Quick fixes and silver bullets…

I find myself reflecting on what I believe is best for my students and best for my students’ beliefs about what mathematics is often.  When I get the opportunity to take a look at my students’ work and time to determine next steps, I can’t help but reflect on how my beliefs inform what next steps I would take.   However, I wonder, given the same students and the same results, if we would all give the same next steps?  Let’s take a look at a few common beliefs about what our students need to be successful and discuss each.


My kids need to know their facts:

Often we see students who make careless mistakes and wonder why they could have gone wrong with something so simple. To some, the belief here is that if we could just memorize more facts, they would be able to transfer those facts to the problems in the assessment. While I agree that we want our students being comfortable with the numbers they are working with, I’m not convinced that memorizing is the answer here. Our Provincial test includes a few computation questions (for grade 3 only) and none of these are timed. Most of our questions involve students making sense of things (across 5 strands), some with contexts and some without.

Instead of spending more time worrying about memorizing facts, I wonder if other strategies have been thought of too?  For example, other than the 4 questions in grade 3, all other questions allow students to use manipulatives or calculators, and all questions have space for students to write in the margins any rough work or visual models might be using.  The question below is one of the few computational questions a grade 3 student is expected to do.  Many who might use the traditional algorithm might accidently pick 41.  How might a number line help our students visualize the space between the two numbers here?

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My kid aren’t reading the questions:

Often we notice that our students understand a concept, but the question itself requires several steps and students don’t end up answering what is asked. For many the solution is having students do some kind of strategy (whether they need to or not) like highlighting key words.

I wonder what answer students might get to the above question?  Will they get the right answer here?  What would you have liked them to do instead?

Highlighting specific words or filling out a standard graphic organizer isn’t the answer for all kids, nor for all questions!

Personally, I think the issue isn’t that our students can’t read the questions, it is that they are jumping to a solution strategy too quickly.  Instead of believing the solution is to have our students highlight or fill out graphic organizers, what might be more appropriate is to help our students slow down and think deeper about the questions they are being asked.  I wonder, however, about how our students’ prior experiences might be a big part of why they jump to solution strategies too quickly?  If students typically receive questions that are simple and closed, and typically follow a lesson directly telling us how to answer those questions, then I wonder if the issue is that our students can’t read or if their experiences have actually been counter-productive? If students don’t experience mathematics in ways that help them make sense of a situation, and instead see math as answering a bunch of questions, then it is no wonder why they aren’t reading the whole question!  They have been trained to believe math is about getting answers quickly, and that we get rewarded (less homework, better grades…) when we are fast.

Instead of more time practicing reading and highlighting questions, or filling out graphic organizers, we might want to spend more time building questions together, asking students to pose their own problems, asking students to notice and wonder. What if we started by showing this:

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What do you notice here?  What do you wonder?

What might our students see?  They might notice things we didn’t realize they were not even aware of (e.g., each row has 7 boxes, some rows are missing numbers, the numbers go in order, the letters at the top probably mean days of the week…..).

What might our students be curious about?  They might wonder why April 1st is on a Monday and not a Sunday.  Or wonder about the “Chapter 1” part.

Then we could continue to show more of the question and again ask what students notice and wonder.

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No matter the grade level or content, my students need to realize that mathematics is about the development of mathematical reasoning, not just quickly jumping to a solution strategy (especially not one that my teacher told me to use all the time).  Taking the time to think deeply about our mathematics is what I want from my students!  Numberless word problems, notice and wonder strategy, contemplate and calculate… any strategy that helps my students slow down, pay attention to visuals, and start to think about the situation more will probably help many of our students given enough opportunities.


They need more practice with questions like these:

Many believe that if students are doing poorly on something, that the best course of action is to continue practicing that thing.  For example, if our students are doing poorly on Provincial testing questions, then giving students more questions like these will answer all of the issues.

To some, the answer to the problem is to give sample questions every week in a package or even more frequently.  While there are times when practice is helpful, if our students are struggling with the content, giving more questions will not be helpful!  Take a look at Daro’s quote:

More often than not, the quick fix solutions like this (noticing our students struggle with something, then providing the same instruction or same types of practice again) will not be successful.  Developing our math knowledge for teaching is probably the most difficult aspect of teaching mathematics, and is definitely NOT a quick fix, but it is probably the answer here!  If our students are struggling with concepts we believe they should be able to do, it is likely that they haven’t had the right experiences to help them learn!  Have we provided experiences for our students to deeply explore a variety of representations?  Have we provided experiences where our students are able to develop reasoning skills?  Have we provided ample opportunities for our students to consolidate their learning?

Remember, questions that are designed to show evidence OF student learning is not necessarily the WAY students learn!  Handing out these questions toward the end of the learning is far more reasonable.


My students need more stamina:

Often when giving young students an extended time to sit and focus independently on an assessment task we have many that struggle to remain focussed.  For some, the solution is to help students build stamina through quiet seat work regularly.  While I do agree that we should have our students work independently at times, I’m not sure this is the answer to the stamina problem.

To me, I think the issue has more to do with how our students experience mathematics.  Do they get lots of short closed questions where the right answer is apparent quickly?  Or do they experience rich problems where they reason through and figure out their own way of making the question make sense?  Do they learn math through independent think time and cooperative problem solving, or are they told material then asked to remember all of the steps and terms.  Is their mathematics class structured in a way where students come to rely on themselves (individually or within their group) to make sense of challenging problems, or do they feel the need to access their teacher every time they don’t know what to do?  When our students are working, are we monitoring all of our students’ thinking, or are we spending a lot of time guiding our students’ thinking?

If stamina is the issue, I wonder if we are allowing our students to productively struggle enough?  If we see a bunch of hands raised around the classroom all wanting us to help, this is a huge red flag moment.  Our students are asking US to think for them!  If we find ourselves sitting beside our students helping out a small group all of the time, this might be another red flag.  Our students are learning that they always have access to us right beside them when they learn, but the unintended problem is that we aren’t allowing our students to struggle enough!

Providing our students with a variety of manipulatives to learn and puzzle through their mathematics on a daily basis might be a big step in the direction of allowing our students to gain the confidence and stamina they need to do well every day.  Notice how these students are using manipulatives to help them make sense of their work:

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When our students learn their mathematics using manipulatives and have access to any manipulative at any time to solve new problems, we start to notice that our students come to realize their role is to slow down and make sense of things.  When our students have had various experiences with manipulatives and can see their role as “thinking tools”, we start to notice fewer hands asking for help, less need to have to sit down with a group, and more time for us to really notice our students’ thinking going on in our classroom.  When this starts to happen, we no longer see stamina as a big issue.


I want you to consider for a moment the differences between the beliefs I’ve mentioned. What messages are we sending to our students about what is important in mathematics?  Strategies that get us to do better on the test, or strategies that help us slow down and think more?  Is math about memorizing or figuring things out?  Is math about removing the context to mathematize a situation, or about using the context to make sense of things?

Sure I want my students to do well on any assessment they are given, but I want them to do well every day!  Quick fixes and silver bullets often don’t help our students in the long run though!!!


So I leave you with a few things to reflect on:

  • What are some of the quick fixes you’ve heard about?  Did you try any of them?  Did they work?
  • Is there a strategy that you see working for all of your students?  Was it actually helpful for everyone, or just some?  Do you expect everyone to use this strategy?
  • Have you been asked or told to use specific strategies?  Do you see it being successful for everyone?  Do you have the autonomy to choose here based on the students you have in front of you?  Do your students have any autonomy over the strategies they use?
  • When looking at your student work, are you determining next steps for your students, or for yourself?

It is far easier to determine what your students can and can’t do well, than it is to figure out what to do next.  While we absolutely need to help our students notice the things they could do to improve, we also need to do the hard work of reflecting on our own practices.  Our beliefs about what is important and how we learn mathematics have a direct effect on how our students will do in our classrooms!

Lessons learned from 3 Mistakes

I make mistakes all the time.  We all do.  As you’ve probably heard, mistakes are an important part of our learning.  However, I don’t think it is always as easy to learn from our mistakes as any catchy statement might make it sound.  So, I thought I’d share three quick stories and then reflect on them.


Story 1

The other day I was leading a string mini-lesson (similar to a number talk) and was modelling on a number line what the students were saying.  The string involved the question 78-29 and I took answers from a few students.

Student 1 explained that they started at 78, went back 30, then forward 1.  However, I modelled it on the numberline incorrectly.  I took what they mentioned, started at 78 and landed at 38 and 39 instead of 48 and 49.


Story 2

A few weeks ago I was co-planning / co-teaching with two teachers starting a new Contexts for Learning unit (Cathy Fosnot’s The T-Shirt Factory).  In our planning we looked at her landscapes of learning, the progression of lessons, did some of the questions together, anticipated student responses, and started gathering needed materials.  I joined the classrooms on day 4 of the unit and came in to hear some issues they had with some group members not doing much of the work.  When asked, they had told me they split the class into groups of 4 students because the resource told them to.  They put each group in charge of figuring out 1 size of T-shirts (as is the context in the unit).  They placed struggling students together to figure out the easier sizes, and “stronger” students together to work on the larger numbers.

After the first few lessons both teachers noted that none of the groups had all students working.  I suggested that it would be far easier for us to work with pairs instead of groups of 4.  I co-taught the same lesson with each teacher that day having students in pairs, but something was still not right.  We read through the unit again and realized that we had grouped students incorrectly and had been assigning problems incorrectly!  We should have placed 4 random students together and given EACH student their own size T-Shirt.  That way each student could work as part of the group, been given their own problem, and we could assign numbers that were appropriate for each student.

We ended up having to redo the previous 2 lessons over again so each student would have proper groups and their own responsibilities!


Story 3

In my third year of teaching I went to a workshop where a teacher shared their practices with the group.  The workshop was about having students create their own math dictionaries where definitions (with examples and pictures) were kept, and lessons and worked examples from each school day would be stored.  I quickly started working on making this a reality.  Over the summer I created the template for these books, I wrote out the notes I wanted every student to copy for each day of the year leaving blanks where their samples would go, and put together a package for each student that would be continually added upon throughout the year.

I was excited because these resources would house all of the definitions, all of the examples, all of the thinking from the entire year, and they would be able to use this resource for studying, for homework, and keep it for future years as a reference!

June quickly came and every student had their very own personal math dictionaries we created from countless hours of work.  Each was neat and complete, full of all of our thinking from the year!  On the last day of school, I watched as students packed up their belongings (forgotten sweaters, old shoes, pencil cases…), handed in their textbooks, and threw out their their unwanted worksheets and duo-tangs.  However, I wasn’t counting on just how happy many of my students were to throw out their math dictionaries.  I tried to pull out of the garbage and recycling bins each book to save anyway.


Some thoughts about the kinds of mistakes we make

We tend to make mistakes at different levels.  Story 1 showed one of the many simple mistakes I make.  They are quick little things that are often due to carelessness, forgetting things, accidents…  When we make mistakes like this, what matters is how we show our students how WE handle mistakes.  If we make a calculation error, do we get embarrassed?  Do we pretend it didn’t happen?  Do we turn it into a teachable moment?  This really matters!  If we show our students that we aren’t comfortable with us making mistakes, what unintended messages will this send?

In my second story I shared a mistake I made by not reading carefully and not being fully prepared for a lesson.  Mistakes that cause us to redo something because we noticed that something isn’t working are great mistakes to make.  When we realize that we have taken a wrong path, we come to see just how important the bigger things are.  We learn to be more intentional because of our mistakes!  In this situation, we realized that we could differentiate the learning in the classroom without ability grouping, and that if we have larger groups (4 students) we still need to give everyone something that they are responsible for.

The third story is a story I don’t share much.  Probably because I had spent an entire year getting my students to memorize, follow procedures, copy out worked examples…  My students eagerly throwing out their books was great feedback for me (I didn’t think it was great at the time though).  Over time, I came to see that I was teaching my students instrumentally!  Not only did my mistake help me to notice just how unhappy my students were in my math class, it helped me realize why my students had done so poorly on their provincial testing (they were the lowest results I’ve ever had).  I was also able to reflect on what it means to learn mathematics and what it means to be engaged in thinking mathematically.  Recognizing we are taking a path that isn’t beneficial for students requires us to see other approaches, understand the research, experience learning mathematics ourselves in ways that help us understand the concepts deeper.  Mistakes like this are not only difficult to recognize, they are difficult to change.  It has been a long road for me to continue to develop my own math knowledge for teaching, but I know that it started when I started realizing math is more than memorizing, more than rules and procedures, more than a collection of unrelated topics.


Some things to reflect on:

  • When you make little errors in front of your students, how do you react?  What do your students think about making mistakes in front of others?  Is there a relationship?
  • Can you think of a time you made a mistake with a lesson, or in teaching a unit?  These types of mistakes are easy to learn from, but we need to take advantage of these opportunities.  How have you learned from your mistakes?
  • Recognizing and reflecting on our practices that reflect our beliefs is probably the most difficult for us to do.  So, how can we find out which teaching practices are helping our students learn?  How can we find opportunities for our students to give US feedback?  What experiences help us reflect on our own beliefs about what is important for our students to do?  What experiences help us reflect on our own beliefs about how students learn mathematics?
  • How do the beliefs I have shared here in this post relate to yours?

 

As always, I encourage you to continue the conversation here or on Twitter (@MarkChubb3)

…a child first has to learn the foundational skills of math, like______?

I’ve been spending a lot of time lately observing students who struggle with mathematics, talking with teachers about their students who struggle, and thinking about how to help.  There are several students in my schools who experience difficulties beyond what we might typically do to help.  And part of my role is trying to think about how to help these students.  It seems that at the heart of this problem, we need to figure out where our students are in their understanding, then think about the experiences they need next.

However, first of all I want to point out just how difficult it is for us to even know where to begin!  If I give a fractions quiz, I might assume that my students’ issues are with fractions and reteach fractions… If I give them a timed multiplication test, I might assume that their issues are with their recall of facts and think continued practice is what they need… If I give them word problems to solve, I might assume that their issue is with reading the problem, or translating a skill, or with the skill itself…….  Whatever assessment I give, if I’m looking for gaps, I’ll find them!

So where do we start?  What are the foundations on which the concepts and skills you are doing in your class rest on?  This is an honest question I have.

Take a look at the following quote.  How would you fill in the blank here?

 

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Really, take a minute to think about this.  Write down your thoughts.  In your opinion, what are the foundational skills of math?  Why do you believe this?  This is something I’ve really been reflecting on and need to continue doing so.


I’ve asked a few groups of teachers to fill in the blank here in an effort to help us consider our own beliefs about what is important.  To be honest, many of the responses have been very thoughtful and showcased many of the important concepts and skills we learn in mathematics.  However, most were surprised to read the full paragraph (taken from an explanation of dyscalculia).


Here is the complete quote:

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Taken from Dyscalculia Headlines

 

Is this what you would have thought?  For many of us, probably not. In fact, as I read this, I was very curious what they meant by visual perception and visual memory.  What does this look like?  When does this begin?  How do we help if these are missing pieces later?

Think about it for a minute.  How might you see these as the building blocks for later math learning?  What specifically do these look like?  Here are two excerpts from Taking Shape that might help:

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Excerpt from Taking Shape
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Excerpt from Taking Shape

Visual perception and visual memory are used when we are:

  • Perceptual Subitizing (seeing a small amount of objects and knowing how many without need of counting)
  • Conceptual subitizing (the ability to organize or reorder objects in your mind. To take perceptual subitizing to make sense of visuals we don’t know)
  • Comparing objects’ sizes, distances, quantities…
  • Composing & decomposing shape (both 2D or 3D)
  • Recognizing, building, copying symmetry designs (line or rotational)
  • Recognizing & performing rotations & reflections.
  • Constructing & recognizing objects from different perspectives
  • Orienting ourselves, giving & following directions from various perspectives.
  • Visualizing 3D figures given 2D nets

While much of this can be quite complicated (if you want, you can take a quick test here), for some of our students the visual perception and visual memory is not yet developed and need time and experiences to help them mature. Hopefully we can see how important these as for future learning.  If we want our students to be able to compose and decompose numbers effectively, we need lots of opportunities to compose and decompose shapes first!  If we want our students to understand quantity, we need to make sure those quantities make sense visually and conceptually. If we want students to be flexible thinkers, we need to start with spatial topics that allow for flexibility to be experienced visually.  If we want students to be successful we can’t ignore just how important developing their spatial reasoning is!


In our schools we have been taking some of the research on Spatial Reasoning and specifically from the wonderful resource Taking Shape and putting it into action.  I will be happy to share our findings and action research soon.  For now, take a look at some of the work we have done to help our students develop their visual perception and visual memory:

Symmetry games:


Composing and decomposing shapes:

Relating nets to 3D figures:

Constructing unique pentominoes:


And the work in various grades continues to help support all of our students!


So I leave you with a few questions:

  • What do you do with students who are really struggling with their mathematics?  Have you considered dyscalculia and the research behind it?
  • How might you incorporate spatial reasoning tasks / problems for all students more regularly?
  • Where in your curriculum / standards are students expected to be able to make sense of things visually?  (There might be much more here than we see at first glance)
  • How does this work relate to our use of manipulatives, visual models and other representations?
  • What do we do when we notice students who have visual perception and visual memory issues beyond what is typical?
  • How can Doug Clements’ trajectories help us here?
  • If we spend more time early with students developing their visual memory and visual perception, will fewer students struggle later???

I’d love to hear your thoughts.  Leave a comment here or on Twitter (@MarkChubb3).

 

The smallest decisions have the biggest impact!

In my role, I have the advantage of seeing many great teachers honing and refining their craft, all to provide the best possible experiences for their students. The dedication and professionalism that the teachers I work with continue to demonstrate is what keeps me going in my role!

One particularly interesting benefit I have is when I can be part of the same lesson multiple times with different teachers.  When I am part of the same lesson several times I have come to notice the differences in the small decisions we make.  It is here in these small decisions that have the biggest impact on the learning in our classrooms. For instance, in any given lesson:

There are so many little decisions we make (linked above are posts discussing several of the decisions).  However, I want to discuss a topic today that isn’t often thought about: Scaffolding.


For the past few months, the teachers / instructional coaches taking my Primary/Junior Mathematics additional qualifications course have been leading lessons. Each of the lessons follow the 3-part lesson format, are designed to help us “spatialize” the curriculum (allow all of us to experience the content in our curriculum via visuals / representations / manipulatives), and have a specific focus on the consolidation phase of the lesson (closing). After each lesson is completed I often lead the group in a discussion either about the content that we experienced together, or the decisions that the leader choose. Below is a brief description of the discussion we had after one particular lesson.


First of all, however, let me share with you a brief overview of how the lesson progressed:

  1. As a warm up we were asked to figure out how many unique ways you can arrange 4 cubes. 
  2. We did a quick gallery walk around the room to see others’ constructed figures.
  3. We shared and discussed the possible unique ways and debated objects that might be rotations of other figures, and those that are reflections (take a look at the 8 figures below).
  4. The 3 pages of problems were given to all (see below).  Everyone had time to work independently, but sharing happened naturally at our tables.
  5. The lesson close included discussions about how we tackled the problems.  Strategies, frustrations, what we noticed about the images… were shared.

Here are the worksheets we were using so you can follow along with the learning (also available online Guide to Effective Instruction: Geometry 4-6, pages 191-212):

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 While the teacher leader made the decision to hand out all 9 problems (3 per sheet) at the same time, I think some teachers might make a different decision. Some might decide to take a more scaffolded approach. Think about it, which would you likely do:

  1. Hand out all 9 problems, move around the room and observe, offer focusing questions as needed, end in a lesson close; or
  2. Ask students to do problem 1, help those that need it, take up problem 1, ask students to do problem 2, help those that need it, take up problem 2…

This decision, while seemingly simple, tells our students a lot about your beliefs about how learning happens, and what you value. 

So as a group of teachers we discussed the benefits and drawbacks of both approaches. Here are our thoughts:

The more scaffolded approach (option 2) is likely easier for us. We can control the class easier and make sure that all students are following along. Some felt like it might be easier for us to make sure that we didn’t miss any of our struggling students. However, many worried that this approach might inhibit those ready to move on, and frustrate those that can’t solve it quickly. Some felt like having everyone work at the same pace wasn’t respectful of the differences we have in our rooms. 

On the other hand, some felt that handing out all 9 puzzles might be intimidating for a few students at first. However, others believed that observing and questioning students might be easier because there would be no time pressure. They felt like we could spend more time with students watching how they tackle the problems. 

Personally, I think our discussion deals with some key pieces of our beliefs:

  • Do we value struggle?  Are we comfortable letting students productively keep trying?
  • Are we considering what is best for us to manage things, or best for our students to learn (teacher-centered vs student-centered)?
  • What is most helpful for those that struggle with a task?  Lots of scaffolding, telling and showing?  Or lots of time to think, then offer assistance if needed?

In reality, neither of these ways will likely actually happen though. Those who start off doing one problem at a time, will likely see disengagement and more behaviour problems because so many will be waiting. When this happens, the teacher will likely let everyone go at their own pace anyway. 

Similarly, if the teacher starts off letting everyone go ahead at their own pace, they might come across several of the same issues and feel like they need to stop the class to discuss something. 

While both groups will likely converge, the initial decision still matters a lot.  Assuming the amount and types of scaffolding seems like the wrong move because there is no way to know how much scaffolding might be needed. So many teachers default by making sure they provide as much scaffolding as possible  however, when we over-scaffold, we purposely attempt to remove any sense of struggle from our students, and when we do this, we remove our students’ need to think!  When we start by allowing our students to think and explore, we are telling our students that their thoughts matter, that we believe they can think, that mathematics is about making sense of things, not following along!

So I leave you with a few thoughts:

  • Do your students expect you to scaffold everything?  Do they give up easily?  How can we change this?
  • When given an assignment do you quickly see a number of hands raise looking for help?  Why is this?  How can we change this?
  • At what point do you offer any help?  What does this “help” look like?  Does it still allow your students opportunities to think and make sense of things?

When we scaffold everything, we might be helping them with today’s work, but we are robbing them of the opportunity of thinking. When we do this, we rob them of the enjoyment and beauty of mathematics itself!

Reflecting on 2016

Last June (2016) I started writing this blog.  I’m not exactly sure what got me started to be honest, probably because I have been inspired by so many others’ blogs, possibly  thanks to @MaryBourassa’s encouragement!  Whatever helped me get started, I am still not sure WHY I am blogging.

Some things I DO know:

  • I started writing last June
  • I wrote 32 blog posts last year
  • People from 123 countries have been reading
  • I try to include pedagogical decisions and mathematical content in every post
  • I tend to write more when I should be working on other things:)

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My 10 most popular posts were:

  1. So you want your students to have a Growth Mindset?
  2. Concept vs Procedure: An anecdote about what it means to be good at math
  3. Questioning the pattern of our questions
  4. Focus on Relational Understanding
  5. Never Skip the Closing of the Lesson
  6. What Does Day 1 Look Like?
  7. Exit Cards – What do your’s look like?
  8. Is This “Real World”?
  9. How do you give feedback?
  10. How do we meet the needs of so many unique students in a mixed-ability classroom?

My least popular posts were:

  1. Purposeful Practice: Happy Numbers
  2. Aiming for Mastery?
  3. “I like math because it’s objective…”
  4. How to change everything and nothing at the same time!
  5. Is That Even A Problem???

So, I’m left wondering, why are some posts more popular and others less so?  Are my least popular posts less read because they are more confrontational?  Do they offer less for others to relate to?

And why do some posts get retweeted or commented on more?  Is it because they offer more chance for reflection, or is it the topic…?

And more importantly, is this what I’m aiming for?  Is the purpose of this blog to share with others and hope it will be read, or is it for me to continue to write so I can reflect on my own thinking/decisions???

Is it about making connections with others?  Or about my own learning?  Or about helping others reflect???


I am left wondering what about blogging is different than reading others’ blogs?  How is this helpful and to whom?

While I don’t think I have the answers to my questions, I do know that I am continuing to learn and that my thoughts are getting others to consider their own teaching.  Hopefully as I continue writing, I will start to find the answers to WHY I do what I do.

Hopefully this blog will continue to be an important aspect to my work in 2017 as well!  Thanks for reading.


I’d love to know why you read math blogs.  Or what it would take to get you started writing your own!

Leave a comment here, or on Twitter (@MarkChubb3)

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
Enter a caption
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!