I’m having a bit of a hard time shutting work down for the Christmas break, so I thought I would see if I could make up these other primary clothesline cards that I have been pondering.

They are a bit addictive… as I make more cards, I keep thinking about more cards I could make…

This set has benchmarks of 0, 5, 10, 15, 20, and then uses ten frames and dominoes. I added the numbers from 11-20 with the ten frames and then mixed them up to make addition cards with the ten frames. I did all the make tens and the doubles and some random other combinations. The dominoes have all the doubles and all the make tens and then some near doubles and some other random combinations…

Does anyone want to try them out? I have no printer at home, so can’t test with my own kiddos and we are on holidays until January now (woohoo!). If you have a chance to try them, I’d love to hear how it goes! Feedback and suggestions welcome 🙂

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:

And outlined how to do an investigation:

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.

We had our fourth meeting of our “Making Number Talks Matter” book club last night. Our focus for this meeting was on Fractions. We spoke a little bit about decimals and percents, but we spent most of our time looking at how we can support our students in developing conceptual understandings of fractions.

This blog post is meant to serve as a recap for those who were there, a fill-in for those who couldn’t make it, and a record for anyone else who is interested!

We started off our meeting with our usual conversation about how things are going in classrooms with number talks. Some participants shared their ideas for how they are keeping track of student thinking. One teacher has tested out incorporating our student self-assessment and another has been using her document camera instead of the whiteboard to record student thinking – she then has a record of strategies being used with student names attached to help inform her Number Talks planning. It is so inspiring to hear about how excited students are about participating in Number Talks. I hope you will all continue to carve out time in your class for them!

We then looked at a “Fractions on the Number Line” activity as a group. We used a double number line for this activity. We placed the benchmarks of 0, 1/2 and 1 on the top number line and then each participant had a number to place on the line. First, we had teachers talk in groups to order the numbers at their table and then one-by-one the tables came up to put their numbers on the bottom number line, re-arranging as necessary to make it make sense.

From this activity, we moved on to talking about the BIG IDEAS for fractional thinking in the new Grade 4-7 curriculum. We used this quote from the book as our jumping-off point:

…for success in high school, there is no avoiding fractions.

We talked about: what do our students struggle with in terms of fractional thinking? And: what do we want our students to understand about fractions?

Some thoughts that came up:

We want our students to understand that the size of the piece changes depending on the size of the whole. It is possible to have a quarter that is bigger than a half if the two wholes are different.

We want our students to understand that fractional pieces have to be the same size but not necessarily the same shape.

We want our students to understand that fractions are numbers that exist on the number line.

We want to help our students make connections between their existing understanding of number and their understanding of fractions.

We then looked at the BIG IDEAS from the curriculum from Grades 3 – 9: where are our students coming from in primary, and where do we want them to go in secondary? Now that all the fraction operations have been moved to Grade 8, we have the opportunity to solidify a conceptual understanding of fractions in elementary school so that students are prepared for fraction operations and linking of fractions to algebra in Grades 8 and 9.

From here, the teachers did another activity that connects a visual representation of a fraction to its place on the number line. (Activity adapted from this blog – printable download of the activity cards are available). Teachers coloured in a section of the given square and determined what fractional part they coloured. They then placed their fraction on the number line again.

As a wrap-up, we briefly reviewed the other three types of Number Talks for fractions that are described in the book: More or less (give a fraction and have students defend whether it is more or less than a half); Closer to 0, Closer to 1/2 or Closer to 1 (give a fraction and have students decide which benchmark it is closer to), and Which is Greater (give two fractions and have students defend which one is greater).

Last but not least, we had a mini “make and take” – teachers took home yarn for a double number line and a package with coloured fraction, decimal and percent cards. I will update this post with a link to the printable package as soon as I add some improper fractions and mixed numbers to it! I will also have these packages at our final meeting for people who were unable to join us this week.

Here are a few useful links that we talked about in our meeting:

The Teacher Studio – this blog has a fabulous series of ideas for teaching fractions conceptually. Scroll down and click on the “fractions” label on the right-hand side to find all the blog posts labelled as fractions.

Is this shape fourths? – this activity by the Teacher Studio has students defending their ideas about fractional parts. A great extension activity from your Number Talks routine.

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:

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:

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 agency: This 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?

This fall, we (the two intermediate math liaisons in my district) have been planning a book study for the book “Making Number Talks Matter” by Cathy Humphreys and Ruth Parker. Our 18 participants teach Grades 4-7 and come from 16 different schools across our district (our district has 31 elementary schools). We will be meeting every second Tuesday until the beginning of December to work our way through the book. We will be talking about number talks, strategies for mental math and doing some planning and practicing of the Number Talks routine. Participation in this book club is totally voluntary and we know how difficult it is as teachers to squeeze in after-school commitments and still have everything ready in the classroom – our book club meetings run from 3:30 – 4:30 and we have committed to getting everyone out of there on time.

Yesterday was our first meeting – we had a few people who had to miss the first meeting because of parent-teacher interviews and staff meetings, so I will do my best to recap our discussion and learning!

We had a few goals for our first meeting –

Understand why we do number talks

Identify the procedures and setup necessary to get Number Talks started

Discuss the underlying values that the routine of Number Talks is based on

Create a plan for doing a dot talk in the classroom before the next meeting

We first showed the teachers a clip from the DVD that comes with Sherry Parrish’s Number Talks book. We asked the teachers to ignore the mathematical strategies (for now) and just to focus on the routine – what is the teacher doing? What are the students doing? What logistics do you notice? (View this YouTube video from 44:50 to 51:30 – this isn’t the exact same clip we watched, but close enough).

Afterwards, we asked each group of teachers to fill in a chart with their ideas from watching the video and from doing the pre-reading. Here are the finished ideas:

We then had teachers do a “gallery walk” to look at all the ideas.

Next, we had intended to pull up the new BC Curriculum website to show all the places that Number Talks fit in both the content elaborations and in the Curricular Competencies from Grades 4-7, but the website was down (*deep breath*), so we skipped this portion. We are planning to do a full workshop on our district’s next ProD day on Number Talks and the Curricular Competencies, so we will have a chance to dive into this further on that day (if you are from SD57 and reading this, you can register on PD Reg for this session!).

From here, we provided groups with some discussion questions from the chapters they read and gave them a few minutes to talk/discuss and plan:

What strikes you as most useful/valuable/exciting about the Number Talks routine?

What parts of the routine are of concern? What do you think will be most difficult for you as the teacher/facilitator?

What norms and structures do you need to have in place to be successful with Number Talks?

What Guiding Principles (from Chapter 3) resonate with you?

Which ones make you feel uncomfortable/concerned?

We provided groups with a blank template to record some guiding principles/norms for Number Talks that they thought they might like to use in their classrooms. There was so much good discussion during this portion of our meeting – I feel so lucky that I get to facilitate and work with groups of teachers on things like this – what a thoughtful group of people! During these conversations, teachers discussed the importance of “wait time” and how difficult that can be, they talked about the difficulties of facilitating if we ourselves are unsure about some of the strategies (hopefully we can clear some of these feelings up in future meetings), they talked about the importance of students listening to one another and how this routine can connect across many math content areas.

Finally, I did a demonstration “dot talk” so that teachers could see what a dot talk would look like in action. I used the same dot pattern from the Chapter explanation and showed teachers briefly how I set up a number talk to get started.

For our next meeting in two weeks, we have asked teachers to try a dot talk (or several) in their classroom, and read Prelude to the Operations, Chapter 4 and Chapter 6 – we are going to dive into addition and subtraction number talks next.

Some resources:

These are the Guiding Principles that I use when I start Number Talks in a classroom – they are adapted from Making Number Talks Matter.

Here is a planning page that Dorianna and I made for a workshop we did last year on Number Talks (also adapted from Making Number Talks Matter).

I borrowed and adapted several ideas for this session from this blog – I am so thankful for teachers who share their ideas and work so graciously online!

This is a great summary of Number Talks if anyone is looking for more information.

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…

… 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:

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…

And a few from the 2/3 class…

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:

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.

As a grade 4/5 teacher for the last 4 years, I generally find that my students enjoy working with fractions. They like working with fraction manipulatives and approach visual representations for fractions with relative ease. Naming and identifying fractions is rarely a problem. But then we get to comparing and ordering fractions… and it all falls apart.

With my tutoring students (Grades 9-11), fractions are generally a disaster. They are chugging along fine with whatever they are working on and then… a FRACTION!!! Reducing fractions and finding common denominators is sometimes OK, but if a fraction is tossed into the middle of an algebraic expression, they don’t know what to do. There seems to be no recognition that fractions are, in fact, numbers.

I have been doing a lot of thinking this year about how and why fractions seem to be a place of struggle for so many students as they advance through math.

Many of my students treat fractions as a completely new set of numbers, with no connection to whole numbers. They are obviously not getting the big idea that all numbers are connected and have their own place on the number line. Part of me wonders if this is because we (I?) over-use certain representations for fractions (pizzas and chocolate bars) and under-use others (number lines). So, this year, I have really been trying to help students connect what they already know about numbers with the new information that they are learning about fractions.

One activity that I have used a few times this year – very successfully – is a fraction clothesline activity. This week, I was invited to visit a Grade 4/5 class to introduce decimals by linking them to what the students already knew about fractions (they have been working on fractions for a couple of weeks). I thought this was a perfect opportunity for me to combine two great things and I set the fraction clothesline up like a number talk (yes, I am obsessed with Number Talks).

I set up the clothesline at the front and we looked at the three benchmark cards I brought: 0, 1/2 and 1. We hung the 0 and the 1 on the number line and then… NO ONE could tell me where 1/2 went. Yikes. This is the part of number talks that still makes me anxious… the WAITING… letting kids think… and HOPING… that SOMEONE… will come up with something to move the conversation forward. My patience paid off… eventually someone suggested – “well, if it’s one-half, couldn’t we just put it in the middle?” And, whew… yes. Yes, we can. We were rolling again.

So, once we had the benchmarks on the line, I handed out a fraction card to each student. Because we were looking at connecting decimals and fractions, I gave each student a “tenth” – nothing tricky. Just 1/10 – 9/10. I asked the students to think for a moment and give me a thumbs up when they were pretty sure they knew where their fraction was supposed to go. And I waited. After a while, a few kids (probably about 1/2) had their thumbs up. So I invited those students who had an idea to come and place their fractions, and others could just watch to see if it helped them figure out where theirs should go. And… this is what I got:

Yikes. Again. But mistakes are SUCH a valuable opportunity to pinpoint misconceptions. The first student started by defending the location of 9/10. He explained that because 9/10 was almost one whole, he put it close to the 1. Whew. Good start.

Then, the student who placed 1/10 wanted to explain. She said she placed her fraction there because 10 is bigger than 2, so 1/10 must be bigger than 1/2. Cool – good explanation, and a good misconception to tackle. But, this is where I am still working on my “thinking-on-the-spot” skills, and trying to find the balance between “teaching” and letting the students help each other and wrestle with their own thinking. I ended up drawing some pictures on the whiteboard behind the number line – I went back to the classic “pizza” shape. I asked the students to help me draw 1/2 behind the 1/2 benchmark card and then 1/10 behind the 1/10 card. Gasps all over the classroom. I asked the student if she was still happy with where her card was and – NO, she was not! She came and moved it to its correct location. I asked if any other students wanted to move theirs and several more came up to make adjustments (accurately). I asked the students to explain why they chose to move their cards and they were able to relate the spot on the number line with what the picture would look like.

I then encouraged the students who had not yet placed their cards to come up and find a reasonable spot for them. This worked well – most of them were able to be successful in the location and were able to explain why. We also talked about the spacing between fractions on the number line and how it is hard on a clothesline to be exact, but we know that fraction pieces all have to be the same size.

I was pretty happy with this number talk. The students were pretty confident with ordering fractions with the same denominator, and I thought we got a good start at thinking about fractions with different denominators. At the end, I gave a few students some “tricky” cards – 0/10, 11/10 and 12/10 and they were able to successfully place and explain these fractions as well.

This lesson really highlighted (again) for me the power of number talks and having opportunities for students to own and explain their thinking. The real power of number talks is in giving students these types of opportunities on a DAILY basis. This is what helps them to build up their number sense over time.

What other activities do you use to help students build a broad, connected understanding of fraction concepts?