I have often rushed through Measurement and Data Management/Probability.  It’s not that I don’t think these are important.  It’s just that I was prioritizing one over the other.  By thinking about how to teach these outside of my regular 60 minute math block, I think I am seeing connections that will help my students build connections and we can all use math in more meaningful contexts.  For science, we were growing plants on the window ledge. For 3 weeks, every couple of days we pulled out the rulers, measured the height of our plants, and recorded that in our journals.  That’s meaningful.  I also recorded the results of a mould growing experiment on a chart as part of our science learning.  But we haven’t taken the next step and graphed any of this, and that is why I’m feeling like I am not ready to report on this yet. I could get there by the first of February if I really wanted to, but I have other plans for this so I’m not going to rush it.

In the coming term, we are going to be learning about Movement, and Strong & Stable Structures.  February and March are really interesting months to track weather in Ontario.  These are things that will give us a context in which to use some data management and probability related math.  I’m not worried about making sure we get enough practice with these concepts.

To get ready to finish first term reports, I guess my math month long plan will look something like this:

Week 1:  Measure things, like temperature & time (January is an interesting time for this, I think.  We’ve talked about time on a clock a fair bit, but need to talk about this human way of measuring the passing of our lives.) (This will also lead us into a social studies connection since we will be learning about Canadian Communities 1780-1850 in Social Studies during the second term.)

Week 2:  Use pattern blocks to measure length, width, area, etc. Talk about why we get different answers when we use different pattern blocks to measure the same thing. (geometry connection…this will give me a chance to check in with a few kids who were having trouble naming attributes of some 2D shapes and see if they’ve met that goal.)

Week 3:  We’ll do this part during our science time: Build 3D shapes using stuff (cardboard, spaghetti & marshmallows, etc.) and start talking about strong, stable structures (science connection for 2nd term)  In math we will start our next Context For Learning math unit (“Measuring for the Art Show”).

Week 4:  By this time I need to be finished with all of my math recording, and should be able to write everyone’s math report card comment.  Should.  🙂  I really want to sit down with each child and ask them some of the questions from our first math assessment in September, but realistically I’m setting an “end of February” deadline for that.  If the Polar Vortex (is that what were are calling it this year?) continues to churn over North America, we’re likely to have some bus cancellation days. This will help me meet that goal since I’ll only have a few students each of those days, but will also hinder me in meeting that goal because I tend to have the same few students each of those days.