Zukei Puzzles

A little more than a year ago now, Sarah Carter shared a set of Japanese puzzles called Zukei Puzzles (see her original post here or access her puzzles here).  After having students try out the original package of 42 puzzles, and being really engaged in conversations about terms, definitions and properties of each of these shapes, I wanted to try to find more.  Having students ask, “what’s a trapezoid again?” (moving beyond the understanding of the traditional red pattern block to a more robust understanding of a trapezoid) or debate about whether a rectangle is a parallelogram and whether a parallelogram is a rectangle is a great way to experience Geometry.  However, after an exhaustive search on the internet resulting in no new puzzles, I decided to create my own samples.

Take a look at the following 3 links for your own copies of Zukie puzzles:

Copy of Sarah’s puzzles

Extension puzzles #1

Extension puzzles #2

Advanced Zukei Puzzles #3

I’d be happy to create more of these, but first I’d like to know what definitions might need more exploring with your students.  Any ideas would be greatly appreciated!

How to complete a Zukei puzzle:

Each puzzle is made up of several dots.  Some of these dots will be used as verticies of the shape named above the puzzle.  For example, the image below shows a trapezoid made of 4 of the dots.  The remaining dots are inconsequential to the puzzle, essentially they are used as distractors.

If you enjoyed these puzzles, I recommend taking a look at Skyscraper puzzles for you to try as well.

An Example of “Doing Mathematics”: Creating Voronoi

I’ve been thinking a lot about how we look at academic standards (what we call curriculum expectations here in Ontario) lately.  Each person who reads a standard seems to read it through their own lens.  That is, as we read a standard, we attach what we believe is important to that standard based upon our prior experiences.  With this in mind, it might be worth looking at a few important parts of what makes up a standard (expectation) in Ontario.  Each of our standards have some/all of the following pieces:

• Content students should be learning
• Verbs clearly indicating the actions our students should be doing to learn the content and demonstrate understanding of the content
• A list of tools and/or strategies students should be using

Each of these three pieces help us know both what constitutes understanding, and potentially, how we can get there. However, while our standards here in Ontario have been written to help us understand these pieces, many of our students experience them in a very disconnected way.  For example, if we see each expectation as an isolated task to accomplish, our students come to see mathematics as a never-ending list of skills to master, not as a rich set of connections and relationships.  There are so many standards to “cover” that what ends up being missed for many students is the development of each standard. The focus of teaching mathematics ends up as the teaching of the standards instead of experiencing mathematics.  We give away the ending of the story before our students even know the characters or the plot. We share the punchline without ever setting up the joke.  We measure our students’ outcomes without considering the reasoning they walk away with… while many students might be able to demonstrate a skill after some practice, it’s quite possible they don’t know how it’s helpful, how it relates to other pieces of math, and because of this, many forget everything by the next time they use the concepts the next year.

To help us think deeper about what it means to experience mathematics from the students’ point of view, Dan Meyer has been discussing building the “intellectual need” with his whole “If Math Is The Aspirin, Then How Do You Create The Headache?”.  Basically, the idea here is that before we teach the content that might be in our standards, we need to consider WHY that content is important and how we can help our students construct a need for the skill.  He has helped us think about how our students could experience the long-cut before our students ever experience the short-cut.

Let’s take a specific example of a specific standard:

construct perpendicular bisectors, using a variety of tools (e.g.,
Mira, dynamic geometry software, compass) and strategies (e.g., paper folding)

If constructing perpendicular bisectors is the Aspirin, then how can we create the headache?  How can we create a situation where our students need to do lots of perpendicular bisectors?  Well, I wonder if creating voronoi could be a possible headache.  Take a look:

A Voronoi diagram is a partitioned plane where the area within each section includes all of the possible points closest to the original “seed” (the point within each section).  So, how might students create these?  If they already knew how to create perpendicular bisectors, they could simply start by placing seeds anywhere on their page, then create perpendicular bisectors between each set of points to find each partition.  However, Dan Meyer points out how important it really is to spend the time to really develop the skill starting from where our students currently are:

“In order for the CONSTRUCTION of the perpendicular bisector to feel like aspirin, I’d want students to feel the pain that comes from using intuition alone to construct the voronoi regions. This idea ties in other talks I’ve given about developing the question and creating full stack lessons. I’d want students to estimate the regions first.

Here is a dream I had awhile ago that I haven’t been able to build anywhere yet. Excited to maybe make it at Desmos some day.”

If you can, I’d recommend you take a look at Dan’s dream.  It really illustrates the idea of building his “full stack” lesson.  If we think back to the original standard again,

constructperpendicular bisectors, using a variety of tools (e.g.,
Mira, dynamic geometry software, compass) and strategies (e.g., paper folding)

it might be worth noting the specific pieces in orange.  I wonder, given a lesson like this, how much time would be spent allowing students opportunities to consider strategies that would make sense?  Or, how likely it would be that our students would be told which strategies/tools to use?

Below you can see the before and after images from a student’s work as they attempt to find perpendicular bisectors for each set of points.

Tasks like this do something else as well, they raise the level of cognitive demand.  Take a look at Stein et., al’s Mathematical Task Analysis Guide below:

While most students might experience a concept like this in a “procedures without connections” manner, allowing students to figure out how to create voronoi brings about the need to accurately find perpendicular bisectors, and consider how long each line would be in relation to all of the other perpendicular bisectors.  This is what Stein calls “Doing Mathematics”!  And hopefully, the students in our mathematics classes are actually “Doing Mathematics” regularly.

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

• How often are your students engaged in “Doing Mathematics” tasks?  Is this a focus for you and your students?
• If you were to ask your students to create voronoi, how much scaffolding would you offer?  If we provide too much scaffolding, would this task no longer be considered a “doing mathematics” task?  How would you introduce a task like this?
• Creating a perpendicular bisector is often seen as a quick simple skill that doesn’t connect much with other standards.  However, the task shared here asks students to make connections.  Can you think of standards like this one that might not connect to other standards nicely?  How can you build a need (or create the headache as Dan says) for that skill?
• Are you and your students “covering” standards, or are you constructing learning together?  What’s the difference here?

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

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How Many Do You See (Part 2 of 2)

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

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

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

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

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

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

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

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

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

What are we looking for?

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

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

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

Convergent vs Divergent Thinking

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

Teacher 1:

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

Teacher 2:

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

Teacher 3:

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

Teacher 4:

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

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

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

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

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

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

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

I want to leave you with a few reflective questions:

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

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

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

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

How many do you see?(Part 1)

A few days ago I had the opportunity to work with a grade 2 teacher as her class was learning about Geometry.  The students started the class with a rich activity comparing and sorting a variety of standard and non-standard shapes, followed by a great discussion about several properties they had noticed.

Shortly after, students started working on following the page as independent work. Take a look:

Take a minute to try to figure out what you think the answers might be.  Scroll up and pick one of the less obvious shapes and count how many you see.

This isn’t one of those Facebook “can you find all the hidden shapes” tasks, it’s meant to be a straightforward activity for grade 2 students. However, I’m not sure what the actual answers are here.  So, I need some help…  I’d love if you could:

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

I’m hoping in my next post that we can discuss more than just this worksheet and make some generalizations for any grade and any topic.

Targeted Instruction

Issues with Individualized / Targeted Instruction

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

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

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

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

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

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

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

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

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

I want to leave you with a few reflective questions:

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

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

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

Co-Teaching in Math Class

For the past few years I have had the privilege of being an instructional coach working with amazing teachers in amazing schools.  It is hard to explain just how much I’ve learned from all of the experiences I’ve had throughout this time.  The position, while still relatively new, has evolved quite a bit into what it is today, but one thing that has remained a focus is the importance of Co-Planning, Co-Teaching and Co-Debriefing.  This is because at the heart of coaching is the belief that teachers are the most important resource we have – far more important than programs or classroom materials – and that developing and empowering teachers is what is best for students.

While the roles of Co-Planning, Co-Teaching and Co-Debriefing are essential parts of coaching, I’m not sure that everyone would agree on what they actually look like in practice?

Take for example co-teaching, what does it mean to co-teach?  Melynee Naegele, Andrew Gael and Tina Cardone shared the following graphic at this year’s Twitter Math Camp to explain what co-teaching might look like:

Above you can see 6 different models described as Co-teaching.  While I completely understand that these 6 models might be common practices in schools when 2 teachers are in the same room, and while I am not speaking out against any of these models, I’m not sure I agree that all of these models are really co-teaching.  Think about it, which of these models would help teachers learn from and with each other?  Which of these promote students learning from 2 teachers who are working together?  Which of these models promotes teachers separating duties / responsibilities in a more isolated approach?

I will admit that after looking at the graphic (without being part of the learning from #TMC17) I was confused.  So, I went on Twitter to ask the experts (Melynee Naegele, Andrew Gael, Tina Cardone and others who were present at the sessions) to find out more about how co-teaching was viewed.  I was interested to find out from reading through their slideshows and from Mary Dooms that often, the “co-teacher” is a Special Education teacher and not an Instructional or Math Coach.

So, I thought it might be worth picking apart a few different roles to think more about what our practices look like in our schools.

Co-Teaching as a Special Education Teacher

Special Education teachers and Interventionists do really important work in our schools.  They have the potential to be a voice for those who are often not advocating for their own education and can offer many great strategies for both classroom teachers and students to help improve educational experiences.  When given the opportunity to co-teach with a classroom teacher though, I would be curious as to which models typically exist?

In my experience, the easiest to prescribe models would be model 3 or 4, parallel teaching / alternative teaching.  Working with a large class of mixed-ability students isn’t easy, so many classroom teachers are quite happy to hear that a special education teacher or interventionist is willing to take half or some of the students and do something different for them.  I wonder though, is this practice promoting exclusion, segregation, integration or inclusion?

While I understand that there are times when students might need to be brought together in a small group for specific help, I think we might be missing some really important learning opportunities.

At the heart of the problem is how difficult it is for classroom teachers to differentiate instruction in ways that allow our students to all be successful without sending fixed mindset messages via ability grouping.  Special Education teachers and interventionists have the ability, however, to have powerful conversations with classroom teachers to help create or modify lessons so they are more open and allow access for all of our students!  Co-teaching models 3 and 4 don’t allow us to have conversations that will help us learn better how to help those who are currently struggling with their mathematics.  Instead, those models ask for someone else to fix whatever problems might be existing.  The beliefs implied with these models are that the students need fixing, we don’t need to change!  Rushing for intervention doesn’t help us consider what ways we can support classroom teachers get better at educating those who have been marginalized.

The more time Special Education teachers and interventionists can spend in classrooms talking to classroom teachers, being part of the learning together and helping plan open tasks/problems that will support a wider group of students… the better the educational experiences will be for ALL of our students!  This raises the expectations of our students, while allowing US as teachers to co-learn together.  I think Special Education teachers and Interventionists need to spend more time doing models 1, 5 or 6, then, when appropriate, use other models on an as-needed basis.

Co-Teaching as a Coach

The role of instructional coaches or math coaches is quite different from that of a Special Education teacher or Interventionist though.  While Special Education teachers and Interventionists focus their thoughts on what is best for specific students who might be struggling in class, Coaches’ are concerned more with content, pedagogy, the beliefs we have about what is important, and the million decisions we make in-the-moment while teaching.  Coaching is a very personal role.  Together, a coach and a classroom teacher make their decision making explicit and together they learn and grow as professionals.  The role of coaches is to help the teachers you work with slow down their thinking processes… and this requires the ability to really listen (something I am continually trying to get better at).

Coaching involves a lot of time co-planning, co-teaching and co-debriefing.  However, in order for co-teaching to be effective, as much as possible, the coach and the classroom teacher need to be together!  Being present in the same place allows opportunities for both professionals to discuss important in-the-moment decisions and notice things the other might not have noticed.  It allows opportunities for reflection after a lesson because you have both experienced the same lesson.  Models 1, 5 and 6 seem to be the only models that would make sense for a coach.  Otherwise, how could a coach possibly coach?

If you haven’t seen how powerful it can be for teachers to learn together, I strongly suggest that you take a look at The Teaching Channel’s video showing Teacher Time Outs here.

To me, the more we as educators can talk about our decisions, the more we can learn together, the more we can try things out together……. the better we will get at our job!  We can’t do this (at least not well) if co-teaching happens in different places and/or with different students!

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

• How would you define co-teaching?  What characteristics do you think are needed in order to differentiate it from teaching?
• If you don’t have someone to co-teach with, how can you make it a priority?  How can your administrator help create conditions that will allow you to have the rich conversations needed for us to learn and grow?
• If you are a Special Education teacher or an interventionist, how receptive are classroom teachers to discuss the needs of those that are struggling with math?  Are conversations about what we need to do differently for a small group, or are conversations about what we can do better for all students?
• If you are a math coach or an instructional coach, what are the expectations from a classroom teacher for you?  How can you build a relationship where the two of you feel comfortable to learn and try things together?  What do conversations sound like after co-teaching?
• Are specific models of co-teaching being suggested to you by others?  By whom?  Do you have the opportunity to have a voice to try something you see as being valuable?
• School boards and districts often aim their sights at short-term goals like standardized testing so many programs are put into place to give specific students extra assistance.  But does your school have long-term goals too?  At the end of the year, has co-teaching helped the classroom teacher better understand how to meet the various needs of students in a mixed ability classroom?

For more on this topic I encourage you to read Unintended Messages  or How Our District Improved

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

A few of my favourite blog posts – to read… or inspire writing

I was asked the other day by another professional to share some blog posts that have inspired me.  She was curious about starting up her own blog and wanted to read through a few different writers’ works to get some inspiration.  To be honest, there are so many great educators posting wonderful blog posts that it is difficult to narrow it down.  Here is my attempt at creating a list of some of my favourite blog posts from the past few years.  Find one you haven’t read and take a look:

Daphne’s DREAM: Drop Everything and Math

http://tjzager.com/2017/08/18/daphnes-dream-drop-everything-and-math/

What We Presume

https://bstockus.wordpress.com/2017/08/21/what-we-presume/

Accessibility and Mathematics

https://andrewgael.com/2016/02/16/accessibility-mathematics/

My Criteria for Fact-Based Apps

https://tjzager.com/2016/01/05/my-criteria-for-fact-based-apps/

Don’t Put the Cart Before the Horse

http://marilynburnsmathblog.com/wordpress/word-problems-dont-put-the-cart-before-the-horse/

Pseudocontexts Kill

http://blog.mrmeyer.com/2010/pseudocontext-saturdays-introduction/

How Not to Start Math Class in the Fall

https://tjzager.com/2016/09/01/how-not-to-start-math-class-in-the-fall/

How to sabotage your classroom culture in 5 seconds

https://logsandreflections.wordpress.com/2016/09/11/how-to-sabotage-your-classroom-culture-in-5-seconds/

Moving Beyond CUBES and Keywords

http://davidwees.com/content/moving-beyond-cubes-and-keywords/

A Few Simple Beliefs

https://buildingmathematicians.wordpress.com/2016/06/29/a-few-simple-beliefs

What Does Day 1 Look Like?

https://buildingmathematicians.wordpress.com/2016/06/09/teaching-approaches-what-does-day-1-look-like

Questioning the pattern of our questions

https://buildingmathematicians.wordpress.com/2016/11/03/questioning-the-pattern-of-our-questions

The Power of Having More than one Right Answer: Ambiguity in Math Class

Is It Enough for Teachers to have a Growth Mindset?

https://medium.com/learning-mindset/is-it-enough-for-teachers-to-have-a-growth-mindset-9093103d0f24#.jwduybfw8

The Difference Between Instrumental and Relational Understanding

https://davidwees.com/content/difference-between-instrumental-and-relational-understanding/

Down the Rounding Rabbit Hole

http://exit10a.blogspot.ca/2016/12/down-rounding-rabbit-hole.html

Strategies are not Algorithms

http://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/Strategies-Are-Not-Algorithms/

Making Sense

https://tjzager.com/2014/10/18/making-sense/

Real World v Real Work

http://blog.mrmeyer.com/2014/developing-the-question-real-work-v-real-world/

My Favorite Thing about Math

A Brief Ode to Blank Paper

https://tjzager.com/2015/02/25/a-brief-ode-to-blank-paper/

My Beginnings with Cuisenaire Rods

https://mathmindsblog.wordpress.com/2016/11/29/my-beginnings-with-cuisenaire-rods/

Unknown Unknowns

http://exit10a.blogspot.ca/2016/10/unknown-unknowns.html

Inclusive Education Part 6: RtI

http://katienovakudl.com/inclusive-education-part-6-rti/

If you are thinking of creating your own blog, my suggestion is to just start!  Write down your thoughts, share something you have done or write about what inspires you…  Nothing has to be perfect and polished.  Write your piece and hit “publish“!

Pick a Quote

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

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

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

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

Differentiated Instruction: comparing 2 subjects

I’ve been thinking a lot about how to meet the various needs of students in our classrooms lately.  If we think about it, we are REALLY good at differentiated instruction in subjects like writing, yet, we struggle to do differentiated instruction well in subjects like math.  Why is this???

In writing class, everyone seems to have an entry point.  The teacher puts a prompt up on the board and everyone writes.  Because the prompt is open, every student has something to write about, yet the writing of every student looks completely different. That is, the product, the process and/or the content differs for each student to some degree.

Teachers who are comfortable teaching students how to write know that they start with having students write something, then they provide feedback or other opportunities for them to improve upon.  From noticing what students do in their writing, teachers can either ask students to fix or improve upon pieces of their work, or they can ask the class to work on specific skills, or ask students to write something new the next day because of what they have learned.  Either way the teacher uses what they noticed from the writing sample and asks students to use what they learned and improve upon it!

In Math class, however, many teachers don’t take the same approach to learning.  Some tell every student exactly what to do, how to do it, and share exactly what the finished product should look like.  OR… in the name of differentiated instruction, some teachers split their class into different groups, those that are excelling, those that are on track, and those that need remediation.  To them, differentiated instruction is about ability grouping – giving everyone different things.  The two teachers’ thinking above are very different aren’t they!

Imagine these practices in writing class again.  Teacher 1 (everyone does the exact same thing, the exact same way) would show students how to write a journal (let’s say), explain about the topic sentence, state the number of sentences needed per paragraph, walk every student through every step.  The end products Teacher 1 would get, would be lifeless replications of the teacher’s thinking!  While this might build some competence, it would not be supporting young creative writers.

Teacher 2 (giving different things to different groups) on the other hand would split the class into 3 or 4 groups and give everyone a different prompt.  “Some of you aren’t ready for this journal writing topic!!!”  Students in the high group would be allowed to be creative… students in the middle group wouldn’t be expected to be creative, but would have to do most of what is expected… and those in the “fix-up” group would be told exactly what to do and how to do it.  While this strategy might seem like targeted instruction, sadly those who might need the most help would be missing out on many of the important pieces of developing writers – including allowing them to be engaged and interested in the creative processes.

Teacher 1 might be helpful for some in the class because they are telling specific things that might be helpful for some.

Teacher 2 might be helpful for some of the students in the class too… especially those that might feel like they are the top group.

But something tells me, that neither are allowing their students to reach their potential!!!

Think again to the writing teacher I described at the beginning.  They weren’t overly prescriptive at first, but became more focused after they knew more about their students.  They provided EVERYONE opportunity to be creative and do the SAME task!

In math, the most effective strategy for differentiating instruction, in my opinion, is using open problems.  When a task is open, it allows all students to access the material, and allows all students to share what they currently understand.  However, this isn’t enough.  We then need to have some students share their thinking in a lesson close (this can include the timely and descriptive feedback everyone in the group needs).  Building the knowledge together is how we learn.  This also means that future problems / tasks should be built on what was just learned.

We know that to differentiate instruction is to allow for differences in the products, content and/or processes of learning… However, I think what might differ between teacher’s ability to use differentiated instruction strategies is if they are Teacher-Centered… or Student-Centered!

When we are teacher-centered we believe that it is our job to tell which students should be working on which things or aim to control which strategies each student will be learning.  However, I’m not convinced that we would ever be able to accurately know which strategies students are ready for (and therefore which ones we wouldn’t want them to hear), nor am I convinced that giving students different things regularly is healthy for our students.

Determining how to place students in groups is an important decision.  Avoid continually grouping by ability.  This kind of grouping, although well-intentioned, perpetuates low levels of learning and actually increases the gap between more and less dependent students.  Instead, consider using flexible grouping in which the size and makeup of small groups vary in a purposeful and strategic manner.  When coupled with the use of differentiation strategies, flexible grouping gives all students the chance to work successfully in groups. Van de Walle – Teaching Student Centered Mathematics

If we were to have students work on a problem in pairs, we need to be aware that grouping by ability as a regular practice can actually lead students to develop fixed mindsets – that is they start to recognize who is and who isn’t a math student.

Obviously there are times when some students need remediation, however, I think we are too quick to jump to remediation of skills instead of attempting to find ways to allow students to make sense of things in their own way followed by bringing the learning / thinking together to learn WITH and FROM each other.

To make these changes, however, I think we need to spend more time thinking about what a good problem or rich task should look like!  Maybe something for a future post?

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

• I chose to compare what differentiated instruction looks like in mathematics to what it looks like in writing class.  However, I often hear more comparisons between reading and mathematics.  Do you see learning mathematics as an expressive subject like writing or a receptive subject like reading?
• Where do you find tasks / problems that offer all of your students both access and challenge (just like a good writing prompt)?  How do these offer opportunities for your students to vary their process, product and/or content?
• Once we provide open problems for our students, how do you leverage the reasoning and representations from some in the room to help others learn and grow?
• Math is very different than Literacy.  Reading and writing, for the most part, are skills, while mathematics is content heavy.  So how do you balance the need to continually learn new things with the need to continually make connections and build on previous understanding?
• What barriers are there to viewing differentiated instruction like this?  How can we help as an online community?

For more on this topic I encourage you to read How do we meet the needs of so many unique students in a mixed-ability classroom?  or take a look at our Ontario Ministry’s vision for Differentiated Instruction in math: Differentiating Mathematics Instruction

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

Unintended Messages

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

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

“This practice can help your children become a better student, but it may make her less creative.” No parent has been given information about effects and side effects of practices in schools.

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

Let’s explore a few possible scenarios:

Practice:

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

Unintended Side Effects:

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

Practice:

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

Unintended Side Effects:

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

Practice:

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

Unintended Side Effects:

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

Practice:

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

Unintended Side Effects:

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

Practice:

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

Unintended Side Effects:

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

Our Decisions:

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

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

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

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

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

Which one has a bigger area?

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

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

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

Student ideas

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

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

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

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

And some students used different shapes to cover the figures.

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

Building Meaningful Conversations

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

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

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

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

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

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

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

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

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

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

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

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

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

A few things to reflect on:

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

The Zone of Optimal Confusion

In the Ontario curriculum we have many expectations (standards) that tell us students are expected to, “Determine through investigation…” or at least contain the phrase “…through investigation…”.  In fact, in every grade there are many expectations with these phrases. While these expectations are weaved throughout our curriculum, and are particularly noticeable throughout concepts that are new for students, the reality is that many teachers might not be familiar with what it looks like for students to determine something on their own.  Probably in part because this was not how we experienced mathematics as students ourselves!

First of all, I believe the reason behind why investigating is included in our curriculum is an important conversation!   I’ve shared this before, but maybe it will help explain why we want our students to investigate:

The chart shows 3 different teaching approaches and details for each (for more thoughts about the chart you might be interested in What does Day 1 Look Like). Hopefully you have made the connection between the Constructivist approach and the act of “determining through investigation.”  Having our students construct their understanding can’t be overstated. For those students you have in your classroom that typically aren’t engaged, or who give up easily, or who typically struggle… this process of determining through investigation is the missing ingredient in their development. Skip this step and start with you explaining procedures, and you lose several students!

Traditionally, however, many teachers’ goal was to scaffold the learning.  They believed that a gradual release of responsibilities would be the most helpful.  Cathy Seeley in her book Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver explains the issue clearly:

In the two pieces above Cathy explains the “upside-down teaching” approach.  This is exactly the approach we believe our curriculum is suggesting when it says “determine through investigation,”  and exactly the approach suggested here:

At the heart of this is the idea of “productive struggle”, we want our students actively constructing their own thinking.  However, I wonder if we could ever explain what “productive struggle” looks / feels like without ever experiencing it ourselves?  How might the following graphic help us reflect on our own understanding of “productive struggle” and “engagement”?

I think it would be a wonderful opportunity for us to share problems and tasks that allow for productive struggle, that have student reasoning as its goal, problems / tasks that fit into this “zone of optimal confusion”.

In the end, we know that these tasks, facilitated well, have the potential for deep learning because the act of being confused, working through this confusion, then consolidating the learning effectively is how lasting learning happens!

Let’s commit to sharing a sample, send a link to a problem / task that offers students to be confused and work through that confusion to deepen understanding.  Let’s continue sharing so that we know what these ideals look like for ourselves, so we can experience them with our own students!