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:

b2

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…

b3

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:

b4

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

b5
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:

b6

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.

b7

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.


b8

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!

b9

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

b10

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!!!

b11

b12

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:

b13


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

b16

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.

b17

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

b18

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:

b19

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.


b20

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

b21

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

b22

…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 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

 

Advertisements

The Importance of Contexts and Visuals

My wife Anne-Marie isn’t always impressed when I talk about mathematics, especially when I ask her to try something out for me, but on occasion I can get her to really think mathematically without her realizing how much math she is actually doing.  Here’s a quick story about one of those times, along with some considerations:


A while back Anne-Marie and I were preparing lunch for our three children.  It was a cold wintery day, so they asked for Lipton Chicken Noodle Soup.  If you’ve ever made Lipton Soup before you would know that you add a package of soup mix into 4 cups of water.

Lipton soup

Typically, my wife would grab the largest of our nesting measuring cups (the one marked 1 cup), filling it four times to get the total required 4 cups, however, on this particular day, the largest cup available was the 3/4 cup.

nesting cups.JPG

Here is how the conversation went:

Anne-Marie:  How many of these (3/4 cups) do I need to make 4 cups?

Me:  I don’t know.  How many do you think?  (attempting to give her time to think)

Anne-Marie:  Well… I know two would make a cup and a half… so… 4 of these would make 3 cups…

Me: OK…

Anne-Marie:  So, 5 would make 3 and 3/4 cups.

Me:  Mmhmm….

Anne-Marie:  So, I’d need a quarter cup more?

Me:  So, how much of that should you fill?  (pointing to the 3/4 cup in her hand)

Anne-Marie:  A quarter of it?  No, wait… I want a quarter of a cup, not a quarter of this…

Me:  Ok…

Anne-Marie:  Should I fill it 1/3 of the way?

Me:  Why do you think 1/3?

Anne-Marie:  Because this is 3/4s, and I only need 1 of the quarters.


The example I shared above illustrates sense making of a difficult concept – division of fractions – a topic that to many is far from our ability of sense making.  My wife, however, quite easily made sense of the situation using her reasoning instead of a formula or an algorithm.  To many students, however, division of fractions is learned first through a set of procedures.

I have wondered for quite some time why so many classrooms start with procedures and algorithms unill I came across Liping Ma’s book Knowing and Teaching Mathematics.  In her book she shares what happened when she asked American and Chinese teachers these 2 problems:

liping ma1

Here were the results:

Liping ma2

Now, keep in mind that the sample sizes for each group were relatively small (23 US teachers and 72 Chinese teachers were asked to complete two tasks), however, it does bring bring about a number of important questions:

  • How does the training of American and Chinese teachers differ?
  • Did both groups of teachers rely on the learning they had received as students, or learning they had received as teachers?
  • What does it mean to “Understand” division of fractions?  Computing correctly?  Beging able to visually represent what is gonig on when fractions are divided? Being able to know when we are being asked to divide?  Being able to create our own division of fraction problems?
  • What experiences do we need as teachers to understand this concept?  What experiences should we be providing our students?

Visual Representations

In order to understand division of fractions, I believe we need to understand what is actually going on.  To do this, visuals are a necessity!  A few examples of visual representations could include:

A number line:

three quarters NL

A volume model:

GlassMeasuringCup32oz117022_x

An area model:


Starting with a Context

Starting with a context is about allowing our students to access a concept using what they already know (it is not about trying to make the math practical or show students when a concept might be used someday).  Starting with a context should be about inviting sense-making and thinking into the conversation before any algorithm or set of procedures are introduced.  I’ve already shared an example of a context (preparing soup) that could be used to launch a discussion about division of fractions, but now it’s your turn:

Design your own problem that others could use to launch a discussion of division of fractions.  Share your problem!  


A few things to reflect on:

As always, I’d love to hear your thoughts.  Leave a reply here on Twitter (@MarkChubb3)

How Many Do You See (Part 2 of 2)

A few weeks ago I shared with you a quick blog post showing a simple worksheet at the grade 2 level – the kind of  simple worksheet that is common to many classrooms.  If you haven’t seen the image, here it is again:

DL5ysx5WkAA4O_x

As you can see, the task asks students to correctly count the number of each shape they notice.  In my first post (Part 1) I asked us a few questions to start a conversation:

  1. Pick one shape (or more if you’re adventurous)
  2. Think about what you believe the teacher’s edition would say
  3. Count how many you see
  4. Share the 3 points above as a comment here or on Twitter

I was quite happy with where some of the conversations led…

Some of the conversations revolved around the issue many have with resources perpetuating stereotypical definitions of shapes:

IMG_E6046

If we look, there are exactly 4 shapes that resemble the diagram at the top of the page labelled as “rectangle”, however, there are several different sized squares as well (a square are a special case of a rectangle).


Other conversations revolved around actually counting the number of each item:

IMG_E6044IMG_E6042IMG_E6053IMG_E6052

trapezoids 21

What interests me here is that we, as a group of math teachers, have answered this grade 2 worksheet with various answers.  Which brings about 2 important conversations:

  1. What are we looking for when students complete a worksheet or textbook questions?
  2. Are we aiming for convergent or divergent thinking?  Which of these is more helpful for our students?

What are we looking for?

Given the conversations I have had with math teachers about the worksheet being shared here, it seems like there are a few different beliefs.  Some teachers believe the activity is aimed at helping students recognize traditional shapes and identify them on the page.  Other teachers believe that this activity could potentially lead to discussions about definitions of shapes (i.e., What is a rectangle?  What is a hexagon?…) if we listen to and notice our students’ thinking about each of the shapes, then bring students together to have rich discussions.

It’s probably worth noting that the Teacher’s Edition for this worksheet includes precise answers.  If a typical teacher were to collect the students’ work and begin marking the assignment using the “answers” from the teacher’s guide, some of the students would have the “correct” answer of 8 trapezoids, but many others would likely have noticed several of the other trapezoids on the page.  If we are looking / listening for students to find the correct answer, we are likely missing out on any opportunity to learn about our students, or offer any opportunity for our students to learn themselves!

I would hope that an activity like this would provide us opportunities for our students to show what they understand, and move beyond getting answers into the territory of developing mathematical reasoning.


Convergent vs Divergent Thinking

Again, many of the teachers I have discussed this activity with have shared their interest in finding the other possible versions of each shape.  However, what we would actually do with this activity seems to be quite different for each educator.  It seems like the decisions different teachers might be making here relate to their interest in students either having convergent thinking, or divergent thinking.  Let’s take a look at a few possible scenarios:

Teacher 1:

Before students start working on the activity, the teacher explains that their job is to find shapes that look exactly like the image in the picture at the top of the page.

Teacher 2:

Before students start working on the activity, the teacher tells the students exactly how many of each shape they found, then asks students to find them.

Teacher 3:

Before students start working on the activity, the teacher explains that their job is to find as many shapes as possible.  Then further explains that there might be ones that are not traditional looking.  Then, together with students, defines criteria for each shape they are about to look for.

Teacher 4:

Before students start working on the activity, the teacher explains that their job is to find as many shapes as possible.  As students are working, they challenge students to continue to think about other possibilities.


In the above scenarios, the teachers’ goals are quite different.  Teacher 1 expects their students to spend time looking at common versions of each shape, then spot them on the page.  Teacher 2’s aim is for students to be able to think deeper about what each shape really means, hoping that they are curious about where the rest of the shapes could possibly be leading their students to challenge themselves.  Teacher 3 believes that in order for students to be successful here, that they need to provide all of the potential pieces before their students get started.  Their goal in the end is for students to use the definitions they create together in the activity.  Finally, teacher 4’s goal is for students to access the mathematics before any terms or definitions are shared.  They believe that they can continue to push students to think by using effective questioning.  The development of reasoning is this teacher’s goal.

Looking back at these 4 teachers’ goals, I notice that 2 basic things differ:

  1. How much scaffolding is provided; and
  2. When scaffolding is provided

Teachers that provide lots of scaffolding prior to a problem typically aim for students to have convergent thinking.  They provide definitions and prompts, they model and tell, they hope that everyone will be able to get the same answers.

Teachers that withhold scaffolding and expect students to do more of the thinking along the way typically aim for divergent thinking.  That is, they hope that students will have different ideas in the hopes for students to share their thinking to create more thinking in others.

Whether you believe that convergent thinking or divergent thinking is best in math, I would really like you to consider how tasks that promote divergent thinking can actually help the group come to a consensus in the end.  If I were to provide this lesson to grade 2s, I would be aiming for students to be thinking as much as possible, to push students to continue to think outside-the-box as much as possible, then make sure that in my lesson close, that we ALL understood what makes a shape a shape.


I want to leave you with a few reflective questions:

  • I provided you with a specific worksheet from a specific grade, however, I want you to now think about what you teach.  How much scaffolding do you provide?  Are you providing too much too soon?
  • Do your lessons start off with convergent thinking or divergent thinking?  Why do you do this?  Is this because you believe it is best?
  • How can you delay scaffolding and convergent thinking so that we are actually promoting our students to be actively thinking?  How can you make this a priority?
  • What lesson or warm-ups or problems have you given that are examples of what we are talking about here?
  • If we do remove some of the scaffolding will some of your students sit there not learning?  Is this a sign of them not understanding the math, or a sign of them used to being spoon-fed thinking?  What do WE need to get better at if we are to delay some of this scaffolding?

I encourage you to continue to think about what it means to help set up situations for your students to actively construct understanding:

I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

P.S.  I’m still not confident how many of each shape are actually here!

Targeted Instruction

The other day I was asked about my opinion about something called entrance slips. Curious about their thoughts first, I asked a few question that helped me understand what they meant by entrance slips, what they would be used for, and how they might believe they would be helpful. The response made me a little worried. Basically, the idea was to give something to students at the beginning of class to determine gaps, then place students into groups based on student “needs”.  I’ll share my issues with this in a moment…  Once I had figured out how they planned on using them, I asked what the different groups would look like.  Specifically, I asked what students in each group would be learning. They explained that the plan was to give an entrance slip at the beginning of a Geometry unit. The first few questions on this entrance slip would involve naming shapes and the next few about identifying isolated properties of shapes. Those who couldn’t name shapes were to be placed into a group that learns about naming shapes, those who could name shapes but didn’t know all of the properties were to go into a second group, and those who did well on both sections would be ready to do activities involving sorting shapes.  In our discussion I continually heard the phrase “Differentiated Instruction”, however, their description of Differentiated Instruction definitely did not match my understanding (I’ve written about that here). What was being discussed here with regards to using entrance slips I would call “Individualized Instruction”.  The difference between the two terms is more than a semantic issue, it gets to the heart of how we believe learning happens, what our roles are in planning and assessing, and ultimately who will be successful.  To be clear, Differentiated Instruction involves students achieving the same expectations/standards via different processes, content and/or product, while individualized or targeted instruction is about expecting different things from different students.


Issues with Individualized / Targeted Instruction

Individualized or targeted instruction makes sense in a lot of ways.  The idea is to figure out what a student’s needs are, then provide opportunities for them to get better in this area.  In practice, however, what often happens is that we end up setting different learning paths for different students which actually creates more inequities than it helps close gaps.  In my experience, having different students learning different things might be helpful to those who are being challenged, but does a significant disservice to those who are deemed “not ready” to learn what others are learning.  For example, in the 3 pathways shared above, it was suggested that the class be split into 3 groups; one working on defining terms, one learning about properties of shapes and the last group would spend time sorting shapes in various ways.  If we thought of this in terms of development, each group of students would be set on a completely different path.  Those working on developing “recognition” tasks (See Van Hiele’s Model below) would be working on low-level tasks.  Instead of providing experiences that might help them make sense of Geometric relationships, they would be stuck working on tasks that focus on memory without meaning.

Figure-1-Examples-of-interview-items-aligned-with-van-Hiele-levels

When we aim to find specific tasks for specific students, we assume that students are not capable of learning things others are learning.  This creates low expectations for our students!  Van de Walle says it best in his book Student Centered Mathematics:

Determining how to place students in groups is an important decision.  Avoid continually grouping by ability.  This kind of grouping, although well-intentioned, perpetuates low levels of learning and actually increases the gap between more and less dependent students.  

Targeted instruction might make sense on paper, but there are several potential flaws:

  • Students enter into tracks that do not actually reflect their ability.  There is plenty of research showing that significant percentages of students are placed in the wrong grouping by their teachers.  Whether they have used some kind of test or not, groupings are regularly flawed in predicting what students are potentially ready for.
  • Pre-determining who is ready for what learning typically results in ability grouping, which is probably the strongest fixed mindset message a school can send students.  Giving an entrance ticket that determines certain students can’t engage in the learning others are doing tells students who is good at math, and who isn’t.  Our students are exquisitely keen at noticing who we believe can be successful, which shapes their own beliefs about themselves.
  • The work given to those in lower groups is typically less cognitively demanding and results in minimal learning.  The intent to “fill gaps” or “catch kids up” ironically increases the gap between struggling students and more independent learners.  Numerous studies have confirmed what Hoffer (1992) found: “Comparing the achievement growth of non-grouped students and high- and low-group students shows that high-group placement generally has a weak positive effect while low-group placement has a stronger negative effect. Ability grouping thus appears to benefit advanced students, to harm slower students.

 

The original conversation I had about Entrance Tickets illustrated a common issue we have.  We notice that there are students in our rooms who come into class in very different places in their understanding of a given topic.  We want to make sure that we provide things that our students will be successful with… However, this individualization of instruction does the exact opposite of what differentiated instruction intends to do.  Differentiated instruction in a mathematics class is realized when we provide experiences for our students where everyone is learning what they need to learn and can demonstrate this learning in different ways.  The assumption, however, is that WE are the ones that should be determining who is learning what and how much.  This just doesn’t make sense to me!  Instead of using entrance tickets, we ended up deciding to use this problem from Van de Walle so we could reach students no matter where they were in their understanding.  Instead of a test to determine who is allowed to learn what, we allowed every student to learn!  This needs to be a focus!

If we are ever going to help all of our students learn mathematics and believe that they are capable of thinking mathematically, then we need to provide learning experiences that ALL of our students can participate in.  These experiences need to:

  • Have multiple entry points for students to access the mathematics
  • Provide challenge for all students (be Problem-Based)
  • Allow students to actively make sense of the mathematics through mathematical reasoning
  • Allow students opportunities to students to express their understanding in different ways or reach an understanding via different strategies

Let’s avoid doing things that narrow our students’ learning like using entrance tickets to target instruction!  Let’s commit to a view of differentiated instruction where our students are the ones who are differentiating themselves (because the tasks allowed for opportunities to do things differently)!  Let’s continue to get better at leveraging students’ thinking in our classrooms to help those who are struggling!  Let’s believe that all of our students can learn!  


I want to leave you with a few reflective questions:

  • Why might conversations about entrance tickets and other ways to determine students ability be more common today?  We need to use our students’ thinking to guide our instruction, but other than entrance cards, how can we do this in ways that actually help those who are struggling?
  • Is a push for data-driven instruction fueling this type of decision making?  If so, who is asking for the data?  Are there other sources of data that you can be gathering that are healthier for you and your students?
  • If you’ve ever used entrance tickets or diagnostics, followed by ability groups, how did those on the bottom group feel?  Do you see the same students regularly in the bottom group?  Do you see a widening gap between those dependent on you and those who are more independent?
  • Where do you look for learning experiences that offer this kind of differentiated instruction?  Is it working for the students in your class that are struggling?

I encourage you to continue to think about what it means to Differentiate your Instruction.  Here are a few pieces that might help:

I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

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:

Slide1Slide2Slide3Slide4Slide5Slide6Slide7Slide8Slide9Slide10Slide11Slide12Slide13Slide14Slide15Slide16Slide17Slide18Slide19


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)

 

Unintended Messages

I read an interesting article by Yong Zhao the other day entitled What Works Can Hurt: Side Effects in Education where he discussed A simple reality that exists in schools and districts all over. Basically, he gives the analogy of education being like the field of medicine (Yes, I know this is an overused comparison, but let’s go with it for a minute).  Yong paints the picture of how careful drug and medicine companies have become in warning “customers” of both the benefits of using a specific drug and the potential side-effects that might result because of its use.

However, Yong continues to explain that the general public has not been given the same cautionary messages for any educational decision or program:

“This program helps improve your students’ reading scores, but it may make them hate reading forever.” No such information is given to teachers or school principals.
“This practice can help your children become a better student, but it may make her less creative.” No parent has been given information about effects and side effects of practices in schools.

Simply put, in education, we tend to discuss the benefits of any program or practice without thinking through how this might affect our students’ well-being in other areas.  The issue here might come as a direct result of teachers, schools and systems narrowing their focus to measure results without considering what is being measured and why, what is not being measured and why, and what the short and long term effects might be of this focus!


Let’s explore a few possible scenarios:

Practice:

In order to help students see the developmental nature of mathematical ideas, some teachers organize their discussions about their problems by starting to share the simplest ideas first then move toward more and more complicated samples.  The idea here is that students with simple or less efficient ideas can make connections with other ideas that will follow.

Unintended Side Effects:

Some students in this class might come to notice that their ideas or thinking is always called upon first, or always used as the model for others to learn from.  Either situation might cause this child to realize that they are or are not a “math person”.  Patterns in our decisions can lead students into the false belief that we value some students’ ideas over the rest.  We need to tailor our decisions and feedback based on what is important mathematically, and based on the students’ peronal needs.


Practice:

In order to meet the needs of a variety of students, teachers / schools / districts organize students by ability.  This can look like streaming (tracking), setting (regrouping of students for a specific subject), or within class ability grouping.

Unintended Side Effects:

A focus on sorting students by their potential moves the focus from helping our students learn, to determining if they are in the right group.  It can become easy as an educator to notice a student who is struggling and assume the issue is that they are not in the right group instead of focusing on a variety of learning opportunities that will help all students be successful.  If the focus remains on making sure students are grouped properly, it can become much more difficult for us to learn and develop new techniques!  To our students, being sorted can either help motivate, or dissuade students from believing they are capable!  Basically, sorting students leads both educators and students to develop fixed mindsets.  Instead of sorting students, understanding what differentiated instruction can look like in a mixed-ability class can help us move all of our students forward, while helping everyone develop a healthy relationship with mathematics.


Practice:

A common practice for some teachers involves working with small groups of students at a time with targeted needs.  Many see that this practice can help their students gain more confidence in specific areas of need.

Unintended Side Effects:

Sitting, working with students in small groups as a regular practice means that the teacher is not present during the learning that happens with the rest of the students.  Some students can become over reliant on the teacher in this scenario and tend to not work as diligently during times when not directly supervised.  If we want patient problem solvers, we need to provide our students with more opportunities for them to figure things out for themselves.


Practice:

Some teachers teach through direct instruction (standing in front of the class, or via slideshow notes, or videos) as their regular means of helping students learn new material.  Many realize it is quicker and easier for a teacher to just tell their students something.

Unintended Side Effects:

Students come to see mathematics as subject where memory and rules are what is valued and what is needed.  When confronted with novel problems, students are far less likely to find an entry point or to make sense of the problem because their teacher hadn’t told them how to do it yet.  These students are also far more likely to rely on memory instead of using mathematical reasoning or sense making strategies.  While direct instruction might be easier and quicker for students to learn things, it is also more likely these students will forget.  If we want our students to develop deep understanding of the material, we need them to help provide experiences where they will make sense of the material.  They need to construct their understanding through thinking and reasoning and by making mistakes followed by more thinking and reasoning.


Practice:

Many “diagnostic” assessments resources help us understand why students who are really struggling to access the mathematics are having issues.  They are designed to help us know specifically where a student is struggling and hopefully they offer next steps for teachers to use.  However, many teachers use these resources with their whole group – even with those who might not be struggling.  The belief here is that we should attempt  to find needs for everyone.

Unintended Side Effects:

When the intention of teachers is to find students’ weaknesses, we start to look at our students from a deficit model.  We start to see “Gaps” in understanding instead of partial understandings.  Teachers start to see themselves as the person helping to “fix” students, instead of providing experiences that will help build students’ understandings.  Students also come to see the subject as one where “mastering” a concept is a short-term goal, instead of the goal being mathematical reasoning and deep understanding of the concepts.  Instead of starting with what our students CAN’T do and DON’T know, we might want to start by providing our students with experiences where they can reason and think and learn through problem solving situations.  Here we can create situations where students learn WITH and FROM each other through rich tasks and problems.


Our Decisions:

Yong Zhao’s article – What Works Can Hurt: Side Effects in Education – is titled really well.  The problem is that some of the practices and programs that can prove to have great results in specific areas, might actually be harmful in other ways.  Because of this, I believe we need consider the benefits, limitations and unintended messages of any product and of any practice… especially if this is a school or system focus.

As a school or a system, this means that we need to be really thoughtful about what we are measuring and why.  Whatever we measure, we need to understand how much weight it has in telling us and our students what we are focused on, and what we value.  Like the saying goes, we measure what we value, and we value what we measure.  For instance:

  • If we measure fact retrieval, what are the unintended side effects?  What does this tell our students math is all about?  Who does this tell us math is for?
  • If we measure via multiple choice or fill-in-the-blank questions as a common practice, what are the unintended side effects?  What does this tell our students math is all about?  How reliable is this information?
  • If we measure items from last year’s standards (expectations), what are the unintended side effects?  Will we spend our classroom time giving experiences from prior grades, help build our students’ understanding of current topics?
  • If we only value standardized measurements, what are the unintended side effects?  Will we see classrooms where development of mathematics is the focus, or “answer getting” strategies?  What will our students think we value?

Some things to reflect on
  • Think about what it is like to be a student in your class for a moment.  What is it like to learn mathematics every day?  Would you want to learn mathematics in your class every day?  What would your students say you value?
  • Think about the students in front of you for a minute.  Who is good at math?  What makes you believe they are good at math?  How are we building up those that don’t see themselves as mathematicians?
  • Consider what your school and your district ask you to measure.  Which of the 5 strands of mathematics proficiency do these measurements focus on?  Which ones have been given less attention?  How can we help make sure we are not narrowing our focus and excluding some of the things that really matter?

baba

As always, I encourage you to leave a message here or on Twitter (@markchubb3)!

Which one has a bigger area?

Many grade 3 teachers in my district, after taking part in some professional development recently (provided by @teatherboard), have tried the same task relating to area.  I’d like to share the task with you and discuss some generalities we can consider for any topic in any grade.


The task:

As an introductory activity to area, students were provided with two images and asked which of the two shapes had the largest area.

Captureb.JPG

A variety of tools and manipulatives were handy, as always, for students to use to help them make sense of the problem.


Student ideas

Given very little direction and lots of time to think about how to solve this problem, we saw a wide range of student thinking.  Take a look at a few:

Some students used circles to help them find area.  What does this say about what they understand?  What issues do you see with this approach though?

Some students used shapes to cover the outline of each shape (perimeter).  Will they be able to find the shape with the greater area?  Is this strategy always / sometimes / never going to work?  What does this strategy say about what they understand?

IMG_4411
Example 3

Some students used identical shapes to cover the inside of each figure.

And some students used different shapes to cover the figures.

ccc
Example 7
IMG_4444
Example 8
C-rods, difference
Example 9

Notice that example 9 here includes different units in both figures, but has reorganized them underneath to show the difference (can you tell which line represents which figure?).


Building Meaningful Conversations

Each of the samples above show the thinking, reasoning and understanding that the students brought to our math class.  They were given a very difficult task and were asked to use their reasoning skills to find an answer and prove it.  In the end, students were split between which figure had the greater area (some believing they were equal, many believing that one of the two was larger).  In the end, students had very different numerical answers as to how much larger or smaller the figures were from each other.  These discrepancies set the stage for a powerful learning opportunity!

For example, asking questions that get at the big ideas of measurement are now possible because of this problem:

“How is it possible some of us believe the left figure has a larger area and some of us believe that the right figure is larger?”

“Has example 8 (scroll up to take a closer look) proven that they both have the same area?”

“Why did example 9 use two pictures?  It looks like many of the cuisenaire rods are missing in the second picture?  What did you think they did here?”

In the end, the conversations should bring about important information for us to understand:

  • We need comparable units if we are to compare 2 or more figures together.  This could mean using same-sized units (like examples 1, 4, 5 & 6 above), or corresponding units (like example 8 above), or units that can be reorganized and appropriately compared (like example 9).
  • If we want to determine the area numerically, we need to use the same-sized piece exclusively.
  • The smaller the unit we use, the more of them we will need to use.
  • It is difficult to find the exact area of figures with rounded parts using the tools we have.  So, our measurements are not precise.

Some generalizations we can make here to help us with any topic in any grade

When our students are being introduced to a new topic, it is always beneficial to start with their ideas first.  This way we can see the ideas they come to us with and engage in rich discussions during the lesson close that helps our students build understanding together.  It is here in the discussions that we can bridge the thinking our students currently have with the thinking needed to understand the concepts you want them to leave with.  In the example above, the students entered this year with many experiences using non-standard measurements, and this year, most of their experiences will be using standard measurements.  However, instead of starting to teach this year’s standards, we need to help our students make some connections, and see the need to learn something new.  Considering what the first few days look like in any unit is essential to make sure our students are adequately prepared to learn something new!  (More on this here: What does day one look like?)

To me, this is what formative assessment should look like in mathematics!  Setting up experiences that will challenge our students, listening and observing our students as they work and think… all to build conversations that will help our students make sense of the “big ideas” or key understandings we will need to learn in the upcoming lessons.  When we view formative assessment as a way to learn more about our students’ thinking, and as a way to bridge their thinking with where we are going, we tend to see our students through an asset lens (what they DO understand) instead of their through the deficit lens (i.e., gaps in understanding… “they can’t”…, “didn’t they learn this last year…?).  When we see our students through an asset lens, we tend to believe they are capable, and our students see themselves and the subject in a much more positive light!

Let’s take a closer look at the features of this lesson:

  • Little to no instruction was given – we wanted to learn about our students’ thinking, not see if they can follow directions
  • The problem was open enough to have multiple possible strategies and offer multiple possible entry points (low floor – high ceiling)
  • Asking students to prove something opens up many possibilities for rich discussions
  • Students needed to begin by using their reasoning skills, not procedural knowledge…
  • Coming up with a response involved students doing and thinking… but the real learning happened afterward – during the consolidation phase

A belief I have is that the deeper we understand the big ideas behind the math our students are learning, the more likely we will know what experiences our students need first!


A few things to reflect on:

  • How often do you give tasks hoping students will solve it a specific way?  And how often you give tasks that allow your students to show you their current thinking?  Which of these approaches do you value?
  • What do your students expect math class to be like on the first few days of a new topic/concept?  Do they expect marks and quizzes?  Or explanations, notes and lessons?  Or problems where students think and share, and eventually come to understand the mathematics deeply through rich discussions?  Is there a disconnect between what you believe is best, and what your students expect?
  • I’ve painted the picture here of formative assessment as a way to help us learn about how our students think – and not about gathering marks, grouping students, filling gaps.  What does formative assessment look like in your classroom?  Are there expectations put on you from others as to what formative assessment should look like?  How might the ideas here agree with or challenge your beliefs or the expectations put upon you?
  • Time is always a concern.  Is there value in building/constructing the learning together as a class, or is covering the curriculum standards good enough?  How might these two differ?  How would you like your students to experience mathematics?

As always, I’d love to hear your thoughts.  Leave a reply here on Twitter (@MarkChubb3)

Starting where our students are….. with THEIR thoughts

A common trend in education is to give students a diagnostic in order for us to know where to start. While I agree we should be starting where our students are, I think this can look very different in each classroom.  Does starting where our students are mean we give a test to determine ability levels, then program based on these differences?  Personally, I don’t think so.

Giving out a test or quiz at the beginning of instruction isn’t the ideal way of learning about our students.  Seeing the product of someone’s thinking often isn’t helpful in seeing HOW that child thinks (Read, What does “assessment drive instruction mean to you” for more on this). Instead, I offer an alternative- starting with a diagnostic task!  Here is an example of a diagnostic task given this week:

Taken from Van de Walle’s Teaching Student Centered Mathematics

This lesson is broken down into 4 parts.  Below are summaries of each:


Part 1 – Tell 1 or 2 interesting things about your shape

Start off in groups of 4.  One student picks up a shape and says something (or 2) interesting about that shape.


Here you will notice how students think about shapes. Will they describe the shape as “looking like a mountain” or “it’s an hourglass” (visualization is level 1 on Van Hiele’s levels of Geometric thought)… or will they describe attributes of that shape (this is level 2 according to Van Hiele)?

As the teacher, we listen to the things our students talk about so we will know how to organize the conversation later.


Part 2 – Pick 2 shapes.  Tell something similar or different about the 2 shapes.

Students randomly pick 2 shapes and either tell the group one thing similar or different about the two shapes. Each person offers their thoughts before 2 new shapes are picked.

Students who might have offered level 1 comments a minute ago will now need to consider thinking about attributes. Again, as the teacher, we listen for the attributes our students understand (i.e., number of sides, right angles, symmetry, number of vertices, number of pairs of parallel sides, angles….), and which attributes our students might be informally describing (i.e., using phrases like “corners”, or using gestures when attempting to describe something they haven’t learned yet).  See chart below for a better description of Van Hiele’s levels:

Van Hiele’s chart shared by NCTM

At this time, it is ideal to hold conversations with the whole group about any disagreements that might exist.  For example, the pairs of shapes above created disagreements about number of sides and number of vertices.  When we have disagreements, we need to bring these forward to the group so we can learn together.


Part 3 – Sorting using a “Target Shape”

Pick a “Target Shape”. Think about one of its attributes.  Sort the rest of the shapes based on the target shape.


The 2 groups above sorted their shapes based on different attributes. Can you figure out what their thinking is?  Were there any shapes that they might have disagreed upon?


Part 4 – Secret sort

Here, we want students to be able to think about shapes that share similar attributes (this can potentially lead our students into level 2 type thinking depending on our sort).  I suggest we provide shapes already sorted for our students, but sorted in a way that no group had just sorted the shapes. Ideally, this sort is something both in your standards and something you believe your students are ready to think about (based on the observations so far in this lesson).


In this lesson, we have noticed how our students think.  We could assess the level of Geometric thought they are currently using, or the attributes they are comfortable describing, or misconceptions that need to be addressed.  But, this lesson isn’t just about us gathering information, it is also about our students being actively engaged in the learning process!  We are intentionally helping our students make connections, reason and prove, learn/ revisit vocabulary, think deeper about specific attributes…


I’ve shared my thoughts about what I think day 1 should look like before for any given topic, and how we can use assessment to drive instruction, however, I wanted to write this blog about the specific topic of diagnostics.

In the above example, we listened to our students and used our understanding of our standards and developmental research to know where to start our conversations. As Van de Walle explains the purpose of formative assessment, we need to make our formative more like a streaming video, not just a test at the beginning!van-de-walle-streaming-video

If its formative, it needs to be ongoing… part of instruction… based on our observations, conversations, and the things students create…  This requires us to start with rich tasks that are open enough to allow everyone an entry point and for us to have a plan to move forward!

I’m reminded of Phil Daro’s quote:

daro-starting-point

For us to make these shifts, we need to consider our mindsets that also need to shift.  Statements like the following stand in the way of allowing our students to be actively engaged in the learning process starting with where they currently are:

  • My students aren’t ready for…
  • I need to start with the basics…
  • My students have gaps in their…
  • They don’t know the vocabulary yet…

These thoughts are counterproductive and lead to the Pygmalion effect (teacher beliefs about ability become students’ self-fulfilling prophecies).  When WE decide which students are ready for what tasks, I worry that we might be holding many of our students back!

If we want to know where to start our instruction, start where your students are in their understanding…with their own thoughts!!!!!  When we listen and observe our students first, we will know how to push their thinking!

How do you give feedback?

There seems to be a lot of research telling us how important feedback is to student performance, however, there’s little discussion about how we give this feedback and what the feedback actually looks like in mathematics. To start with, here are a few important points research says about feedback:

  • The timing of feedback is really important
  • The recipient of the feedback needs to do more work than the person giving the feedback
  • Students need opportunities to do something with the feedback
  • Feedback is not the same thing as giving advice

I will talk about each of these toward the end of this post.  First, I want to explain a piece about feedback that isn’t mentioned enough…  Providing students with feedback positions us and our students as learners.  Think about it for a second, when we “mark” things our attention starts with what students get right, but our attention moves quickly to trying to spot errors. Basically, when marking, we are looking for deficits. On the other hand, when we are giving feedback, we instead look for our students’ actual thinking.  We notice things as almost right, we notice misconceptions or overgeneralization…then think about how to help our students move forward.  When giving feedback, we are looking for our students strengths and readiness.  Asset thinking is FAR more productive, FAR more healthy, FAR more meaningful than grades!


Feedback Doesn’t Just Happen at the End!

Let’s take an example of a lesson involving creating, identifying, and extending linear growing patterns.  This is the 4th day in a series of lessons from a wonderful resource called From Patterns to Algebra.  Today, the students here were asked to create their own design that follows the pattern given to them on their card.

IMG_1395
Their pattern card read: Output number = Input number x3+2
IMG_1350
Their pattern card read:  Output number = Input number x7
IMG_1347
Their pattern card read:  Output number = Input number x4

 

IMG_1175
Their pattern card read:  Output number = Input number x3+1
IMG_1355
Their pattern card read:  Output number = Input number x8+2
IMG_1361
Their pattern card read: Output number = Input number x5+2

Once students made their designs, they were instructed to place their card upside down on their desk, and to circulate around the room quietly looking at others’ patterns.  Once they believed they knew the “pattern rule” they were allowed to check to see if they were correct by flipping over the card.

After several minutes of quiet thinking, and rotating around the room, the teacher stopped everyone and led the class in a lesson close that involved rich discussions about specific samples around the room.  Here is a brief explanation of this close:

Teacher:  Everyone think of a pattern that was really easy to tell what the pattern rule was.  Everyone point to one.  (Class walks over to the last picture above – picture 6).  What makes this pattern easy for others to recognize the pattern rule?  (Students respond and engage in dialogue about the shapes, colours, orientation, groupings…).

Teacher:  Can anyone tell the class what the 10th position would look like?  Turn to your partner and describe what you would see.  (Students share with neighbor, then with the class)

Teacher:  Think of one of the patterns around the room that might have been more difficult for you to figure out.  Point to one you want to talk about with the class.  (Students point to many different ones around the room.  The class visits several and engages in discussions about each.  Students notice some patterns are harder to count… some patterns follow the right number of tiles – but don’t follow a geometric pattern, some patterns don’t reflect the pattern listed on the card.  Each of these noticings are given time to discuss, in an environment that is about learning… not producing.  Everyone understands that mistakes are part of the learning process here and are eager to take their new knowledge and apply it.

The teacher then asks students to go back to their desks and gives each student a new card.  The instructions are similar, except, now she asks students to make it in a way that will help others recognize the patterns easily.

The process of creating, walking around the room silently, then discussing happens a second time.

To end the class, the teacher hands out an exit card asking students to articulate why some patterns are easier than others to recognize.  Examples were expected from students.


At the beginning of this post I shared 4 points from research about feedback.  I want to briefly talk about each:

The timing of feedback is really important

Feedback is best when it happens during the learning.  While I can see when it would be appropriate for us to collect items and write feedback for students, having the feedback happen in-the-moment is ideal!   Dan Meyer reminds us that instant feedback isn’t ideal.  Students need enough time to think about what they did right/wrong… what needs to be corrected.  On the other hand, having students submit items, then us giving them back a week later isn’t ideal either!  Having this time to think and receive feedback DURING the learning experience is ideal.  In the example above, feedback happened several times:

  1. As students walked around looking at patterns.  After they thought they knew the pattern, they peeked at the card.
  2. As students discuss several samples they are given time to give each other feedback about which patterns make sense… which ones visually represented the numeric value… which patterns could help us predict future visuals/values
  3. Afterward once the teacher collected the exit cards.

The recipient of the feedback needs to do more work than the person giving the feedback

Often we as teachers spend too much time writing detailed notes offering pieces of wisdom.  While this is often helpful, it isn’t a feasible thing to do on a daily basis. In fact, us doing all of the thinking doesn’t equate to students improving!  In the example above, students were expected to notice patterns that made sense to them, they engaged in conversations about the patterns.  Each student had to recognize how to make their pattern better because of the conversations.  The work of the feedback belonged, for the most part, within each student.

Students need opportunities to do something with the feedback

Once students receive feedback, they need to use that feedback to continue to improve.  In the above example, the students had an opportunity to create new patterns after the discussions.  After viewing the 2nd creations and seeing the exit cards, verbal or written feedback could be given to those that would benefit from it.


Feedback is not the same thing as giving advice

This last piece is an interesting one.  Feedback, by definition, is about seeing how well you have come to achieving your goal.  It is about what you did, not about what you need to do next.  “I noticed that you have switched the multiplicative and additive pieces in each of your patterns” is feedback.  “I am not sure what the next position would look like because I don’t see a pattern here” is feedback.  “The additive parts need to remain constant in each position” is not feedback… it is advice (or feedforward).

In the example above, the discussions allowed for ample time for feedback to happen.  If students were still struggling, it is appropriate to give direct advice.  But I’m not sure students would have understood any advice, or retained WHY they needed to take advice if we offered it too soon.


So I leave you with some final questions for you:

  • When do your students receive feedback?  How often?
  • Who gives your students their feedback?
  • Is it written?  Or verbal?
  • Which of these do you see as the most practical?  Meaningful for your students?  Productive?
  • How do you make time for feedback?
  • Who is doing the majority of the work… the person giving or the person receiving the feedback?
  • Do your students engage in tasks that allow for multiple opportunities for feedback to happen naturally?

PS.   Did you notice which of the students’ examples above had made an error.  What feedback would you give?  How would they receive this feedback?

 

 

 

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!

assidere


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:

https://player.vimeo.com/video/136761933?color=a185ac&title=0&byline=0&portrait=0

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.