NCTM 2018 Presentation Resources ‘Engaging Students in Seeing Structure’.

NOTE:  This is probably my longest post ever.  I tried to write up my one hour session in DC.  What will not come across in this post is the 3D version of me.  As you read this post – imagine a really loud voice passionately preaching at you.  I’ve been compared to a Southern-Baptist preacher – though I am not southern or Baptist.  Enjoy.

How do you engage every single – I mean every single student –  in mathematics every single day – yes, you heard me – engage them in mathematics?  I shared some ideas on this at the 2018 NCTM conference in Washington DC April 25-28th in a session titled “Engaging Students in Seeing Mathematical Structure”  I’ll give you a taste of my session.  At the bottom of the post you will find resources including my power point.  Here is the question I started the session with.

Can you see it?

Stop.  Look at the picture for a few minutes?  Can you see it?  I’ll answer the question at the end of the post – but you’ll get more out of this post if you look first for your self – read the blog and get the answer at the end (don’t cheat).

So, Can you see it?

1st, what I value as a teacher. 

#1:  My school year starts with defining mathematicsCheck this post out if you really want to go deep on this topic.  

#2:  Students will talk out loud about mathematics everyday in my classroom.  This means I will prioritize in my lesson planning instructional routines that support student discourse.  If they are not talking about mathematics out loud then I must do something different.

#3:  One way I’ve found that values talking is an Instructional Routine I call ‘Stand & Talk‘.  Most of the suggestions in this post are how how to give students interesting ideas to discuss.  ‘Stand and talks‘ are a mode that encourages even more conversation with what you create to talk about.

I have other values that I bring to lesson planning (many you can find on my blog -but these are the ones most connected to this blog.

A few things to know about me

I am from Minneapolis, Minnesota.  Knowing where I am is integral to my next story.  My state math conference is in Duluth (shores of lake superior) each year – again you will need to know this in a minute.

I grew up in a family that went on lots of car trips – as the cost of a family on a plane was prohibitive.  In the 70’s and 80’s we did these car trips with maps.  I LOVED looking at maps and so did my siblings.  We mapped things out and looked at locations that we would love to visit someday..  (By the way – this photo is not me.  It can’t be – people my age did not have devices like this to listen to music – we had the radio and at some point cassette tapes)

I am from one of ‘those families’ that plays board games all the time. – especially on holidays. In this picture is my brothers roommate from grad school, Volker.  Volker is from Germany and introduced us to cool games from his country a 15 years ago.  One of our favorites is Setters of Catan.  The first 50+ times I played it it was on a board all in German as that was the only way you could buy it.  Now days you can find this game everywhere.

Board Games and how we teach Mathematics

Another game produced by the same company as Settlers of Catan is Ticket to Ride.  Another favorite of our family.

Here is how the game works.  You get a game board which is a map of Europe.  Your task is to build railroads across the country.

You get tickets with cities in Europe.  The further the distance between the cities the more points you get if you build the route.

Finding the cities on the European maps is easy if they are from cities like London or Amsterdam.  But is a challenge when you need to find cities with names you don’t recognize – for me that was towns like ‘Smyrna’ or ‘Sevastopoi’.

Luckily ‘Ticket to Ride’ is there to help.  The route cards have stars showing you where to find the cities on the map.

Some years later I was excited when the USA version of the game came out.  Finally I could find all the cities on the map quickly.

Within 5 minutes of opening the board. I was extremely frustrated with the map.

I know you can’t see what frustrated me and blow it up.  Do you see it?

Not from Minnesota?  Can you see it?  Remember the map of my state I showed you earlier?

DULUTH is on the game board in the WRONG location.  They put Duluth where I live – Minneapolis.  Not up by Lake Superior where it is really located.

Honestly, this annoys me so much that every time I play on this game board I get re-annoyed and need to vent for several minutes.  If you look at the board there are other annoying things.  For example Chicago looks like it is in Indiana.

Ugh….That got me thinking about the European Game Board again.  What cities on that game board are in the wrong location.  I’ve played the game 50-100 times and never once did I question the location of the cities.  Not once.

The idea that I could not recognize errors on the European board made me think about how we teach mathematics.

Think of ‘Ticket to Ride’ European version as the new mathematics we teach our students.

Because we most of our students struggle with new mathematics and because we are nice teachers who don’t want our students to struggle we jump in and help them with supports to show them the way.  We give them cards that show them the location of cities – or in math we write learning targets that give away what the learning for the day will be – ……

and then we map out all the steps to get from one city one by one for them.  In math class we map out all the steps for them to solve a problem.

Matt Larson, the past-president of NCTM, talked about how “Caring teachers everywhere jump in and rescue students when they don’t get it.”.

My biggest mistake as a teacher in my career was jumping in and rescuing students when they needed it.  We math teachers have good intentions.  We want our students to feel successful and so we scaffold to the point of mathematical death most everything we put in front of students.  We do the heavy lifting in learning.  Students often just copy or mimic what we ask them to do.  Then this happens.  We ask them to do to different cities on their own (on assessments for example) and they can’t.  We’ve handicapped them to do math when they leave our presence.

Worse – not only can they not find the cities and map out a route in math.  Students can’t see errors that surround the contexts we’ve given them.  Ugh.  Something needs to change.

The USA version of the game represents previously learned material.

In this game board of mathematics I can recognize mistakes.

This realization for a board game left me with 2 more questions.

One – I could recognize mistakes on the USA game board because I grew up in a family that looked at maps all the time on car trips in the 80’s and 90’s…..before GPS.  What about the students now days who learn directions from a map on GPS that steps out the route for them?  Can they see the mistakes I see on maps?

Two if you are a student who enters a course missing prior learning – can you even see anything on the European/New Math game board?

A lesson planning mantra 

These 2 questions have lead me to a new mantra in how I plan lessons and what I ask students to do in my classroom.  Here is the mantra I ask myself over and over again as I plan…..

Since beginning to use this lens, I have seen a radical change in students owning the mathematics they are learning in my classroom.  In the past when I rescued them when they struggled, I owned their learning.  Now when they see (notice) a pattern and we, the class, discuss it.  Their brain has a place to put the math learning, because they prepared a place for it to go in seeing something first.  In class we add mathematical vocabulary to what they see and say.  We also clean up misconceptions that can come from seeing something incorrectly – even in these times, they own the learning.  This is my goal.

So how do I do this?

The short answer is lots of ways.  Here are a few.  I shared more in my session.

Lots of Noticing and Wondering

If you have not yet watched Annie Fetter’s 5 minute ignite talk from several years ago on ‘Noticing and Wondering’ stop reading my blog post and do this now…..

Notice and Wonder is an Instructional Routine we should be doing every single day in mathematics classrooms K-12.  It has radically changed students owning mathematical learning in my classroom. Let me give you an example of how I use this.

Look at his equation from my Algebra 2 classroom this year.

How would you find the VERTEX using this equations?  You could put the equation into standard form and use this…

In fact, I was at a session at the NCTM national conference this week and a teacher yelled out b/2a for finding the vertex for a problem we were solving.  I would argue though that outside of teacher conferences, most of us do not have this information on the tip of our tongues.  So, I so no to this as a students first interaction to finding vertices.

What my students do know about the original equation at the Algebra 2 level is that the equation shows us the roots of the equation.

Because of this, I gave them this visual to talk about during a stand and talk (more on this below)…. What do you notice?

I know this is a bit hard to read, so let me make the 3 images a bit larger for you.  Again, what do you notice?  What do you wonder?

My students, without me having to say it or show them, immediately talk about he vertex being in the middle of the 2 roots.  They may not always use the word ‘symmetry’ – but our class conversation can add that academic language to their use of ‘middle’ in their initial noticings.  They talk about finding the average of the roots before I say anything to them about this.  They use the x-value of the vertex and the function to find the y-value – all without me to help them do this.  All I did is give them a visual I had made on Desmos to look for patterns – something to notice and ask questions about.  This is the new norm in my class.  What can I make that will give students something to notice and wonder about?  How can I help them see and talk about a math concept first?

Here is another example of something I ask students to notice and wonder about:

I love using visuals with noticing and wondering for academic vocabulary.  Here is another one- What do you notice?  What do you wonder?

I believe the reason MN’s math test scores continue to be at the top of the nations is that our students must be familiar with integers to live here – just sayin….

Here is one last example of something my students notice and wonder about….this one is hanging in my  classroom all year.  You can read all about what I do HERE.  Where is this orange golf ball located?

Stand & Talks

I do a ‘stand & talk’ nearly every day in my classroom.  I HIGHLY recommend you do the same.  They have greatly increased the quantity and quality of classroom discourse in my classroom.  I wrote about them HERE – I recommend reading this post if you plan to do them in your classroom, but here is a short synapses here with an example.  Here is what I would say to my class.

Put everything down.  No pens/pencils, calculators or phones in your hand.  When I am done talking I want you to walk across the room and find a partner.  Find a partner, not at your table.  Partners of 2 only.  I am going to give you something to notice.  I want you and your partner to notice at least 16 things about what I give you.  Then ask yourself, what do you wonder.  Ready?  Go.”   Here is what I gave them on day 1 of my probability unit.

What do you notice?  What do you wonder?

I circulate and give partners one copy of this visual for the 2 of them on a 1/2 page of card stock. Here is my original.  Intro probability stand & talk marbles & this one is the small one students glue into their notebook zero to 1 prob intro for notebook  This is what I see as I walk around the room – every partner looks like this….

Every student, without much work from me is talking out-loud about mathematics.  This does not happen with every student when I have them sit and ‘turn and talk’.  Stand and talks take the same amount of time and result in more discourse.  In the photos they are looking at different visuals – but every day in my class, students are pointing – talking and wondering without me having to stand next to them.  After 1-3 minutes, I ask them ‘what do you think the title to this visual is?’  Students say the title before I show them.  We do some noticings while they are standing, but then I tell students to come up and grab a smaller copy of what they were looking at and glue it into their notebook.  We then annotate the visual. 

In 10 minutes we have reviewed a lot of probability’s basics from previous years.  I have several students who have interrupted educations and never learned this information before -but even they in 10 minutes had a lot to say about probability.  What is important is that students saw it and said it before I did.  If you can’t tell, I love, love, love stand and talks.  You need to do these.  The spring is the perfect time to test the out in your classroom.  This fall though, set them as a norm in your classroom from day 1, week 1 and see how the culture of your classroom changes for the better.

Remove Details.

In the example above I started with a visual that I found on google images.  When I used it first with students though, I removed the labels and and titles.   I wanted students to say the title of this visual before I told them – and every class l have ever used this with has named the title – therefore they own the title – before i told them what it was.

The Math Forum people (max, annie, steve, suzanne and others) have been advocating for this for years.  they call it ‘Creating Scenarios’.  They advocate removing the question and other details so students are not in a race to solve something.  I love this so that students will stop and notice mathematical structure before racing towards a solution.  As students name things or ask for things we give them the information they need.  Brian Bushart does something like this at the elementary level with ‘Numberless Word – Problems’.  He has written about this a lot.  Check some of it out HERE.

Here are a few of the visuals I’ve used where I removed information before using them. ….

When finding measures of center I have used this…

Students notice the numbers on the jerseys and talk about that some students are taller and some are shorter.  I ask them what math question we could ask about this picture and students say lots of things, but always someone says, I wonder what the average height of the players on the team is…then I reveal the information we need. to answer this question.

One of my favorite tasks to use week 1 with students is Fawn Nguyen’s Noah’s Ark task.  Before giving them the task I remove the question.  What do you notice?  What do you think the question will be?

What did you notice?  I know, I know…you are wondering if you are correct – I can’t do everything for you- to find the answer you need to do a little work so you can see what is out there – either search my site for ‘noah’ or google ‘fawn nguyen noah’s ark’ to find out for yourself.

Here is a visual I use at the start of my systems of inequalities unit.  There is no question or directions other than ‘What do you notice?’  Students notice the solid and dashed lines  They ask about ‘test points’.  They notice regions.  Since it is color coded they notice things about graphing each inequality.  We glue this into our notebook and write all those things down.  Throughout the unit we come back to this visual again and again as we learn new things.

Here is a visual from my quadratics unit.  I remove questions, but then I ask…  What math questions could we ask about this visual?  Remove and ask students for the question.  It is powerful.  You will be amazed at how they look at the structure of the graph first and make sense of this.  When a question is asked by another student, they already have seen things that will be useful in answering the question.  What math questions could you ask about this picture?

Here is a real life dilemma from my classroom.  My students – despite using graphs/tables – were continually struggling to see the symmetry and relationship of points on quadratic graph.  Some students always saw this – but many did not no matter what I did.

When I became addicted to removing information from graphs I tried something new.  I created this visual for day 1 of my transformation unit.  I removed axis, curves, lines and numbers and asked students to notice/wonder.

It was amazing.  Students noticed all kinds of things.  They even drew in curves without me asking them too.  Without the distraction of axes and numbers – they saw the pattern in tables throughout he unit in deeper ways.  Because we glued this into our notebook – students kept coming back to this over and over again throughout the unit to help them as they transformed equations and graphs of a variety of functions.

Here is what I learned from my students….

It is not the students fault if they do not understand the mathematics I am asking them to struggle with.  If they can’t figure things out – of course struggle is good and necessary -but I am talking about the struggle of having no access point -then it is me who must change.  It is me who must figure out a way for them to see and say the relationships and concepts we are studying.

Visualize Numbers.

I love to turn numbers into visuals for students so they can see the pattern before I tell them.  Here is one I use with an introduction to exponents.  Students say ‘four to the fourth power’ before I tell them to.  These visuals are great for math talks.

Here are a few others I use.  This one is my visual multiplication chart connecting multiplication facts to area arrays.

Or this for square root.

Or this when we are going to talk about he commutative property.

Introducing something new

I love starting all units with a Stand & Talk – I love to give them something to look at in the unit to get them talking about what we will learn.  I am always looking for something that levels the playing field for all students regardless of previous learning.  Here is one I use at the start of a Pythagorean Theorem Unit.  I love this one, because students think they understand the theorem because somewhere someone has told them a squared plus b squared is c squared.  This picture eliminates this memory and starts getting at conceptually what is going on.

Here is another visual – I love love love this one.  In one visual my students connect exponential functions to power functions.  Look at it.  What do you notice – spend some time with this one – there is so much there.    Note:  I get asked a lot how I make these – often I steal something I find on twitter or google images.  This one I made using this cool site that animates the factors of numbers.…a few screen shots later I made this.

I also use notice and wonderings to introduce a new concept – like ‘what is a number raised to the zero power?’ or ‘what is a number raised to a negative power?’….love this one too.

Convince Me

To encourage student discourse and to help them see mathematical structure in ways they might not otherwise I have a new favorite phrase i use all the time in class…..’CONVINCE ME‘……  For example, with the visual below, I would say, “Convince me that the equations below are true or false.”  Then i give them a visual that is provocative and gives them something to argue about.  I ask them to provide evidence in their arguments.

Try asking a ‘Convince me’ question/statement this week in your classroom and notice how it improves the quality of the discourse.  Here is another example I love when teaching or reviewing order of operations.  ‘Convince me which calculator is correct’.

Favorite way to teach NOTATION

I love using Stand & Talks with notice/wonder to teach new notation.  For example.

I use the visual above to introduce Sigma notation to my students.  When I use this almost zero percent of my students have ever seen this.  Within 4 minutes of noticing with other students they basically tell me everything I want to hear about how this works.  For example the words in green are what I hear first.

Then students go on to tell me that we should substitute the starting value into the rule until we get to the stopping value.  They then tell the class to add each of these expressions together.  Again – my goal has been met.  Students tell me how it works before I tell/show them.  In total the process took 10 minutes and I have almost no students understand it incorrectly the rest of the year.  If they do, they go back to this visual to remind themselves what to do.

I also used this technique with teaching ‘augmented matrices’ and ‘matrix equations’.

Two-way tables is another notation I use this technique with.

After doing some notice wonder with the visual above, I ask students to draw the bag in the 2-way table.  Without me showing them, almost every student draws something that looks like this:

Same but Different

I love the ‘same but different’ visuals that have been appearing on #MTBoS twitter – led by Brian Bushart –  Here is an example of one I used this week in my class as an introduction to tree diagrams.  Take a few moments before reading and ask yourself – What is the same?  What is Different?  

The beauty of this is from what we know from research is best practice.  Students need to be comparing and contrasting in mathematics more often.  My students were able to name the difference between Independent vs. Dependent events before I told them.  Here are 2 other examples – one related to fractions and one related to graphs of quadratic functions.  Beautiful, right.  I did not create the ones below.  They came from twitter and Brian’s site HERE and HERE.

Here is one I made and use for defining polynomials:

So, you ask, how do I make these?  How can you make your own.  I ask myself 2 questions….

The arrows below represent my answers to the questions above and also represent what my students said to one another before I told them in class. They were able with this visual to define polynomials without my help.


The students I teach often struggle with the context of the problem vs the mathematics of the problem.  I use Stand & Talks with contextual information all the time.  For example – this visual..

Instead of me telling my students there are 52 cards, I have them say it.  They figure out their are 4 types of cards (I often need to give the academic term ‘suits’).  They count 13 of each type….and so on.  Again my students say and see it before I tell and show them.


Above are lots and lots of tips for engaging students in seeing structure.  I want to end with 3 more tips and as well as some closing thoughts.  Here is extra TIP #1:

One of the reasons your students are struggling to see structure is they stop thinking once they think someone else has the answer.  One structure than many of us need to rethink in our classroom is ‘hands up’.  Once a student raises their hand to think – many of your most vulnerable students stop thinking and self-define themselves once again as not smart at math.  Many of these students are only 30 seconds or 5 minutes behind their peers yet they feel miles behind.  I talked about this fact in THIS post recently.  Think about what structures in your class enforce the idea of ‘being good at math is being fast at math’ and change these structures.  Do it now.

One way I dismantle quickness in my classroom is to teach my students the word ‘Contemplate’.  I teach them the meaning of the word and use it often. 

I stole this from one of the Instructional Routines David Wee’s uses in his work “Contemplate then Calculate”.  Here is an example.  In order for students to see how numbers and operations relate to one another I would say “I want nothing in your hands.  Everyone look up here.  No hands up.  I want you to think about, contemplate, what I am going to show you next. Don’t say anything out loud”  Then I would show this visual…

Giving students adequate think time allows more students to see 135-10 as 125 which is 5 times 5 times 5.  It gives them a chance to decompose 36 into 6 times 6.  This expression is fairly easy to do, if you stop and think for 30 seconds.  If you rush in and try and solve it fast you panic and find it difficult.  You could certainly do this same task by saying ‘think’, but something about a fancy word you’ve taught like ‘contemplate’ encourages students to think a little longer giving their brains the chance to see the decomposed parts.

Here is extra TIP #2:

Seriously secondary teachers – especially those of us that teach high school – we need to use more visual representations of student thinking.  When is the last time you used open number lines with your students?  Open number lines are not just for arithmetic.  I use them with expressions and variables (but that is an entire post in itself).  Are you using area models and bar models (don’t know what these are, google them and learn)?  Are you daily connecting graphs to tables to equations to academic vocabulary?  Are you having students make these connections first?  This is so easy to do today on my favorite way to graph – Desmos.

Here is extra TIP #3:

Again, this could be a post all by itself.  Ask yourself, what mathematics is visible to me, but invisible on the paper to students?  Ask yourself, how can I make the invisible, visible.  The hidden zeros and ones are just a start.  There is so many other hidden structures students don’t see that we do.

Closing Thoughts..

I know, I know.  This was the worlds longest blog post.  Can you believe I did all of the above and more in an hour?  Whew.  I hang the following visual by my desk.  I use it to remind me of how I don’t want to teach….

I think all of us have at one time or another valued teaching skills over concepts.  I think every teacher gets into the profession because we want to help students.  Some of your help of students though is causing them to not be the owners of their learning.  We jump in and rescue them – give them answers -scaffold too much – when we see them struggle.  This has been my biggest mistake as a classroom teacher in my career.  

I could beat myself up about his – I get things wrong all the time – but the frequent readers of my blog already know how I handle the negative thoughts in my head.  I always go back to a quote from Maya Angelou….

After reading this post, you know better, so I expect me and you to do better moving forward.  Please do me the honor of using these 2 questions to guide your lesson planning.



These 2 questions get at the 2 part goal I shared at the start of this post.  Print this goal out and hang it by your desk.

Lastly – remember this photo I started the entire blog with?  If you still can’t see it.  Take 2 minutes to look at again…..

Can you see it?  Did you cheat and look down below for the answer?  If you can’t see it, I’ve highlighted what is right in front of your face in yellow…

Sticking out perpendicularly from the wall is a cigar.  Crazy right?  it was there the entire time.  Right in front of you.  Here is my last thought and the good news of valuing students seeing structure in mathematics.  Once you see the cigar – once students see the structure in mathematics for themselves, one can never unsee it again.  It is now obvious and right in front of you….remember that.  Use this fact to encourage yourself.  Helping students see and talk about mathematics first is hard work.  It takes planning.  You will not get it right the first time, maybe even the first several times, but you will get it right.  Students deserve your efforts.  It is worth it, I promise.


Let’s start with my POWER POINT  Note:  I am not sure how helpful it will be for those of you who did not attend because my slides are way more visuals and less writing – but for everyone please note that some slides have links in the notes with more information.

As always, I love hearing from you.  Please comment below with thoughts or questions.  If you were at my presentation and I am missing something, please let me know and I will update the post as soon as I can.  Thanks, my new friends.  I appreciate everyone that gave me such kind feedback when I presented.

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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