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:

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.

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.

Silence.

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!

You’re back, Mona! I can slide off the edge of my seat now! I was a little worried you might not post again!

I haven’t really got an answer to your noticing question. Like you, sometimes I get a torrent of things noticed, other times a trickle. Like you, I sometimes prompt them, what mathematical things do you notice? There’s always the option of moving on to the next thing!

I’m going to try Pascal’s triangle (I don’t think the one you saw was his; something similar). I’ve used it with older students (Year 6 / Grade 5) but I can see younger students can get lots out of it. And if there are candidates for “no ceiling” lessons, this must be one of them!

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Hi Mona — I love the stories and the prompts here, and hadn’t seen the Family Math Night posters. They’re great!

Two things I’ve been playing with recently with noticing and wondering are:

1) The slow reveal — when noticing and wondering about patterns, starting with one iteration, adding a 2nd, etc. Like with Pascal’s triangle, revealing one row at a time.

2) Asking questions borrowed from other “math talks” such as “how do you see the dots?” or “how do you count the dots?”

I love the “what would mathematicians notice” question and find it’s often super useful!

Thanks for sharing,

Max

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