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,

construct perpendicular 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:

math_task_analysis_guide - Level of Cognitive Demand

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:

DL5ysx5WkAA4O_x

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

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

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

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

IMG_E6046

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


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

IMG_E6044IMG_E6042IMG_E6053IMG_E6052

trapezoids 21

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

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

What are we looking for?

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

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

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


Convergent vs Divergent Thinking

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

Teacher 1:

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

Teacher 2:

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

Teacher 3:

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

Teacher 4:

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


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

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

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

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

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

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


I want to leave you with a few reflective questions:

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

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

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

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

Targeted Instruction

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


Issues with Individualized / Targeted Instruction

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

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

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

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

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

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

 

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

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

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

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


I want to leave you with a few reflective questions:

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

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

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

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:
co-teaching models

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.

coteaching1

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?

co-teaching models

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?

 

img_5080
See Tim Villegas’ article

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.

intervention3

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

 

RTI for Adults

https://mathmindsblog.wordpress.com/2016/04/24/rti-for-adults/

 

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

http://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/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

http://www.elementarymathaddict.com/2016/05/my-favorite-thing-about-math.html

 

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“!blog-button.png

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:

Teaching Approaches - New

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:Gradual Release

Cathy Seeley quote

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:

Page 24 - paragraph 2
Page 24 of the Ontario Curriculum

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”?

img_3336
Research Gate, Confusion can be Beneficial for Learning

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!