How Big is “Big”?

In the last few weeks I have asked several groups of teachers to indicate where 1 billion would go on this number line:

It has been really interesting to me that many have placed the 1 billion mark in a variety of areas and have had a variety of reasons why.  Many have attempted to use their understanding of place value digits  (there are 12 zeros in 1 trillion and only 9 zeros in 1 billion, so 1 billion should be 3/4 of the way toward a trillion) or their knowledge of prefixes to help (million, billion, trillion… so it must be 2/3 the way along the line). Others thought about how many billions are in a trillion asking themselves, “Is their one-hundred or one-thousand billions in a trillion?” Using this strategy, everyone picked a spot toward the left, but some much closer to zero than others.

Others did something interesting though. They started placing other numbers on the number line to help them make sense of the question. Often placing 500 billion in the middle, then 250 billion at the 1/4 mark and so on until they realized just how close to 0 a billion is when we are considering 1 trillion.

What’s the point?

Really big numbers, and really small numbers (decimal numbers), are difficult to conceptualize!  They are hard to imagine their size!  Think about this:

How long is 1 million seconds?  Without doing ANY calculations would you guess the answer is several minutes, hours, days, weeks, months, years, decades, centuries…?  Can you even imagine a million seconds without calculating anything?

How about 1 billion seconds?  Or 1 trillion seconds?

I bet you’ve started trying to calculate right!  That’s because these numbers are so abstract for us that we can’t imagine them.

Because of this little experiment, I am left wondering three things:

  1. What numbers can/can’t the students in our classrooms conceptualize?
  2. What practices do we do that gets kids to think about digits more than magnitude?
  3. What practices could/should we be including that helps our students make these connections?

What numbers can/can’t the students in our classrooms conceptualize?

Before we start working with operations of any given size, I think we need to spend time making sure our students can visualize and estimate the size of the number.  Working with numbers we can’t imagine doesn’t seem productive for our young students!  In our rush to move our kids into more “complicated” mathematics, we often move too quickly through numbers to include numbers that are too abstract for our students!  We think that if a student can accurately carry out a procedure that they understand the numbers they are working with. However, I’m sure we have all seen many students who produce answers that are completely unreasonable without them noticing. Is this carelessness, or is it a lack of understanding of the magnitude of the numbers involved?  Or possibly that our students aren’t visualizing the size of and relationship between the numbers???


What practices do we do that gets kids to think about digits more than magnitude?

The other day, Jamie Garner shared her frustration on Twitter:


Think about the question from the textbook for a second. Students trying to think about 342 pencils (not sure why they would want that many) should be considering a strategy that makes sense. For example, if you had 342 pencils how many boxes of 10 would that be?  Thinking this way, student should answer 34 or 34.2, or maybe 35 boxes (if you wanted to purchase enough boxes).  However, the teacher’s edition tells us that none of these are the right answer. Take a look:


If our students attempt to make sense of the problem, they will be completely wrong!  In fact, many students will likely answer 4 because they’ve been trained not to think at all about the mathematics, and instead focus their attention on what they think the text wants them to do.

This is one of MANY cases where elementary mathematics focuses on digits over understanding magnitude or relative size.  Here are a few others:


These, along with pretty much any standard algorithm (see Christopher Danielson’s post: Standard Algorithms Unteach Place Value) tell our kids to stop thinking about what makes sense, and instead focus on steps that help kids get an answer without understanding.


What practices could/should we be including that helps our students make these connections?

If we want our students to understand numbers, and their relative size… if we want to help our students develop a conceptual understanding of operations… if we want our students make sense of the math they are learning… then we need to:

  • Use contexts that make sense to our students (not pseudo-contexts like the pencil question above).
  • Provide plenty of experiences where students are making sense of numbers visually.  When we allow our students to access their Spatial Reasoning we are allowing them to see the relationship between numbers and help them make connections between concepts.
  • Provide plenty of experiences estimating with numbers

Below are 2 activities taken from Van de Walle’s Student Centered Mathematics.  Think about how you could adapt these to work with numbers your students are starting to explore (really big or really small numbers).

vdw-number-line1vdw-number-line2


A few questions for you to reflect on:

  • How might you see how well your students understand the numbers that are really large or really small?
  • How are you helping your students develop reasonableness when working with numbers?
  • What visuals are you using in your class that help your students visualize the numbers you are working with?
  • What practices do you use regularly that help with any of the 3 above?

P.S. Here are the answers to the seconds problem I posted earlier:

Subtracting Integers – Do you see it as removal or difference???

If you are thinking about how we should develop an understanding of adding and subtracting integers, it is very important to first consider how primary students should be developing adding and subtracting natural numbers (positive integers).

Of the two operations, subtraction seems to be the one that many of us struggle to know the types of experiences our students need to be successful.  So let’s take a look at subtraction for a moment.


Subtraction can be thought of as removal…

We had 43 apples in a basket. The group ate 7.  How many are left?  (43-7 =___)

Or subtraction can be thought of as difference…  

I had 43 apples in a basket this morning.  Now I only have 38.  How may were eaten?  (43-___=38  or 38+____=43)


Each of these situations requires different thinking.  Removal is the most common method of teaching subtraction (it is simpler for many of us because it follows the standard algorithm – but it is not more common nor more conceptual as difference).  Typically, those who have an internal number line (can think based on understanding of closeness of numbers…) can pick which strategy they use based on the question.

For instance, removing the contexts completely, think about how you would solve these 2 problems:

25-4 =

25-22=

Think for a moment like a primary student.  The first problem is much easier for many!  If the only strategy a student has is counting backwards, the second method is quite complicated!  In the end, we want all of our students to be able to make sense of subtraction as both the removal and difference!  (We want our students to gain a relational understanding of subtraction).


Take a minute to watch Dr. Alex Lawson discuss the power of numberlines:

https://player.vimeo.com/video/88069524?color=a185ac&title=0&byline=0&portrait=0

Alex Lawson: The Power of the Number Line from LearnTeachLead on Vimeo.


How does this relate to Integers?

Subtraction being thought of as removal is often taught using integer chips (making zero pairs…).  Take a look at the examples below.  Can you figure out what is happening here?  What do the boxes mean?

integers subtraction.png
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Number lines are often used with Integer operations too, but the method of using them is typically removal as well.  Think about this problem for a minute:

int_addsubtract08
Enter a caption
Again, students view of subtraction is removal here (or with the case of subtracting negative numbers here, students will be adding).


However, now let’s think about how a number line could help us understand integers more deeply if we thought of subtraction as difference:

Hopefully for many students, they might be able to see how far apart the numbers (+5) and (-2) are.  Without going through a bunch of procedures, many might already understand the difference between these numbers.  Many students come to understand something potentially difficult as (+5) – (-2) = 7 quite easily!

integers-difference


I encourage you to try to create 2 different number line representations of the following question, one using removal and the other using difference:  

(-4) – (-7) = 

 


Some final thoughts:

  • When is it appropriate for us to use difference?  When is it appropriate for us to use removal? 
  • Should students explore 1 first?  Which one?
  • Which is easier for you?  Are you sure it is also the easiest strategy for all of your students?
  • The questions above have no context of any kind.  I wonder if this makes this concept more or less difficult for our students?
  • How do you provide experiences for your students to make sense of things, not just follow rules and procedures?
  • How can we avoid the typical tricks used during this unit?

As always, I’d love to continue the conversation. Send me your number lines or share your thoughts / questions below or on Twitter (@markchubb3)

Questioning the pattern of our questions

I find myself spending more and more time trying to get better at two things.  Listening and asking the right kinds of questions that will push thinking.  While I find that resources have helped me get better at asking the right questions, I have learned that listening is actually quite difficult.  The quote below is something that made me really think and reflect on my own listening skills:listening

More about this in a minute…


A while ago I had the pleasure to work with a second grade teacher as we were learning how to do String mini-lessons (similar to Number Talks) to help her students reason about subtraction.  After a few weeks of getting comfortable with the routine, and her students getting comfortable with mental subtraction, I walked into the class and saw a student write this:

img_2854

What would you have asked?

What would you have done?

Did she get the right answer?


My initial instincts told me to correct her thinking and show her how to correctly subtract, however, I instead decided to ask a few questions and listen to her reasoning.  When asked how she knew the answer was 13 she quickly started explaining by drawing a number line.  Take a look at her second representation:

She explained that 58 and 78 were 20 away from each other, but 58 and 71 weren’t quite 20 away, so she needed to subtract.

I asked her a few questions to push her thinking with different numbers to see if her reasoning would always work.

Is her reasoning sound?  Will this always work?  Try a few yourself to see!

Typically, we look at subtraction as REMOVAL (taking something away from something else), however, this student saw this subtraction question as DIFFERENCE (the space between two numbers).

I wonder what would have happened if I “corrected” her mathematics?  I wonder what would have happened if I neglected to listen to her thinking?  Would she have attempted to figure things out on her own next time, or would she have waited until she was shown the “correct” way first?

I also wonder, how often we do this as teachers?  All it takes is a few times for a student’s thinking to be dismissed before they realize their role isn’t to think… but to copy the teacher’s thinking.


Funneling vs. Focusing Questions

As part of my own learning, I have really started to notice the types of questions I ask.  There is a really big difference here between funneling and focusing questions:

slide_12.jpg

Think about this from the students’ perspective.  What happens when we start to question them?

Screen Shot 2013-11-07 at 1.49.12 PM.png
Summarized by Annie Forest in her Blog

Please make sure you continue to read more about we can get better at paying attention to the pattern of our questions:

Questioning Our Patterns of Questioning by Herbel-Eisenmann and Breyfogle

Starting where our students are….. with THEIR thoughts


So I leave you with some final thoughts:

  • Do you tend to ask funneling questions or focusing questions?
  • How do we get better at asking questions and listening to our students’ thinking?
  • What barriers are there to getting better at asking questions and listening?  How can we remove these barriers?
  • Is there a time for asking funneling questions?  Or is this to be avoided?
  • What unintended messages are we sending our students when we funnel their thinking?  … or when we help them focus their thinking?
  • What if our students’ reasoning makes sense, but WE don’t understand?

 

I’d love to continue the conversation about the subtraction question above, or about questioning and listening in general.  Leave a comment here or on Twitter @MarkChubb3

What are your thoughts?

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.

Thinking Mathematically

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…

View original post 571 more words

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 our district improved…

Earlier this week news was made public about the Province’s and our district’s grade 3 and grade 6 math results.  For the past several years, there has been a negative trend both here and across the Province in grade 6 math, and that trend continued again this year across the Province… with our district being the outlier… we made significant gains.  In fact, our school board jumped 12% since the last testing (I’m not sure, but I don’t believe there has been such a spike for a school board our size before).  So I think it might be worth discussing WHY this might have happened.

If you are reading this article, you are probably a teacher, or administrator, or have some other role in education.  Before you continue, I want you to think for a minute about why you became an educator?  It was likely because you cared about students, saw value in providing children with opportunities that would positively affect their lives… probably it wasn’t to try to get scores of some test to go up.  I want you to keep this in mind as you continue to read.


Our district has put a concerted effort into mathematics teaching and learning over the past few years.  While we might want to look at what the changes were last year that made the difference, I want to reach back a little further to give you the bigger picture.


2006-2010

Our school board engaged in intensive training for several teachers in a program called SUM (Supporting Understanding in Mathematics).  This training involved willing teachers who wanted to learn more about the process of learning and teaching of mathematics.  These teachers delved into well-researched resources, co-taught lessons together, viewed others’ classrooms, and deepened their own understanding of mathematics in the process.  The goal for these sessions was to help build teachers’ capacity, help develop their math knowledge for teaching.  The drawback, was that few teachers across the board could participate, but the teachers who did participate, quickly became leaders in their buildings and within the system.  For many, these experiences changed their view of mathematics education significantly.


2011-2013

Our school board invested heavily in Cathy Fosnot’s Contexts for Learning.  Cathy herself trained many teachers.  Teachers had many opportunities to learn together in co-planning, co-teaching, co-debriefing sessions as they learned through using these resources.  The benefit from these sessions is that we learned what teaching THROUGH problem solving looks like, how we can assess using developmental landscapes, how we can build procedural fluency from conceptual development.  The teachers who participated learned mathematics in ways they had never experienced as students.  Personally, I learned to think mathematically because of these experiences.  The drawback, was that these units only covered 1 of the 5 strands in our curriculum, but it was a great way to help all of us see that mathematics could look different than it did when we were in school.  During this time, our board also invested heavily by purchasing a copy of Van de Walle’s Teaching Student Centered Mathematics for every teacher.

Discussions from this resource helped start conversations about other strands (lots of content learning and ideas), and helped foster changes in how we viewed the subject (relational understanding, assessment, differentiated instruction…).


2014-2016

Over the 2014-2015 years, our board increased the number of instructional coaches in the system, we implemented a flexible scope-and-sequence to allow conversations between teachers to happen about the same topic, we released math newsletters helping us deepen our understanding of math concepts, we gave every student a Dreambox account, and we started offering free Additional Qualifications (AQ) courses to any teacher who was willing to take one.  While each of these are important, I want to address the last point.  Between 2014 and 2016 we have provided over 550 AQ courses to the teachers in our school board (our board has approximately 1500-2000 teachers).  These courses provided experiences for teachers to deepen their understanding in mathematics THROUGH problem solving opportunities.  I believe that the reason why so many teachers in our board have invested their time (125+ hours after school per course) and energy in these courses is because of the initial investment our board put into its teachers.  Our teachers have seen just how important OUR learning is, and because of this they are willing to continue their learning.


As a system, over the past several years, the goals of our school board have been very clear.  As a system, we have continued to work towards:

 

These 3 goals are aimed at helping us reach our board goal for students:

d.jpg

I want you to notice which words/phrases you like above?  Which ones catch your attention?  Are each of these important to you?


So, I assume that there are going to be a lot of questions about why scores improved so much, and I hope 2 stories become front and center:

  1. We have invested in our teachers!  From SUM groups, to site-based instructional coaches, to providing AQ courses, we have put OUR learning as the focus.  If we want to provide a better education for our students, we need to understand the mathematics deeply, understand how mathematics concepts develop over time, we need to understand our curriculum deeply, we need to understand pedagogical moves that will help our students learn…  Changing the culture to help us become learners has made a huge difference!
  2. We are using researched-based resources.  From Cathy Fosnot’s Context for Learning units, to Beaty/Bruce’s From Patterns to Algebra resources, to Jo Boaler’s research, to Marian Small’s resources, to Cathy Bruce’s Fractions research, to Fosnot’s Dreambox and String mini-lessons, to our Province’s Guide to Effective Instruction work, to Van de Walle’s Teaching Student Centered Mathematics……..  We have delved into a lot of resources, and it is paying off.  While resources typically aren’t the answer, much of these resources have helped us understand mathematics relationally, they have helped us see and understand mathematics in ways that we never experienced as students.  They have helped us visualize, and conceptualize in ways that help us help our students.  More than just a resource to follow, these have become platforms for which we have been learning.

These two pieces, in my mind, are the big reasons we have started to make gains.  Yes, I am sure there are countless other factors, but none of them could possibly help without making sure we have invested in our teachers, and have provided appropriate resources that will help us learn.


I wanted to write this post, not as a “how-to…” for districts, but as a reminder for what is important.  When we focus on deepening OUR understanding, our students benefit… when WE learn through problem solving, we will likely see how we can help our students do the same… when WE see how to provide experiences for our students that are powerful learning opportunities, we won’t rely on gimmicks and fads…

Yes, our scores went up, but that’s not what is important.  What IS important is that our students learn mathematics in ways that allow them to gain a relational understanding.  Our students deserve lessons that are interactive and experiential… they deserve to learn mathematics through thinking and doing and building and exploring, not listening and copying… they deserve to have teachers who understand the mathematics they are teaching and are passionate about the subject… they deserve to have classrooms that are vibrant and full of rich mathematical discussions… they deserve to learn in safe classrooms that promote growth mindset messages… they deserve teachers that view games and puzzles as potential sources of learning or practice… they deserve to see mathematics in ways that makes sense, not through “answer getting” tricks… they deserve teachers who care about mathematics and see mathematics as valuable and fun and creative and beautiful… and they deserve a system that cares about their teachers, because a system that cares about their teachers has teachers who care about their students!

Yes, our scores have gone up, but what really matters is that our students are liking mathematics better… they see themselves as mathematicians… they believe that can succeed… they are starting to see mathematics as something that makes sense!  My hope is this doesn’t mean we are done learning.  We have a lot of work still to do with the list above!  Real change takes time!


So I leave you with a few thoughts:

  • How is your district supporting you?  Are the initiatives similar to the ones I’ve written about above?
  • Is your district interested in providing the best mathematics education for your students, or trying to get scores up?  Is this the same thing?
  • How are you deepening your own understanding of the concepts you teach?
  • What ways have you /could you collaborate with others to deepen your own understanding of the concepts you teach?
  • What opportunities are out there for you to continue to learn?
  • What resources have you used that have pushed your own thinking?
  • If we focus on scores will teaching and learning improve?  If we focus on teaching and learning will scores improve?  Are we mixing up the goal and the evidence of that goal?

 

As always, I’d love to continue the conversation here, or on Twitter (@MarkChubb3).

How do we meet the needs of so many unique students in a mixed-ability classroom?

Explaining what something is can be really hard to do without that person actually experiencing the same thing as you.  One strategy that we often use to explain difficult concepts in math is to discuss non-examples.  Consider how the frayer model below could be used with any difficult concept you are discussing in class.

frayer-model

If we discussed fractions in class, many students might believe that they understand the concept, however, they might be over-generalizing.  Seeing non-examples would help all gain a much clearer idea of what fractions are.  Of the shapes below, which ones have 1/2 or 1/4 of the area shaded blue?  Which ones do not represent 1/2 or 1/4 of the shapes’ area?

Having students complete activities like this would be an excellent way for you to originally see students’ understanding, and then see students refine and develop ideas throughout the unit (they can continually add different models and correct misconceptions)


The purpose of this post, however, isn’t about fractions or even a Frayer Model.  I am actually writing about the often used phrase “Differentiated Instruction” (DI).  Hopefully we can think more about what DI looks like in our math classes by thinking about what DI is and isn’t.


How would you define Differentiated Instruction?

Take a look at the following graphic created by ASCD helping us think about what DI is and isn’t.

di

In many places, DI is looked at as grouping students by ability, or providing individualized instruction.  However, if you look at the graphic above, these are in the non-example section.  These views are probably more common in highly teacher-centered classrooms, where the teacher feels like they need to be in control of every student’s process, products and content.  For me, I don’t see how it is possible to have every student doing the same thing at the same time, or how productive it would be if we assigned different students to do different work (sounds like an access and equity issue here).


So how do we help all of our students in a mixed ability classroom???

First of all, a few words from Van de Walle’s Teaching Student Centered Mathematics p.43:

All [students} do not learn the same thing in the same way at the same rate.  In fact, every classroom at every grade level contains a range of students with varying abilities and backgrounds.  Perhaps the most important work of teachers today is to be able to plan (and teach) lessons that support and challenge ALL students to learn important mathematics.

Teachers have for some time embraced the notion that students vary in reading ability, but the idea that students can and do vary in mathematical development may be new.  Mathematics education research reveals a great deal of evidence demonstrating that students vary in their understanding of specific mathematical ideas.  Attending to these differences in students’ mathematical development is key to differentiating mathematics instruction for your students.

Interestingly, the problem-based approach to teaching is the best way to teach mathematics while attending to the range of students in your classroom.  In a traditional, highly directed lesson, it is often assumed that all students will understand and use the same approach and the same ideas as determined by the teacher.  Students not ready to understand the ideas presented by the teacher must focus their attention on following rules or directions without developing a conceptual or relational understanding (Skemp, 1978).  This, of course, leads to endless difficulties and can leave students with misunderstandings or in need of significant remediation.  In contrast, in a problem-based classroom, students are expected to approach problems in a variety of ways that make sense to them, bringing to each problem the skills and ideas that they own.  So, with a problem-based approach to teaching mathematics, differentiation is already built in to some degree.


When we take a student centered view of Differentiated Instruction, we start to see that all students can be given the SAME work, yet each individual student will be able to adjust the process, product and/or content naturally.  However, this requires us to start with things where students are going to make sense of them.  It requires us to move our instruction from a teaching FOR problem solving approach to a teaching THROUGH problem solving approach.  It requires us to offer things that are actually problems, not just practicing skills in contexts.


3 Strategies for Differentiating Instruction:

There seem to be 3 different ways we can help all of our students access to the same curriculum expectations, while attending to the various differences in our students:

  1. Open-Middle Problems
  2. Open-Ended Problems
  3. Parallel Problems

Open-middle problems, or open routed problems as they are sometimes called, typically have 1 possible solution, however, there are several different strategies or pathways to reach the solution.  These are a great way for us to offer something that everyone will have access to.  Ideally, these problems need to have an entry point that all students enter into the problem with, yet offer extensions for all.  The benefit of this type of problem is that we can listen to and learn from our students about the strategies they use.  We can then use the 5 Practices as a way to move instruction forward based on our assessments.

Open-ended problems, on the other hand, typically have multiple plausible answers.   These problems, in contrast, offer a much wider range of content.  Again, the benefit of this type of problem is that we can listen to and learn from our students about the types of thinking they are currently using, and from there, consider what they are ready for next.  Again, we can then use the 5 Practices as a way to move instruction forward based on our assessments.

Parallel Problems differ from the other two types in that we are actually offering different things for our students to work on.  Hopefully, our students are given choice here as to which path they are taking, so we don’t run into the issue earlier posted about DI not being about ability grouping.  Parallel problems are aptly named because while some of the pathways are easier than others, all pathways are designed to meet the same curricular expectations (content standards).  Again though, we should be using the 5 Practices as a way to share student thinking with each other so our students can learn WITH and FROM each other and so we can move instruction forward based on our assessments.


For an in depth understanding of how these help us, please read the article entitled Differentiating Mathematics Instruction.


As always, I want to leave you with a few reflective questions:

  • How do you balance the need to teach 1 set of standards with the responsibility of meeting your students where they are?
  • Do you hear student centered messages like the ones I’ve posted here about DI, or more teacher centered messages?  Which set of messages do you believe is easier for you to attempt as a teacher?  Which set of messages would you believe would make the learning in your classroom richer?
  • Who tends to participate in your classrooms?  Who tends to not participate?  How can the DI strategies above help change this dynamic so that the voices being heard in your classroom are more distributed?
  • What issues do you see being a barrier to DI looking like this?  How can the online community help?

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

Aiming for Mastery?

The other day, a good friend of mine shared this picture with me asking me what my thoughts were:

mastery

On first glance, my thoughts were mixed.  On one hand, I really like that students are receiving messages about mistakes being part of the process and that there is more than one opportunity for students to show that they understand something.

However, there is something about this chart that seems missing to me…  If you look at the title, it says “How to Learn Math” and the first step is “learn a new skill”.  Hmmm…… am I missing something here?  If the bulletin board is all about how we learn math, the first step can’t be “learn a new skill”.  For me, I’m curious about HOW the students learn their math?  While this is probably the most important piece (at least to me), it seems to be completely brushed aside here.

I want you to take a look at the chart below.  Which teaching approach do you think is implied with this bulletin board?  What do you notice in the “Goals” row?  What do you notice in the “Roles” row?  What do you notice in the “Process” row?

teaching-approaches
Taken from PRIME Leadership

If “Mastery” is the goal, and most of the time is spent on practicing and being quizzed on “skills” then I assume that the process of learning isn’t even considered here.  Students are likely passively listening and watching someone else show them how… and the process of learning will likely be drills and practice activities where students aim at getting everything right.


I’d like to offer another view…

Learning is an active process.  To learn math means to be actively involved in this process:

  • It requires us to think and reason…
  • To pose problems and make conjectures…
  • To use manipulatives and visuals to represent our thinking…
  • To communicate in a variety of ways to others our thinking and our questions…
  • To solve new problems using what we already know
  • To listen to others’ solutions and consider how their solutions are similar or different than our own…
  • To reflect on our learning and make connections between concepts…

It is this process of learning that is often neglected, and often brushed aside.


 

While I do understand that there are many skills in math, seeing mathematics as a bunch of isolated skills that need to be mastered is probably neither a positive way to experience mathematics, nor will it help us get through all of the material our curriculum expects.  What is needed is deeper connections, not isolating each individual piece… more time spent on reasoning, not memorizing each skill… richer tasks and learning opportunities, not more quizzes…

“One way to think of a person’s understanding of mathematics is that it exists along a continuum. At one end is a rich set of connections. At the other end of the continuum, ideas are isolated or viewed as disconnected bits of information. A sound understanding of mathematics is one that sees the connections within mathematics and between mathematics and the world”
TIPS4RM: Developing Mathematical Literacy, 2005

So, while I don’t think putting up a bulletin board (or not) is really going to do much, I really do hope we are spending more of our time thinking about HOW our students learn and WHAT we are goals are for our students (see the chart above again).

I still wonder what it means to be good at math?  I wrote about my questions here, but I am still looking for ways to show others how important mathematical reasoning is for students to develop.  Skills without reasoning won’t get you very far.  Maybe more about this another time…

Parallel Tasks

Open problems are probably my most used strategy to help meet each student where they are.  Problems that offer a low floor and high ceiling are great because all students engage in the learning, then can participate and learn from each other.  However, some teachers also like to offer Parallel Tasks as a way to differentiate instruction.  The idea here is that students can be given a choice of a task/problem, some being more difficult than others, yet all of the tasks/problems deal with the same standard (curriculum expectation).  Let’s take a look at an example of how a quality parallel task can work:

Take a look at the problem below.  What is it asking us to do?

Pick ONE of the choices… build your design worth “B”.  Be ready to share how you know your answer is correct.

I’d love some actual responses here.  Build it using actual manipulatives (ideally) or using virtual manipulatives (This Illuminations Applet might help).


 

Notice that each choice allows students to do the same expectation related to proportional thinking, however, students are given choice about what numbers they want to think about.

Think about what the answers would look like?  When we discuss designs afterward, we should be able to discuss the solutions to each problem and compare the similarities and differences.


Here are some designs students made.  Can you tell which option each student chose?

pattern-blocks-1
Student #1
pattern-blocks-4
Student #2

 

pattern-blocks-3
Student #3
pattern-blocks-5
Student #4

 

Actually try to match the designs to each of the tasks/problems.  Take a moment to think this through.  What do you notice about the 4 images above?

This task was designed very cleverly to help make a point… to help us bring ALL of our students together to have a conversation  (Even when we ask our students to do different things from each other, we still need to make sure we come together and have shared experiences).


 

Did you notice anything about the 3 options?  Did you try decimals or fractions to solve any of the students’ designs?  If you did, you would notice that all 3 options used the same proportions.


 

A great parallel task helps us to learn things together…  It helps us see others’ thinking…  It allows every student to start to think where they are comfortable, yet be able to learn and grow from the ideas of others.


 

If you were to offer a parallel problem/task for your students would you:

  • Choose which students get each choice, or allow students to pick themselves? (Does this matter?)
  • Expect all students to create the design using blocks or digitally?  (Does this matter?)
  • Ask students to work independently or in a small group?  (Does this matter?)
  • Offer calculators or not?  (Does this matter?)
  • Engage in 3 different conversations – 1 per group – or 1 conversation all together?  (Does this matter?)

The small decisions we make tell a lot about what we value!    Personally, IF I want to offer Parallel tasks/problems, I want to make sure that all of my students feel successful, that everyone realize their ideas are valued, that there isn’t a hierarchy of ability in the room… and of course, that the mathematics we are engaged is important.  

I’d love some feedback about Parallel tasks in general, or the task itself.

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?