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.)


The Math Pod: Week 1 Reflection

This weeks Math Pod podcast with Cathy Fosnot was about teaching math in context – making it relevant to students and teaching it as something they will need to use, rather than a bunch of skills they might find helpful in certain jobs or when they have to balance a chequebook or figure out their discount at the clothing store sale, and thank goodness for fractions if we want to double our cookie recipe!

In August of this year I was at a Summer Institute session put on by leaders in my board. Someone (I want to say it was the Director, but I might be wrong) said, “If kids want to know why and the only answer you have for this is ‘Because I said so.’, then you are not prepared enough.”  Or something like that anyway – that’s not necessarily the exact quote.  I’ve thought a lot about this since and was thinking about it again while I was listening to this podcast.  I’m noticing more and more how important it is to have a “why” attached to every single thing we teach.  In my first real classroom teaching assignment (in 2000), I taught grade 5.  I was the math teacher on a rotary team, which meant I taught math 3 times each day.  It was a great experience.  I can picture a boy in my first class so clearly saying to me, “But why do I need to learn to ‘Guess and Check’ to solve problems?  I hate that strategy!  I don’t like writing down guesses that are wrong and having to start again.  I want to figure it out in my brain and then write down the correct answer.” That is a direct quote.  He inspired me to go looking for the “why” of Guess and Check, which was a favourite problem solving strategy for the math textbook publisher whose books we used. Because I needed to give him a why (and thank goodness I realized I needed to!) I spent time figuring out this strategy, and realized it was more of an estimation strategy than a real guess.  That changed my teaching of this strategy because I realized what it meant, and how and when to use it.  Can you believe I was a teacher before I ever learned this? Well, I was.  (I’m putting the blame for that on my math teachers, who are all probably enjoying a lovely pension cheque right about now, and congratulating themselves on a fabulous career.) Later in the year, I was teaching multiplication of fractions.  Before multiplying, we would “cross reduce” the fractions so we didn’t have to reduce big numbers in the end.  The same boy wanted to know why this works.  I couldn’t tell him. It was a big school and there were 13 other math teachers in the building, including one working on her doctorate in math education, and not one of them could tell me why this works. There we all were, university educated teachers, teaching this “strategy” (or is it a trick?) to hundreds of kids year after year, and nobody could actually explain it. Weird, right?  But those two experiences, and a few others that year, and the year before when I was on a short term assignment in grade 7 and grade 8 math classes, showed me how important it is to know the math deeply before teaching it, or at least be willing to learn it along the way.

Early on I found that, especially when teaching math, I really had to consider what my students already knew, or already needed to know, before we could launch into an activity. I taught grade 5 for a while, with a really nice textbook to follow, and I taught kindergarten for one super long year.  But most of my career has been in grade 3/4 split classes.  Every year I’d find myself thinking, “Why don’t they know this?  Why do I feel like I am starting from scratch?” Then I taught a grade 2/3 split for the first time and really looked carefully at the grade 2 curriculum for the first time.  I realized my grade 3 students didn’t know certain things because I was the first person teaching it to them. I knew that was true about multiplication and division.   However, I didn’t realize how true this was when I was teaching fractions, or making graphs. The amount depth of work in these two subjects in grade 3 compared to grade 2 is huge.  This seems so obvious now, and maybe a bit embarrassing to admit, but I spent so much time looking at my own curriculum requirements that I didn’t have time to look at a different grade. That’s what learning the Landscape of Learning has done for me.  I’ve stopped thinking only about what the curriculum is asking, and looking more at what the students are doing, what skills they have and which strategies they use, and then going from there.  I know that I need to pay attention to curriculum.  I know that!  And I do that. But I don’t feel constrained by that.  Are they meeting those standards?  This is a question I have to ask as I prepare report cards.  Have I covered that material?  Of course I have to look at that. But as with reading and writing, I feel like I am moving students toward their personal next steps more and more, and we are doing this along a nice trajectory instead of trying to jump to a level that we are not ready for, and then floundering.

My biggest take-away from the last round of VoiceEd podcasts with Cathy Fosnot is that I don’t need to pre-teach skills before starting a unit. I will admit to having thought that before.  But I am not doing that at all this year, and I find it so interesting to see students really build the strategy for themselves.  I think this gives them a purpose for using the strategy.  They have seen it work, they have used it to solve a big problem so it makes sense to them.  This shift has really changed how I am approaching the Math in Context units.  I love how Cathy says that our math teaching shouldn’t be about some rich tasks, but rather a series of tasks that build one upon the other to help students progress in their understanding of math concepts.

I also love how Cathy Fosnot keeps talking about opening students up to the aesthetic of math, rather than just teaching them some useful skills. The useful skills are important, but we have to go a bit further than that.

I’m reaching the end here and want to have a really good summary paragraph that pulls all of my points together.  But this is more of a rambly kind of brain-purge. Hopefully not in an “Oh, dear.  It’s Sunday night and this woman has to be in charge of 22 little people tomorrow.  Someone do something!!” kind of way, but more of a “She’s experiencing some cognitive dissonance. Excellent!”  kind of way.