‘fessing up – Number Talks gone awry

One of my professional goals this fall as a Math Liaison in my district is to spread the message of Number Talks far and wide in intermediate classrooms in my district.  Between the readings I have done (Making Number Talks Matter and Number Talks, blog posts, articles), the Pro-D workshops I have led (Number Talks book club, Number Talks and the Curricular Competencies, Intro to Number Talks), and the various Gr. 4-7 classrooms I have visited, I am starting to feel like a bit of an “expert” on the subject… and yet, Number Talks are still complicated and challenging.  I think that’s one of the things I like most about the Number Talks routine – it is simple enough to be accessible, but challenging enough to keep both students and teachers engaged.  So, today I thought I would share a blog post about a “failed” number talk that I have been pondering and what I learned from the experience.

The setup:

I was visiting a Grade 4 class – This was my third visit to this classroom this year, and I have done Number Talks with them on each visit.  We have done some dot talks and some addition number talks, and on this visit, we were going to be working on subtraction. Students in this particular class (and at this school in general) are very capable but tend to just do the traditional algorithm in their heads – this happens much more frequently here than at other schools that I visit (my theory is that it is related to high levels of parental involvement).

The Problems:

Rather than just one number talk, I brought a number string with me… My plan:

img_1745

I really thought that these (especially the first one) were going to be easy, but when we got going on the first question, I ended up with 4 different answers.  I have led a lot of number talks with multiple answers, and 99% of the time (100% of the time before this particular visit), the errors work themselves out nicely during the discussion of the problem.

So, for this particular problem, I got the following 4 answers:

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Take a minute and see if you can figure out where the errors come from.

I wish I had thought to take a photo of the board after we were finished (need to get better at documenting things for the blog!).  First, I had a few students who defended the correct answer of 6 with some good strategies – adding on, counting back, making 44 into 45 etc.  If I had taken a picture, you could note my nice use of number lines and whatever else I did to record student thinking…

But what I really want to discuss is the student who wanted to defend the answer of 14.

She said something like:

“I did the 5 minus the 4 to get 1 and the 4 minus the 0 to get 4.  The answer is 14.”

This is not really earth-shattering… probably the most common mistake made in subtraction by Grade 4 students.  Here is the interesting thing: this student had just listened to 3 or 4 of her peers defend (very clearly) their answer of 6 with very good strategies (and I’m quite sure she was listening).  This is the first time I have had a student sincerely defend a mistake after multiple other students have made their case for the right answer – she had no recognition that her answer might not be correct.  In hindsight, I can think of quite a few good ways to respond, but in the moment, I was caught off-guard.  I wish I could tell you that I referred her back to the original problem to see if her answer made sense… or that I asked a classmate to respond to her thinking… or that I asked her to explain why what she did made sense to her.  But… I just told her that you couldn’t flip the numbers around and subtract from bottom to top.  Sigh. Fail.

Even in the moment, I knew that my response was woefully inadequate… I could tell from the look on her face that I had done nothing to convince her.  I think she believed me that her answer was wrong, but she had gained no understanding to move her thinking forward.  Other students had not learned anything useful from our exchange.  And, possibly (hopefully not!), the experience has discouraged her from taking another risk to share her thinking.

Moving Forward

So, what have I taken away from this experience?  I went back to Making Number Talks Matter and reminded myself of some guiding principles…

  • Through our questions we seek to understand students’ thinking: It is not my role to be the judge of student answers, or even to correct mistakes.  It is my role to try to understand why students are thinking the way they are.  I need to focus my responses on questioning with the genuine desire to understand student thinking.
  • One of our most important goals is to help students develop social and mathematical agencyThis exchange would have been a great opportunity to encourage students to respond to each other.  By “explaining” the right answer, I removed the opportunity for students to be the thinkers and brought the responsibility for “correct mathematical knowledge” back to the teacher.  My new #1 goal for number talks: stop talking so much and LISTEN.
  • Confusion and struggle are natural, necessary, and even desirable parts of learning mathematics: In hindsight, it is really interesting how uncomfortable I felt dealing with this mistake… as teachers, it is very hard to let go of our instincts to help our students through their struggles.  I am totally on board with the IDEA of stepping back and letting my students wrestle with mistakes, but in the moment, it is still a challenge to stop the “traditional teacher” who hides out in the back of my brain.

I am thankful that teaching is such an interesting job – regardless of how much experience we have, there is always more to learn.

I am going to better prepare for my number talks… I have been lazy about anticipating student responses. For our last workshop, we prepared a “cheat sheet” of phrases and sentence stems, and I have printed a copy to refer to.

I am giving myself some grace… making mistakes is the best way to learn, even for teachers!

And I found this lovely quote from Ruth Parker to help me remember why I am so excited about doing Number Talks in the first place:

I’ve come to believe that my job is not to teach my students to see what I see.  My job is to teach them to see.

So… who else wants to ‘fess up?  What surprises have you been faced with during a Number Talk?

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Noticing and Wondering Across the Grades

I have been reading a lot lately about having kids “notice and wonder” to start off a math task.  This seemed to be a nice extension from the Number Talks that I have been doing lately, so I was looking for an opportunity to visit some classes to try it out.  Then, last week, a friend of mine posted this picture to her Facebook page…

Egg Array
An egg array – beautiful!

… and I knew I had to use it!  There are sooo many awesome things to notice and wonder about in this picture!

 

So, I “invited” myself into some classrooms at my school.  I was especially curious about how kids at different grades would respond similarly/differently to this photo.  I visited a Grade 1/2, a Grade 2/3 and a Grade 3 class with the same activity.  First, I showed the whole class the picture and gave them a few minutes to observe and think about what they noticed and wondered.  Then, I collected all their ideas onto the whiteboards at the front.  I was so impressed!  I love how curious kids are at this age, and I love the variety of things that they noticed and wondered about:

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Grade 1/2 noticings and wonderings – lots of math “noticing” already
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Grade 2/3 noticing and wondering
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Grade 3 noticing (I JUST realized that I put a “wonder” in the “notice” list – oops!)
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Grade 3 wonderings

I love how many things the Grade 3’s wondered about before they wondered how many eggs there were! I think my favourite “noticing” is “it looks like they just came out of a chicken!” And I love how much real-world knowledge is being talked about here – in addition to the math.  I thought it was so interesting to hear the variety of background knowledge that the kids had about chickens, farms, eggs, and where their food comes from.  They were all very enthralled by that tiny egg in the middle (which, by the way, had no yolk according to my “farmer” friend).

Next, we did some “math” with the picture.  I gave all the kids a black and white copy of the picture in a sheet protector and a dry erase marker to use.  I challenged them to figure out how many eggs were in the picture WITHOUT counting one-by-one.

Here are some samples from the 1/2 class…

Grade 12-3
Grade 1/2 sample – I have never taught these grades, so wasn’t sure how easily kids would be able to count by grouping.  I was expecting to see a lot of this, but only had a few that ended up counting one-by-one.
Grade 12-4
Grade 1/2 – A slightly more sophisticated version of one-by-one counting.
Grade 12-1
Grade 1/2 – interesting! This student started counting by 2’s but got to 22 and couldn’t continue, so she switched to counting by 1’s to finish off.

 

Grade 12-2
Grade 1/2 – counting by 3’s, but a little mix-up at the end.  This reminds me of a hundreds-chart layout for counting by 3’s (row-by-row).
Grade 12-5
Grade 1/2 – this was the most sophisticated version from the 1/2 class.  It looks like he counted 1 by 1 but when I asked him about his picture, he explained that he did 9 groups of 4.  I like how he arranged it as a grid.

And a few from the 2/3 class…

Grade 23-1
Grade 2/3: Counting by 3’s
Grade 23-5
Grade 2/3: Counting by 4’s

 

Grade 23-3
Grade 2/3: Counting by 4’s a different way
Grade 23-4
Grade 2/3: Counting by 6’s
Grade 23-2
Grade 2/3: Hmmm… interesting.  I’m just guessing here, but maybe the student decided that counting by 2’s would take too long?  In any case, this is probably the most unique grouping I saw!

With the Grade 2/3’s we ended up discussing that 36 is a really interesting number because there are a lot of ways that you can group the eggs and still get to 36.  We talked about the word factor and how we could use it to describe the way we grouped the eggs (ie. my picture shows 9 groups of 4 – 9 and 4 are factors of 36).

And the Grade 3’s:

Grade 3-3
Grade 3: Counting by 3’s – an interesting way of grouping
Grade 3-1
Grade 3: This student actually grouped them a few different ways.  He counted by 2’s and then by 3’s and then had a great idea: “I bet I can put them in dozens!” Kind of a cool connection to real-life knowledge about how we buy eggs.

So… after all that… what did I notice and wonder?

I noticed that the kids were all really engaged in this activity.
I noticed that all the students (even the lowest Grade 1/2’s) were able to meaningfully engage with this activity.
I noticed that many kids wanted to try different ways of grouping and were getting ideas to try from their neighbours.
I noticed that all the students were keen to explain their thinking.

AND… I noticed… that not ONE student in any of the classes grouped the eggs in a “traditional” array pattern.  There were some kids who counted by 4’s, but none made columns of 4.  And nobody thought to make rows of 9 or to turn the page and make columns of 9.  This is a big ??? for me, because I would think it would be natural to group things in rows and columns, and this is where we want kids to access multiplication.  So, this leaves me wondering… do the students naturally group things the way that they did because of experience using hundreds charts?  What kinds of activities can we do to help them see things in arrays?  Should I be “encouraging” kids to see an image like this as an array, or will that representation naturally develop over time?

So much to think about!!  I love activities that make me wonder about my teaching and student learning.  I will definitely be doing more noticing/wondering with kids…

If you are interested in doing activities like this one with your students, there are many more images like these available on the Number Talk Images website and I have submitted the picture from this activity there as well.