Subtraction is Like Australia

I have a question for all you math bloggers out there. HOW DO YOU DO IT?! How in the world do you find the time to sit down and write a post at least once a week? More importantly, how do you make it sound so perfect?! UGH! You’re like those people I went to school with– the ones who could whip out an flawless, on-demand five-page literary analysis in 3.2 seconds. How do you do it? How do your words just come out sounding smart and perfect? I honestly don’t know which one is harder for me– finding the time to sit down and write a post or the actual writing of the post. The former is near impossible, while the latter is simply excruciating. I wish I could make the ideas just flow and not have to rethink every second word.

You’ve probably used or at least heard of the voice-to-text app, right? Oh, what I wouldn’t give for a brain-to-blog app– an app that automatically records your thoughts into a blog. Yep, that needs to be invented yesterday. I’d have about 20 thousand posts by now because I’m constantly thinking about what I’d write in a post. Sitting down and actually translating those thoughts to coherent sentences is another story…

This is the second day (well, night, actually) that I’ve attempted to compose this post. Let’s hope that baby stays asleep long enough this time for me to finish. So a few nights ago I had the tremendous pleasure and privilege of learning from the brilliant Tracy Zager in her webinar, How Do They Relate? Teaching Students to Make Mathematical Connections, hosted by Global Math Department. Wow. That’s about all I can say. Wow wow wow. I am not articulate enough to describe the awesomeness of her presentation. I can, however, share a few humble thoughts.

In her presentation, Tracy emphasized the importance of relational thinking in mathematics. Oftentimes, students perceive math concepts as discrete topics rather than as a “landscape of interconnected concepts.” The term landscape immediately made me think of Fosnot’s landscape of learning (click here for an example). Similar, yet different. Anyway, Tracy asked us to think about two concepts that our students might have trouble relating i.e. fractions and division. Having just wrapped up a unit on developing multiplication with my 2nd graders, I thought perhaps addition and multiplication might be a good pair, although I felt like my students made a fairly strong connection between the two and wouldn’t really have trouble relating. Tracy then gave us the sentence stem, “What does ____ have to do with ____?” where the blanks are intended to be math concepts. In my opinion, this is a fabulous way to guage the students’ depth of understanding of interrelated concepts. I will definitely add this to my routine next year!

So Tracy challenged us to go back to our classrooms and pose this question to our students. Yikes! I felt both eager and apprehensive about how they would respond. Eager because, well, my kids have a way with words that can either bring you to tears with laughter or stun you to silence (in a good way). Apprehensive because…what if they didn’t know how to respond?! What if they didn’t see the relationship?! How would that reflect on my teaching???

So, a few days later, I posed the following question to my students– What does addition have to do with multiplication? I sent them off to reflect individually and record their thoughts on their iPads. Some wrote equations. Some drew pictures. One student told me, “You’re adding on for both.” Then he proceeded to draw an open number line, and illustrated ‘adding on’ by taking a jump from left to right. “See? When you add, you’re adding on. Multiplication, it’s like you keep adding on. So with both, you’re adding on.”

I brought the group back together to share their responses. Here’s what they came up with:IMG_1415

I thought the ideas of ‘remembering’ and memorizing’ were interesting, however my favorite response described addition as a ‘backup’ for when you don’t have a certain multiplication fact memorized. This student didn’t stop there. He went on to state matter-of-factly…

“…they’re kind of like continents…”

In my mind I was thinking, “WHAT?!” “Tell me more,” I said.

Slowly and thoughtfully, he continued, “So, addition is like a continent, and multiplication is a country in that continent…”

I couldn’t make this up if I tried. Read on.

“…and subtraction is the opposite, so it’s another continent…subtraction is like… Australia.”

I couldn’t help but laugh out loud. After all, I’ve never heard of a mathematical operation being likened to a massive landmass. Of course, I had to ask,”What about addition? Which continent is addition?”

He thought for a moment, then said,”Addition is like…like Asia.”

I was used to this student’s sagacious remarks, but regardless of his poetic analogy, I wondered if this is what Tracy was referring to when she spoke of students overgeneralizing, or attempting connections that don’t hold mathematically.

So what do you think? If subtraction was a continent, which continent would it be?


Nuances in Noticing

Well, I’m back. After two months of being MIA, I return to my much neglected, yet beloved blog-baby. Not that my readers (all two of them) have been sitting at the edge of their seats waiting earnestly for my next post, but it was quite motivating to believe that such was the case.

Back in January, I saw an interesting tweet from my MTBoS mentor, Simon Gregg, who teaches in Toulouse, France. He showed his Year 4 students a part of Pascal’s Triangle and asked, “What do you notice?” I was intrigued by the simplicity of this question, and was curious to find out how my own 2nd grade students would respond to such a task. Here’s what they came up with:

noticing I loved this task for so many reasons. First and foremost, the students loved it. Their level of engagement was sky-high as they discussed and debated with one another. The more they shared, the more noticings they made. Another reason I liked this task has to do with what those in the math education world describe as a “low floor, high ceiling” task. I interpret this as meaning a task that is easily accessible or initiated by all and that has numerous possibilities to challenge students individually. I chose the word ‘numerous’ rather than ‘endless’ because if the possibilities were endless, then I suppose the task would be considered “low floor, no ceiling?” Anyway, I digress…

Just before spring break one morning, I was looking for a little spice to add to my morning message– something unique, something math-y, something similar to the task above. Having recently stumbled upon the brilliance of Robert Kaplinsky, Graham Fletcher, and other like-minded geniuses, my new favorite questions as a math teacher have become, “What do you notice? What are you wondering?” So, I performed a highly sophisticated research study (i.e. I Googled the phrase ‘what do you notice math’) and came across this page. It was like a treasure trove of “What do you notice?” tasks. Needless to say, my heart was happy.

I’ve used a few of the tasks with my students, but here is the one that I found most interesting. This first photo shows the original from the website so you can clearly see the numbers.


Here’s my version of the task– same thing, just less clear than the photo above.

photo 2


I had high expectations for my students on this task. Again, I was very eager to hear what they came up with. As I read their noticings, however, I was slightly (okay, more like extremely) disappointed. “The dot in the middle is pink,” one note said. Another read, “The dots on the outside are green, the middle are orange, and the middle is pink.” My heart sank. Um. Thanks for pointing out the obvious, but…

I was stuck. Part of me wanted to ask them, “Seriously?!” but the other, better part of me chided, “Tsk, tsk, you know they can do this! Just give them a nudge…” So, I took a deep breath and rephrased my question. “Friends, you made a lot of, ahem, interesting noticings, but what would a mathematician notice?” I braced myself.


More silence.

A hand went up.

Then three hands.

Four. Five. Ten.

Then, just like with Pascal’s Triangle, they kept noticing. Several went off to make some calculations, then came back a few minutes later with their findings. As each student made an observation, the rest of the class would respond with a chorus of “Oh, yeaaaahhh!” as they realized what they had previously overlooked.

And then came the next question, “What are you wondering?”

Like a hundred fireworks going off one after the other, the already-energized conversation exploded into an exposition of questions and wonderings. It was hard to believe that these were the same students who were trying to get by with shallow statements about the colors of the dots they saw!

My question, then, to all you wise, mathematically-inclined educators is this– In tasks like this where students are asked to notice things, what do you do when they provide surface-level answers? What words do you use to veer them towards deeper-level thinking without noticing things for them? I’d love to hear your thoughts! Also, if you’ve used any ‘noticing/wondering’ tasks with your students, I’d love to hear about them, as well!



No Such Thing as One Good Thing

Our blogging challenge for Week 1 was fairly simple. Bloggers were given a choice of the following:

  1. Blog about the small good moments during our teaching day.
  2. Blog about a day in our teaching life– from start to finish.

At first, I thought I would choose the latter option, so I started keeping a mental journal of my day which usually begins around 5 a.m. and tapers off around midnight (or later). As I began drafting, I started wondering if my day looked any different than another teacher’s day. I read the blogs of my #MTBoS colleagues and realized that indeed, our days looked quite similar. The last thing I wanted to do was add another “Day in the Life” post to a blogosphere that was already full of them.

So, I started looking for small moments– things that bring me joy– in my classroom. My mind was filled with ideas based on what I already knew of my students. The 21 little human beings who have only been on the planet seven or eight years bring me joy every single day. Even on the hardest, most trying days, there’s always something that makes me smile. Maybe not in the moment, but later. Perhaps during my drive home or at night after everyone has gone to sleep, the memory of something that someone said or did tempers my frustration with a happiness that only the heart of a teacher can understand.

One morning this week as I pondered the content of my morning message to students, I thought, “Wouldn’t it be fun to find out what they think brings me joy?” So, that’s exactly what I did. Here’s what they came up with:

photo 1

I know it’s hard to read their responses, so I’ll list them below (spelled correctly) 🙂

  • purple
  • flowers
  • smiles, hugs, and when we do the right thing
  • making cakes
  • me
  • your mom and dad
  • focused brains
  • questions and mistakes
  • teaching and learning
  • strawberries
  • us
  • flower perfume
  • people
  • a little of everything but not sports
  • pictures that we draw
  • when we have growth mindsets
  • people caring about each other
  • design
  • looking up cake recipes on the internet
  • the sea
  • sandalwood (This one was completely random–I have absolutely no clue where she got this from!)

Even more delightful than reading their responses was watching them think of their responses. I regret not taking a picture or video of them huddled around the easel discussing/debating my passions 🙂

You can easily understand why I can never see myself doing anything but this. I readily admit that I couldn’t find “one good thing” to blog about.

I found twenty-one.


Of all the things I could choose to blog about– my favorite cheesecake recipe, life as a full-time teacher and mother of five, my travels, favorite books that I’ve read– math was an unlikely choice. If you asked me five years ago to do this, start blogging about math, I would have a) asked, “What’s a blog?” and then b) run the other direction at the mere mentioning of math.

You see, from the time I was in middle school (or junior high as they called it back in the day), math has always evoked a sense of unease distress dread terror anxiety in me. Seeing test after test with my teachers’ glaring red marks were a constant reminder of my mathematical ineptness. As my aversion to math increased, my confidence withered away to almost nonexistence. My brain, I believed, was simply not made to understand, much less excel in, math. I had the epitome of a fixed mindset.

Fast forward twenty-some years. Reflecting upon the work of Carol Dweck on mindset and more recently, the work of Jo Boaler on math achievement, coupled with my experience teaching Cathy Fosnot’s Units, Contexts for Learning Mathematics, I experienced an epiphany of sorts. I began to see math not as something to do but something to understand. Mistakes didn’t mean that I was dumb, but that my synapses were firing like crazy due to new learning. I was beginning to see math as more than simply a fork in the road– one path leading to “right,” the other to “wrong”– but rather as an endless sea of possibilities.

This transformation not only changed my mindset, it revolutionized my approach to teaching math. The fact that I find joy in the daily 90 minutes of Math Workshop speaks volumes. Thankfully, my enthusiasm has been contagious. But what if I had not undergone this transformation? What if I continued to secretly abhor the very subject I was trying to teach my students? The results, I suspect, would be disastrous. Math Workshop, for them, would be a time to perform, where only an elite few would be deemed worthy of being true math achievers, and the rest…Well, would not.

I would have continued to teach the way I was taught and more frighteningly, I would have found no fault in doing so. While I regret my pre-transformation days, I cannot dwell on them because I am too excited in the present, and for the future. While I’m happy with the changes I’ve made, I’m far from being satisfied. There is, I believe, an inherent danger in being satisfied with oneself (growth mindset speaking!). I want to be better. I need to be better. My kids deserve better than the math teacher they have today, and that’s why I’m doing this.

And so begins my journey into the MathBlogosphere. Equipped with nothing more than ambition, an oh-so-patient mentor (Thanks, Simon!), and an aging laptop, I take my first steps into the blogging world with a sense of trepidation.

But I don’t have anything brilliant to share. 

I will share the brilliance of my colleagues (like this Queen of Teaching/Educator Extraordinaire a.k.a. the unbelievably talented and lovely Jen Bearden) and of those who continue to enrich the world of math with passion, wonder, and creativity.

What if no one reads my blog?

I will write for myself– to reflect, plan, and improve.

What if they don’t like what I write? 

I will continue to write– my work is not perfect and I seek to improve.

Consider this a personal invitation to join me on this important journey by sharing your insights, experiences, and knowledge. I look forward to our learning together!