Secondary Number Talks (I’ll convince you with ducks)

NOTE:  This blog post contains resources from a training I did on June 22, 2016 on Number Talks in the Secondary Classroom in Minneapolis, Minnesota.

Let me clear from the start of this post about my beliefs about Number Talks.  I have strong opinions on this.  I believe in 2016 – Number Talks are an issue of equity and access for all students.  Every, yes every, 6-12 Math Teacher should be doing Number Talks regularly in the core mathematics classes they teach.  If a secondary teacher wants all students to make growth in their classrooms then Number Talks MUST be a part of what they are doing in their classrooms.

In this post of my top ideas for support classes I advocated doing a number talk every single day if a student has a second math class.  I’d love to advocate for the same frequency in a core math class but I know this is unrealistic for many.  So here is my challenge to you for the 2016-17 school year in your 6-12 core math classrooms.  Commit to doing at least 30 number talks over the course of the year.  Commit that at least 10 of the 30 will be done 1 per day for a two week time.  

I know, I know what you are going to say.  I don’t have time.  I have so much material to cover.  I can’t do this.  Yes you can.  We can do anything in our classrooms if we prioritize it.  We will do anything in our classroom once we are convinced it has value.  You will be convinced this has value once you become educated on this tool AND you do a few of them. Don’t complain yet.  Try it.  I implore you.  I had too had the same internal reservations.  I am now a Number Talk evangelist.  Once I figured out how to deliver them to a secondary audience, they have turned out to be great for all students regardless of background.  They have not taken time away from what I want to accomplish, they have improved the quality of students participating in my class.  Below I will give you a taste of why this is and try to convince you in writing – but if you really want to be convinced – talk to me in person.

i don't want

For my entire 25 years of teaching I have been surrounded by 6-12 math teachers (particularly in HS) that continually complain  about our students lack of numeracy skills. I too have complained.  Almost never in my 25 years have I seen secondary teachers address this complaint with effective actions.  We wasted time blaming.  We were secretly afraid to act, because we really have had no idea to help our students have a flexible understanding of the base 10 number system.  We were untrained in our education programs for addressing our students needs in this area.  We’ve needed to look to our Elementary peers for answers.  Let’s stop complaining about what students can’t do and do something about it.  In my opinion, Number Talks, are one of the best ways to support our secondary students.

I am calling every secondary math teacher to spend some time learning how students form the concept of number and their operations and then use Number Talks as the vehicle for acting on our students lack of numeracy skills.  If you reread this sentence, you will see it starts with US – the teachers.  We need to learn.  Here is a quick way of knowing if you have learning to do….Do you know what all of the following words and phrases mean?  If not, you have some learning to do…..

  • doubling/halving
  • counting on
  • partial products
  • decomposing numbers
  • making 10’s
  • friendly numbers
  • open number lines
  • area model
  • relational thinking
  • compensation

As secondary teachers we demand that our students have algebraic reasoning when many of them are further back in their sophistication of number.  We expect all students to perform at high levels of thinking when they need us as teachers to back up and be flexible enough in our teaching to meet them where they are, not where we want them to be.  If you want students to improve their numeracy skills, then we as teachers need to do some deep learning on how students develop algebraic reasoning skills which has its roots in teh earliest of elementary math expierences.  Again, if you don’t know what the sequence of learning below looks like or how to diagnose where your students are in this sequence, you need to stop complaining about your students lack of skills and fix your own skills as a teacher first.  (Note:  I say this not to blame teachers – I was never taught this in college – I did not know this until well into my 10th-20th years of teaching.  I know better now, so I can do better for my students.  I say this simply to waken us up.)

  1. Counting Strategies
  2. Additive Thinking
  3. Relational Thinking
  4. Proportional Reasoning
  5. Algebraic Reasoning

If you, like the vast majority of secondary math teachers, feel like you have some learning to do, I encourage you to start by reading one or more of the following books:

Four Books you should read:

  1. Making Number Talks Matter by Cathy Humphreys & Ruth Parker (Seriously, what are you waiting for read it today – by far the most useful book I’ve read as a secondary teacher in the last 10+ years)
  2. Building Powerful Numeracy for Middle and High School Students by Pamela Weber Harris (For you MN peeps, Pam Harris will be the Symposium AND Keynote speaker at the 2017 MN Math Conference in Duluth in April)
  3. Number Talks K-5 by Sherry Parrish (this is the bible of number talks world – don’t be scared if you teach secondary)
  4. Number Talks: Fractions, Decimals and Percents by Sherry Parrish (Available October 2016 – you can pre-order from Math Solutions)  I WILL be reading this one when it comes out.


At my recent training of highly motivated amazing secondary teachers in MN, I put up the question to the left (from Jo Morgan via James Tanton) to start our time together.  When the secondary teachers started talking about the first expression they said things like “I factored out the 37 and 23-13 is 10 so 10 times 37 is 370”  the secondary teachers had difficulty (really no experience with) representing the first expression as ‘the difference between 23 groups of 37 and 13 groups of 37 giving 10 groups of 37’.  They struggled to see multiplication as groups of something.  In their heads they were so proficient with algorithms and procedures that they struggled to back up and represent this information in ways students could make sense of.  If we want to help our students, we have to be open to their ways of naming what is going on and not rush them to our ‘right way’.

So what does a number talk (or math talk or problem string or…any name you may go by) look like in Sara VDW’s core mathematics class?  

First a couple of rules I hold to (these are taken from the books/blogs/trainings I’ve read/attended):

  • Number Talks will happen regularly.  Especially at the start of the year it is better to do 10 number talks in a row than one number talk per week for 10 weeks.
  • Number Talks will last 5-10 minutes.
  • I will start the year doing number talks with visual patterns.  Once I start doing number talks with numbers I will start with subtraction.  (Read Ruth Parker’s book for why I do this).
  • I will use the following norms with my students:

For the most part I follow the script for number talks found in ‘Making Number Talks Matter’ book.  Read this book to see what I do.  I establish the norms with students every time we do this.  I do make my students clear their desk and hands of everything.  I am a HUGE believer in having them put their fist to their chest to communicate when they have at least one way subtractionof thinking about the problem.  I also ask my students to turn their knees so they are focused up front. (this slight tweak of their physical orientation has resulted in greater engagement).  I do all the norms and give directions BEFORE I show the number talk prompt.  I show the prompt.  Wait silently as students think and encourage students to use their fingers next to their chests to show me how many ways they have to think about the solution to the prompt.  Once I see most student’s thumbs up.  I call randomly on 4-5 students to give me their solution which I record with as much non-judgement as I can muster.  I then record 2-5 different ways to solve the problem and for each try and represent the thinking of the students with some type of model (open number line, area model, bar model…).

One way I’ve become sold on Number Talks in my classroom is to connect them to what we are learning in class.  (although i recently watched a youtube interview of Ruth Parker who said she does not do this).  The day I did this in my quadratics unit this year I was SOLD ON NUMBER TALKS FOR LIFE.  My coworker, Morgan, and I had such a great experience with all our students engaging in such deep math we committed to doing more and more of these.  Here is the quick story of how this played out in our classrooms.

ducks in 3's

Morgan and I teach Advanced Algebra.  At the start of our quadratic unit, I told her that I wanted to start with one of the amazing visual Patterns at Fawn Nguyen’s site (pattern goals#147 to be exact).  Here is where Fawn blogged about the duck pattern with her 6th graders.    Like everything in our planning – Morgan and I started by naming the intended learning of our students with this activity.  Fawn’s visual patterns are a set of 3 images.  For the number talk, I took just one of the images.  Morgan and I started our unit with the following prompt for our Number Talk:


After giving students 60 seconds to look at this pattern, these were the 4 most common ways of seeing this in the 10 classes we did this with this year (though each class was a bit different)…

student work

When we gave students the visual with 3 images, they wrote a different generalization for each way of counting ducks shown above and over the next 2-3 days we compared the generalizations they wrote and asked them to convince us that the generalizations were equivalent.  Here is some of their work:

We worked hard to connect the algebraic generalizations to the visuals.  We also used Desmos to have students see that each generalization was equivalent.(seriously, how great is Desmos for this?)  We then compared the equations we wrote to the different forms of quadratic equations we would study in this unit and had great discussions about if any of the expressions we wrote were in these forms.

In past years I made the mistake of skipping doing a number talk on just one duck image and instead handed out the 3 images and asking students to jump in and write a rule for the number of ducks in any part of this sequence of images.  When we did this – about a third of the class was able to write ONE rule for the pattern.  This year when we started by doing a number talk ALL students wrote at least one rule/generalization and most write 2 or 3 or more.  When is the last time you did something in class when EVERY student had the joy of feeling successful in writing a generalization for a pattern?  EVERY student acted as a mathematician on this day.  They noticed patterns.  They described the pattern and they used this information to generalize what they notices.  Slowing down 5 minutes to do a number talk meant all my students had the time to see the structure of the pattern in a variety of ways and then apply this structure to algebra.  It was so beautiful to see, I wanted to cry.  This is how I can work to close the gaps in achievement in my classroom.

After doing the duck p2016attern I tweeted out to the world “I will never use another visual pattern without using just one image first and doing a number talk”.  I became addicted in this experience because students of all levels were able to engage in ways they would not have had I not done the number

So at the start of this next fall – 2016 – if you need a way to stGaussart the year, I suggest you start with this image and ask students what they notice.  Why?  Most years one of my opening year tasks is the story of Carl Friedrich Gauss in Elementary School about adding the numbers 1-100.  If you
don’t know the story, google it.  Often students see a pattern to build a generalization that looks something like this….

gauss 2

This year – I am going to tackle this a bit differently with footballs & a number talk (something I used midyear last year).


After recording the ways students see the number of footballs we will look at the set of 3 images and build a generalization….

This pattern is a nice one for talking about triangular numbers.  The year 2016 is a triangular number.  my first triangle image has 2016 little dots.  How could we use the rule we wrote to quickly figure out how many rows of dots make up the number 2016?  Desmos to the rescue.

triangular numbers

So the start of my school year will include lots of number talks with visual patterns – but then it will move on to doing number talks with naked operations.  I also plan to do a variety of other math talks.  Download the pdf my presentation to see examples of other forms of number talks in a secondary classroom.  I am particularly loving David Wee’s ‘Contemplate & Calculate’ Routine in his A2I curriculum to use as number talks.

Here are the handouts from our training:

In my experience of asking secondary teachers to begin using number talks most are excited to have a possible way to address their students lack of numeracy.  They begin with excitement, but when they jump into it, it does not go so well right away and they give up after very few attempts.  Teachers who ask their students to have a growth mindset have a very fixed mindset of their own when it comes to what we can do in our classes to support our students with numeracy skills.  I am going to ask you to be brave and commit to taking what you learn work through the stumbles that will certainly come from teaching something we’ve not taught.  For many teachers, the hardest part of using number talks is being able to record students thinking and visually representing their words.  I promise this gets better with practice.


Resources for learning more:

Join an online summer reading group for ‘Making Number Talks Matter’.  More information can be found by clicking HERE!

  • My friend Ole Rapson introduced me to this site: Number Talk Images (in French)
  • Fawn Nguyen’s Math Talks  (Fawn teaches middle school and has documented her Number Talks and Pattern Talks with students).
  • Nat Banting created the crowd sourced Fraction Talks this year.  Love.
  • Follow @Fraction Talks online.

Video Clips

Blogs on Math Talks:

For more Number Talk Resources – go to the MTBoS search engine and search ‘Number Talk’.  



I am going to end this post in the same way I started this post – to repeat my challenge to EVERY 6-12 mathematics teacher…

I challenge you to commit to doing

at least 30 number talks over the course of the year.

 Commit that at least 10 of the 30 will be done 1 per day for a two week time.  

I challenge you to find a teacher buddy to do number talks with.

Join me this summer in learning more and planning out the number talks you will use in your classroom this fall and throughout the classroom.  Stick with it.  Work through the challenges.  

During my training, MN tweeter Andy Martianson sent out the following tweets.  They nicely sum up my deep passion for why number talks need to be part of the secondary math classroom.  Happy Summer all.




Sara VanDerWerf

I am Sara Van Der Werf, a 24-year mathematics teacher in Minneapolis Public Schools. I have taught math in grades 7-12 as well as spent several years leading mathematics at the district office. I currently teach Advanced Algebra at South High School and I'm also the current President of the Minnesota Council of Teachers of Mathematics (MCTM). I am passionate about encouraging and connecting with mathematics teachers. I'd love to connect via twitter.  Join the community.  Tweet me @saravdwerf.

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