Monthly Archives: October 2016

5 Ways to Represent a Two-digit Number in Grade 2

I typically teach grade 5 and 6, so when I helped out in a grade one/two class and the students were asked to represent a 2-digit number four different ways, I drew a little bit of a blank.  I immediately thought of representing the number with base 10 materials, and then thought of using tally marks, but that was as far as I got!   What are four different ways to represent a two-digit number?

Using some of my own ideas, and watching and learning with the kids, here are some representations we came up with:

1.  Drawing Shapes in a One-to-One Correspondence

Most students started with representing their number using shapes; one picture or shape until they reached their two-digit number.

It was a lot of drawing, and a lot of counting, and some kids found it challenging to count what they were drawing accurately.  Most students drew their shapes in rows, which was easier than counting 30 or more shapes randomly arranged on the page, but there were still some counting errors.  We talked about strategies for counting accurately, but some students still found this difficult.

There were also a number of students who thought that drawing squares and then drawing triangles counted as two different ways to represent their number.  It opened up discussion that representing a number with a one-to-one shapes or drawings was one strategy, and choosing a different shape did not change the strategy.

2.  Tally Charts

The second-most popular representation students used was the tally chart.  I could tell that this was a representation they had been practicing.  Almost all students used this representation, and used it confidently.

Many students checked their work by counting the groups and skip counting by 5s, but some counted each tick individually if they lost where they were or wanted to double check.

3.  Ten Frames

Students had only very recently been introduced to the ten frames.  Some students shied away from using it because they weren’t confident enough to try it even thought they identified as a representation that could be used.  Others got stuck on trying to draw the ten frames, drawing a 10 frame with only 8 squares.

If they made mistakes drawing their ten frames, they typically did not catch their error because they checked their work by counting by ones instead of tens.  They did not realize that a 10 frame holds ten.  A rather interesting error.  I thought some students might count their 10-frame as 10, but not a single student did.

4.  Groups of Ten

It had been suggested to the students in an earlier math activity to organize things into groups of 10 to assist with accuracy.

Students made the connection to this activity and grouped shapes into sets of 10.





5.  Number Sentences

Before students set off to task, the children participated in a class brainstorm and discussion to come up with some ideas about the different ways a number could be represented.  One of the ideas that was shared was that number could be represented as an addition or subtraction sentence.   Very few students used this representation, although many students had not finished the task before it was time for me to go.  They got so caught up in the counting I think, that they forgot about number sentences.

Final Thoughts

Some students were not able to come up with 4 different representations.  Some students represented them inaccurately because they mis-counted, or drew their 10-frames with only 8 squares.

How do we get students comfortable with all of these ways of representing number?  And how do we get them representing number accurately so they don’t make mistakes counting or drawing 10 frames?

I created a chart to summarize the ways that grade 2 students can represent number.  Feel free to use it in your classroom.



Creating a Positive Classroom Culture: 4 Practices to Help Students Meet High Expectations



As part of the positive classroom culture that I want to create for my students, I am working on creating a place where meeting high expectations is the norm.

This is part of my over all vision of a positive classroom culture, which has four components:

Culture of Risk-Taking and Mistake-Making:  Students feel safe to take risks and make mistakes.

Culture of Belonging:  Students feel a strong sense of community and identify themselves as an important member of the community.

A Culture of High Expectations:  Students are supported in meeting high expectations, both academic and behavioural.

A Culture of Perseverance:  Students develop a growth mindset.

In this blog post I will focus on the practices and I have been implementing to support the third component, creating a culture of high expectations.

1)   Students Know How and When to Apologize

One of the first things we talk about at the beginning of the year, is that it’s ok to make mistakes.  In fact, mistake-making and risk-taking is part of the class culture I am trying to create for my students.

But how does that apply to behavioural expectations?  I make behavioural expectations clear at the beginning of the year, but I know they are going to make mistakes.  It’s not realistic to think that students will follow them perfectly all the time.  To keep expectations high without expecting perfection, we need a plan in place for when it does.

Essentially, students are expected to ‘make things right’.  This can mean a number of different things, and one of the ways to make things right is to apologize.  We discussed how and when to apologize, and what to do when on the receiving end of one.  We use a four-step apology in our classroom, one where they identify specifically what they are apologizing for, acknowledge why it was wrong, identify what they are going to do differently next time, and asking if the person if they will accept their apology.

Having clear expectations about what is to be done when we make a mistake, break a rule, or hurt a classmate helps to promote a culture of high behavioural expectations in a positive way since students since students aren’t expected to be be perfect, but they expected to take responsibility for their actions.

2)  Post Success Criteria for Assignments

In order to promote a culture of high expectations, students need to know what success looks like.

I have been trying to post a checklist citing the success criteria that will be used to assess their work for every activity, lesson, or assignment.  Sometimes I will post rubric, but for simplicity I often use a checklist since most students find it easier to assess their own work against a checklist than a rubric.

3)  Post the Classroom Expectations Everyday

It probably sounds strange to take down the poster that has our classroom rules written on it at the end of each day.  But because I do, it means that everyday we must post them back up again.  It keeps the classroom expectations front and centre.

If I hang the poster in September and leave it up, after a while, the poster seems to almost disappear.  It’s been there so long we don’t even see it anymore.  Having to re-post it everyday, or every week,  keeps it relevant.

I ask a different student to read the expectations to the class, and find a place to post it.  I thought this might be a little cheesy for my grade 5 and 6 students, but hanging the poster has become a regular part of our morning routine and they don’t think anything of it.  Not only does it serve as a reminder to the students, but it serves as a reminder to me to refer to the classroom expectations when addressing student behaviour.  I find it helpful if I can tie the issue back to the expectations that they had agreed upon at the beginning of the year.

4)  Provide Feedback

Students need to know what they are doing well and what they need to work on if students are going to meet high expectations.    Feedback is essential to make this happen.

We talked about feedback early in the year, and students know that it is an important part of learning and improving.  I helped to prepare students for regular feedback by letting them know that feedback would be kind, specific, and would be in small doses.  Students understand that it is given in small doses because they are expected to improve their work in response to the feedback for next time.

Classroom Culture

Next time I will talk about what I am doing to help create a culture of perseverance.

Happy Learning.

#MyMathStory: How I Fell In Love With Mathematics

About 10 years ago, I fell in love with my arch nemesis.

I fell in love with math.

Nemesis:   something that a person cannot conquer, achieve, etc., an opponent or rival whom a person cannot best or overcome.


The Primary Years:  The Friend with Some Annoying Habits

In grade 1, my only math memory is being mildly jealous of the kids who could quickly and easily skip count by two.  The teacher asked students what some of their strategies were for being able to skip count by twos.  One of the strategies that sounded good to me was to say every other number in your head, but that ended up being a terrible strategy because I didn’t have time to say a number in my head when we were doing choral skip counting as a class.  I was behind by the time we got to 6, so I reverted back to just moving my mouth and hoping my teacher didn’t notice I wasn’t saying anything.

In grade 3, we started learning multiplication.  I had been warned by my older sister and my parents I would have to be quick with my multiplication tables.   I wasn’t quick even with things I did know, so I had resigned myself to the fact that this was going to be unpleasant.

We didn’t do timed tests that I recall, but I remember being stressed and having feelings of mild dread about being called on in class.  Times tables were tough, and some of them, like the 6, 7, and 8 times tables were really hard.  At least with 3’s you could skip count reasonably quickly by uttering 2 numbers under breath before saying every third number louder in your head until you got to the answer you wanted… unless of course you got so distracted with the skip counting that you forgot to stop in which case you had to start all over.  That was the worst.

I was very relieved part way through the unit when the teacher let us keep a multiplication chart taped to the corner of our desks.

This was apparently something new.  I recall having the sense that the teacher had a logical reason or two for doing it, but also that her argument for doing so would fall apart under too much scrutiny.  It was probably bending the rules a little bit.  I knew better than to tell my Dad about it.  He would probably think it was a bad idea, and if enough people who thought it was a bad idea found out we might have them taken away.

I remember the teacher telling us that they were there to help us learn, but I wondered if that was true or if she had really just given up.  Maybe trying to teach us multiplication was just as frustrating of an experience for her as it was for us.  I didn’t really care though; I was very glad they were there.

The Middle School Years:  The Bully that’s Not So Bad If You Stay Under the Radar

In Grade 4, I remember my brain hurting a little doing 3 digit multiplication, and long division.  It wasn’t difficult, I knew the steps, but it was a lot of skip counting by a lot of unpleasant numbers.  Cumbersome, if not tedious.  It took a long time to do questions and homework, probably because I daydreamed about more interesting things for most of the time.

It was also in grade 4 that we were encouraged to try some of the “brain teasers” in our math books if we finished early.  They were impossible.  After the occasional enthusiastic start I knew better than to even attempt them.

I continued to do well for the remainder of my elementary schooling by following procedures to solve math problems.  If I did get stuck i would ask for help from my Mom, but she would often refer us kids to Dad… But asking my Dad usually ended in tears.  I was getting A’s for the most part, and while there may have been the odd test that came home with a low score that my parents weren’t thrilled about, my marks were no reason for any major concern.

My High School Years:  The Bully that’s Discovered You Exist

Once I reached high school my experiences with math deteriorated with every passing year.  I got a C in grade 9 math and my parents made me repeat it in summer school.  The “solid” foundation I got in summer school didn’t pay off; my marks stayed in the C-range until they dropped off drastically in Grade 12 and OAC (grade 13).

Every year I struggled to find the rules and procedures that would consistently  bring me to the right answer, and every year there seemed to be fewer and fewer.  I wasn’t about to raise my hand and look stupid, so I rarely, if ever, participated, and never asked questions.  Once, when called upon in trigonometry my teacher, unimpressed with “I don’t know” as an answer to his question, told me I didn’t belong in his class.  I kept trudging along trying to crack the code on my own.  When random exceptions to a rule came up I tried to memorize when the exception applied.

Math was something that I was just going to have to endure.  Like my trig teacher said, I didn’t really belong.  I was there because it was a hoop I had to jump through, meant to weed out most of the applicants applying to programs under the ‘Bachelor of Arts’ umbrella .  A necessary evil.

I tried to figure out the procedure, the exceptions, and the exceptions to the exceptions, and it all got to be too much.  By grade 12 and OAC (grade 13) I squeaked by with a D in Trigonometry and a generous 51% in calculus.

The only way you get a 51% in a course is if you fail but the teacher passes you anyway.  Mr. Khan passed me out of the goodness of his heart.

It was during the term that I was taking Calculus with Mr. Khan that the teachers went on strike.  We must have been only days into the school year when the strike started.  I started going to a school in another board so that if the strike went on a long time I wouldn’t lose the year since it was my last year of high school.

It took a week or two to get registered and put in classes, so my first day in the calculus class the teacher informed me that a test was scheduled for the following day and that I would be expected to write it.

How on earth I was going to write a calculus test when I had not had a day of calculus in my life was beyond me.  A friend, who was good at math, tried to teach me what he could about calculus in a couple of hours that evening.  It was not an easy task since I couldn’t even tell him what was going to be on the test.  Telling him “calculus” was apparently too vague.  I didn’t have a single homework assignment or a single note to study from.

He explained to me what he could, and having nothing to lose since I couldn’t possibly look stupid having not had any calculus instruction whatsoever, I began to be vocal about what didn’t make sense.  I asked why something worked and why something else didn’t, and why things worked in one case, but not in another.  Sometimes he had an answer, but often he didn’t.  Sometimes he would say that something I suggested wouldn’t work because it didn’t make sense, or because  that’s not how it works, but couldn’t really explain why.

I didn’t realize it at the time, but I wasn’t really looking for answers to my questions about why something did or didn’t work.  I was really looking to make the point that math was random and didn’t make sense.   This is why math was horrible.

And ridiculous really.

He sympathized, but ultimately I had to accept a lot of random things without any understanding just so I get through the test.

I wrote it, and I think I almost past.  I wasn’t embarrassed about this mark. I had 3 hours to study for a test that took the teacher three weeks to cover.  In hindsight, I wish I had kept it.  Today, I would have it framed.  It said everything I thought about math but didn’t have the words or courage to say.  I showed my parents, knowing they couldn’t possibly be mad.  I found it amusing that I could almost pass a test with zero understanding, not realizing that, in fact,  I had been doing that for years.

My University Years:  Surviving the Hurricane

You know it’s coming.  You can’t escape it.  So you bunker down and hope you come out the other side.

That’s how I got through my university calculus.  I failed it the first time I took it.  The second time I got a D.

Done.  I’d never have to take another math course again.

My First Years Teaching:  Hanging Out with that Old, Familiar, Comfortable Friend

My first few years of teaching elementary math were spent getting re-acquainted with the math I was comfortable with.  I even relished the role of showing my students that math was really not so bad.  Nice in fact! Predictable.  At this level you follow the process and everything works out.  It made sense.  Input.  Output.

Perhaps it would be more accurate to say it wasn’t so much getting familiar with an old friend as it was Stockholm Syndrome.  You know, math is not so bad so long as you don’t poke the bear…. math is nice if you follow the rules and the rules still work.  I didn’t have to tell them about what was coming down the pike.

I made sure my students were good at following the rules and procedures.  I thought they should appreciate how I didn’t let math beat the life out of them with those impossible brain teasers.

Yet, students complained math was boring.  I really didn’t know how to respond to this.

Initially, I was defensive.  I told myself they had no idea of the beast I was saving them from.  Boring?  That was like complaining that putting on a seat belt is boring.  Boring-ness has nothing to do with it.  This “boring” math could save their life one day.  If they could just get really good at knowing what to do and when, and then maybe get really good at figuring out what to do and when, they might one day crack the codes I wasn’t able to crack in their math courses later on.  Or at least survive math until such time as they didn’t have to ever take it again.

When I wasn’t defensive, I was sad.  I was sad because I believed that this — what I was teaching and doing with my students — was as good as math got.

My students were in grade 6 and were past the pain of learning multiplication facts and tedious long division that makes your head hurt.  I let them have their multiplication charts.  In middle school, there is math that looks pretty daunting, like exponents and equations that take up the width of the page, yet can still be figured out.  While far from a favourite past-time, there was a sense of accomplishment when I was in middle school and figured out a question that looked scary and took a page of work to solve.

But moreover, I was sad because I had the sense that math was lovely when it wasn’t horrible.  I believed, despite my experiences with math, that math was supposed to make sense.  I continued teaching procedures not knowing what else to do.  If that was all I could do, I would at least do it well.  Still, my students struggled.

The Day Everything Change 

I can tell you with certainty that the day I fell in love with math was the day I took an in-service with John Mighton, the founder of J.U.M.P. Math.

Something clicked.  The way he taught the concepts in math made all the codes I had learned in elementary school make sense.  The way I remember it was he showed us a number of different rules and procedures you typically learn in elementary school and explained why they worked.  We were shown how to uncover these to our students.  He covered a fractions unit, but I took what I learned and was unlocking things across all topics in mathematics.

His approach was the most genius thing I had ever heard.  I left, not only understanding math, but also feeling pretty sure I could help my students understand it too.  I left feeling like I was floating.

Now, I hadn’t found the holy grail.  I knew that there was still a lot missing from my teaching of mathematics, but a large piece the puzzle had been found that was the key to a lot of other pieces.   I wasn’t the best at teaching math after that, but boy, did I love teaching it.  Middle school math was the beautiful thing I knew it could be and I did my best to share that with my students.

New Love

I spent the next year to two teaching what I understood, and if I didn’t understand something I would tell my students so.  I assured them that sense could indeed be made of it, even if it wasn’t by me.  I told them that if they ever figure it out to please come back and tell me. With every passing year I learned a little bit more about math, which allowed me to teach it a little bit better.  I was feeling my way through the dark, but progress was being made.

I quickly realized though that loving math, or even loving just the teaching of it, did not leave me with much company.   People thought I was weird.  Or they thought I was a math person.  Yeah, that’s a funny one.

Eventually, I was loving teaching and learning about math so much that teaching one period of math a day was not enough.  I found other ways to get my “fix”.  I took teacher Continuing Education (CE) course in mathematics, and tutored high school math (yes, high school).  I even took university calculus over again as a mature student in case I ever wanted to upgrade my 3-year degree to a 4-year.  I joined the NCTM which I had heard about from taking the CE course, and I was a member for a couple of years before I finally took notice of the endless number of reminders that I was inundated with to register for their annual conference.

I usually just deleted them without reading, but one February I guess I had seen one too many and thought I had better open one to feel better about discarding it.  I don’t know what it said, but it must have been good because I was off to the NCTM Annual Conference that April.  I went again the following year.

To back up for a moment, not one of these things I did without agonizing over the decision to do it.  My husband can attest to that.  I entered each decision knowing that it was really a disaster waiting to happen.  The tutoring, the CE course, retaking university calculus, going to the NCTM conference, all of it.  Every step was like jumping off a cliff in the dark.

Things worked out though.  That university math course I re-took?  I got an A+.  I took the second half of the course and, admittedly I understood a lot less, but I still got an A-.  It was madness.

It wasn’t pretty though.  I spent hours studying.  I was not studying to get that A I ended up getting, but just to make any sense of the material whatsoever.  And that tutoring?  I am not exaggerating when I say that anywhere between 4 and up to 12 hours of work went into just one hour of tutoring.  So I find it funny when anyone calls me a “math person”.

Truly, I am not.

I tell people this, and explain that after hours of asking the right questions and committing to not doing anything without understanding why, I can usually figure it out.

They still call me a math person.  I don’t think they understand how hard it is, the time it takes me, to understand math.

But I love the beauty found in understanding mathematics.  And that, if anything, is what makes me a “math person”.  Everyone has the potential to be a math person or become one at anytime.  Anyone can love math.

So I guess that’s #MyMathStory:  I am a teacher with no special talent, but who has a passion, an appreciation, and a respect for mathematics.  I want to learn everything I can.

Thanks for reading this long post.  I learned a lot just from writing it.  I’d love to hear if anything resonated.

Happy Learning.

Reflections on the First Month Back to School


September has somehow come and gone.  I’ve seen a few posts of teachers sharing their reflections about how the first month of school has gone, and I thought it might be a good exercise for me to go through as well.  Here is what I have learned so far:

1)  Simplify with Fewer Routines. 

As usual, I started the year brimming with ideas of all the amazing, pedagogically-sound routines I was going to do with my students.  Most of my excitement was in the area of mathematics and I had plans for Journaling, Number Talks, Think/Pair/Shares, Depth of Knowledge Questions, Estimating activities, Fluency Stations,  problem-based learning, regular 3-act math lessons, and many more.

It’s been too much.  I have realized that I need to streamline my routines and just implement a few consistently.  I have chosen to focus on Depth of Knowledge Questions, Journaling, and Number Talks.

I know that I can introduce more routines as students become ready, but for now it’s enough for my students to work on these 3 routines.

Still, it feels like I am committing some sort of math sin not to include some of the other routines that I know about and see the value in.  To be sure, I didn’t cut them because my students wouldn’t benefit from them, but because it is proving too ambitious to implement too many routines at once.

Now that I write that, it seems kind of obvious…  It’s not realistic to implement 20 new routines during the first month of school, and well this year the number isn’t 20, it’s more like four.

And that’s ok.

Fortunately, we are running a marathon, not a sprint, and while the year goes by fast, and it’s tough to fit everything in, the solution is not to cram it all in but rather to be selective.

I am trying to think about it, not in terms of what I need to cut out, but rather as an issue of timing;  it’s a matter of when I will introduce some of the other routines and activities, not a question of if.

I thought deeply about what routines I would continue implementing and which ones will have to wait, and I will eagerly share my students’ thinking with the work we do tackle this term, and look forward to implementing some additional routines when the time is right.  Most of all, the 3-act math lessons.  I’ve been itching to try one.

2)  I Need More Groupings. 

I have a small class and I was hopeful that I would be able to have to make use of low-floor, high-ceiling questions as a way to differentiate and keep my groupings to two.  I have a small class, so number-wise, two skill-based groups not unrealistic.

This is proving not to work.  I teach a split grade, and in mathematics for example, I have students working at grade level (Grade 7), and some struggling to use “counting on” as a strategy to add numbers like 35 + 8.

I know I will be able to serve the students better by working with three groups instead of two even though it means I am spread a little thinner.

3)  Keep Going with the Meditation and Mindfulness Work

I have included a time of practicing meditation and mindfulness with the students  during a time of the day that has been a struggle for the past few years.  On Mondays,  I essentially lose an entire period between gym and their afternoon nutrition break when the students return from gym class.

In years past, I have experimented with what subject I taught during that time, as well as with the type of activity we do during that period such as quiet seat work, hands-on activities, and team-building activities, to name a few.

This year, I spend between 5 and 15 minutes teaching my students about meditation and practicing it.

It is still remains a challenging time of the day,  but some students are responding well,  and learning a valuable skill.  Despite the challenges, I am realizing that the students who are most disruptive during the meditation are likely the ones who need it most.  I also need to keep in mind that this is the first year I am teaching meditation and mindfulness.  There is going to be a learning curve as I figure out what works and what doesn’t.

Final Words

Reflecting on my practice tends to happen on it’s own as I think back on how a day, a week, or a unit has gone.  But rarely do I write my reflections down.  Articulating my thoughts in print, however, has me reflecting a little more deeply.

Thanks to those who have reflected on their September and inspired me to do the same.

Happy Learning.