Planning for Discovery

As an educator, I have been very lucky.  I have been blessed with an abundance of game changing opportunities:  planning time with teachers of the same subject and grade built into my regular schedule, access to mentors and math coaches who have worked with me side-by-side in my classroom, and plenty of choices in accessing meaningful and timely professional development.  

In that common, scheduled time with my school-based professional learning community or PLC, teachers meet and plan together with student achievement at the core of the work.  We have guiding questions that frame what we do so everyone comes to the meetings with similar intentions and a common purpose.  These questions are:

What am I teaching?
Why am I teaching it?
How will I teach it?
How will I know if students have learned it or not?
What is next…if this works?  If it doesn’t work?
Where do I go for help?

At the start of a professional development session, we might also hear or be reminded of these same guiding questions.  During staff meetings we are sure to hear these questions again, and without a doubt when we meet to problem solve ways to engage and inspire students that have yet to meet targets, you better believe these questions will show up once more.

It was no surprise then, that on the first day of a new course at Mount Holyoke College these guiding questions seemed to resurface.  Jan Szymaszek, one of my favourite teachers of all time, started asking those same questions that have been at the center of my planning, and the focus of my professional learning community:

What do I want my students to learn?
Why do I want them to learn that?
How am I going to make it possible for them to figure it out for themselves?

Wait what?  What was that last line again?

I stopped…literally frozen in time.  It is not an exaggeration for me to say that I felt like everything around me took on a new and exciting glow.  That guiding question was different…and in such an important way.  While I know and have known that offering chances for discovery have lasting effects on understanding and retaining new ideas, I did not approach planning with this idea as a core component.  Discovery happened for sure, but planning for discovery was different. It was intentional.  Impactful.  Powerful.  This was the focus question that I needed to bring to my planning each and every single day.  This was the question that would change everything for me.

Jan kept talking in that lovely soft, curious, warm and inviting way that she does, oblivious to the fact that she had just blown up my whole thought process and changed my mindset forever.  I am sure it looked like my screen was frozen for the next ten minutes with a dazed and crazed look on my face as I thought about how this idea would fundamentally change how I operated, how I planned, and how my students learned.  I did not hear the rest of her list, or really anything else for a solid chunk of time as I let that idea and everything it entailed wash over me and take hold.  When I finally refocused and tuned into the class conversation, our other amazing facilitator, Zak Champagne, was in the middle of his humble brag that his previous class claimed, “…he didn’t teach them anything all year!  He just happened to be there when they figured things out.”  Yasssssssssssss!  That’s it.  That’s my new goal, my new focus:  planning for discovery.

I started thinking about this amazing program I was in at Mount Holyoke and realized that so much of the content and organization of my classes were conducted in this manner.  Course creators and facilitators designed and selected activities that would lead participants to notice and wonder.  We could make conjectures, test theories, prove with representations, extend to other number domains, then reword generalizations as new information gets consolidated with previous knowledge.  Facilitators did not tell us the answers to problems.  We examined and discussed student work, ideas and conversations but they didn’t tell us how to teach content.  They were planning for discovery. 

So how does this translate into my daily practice?  Well here’s a quick example from something on deck for this week.  Halving and doubling.  A great computational strategy when opportunities to use it are recognized.  

One way to introduce this concept to students is to simply show them: 
Hey kids!  Did you know that 18 x 5 has the same product as 9 x 10?  It does!  In fact anytime that you have a multiplication problem to do you can take half of one factor and double the other factor and the product stays the same.  Isn’t that cool?  Let’s test it out and practice with these examples!  

If I am planning for discovery, however, I would approach this idea differently.  I might have students give all the factor pairs that have a product of 100.  (…or another product. I like 100 since the factor pairs might be easy to access for many and we can focus on the upcoming strategy without getting bogged down with unfamiliar facts)

1 x 100
2 x 50
4 x 25

5 x 20
10 x 10

I might record what students share this way so that the pattern is easier to notice.  Then I might ask if they notice anything?  Is there a pattern? Students will notice that, “one side doubles and the other side gets cut in half…and the answer stays the same!”  I’ll record their noticings using their words.  I might wonder if there is a way to represent what is happening with a picture or context.  Or wonder if this only works for a product of 100.  What do we think?  Some students will start scrambling to test other products and draw pictures while some might need more direction.  Why don’t we test other possibilities!  I’m going to test a product of 36.  You can choose another number if you like…

As we go, I might ask if this works for other numbers as well, or only certain products.  I might ask if this only works for multiplication. Does this only work with halving and doubling…does quartering and quadrupling work? What about taking a third then tripling? Whole numbers only or will fractions and decimals work? My questions might be for the whole class or might be for particular students depending on what they are ready to investigate. I might wonder when or if this information is useful.  I might present some carefully selected examples and ask if or how we could use our generalization to change these problems to equivalent problems.  Why might we want to change them?  Letting the students test and check and play.  Letting them discover.  Being flexible with how far we get that day in the process and being open to letting students take an unproductive path before embarking on a new route.  Letting kids share what they learned and what they still wonder about. Planning for discovery.

It may seem as though planning for discovery is more time consuming than simply telling students how to do something and hoping they trust your expertise and take your word at face value.  But do we want that?  Compliance over engagement? Will students be able to recognize opportunities to use this strategy without the notice and wonder?  Maybe…maybe not.  What I do know is that there is no better feeling than being present when a student makes a discovery for themselves.  That satisfied, confident look when they notice, test, and confirm the patterns they see in their own examples.  When they compare notes with their buddies and decide something works every time – I can prove it!  Lighting kids up with discovery beats any regurgitation satisfaction every day of the week.  And deep understanding that comes from digging into the content and testing your own examples?  Well that’s the stuff that sticks.  

My new guiding question is really just an important edit of what I have been using for years.  Instead of, “How will I teach it”, I think, “How am I going to make it possible for students to figure it out for themselves?”  You know – planning for discovery.

I laugh when I think about that moment when Jan said those words that shifted my focus. I have yet to find out how she finished that list of questions.  But that gave me another insight.  When your words have a deep impact – your students might not hear another word that day.  Big ideas need time and space.  You will repeat yourself – and not because students weren’t listening.  Maybe…just maybe (and this is what I choose to believe) you are repeating yourself because the impact of your words just changed everything.

My big moment

I meant to do that…

A few years ago I had my ideal teaching position.  My official assignment was Grade 7 and 8 Math but I also had the semi-official title of Math Leader in our biggish Grade 6-8 middle school.  In that role, I could share, guide, and wonder out loud from amongst the group – a slight but powerful difference from my current role as a Math Coach.  As a fellow classroom teacher, I know the challenges and stress of a typical day.  I might have a warmer reception when offering ideas in lesson planning than someone who “doesn’t get what happens in a real classroom”.  In that role, I would occasionally go to professional development sessions at the central office and bring back news, ideas, initiatives and the like to the teachers of math at home base.  I was the contact at the school to receive math mailings and resources, and would always be called on to host an informal professional development session or facilitate discussion in-house when school-based PD days happened several times a year.  It was in the planning and execution of one of these sessions for teachers that I discovered a new teaching move to add to my tool kit…

I was swamped with work.  (Aren’t we always!?!)  The school-based PD session was next week and it was on my mind.  My principal hadn’t asked yet if I would lead a conversation with the math group – but I knew it was coming.  If I was exhausted and overwhelmed and I only had two different classes to prepare for – how were the other teachers feeling right now?  Some taught Grade 6 – a level of preparation that is impossible for me to wrap my head around – all core subjects with the same group of children for the whole day…I get tired thinking about that work load.  Others in our group taught math, science, healthy living, physical education, core french, or any combination of these.  If I was taking some of their valuable time and hoping to make an impact, this session had to be relevant, had to be something they could and would want to use in their classrooms immediately, and had to help unite us as a group of learners on equal footing.  I wanted us all to work on a question – but not in an ‘I’m the teacher – look at what you can learn from me’  way.  Something interesting but not intimidating, satisfying but not stressful. Do some math together to have that shared experience but an idea applicable to every or any math topic.  And I wanted to show and talk about some actual student work too…Hmmmmmmm.

Luckily a problem I was working on with my students that week gave me an idea.  Check it out:

Taken from University of Waterloo’s POW Archive found here.

We could work out this problem, discuss and share our work and reasoning, then have a look at some carefully selected student work that would generate some great conversation.  Not groundbreaking I know but I wanted to model the way we discuss, revise, and edit in my class. Students often review each others’ solutions, offering feedback for clearer communication, then we incorporate some of those ideas, rework, and share. My students and I saw so many cool ideas this way:  different methods to solve and different methods to organize.  It was a newer practice for me as well and was so successful in my classroom that I wanted to share it – but not in a show-and-tell way.  I wanted the session to mirror what happened in class. And by using a cool problem from the University of Waterloo’s Problem of the Week archives, I could show teachers a spot to look for more awesome problems they could use in the future.  While there was only one correct answer for this problem – there were lots of ways to go about arriving at the solution and showing and organizing your work.  Teachers might then see that offering just their own solution at the board in a this-is-how-you-do-it kids!  kinda way might be lacking.  There.  Decided.  A quick prep for me – just select some sufficiently diverse student solutions to decipher and copy problem sheets for the teachers to use first.  I was excited. They were going to love this!

The morning of the session, the teachers filed into the library.  I decided to host our session there instead of my classroom.  Comfy chairs, softer lighting, we could all sit together around a big table.  I had some coffee and muffins, my fun sharp pencils, and my fun sharp problem.  I laid out my plan with enthusiasm:  let’s all try this great problem first and compare our solutions.  Then we can have a look at what some students did with the problem and see where that discussion takes us.  I passed out the question, my new pencils, and paused.  The vibe just wasn’t right.  We were too quiet.  Was there nervous tension?  I remembered that I wanted us to be conversational, on equal footing.  While I taught math all day and had for years, some of these teachers would describe themselves as language experts or science guys.  In a split second decision I found a way to equalize and maybe remove some stress.  “I think the answer we found in class was 74 square meters – but we can see if that matches what you find and compare how we found it and how we organized our work.”  Almost immediately the atmosphere changed.  Some teachers started in on their own, others grouped up casually and talked it through together.  When people were finished they grabbed a coffee or snack sending the message they were ready for whatever came next.  The rest of the session went great.  Feedback from teachers revealed that doing the problem themselves first was really helpful.  Seeing the different ways teachers and students handled their communication highlighted that there may not be one best way to record your work.  Different teachers shared different ideas of what they liked or how they might incorporate what we did that morning.  But the best takeaway for me was the power of giving the answer.  Could I use this idea with my students?

This really got me thinking.  How many times have I lamented that students just wrote down an answer – they didn’t show their work?  How many times did my students see a problem and stress about being the last to figure it out or worry that the answer they shared was incorrect?  How many times did I SAY we were focused on showing our work but ended up just solving and moving on?  Through this reflection one of my favourite number routines was born:  Prove it! I give any word problem or computation and supply the answer.  Then the task was clear.  The solution was secondary to the process.  Stress removed.  Creativity flowed.  Variety was expected.  Conversations, comparing, and revising were necessary in order to be concise and clear.  I don’t do this all the time – but when I am focusing our energy on communication – I do.

I made another realization as well.  I was so careful with my colleagues in thinking through the session.  What did they need?  What was relevant to them?  Would this activity have an access point for everyone?  Could everyone find something here to move their practice forward?  Did we have supplies?  Snacks?  Could we all sit together?  I realized that I needed to be this intentional when planning for my students as well.  What did each of my lovely kids need each day to be successful?  What could I give them to do so everyone, regardless of their starting point, came out of the lesson with some growth?  

While I have had some classes where the lessons were just stellar I have also had classes where the best I could do that day was give 10 problems to try from a worksheet printed from the internet, found and copied minutes before students arrived.  I’m human like everyone else.  I remember one such occasion passing out a boring worksheet and realizing when students had it in their hands that I copied the sheet with the solutions given instead of the front side.  One student said (a little too triumphantly), “Uh Mrs Sandford – you made a mistake!  The answers are all here!”  

“Nope kids”, I countered, “I meant to do that”.

Surviving the Countdown

Don’t get me wrong…I love being a teacher.  But during the third week in June an unbelievable exhaustion sets in.  At the end of the school year, teachers have report cards to complete, final grades to justify, transition notes to update, end of year celebrations to plan, materials to collect, classrooms to pack up, textbooks and final assignments to chase down…

Teachers:  Congratulations on making it this far in an incredibly challenging year.  Only one more week of classes to plan. Have you ever noticed that once you start that countdown – time stands still?

Students in middle school know what’s going on…exams or final projects are handed in.  Nothing they do now “counts towards their mark”.  The Principal comes over the PA system reminding everyone that if you have to come to school you will be engaged in meaningful activities even if they will not be graded.  Serenity now!  So what do you do?  What are those worthwhile activities that can be done with minimal prep, that maximize engagement, that are adaptable and accessible for all your learners and are possible to do if you have a whole class or just a handful of students?

Here are some of my go-to activities when I am in survival mode:

Skyscraper Puzzles:
There is something about having fun things to play with that gets students excited.  I love these puzzles because it usually highlights the strengths of a different set of students.  And if you ask me – we don’t do nearly enough spatial awareness tasks.  I can confidently say that after years of sharing this activity with my classes – kids love these puzzles.   And I mean all kids love these puzzles.  Find the directions, templates, and how-to pictures here.  I have also used the pictures from this link and put together a little slide deck to review the rules with my students as a group.  Unifix cubes work great with this task as you see in the pictures but I also saw recently on Twitter that Mrs Murray’s Class used cuisenaire rods instead – genius!  Thanks Mark Chubb! (Follow him on Twitter @MarkChubb3)

Tangram Puzzles:
Again, having materials to play with is just fun for everyone! I do this type of puzzle just before Halloween, Christmas, March Break, and June….you know those moments when you need guaranteed fun! My favourite source for puzzles is I usually make a slide deck with the puzzle in black, followed by a slide with the solution. If I notice that the measure of frustration passes from an acceptable I-can-do-it-but-not-yet engagement to an I-am-ready-to-give-up level I ask if anyone needs a “flash”.  I advance the slideshow to bring up the solution and then backspace quickly just to flash a little hint and get kids back in the game.

Mash-Up Math Puzzles
These free algebraic reasoning picture puzzles are beautiful and fun.
My students love racing to figure them out…then we slow it down to discuss how we ordered the equations to solve them and prove why and how our solution works.  Maybe it is the colorful pictures that makes them appealing – I’m not sure.  But I do know that my students are always engaged and determined when I bring these out!  I usually create a slide deck and pop each puzzle on a slide.  I display it on the whiteboard and give everyone a minute or two to work it out on their own.  Then a student comes up to share their reasoning.  If they get stuck or confused they can hand it off to a friend.  Easy to set up and minimal prep.  Find lots of great free puzzles here!

Connect Four
Want to gamify some math problems?  This format has worked well in my classes.  Draw a big grid on the whiteboard, maybe 6 by 6.  Make teams of 4 (2-3 would also be fine).  Each team has 5 or so magnets of a certain colour.  (I use business card magnets and stick colored paper to one side).  Display a math problem or puzzle on the whiteboard.  When a team gets a question correct (or maybe the first 3 teams to solve a tangram puzzle for example) they can take one of their magnets and place it in the grid.  As play continues, each time a team is first to get a question correct they can bring a magnet into play or can change the location of one of their magnets in the grid.  The first team to connect four magnets in a row wins.  Here is an example that uses a mix of mash-up math puzzles, tangram puzzles, and questions on constructing 3D objects from their views. Feel free to make a copy and edit to fit your needs.  This “connect four” idea could be used with any type of math problem.  But for me, the combo of mash-up puzzles, and tangram challenges had the widest appeal.

June Team Challenge

My Connect Four game board in action. What team will get four in a row first?

Math Art
I first discovered Annie Perkins’ Math Art Challenge during our learning from home period last year.  It was just what I needed to de-stress and relax while still working with some cool geometric ideas and patterns.  There are 100 different challenges to check out.  I gravitated to the Isometric Cube Cut-Outs, Celtic Knots, and Apollonian Gaskets  since they were kinda connected to what we were learning and seemed manageable even from a distance.  I am also excited to try the Decagon Pride Flag and so many other projects given at

Scavenger Hunts
Take the kids outside!
This year I have made a few scavenger hunts that will get kids on the move while still engaging in some worthwhile problem solving.  Students can choose from three levels of difficulty.  Just print, post, break out the clipboards and go for it!

Central Tendency


Adding & Subtracting Fractions

Area and Perimeter Basics


Math Libs
This activity uses multiple choice questions where the solutions are connected to a word or phrase.  When the activity concludes, we string the words together to form a sentence or paragraph that we read out loud to confirm our answers.  Somehow arranging typical questions in this format has kids engaged.  I don’t claim to know why or how that happens but I have seen year after year that students enjoy figuring out the sentence at the end.  I change up answers so it includes their teachers’ names, or an inside joke.  In one math libs example, I had their Star Wars obsessed teacher sharing a coffee in the Tim’s drive thru with Yoda…they loved it!  It also helps if the sentence says there are popsicles – that you actually do have popsicles!  This activity is flexible.  Post questions around your space or outside to get students up and moving.  Or give each student or team a copy to complete electronically.
Here are a few I made this year:

Grade 7 Foundational Outcomes Math Libs

Divisibility Math Libs

I’ve Got One!
Lastly I’m playing around with a new idea…I’ve Got One!  In this activity, the prompts have many possible answers.  As students make their way from station to station, they view what has been given so far and try to find an original response.  Many activities that try to gamify math emphasize speed.  This could be a way to promote creativity instead.  I’m not satisfied with the name though – please send suggestions! Here are 20 questions that I pulled together as an example, but this idea could be applied to any topic in your math program.

Wow.  This has been quite the year.  We are almost there!  Try to take a moment to reflect on the challenges you have overcome.  Then finish with a little joyful math.  I hope what I have shared will help in surviving the countdown.

Tangram Challenges…so fun!

A Choice Engagement

Yesterday on Twitter, The Modest Teacher asked:  “Veteran teachers, what is something you did your first year of teaching that makes you cringe when you think about it?”

It got me thinking.  Well I had already been thinking…but it had me reflecting specifically on the challenges teachers are up against and how the management of those challenges can have unintended, and disastrous results.  

Teachers face deadlines, unmanageable amounts of curriculum standards, and overcrowded classrooms with children of varying readiness for the learning targets.  Decisions we make are an attempt to cope.  But think of the messages we may be sending.

My cringeworthy mistake didn’t happen my first year – I had too many hovering veterans to veer too far off course.  It happened maybe my fifth year when it became clear that in my classroom I could not assign the same work to everyone.  I had students with big gaps in their prerequisites and others that had mastered the outcomes already.  Some students were super strong in math reasoning but struggled with reading, while others were so traumatized by their previous math experiences that getting them to try – to just take a risk – was going to take a lot of encouragement and a focus on building their confidence gradually.  My solution:  (oh this is hard to write) I gave out folders to all ninety students that I taught, that were colored coded to indicate the “level” of math they contained.  (Cringe) The goal:  differentiate so that students get the math they need, at that moment, to move their thinking forward.  The message received:  You can only do this math.  You can’t do what this student can do.  This student is smarter or more capable than you.  I had the best intentions but the wrong method. (Cringe)

I knew instantly I had made a mistake.  I was ready then for mistake number two.  I spent the next weeks and months carefully crafting and editing tasks in an attempt to meet the needs of everyone.  I tried to predict the current understanding of all my students, then make all the worksheets I distributed look the same as possible, so that students couldn’t readily decipher where I had ranked them…still cringeworthy but what was the alternative?

Thankfully I attended a professional development session that year that literally changed everything.  Thank you Marian Small!  Dr Small spoke of opening up questions, allowing students to make choices about what numbers to use and what direction the task might take.  She reassured the crowd that students make choices that are appropriate for their current understanding.  That if we are confident enough to predict some likely avenues a task might take, we could engage students in new and exciting ways. 

I was skeptical.  Aren’t we all a little skeptical at professional development sessions?  Thoughts creep in when experts try to set you off on a new path.  In my head I was thinking, “This would never work with my students.”  and “This expert is too far from the classroom to really get what  happens here, in the trenches, day-to-day”.  But I pushed those thoughts aside knowing that what I was trying was not manageable and more importantly, it was not working.  I had a lot of questions for Dr Small, but the main one was, “Could students actually make their own decisions about the math that was right for them?”  I was doubtful.  But I was willing to try.  

One of my first attempts at inclusion and differentiation was a warm-up math routine I called Number of the Day. It is still one of my favourites. I give the answer, the students write the questions.  This was a big departure from my previous practice of 10 mental math questions advancing on a timer where only one correct answer was possible.  (Cringe)  I could say, “The Number of the Day is 5!”  Then students had ten minutes or so to craft their own questions.  I can hear you skeptics out there!  “But students won’t challenge themselves to offer something difficult!”  “Students will do the least amount of work possible to just get it done!”  Nope.  You are incorrect.  Students will surprise you with their motivation and creativity.  A well placed, “Oh…I see an example with more than one operation!” or “Wow!  I see some fractions and decimals in the mix!” will encourage and excite students to extend into avenues they hadn’t previously considered.  Everyone can participate.  And everyone will have the benefit of hearing and seeing what their classmates offer.  I might ask when we start to share, “Tell me something that you think everyone has on their list”.  This phrasing helps students that may fear their work is not difficult enough a way to engage and contribute without feeling less capable.  Then I might ask for contributions they think no one has – so much fun!  Students can and will learn from each other.

My directions during an independent practice day is another success story.  You know the struggle.  After a few classes of exploring a new concept it’s time for students to do some work on their own.  But what do you assign?  Some kids are still just scratching the surface.  Others might be ready to move on to something new already, having mastered grade level expectations.  And of course – many students are somewhere between these two extremes.  There is nothing worse than overwhelming a fragile, nervous learner with a mountain of questions to do unless it is giving a stack of repetitive basics to a kid that has mastered the concept already.  My solution?  Choose your own practice.  Now don’t wring your hands worrying, “What if students just do the easiest questions!?!”  They don’t.  I had the same concern.  I tried it.  I’m convinced.  Here’s an example of what I might post on an independent working day. This one is from our work with the Pythagorean Relationship.

Aside from the obvious benefits of each student getting manageable and appropriate work to complete, there are amazing results that I had not anticipated.  Engagement.  Conversations.  Not compliance which I had mistaken for engagement at the start of my career.  Previously I had students sitting quietly, working on the same questions, then correcting them together after an arbitrary amount of time.  (Cringe)  Now I had engagement.  A nice buzz of conversations (and mathematical arguments) as students figure out problems together.  Friends separate in favour of finding others working on the same tasks.  There is something very powerful about having a choice.  Because they chose it – they do it.  Sometimes students choose an inappropriate category but quickly adjust.  “I chose Practitioner, Mrs Sandford but after a few questions I knew I had to go back to try a few Apprentice questions”.   Awesome.  

Let the students choose.  They can handle it.  They know where they are right now in their learning.  And they know it is just a snapshot in time.  This current condition is not a permanent state – they will keep growing!  Make that the message they receive.

I am pretty sure that Dr Small offered much more in her two day workshop than I took away.  But just like students in the classroom, everyone in attendance was at different stages in their learning journey.  I was just a Novice – at the start of my work on differentiation.  All I knew was what didn’t work.  Her opening remarks gave me enough to think about and act on for a decade.  If I had the same talk today, I would take away something new again.  Hopefully my students will take away at least one message from my lessons:  I trust you as a learner.

The Secret Ingredient

Teachers often ask me for lesson ideas, projects or warm-ups.  When you’ve been teaching as long as I have you start to accumulate lots of stuff.  I’ll get a text or email saying, “You have anything I could use to introduce operations with fractions?” or “Have something for the end of the polynomial unit?”  The answer is often…yes!  Then I pick out those great activities and send them along.  You know the ones.  Everyone remembers those classes where the kids were not just engaged – they were having fun.  Not just understanding concepts – they were talking about what they noticed, proving their ideas, and feeling satisfied, confident, competent.  When those moments happen – you remember them.  And you want to share the great news – it IS possible to explore the required content and have a fun and rewarding experience doing so.  But sometimes an activity comes highly recommended and falls flat.  Why?  What’s the missing secret ingredient?

Yesterday a teacher messaged me saying, “I heard you had a really great activity for composite shapes – care to share?”  I lit up.  I DID have a great activity.  Check out this beautiful alphabet I found as a free download here

Found at

As I passed along this cool resource I found myself explaining what I did with it in one class and how I used it very differently in another…

In one class, this alphabet became The Measurement Challenge!  Find the three letter word with the greatest area.  Kids teamed up and got down to business.  Voices in groups were hushed so they wouldn’t give their strategies away to other teams.  Check this out Mrs Sandford – no one’s going to do better than this!  I remember there was a fire drill in the middle of the class.  Out on the field students were discussing the challenge.  Instead of trying to maximize their time out of class – they were itching to get back in so they could try out all their ideas.  Can we do this again tomorrow?  We want to keep trying!  Totally awesome!  The discussion later was even better.  I heard things like, “At first we thought…., then we realized…” Students told me about estimating strategies that they used because “it was a way to save time”.  Estimating was useful!  Who knew!?!  Getting students to estimate has always been a struggle for me.  In this challenge they did it independently – whaaat!?!

In my other class I used the letters a different way.  Students were asked to use measurement tools to accurately draw their name or favourite word or phrase.  We broke out the protractors, rulers, compasses, and bullseyes. We took our time and made our work beautiful. Oh the lovely math art we created! We found the areas as well and noticed interesting things about how the letters looked in the picture and how they appeared when we constructed them according to the given measurements.  “Is that what it means when we see ‘not drawn to scale’ on pictures?”  So much satisfying learning!  

So why the two activities?  That’s when the secret ingredient to a successful lesson became crystal clear.  Consider the students.  My first class was given a challenge because that’s what they liked.  They loved to compete.  They preferred working in teams.  Whatever the task, they were always trying to win.  I took a cool resource and framed it in a way that would appeal to their competitive nature.  My second class were artists.  Precise and detail oriented; slow and exacting.  They may have shut down in a competition scenario but thrived when given the same math work to do with a different spin.

Then, by displaying the work from both classes, I had a nice cross-over effect.  Some students from the challenge class created their own art when they saw the beautiful creations by my artists.  Students from the artist group wondered if the largest area found in the competition was the biggest one possible.  “I bet I could find a three letter word with an even bigger area Mrs Sandford!”  Me:  “Really?  Do it!”  Begin with their preferred approach, then spark a little curiosity and interest in other connections.  Kids start exploring on their own. Yes – On their own!

Now when teachers inquire about a lesson idea or resource, that’s the information I need first: Tell me about your students. The kids are the difference.  No matter how much you know the content, you must also know your students.  They are the most important (and shouldn’t be secret) ingredient.

What word or phrase will you draw?


This week I registered for my last two courses in my Masters of Arts in Teaching Mathematics, from Mount Holyoke College.  Realizing that this program was almost over brought up a lot of emotions.  Pride and satisfaction for sure but also disappointment – this amazing journey was coming to an end.  I miss Cohort 7 already!  A collection of educators from all over, varying in age, assignments, experiences and teaching environments, all working together to improve our understanding of mathematics and our ability to create the conditions that help children succeed in this amazing subject. I will never meet most of these educators in person – but I feel that the experiences we have shared will bind us together as a lasting learning community.  I started looking back over some of the reflections I wrote after different classes and courses.  Here is one reflection in a group I call, Play, Persist, Prove.

Today was another awesome experience in my mathematics journey.  Hours later I am still buzzing about how transformative these experiences have been.  So transformative in fact, that I feel compelled to write about it.  

Being a math teacher implies a certain level of success and know-how from my early days as a student.  Why else would I or could I become a math teacher?  Being of a certain age also implies the type of math education I received:  memorize your facts, watch what I do, now you do the same.  What most people know very little about is that math is taught very differently now.  Or they may know math is taught differently but may not know much, if anything, about the how or the why.  While some people (and even some math teachers) may think, “I learned it this way – and I got it!  Why can’t everyone still learn it this way?”  Those truly interested in effective math education know that, like everything else, when practice follows research, everyone wins!

Today, like most days with Cohort 7, we dug deep into a math investigation.  The context wasn’t beyond my grasp – we were discussing the sum of the interior angles in polygons.  But, like so many other math topics, I had a long before memorized solution, formula, or fact.  Today, the formula was just out of reach.  I hadn’t used it in a while.  It wasn’t something I ever thought about.  When I learned about this concept way back in junior high, the teacher just told me the formula.  I believed them.  I applied it.  We moved on.  What was it again?  I knew it was something to do with 180°.  Maybe the side measure of the polygon was in there?  Minus 2?  Guess.  Check.  Confirm.  Got it!  180(n-2).  But I wasn’t going to get away with reciting a formula today!  And I didn’t want to.  

Out came the pattern blocks, the power polygons, and the virtual manipulatives.  The directions:  figure out the interior angles of these polygons, be convincing, don’t use a protractor.  I was excited.  Time to play.  Time to persist.  Then time to prove.  There was no rush.  Work on your own, discuss with your group as you go or when you are ready.  I created a couple of models to assist in my investigation.  Can you see what I was trying to do here?  How could these models help or do they?  What do you think my explanation was that accompanied these models?  What would you create?

Knowing a full circle is 360 degrees, placing congruent angles together is helpful.
To increase the number of sides in these polygons by 1, I added a triangle.

The investigation alone was powerful enough.  But the seasoned facilitators, Jan Szymaszek and Zak Champagne, did not stop there.  After comparing notes and models in small groups, each group created a few slides to share with everyone.  Together as a class we scrolled through the gallery of representations, asked questions of the creators, noticed what was common, and combined any new information with our own.  It was awesome.  I saw another group’s model and relief and understanding washed over me. I was finally able to connect the formula I have learned and relearned so many times to a picture for figuring out interior angles.  Because that formula corresponds to a picture – that connection is solid.  I will never forget it again.  It is so powerful to be in that student role and make a discovery that is deeply meaningful to me.  I have that wonderful satisfaction and contentment that things do, in fact, make sense.  That everything is connected.  I don’t need to memorize it – I KNOW it.  What was interesting too is that my colleagues had similar experiences but not about the same models. Seeing and hearing about lots of ways was neccessary for the group to be satisfied.

How can I make sure my students have these experiences too?  Have the feeling that they know something – not that they were told it or heard it;  the knowledge and connection has to be theirs.  That is what I am left to ponder.  While I don’t have that all figured out just yet…I did take notes on the “teacher moves” of our facilitators.  What did they do that made this lesson such a success?  Here’s what I wrote as tips to myself about being a better teacher:  Listen and listen hard.  Stop talking so much and really listen.  Don’t be so quick to jump to conclusions about what kids do and do not know. Ask questions.  Put the life preserver away.  They’ve got this.  Give students a safe space to play, lots of tools and encouragement so they can persist when something doesn’t work right away and the opportunity to share, discuss and defend their proof.  Your job is to find cool problems to investigate with lots of hands-on irregularly wonderful mathy things – then be present at the moments of insight and an enthusiastic witness to their awesomeness.  

Hmmmm.  Let my students Play, Persist, Prove.  Yup…I can do that.

The Math Challenge Lady

Ok class, do you remember who this is?  
“Mrs Sandford – the Math Challenge Lady!” the kiddos chant in singsong unison.
Yup – that’s me.  I smile. 

I wasn’t always the Math Challenge Lady but this has been a strange year for everyone.  For me, it has been different since I am in a new role as a Junior High Math Coach.  Instead of focusing my attention on my own classes of thirty or so grade 7 and 8 math students as I had for two decades as a classroom teacher, I am assigned to schools that request a coach.  For about 5 weeks at a time, I work alongside three math teachers in their classrooms, with their students, toward their goals.  My new role, collaborating with other educators in pursuit of student achievement, has been incredible.  I have learned from and with some amazing teachers and been welcomed into rich, exciting, unique, and special learning communities.  I feel like I am making an impact – but it is different.

Those who know me will recognize that not having my own class this year is a difficult adjustment.  I miss both the room and the wonderful personalities that fill it.  I spend a lot of time and energy creating a physical space that is welcoming, comfortable, creative and interesting.  I want students to relax when they enter, have something fun and inspiring to look at, and feel like it is their room too.  Then comes the atmosphere.  Many kids have a visceral reaction to mathematics.  Love or hate or fear.  I want to expand that to include joy and curiosity, and provide experiences that bring intense satisfaction.  And I want students to want to be there.  I work really hard to build trust and create a supportive and collaborative community of learners.  This effort and attention pays huge dividends. 
I get to witness awesomeness, growth, independence, and persistence.  I get to observe vibrant youngsters figuring out who they are and maybe even play a role in how they see themselves.  And ok – I teach junior high so I witness a lot of other things too!  But the overall feeling I pursue and relish is that quiet satisfaction that comes when you see young minds buying in; when kids feel safe enough to try, to persist, to give a task their best shot. When you are present for that moment when a child has figured something out; when the lightbulb goes on…it’s pretty special. 
I need those moments. It’s why I became a teacher.  They come because I work hard to create the conditions under which they are possible.  My new position found me in a supporting role.  Would I still have those moments?  

As a classroom teacher, my very first day had to have a special math activity.  Something to set the tone for how things were going to operate on my watch.  Lots of deep thinking, conversations, revising, connecting.  An activity that guaranteed participation.  Easy enough to begin for those tentative or reluctant students but perplexing enough to be challenging for the confident.  We’d make mistakes together and learn from mistakes together.  We would have this first shared experience as a new community of learners and begin our year-long math journey.  Together. 
I didn’t know how to begin a school year without it.  Maybe I didn’t have to…maybe I just had to think bigger.  Expand that community of learners school wide.

I was lucky enough to have my first coaching placement at the school I have worked at for a gazillion years – Sackville Heights Junior High.  Best school there is.  When I asked teachers if a weekly school wide math challenge was something they might let me pursue – it was greeted with open arms.  I set right to work.  Each week, I created a math challenge, introduced it to classes with my coaching partner, and students had the week to complete their best work.   Winning classroom teams were awarded, we showcased some math awesomeness and tried to promote some math play and engagement on a larger scale.  I started to feel it – some sparks!  Flickers of interest in teachers as well as in students.  Joyful math moments!  It was still possible in this role!  The concept, however, wasn’t without issues.  Making time to introduce and work on the challenges was tricky for teachers.  So was the team idea.  With the stress of teaching during a pandemic, the last thing I wanted was for teachers to feel pressure to participate or to have one more item to juggle, schedule, or consider.  When my coaching block wrapped up and I moved on to another school, I decided to let the weekly challenge idea go.

Months later, I found myself back at my home away from home:  Sackville Heights Junior High.  As I walked into the building that first day back, students were stopping me asking if the weekly math challenges were going to start up again. I was surprised and excited.  Impact.  Interest.  Curiosity.  I didn’t imagine it!  But how could I promote this joyful math without encroaching on time needed for curriculum outcomes?  How could I make it optional, no pressure, but open and welcome to all?  

My solution?  I created a Weekly Math Challenge Google Classroom.  Kids – join if you like!  I remember sitting in front of the computer screen after I put in the morning announcements word of its existence.  I stared at the students section of my new Google Classroom. Would anyone join?  (Refresh)  How much of this math joy was my own imagination?  (Refresh) I needn’t have worried.  Currently 207 young mathematicians check out my weekly challenges and many submit ideas for feedback.  I get random inquiries about possible solutions and requests for hints on Saturday nights, Sunday mornings before dawn, and even during our Spring Break.  Math for fun.  Participation by choice.  Best marking I have ever done.

I’m still figuring out this coaching gig.  I’m not as great at working with grown-ups (yet) but I’m a learner too.  I’ll get there…I have to be patient with myself.  And my need to create and be present for those magical math moments?  Maybe it will be enough to help other teachers spark a little math joy in their students.  Maybe not.  But until I can figure that out  – I’ll be the Math Challenge Lady.  Wondering where to find me?  Just look for the sparks.

In the Beginning…

I remember fondly, my first real teaching job.  With an education degree and teaching practice under my belt, I was ready.  I had accumulated some student loan debt and needed a job.  Immediately.  And a harsh reality had set in during my short stint substitute teaching in my home town of New Waterford. The road to a full time position here would be long and bumpy. It seemed to be based more on years waiting than skill set, and steeped in politics and who knows who.  When I was passed over for a high school math term even though I was the only qualified applicant, I decided to accept the full time position of Grade 7 Math and Science Teacher in a small town in Alberta.  My first full time teaching position – woohoo!  I was excited, optimistic, nervous, naive.  I headed west.  

That year I was one of six brand new teachers at St Anthony’s, a K-9 school in Drumheller, a small town an hour and a half northeast of Calgary.  Drumheller is an amazing place.  Home of the famous Tyrrell Dinosaur Museum, and the friendliest bunch of outdoor adventure-seekers ever.  I knew instantly this would be a chapter to remember. 

The school board hosted an orientation session for new teachers in Okotoks.  I remember distinctly the message I took away with me.  As a teacher you are a role model for the students in your care.  Consider carefully the gravity of that responsibility.  If you need help, ask for help.  Plan carefully and with purpose.  What will the teacher be doing? What will the students be doing?  What routines will you put in place to make that a reality?  Those in charge had high expectations and they wanted us to have high expectations as well.  

Back at St Anthony’s the administrators wanted to know more about me.  What were my talents, interests, and special skills?  Assuming we were just getting to know each other I spoke of a youth spent as a competitive figure skater, more recent experience as a track and field athlete, and interest (but not necessarily talent) in art.  Later that day I received my schedule.  Along with the Grade 7 Math and Science that I expected, I had a few extras:  Art, Drama, and Fitness For Girls.  Awesome.  Along with the core subjects, students chose “Options” classes as part of their schedule.  Courses varied year to year and term to term since staff designed their own programs around their passions and student interests.  Pretty great right?

Staff were paid to be in school for over a week before students arrived and not just for content planning.  We had a fascinating talk about colour and environment.  What colours promoted what moods?  Creative and calm classrooms had lots of blues and greens – we decorated accordingly.  I remember the admin coming around to check in and see if our rooms were welcoming enough for our new little families.  Too much red?  Ok..yes of course I will change it.

I remember fondly and in detail each and every student I taught that year.  Their personalities, likes and dislikes, the sound of their laughter.  Yes – I had a lot of work.  Hours of meticulous planning.  Lessons and support materials were expected for every class so admin could pop in and join in with the lesson.  But I always felt as though we were in it together.  All of us.  The staff, the students, even the parents. From the Monday Morning Muffin Meeting to the wind down Friday dinner and drinks at Stavros Pub, there was definitely a family vibe.  It also helped that I only had 17 students in my classes.  SEVENTEEN!!!  Can you imagine how well you can get to know your students, how much easier it is to do projects, field trips, and literally anything and everything when it is 17 and not 30+?

Yesterday I reached out to a few of my former students and connected with a couple.  Was it my imagination?  Was my first year of teaching really as awesome as I remembered it?  I was so pleased (and relieved) to hear that my students (now in their thirties!) also remember the laughter.  It’s amazing the fun you can have learning together with the right set of priorities.  Create a welcoming environment and share your passions with your family.  Work hard, plan carefully and have high expectations of your students and of yourself.  Then enjoy the fruits of your labour. Even after all these years I find myself striving for those connections, that teamwork that was prioritized my first year. I feel so blessed to have begun my career the way that I did.

The Realy Teacher

My mom was visiting this week and we got to reminiscing.  A math teacher herself, we always have a lot to chat about.  Today we were laughing about my first day as a “real” teacher.  Well maybe not a real teacher – I was reminded enough that first day that I was not, in fact, the real teacher.  But it was my first day as a substitute – a paid gig.  I was excited and nervous, but mainly excited.  In for the day in a grade four class in my hometown of New Waterford, Nova Scotia.  Grade 4’s were scary.  Younger than my comfort zone and probably super attached to their “real” teacher and their routines.  I was going to mess it up.  I was going to mess it up huge.

I was shown a bright, beautiful classroom with rows of tiny furniture.  Excited.  That’s how I felt.  This is what I wanted: my own class to decorate and organize and breathe life into.  Then  I found the sub plan.  It was a book.  I could feel my pulse start to race…back to nervous.  There was no time to read any of this.  I saw the schedule, a list of spelling words, found a novel, and some math questions.  We would be fine?  We would be fine.  I took a few deep breaths and practiced my opening introduction and reviewed my expectations out loud to the empty desks.  A throat cleared.  A teacher from the neighboring room was there to show me where to collect my students from the line up outside.  She grinned as I turned a deep purple.  “You’ll do fine,” she reassured me.  

The day was a roller coaster of ups and downs.  I used the wrong colour star stickers on the spelling chart and “ruined the whole thing” according to a little blond girl with curls.  A few minutes before recess I tried to get the kiddos into their outside clothes and out to the playground only to discover that they needed easily five times the minutes I allotted to get ready.  The bell rang to come in as we were making our way out.  “Substitutes!” one little boy sighed rolling his eyes.  Hilarious.  I decided to take them out anyway and we had the playground to ourselves.  “Best recess eva!” a few said.  Back inside, the students couldn’t agree on where the teacher left off in their read aloud book so I read one that I had brought, Socks for Supper by Jack Kent.  We talked for a long time about the story and drew pictures until lunch time.  Just when I thought I was winning them over the same know-it-all girl told me, “You were supposed to take us to the bathroom 15 minutes ago to wash our hands!”  Yikes.  Tough crowd.

The afternoon was better.  The kids had music and gym and I went along with them as part of the class.  They thought that was hilarious and awesome.  I thought they were hilarious and awesome so of course I wanted to hang out with them.  “You’re better as a student”, one of the kids said.  I decided to take that as a compliment.  The last class of the day was math.  That seemed odd to me.  I probably had that wrong.  No – that was the schedule.  A closer look revealed that the end of every day was math.  Interesting.  I told the kids that the best part of the day had finally arrived – math!  They groaned.  Some even looked nervous.  Wow.  Weren’t these kids too young to have an opinion (never mind a negative one) about math?  Apparently not.  I powered through.  I decided to ignore the lesson plan of drilling times tables and played a math game, had them guess my “Mystery Number” and then write as many questions as they could that had an answer of 5.  We ended the day on a high note.  Lots of sharing, happy faces, relaxed postures.  “You are better at math than the other things”, my biggest critic relented.  

Learning from my earlier recess debacle I had those kids packed up and ready for the bus in record time.  As they filed past me one quiet little girl shoved a note in my hands.  It read, “Thank you for teaching me.  You make my work easier.  Too bad you weren’t the realy teacher.  From Sarah”. 
Yup – a pretty perfect day.

On Becoming a Math Person…

I may be a middle school math teacher now, but my math story began a long time ago.  It didn’t begin as a love story, but it could have.  It should have.  I am one of the lucky ones.  My experience with math keeps getting sweeter.  While it admittedly took a few decades longer than ideal to ripen, I stuck around long enough to reap the rewards.  There are others out there that did not make it:  parents, teachers of other subjects, friends.  We all know them.  They say things like, ”I’m no good with numbers”, or “I’m not a math person”.  Others nod knowingly like “math people” actually exist and sympathize that they too weren’t lucky enough to get the math gene or whatever the latest urban myth is about being capable of curiosity, reasoning and relationships.  So what can I do to change perceptions?  Perhaps by sharing my story and trying to connect with other elusive “math people” we can figure this out together.

 I have always been good at math.  My teachers and friends told me this was because my mom was a math teacher.  I believed them.  It didn’t matter that my math work at home was probably the standard forced homework that everyone did at that point in their lives.  I was a math person – it was settled.  Grade 3, Mrs Edwards class:  I remember taking my turn, standing in front of the class, reciting my multiplication tables and getting a gold star on the classroom chart.  I did so well in my classwork I got rewarded by being allowed to tidy up the coatroom.  Message received:  If you can memorize some facts, follow some steps and get through this drudgery you can have some actual fun like straightening the shoes!

Junior high was more of the same.  I knew I could do the work.  But it was just that – work.  In high school I took the advanced classes.  I analyzed graphs, knew my trig identities, applied the right procedures and found the correct solutions.  I checked the necessary math courses off my list so I could move on to the fun stuff.

Fast forward a few more years than I care to count and I am a math coach and teacher.  I am still a math person.  Or maybe finally a math person.  Not because of my mom, my multiplication facts or my memorizing skills.  I am happy and proud to say that I have found the joy, satisfaction, and beauty in math.  I now know there is so much more to mathematics than what I had learned as a young student.  What happened between then and now?  Well lots!  I have had some experiences that have fundamentally changed what I believe math education could and should be.  I guess you could say I have actually become the math person everyone believed me to be from the start.  In the posts that follow I’ll share some of those experiences.

I plan to use this venue to put some ideas out there, share some lessons, and reflect on their success.  The best collaborations that I have been a part of have been motivated educators figuring things out together.  Want to work on something?  Reach out!  I hope you will stick with me – give me some feedback.  Share your ideas and experiences too.  Let me know what I should still consider in the lessons that I share, or what improvements or adjustments should be made.  I am committed to getting better.  Committed to learning.  What you will find here are lessons that offer students choices and challenge learners to prove, discuss, and persist.  And of course, perhaps most importantly, my creations will promote joyful mathematics through play.