Problem-Based Learning: A Vehicle for Trauma-Effected Students to Heal?

mountains with crepuscular ray
Could experiencing PBL be a place to safely confront personal trauma?   Photo by Min An on



As I was writing my personal blog post here about my experiences with trauma, both as a student and a teacher, I was simultaneously taking (this is my second) course on Problem-Based Learning. This course is Higher Ed focused, but I have a lot of thoughts, wonderings, and quite frankly, more questions than actual well-thought out and researched ideas, around how Problem-Based Learning could be a vehicle for students who are dealing with trauma.

From my understanding, Problem-Based Learning, or PBL, students, or the learner, would be deciding on what the problem they were trying to solve was. This, to me is an excellent example of using an authentic experience. In my experience, those who are experiencing trauma are problem-solving/decision making constantly, especially in instances of ‘fight or flight’ in their current state.

Having students tap into their already present survival skills, but seeing them used widely, could assist in the slow healing process. For example, if a student is experiencing neglect and poverty, they might become interested in looking at poverty as a social condition on a wider scale. They could empathize directly with poverty and neglect, and therefore, bring personal perspective and experience to their research. If students were given the opportunity to outreach into their community organizations around poverty, perhaps this would help them macro lens their experiences with poverty, and help them see the complexities and systems that are in place to hold people in poverty, rather than ‘blaming’ those in their care.  My hope is a wider understanding will be developed about their specific trauma experiences, and, the student can then begin to see the various perspectives in their own trauma, as, in PBL, would be the goal if researching the problem.

In my experience, my memory of my personal traumatic experiences has gaps, incompleteness, and is discontinuous. Students who have experienced trauma may feel as though they only have ‘parts’ of their experience at their disposal. Being able to look at the ‘big idea’ of their trauma may help them zoom out of their experiences to start to make connections, and begin healing.

This is also to say that a student would identify one of their traumas as a community/global problem to being researching and learning about. They may, and in fact, might avoid anything to do with their trauma experiences. However, giving the student the power to choose an area of curiosity to look into, and begin to see multiple perspectives, and start to dig into understanding.

Reasons Why I think PBL Would Work for Trauma-Effected Students:

  • Students will be able to explore their own ideas/interests with guidance from an educator who will ask questions, help them navigate, and provide feedback for them as a learner, not just a ‘student in trauma’;
  • I believe (and I realize this needs more research) that students who are/have experiencing/experienced trauma use problem solving skills daily, they just not might be fully aware of the types/styles of problem solving they are doing, and not all of it healthy (example: I used anxiety and putting undue pressure on myself for all tasks, since I had no external guidance with how to prioritize issues – everything was equally weighted in my mind as urgent, as if I weren’t ‘on top’ of even the small things, negative events would occur. A general idea that I could control my choices whereas most aspects of my life were chaotic). I feel students using these problem solving strategies to work through an issue of interest may help with navigating and making better problem-solving choices for them;
  • The ability for the trauma-affected persons to engage in learning, questioning in feeedback with both other learners and a teacher may help them gain strengths and skills, and to be able to accomplish something ‘beyond their trauma’.

I feel that healing includes understanding and approaching trauma from different perspectives, creative outlets, and time. As an educator, giving the student feedback needed to encourage their curiosity, explore who they are with and without their trauma will help them take a more global approach their learning, and defining who they are.

Research Needed/Questions:

In general, most PBL research is done in the Higher Ed community, with less on in the K-12 realm. This leads to the following questions:

  • How would an educator set up PBL effectively in the classroom? What resources would they need?
  • Would the entire school-community need to become PBL, or PBL focused, so that teachers can co-plan and share resources, etc.?
  • Avoidance might be a strategy for trauma persons, and therefore, a lack of problem-solving skills. How could researching a topic, delving into building their knowledge, help parallel their experiences as students, and their experience in trauma?
  • What would assessment look like? I personally imagine consistent conferences and student to educator check-ins, discussions, and immediate feedback of their learning. Students would be able to find a creative way to visually show their research.

The idea here is that strong teaching, building students’ experiences through their interests, and guiding them through their learning experiences, could provide self-insight into trauma.

What Do You Think?

Educators, do you have students who you think would benefit from PBL? Would working through them identifying a problem, researching perspectives, identify possible solutions perhaps lead to students examining themselves as learners, and therefore, tap into their experiences with trauma?

References — places I have started looking:

Demirbas, H. (n.d.).EXAMINATION OF PROBLEM SOLVING SKILLS AMONG UNIVERSITY… Retrieved January 19, 2020, from

Ertmer, P. A., & Simons, K. D. (2006). Jumping the PBL Implementation Hurdle: Supporting the Efforts of K–12 TeachersInterdisciplinary Journal of Problem-Based Learning1(1). doi: 10.7771/1541-5015.1005

Savin-Baden, M. (2000). Chapter 9 Critical Perspecitves on Problem-based Learning. In Savin-Baden, M. (2000). Problem-based Learning in Higher Education: Untold Stories. The Society for Research into Higher Education & Open University Press: Philadelphia.
Ravitz, J. (2009). Introduction: Summarizing Findings and Looking Ahead to a New Generation of PBL Research. Interdisciplinary Journal of Problem-Based Learning3(1). doi: 10.7771/1541-5015.1088




Am I Effective? Data in the Coaching Role

I walk into the classroom for the first time. The teacher and I, excitedly, have been planning to focus on data management. I suggested Google Sheets at our planning meeting as a way for students to focus on the types of charts/graphs they can create with simple data collection. Sheets will pump out a graph quickly, rather than students drawing their own. The benefit of this speed is so that students can create multiple graphs and charts, and be able to discuss and determine which graph is better at giving information that another, and why this might be so. The teacher agreed, and so we set the dates to co-teach the lesson.

It was Hallowe’en, so the class was excited. I walked in with 100% energy and began the lesson. The teacher supported by asking me and students’ questions and played with the program as I worked with students so they could become more comfortable with Google Sheets.

The room was loud as students stacked cups, built linking cube towers, or did jumping jacks. I gave simple data input instructions, told them how to find a timer on the internet, and I let them go. Students started immediately, the engagement was palpable in the room.

The kids ran the show.

Some had time to create a few different graphs, and I lead conversations around what graphs were more effective (and, first, with a few groups, what an effective graph meant), and why. We managed to share our graphs in a shared slide. We had a quick whole group summation of our data — hey! Data collection can be more than me walking around the class asking questions. It can be helpful when we have chosen an effective graph because the reader can interpret our data quickly and efficiently.

Sharing our graphs/charts and determining why graphs are important; and which ones are more effective for our purposes.

Then I left.

My time was up with those students and that teacher for that day…

And, since I support 19 schools, I wouldn’t be back for a month.

Sure, I emailed the teacher all of our work and gave them a chance to ask any follow-up questions as needed.

But that was it.

But did I do anything? Did I create change? Would that teacher now move ahead in two ways:

  1. Consider ways to make the curriculum come alive creatively?
  2. Continue to use a piece of technology meaningfully and with purpose?

Metrics: Collecting Data

The first response to this is to collect tangible data. In this case, when I am completing a cycle of schools, I send out a survey to collect data from teachers. This is dependant on if teachers fill out the survey, and I always wonder if they are being completely honest. I think many fill it out to be kind, which doesn’t give me a sense of how my time with them will be beneficial in the future.

 Another tangible piece is total bookings. If I am booked a lot, in theory, I am reaching more teachers, and assisting in curriculum and innovative connections. I have some of this data — it is quantative, but is it really showing what is going on in the daily use of instructional practices by the teacher?

How do I, without a doubt, know that my time with that teacher has contributed to long-lasting educational change? Are teachers really thinking about their practice after I work with them? Or am I stand alone show? Sure, metrics can show some data, but does this tell me if I have really created long-lasting, sustainable change?

Student Carry Over

I also wonder about the students: Sure, a teacher may not have had sustainably changed practice after co-planning and teaching with me. But, perhaps, a student took the tool that was used and is now using it for another purpose. Perhaps when another math task comes up, they will open up Google Sheets, for example, if they feel it is the tool for them. How do I measure student success — because maybe the student is the one taking leadership, and encouraging other students (and teachers) to use a particular tool, or try a certain task.

Stages of Coaching

Perhaps it is in the next steps. For this particular example, maybe the change isn’t happening yet. It will come when I check in with that teacher the next time I arrive, and have a face-to-face conversation. Or, theywill book me for a second round of co-planning and teaching, this time wanting to explore at different angle of math, or a different tool that can bring, perhaps, student reflection about their math (the first tool assisted the students completing the math task, the next step might be for a tool to help students speak to their understanding).

 I am learning that this means that I am going to be having a different conversation with this teacher the next time around. I am not completely sure what that conversation will be. It may be as open as “What have you done since the last time we were together?”

Teachers, this is a great chance for you to let me know: What do you hope your Coach does with you? How do you know that your Coach has been effective? How do you like to be approached? Feel free to continue to talk with me here, or on Twitter (@heidi_allum).

I am No Chef, but Here are Some Ways That Might Bring More Flavour to your Math Classroom to Start the Year

It’s about that time of year when teachers are making decisions, whether real or imagined, about how they want their classroom to feel.  I say feel because I think that we underestimate how our students will feel walking in.  There is such nervous excitement, that to be honest, I think many kids forget their first day.  I cannot really remember any single ‘first day’ of school.

What I do remember is how the first few weeks set the tone.  I remember the moment I could feel comfortable, or I got a sense of the teacher’s expectations, and what the year was going to be about.  Super organized and structured, more relaxed and flexible — once I figured this out, I could navigate my way through the classroom.

Sure, the first day brought on way too hot ‘fall’ clothes, a new hair cut, and clean sneakers – but it’s the following few weeks that I tasted my classroom.

That’s right:  A very visceral experience that one could taste.  For me, it was a mix of chalk dust, someone’s wilting lunch, and polished floors.  Whatever flavour or recipe was set out in the beginning generally told me how the rest of the year would follow out.

Strangely enough, I found when we had our first math lesson that I really got a sense of what was going to happen — was this teacher patient and valued thinking, or wanted a right answer now?  Willing to listen, or plow through?

Now as a teacher, I know better.  We know better.  We know to give students voice – listen and respond to their learning needs.  So what are some ways to make our mathematics classroom the ‘it’ place to be – which, in turn, will set the tone for the rest of the school year (being an elementary teacher, I teach it all)?

As Sam Shah prompted the Twitter world — How can we create a classroom with some mathematical flavour?, I decided I could talk about few things that have added to the spice of mathematics.

Of course, these are just some suggestions that I have personally used and found open the doors to students engaging with mathematics.  There is so much out there – so explore and find out what works for your curriculum and your students.

Children’s Rights to Mathematics:

Before we talk about specific mathematical ways we can create the flavour of our room, we have to decide what we want students to feel, do, and think about mathematics.


This tweet was posted, and it got me thinking.  Rather than decide on ‘rules’ for the math classroom, I thought about how we could decide on what students need in order to flourish in mathematics.  I started a conversation with Matthew Oldridge, and between us, here is what I think, overall, you should consider when starting your math classroom, or, the student’s right to mathematics:

  • Let students work/struggle on interesting math problems
  • Math is action – make sure math is being done and investigated
  • Encourage talk in mathematics – it should be loud
  • Play with math concepts
  • Go slow
  • Process and reflect while doing the math

and, as the teacher, listen more.

Perhaps rather than thinking in terms of what you want students to finish with, consider the experience with math you want to give them.

Sure, the final meal is rewarding, but the dinner company, conversation, atmosphere, and anticipation of the meal is what you walk away with.

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Do and Play with Math Right Away:

Remember Nike’s “Just Do It” slogan?  Sure you do.  just do it

Just jump in and start with math.  Make it fun and playful.


  1.  Do an interesting math problem:  Matthew Oldridge has a great post here of problems that are engaging, fun, and interesting to solve.  Start with one of these, and let students play.
  2. Create a math play table:  Below are a few photos of mine.  Read more about this in Sarah VanDerWerf‘s Blog about math play tables.  I purchased Trianglemancsd‘s tiling turtles and spiralling pentagons to make my play table amazing.  My students loved it.



3.  Do Dot Number Talks:  These are fun, non-threatening, and get kids talking about math.  Check out my blog post here.  They encourage and validate all ideas, force processing and thinking time, and facilitate talk between students.  This can start your conversation about prompts to continue math conversations.

4.  Estimate:  Estimating is fun, especially with a website like estimate 180, and again, this starts the math conversation flowing, and justification happening.  See more of how I got my kids engaged (human number lines) and up and active around estimation here.

5.  Play Sarah VanDerWerf’s 100 Game:  Can you tell I am a fan of Ms. VanDerWerf?  This game worked exactly as her blog said — this got my kids yelling, working, and defining what collaboration in our classroom means.  Here are photos of my students engaged, on the first day, in this task.


Sarah also has a first day blog post with lots of ideas — check it here.

Making Math More Visual:

Awhile ago, I blogged about how my daughter ‘wrote’ in the sky with her finger to help her fall asleep.  I noticed that we could follow each other’s ideas and thinking by placing our mental ideas in the air.

We need to embrace, and help students embrace, that math is visual.  It is something that can be imagined and experimented with.  I love the ideas that these prompts below create situations where students are forced to see math in their mind’s eye.  Tracy Johnston Zager brought forth some exciting work from Christof Weber — I am so eager to read more of Mathematical Imagining!  How are you seeing this brought to life in your classrooms?

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At the beginning of the year, if we give students opportunities to flex their mathematical imagination, then we are demonstrating how math is a human endeavour – creative and dynamic.

So, as you think, plan, and consider your new year with new students — make math human again.  Take into consideration student backgrounds, experiences with and knowledge of mathematics. Let them dive in and talk about the math.  Be the guide of wonder and curiosity.  Above all:  Have fun with the math curriculum you are using, and enjoy learning math with your students.  You may want to read Sunil Singh‘s post about this.

So, let your students be chefs with math this year.  Sure, your end goal is a good meal, but let them add too much child powder and have the experience of figuring out how to fix it, let them make a mess.  Math is complex and messy, too.


You sang it, didn’t you?

How many other blogs have you seen with that title?


Summer = change.

Reflection.  Wondering.  Thinking.

Thinking, and, let’s be honest — stressing.

Am I going to be enough this year?  What should my focus be, and why?  Where should I put my emotional and intellectual energies?

All of these questions swirl around my mind like paper caught in a dust storm.  I spend my summer trying desperately to grab hold of one.  This almost never happens.  Generally, I let my fingers skim each floating idea, never settling on one for fear another could be more important.

To add to this state of “hmmm”, I am starting a new role this year.

Change Is Good – but I am not good with change:

I believe that educators should change when the opportunity arises.  I also think they should change when they don’t think they are ready.  Actually, this is not educator specific:  Change happens.

However, if you are like me (which I assume many of you are), change is not your strong suit.  Sure, I can smile during change, press on through change — but deep down I am constantly wondering if the change is doing any overall good.  Is anyone benefiting from the change?  Is the change moving anything or anyone forward?

Essentially, change makes you the newbie.  And hanging out in newbie status is always super hard.  Because you are going to fail, and people will see you fail.

I don’t think I truly innately have a growth mindset.

I think in terms of all the mistakes I will make in my new role, and yes, I will learn from them and grow from them, but before that happens I beat myself up about them, stress on them, and replay them over and over.

My first plan is to, rather than harp on a mistake, sit with it, take responsibility for it, and  move on from it.  Change is always a process.

Educational Technology Coach

This is the title of my new role.  I am honestly not 100% sure of what that role means, but I know I am excited at the prospect of learning, and learning with others.  The first word in the title is educational, after all.

What I Envision/Goals for myself:

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Photo by Pixabay on
  • Listening to teachers  – what are they seeing?  Doing?  Wondering?  What/How can I help that will be most beneficial?
  • Connecting – making the curriculum come alive for teachers, and help them continue to make it engaging for students.
  • Collaborating – with teachers and students; how can we use technology to make education more fluid and dynamic?  Access experts?  Share our ideas?  Have students share their ideas with each other/teachers to gain immediate feedback?
  • Focusing on the Global Competencies and making them come alive for teachers and students.

Big Questions:

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Photo by Pixabay on

I have this goals and visions, and hopes, but right now it is all single minded.  I am going into an unknown role, and so on top of goals, I want to keep focus questions in the back of my mind.  Here are some that have been floating around:

  • How can I help teachers explore and expand their experience with the curriculum? What pieces of technology will compliment the content?  And bring out the global competencies? (I selfishly admit, I can’t wait to get into classrooms and do some math — and add tech when it enhances!)
  • If a teacher isn’t sure of what is next, what questions can I ask to help teachers discover what is next for them?
  • How can I create a positive attitude toward technology?
  • How can I help teachers find thinking technology vs. skills?

These are just a few to start my thinking journey.


Anyone else out there been in a coaching role?  What did you find most helpful?  Any books, resources, etc. that you would suggest I look into to help me visualize and refine my goals for myself as a coach?  I would love to hear from you.

My mind is swimming with ideas and thoughts — and I believe this is all part of the process of moving into a new role.  I hope to document with photos and blogs how the year is going, and discuss all of the ups and downs.  At points in the year, I hope to refine my questions and goals — and see if I have measurable data to support my learning, as well as teacher and student learning.

Thank you for coming on this adventure with me!

Student Collaboration on Google Slides: Start Here.

One of the most powerful educational tools that I have loved this year is Google Slides.  Recently our Board started using Google Education Suite, and Google Classroom.  I happily tried it out, and have not have looked back.

I am going to go through with what worked for me and my class, and many of you may find these tips intuitive and repetitive, but I feel knowing this, or considering these, ahead of time for those who have not tried making Google Slides collaborative (or any Google Suite program) somewhat helpful.

At first, it’s easy to click that little triangle and assign “make a copy for each student” — but let me try and steer you into choosing “each student can edit”. Screen Shot 2018-05-21 at 1.08.19 PM

However, I can understand how this could be a scary thing.  What if students ruin each other’s work?

So, though some are predictable, I have created a list of the most helpful ideas in order to make Google Slides collaborative (and less work for you — looking at student work on one slide is so much faster!).

Here are some tips – based solely on my experiences – in no particular order:

Tip One:  Let Them Play

Sure, at first, you may want to give a copy to each student.  Let them explore the features and muck around in a low-stakes setting.  One of my colleagues, Michael McClenaghan, actually had his students ‘ruin’ a slide.

Tell students “I am not sure”, shrug, and walk away a lot.  They may work on one slide to start, but they will inevitably start talking, asking, and helping each other solve problems.

The key hear is to consolidate.  Have students discuss what went well, what didn’t and what they need to know for next time.

Tip Two:  Start With Math

The first time I gave a shared google slide, it was a group problem that students could read together, and work on.  Students could do the math in Google Drawings, and then import the photo of their work into Google Slides.

Students are generally more used to working collaboratively in math — be it over a white board, chart paper, or manipulatives.  Students may ‘ease’ into working on shared slide a little faster through a rich math task.

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Tip Three:  Teach Version History/Restore

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Show them under file they can see version history.  If everything, or things, get erased (and they will), they can restore to a previous version.  This has saved many a shared slide — and once students know this, they are more willing to want to share slides without the fear of ‘their work being erased’.

Tip Four:  Consolidate After Each Session

This may seem redundant, but so important.  Each time your students are working on a shared slide, talk after about how it went.  Troubleshoot together.  Ask:

  • What went well?
  • What needs to change?
  • What should we do next time?

Create a classroom norms of collaborative spaces chart — adding new information as you go (my students taught themselves how to share a slide amongst themselves – So they taught the class how.  Next time I would have a working ‘chart’ or space to have definitions, directions, etc. so we students can reference as needed).

Students, who are researching Canadian NGOs, created a slide, and shared with each other without instruction.  They felt comfortable enough to do this and could each work on slides together, and give each other feedback (one student added links to the comments section so others could access as needed).

Tip Five:  Use the Comment Feature All of the Time

View the slides in grid view, and comment as students work. Circulate and conference with your laptop, and comment on the slides as to what you conferenced.  Add links, videos, etc. so students can access together.  Teach them how to read and apply your comments, and write back to you.  Comment, and comment widely (Trust me:  You feel very accomplished when you write feedback on each slide).

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Now that my students are comfortable, not only do they respond to my feedback, they are able to give feedback to each other in a productive way.

Tip Six:  Some Organization Ideas

  • Write the partner/triad/group name at the top of the slides you want them to work on as they go off to work;
  • Make the last slide the interesting links/videos slide that students can ‘dump’ ideas/links into;
  • Put your learning goal/success criteria on the first slide(s).  Tell students not to edit this, but use it for referral.

In general, once collaborative Google Slides becomes embedded in your classroom, you will use Google Slides as a way for students to demonstrate their learning, and a place for you to give feedback in.  It becomes a loop of learning and assessment.

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These tips also work with any of the other Google Suite programs that can become collaborative — Google Docs, Sheets, and Drawings.  These only become powerful when students see the power in using them to work together.

One final note:  There will be problems, but they will be overcome.  When you consolidate and reflect, the experience becomes even more valuable – the learning from each other’s mistakes leads to a better understanding of Google Slides, and the content itself.

Give it a try!  Let me know how it goes.  Share any other tips – I would love to hear how you are using Google Slides (Docs, Sheets…anything collaborative!) in your classroom.

Probability is Underrated

Ah, probability.  The fun strand.

And, truly – it is fun!  This strand screams play.  Dice, cards, games, odds – esepcially in play off season.

However, I can attest to overly simplifying probability:  Guilty.  I have left it until June, played a few games, and called it a day.

There is nothing wrong with playing games with probability – in fact, I can’t think of anything else more engaging and enticing for students.  What is missing is the rich conversation – and connections – you can make within other strands of mathematics.

Let’s take a closer look at one of the specific expectations in the Ontario Curriculum.  I am choosing a Grade 6 expectation because this is where I live:

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There are some key terms in here that may be focused on, and other parts missed.

  • Dice, favourable outcomes, and equally likely.  Skimming, a teacher may have a quick conversation about the odds of getting even vs. odd numbers, out of a total of 6 – since there are 6 sides on the dice.  What may get missed is ratio – how can we turn dice rolling into a fractional conversation – and help students make these connections.


Now, in this would you rather, students are going to actually think first by making an assumption before they roll any dice.  I used a shared google doc to show their thinking.  Then students got a chance to try it out – what should happen (theoretical) compared to what actually happened (experimental) is now defined.

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Students are now forced to explain why – and how- they would choose option A or B.  Now it’s a great time to describe their odds in a ratio – to pull how the game leads to mathematical explanation.

It’s also amazing how simplicity – rolling some dice, and letting them explore what happens, and challenging students to represent their findings – leads not only to rich conversations about probability, but dissecting why everyone had different outcomes.

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Another expectation that may be skimmed and glossed – some may focus on the words game, and the simple fraction (1/4).  The richness comes out of this expectation is extending to what would happen if the spinner was spun 100 times – and the comparison of predicted and experimented outcomes.

Probability can also become richer when you look at individual results versus group results.  I used small Smarties packets and we estimated all of the Smarties we had altogether as a class.  We talked about which colours we might continue to pull once we pulled out 3-4 from a single package.  When would get our first ‘double’ colour, and why?  What should happen with colour distribution if we opened all of the small packages?  We recorded our class predicitions on one Google Sheet, and then students recorded their own findings in another.  Now we have data to compare.


Probability can also be extremely visual.  This amazing visual by Sara Van Der Werf brought fantastic conversation.  Using visuals in probability was not a natural link for me, but it proved to be for my students.

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This is a powerful visual that brought in fantastic fraction, percent, and decimal conversation around a probability framework.
I love how my students ‘flipped’ the number line — it could work both ways, and therefore, was it the item that really mattered, or the concept of fractional probability of the distribution of colours?

So, pull out the probability games.

I am learning to make the connections with students to the deep math connections that happen through the games.  There is a lot of rich, amazing math here that is worth more than just June.

My next go to is to try these rich, visual probability prompts that invite wondering and conversation about the complexity of probability, and connect to other strands in mathematics.

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Care of Matthew Oldridge.  I love how students have to define “how hard” – what would “how hard” look like fractionally? Compared to something ‘easy’ you could roll?  Some great places to go here that deepen probability.
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How would this differ if everyone in the class pushed one of the buttons?  Would that change which you pressed?  Why?  Now students are invited to break down the amount of money (100 million times half of the class!).  Thank you to @Mathgarden for this one!

What are some ways you are adding complexity and richness to your probability tasks?  I would love to hear your thoughts!

From Patterns to Algebra: The Genius of Dr. Beatty and Dr. Bruce

From Patterns to Algebra, the second that I was given the chance to explore, play, and investigate, stands at the forefront of an amazing patterning and algebra resource.

This blog is essentially a love letter to Dr. Cathy Bruce and Dr. Ruth Beatty — and how they seamlessly weave play, patterns, and rich opportunities for students and teachers to immerse themselves in the joy of patterning (Hey!  I said I was going to write the blog, so here I am):

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I’ll let the introduction of the book speak for itself (as seen on page 1, and in the pdf sample you can access here):

Mathematics has been called “the science of patterns” (Steen, 1988). Young children enjoy working with patterns, and older students enjoy discovering and manipulating patterns. In fact, it is human nature to nd patterns in our everyday experiences. Some educators and mathematicians would go so far as to say that patterning is the foundation of mathematics (Lee, 1996; Mason, 1996)

Let’s start with how the first lesson revolves around playing a game:  Guess My Rule.  Right away this game is A) super fun, and inviting to students — all of students really loved playing it, and B) gets students to make the relationship between term number and the output — and how the two interact.

My students loved this game so much that this happened:

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(also cool is how this student used Google Draw to create the table, screenshot it, and add it to Kahoot).

My students willingly played this game — and often.  I added some elements by using cards- whatever you flipped up became the pattern rule — and putting the black line master in page protectors so my students could play when they were finishing other work – or for recess (yes, a lot of my students like to stay in and play math for recess).


Playing Guess My Rule was fantastic, because, when we got to creating and looking at patterns, we already had some understanding of the relationship between input and output.

This is another reason why this resource is so great — the patterns are visual.  Everything students do with this resource is creating and analyzing visual patterns; making students see how accessible and tangible patterns can be.  Students can manipulate patterns, and can play with them – and feel like anything they create can easily be added to, changed, or redirected without losing the visual representation.  Students were able to make connections quickly – and talk about the patterns we were examining.  Students could also identify the pattern quickly, move to graphing linear patterns, and determining the nth term.

Below are some of the awesome ways we used the resource — and really, we’ve only just touched on it.  Dr. Beatty and Dr. Bruce have created a place where non-intimidating patterning play can happen; and deep connections can be made.

See the images below – -some have captions so you can see a little more about what we did as a class.


The Four Step Problem Solving Makeover

There is a lot to be said about the problem solving model that sits at the front of the Ontario Math Curriculum.  It’s common place to hear and see teachers discussing and using a model, or a similar model, in their classrooms.  It seems for some, it works, or, at least, it gives a framework for teachers to assist students to work through a math problem.

However, it never fully sat with me quite comfortably.  First of all, I will make many non-friends writing this, but, it created zombie-like problem solvers.  I witnessed many students who would be messy, take risks, and think during their math problem solving.  The four step problem solving model was introduced, and suddenly, I found these creative math thinkers felt like they had to follow a path, or a formula, to solve a problem.  They weren’t thinking as richly and deeply, and, in fact, were rushing to get through the steps rather than taking their time, and really questioning their math choices.

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The Polya Four Step Problem Solving, as seen on page 13 of the Ontario Math Curriculum.

I have re-looked, re-examined, and re-thought of this model many a time.  In fact, it has been the focus of math workshops and district meetings I have attended, and I not once could say that following this model in a linear manner made any of my students any better at math.

Don’t get me wrong:  I see it’s purpose.  I have found it most helpful with students who struggle to organize their mathematical thinking, and I present it more of a way of thinking, or a framework, they might want to consider if they find they can’t focus on the math, and are getting jumbled in the process of their mathematics.  I never ‘forced’ a student to use it, as I always felt that solving a math problem – or any real problem in life, really – isn’t linear.  Many times we might jump to carrying out what we think is right, find it didn’t work, then go back to realize we probably didn’t quite understand.

I recently revisited the front matter of the the Ontario Curriculum to really investigate the process expectations, as I was thinking to myself:

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An overview of the 7 Mathematical Processes as highlighted in the Ontario Math Curriculum (page 12)

Don’t we go through processes of thinking while we are solving mathematics tasks, and also taking steps to do so?  Could the two not be intertwined?

But, then I saw this:

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A problem solving model hasn’t been re-investigated since 1945?

So now my head is really spinning.  I wanted to know a few things:

  1. What do students really do and think while they are solving a complex mathematical problem?  Do they naturally go through ‘steps’?
  2. What processes do they consider, or act through, when solving a math problem?
  3. How do we bring a problem solving model back to the actual mathematics?

In my mind, I started to visualize what I had noticed my students were doing during math.  Over the months, I have been able to listen to many math conversations, and sat in on many math problem solving sessions with my students.

Right away, I wanted to depict how mathematics is a flow; people move in and out of their thinking process depending on what they are working on.

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I feel that this is what my students do in the classroom while they are working on math.  More than that, this is what I would like my students to consider as they solve a math problem.

I decided to take it one step further:  I asked them to consider their thinking while they were working on some math.

I created a Google Form, and asked students to reflect what they were thinking before, during, and after their math task.  Here are some results.


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You can see the top two for before were noticing and wondering the math information, and choosing a tool.  This was great — this means they were actively processing the problem, and selecting a tool (which is also one of the aforementioned process expectations).

Screen Shot 2018-03-19 at 8.09.12 PMScreen Shot 2018-03-19 at 8.08.59 PMDuring the problem, students were fixing mistakes and changing their ideas.  This means students were actively seeking math connections, and finding and fixing their misunderstandings.  Using a tool is still high on their choice, which means that tool must help them represent their thinking.

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Students did reflect at the end, as the Polya Problem Solving Model states, which in the re-vamped model is embedded in the questioning themselves area — students, even at the end, are ensuring their mathematical ideas are complete.

I was really excited to see my student responses, and I am going to continue to check in with them as they work through math.

Feedback, please!

What do you think of my re-vamped Mathematical Thinking Flow Chart (because I am not sure what else to call it)?

Would you use this in your classroom?  Do you see your students doing this?

Walking the Fault Line: The Anxiety of Straddling Slow Math with Preparing for Standardized Testing


Earthquake Regions

I went to UBC for my undergrad, an experience that changed my life for many reasons.

The biggest shift was the immense beauty of ocean meeting mountains.  Never had I experienced this before, and to be thrown in and living in this beauty was something I never really became totally comfortable with.  It was almost always to good for me.

I grew up in Southwestern Ontario, where the Canadian Shield grounded me in safety and security in terms of natural disaster worries.  Suddenly I had to learn and be prepared for tsunamis and earthquakes.  It was strange and different for me, and although I loved and appreciated the beauty of B.C., I was always constantly thinking about the next possible earthquake.  Would I be in B.C. when the “big one” hit?  What would my reaction be?  Was there really any ‘safe’ place?

Perhaps silly, I had to mentally find a balance between appreciating where I, at the time, lived, and not constantly worrying – that small nagging voice in the back of my mind – of an earthquake of epic proportions.

The Fault Line in Teaching

At this point in my teaching career, I am having this giant existential-like questioning about why and how I am teaching.  In my mind, I am constantly not ‘doing it right’.  There is always something better I could present to my students, always a question I could have asked, and always a connection that was not made.

It’s like I am back living in B.C.:  I love teaching.  I enjoy the beauty and energy of teaching.  But I am standing on a fault line where that huge earthquake could come at any moment.  I am never truly comfortable.

Tectonic Math Plate Movement

In Mathematics, I feel that this is especially true for me.  As I learn more about Mathematics, as I do my best to bare my misunderstandings and misconceptions, as my personal philosophy of Math is developing (math is play!), I know that my ideals in education are shifting.  There is movement below the surface, and I just don’t know where to take this movement, and how to channel it.

What I mean is this:  I am a bigger picture teacher.  I teach in order for students to find meaning for themselves; to develop a sense of self in this overbearing world.  I want my students to be ok with not having the right answer; to not have to be perfect.  I want math for them to be interwoven in their lives, to be problem solving — to be an action they take with pride and honesty.

The shift is happening as I stand on the fault line of giving my students the bigger picture experience, and keeping my instruction focused and channeled.

Can really doing the math create successful EQAO results?


I asked the question.

Here is where I am at, and what I mean by that question:  My students will be writing EQAO this year.  I am not about to discuss EQAO in itself, I am going to discuss how the act of performing EQAO is generally approached.

Here is an example.

Teachers engage students in using EQAO released questions (or EQAO like questions – focusing on application and thinking) to guide their instruction in a variety of ways, for a variety of reasons.  For example, to see if their students are able to understand the problem, if they can apply a workable strategy to solving the problem (was the student able to identify that this problem required more than one mathematical ‘step’ in order to be successful?  If not, what is our next step as educators?).  Are students able to reason and work their way through multiple choice, and have strategies if they are not sure of their choices?

Yet, my question is always this:  Will I be able to let students go deeply into all the math content so that they are ready for EQAO?

This is where I stand.  On this fault line of giving my students a rich, immersive math experience (and, I am still working and learning this!), and wanting them to feel like they can apply their learning in a testing setting.

Richter Scale

I tried something different this year.  Rather than make EQAO something students would “have to do, so we had better be prepared”, I had them examine the math in an EQAO released question.  I had them determine which tasks and problems in the past we had worked on and make connections.

This was me trying to bridge the fault line.

An example of this would be proportionality, and unit rate.  I started by asking students what was the best deal in a morning activity through grocery store photos I had taken (nothing new here), and to my surprise, students self taught unit rate — through no instruction.  Obviously students used other strategies (adding/multiplying the number of items until they came to the total cost, or estimating a reasonable cost of what one would be, then multiplying until they came to a close total), was one strategy, but many figured out the price of one by dividing number of items/weight and cost.

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I gave them time to talk and share strategies, and discover if this was always true.  We played with the idea of proportion and rate, and compared it to ratio.  We made connections between ratio and rate — and how results can be effected by ratio.

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But the nagging fault line struck me again:  Would students connect this new, and, believe me, we could still go a lot deeper into proportionality, learning to an EQAO question?

For a Board/Ministry Initiative I am involved in (Renewed Math Strategy), we were asked to give students a unit rate (well, a problem that can be solved through unit rate) problem, and at first my students had no idea how to go about solving it.

Even though they were disappointed at first, they begged me to go over the problem.  They wanted me to walk them through how it could be solved.  They wanted to know.  This struck me — if anything, I had created a classroom where, sure, they didn’t know what to do, but they had a desire to know.  They didn’t get wallowed in defeat.  They understood the yet — they weren’t sure how to go about solving it…yet.

Weeks later, after some exploration into unit rate, I decided I would have them own this problem.  I would let them figure it out – and on their own.  They would try the problem again, and they would self evaluate.  I gave them the codes; I had them lead the conversation, I had them determine and compare their previous work to their current understanding.  I had them discuss and identify what math they would need to use and apply in order to be successful.

I admit I did this rather randomly.  My purpose was for them to own the math.  I wanted to walk on that fault line.  I wanted to prove to myself that them as a team experiencing and feeling and playing with proportionality would transfer to a place where they would have to explain and justify in a testing manner.


Amazingly, the dialogue was rich.  My students were saying things like, “Oh! I see how when we found the best deal with the store prices, we had found the unit rate – we can do this here!”

“Oh, I see how you and I came up with different strategies, and I see where I went wrong – I forgot the last step of subtracting.  I see now how that makes the total cost make more sense.”

My students proved something to me — that really immersing in the content of mathematics, even though it is slow, and I feel a major pressure to go fast and cover more, is important.

My fault line fear — the earthquake of pressure to make sure I cover it all, and well enough that they can do well on EQAO, is still nagging me, but I can breathe a little more knowing that I am at least giving my students positive math experiences.



Math Self Talk: How can Collaborative Math Talk turn into Math Self Talk?

If you are a student in my classroom, every morning when you walk in, you are greeted with math talk and thinking right away. I greet you by name at the door, ask how you are, and then you move to the whiteboards or collaborative tables. There might be images for you to notice and wonder on, there might be a problem I want you and your team to tackle. There could be math tools laying around, and tasks for you to think through and explore. I have prompts for you to start the conversation.

Either way, before the bell, before O Canada, math is happening.

I purposely don’t interact right away. This is my time to check in with each student, quickly do attendance, and let the math conversation flow. Let’s be honest, the first thing you and I do at a PD session is catch up — talk about what happened since the last time we talked, and the kids do it, too. And it’s ok.

Because guess what? The math comes. Just wait. Watch for it. Listen for it.

It’s so nerdy to say, but it is my favourite part of the day. I come in to work to hear what they will do with what I put in front of them. I am excited to learn from them. To get a small glimpse into their beautiful brains.

Once attendance is done, I sneak around to the groups who are working. I listen in to the conversations. This is when I decide if I need to intervene or not.

I don’t tell them they are right or wrong. I don’t tell them how to solve, or where to take their thinking, I ask questions.

My goal through math talk is this: That students become aware of what they are thinking while they are doing math. They can do math self talk. I want my students to be good self math talkers.


It’s pretty simple.

I feel like there is still a shame when someone has to think and process math. Math is still awarded for speed and accuracy. Worse, when a math mistake does happen, many people flounder, shrug, and then give up.

I at least want my students to have the perseverance to reason through their mistake. I want them to live in a world where we are patient math thinkers. This will only happen if I teach and value patient math thinking. I want them to value if they fall off of a math track, they get back on. Even better — if they encourage and assist someone who is struggling by asking them to share their thinking, and help process through it. I want to create the next generation where it is socially acceptable to ask for help with math.

So, this is why I spend so much time talking math. I want math talk to be promoted, valued, and second nature.

To give you a feel for my room, here is an example of a math conversation I was honoured to overhear recently:

I took an EQAO MC and had students attack it without numbers first. I then added in the numbers later.

Student 1: OK — well, the books are in a box, in a container — so the amount of books is going to get higher. We are going to have to multiply at some time.

Student 2: I see that — but we need to find the cost of one book. We are given the total cost. What would be reasonable here?

Student 1: Books are kinda expensive — I think around 10 bucks? Should we play around with that?

Student 2: Hmmm…the total cost is $2592; I think a lot lower? Maybe books are cheaper in huge quantities like that?

Student 1: Could we try a cost then keep multiplying up? There are 12 in box — what if we started to 12? Would that get us the cost of a box?

Student 2: I don’t know if that would work every time, I think we need to divide somewhere too…

Wow. So much going on here. Student 1 and 2 are working off of each other’s ideas, questioning, and giving context to the problem, and playing with the idea of reasonableness.

I was thrilled!


You see, here is where I am at. My students are getting pretty good at talking through their math with each other. I hear them correcting their thinking, changing their minds and their trajectories, without blame or negativity.

The more they do this, I assumed, the more it would become an innate independent practice. I assumed that since they were doing it with each other, they would be doing it in their own minds when given an independent task.

But it doesn’t seem so.

So, yes — I want math to be a social and collaborative language that everyone is fluent in. It seems like we are heading that way.

But now my goal is shifting. I want that math talk to turn into self talk that happens as students are working through the math task. I want them to imagine all of the questions and visuals and ideas in their minds as they solve math.

To start to move toward this, I have started consolidating our math talks with conversations like this:

  • How will the math talk you just particpated in help when you are thinking on your own?
  • What math words/visuals/strategies were easy to hold in your head?
  • How was working through this problem going to help you through future problems?
  • What did someone else say that stuck with you, made sense to you, or you want to try?

The fluidity of self and peer talk is so important — it’s where thinking becomes learning, and learning turns into new thinking.