Investigations with Number Talks

Our last *sniff, sniff* book club meeting was held this week.  We are so appreciative of the teachers who signed up to participate with us.  We are so grateful for the rich and thoughtful conversations and collaborative trouble-shooting that went along with our study of Making Number Talks Matter.  We really believe that professional learning is so much better with colleagues and that setting aside time for professional learning is great for our students, but also helps us (as teachers) stay excited about our work on a day-to-day basis.

This week, we looked at Chapter 9 – Investigations.

We started our meeting with our usual check-in about how Number Talks have been going in the classroom.  It sounds like most people have established a good routine with Number Talks.  Some folks are taking a short break and shifting focus but planning to come back to Number Talks in the New Year.   We did some trouble-shooting discussion about how to deal with students who offer silly answers or make up strange answers that don’t relate to the question posed.  We talked about using phrases that help the student connect their answer with the question (can you explain to me where in the question you got the numbers that you are using in your strategy?).  We also talked about moving on from a student who is having trouble explaining his/her thinking clearly with a statement like: I’m having a hard time understanding your explanation and I would like us both to think about it some more – can I check in with you about your strategy after the number talk is over?

Next, we talked about our BIG IDEAS for the day:

investigations-big-ideas

And outlined how to do an investigation:

investigation-procedure

We then delved into exploring the multiplication strategy of halving and doubling using the general procedure for an investigation.  We started with the question 8 x 26 to try to elicit the strategy of doubling and halving.  Once we looked at all the suggested strategies, we focused in on doubling and halving and talked about the big question of “Does it always work?”  The group split up into partnerships to explore this question – we provided graph paper, colour tiles, rulers, paper and scissors and then circulated to try to see how the investigation went.

It was interesting to note that it is really difficult to be a skeptic in Math – the strategy might make sense to us, but actually thinking about what it takes to PROVE that it works requires much more depth to our thinking.  Many groups got started by discussing WHEN it would be good to use this strategy (ie. what circumstances/numbers make it an efficient strategy).  Some groups explored odd vs. even numbers, some explored big and small numbers, some tried to delve into fractions to see if it worked there.  Some groups worked with the colour tiles to make arrays and others used the graph paper to show the strategy visually.

Then, we wrapped up with a discussion – different groups shared their approaches and it was interesting to note how varied the ideas were.  We looked briefly at the questions offered in the book to guide small group work for this investigation:

  • Will it only work with even numbers?
  • What would happen if, instead of halving, you took a third of one factor?
  • Can you represent this strategy visually/geometrically?
  • What generalizations can you make?
  • Would this work for division?

We had hoped to have time to also have teachers choose another investigation: either the same difference strategy for subtraction or the halving-halving strategy for division, but we ran out of time.  I think participating in an investigation was really valuable.  I enjoyed seeing how much mathematical thinking (curricular competencies) is involved in this type of activity.  Of note – mathematical investigations take time, and it is worth setting aside some time to do activities like this in class.

Once again, a HUGE thank-you to our teachers who participated!  We will be running some more after school PD for intermediate math teachers in the New Year – likely a Mathematical Mindsets book study at some point – so keep an eye on your email in January for information on how to join in.

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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:

P1000152
Grade 1/2 noticings and wonderings – lots of math “noticing” already
Branigan
Grade 2/3 noticing and wondering
Dunn-notice
Grade 3 noticing (I JUST realized that I put a “wonder” in the “notice” list – oops!)
Dunn
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.