What Makes Math Interesting Anyway?

What does it mean to be “creative” in math?
What makes math interesting anyway?

Questions I think we all need to dive into!

Many teachers are comfortable allowing their students to read for pleasure at school and encourage reading at home for pleasure too. Writing is often seen as a creative activity. Our society appreciates Literacy as having both creative and purposeful aspects. Yet mathematics as a source of enjoyment or creativity is often not considered by many.

I want you to reflect on your own thinking here. How important do you see creativity in mathematics? What does creativity in mathematics even mean to you?

Marian Small might explain the notion of creativity in mathematics best. Take a look:

Type 1 and Type 2 Questions

Several years ago, Marian Small tried to help us as math teachers see what it means to think and be creative in mathematics by sharing 2 different ways for our students to experience the same content. She called them “type 1” and “type 2” questions.

Type 1 problems typically ask students to give us the answer.  There might be several different strategies used… There might be many steps or parts to the problem.  Pretty much every Textbook problem would fit under Type 1.  Every standardized test question would fit here.  Many “problem solving” type questions might fit here too.

Type 2 problems are a little tricky to define here. They aren’t necessarily more difficult, they don’t need a context, nor do they need to have more steps.  A Type 2 problem asks students to get to relationships about the concepts involved.  Essentially, Type 2 problems are about asking something where students could have plenty of possible answers (open ended). Again, here is Marian Small describing some examples:

Examples of Type 1 & 2 Questions

Notice that a type 2 problem is more than just open, it encourages you to keep thinking and try other possibilities!  The constraints are part of what makes this a “type 2” problem! The creativity and interest comes from trying to reach your goal!

Where do you look for “Type 2” Problems?

If you haven’t seen it before, the website called OpenMiddle.com is a great source of Type 2 problems.  Each involve students being creative to solve a potential problem AND start to notice mathematical relationships.

Remember, mathematically interesting problems (Type 2 problems) are interesting because of the mathematical connections, the relationships involved, the deepening of learning that occurs, not just a fancy context.

Questions to Reflect on:

• When do you include creativity in your math class? All the time? Daily? Toward the beginning of a unit? The end? What does this say about your program? (See A Few Simple Beliefs)
• If you find it difficult to create these types of questions, where do you look? Marian Small is a great start, but there are many places!
• How might “Type 2” problems like these offer your students practice for the skills they have been learning? (See purposeful practice)
• What is the current balance of q]Type 1 and Type 2 problems in your class? Are your students spending more time calculating, or deciding on which calculations are important? What balance would you like?
• How might problems like these help you meet the varied needs within a mixed ability classroom?
• If students start to understand how to solve type 2 problems, would you consider asking your students to make up their own problems? (Ideas for making your own problems here).
• Would you want students to work alone, in pairs, in groups? Why?
• If you have struggled with developing rich discussions in your class, how might these types of problems help you bring a need for discussions? How might this change class conversations afterward?
• How will you consolidate the learning afterward? (See Never Skip the Closing of the Lesson)
• As the teacher, what will you be doing when students are being creative? How might listening to student thinking help you learn more about your students? (See: Noticing and Wondering: A powerful tool for assessment)

Can you visualize this?

Many mathematicians are good at searching for patterns in numbers, however, an area that I think we all need to continue to explore is Visualizing.

Instead of just looking for procedural rules, or numeric patterns I encourage you to take one of the following and actually VISUALIZE what is going on.

Pick one of the above that interests you. Answer some of these questions:

• What relationships do you notice here?
• What are you curious about?
• What visual might be helpful to represent this/these relationships?
• Will these relationships work in other instances? When will it work/ when won’t it work?
• How might a visual help others see the relationships you’ve noticed?

I’d love to hear some answers. You can respond here below, or via Twitter @MarkChubb3

The Types of Questions we Ask: which categories of questions should we focus on?

I think we can all agree that there are many different ways for our students to show what they know or understand, and that some problems ask for deeper understanding than others. In fact, many standardized math assessments, like PISA, aim to ask students questions at varying difficult levels (PISA uses 6 difficulty levels) to assess the same concept/skill. If we can learn one thing from assessments like these hopefully it is how to expect more of our students by going deeper… and in math class, this means asking better questions.

Robert Kaplinsky is a great example of an educator who has helped us better understand how to ask better questions. His work on Depth of Knowledge (DOK) has helped many teachers reflect on the questions they ask and has offered teachers examples of what higher DOK questions/problems look like.

In Ontario though we actually have an achievement chart that is aimed to help us think more about the types of questions/problems we expect our students be able to do. Basically, it is a rubric showing 4 levels of achievement across 4 categories. In Ontario it is expected that every teacher evaluate their students based on each the these categories. Many teachers, however, struggle to see the differences between these categories. Marian Small recently was the keynote speaker at OAME where she helped us think more about the categories by showing us how to delineate between the different categories of questions/problems:

• Knowledge
• Understanding
• Application
• Thinking

Knowledge vs. Understanding

Below are a few of Marian Small’s examples of questions that are designed to help us see the difference between questions aimed at knowledge and questions aimed at understanding:

As you can see from the above examples, each of the knowledge questions ask students to provide a correct answer. However, each of the understanding questions require students to both get a correct answer AND be able to show that they understand some of the key relationships involved. Marian’s point in showing us these comparisons was to tell us that we need to spend much more time and attention making sure our students understand the math they are learning.

Each of the questions that asks students to show their understanding also help us see what knowledge our students have, but the other way around is not true!

Hopefully you can see the potential benefits of striving for understanding, but I do believe these shifts need to be deliberate. My recommendation to help us aim for understanding is to ask more questions that ask students to:

• Draw a visual representation to show why something works
• Provide an example that fits given criteria
• Explain when examples will or won’t work
• Make choices (i.e., which numbers, visual representations… will be best to show proof)
• show their understanding of key “Big Ideas” and relationships

Application vs. Thinking

Below are a few examples that can help us delineate the differences between application and thinking:

These examples might be particularly important for us to think about. To begin with, application questions often use some or all of the following:

• use a context
• require students to use things they already should know
• provide a picture(s) or example(s) for students to see
• provide almost all of the information and ask the student to find what is missing

Thinking questions, on the other hand, are the basis for what Stein et. al called “Doing Mathematics“. In Marian’s presentation, she discussed with us that these types of questions are why those who enjoy mathematics like doing mathematics. Thinking and reasoning are at the heart of what mathematics is all about! Thinking questions typically require the student to:

• use non-algorithmic thinking
• make sense of the problem
• use relevant knowledge
• notice important features of the problem
• choose a possible solution path and possibly adjust if needed
• persevere to monitor their own progress

Let’s take a minute to compare questions aimed at application and questions aimed at thinking. Application questions, while quite helpful in learning mathematics concepts (contexts should be used AS students learn), they typically offer less depth than thinking questions. In each of the above application questions, a student could easily ignore the context and fall back on learned procedures. On the other hand, each of the thinking questions might require the student to make and test conjectures, using the same procedures repeatedly to find a possible solution.

Ideally, we need to spend more time where our students are thinking… more time discussing thinking questions… and focus more on the important relationships/connections that will arise through working on these problems.

Final Thoughts

Somehow we need to find the right balance between using the 4 types of questions above, however, we need to recognize that most textbooks, most teacher-made assessments, and most online resources focus heavily (if not exclusively) on knowledge and occasionally application. The balance is way off!

Focusing on being able to monitor our own types of questions isn’t enough though. We need to recognize that relationships/connections between concepts/representations are at the heart of expecting more from our students. We need to know that thinking and reasoning are HOW our students should be learning. We need to confront practices that stand in the way of us moving toward understanding and thinking, and set aside resources that focus mainly on knowledge or application. If we want to make strides forward, we need to find resources that will help US understand the material deeper and provide us with good examples.

Questions to Reflect on:

• What did your last quiz or test or exit card look like? What is your current balance of question types?
• What resources do you use? What balance do they have?
• Where do you go to find better Understanding or Thinking questions?
• What was the last problem you did that made you interested in solving it? What was it about that problem that made you interested? Likely it was a Thinking question. What was it about that problem that made it interesting?
• Much of the work related to filling gaps, intervention, assessment driving learning… points teachers toward students’ missing knowledge. How can we focus our attention more toward understanding and thinking given this reality?
• How can we better define “mastery” given the 4 categories above? Mastery must be seen as more than getting a bunch of simple knowledge questions correct!

If you haven’t already, please take a look at Marian Small’s entire presentation where she labels understanding and thinking as the “fundamentals of mathematics”

The role of “practice” in mathematics class

A few weeks ago a NYTimes published an article titled, Make Your Daughter Practice Math. She’ll Thank You Later, an opinion piece that, basically, asserts that girls would benefit from “extra required practice”.  I took a few minutes to look through the comments (which there are over 600) and noticed a polarizing set of personal comments related to what has worked or hasn’t worked for each person, or their own children.  Some sharing how practicing was an essential component for making them/their kids successful at mathematics, and others discussing stories related to frustration, humiliation and the need for children to enjoy and be interested in the subject.

Instead of picking apart the article and sharing the various issues I have with it (like the notion of “extra practice” should be given based on gender), or simply stating my own opinions, I think it would be far more productive to consider why practice might be important and specifically consider some key elements of what might make practice beneficial to more students.

To many, the term “practice” brings about childhood memories of completing pages of repeated random questions, or drills sheets where the same algorithm is used over and over again.  Students who successfully completed the first few questions typically had no issues completing each and every question.  For those who were successful, the belief is that the repetition helped.  For those who were less successful, the belief is that repeating an algorithm that didn’t make sense in the first place wasn’t helpful…  even if they can get an answer, they might still not understand (*Defining 2 opposing definitions of “understanding” here).

“Practice” for both of the views above is often thought of as rote tasks that are devoid of thinking, choices or sense making.  Before I share with you an alternative view of practice, I’d like to first consider how we have tackled “practice” for students who are developing as readers.

If we were to consider reading instruction for a moment, everyone would agree that it would be important to practice reading, however, most of us wouldn’t have thoughts of reading pages of random words on a page, we would likely think about picture books.  Books offer many important factors for young readers.  Pictures might help give clues to difficult words, the storyline offers interest and motivation to continue, and the messages within the book might bring about rich discussions related to the purpose of the book.  This kind of practice is both encourages students to continue reading, and helps them continue to get better at the same time.  However, this is very different from what we view as math “practice”.

In Dan Finkel’s Ted Talk (Five Principles of Extraordinary Math Teaching) he has attempted to help teachers and parents see the equivalent kind of practice for mathematics:

Below is a chart explaining the role of practice as it relates to what Dan Finkel calls play:

Take a look at the “Process” row for a moment.  Here you can see the difference between a repetitive drill kind of practice and the “playful experiences” kind of  practice Dan had called for.  Let’s take a quick example of how practice can be playful.

Students learning to add 2-digit numbers were asked to “practice” their understanding of addition by playing a game called “How Close to 100?”.  The rules:

• Roll 2 dice to create a 2-digit number (your choice of 41 or 14)
• Use base-10 materials as appropriate
• Try to get as close to 100 as possible
• 4th role you are allowed eliminating any 1 number IF you want

What choice would you make???  Some students might want to keep all 4 roles and use the 14 to get close to 100, while other students might take the 41 and try to eliminate one of the roles to see if they can get closer.

When practice involves active thinking and reasoning, our students get the practice they need and the motivation to sustain learning!  When practice allows students to gain a deeper understanding (in this case the visual of the base-10 materials) or make connections between concepts, our students are doing more than passive rule following – they are engaging in thinking mathematically!

In the end, we need to take greater care in making sure that the experiences we provide our students are aimed at the 5 strands shown below:

You might also be interested in thinking about how we might practice Geometrical terms/properties, or spatial reasoning, or exponents, or Bisectors

So I will leave you with some final thoughts:

• What does “practice” look like in your classroom?  Does it involve thinking or decisions?  Would it be more engaging for your students to make practice involve more thinking?
• How does this topic relate to the topic of “engagement”?  Is engagement about making tasks more fun or about making tasks require more thought?  Which view of engagement do you and your students subscribe to?
• What does practice look like for your students outside of school?  Is there a place for practice at home?
• Which of the 5 strands (shown above) are regularly present in your “practice” activities?  Are there strands you would like to make sure are embedded more regularly?

Noticing and Wondering: A powerful tool for assessment

Last week I had the privilege of presenting with Nehlan Binfield at OAME on the topic of assessment in mathematics.  We aimed to position assessment as both a crucial aspect of teaching, yet simplify what it means for us to assess effectively and how we might use our assessments to help our students and class learn.  If interested, here is an abreviated version of our presentation:

We started off by running through a Notice and Wonder with the group.  Given the image above, we noticed colours, sizes, patterns, symmetries (line symmetry and rotational symmetry), some pieces that looked like “trees” and other pieces that looked like “trees without stumps”…

Followed by us wondering about how many this image would be worth if a white was equal to 1, and what the next term in a pattern would look like if this was part of a growing pattern…

We didn’t have time, but if you are interested you can see the whole exchange of how the images were originally created in Daniel Finkel’s quick video.

We then continued down the path of noticing and wondering about the image above.  After several minutes, we had come together to really understand the strategy called Notice and Wonder:

As well as taking a quick look at how we can record our students’ thinking:

At this point in our session, we changed our focus from Noticing and Wondering about images of mathematics, to noticing and wondering about our students’ thinking.  To do this, we viewed the following video (click here to view) of a student attempting to find the answer of what eight, nine-cent stamps would be worth:

The group noticed the student in the video counting, pausing before each new decade, using two hands to “track” her thinking…  The group noticed that she used most of a 10-frame to think about counting by ones into groups of 9.

We then asked the group to consider the wonders about this student or her thinking and use these wonders to think about what they would say or do next.

• Would you show her a strategy?
• Would you suggest a tool?
Would you give her a different question?

It seemed to us, that the most common next steps might not be the ones that were effectively using our assessment of what this child was actually doing.

Looking through Fosnot’s landscape we noticed that this student was using a “counting by ones” strategy (at least when confronted with 9s), and that skip-counting and repeated addition were the next strategies on her horizon.

While many teachers might want to jump into helping and showing, we invited teachers to first consider whether or not we were paying attention to what she WAS actually doing, as opposed to what she wasn’t doing.

This led nicely into a conversation about the difference between Assessment and Evaluation.  We noticed that we many talk to us about “assessment”, they actually are thinking about “evaluation”.  Yet, if we are to better understand teaching and learning of mathematics, assessment seems like a far better option!

So, if we want to get better at listening interpretively, then we need to be noticing more:

Yet still… it is far too common for schools to use evaluative comments.  The phrases below do not sit right with me… and together we need to find ways to change the current narrative in our schools!!!

Evaluation practices, ranking kids, benchmarking tests… all seem to be aimed at perpetuating the narrative that some kids can’t do math… and distracts us from understanding our students’ current thinking.

So, we aimed our presentation at seeing other possibilities:

To continue the presentation, we shared a few other videos of student in the processs of thinking (click here to view the video).  We paused the video directly after this student said “30ish” and asked the group again to notice and wonder… followed by thinking about what we would say/do next.

Followed by another quick video (click here to view).  We watched the video up until she says “so it’s like 14…”.  Again, we noticed and wondered about this students’ thinking… and asked the group what they would say or do next.

After watching the whole video, we discussed the kinds of questions we ask students:

If we are truly aimed at “assessment”, which basically is the process of understanding our students’ thinking, then we need to be aware of the kinds of questions we ask, and our purpose for asking those questions!  (For more about this see link).

We finished our presentation off with a framework that is helpful for us to use when thinking about how our assessment data can move our class forward:

We shared a selection of student work and asked the group to think about what they noticed… what they wonderered… then what they would do next.

So, let’s remember what is really meant by “assessing” our students…

…and be aware that this might be challenging for us…

…but in the end, if we continue to listen to our students’ thinking, ask questions that will help us understand their thoughts, continue to press our students’ thinking, and bring the learning together in ways where our students are learning WITH and FROM each other, then we will be taking “a giant step toward becoming a master teacher”!

So I’ll leave you with some final thoughts:

• What do comments sound like in your school(s)?  Are they asset based (examples of what your students ARE doing) or deficit based (“they can’t multiply”… “my low kids don’t get it…”)?
• What do you do if you are interested in getting better at improving your assessment practices like we’ve discussed here, but your district is asking for data on spreadsheets that are designed to rank kids evaluatively?
• What do we need to do to change the conversation from “level 2 kids” (evaluative statements that negatively impact our students) to conversations about what our students CAN do and ARE currently doing?
•  What math knowledge is needed for us to be able to notice mathematicially important milestones in our students?  Can trajectories or landscapes or continua help us know what to notice better?

If you are interested in reading more on similar topics, might I suggest:

Or take a look at the whole slide show here

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 ).

This slideshow requires JavaScript.

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 ).

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 ).

The Same… or Different?

For many math teachers, the single most difficult issue they face on a daily basis is how to meet the needs of so many students that vary greatly in terms of what they currently know, what they can do, their motivation, their personalities…  While there are many strategies to help here, most of the strategies used seem to lean in one of two directions:

1. Build knowledge together as a group; or
2. Provide individualized instruction based on where students currently are

Let’s take a closer look at each of these beliefs:

Those that believe the answer is providing all students with same tasks and experiences often do so because of their focus on their curriculum standards.  They believe the teacher’s role is to provide their students with tasks and experiences that will help all of their students learn the material.  There are a few potential issues with this approach though (i.e., what to do with students who are struggling, timing the lesson when some students might take much more time than others…).

On the other hand, others believe that the best answer is individualized instruction.  They believe that students are in different places in their understand and because of this, the teacher’s role is to continually evaluate students and provide them with opportunities to learn that are “just right” based on those evaluations.  It is quite possible that students in these classrooms are doing very different tasks or possibly the same piece of learning, but completely different versions depending on each student’s ability.  There are a few difficulties with this approach though (i.e., making sure all students are doing the right tasks, constantly figuring out various tasks each day, the teacher dividing their time between various different groups…).

There are two seemingly opposite educational ideals that some might see as competing when we consider the two approaches above:  Differentiated Instruction and Complexity Science.  However, I’m actually not sure they are that different at all!

For instance, the term “differentiated instruction” in relation to mathematics can look like different things in different rooms.  Rooms that are more traditional or “teacher centered” (let’s call it a “Skills Approach” to teaching) will likely sort students by ability and give different things to different students.

If the focus is on mastery of basic skills, and memorization of facts/procedures… it only makes sense to do Differentiated Instruction this way.  DI becomes more like “modifications” in these classrooms (giving different students different work).  The problem is, that everyone in the class not on an IEP needs to be doing the current grade’s curriculum.  Really though, this isn’t differentiated instruction at all… it is “individualized instruction”.  Take a look again at the Monograph: Differentiating Mathematics Instruction.

Differentiated instruction is different than this.  Instead of US giving different things to different students, a student-centered way of making this make sense is to provide our students with tasks that will allow ALL of our students have success.  By understanding Trajectories/Continuum/Landscapes of learning (See Cathy Fosnot for a fractions Landscape), and by providing OPEN problems and Parallel Tasks, we can move to a more conceptual/Constructivist model of learning!

Think about Writing for a moment.  We are really good at providing Differentiated Instruction in Writing.  We start by giving a prompt that allows everyone to be interested in the topic, students then write, we then provide feedback, and students continue to improve!  This is how math class can be when we start with problems and investigations that allow students to construct their own understanding with others!

The other theory at play here is Complexity Science.  This theory suggest that the best way for us to manage the needs of individual students is to focus on the learning of the class.

The whole article is linked here if you are interested.  But basically, it outlines a few principles to help us see how being less prescriptive in our teaching, and being more purposeful in our awareness of the learning that is actually happening in our classrooms  will help us improve the learning in our classrooms.  Complexity Science tells us to think about how to build SHARED UNDERSTANDING as a group through SHARED EXPERIENCES.  Ideally we should start any new concept with problem solving opportunities so we can have the entire group learn WITH and FROM each other.  Then we should continue to provide more experiences for the group that will build on these experiences.

Helping all of the students in a mixed ability classroom thrive isn’t about students having choice to do DIFFERENT THINGS all of the time, nor is it about US choosing the learning for them… it should often be about students all doing the SAME THING in DIFFERENT WAYS.  When we share our differences, we learn FROM and WITH each other.  Learning in the math classroom should be about providing rich learning experiences, where the students are doing the thinking/problem solving.  Of course there are opportunities for students to consolidate and practice their learning independently, but that isn’t where we start.  We need to start with the ideas from our students.  We need to have SHARED EXPERIENCES (rich problems) for us to all learn from.

As always, I leave you with a few questions for you to consider:

• How do you make sure all of your students are learning?
• Who makes the decisions about the difficulty or complexity of the work students are doing?
• Are your students learning from each other?  How can you capitalize on various students’ strengths and ideas so your students can learn WITH and FROM each other?
• How can we continue to help our students make choices about what they learn and how they demonstrate their understanding?
• Do you see the relationship between Differentiated Instruction and the development of mathematical reasoning / creative thinking?  How can we help our students see mathematics as a subject where reasoning is the primary goal?
• How can we foster playful experience for our students to learn important mathematics and effectively help all of our students develop at the same time?
• What is the same for your students?  What’s different?

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

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

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

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

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

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

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

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

Here is the complete quote:

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

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

Visual perception and visual memory are used when we are:

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

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

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

Symmetry games:

Composing and decomposing shapes:

Relating nets to 3D figures:

Constructing unique pentominoes:

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

So I leave you with a few questions:

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

Reflecting on 2016

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

Some things I DO know:

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

My 10 most popular posts were:

My least popular posts were:

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

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

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

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

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

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

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

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

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

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

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

Subtraction can be thought of as removal…

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

Or subtraction can be thought of as difference…

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

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

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

25-4 =

25-22=

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

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

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

How does this relate to Integers?

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

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

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

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

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

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

(-4) – (-7) =

Some final thoughts:

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

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