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?

 

 

 

Seeking Challenges in Math

I was working with a grade 7 teacher and his students a while back.  The teacher came to me with an interesting problem, his students were doing quite well in math (in general) but only wanted to do work out of textbooks, only wanted to work independently, and were very mark-driven. The teacher wanted his students to start being able to solve non-routine problems, not just be able to follow the directions from the textbook, and he wanted his students to see the value in working collaboratively and to listen to each other’s thoughts.

Our conversations quickly moved to the topic of mindsets. It sounded like many of his students had fixed mindsets, and didn’t want to take any risks.


For those of you who are not familiar with growth and fixed mindsets, students with fixed mindsets believe that their ability (in math for example) is an inborn trait.  They believe how smart they are in math is either a gift or a curse they are born with.  Those with growth mindsets, however, believe that their ability improves over time with the right experiences, attitude and effort.

When confronted with challenges, those with growth mindsets are willing to struggle, willing to make mistakes, knowing that they will continue to learn and grow throughout the learning process.  On the other hand, those that have fixed mindsets tend to avoid challenges.  They believe that struggle, making mistakes, and being challenged are signs of weakness.  Psychologically, they will avoid the feeling of discomfort in not knowing, as this threatens their belief about how smart they are.


Knowing this, we devised a plan to see whether or not his students were able to take on challenges.  We started the class by giving each student their own unique 24 card (see below).

24 card.jpeg

We explained that each card had 4 numbers that could be manipulated to equal 24.  For instance, the card above could be solved by doing 5 x 4 x 1 + 4 = 24.


We then explained that we would give them time to solve their own card (which had a front and a back), and that we would give them additional cards if they completed both problems.  We also explained the little white dots in the center of the card, 1 dot being an easy card, 2 dots being more complicated, and 3 dots being the most difficult.

As students continued to work, we noticed some students eagerly trying to solve the cards, and others starting to become frustrated by others’ successes.  After a few minutes, the first few students had completed both problems and asked for their next card.  We asked, “Would you like another easy card, or would you like to challenge yourself?” to which the vast majority asked for another easy card.  In fact, some students completed many cards, front and back, all at the easy level, never accepting a more challenging card (even bragging to others about how many they had completed).  Others, after giving up pretty quickly, asked if they could work with a classmate to make a pair.  While we were happy at first with this, none of the pairs had students working cooperatively together for most of the time.

Take a look at some of the challenging cards.  What do you do when confronted with something challenging?  Do you skip it and move on, or do you keep trying?

 


As soon as we were finished, we showed the class this video:

Watch the 3 minute video above as it ties in perfectly with the 24 problem from above.  We had a quick discussion about the video and why some of the students wanted to choose the easier puzzles.  The class quickly saw the parallels between the problems we had just done and the video.

While we had a great discussion about fixed and growth mindsets, it took most of the year to be able to get this group to see the value in collaboration, to focus on their learning instead of their marks, to be able to take on challenges and not get frustrated when they didn’t have immediate success.

Changing our mindset takes time and the right experiences!


I am really interested in why students who believe themselves to be “smart” at math would opt out of challenging themsleves.

Do any of your students exhibit any of the same signs as these students:

  • Not comfortable with tasks that require thinking
  • Eager for formulas and procedures
  • Competitive with others to show they are “smart”
  • Preference to work alone
  • Preference to work out of textbooks/ worksheets instead of on rich problems/tasks
  • See math as about being fast / right, not about thinking / creativity
  • Eager to do easy work that is repetative

 

So I leave you with some reflective questions:

What previous experiences must these students have had to create such fixed mindsets?

What would you do if your students avoided challenges?

What would you do if your students groaned each time you asked them to work with a partner?

How are you helping your students gain a growth mindset in math?

Can you recognize those in your class that have fixed mindsets?  Are you noticing those from different achievement levels, or just those who are struggling?

If our students find everything we do “easy” what will happen to them when they get to a math course that actually does offer them some challenge???


 

 

P.S.  Did you solve any of the 24 cards above?  Did you skip over them?  What do you typically do when confronted with challenges?

An Unsolved Problem your Students Should Attempt

There are several great unsolved math problems that are perfect for elementary students to explore.  One of my favourites is the palindrome sums problem.
In case you aren’t familiar, a palindrome is a word, phrase, sentence or number that reads the same forward and backward.


The problem itself comes out of this conjecture:

If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome.

Let’s take a look at the numbers 12, 46 and 95:

Palindrome adding1


Palindrome adding2


Palindrome adding3


As you can see with the above examples, some numbers can become palindromes with 1 simple addition, like the number 12.  12 + 21 = 33.  Can you think of others that should take 1 step?  How did you know?

Other numbers when added to their inverse will not immediately become a palindrome, but by continuing the process, will eventually, like the numbers 46 and 95.  Which numbers do you think will take more than 1 step?


After going through a few examples with students about the process of creating palindromes, ask your students to attempt to see if the conjecture is true (If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome).  Have them find out if each number from 0-99 will eventually become a palindrome. Ask students if they need to find the answer for each number?  Encourage them to make their own conjectures so they don’t need to do all of the calculations for each number.


Mathematically proficient students look closely to discern a pattern or structure. They notice if calculations are repeated, and look both for general methods and for shortcuts. (SMP 7)

Some students will notice:

  • Some numbers will already be a palindrome
  • If they have figured out 12 + 21… they will know 21 + 12.  So they don’t have to do all of the calculations.  Nearly half of the work will have already been completed.
  • A pattern emerging in their answers … 44, 55, 66, 77, 88, 99, 121… and see this pattern regularly (well, almost)
  • A pattern about when numbers will have 99 as a palindrome, or 121… 18+81 = 27+72 = 36+63…

Thinking about our decisions:
  • What is the goal of this lesson?  (Practice with addition?  Looking for patterns?  Perseverance?  Making conjectures? …)
  • Do we share some of the conjectures students are making during the time when students are working, or later in the closing of the lesson?  
  • How will my students record their work?  Keep track of their answers?  Do I provide a 0-99 chart or ask them to keep track somehow?
  • Will students work independently / in pairs / in small groups?  Why?
  • Do I allow calculators?  Why or why not?  (think back to your goal)
  • How will I share the conjectures or patterns noticed with the class?
  • Are my students gaining practice DOING (calculations) or THINKING (noticing patterns and making conjectures)?  Which do you value?

The smallest of decisions can make the biggest of impacts for our students!  


So, at the beginning of this post I shared with you that this problem is currently unsolved.  While it is true that the vast majority of numbers have been proven to easily become palindromes, there are some numbers that require many steps (89 and 98 require 24 steps), and others that have never been proven to either work or not work (198 is the smallest number never proven either way).


Some final thoughts about this problem…

After using this problem with many different students I have noticed that many start to see that mathematics can be a much more intellectually interesting subject than they had previously experienced.  This problem asks students to notice things, make conjectures, try to prove their conjectures and be able to communicate their conjectures with others…  The problem provides students with the opportunity to both think and do.  It offers students from various ability levels access to the problem (low floor), and many different avenues to challenge those ready for it (high ceiling).  It tells students that math is still a living, growing subject… that all of the problems have not yet been figured out!  And probably the most important for me, it sends students messages about what it means to really do mathematics!

 

palindrome2
Taken from Marilyn Burns’ 50 Problem Solving Lessons resource

 

How to Set Up Positive Norms…

How will your students view mathematics this year?  What norms will you set up to help your students share your views of mathematics?

If you haven’t already seen YouCubed list of norms, you will probably want to take a close look:

Norms1Norms2

Take a look at the following resource about how to set up positive norms in your class…Setting up Positive Norms in Math Class (linked is an expansion of each norm).  Which one(s) of these is really important to you?  Which one(s) challenges your thinking?

  1. Everyone Can Learn Math to the Highest Levels. Encourage students to believe in themselves. There is no such thing as a “math” person. Everyone can reach the highest levels they want to, with hard work.
  2. Mistakes are valuable. Mistakes grow your brain! It is good to struggle and make mistakes.
  3. Questions are Really Important.  Always ask questions, always answer questions. Ask yourself: why does that make sense?
  4. Math is about Creativity and Making Sense. Math is a very creative subject that is, at its core, about visualizing patterns and creating solution paths that others can see, discuss and critique.
  5. Math is about Connections and Communicating.  Math is a connected subject, and a form of communication. Represent math in different forms eg words, a picture, a graph, an equation, and link them. Color code!
  6. Depth is much more important than speed. Top mathematicians, such as Laurent Schwartz, think slowly and deeply.
  7. Math Class is about Learning not Performing.  Math is a growth subject, it takes time to learn and it is all about effort.

Thinking about what contradicts our current thinking is where learning can really occur!


While setting up classroom norms is a great thing to do, I think it is far more important to start the year off by enacting these norms rather than just discussing them or posting them.

The first few things you do in your year sets the tone for what you believe is important in mathematics!

My suggestion:

  • Start with something that allows your students to see their strengths
  • Start with something that is open enough to allow for differences in strategies and/or their answers
  • Start with something that focuses on the SMPs
  • Start with something that helps your students look for patterns, notice things, explore…
  • Start with something that allows your students to be creative
  • Start with something where your students can see the beauty in mathematics

If you want to share the norms listed at the beginning of the article, maybe experiencing a rich task/problem first might help students see what you mean and start to get excited about learning this year!


 

In the next few days I’ll share a possible example of what this could look like.

 

Never Skip the Closing of the Lesson

Once again, Tracy Zager has pushed us to think about our teaching.  In her recent talk at #TMC16 Tracy asked us to consider what it means to “close the lesson”.  Here is an example of a problem and a potential close, followed by some of my thoughts about how we should close any lesson.


First of all, give a problem that will help you achieve a specific goal.

Take this problem published in Marilyn Burn’s 50 Problem Solving Lessons resource:

If rounds 1 & 2 of a tug-of-war contest are a draw, who will win the final round?


Here is the full problem. Please take a minute to read through the problem and try to solve it for yourself. Which side will win round 3?  How do you know?  Are you sure?


Once students understand the problem and are given time to write their solution (individually or in pairs) the learning isn’t over. In fact, while answering the problem might require thinking and writing an answer requires decisions about representations, much of the learning hasn’t happened yet!  Really, students have just shown what they already understand…new learning happens in the close!

Closing the lesson:

Step 1 – Sharing Different Solutions

Since this problem is open, allowing for different strategies, there is a huge potential for interesting discussions.  For example, some students will create an answer similar to the one below.  Using this sample there are several things that we can bring up in conversation with the class.  Notice the student(s) created equations using symbols and equal signs.  Also, they did something really interesting in round three (notice the brackets and the arrows).  Having a conversation about substituting would be really helpful for many students.  Many might not realize that this problem would be easier to solve if only acrobats and grandmas were considered.  Through substituting Ivan for 2 grandmas and 1 acrobat, the final round ends up with 5 grandmas and an acrobat against 4 acrobats.  Substitution requires us to really understand the equal sign and what balance means!

Acrobats6.jpg

Other students will create things more like the picture below.  They will use numbers to represent values for each of the figures.  Again, conversations with the class could be about how they chose the 1 = 1.25 at the top.  Or, how this information could be used later.  Again, the importance of balance and the equal sign could be brought up.

Others might do things like the sample below.  Here the students have decided not to draw a picture, but instead to represent the players with letters.  For many students, their first introduction with algebra is where letters are all of a sudden thrown into some equations.  However, these students have realized that it is far easier to represent the items with letters than draw the pictures.  The conversation here could be quite useful in bringing about the need for letters!  Also, noticing that their strategy is similar to the 2nd sample above might be helpful since while the strategy is the same, the values are not.  How are the two student samples’ values different?  How are they the same?  Is one right or are they both right?

Acrobats1.jpg


Step 2 – Making connections / discussing big ideas

Once a few samples have been shared, we need to make sure our students are making connections between the samples.  The learning from the problem needs to made clear for all students.

We can do this by asking students to notice similarities and differences between samples, or by taking a few minutes writing down a few simple things we can take away from the problem.  Either way, our students need to be involved here and the generalities we can draw from the problem need to be clear.

For example, things that can be discussed here:

  • The equal sign represents balance
  • We can substitute things in equations we don’t know with things we do know
  • Symbols or letters can be used in equations to represent either a variable or a constant

Step 3 – individual practice

Now that our students have had time to grapple with this problem, and have discussed what we can learn from it together, we need to continue the learning.  Part of the close includes time for students to do something with their learning.  Here are a few possible ways we can include individual time to practice what has been learned.

  • Provide an Exit Card or journal prompt that asks your students to show what they have learned (linked are a variety of types of exit cards you can choose between)
  •  Have your students create their own problem using 3 rounds where the first 2 are a tie and the 3rd round needs to be figured out (possibly have students switch problems with a classmate and find the solution)
  • Relate the problem to some practice questions that will help continue your students’ thinking.  Practice with some of the early Solve Me Mobile puzzles might work nicely here.

Some General Advice about the Close

  • Predetermine which students you want to share.
  • Have a goal in mind for each person/group sharing.
  • Ask them specific questions, or ask questions of the rest of the class that helps you achieve your goal.
  • Avoid general questions like “tell us what you did” (we don’t want the sharing to become a show-and-tell, we want rich discussions about specific things from their work).
  • Use the 5 Practices for Orchestrating Productive Conversations (found here, here, and here) to make sure your close involves real discussion of the meaningful mathematics.
  • Practice is important, but students still need to think and make choices.
  • Closing the lesson is about bringing the learning together as a group, then individually.  The SHARED experiences here are where our students can learn WITH and FROM each other so they will be ready to work independently.
  • The majority of the learning takes place in the close!
  • Closing a lesson takes time, but skipping the close is the biggest waste of time!

Want more information about the close?  Take a look at this monograph: Communication in the Mathematics Classroom

So I leave you with some final reflective questions:

  • How often do you close the lesson?
  • What obstacles stand in your way to closing your lessons regularly?
  • Does your close look like a show-and-tell where students are not listening to those talking, or do you have rich discussions going on between many members of your classroom?  How can you help increase the level of discourse in your classroom?
  • How is this different than closing the lesson by taking up homework?  Does this allow more opportunities for those that might struggle, or who don’t identify with mathematics yet?
  • What goals do have this year with regards to closing your lessons?

Some say they can’t afford the time to close the lesson.  I say you can’t afford not to!

 

 

Focus on Relational Understanding

Forty years ago, Richard Skemp wrote one of the most important articles, in my opinion, about mathematics, and the teaching and learning of mathematics called Relational Understanding and Instrumental Understanding.  If you haven’t already read the article, I think you need to add this to your summer reading (It’s linked above).

Skemp quite nicely illustrates the fact that many of us have completely different, even contradictory definitions, of the term “understanding”.   Here are the 2 opposing definitions of the word “understanding”:

“Instrumental understanding” can be thought of as knowing the rules and procedures without understanding why those rules or procedures work. Students who have been taught instrumentally can perform calculations, apply procedures… but do not necessarily understand the mathematics behind the rules or procedures.

Instrumental2.png

“Relational understanding”, on the other hand, can be thought of as understanding how and why the rules and procedures work.  Students who are taught relationally are more likely to remember the procedures because they have truly understood why they work, they are more likely to retain their understanding longer, more likely to connect new learning with previous learning, and they are less likely to make careless mistakes.

Relational2.png

Think of the two types of understanding like this:

 

Shared by David Wees

Students who are taught instrumentally come to see mathematics as isolated pieces of knowledge. They are expected to remember procedures for each and every concept/skill.  Each new skill requires a new set of procedures.  However, those who are taught relationally make connections between and within concepts and skills.  Those with a relational understanding can learn new concepts easier, retain previous concepts, and are able to deviate from formulas/rules given different problems easier because of the connections they have made.

 

While it might seem obvious that relational understanding is best, it requires us to understand the mathematics in ways that we were never taught in order for us to provide the best experiences for our students. It also means that we need to start with our students’ current understandings instead of starting with the rules and procedures.

Skemp articulates how much of an issue this really is in our educational system when he explains the different types of mismatches that can occur between how students are taught, and how students learn.  Take a look:

Instrumental vs Relational

Notice the top right quadrant for a second.  If a child wants to learn instrumentally (they only want to know the steps/rules to solve today’s problem) and the teacher instead offers tasks/problems that asks the child to think or reason mathematically, the student will likely be frustrated for the short term.  You might see students that lack perseverance, or are eager for assistance because they are not used to thinking for themselves.  However, as their learning progresses, they will come to make sense of their mathematics and their initial frustration will fade.

On the other hand, if a teacher teaches instrumentally but a child wants to learn relationally (they want/need to understand why procedures work) a more serious mismatch will exist.  Students who want to make sense of the concepts they are learning, but are not given the time and conditions to experience mathematics in this way will come to believe that they are not good at mathematics.  These students soon disassociate with mathematics and will stop taking math classes as soon as they can.  These students view themselves as “not a math person” because their experiences have not helped them make sense of the mathematics they were learning.

While the first mismatch might seem frustrating for us as teachers, the frustration is short lived. On the other hand, the second mismatch has life-long consequences!


I’ve been thinking about the various initiatives/ professional development opportunities… that I have been part of, or have been available online or through print that might help us think about how to move from an instrumental understanding to a relational understanding of mathematics.  Here are a few I want to share with you:

Phil Daro’s Against Answer Getting video highlights a few instrumental practices that might be common in some schools.  Below is the “Butterfly” method of adding fractions he shares as an “answer getting” strategy.  While following these simple steps might help our students get the answer to this question, Phil points out that these students will be unable to solve an addition problem with 3 fractions.  These students “understand” how to get the answer, but in no way understand how the answer relates to addends.

Daro - Butterfly.jpg

On the other hand, teachers who teach relationally provide their students with contexts, models (i.e, number lines, arrays…), manipulatives (i.e., cuisenaire rods, pattern blocks…) and visuals to help their students develop a relational understanding.  If you are interested in learning more about helping your students develop a relational understanding of fractions, take a look at a few resources that will help:


Tina Cardone and a group of math teachers across Twitter (part of the #MTBoS) created a document called Nix The Tricks that points out several instrumental “tricks” that do not lead to relational understanding.  For example, “turtle multiplication” is an instrumental strategy that will not help our students understand the mathematics that is happening.  Students can draw a collar and place an egg below, but in no way will this help with future concepts!Turtle mult.jpg

Teachers focused on relational understanding again use representations that allow their students to visualize what is happening.  Connections between representations, strategies and the big ideas behind multiplication are developed over time.

Take a look at some wonderful resources that promote a relational understanding of multiplication:

Each of the above are developmental in nature, they focus on representations and connections.


So how do we make these shifts?  Here are a few of my thoughts:

  1. Notice instrumental teaching practices.
  2. Learn more about how to move from instrumental to relational teaching.
  3. Align assessment practices to expect relational understanding.

Goal 1 – Notice instrumental teaching practices.

Many of these are easy to spot.  Here is a small sample from Pinterest:

 

The rules/procedures shared here ask students to DO without understanding.  The issue is that there are actually countless instrumental practices out there, so my goal is actually much harder than it seems.  Think about something you teach that involves rules or procedures.  How can you help your students develop a relational understanding of this concept?


Goal 2 – Learn more about how to move from instrumental to relational teaching.

I don’t think this is something we can do on our own.  We need the help of professional resources (Marian Small, Van de Walle, Fosnot, and countless others have helped produce resources that are classroom ready, yet help us to see mathematics in ways that we probably didn’t experience as students), mathematics coaches, and the insights of teachers across the world (there is a wonderful community on Twitter waiting to share and learn with you).

I strongly encourage you to look at chapter 1 of Van de Walle’s Teaching Student Centered Mathematics where it will give a clearer view of relational understanding and how to teach so our students can learn relationally.

However we are learning, we need to be able to make new connections, see the concepts in different ways in order for us to know how to provide relational teaching for our students.


Goal 3 – Align assessment practices to expect relational understanding.

This is something I hope to continue to blog about.  If we want our students to have a relational understanding, we need to be clear about what we expect our students to be able to do and understand.  Looking at developmental landscapes, continuums and trajectories will help here.  Below is Cathy Fosnot’s landscape of learning for multiplication and division.  While this might look complicated, there are many different representations, strategies and big ideas that our students need to experience to gain a relational understanding.

Fosnot landscape2
Investigating Multiplication and Division Grades 3-5

Asking questions or problems that expect relational understanding is key as well.  Take a look at one of Marian Small’s slideshows below.  Toward the end of each she shares the difference between questions that focus on knowledge and questions that focus on understanding.


I hope whatever your professional learning looks like this year (at school, on Twitter, professional reading…) there is a focus on helping build your relational understanding of the concepts you teach, and a better understanding of how to build a relational understanding for your students.  This will continue to be my priority this year!