Noticing and Wondering: A powerful tool for assessment

Last week I had the privilege of presenting with Nehlan Binfield at OAME on the topic of assessment in mathematics.  We aimed to position assessment as both a crucial aspect of teaching, yet simplify what it means for us to assess effectively and how we might use our assessments to help our students and class learn.  If interested, here is an abreviated version of our presentation:


We started off by running through a Notice and Wonder with the group.  Given the image above, we noticed colours, sizes, patterns, symmetries (line symmetry and rotational symmetry), some pieces that looked like “trees” and other pieces that looked like “trees without stumps”…

Followed by us wondering about how many this image would be worth if a white was equal to 1, and what the next term in a pattern would look like if this was part of a growing pattern…


We didn’t have time, but if you are interested you can see the whole exchange of how the images were originally created in Daniel Finkel’s quick video.

We then continued down the path of noticing and wondering about the image above.  After several minutes, we had come together to really understand the strategy called Notice and Wonder:


As well as taking a quick look at how we can record our students’ thinking:

Shared by Jamie Duncan

At this point in our session, we changed our focus from Noticing and Wondering about images of mathematics, to noticing and wondering about our students’ thinking.  To do this, we viewed the following video (click here to view) of a student attempting to find the answer of what eight, nine-cent stamps would be worth:


The group noticed the student in the video counting, pausing before each new decade, using two hands to “track” her thinking…  The group noticed that she used most of a 10-frame to think about counting by ones into groups of 9.

We then asked the group to consider the wonders about this student or her thinking and use these wonders to think about what they would say or do next.

  • Would you show her a strategy?
  • Would you ask a question to help you understand their thinking better?
  • Would you suggest a tool?
    Would you give her a different question?

It seemed to us, that the most common next steps might not be the ones that were effectively using our assessment of what this child was actually doing.


Looking through Fosnot’s landscape we noticed that this student was using a “counting by ones” strategy (at least when confronted with 9s), and that skip-counting and repeated addition were the next strategies on her horizon.

While many teachers might want to jump into helping and showing, we invited teachers to first consider whether or not we were paying attention to what she WAS actually doing, as opposed to what she wasn’t doing.


This led nicely into a conversation about the difference between Assessment and Evaluation.  We noticed that we many talk to us about “assessment”, they actually are thinking about “evaluation”.  Yet, if we are to better understand teaching and learning of mathematics, assessment seems like a far better option!


So, if we want to get better at listening interpretively, then we need to be noticing more:


Yet still… it is far too common for schools to use evaluative comments.  The phrases below do not sit right with me… and together we need to find ways to change the current narrative in our schools!!!



Evaluation practices, ranking kids, benchmarking tests… all seem to be aimed at perpetuating the narrative that some kids can’t do math… and distracts us from understanding our students’ current thinking.

So, we aimed our presentation at seeing other possibilities:


To continue the presentation, we shared a few other videos of student in the processs of thinking (click here to view the video).  We paused the video directly after this student said “30ish” and asked the group again to notice and wonder… followed by thinking about what we would say/do next. b15


Followed by another quick video (click here to view).  We watched the video up until she says “so it’s like 14…”.  Again, we noticed and wondered about this students’ thinking… and asked the group what they would say or do next.


After watching the whole video, we discussed the kinds of questions we ask students:


If we are truly aimed at “assessment”, which basically is the process of understanding our students’ thinking, then we need to be aware of the kinds of questions we ask, and our purpose for asking those questions!  (For more about this see link).

We finished our presentation off with a framework that is helpful for us to use when thinking about how our assessment data can move our class forward:


We shared a selection of student work and asked the group to think about what they noticed… what they wonderered… then what they would do next.

For more about how the 5 Practices can be helpful to drive your instruction, see here.


So, let’s remember what is really meant by “assessing” our students…


…and be aware that this might be challenging for us…


…but in the end, if we continue to listen to our students’ thinking, ask questions that will help us understand their thoughts, continue to press our students’ thinking, and bring the learning together in ways where our students are learning WITH and FROM each other, then we will be taking “a giant step toward becoming a master teacher”!

So I’ll leave you with some final thoughts:

  • What do comments sound like in your school(s)?  Are they asset based (examples of what your students ARE doing) or deficit based (“they can’t multiply”… “my low kids don’t get it…”)?
  • What do you do if you are interested in getting better at improving your assessment practices like we’ve discussed here, but your district is asking for data on spreadsheets that are designed to rank kids evaluatively?
  • What do we need to do to change the conversation from “level 2 kids” (evaluative statements that negatively impact our students) to conversations about what our students CAN do and ARE currently doing?
  •  What math knowledge is needed for us to be able to notice mathematicially important milestones in our students?  Can trajectories or landscapes or continua help us know what to notice better?

I’d love to continue the conversation about assessment in mathematics.  Leave a comment here or on Twitter @MarkChubb3 @MrBinfield

If you are interested in reading more on similar topics, might I suggest:

Or take a look at the whole slide show here


Pick a Quote

Seems to me that many schools and districts are asking questions about assessment in mathematics.  So, I thought I would share a few quotes that might get you to think and reflect on your views about what it means to assess, why there might be a focus on assessment, and what our goals and ideals might look like.  I want you to take a look at the following quotes.  Pick 1 or 2 that stands out to you:


A few things to reflect on as you think about the quotes above:

  • Which quotes caught your eye?  Did you pick one(s) that confirm things you already believe or perhaps ones that you hadn’t spent much time thinking about before?
  • Some of the above quotes speak to “assessment” while others speak to evaluation practices.  Do you know the difference?
  • Take a look again at the list of quotes and find one that challenges your thinking.  I’ve probably written about the topic somewhere.  Take a look in the Links to read more about that topic.
  • Why do you think so many discuss assessment as a focus in mathematics?  Maybe Linda Gojak’s article Are We Obsessed with Assessment? might provide some ideas.
  • Instead of talking in generalities about topics like assessment, maybe we need to start thinking about better questions to ask, or thinking deeper about what is mathematically important, or understanding how mathematics develops!

Please pick a quote that stands out for you and share your thoughts about it.

Leave a reply here or on Twitter (@MarkChubb3)


Who makes the biggest impact?

A few years ago I had the opportunity to listen to Damian Cooper (expert on assessment and evaluation here in Ontario). He shared with us an analogy talking to us about the Olympic athletes that had just competed in Sochi.  He asked us to think specifically about the Olympic Ice Skaters…

He asked us, who we thought made the biggest difference in the skaters’ careers:  The scoring judges or their coaches?

Think about this for a second…  An ice skater trying to become the best at their sport has many influences on their life…  But who makes the biggest difference?  The scoring judges along the way, or their coaches?  Or is it a mix of both???

Damian told us something like this:

The scoring judge tells the skater how well they did… However, the skater already knows if they did well or not.  The scoring judge just CONFIRMS if they did well or not.  In fact, many skaters might be turned off of skating because of low scores!  The scoring judge is about COMPETITION.  Being accurate about the right score is their goal.

On the other hand, the coach’s role is only to help the skater improve. They watch, give feedback, ask them to repeat necessary steps… The coach knows exactly what you are good at, and where you need help. They know what to say when you do well, and how to get you to pick yourself up. Their goal is for you to become the very best you can be!  They want you to succeed!

In the everyday busyness of teaching, I think we often confuse the terms “assessment” with “evaluation”   Evaluating is about marking, levelling, grading… While the word assessment comes from the Latin “Assidere” which means “to sit beside”.  Assessment is kind of like learning about our students’ thinking processes, seeing how deeply they understand something…   These two things, while related, are very different processes!


I have shared this analogy with a number of teachers.   While most agree with the premise, many of us recognize that our job requires us to be the scoring judges… and while I understand the reality of our roles and responsibilities as teachers, I believe that if we want to make a difference, we need to be focusing on the right things.  Take a look at Marian Small’s explanation of this below.  I wonder if the focus in our schools is on the “big” stuff, or the “little” stuff?  Take a look:

Marian Small – It’s About Learning from LearnTeachLead on Vimeo.

Thinking again to Damian’s analogy of the ice skaters, I can’t help but think about one issue that wasn’t discussed.  We talked about what made the best skaters, even better, but I often spend much of my thoughts with those who struggle.  Most of our classrooms have a mix of students who are motivated to do well, and those who either don’t believe they can be successful, or don’t care if they are achieving.

If we focus our attention on scoring, rating, judging… basically providing tasks and then marking them… I believe we will likely be sending our struggling students messages that math isn’t for them.  On the other hand, if we focus on providing experiences where our students can learn, and we can observe them as they learn, then use our assessments to provide feedback or know which experiences we need to do next, we will send messages to our students that we will all improve.

Hopefully this sounds a lot like the Growth Mindset messages you have been hearing about!

Take a quick look at the video above where Jo Boaler shows us the results of a study comparing marks vs feedback vs marks & feedback.

So, how do you provide your students with the feedback they need to learn and grow?

How do you provide opportunities for your students to try things, to explore, make sense of things in an environment that is about learning, not performing?

What does it mean for you to provide feedback?  Is it only written?

How do you use these learning opportunities to provide feedback on your own teaching?

As  always, I try to ask a few questions to help us reflect on our own beliefs.  Hopefully we can continue the conversation here or on Twitter.


What does “Assessment Drives Learning” mean to you?

There are so many “head nod” phrases in education.  You know, the kind of phrases we talk about and all of us easily agree upon that whatever the thing is we are talking about is a good thing.  For instance, someone says that “assessment should drive the learning” in our classroom, and we all easily accept that this is a good practice.  Yet, everyone is likely to have a completely different vision as to what is meant by the phrase.

In this post, I want to illustrate 3 very different ways our assessments can drive our instruction, and how these practices lead to very different learning opportunities for our students.

Assessment Drives Learning

Unit Sized Assessments

Some teachers start their year or their unit with a test to find out the skills their students need or struggle with.  These little tests (sometimes not so little) typically consist of a number of short, closed questions.  The idea here is that if we can find out where our students struggle, we will be able to better determine how to spend our time.

But let’s take a look at exactly how we do this.  The type of questions, the format of the test and the content involved not only have an effect on how our students view the subject and themselves as learners of math, they also have a dramatic effect on the direction of learning in our classrooms.  

For example, do the questions on the test refer to the types of questions you worked on last year, according to previous Standards, or are they based on the things you are about to learn this year (this year’s Standards)?  If you provide questions that are 1 grade below, your assessment data will tell you that your students struggle with last years’ topics… and your instruction for the next few days will likely be to try to fill in the gaps from last year.  On the other hand, if you ask questions that are based on this year’s content, most of your students will likely do very poorly, and your data will tell you to teach the stuff you would have anyway without giving the test at all.  Either way, the messages our students receive are about their deficits… and our instruction for the next few days will likely relate to the things we just told our students they aren’t good at.  I can’t help but wonder how our students who struggle feel when given these messages.  Day 1 and they already see themselves as behind.

I also can’t help but wonder if this is helpful even for their skills anyway?  As Daro points out below, when this is our main view of assessment guiding our instruction, we often end up providing experiences for our students that continue to keep those who struggle struggling.

Assessment Drives Learning (2)

Daily Assessments

On the other hand, many teachers view assessment guiding their practice through the use of daily assessment practices like math journals, exit cards or other ways of collecting information while the learning is still happening.  It is really important to note that these forms of assessment can look very different from teacher to teacher, or from lesson to lesson.  In my post titled Exit Cards: What do yours look like?  I shared 4 different types of information we often collect between lessons.  I really think the type of information we collect says a lot about our own beliefs and our reflections on this evidence will likely form the type of experiences we have the next day.

When we use assessments like these regularly, we are probably more likely to stay on track with our curriculum Standards, however, what we do with this the information the next day will completely depend on the type of information we collect.

In-the-Moment Assessments

A third way to think of “assessment driving instruction” is to think of the in-the-moment decisions we make.  For example, classrooms that teach THROUGH problem solving will likely use instructional practices that help us use in-the-moment assessment decisions.  Take for example The 5 Practices: for Orchestrating Productive Mathematics Discussions listed below, might be useful as part of the assessment of our students.

1. Anticipating
• Do the problem yourself.
• What are students likely to produce?
• Which problems will most likely be the most useful in addressing the mathematics?

The first practice helps us prepare for WHAT we will be noticing.  Being prepared for the problem ahead is a really important place to start.
2. Monitoring
• Listen, observe students as they work
• Keep track of students’ thinking
• Ask questions of students to get them back on track or to think more deeply (without rescuing or funneling information)

The second practice helps us notice how students are thinking, what representations they might be using.  The observations and conversations we make here can be very powerful pieces of assessment data for us!

3. Selecting
• What do you want to highlight?
• Purposefully select those that will advance mathematical ideas of the group.

The third practice asks us to assess each of the students’ work, and determine which samples will be beneficial for the class.  Using our observations and conversations from practice 2, we can now make informed decisions.
4. Sequencing
• In what order do you want to present the student work samples?  (Typically only a few share)
• Do you want the most common to start first? Would you present misconceptions first?  Or would you start with the simplest sample first?
• How will the learning from the first solution help us better understand the next solution?• Here we ask students specific questions, or ask the group to ask specific questions, we might ask students what they notice from their work…

The 4th practice asks us to sequence a few student samples in order to construct a conversation that will help all of our students understand the mathematics that can be learned from the problem.  This requires us to use our understanding of the mathematics our students are learning in relation to previous learning and where the concepts will eventually lead (a developmental continuum or landscape or trajectory is useful here)
5. Connecting
• Craft questions or allow for students to discuss the mathematics being learned to make the mathematics visible (this isn’t about sharing how you did the problem, but learning what math we can learn from the problem).
• Compare and contrast 2 or 3 students’ work – what are the mathematical relationships?  We often state how great it is that we are different, but it is really important to show how the math each student is doing connects!

In the 5th and final practice, we orchestrate the conversation to help our class make connections between concepts, representations, strategies, big ideas…  Our role here is to assess where the conversation should go based on the conversations, observations and products we have seen so far.

So, I’m left wondering which of these 3 views of “assessment driving learning” makes the most sense?  Which one is going to help me keep on track?  Which one will help my students see themselves as capable mathematicians?  Which one will help my students learn the mathematics we are learning?

Whether we look at data from a unit, or from the day, or throughout each step in a lesson, Daro has 2 quotes that have helped form my opinion on the topic:

Assessment Drives Learning (3)

I can’t help but think that when we look for gaps in our students’ learning, we are going to find them.  When our focus in on these gaps, our instruction is likely more skills oriented, more procedural…. Our view of our students becomes about what they CAN’T do.  And our students’ view of themselves and the subject diminishes.

Assessment Drives Learning (4)

“Need names a sled to low expectations”.  I believe when we boil down mathematics into the tiniest pieces then attempt to provide students with exactly the things they need, we lose out on the richness of the subject, we rob our students of the experiences that are empowering, we deny them the opportunity to think and engage in real discourse, or become interested and invested in what they are learning.  If our goal is to constantly find needs, then spend our time filling these needs, we are doing our students a huge disservice.

On the other hand, if we provide problems that offer every student access to the mathematics, and allow our students to answer in ways that makes sense to them, we open up the subject for everyone.  However, we still need to use our assessment data to drive our instruction.

As a little experiment, I wonder what it would look like if other subjects gave a skills test at the beginning of a unit to guide their instruction.  Humor me for a minute:

What if an English teacher used a spelling test as their assessment piece right before their unit on narratives?  Well, their assessment would likely tell them that the students’ deficits are in their spelling.  They couldn’t possibly start writing stories until their spelling improved!  What will their instruction look like for the next few days?  Lots of  memorization of spelling words… very little writing!

What if a Science teacher took a list of all of the vocabulary from a unit on Simple Machines and asked each student to match each term with its definition as their initial assessment?  What would this teacher figure our their students needed more of?  Obviously they would find that their students need more work with defining terms. What will their instruction look like for the next few days?  Lots of definitions and memorizing terms… very little experiments!

What if a physical education teacher gave a quiz on soccer positions, rules, terms to start a unit on playing soccer.  What would this teacher figure out?  Obviously they would find out that many of their students didn’t know as much about soccer as they expected.  What would their next few days look like?  Lots of reading of terms, rules, positions… very little physical activity!

A Few Things to Reflect on:
  • How do you see “assessment guiding instruction”?
  • Is there room for all 3 versions?
  • Which pieces of data are collected in your school by others?  Why?  Do you see thes as helpful?
  • Which one(s) do you use well?
  • Do you see any negative consequences from your assessment practices?
  • How do your students identify with mathematics?  Does this relate to your assessment practices?

Being reflective is so key in our job!  Hopefully I’ve given you something to think about here.

Please respond with a comment, especially if you disagree (respectfully).  I’d love to keep the conversation going.

Learning Goals… Success Criteria… and Creativity?

I think in the everyday life of being a teacher, we often talk about the word “grading” instead of more specific terms like assessment or evaluation  (these are very different things though).  I often hear conversations about assessment level 2 or level 4… and this makes me wonder about how often we confuse “assessment” with “evaluation”?

Assessment comes from the Latin “assidere” which literally translates to “sit with” or “sit beside”.  The process of assessment is about learning how our students think, how well they understand.  To do this, we need to observe students as they are thinking… listen as they are working collaboratively… ask them questions to both push their thinking and learn more about their thoughts.


Evaluation, on the other hand, is the process where we attach a value to our students’ understanding or thinking.  This can be done through levels, grades, or percents.

Personally, I believe we need to do far more assessing and far less evaluating if we want to make sure we are really helping our students learn mathematics, however, for this post I thought I would talk about evaluating and not assessing.


A group of teachers I work with were asked to create a rubric they would use if their students were making chocolate chip cookies as a little experiment.  Think about this task for a second.  If every student in your class were making chocolate chip cookies, and it was your responsibility to evaluate their cookies based on a rubric, what criteria would you use?  What would the rubric look like?

Some of the rubrics looked like this:

Rubric 3

What do you notice here?  It becomes easy to judge a cookie when we make the diameters clear… or judge a cookie based on the number of chocolate chips… or set a specific thickness… or find an exact amount for its sugar content (this last one might be harder by looking at the final product).

While I am aware that setting clear standards are important, making sure we communicate our learning goals with students, co-creating success criteria… and that these have been shown to increase student achievement, I can’t help but wonder how often we take away our students’ thinking and decision making when we do this before students have had time to explore their own thoughts first.


What if we didn’t tell our students what a good chocolate chip cookie looked like before we began trying things out?  Some might make things like this:

or this?

or this?

But what if we have students that want to make things like this:

or this?

or this?

Or this?

I think sometimes we want to explain everything SO CLEARLY so that everyone can be successful, but this can have the opposite effect.  Being really clear can take away from the thinking of our students.  Our rubrics need to allow for differences, but still hold high standards!  Ambiguity is completely OK in a rubric as long as we have parameters (saying 1 chip per bite limits what I can do).

What about the rubric below?  Is it helpful?  While the first rubric above showed exact specs that the cookies might include, this one is very vague.  So is this better or worse?

Rubric 4


As we dig deeper into what quality math education looks like, we need to think deeper about the evidence we will accept for the word “understanding”!

…and by the way, are we evaluating  the student’s ability to bake or their final product?  If we are assessing baking skills, shouldn’t we include the process of baking?  Is following a recipe indicative of a “level 4” or an “A”?  Or should the student be baking, using trial and error and developing their own skills?  Then co-creating success criteria from the samples made…

If we show students the exact thing our cookies should look like, then there really isn’t any thinking involved… students might be able to make a perfect batch of cookies, and then not make another batch until next year during the “cookie unit” and totally forget everything they did last year (I think this is currently what a lot of math classes looks like).

Learning isn’t about following rules though!  It’s about figuring things out and making sense of it in your own way, hearing others’ ideas after you have already had a try at it, learning after trying, being motivated to continue to perfect the thing you are trying to do.  We learn more from our failures, from constructing our understanding than we ever will from following directions!

Creativity happens in math when we give room for it.  Many don’t see math as being creative though… I wish they did!