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As close as it gets

I’ve never been great at estimating…I can admit it.
I remember my Dad asking typical Dad stuff like, “How many kids were at the dance?” and I would literally have no idea.  Over the years, I have noticed this weakness in my math game and wondered a lot about what experiences are needed to feel good about estimating.  How close is close enough?  How important is estimating anyway?  When do we need an exact answer and when will an estimate do?   The other thing I have noticed is how satisfying it is when you can check and see how close your estimate is and how frustrating it can be without that verification.  

In the Grade 8 Mathematics Curriculum used here in Nova Scotia, estimating and determining reasonableness of answers is part of everything we do.  However, there is one outcome in particular that is focused on estimation:  Find the approximate square root of a whole number that is not a perfect square.  I’ve spent a lot of time trying to get this topic right.  Maybe it’s because estimating has been a struggle for me personally or maybe it’s because I’ve seen this topic become calculator driven without a solid understanding first.  In any case, my focus, as with all math topics, is to make this one very visual.  We stick with models and manipulatives first to make sense of the concept.  Knowing what I do about the satisfaction of verifying the accuracy of estimates, I design tasks that allow students different ways to test.

At the start of the year, students spend a lot of time investigating squares.  We break out the square tiles to find what numbers can be represented as a square area.  We discuss terms like perfect square and square root and label pictures to match our concrete models.  We look for patterns in multiplication charts, and patterns in the differences between consecutive perfect squares.  We notice common traits in terms of numbers of factors and how the prime factorization might look for a perfect square compared to other numbers.  We dive deep.

With that base of knowledge firmly in place, we are ready to find the square roots of numbers that are not perfect squares.  My first investigation is to find the square root of 18.  I chose 18 intentionally.  It is a number that is manageable with manipulatives and is not bang in the middle of two perfect squares. (This avoids the knee jerk 4.5 response.)

I ask students, “What is the square root of 18?”

Over the years this opening question has taken on a number of formats based on the students in front of me.  I could choose to facilitate a whole class discussion with a chatty, comfortable bunch.  In some classes, I have suggested that we think on our own for a minute then turn and talk to a partner before we share out as a class.  These days I might send random groups of three to a vertical non-permanent surface to see what they can think through together.  After many years of asking this question, I always have the same responses:

“There is no square root – 18 is not a perfect square!”
“Well…there has to be a square root – it’s just not a whole number.”
Then “Ohhhhhhhh!”
(more scribbles and chatting…)
“It has to be between 4 and 5!”
Me:  Why?

Then with their picture or square tiles, students show me their 4 by 4 square with 2 squares left over.  
Me:  So how can we get those two into the picture?

At this point, someone usually suggests that the other two tiles could be spread on the length and width.  (More “Ohhhhhhh!” and chatter and scrambling)  One student usually tries to be “super helpful” and break apart, bust up, or tear my square tiles to make the model while I stand lamenting out loud, “It’s too bad we didn’t have a better manipulative for this….squares we could just rip”.  At this point I am usually holding a stack of square post-it notes, absentmindedly flipping through them as I search for the answer to this problem…heehee.  The students clue in, come grab some post-its and the investigation continues.

Modeling the square root of 18. I can make a square with 16…now what?

Here we see the largest perfect square we can create with 18 post-its – 16, represented in a 4 by 4 grid.  We have 2 post-its left…what should we do? Students figure out what to do with the extras.
They break up the squares so they can be evenly distributed on the length and width. 
Me:  “Why should we spread these equal pieces on both the length and the width?”
Students:  “So it stays a square!”

Post-its can be cut easily to create a model.


Some students spread the pieces around the whole square, or cut up the post-its randomly and not equally.  Let them.  They will discover on their own what methods work best and make adjustments as they go. Lastly we check out our models.  We know from our work with perfect squares that the side length is the square root.  What is our new side length?  How can we tell?  Label it.  Is this a good estimate?  Convince me!  How can we tell from our picture that this is an estimate and not an exact answer?

Complete!

I love this investigation.  I see a lot of light bulbs turning on as we work.  I am mindful too that students don’t have the same comfort, ease, and experience with scissors and glue that students their age used to have, so we pair up if we need to and really take our time.

When we are finished with this example I ask if this method of modeling would work every time for every number.  And “…how many more examples do we need to do before we are comfortable estimating with models?”   At this point I’ve also been known to exclaim, “Man…if we have to do this every time we estimate square roots, I am going to need to buy a lot more post-it notes!”  Some students look at my stacks of post-it notes (I do have a lot!  I use them for everything!) and say we should be good for a bit, while others tell me they could probably draw it without using post-its or even picture the model in their head and get an estimate.  They often recognize and point out the limitations of the manipulative, knowing that larger numbers would present a problem.  Yasssssssss!   Students are then directed to choose some square roots to find.  I may suggest some numbers like 20 and 13 so we see some different fractions but leave it up to students to decide what they need to do to feel confident with this method.  Some use post-its, some sketch, some picture their answer and write a description – all good!  I like them to test by squaring their estimate to see if their solution is valid and we consolidate by sharing what we have learned.  Sometimes we just talk about the examples we tried, while other times students may be ready to talk in general terms.  With some confident learners I might further nudge with, “How could we be sure of the fractional portion of our estimate without drawing or constructing a model?”  Some students note that the numerator is the number of leftover squares while the denominator is the number of spots they need to be spread along or the sum of the length and width.  Others won’t make that connection yet but we try lots more examples to test theories. 

In the days that follow, we might do a few more typical things like using number lines for estimating and using systematic trials to get closer and closer to an answer while discussing that an exact answer may be out of reach.  

Another favorite activity to guide students in confirming their estimates is the one I do with tilted squares.  For this you will need 1 cm square dot paper.  I use this link.  When you print it, confirm with a ruler that it is in fact 1 cm dot paper.  Otherwise the lesson will not work.

I start by asking students to pick a perfect square under 25 and draw it.  We discuss again that the perfect square is seen in the area and the square root is the length of the side.  We use our rulers to confirm what we already know.  Next I remind students that we are estimating square roots of non-perfect squares, so we should probably draw some of them…can we?  Let’s draw a square with an area of 8 square centimeters.  I say, “Be exact!  We want to be able to check our estimates by measuring like we did with our first example!”  Students usually try their luck drawing squares with the same orientation.  To their credit, they try drawing squares with side measures just under 3 cm (yay!) but say that it won’t be exact or it is impossible to be exact if the number is not a perfect square.  That’s when I blow their minds with my picture…

Doesn’t this square have an exact area?

What area does this model represent?  8?!?  What?!?  How can that be? 8 isn’t a perfect square!?!
How can we confirm the area is 8? Here a student will come up and box out or count up the squares for the students that don’t see it yet.  They show that partial squares can be combined to have an exact total area of 8 square centimeters.  Can we estimate the square root of 8?  How could we use our model to check our estimate?  

Estimate the square root of 8…then check by measuring.

I love how simple this is. I love how concrete this is.  I love how the looks on student faces go from perplexed to curious to confident.  I often do not have to prompt students to try other tilted squares – they just do.  Students can estimate the square root of their number in any way that makes sense to them, then put a ruler on the side length of their square and see how good their estimate actually is.  I love that the immediate feedback they receive comes from their own measuring.  No need to always resort to a calculator for “the correct answer”.  I also love the added challenge of drawing titled squares and confirming their areas.  What other squares can we draw?  Are there numbers that can’t be drawn?  Great estimating practice (which we need for this outcome) with a different spin.  Try it!

I have used this presentation to consolidate both activities after the fact . I find a digital version especially useful for those students whose lack of precision may have impacted their understanding. These lessons are, of course, just two within a bigger picture that focuses on problem solving using estimating square roots in context.  We figure out the best fencing prices for yards, determine what square rugs fit in what rooms, and all sorts of other scenarios.  But wherever our work with estimating square roots takes us,  these two investigations are the ones that I make sure to include. So what’s the appeal?

In the case of the post-its, perhaps the appeal is the concrete nature of the model – so obviously an estimate with that missing corner piece.  Maybe it’s the chance to use fractions as the more obvious choice for side measure instead of the regular decimal default.  

The draw of the tilted squares is its simplicity.  And of course the chance to do some measuring.  This IS a measurement outcome after all!  The idea that your own measuring provides the immediate feedback on your estimating skills – well that is just pure gold.

Lastly, I love how satisfied and confident students are after these tasks.  They can often explain, show and verify and are willing to create their own tests to discuss with their peers.  It’s one of my favorite examples of the Play…Persist…Prove ideal that I try to bring to each and every math class. Students are completely in the driver’s seat, challenging each other to draw a certain square area or seeing who can get the closest estimate.   During these moments I often ask, “Are we there yet?” to see how students feel about their estimate.  With these activities students often reply with confidence, “This is as close as it gets!”

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First Week Frenzy

Preparing for that first week of school is no small task.  There are so many things to consider!
As teachers, we want all students to feel safe, welcome and valued.  We want to set the tone for how our classes will operate, and work to build a community of learners that collaborate, create, and problem solve.  We want to model respect for self, the space, and for others and do all of this while building relationships, promoting curiosity and having a little fun.  A tall order indeed!  

In spite of being in the business for more than two decades, this start-up process remains incredibly hectic.  Classroom teachers scramble to secure the appropriate number of tables and chairs that were moved out during summer cleaning.  Cords to technology, remotes, and whiteboard erasers need to be relocated.  Devices, textbooks, manipulatives and whiteboard markers must be counted and distributed. Rules, procedures, and policies are revisited and clarified so that staff share a common vision . Staff changes mean you may have different courses to teach, different partners to collaborate with, and different administrators in your corner.  Beyond the school walls are the school communities to consider.  Connecting with families and sharing your hopes, expectations, and enthusiasm for the year ahead is yet another essential item on the things-to-do list.

In order to navigate this first week frenzy with my sanity intact, I have always attempted to checklist my way to readiness.  Once my physical space is organized, technology tested, student files perused, Letter to Families updated and course summaries reviewed…Whew!  I revise and ready my list of first week activities.  While I often test out something new every year, there are a few tried and true lessons that always make the cut.  Here are a few of those favorites and why I love them so much.

My super fun name tents!

Welcoming Students

When students arrive that first day they may be nervous.  Behind that too-cool-for-school exterior, is often someone uneasy about being in the wrong spot, not being with their friends, and not having a seat.  My solution:  I try to greet students at the door, list in hand for confirmations.  I have another large class list on my door and assign seats inside with name tents.  Everyone has a designated spot and can confirm quickly that they are in the right place.  

Now let me talk about the name tents!  I use https://mathonyms.xyz to create math-y versions of student names.  Just type in a name and download the image.  This is time consuming if you create them for multiple classes but the response I get from students makes it so worthwhile! On the inside of the name tent I have tried different things.  I think my favorite is the student comment/teacher feedback idea from Sara VanDerWerf found here.  It can really help you connect with individual students daily that first week.  Here is my template.  I usually print these on cardstock in bright but readable colors.  Easy for me to get to know student names and easy for students to find their seat.  While the prompts on the inside often change, name tents are a must for me every single year.

When students get to their seat they often have a welcome letter to bring home to their families, a student information sheet (contact information, strengths and challenges, interests and hobbies, etc), a few copies of their schedule (one for their binder, one for their locker, and small one for inside their phone) and a bunch of other forms from the office.  Having something on the desk to look at seems helpful for those students that may not have a friend in the class.  They can busy themselves with shuffling through the contents while waiting for their teacher to begin.  Students often tell me later that they could tell right away that I was organized – a good message to send right away too!  I try to make my room a colorful but calm space.  I post schedules, a calendar, and my learning targets. I write the agenda for the day including times on the whiteboard so students can see when they get their lockers, when they may have specialists, and exactly how long they can expect to stay with me and precisely what we will do. (In junior high we often have a different schedule those first few days to allow for extra time with homerooms). I also list daily what supplies students need for my class.  That way I can start one routine right away by saying, “I see some students have noticed my Be Ready List and have only what they need on their desk.  What a great way to get started!  If you haven’t noticed yet, please look over here and take the next 2 minutes to get out just what you need.  I have extra supplies here in case you need ‘em.”  This gives me the time I need to do attendance.  I usually circulate through the class and check student names off my class list on my clipboard instead of reading off names in front of the class.  That way I can confirm preferred names, double check pronunciation and pronouns and avoid making mistakes that may embarrass or offend. I am hopeful that these few little measures help students feel considered, safe, and welcome.  Next we start the get to know you process…

Getting to Know Each Other

Introductions and Setting Norms
I often begin with a slide deck to introduce myself, and highlight just a few items that are really important.  I talk about what I need from them in order to do my best work as their teacher:  basically I outline my expectations.  Then I ask them a few prompts so I can find out what they need from me and from each other.  I use post-it notes – about 3-4 for each student.  I ask, “What is something Mrs Sandford can do to help you be successful?”  Another prompt is often, “Everyday our class should be…?”  I ask, “What can your classmates do to help you be successful?” and “What is something you wish your teacher knew about you?”  These prompts consistently help me learn what students value, what they believe worked (or didn’t work) in the past and what they are hoping for (or dreading) in terms of class climate.  I have also asked questions like how they like to celebrate their achievements, and their favorite class appropriate treat, but these two could be better as prompts inside the name tents or as part of my student information sheet.  The post-its I collect on a poster board giving students a chance to get up and move around a little.  Later, I use the main themes to create a list, drop them in a slide deck and open with them on day two.  I ask students if there is anything we should clarify, add, or subtract.  We edit and pull it together to make our class expectations.  Once finalized, I write ones we have endorsed on my thought bubble post-its to create a poster to display in class.  Creating our expectations as a group really helps to bring us together as a class.  

Figure Me Out
In my introduction slide deck I have a slide or slides that asks students to Figure Me Out.  This requires that students do some math but offers choices in that they could start at any section they like and may not have time to get to all of them…that’s ok!  Students learn a little more about me and we can review a few math basics along the way.  I learn a few things too by noticing who volunteers to answer and who doesn’t.  I notice whether or not students are used to just giving an answer or are accustomed to also explaining their reasoning.  Then I ask students to create a similar page about themselves.  This could be a slide in a class slide deck, but I usually stick with paper on day one.  Chromebooks are often not ready for student use, and logins and passwords may or may not be remembered.  In this case, paper has been easier.  I may challenge students to use multiple operations, fractions and decimals, and showcase some math terms as they share some facts about themselves.  Facts range from lucky numbers to number of broken bones, provinces visited and number of pets.  It is amazing how creative students can be when given lots of choices.  Students pass these in and I use them for warm-ups (or cool-downs) for the next week or two.  I ask students the day before if they can host the warm-up.  We review the two or three prompts that they like the best and make sure the answers are what they expect them to be.  On the day they present I project their prompts using a document camera or by taking a picture of their work and dropping it into a class slide deck.  It becomes a low stress, fun, and personalized activity that gets students presenting and up in front of the class early in the year.  These Figure Me Out prompts come in handy when you have an extra 10 minutes here and there to fill as many students don’t mind presenting at a moment’s notice, especially when they have confirmed their answers in advance.

One of my Figure Me Out prompts

Venn diagrams
This is a fun activity to get students up and talking at the whiteboards.  I randomly create groups of three set up at a whiteboard with a marker per group.  There they create a Venn diagram with a student representing each of three circles.  They try to find entries for each section of the diagram.  Maybe everyone loves pizza but only two students have pets, etc. Later we circulate to see the work from each group and hopefully learn a few new things about each other:  what we share and what makes us unique.  This activity gets students up, moving, and talking.  It serves as a way to introduce a group work method that will happen frequently in my class but is not stressful in terms of content.  I usually take pictures of each diagram so I can save it and take a closer look later.

Modeling Group Work

I can’t say enough about this amazing activity by Sara VavDerWerf.  Not intimidating and highly effective.  This link provides the how to and the print-out required to complete the task.  Take the time to read it all.  I’ve seen this done well and not so well and reading the post makes all the difference.
Read more here:

Trust me…you will love it!

Highlighting Persistence

Dividing into Squares
I do this activity during my first grade 8 math class every year.  It requires persistence, multiple attempts, lots of erasing, etc.  It highlights for students that mistakes are expected and can be a welcome path to satisfaction.  I engage students with this activity after gifting them a fun eraser and talking about the importance of making (and being open to making) mistakes.  Their new eraser in hand, we begin.  Check out my explanation video and find the handout I use here.

Tangram Challenges
I love using tangrams.  Students have a chance to engage with an area of math that does not require formal calculations or number manipulations. Puzzles often showcase the math skills and reasoning of a different set of students.  I use images and solutions from https://www.tangram-channel.com/ to create a slide deck.  Advancing then backspacing quickly can flash the answer on the screen for a hint when or if you feel the persistence is wearing thin.  Tangram puzzles are engaging and appealing.  After testing this activity with hundreds of students I rarely have anyone reluctant to participate.  I love that students have math to hold in their hands.  This activity requires persistence and getting to a solution is intensely satisfying. When I travel with one bag to school, my personal class set of tangrams is always included.  It is my go-to math activity when I am covering a class unexpectedly and a welcome change during those predictably hectic times during the school year (first week, week of Halloween, week before Christmas, etc).  Here is one of my many slide decks that I use to engage students in solving tangram puzzles.

Solved it…yes!

Promoting Organization

As mentioned, I make sure to post a schedule, agenda, learning targets, and materials needed each day.  At the start of the year, I also try to assist students in organizing their binders.  Organization comes naturally to some students but must be taught and retaught to others.  I usually give students some options, knowing that there isn’t one best way to do things, but stress that finding a way that works for them can really make a difference in their success.  One thing that students really seem to like is the coloring sheets I offer as subject dividers.  Found here, I print them on white cardstock and hand them out as part of the set-up process.  We may take some time to color them…we may do a little here and there during the week as time allows.  I love coloring…and it is great for reducing first week jitters.  

Wouldn’t this make a great divider or cover page?

Accepting Feedback

The last thing I will highlight today (or I will never get this post posted!) is the appreciation I get for asking, accepting, and acting on student feedback.  Aside from the back-and-forth comment/response given by the name tents mentioned earlier, I administer quick exit tickets (electronic using google forms or just a scrap of paper) that ask students simple things: What went well today?  What should I or we do differently tomorrow?  What was the most important takeaway message from today?  Is there something you are unsure about?  Is there something else you think I should know?  I share a few common themes the next day and discuss what I might alter or try in response.  Once students see that their opinions are seriously considered, the responses become more detailed, relevant and helpful.  This goes a long way in students believing that we are in this together and that their voices matter.

I know this post is a little scattered and I am sure there is a lot I am forgetting.  Like every educator at this time of year, my mind is busy with ideas, checklists, and other concerns.  I’ll try to look back on this entry and keep revising as I learn and remember ways to start the year off right.  As for my first day today…it was amazing!  I had the chance to reconnect with some of my former students, now in Grades 11 and 12 at Millwood High.  I have welled up with pride and excitement so many times today seeing incredible young people enthusiastically starting off a new school year.  Students were happy to see me…yay!  They shared their hopes and fears, dreams and plans. We chatted…we laughed.  My heart is full.  My students have positive memories of our time together back in Grade 8 – I can’t ask for much more than that. 

My advice for educators?  Take deep breaths.  Drink lots of water – but not too much!  Remember you can’t really go to the bathroom whenever you want.  Don’t be too concerned with covering content just yet.  Select a few meaningful activities. Find ones that build relationships and showcase how you operate. Choose those that highlight what you value and help reveal what matters to your new communities.  Hopefully there is something here that might take the frenzy out of your first week.  Have a wonderful start to your year everyone!

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Battling Butterflies

In just a few days I’ll be back to school.  Butterflies?  You betcha!
Usually this is a busy, fun and exciting time for me…usually.

As a classroom teacher, I had my process.  That process often began August 1st when the class lists became live in Powerschool and I could access my schedule.  I’d look hopefully for that Friday afternoon prep period (then console myself when I didn’t get it) as I copied my schedule in my new teacher planner.  I’d read up on my new students and make class lists knowing that they would likely change a bunch of times before opening. I’d start on my name tents, update my Figure Me Out Activity and edit my introduction letter to families.  I’d round up my post-it notes to prepare for my conversations to establish classroom norms, print my Dividing into Squares handouts, and hit every dollar store in the greater Halifax area looking for fun colorful erasers and super cool supplies. 

I’d reread my favorite articles and rewatch inspirational videos – the ones that help me focus, the ones that get me psyched, and those that remind me of the important best practices that I need to have at the forefront of my mind.  By now I would have been in my classroom several times setting up.  I’d likely be checking the forecast to try and find another rainy (or at least cloudy) day to head back in to continue the process.  I’d have my girls in tow, lugging in bags and boxes and “helping” by writing welcome notes and math words on my whiteboards.  My cart of math paper would be ready with stacks of lined, unlined, graph, square dot and isometric paper all good to go.  I’d try to reimagine how I might display my math tools and manipulatives for easy access so that students can see the possibilities but make their own choice about what tool fits the job.  I’d arrange and rearrange my furniture and scour kijiji and FB Marketplace for new stools, tables and benches. I’d be checking in and catching up with friends and colleagues while making lists with my new pens on my new stationary and checking things off my new things to do list one by one.  While completing my housekeeping duties like testing student locks for lockers and pre-writing receipts for student fees, I’d be thinking about and reflecting on the first few days from years gone by.  What ideas and activities worked to build our class community and set the tone?  What could I or should I do differently?  As a classroom teacher – I have my process:  a way to prepare that makes me feel ready, confident, and excited.  Butterflies?  Yes!  But the good kind.  

Times have changed. This September marks my third year as Junior High Mathematics Coach.  I don’t have a classroom.  I’m a guest in several classrooms for 5 weeks at a time.  My usual prep process does not apply.  (Here come the butterflies again!)  In a desperate attempt to feel ready, I find myself panic buying random things that I would love for my classroom.  New stackable containers for manipulatives, pencils with fun messages, magazine holders for student journals…until one of my girls reminds me I don’t have a classroom anymore.  My husband suggested I could get my need for organizing and preparedness satisfied by tackling the corner of our unfinished basement where I dumped my entire classroom packed into bins in June 2020.  Nope – not ready for that yet.  (Nice try Kyle!) What to do…what to do…

Thinking back I didn’t feel this antsy and unsettled starting off my last two years.  I wonder why that is?  I guess circumstances were a little different.  My first year as a coach I started at Sackville Heights – my professional home for over a decade.  I knew the place, the people, the content, the bell schedule.  The butterflies were good.  Year two I was starting at Cunard Junior High.  A lovely welcoming school I had the pleasure of spending time in the year before.  Scheduled to work with a few teachers I knew, in grades where I had a lot of experience and a ton of resources to share – good butterflies.  This year is very different.  New school.  New teachers.  New grade.  New course.  Butterflies.  Lots of butterflies.  Are they good butterflies?  

I fought the urge to reach out to a teacher I am working with (no one wants to receive emails about work before the summer is over) and instead tried to distract myself with just a peek inside one of those overflowing classroom bins in the basement.  I’m glad I did.  Right on top, the last thing that got packed away, was my folder of favorite notes and cards.  Proof that I did make a difference!

My folder of pick-me-ups

I have cards from students, letters from parents, and emails from colleagues that I printed before they got accidentally deleted.  While I don’t save every note I receive, the ones I do have really matter to me.  A mismatch of formal and fun, I have a few gems that are truly special.

The first thing I pick up is a paper copy of an email.  A teacher I don’t know took the time to write to me about my most challenging student from the year before.  This student moved to another area and mentioned to his new teacher how much he enjoyed my class.  This note mattered so much to me – I remember how emotional I got when I received it.  I poured my heart into connecting with this kid and wasn’t sure that I had succeeded until that email arrived.  Other treasures I saved included silly notes from students.  I laughed when I found one saying, “I am always nervous about a new year…especially a new math class.   But once I saw you with that dumb eraser ring I knew it would be ok”.  Side note:  This was about my eraser ring I wore on Day One a few years back.  When a student finally asked about it I said dramatically, “…I’m wearing this ring to celebrate my commitment to making mistakes”.  Groan.  I mean kids expect bad jokes from math teachers right?!?  I opened another enthusiastic note from a math coach that worked with me in my classroom, and a copy of a reference letter from one of my math idols that made me feel like a rockstar.  Reading through my folder was the best butterfly medicine ever.  I laughed…I teared up a little…then I put them away.  But not back in the bins.  I brought my folder to my desk so I can easily shuffle through it again when the butterflies strike.

I am committed to making mistakes!

I have a new realization too.  It took me a long time to get to a place where I was proud and satisfied with the work I was doing in my classroom.  It will likely take me a few more years to feel confident and impactful in my new role, and that’s ok.  Until then I need to revamp my get-ready process.  To that end I have joined a Math Coaches Professional Learning Network and started connecting with more professionals in similar roles on Twitter.  I have dug out some favorite professional readings for coaches and reread my highlighted portions.  I’ve shared some first week back ideas with teachers that have reached out and focused my love of buying fun supplies on purchasing what my two girls may need, want, and love. I am also actively avoiding Chapters since their back-to-school organizers, stickers, pens, post-its, and planners are definitely my favs.  I will let go (or continue to attempt to let go) of the need to have everything organized ahead of time and embrace the collaborative nature of the coaching process.  As for my folder of uplifting positive memories and messages – it may be time to pay it forward. This year I plan to make the time to write a few notes to educators that they might hang on to and reread when in need.  Teaching is a tough gig.  Maybe one of the benefits of my new role is that I have a little more time to remind teachers of their wins and support them when they need it.

Do I still have butterflies?  Yup.  No matter what my role is as an educator, I will likely battle butterflies at this time of year until I retire.  But maybe now there’s more good butterflies than bad.  And I’m pretty sure that a beverage on the beach might settle a few of those.  Cheers to the final days of summer! Happy New Year teacher friends!

Paying it forward with these fun post-its…
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Just another day as an oldie but goodie.

I’m an oldie.  An oldie but goodie.  I think.  I hope.  I try to be.

I remember the day that I realized I was now one of the veteran teachers on staff…one that some newbies looked to for advice…maybe even answers.  Yikes.

It was a Thursday after school.  I was sweaty, exhausted, relieved, and satisfied.  A common range of emotions – ways of being in this topsy turvy job of teaching math to middle schoolers.  I had just finished my positive email home which may have been responsible for the satisfied feeling.

Sidebar:  I did this every single day to make sure I ended on a high note.  One positive email home to share something awesome about one of my ~120 students with their family.  A great way to end the day.  I highly recommend it.

I sat twirling on my Ikea desk chair going over the events of the day in my head.  I had reviewed my exit slips so I was pretty much ready for tomorrow…but still I sat.  I had a particularly great class that day but the not-so-productive conversation with a colleague later had me wondering what my next step should be.  Could I, should I, share what happened?  With who?  To what end?  I opted instead to do what I often do at the end of the day…I journaled a little to reflect and remember.  Here is some of what I wrote.

Good day today?  I think so…maybe?  I think it might be the start of something great…fingers crossed!  It was the period before lunch where it all came apart before coming together.  10:50-12:00.  That extra ten minutes always felt like fifty.  My least favorite period.  On top of that, today I was frustrated.  I had a great lesson planned and it was not going well.  I knew it was a great cause because the two classes prior had the same lesson.  While it did go a different direction in each class, it landed.  It was productive.  Engaging.  This class?  Not so much.  Frustrating too since this was 8-5, my favorite class.  I could usually count on participation, great conversations, and thoughtful questions.  Not today.  We had no chemistry.  Kids were talking when I was talking, others were tuned out.  I am sure I saw one boy check his watch, roll his eyes, and sigh at least three times.  My voice was getting louder, slower, and snarkier.  Yikes.  I stopped mid sentence and got a drink of water.  I took a deep breath and sat down.  I didn’t have a fight in me – not today.  The students noticed and waited for what was next.  Maybe it was the sitting down that piqued their interest…I never sit down.

I took another deep breath.  Instead of a fight we had a conversation.  I started by saying, “Well this is not working.”  Instead of telling them how awesome the other two classes were and how uncooperative they were being – a rant that was literally playing out in my head at that very moment, I said it again.  “This is not working.”  And then “…what’s going on today everybody?”

There was almost a full minute of silence where students debated what I meant, what was safe to say, and what I would want to hear.  This may be because my vein was likely visible in my neck.  Heehee.  According to my students this is a sure sign that my frustration may have reached a breaking point.  Then, one of my lovely students offered their perspective.  “This is the worst period – no offense.  We just had gym.  We are hungry, tired, and can’t concentrate.  Well that’s true for me anyway.”  This was followed by cautious nodding around the room.  

“Well I can’t change the schedule”, I said.  “We have to have math now, on this day, this period every Thursday.  So what should we do?”  Crickets.  “No – please tell me.  Help me.  I really want some solid advice.  We have to make this work.  Let’s figure it out.”

The next 15 minutes or so I listened and recorded their suggestions.  Some good, some great, some terrible.  It was a real conversation.  An attempt at problem solving.  I tried to model how to respectfully disagree with someone and how to value all voices.  I distributed then collected index cards so some suggestions could be anonymous. After hashing it out, we decided together that we would try a few things in this class the following week.  

  • Snacks.  
  • A detailed and visible class agenda.  
  • Independent work time with choices.  
  • Music.  
  • Fun challenge to close.  

When we had our plan one kid said, “Isn’t this what we already do?”  I laughed to myself.  Well yes and no.

This is what we decided could work for us.

Not what I decided would work for them.  

It felt good.  Something shifted.  

It’s hard to describe if you were not there – not part of it.  Which probably led to the second part of my reflection that day.

When lunchtime finally arrived, I retreated to my happy place in a friend’s classroom.  Laughing, venting, and gearing up for the afternoon with “my people” on staff when one of the newbies shared what they had overheard that morning.  He had stopped by my neighbor’s class with a question for her during her prep period.  My neighbor and I shared a door, and through it they could hear my class pre-breakthrough conversation.  They heard kids not listening and the frustration in my voice.  “JoAnn – they were not listening to you!?!” (Surprise in his voice.)

It was not a question but yet a question. 

I understood.  I have a good reputation.  I am organized, prepared and I know my stuff.  I have solid classroom management, build good relationships with my students and their parents, and share resources with staff.  I am positive but realistic, confident not cocky.  I love my job and I am sure that it shows.  I could see he was trying to reconcile this knowledge with what he overheard today.  Ha!  Newbies!  

I started to share with him the condensed version of events.  “Yeah, it definitely started off rough!  I have those days too ya know!  The class was tired.  I was frustrated.  But we had a good conversation and we decided that next week I’m going to try …”

Nope.  He wasn’t having it.

“What?  No!  They need to be quiet.  They need to do what you say no matter what!  They need to stay in for detention!  They need…”

He had lots more us versus them ideas and I let him rant for a while.  He was trying to be supportive.  Who am I to say what is right or what will work best for him?

What I know is I left school that day feeling really good.  I remember that day, that feeling.

What I didn’t remember and what still makes me laugh out loud are my scribbly reflections with their hand drawn emojis.  This one ended with “A  good day.  Don’t try to give advice to newbies that know everything…They need to make their own mistakes!”  

I don’t remember having another class with that group that escalated to a vein popping crescendo.  And I don’t remember altering or designing specifically with that post phys ed class in mind.  Just having a conversation with my students and really listening and problem solving together seemed to make a difference.  Or maybe I’m remembering things the way I want to remember them – who knows?

As for offering advice to new teachers?  I didn’t get it right that day but I’m not sure, even now, how I could have or if I should have tried to steer that conversation in a different and more productive direction.  I may not be qualified to give advice at all.  I think the best thing I can do is point out that I have ideas and lessons that work and ones that fail just like everyone else.  If there is a point in an educator’s career where they have it all figured out – I haven’t gotten there yet and I’ve been at it for quite a while.  

I do have some strengths that have gotten me this far.  I love my job and I love what I teach.  I truly believe that my enthusiasm for learning, trying new things and sharing ideas is the good kind of contagious.  I come super prepared but I try to be flexible enough to let the conversations go where the learners need them to go.  I work really hard to build a safe and positive learning community.  I listen more than I talk.  I focus on the bright spots.  I give and receive timely feedback.  I seek out, learn from and collaborate with other positive people.  I reflect and revise and reflect and revise.  And stickers!  Middle schoolers still love stickers!

This is not advice…I don’t give advice.
Just me sharing what has worked for me so far.  

Me:  an oldie but goodie.  

Maybe my next post is about my stickers and post-its!
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Piecing it Together…

For the past two weeks I’ve been doing some really deep thinking.  Enrolled in a curriculum foundations course, my class of 20 or so educators has been working hard to understand and reimagine the perplexing puzzle of curriculum design.  There were times I was grateful for the chance to hear so many perspectives, times I lamented not taking a course like this earlier in my career, and times I was confident with my conclusions only because of my years of experience.  Truthfully, there were moments I wondered why I was spending my sunny summer days in a classroom instead of at the beach – but as the final pieces fall into place – I am proud of my effort and intensely satisfied with the journey.

The Task
Early in the week our facilitator gave us an overview of the course requirements and projects.  One stood out:  The Artifact. We were asked to create something:  a product that, when viewed later, would trigger the learning from this course. Eventually displayed in our classrooms, it would be a reminder of the most powerful take-aways and aha moments we experienced these few weeks. Interesting. I liked the sound of this one.  

Our learning was structured around the examination of curriculum design through eight lenses:  curriculum theory, postmodernism, critical theory, methods and models, tools and techniques, persona, motivation, and assessment.  We completed activities to gain a deeper understanding of what these lenses entail.  We identified challenges in our profession and searched for the root causes from each vantage point.  Together we discussed solutions, reflected on our practice, and made plans for our own redesign with these lenses in mind.

As we made our way through the week, the artifact idea swirled around in my head.  What would I make?  Can I focus on these lenses and also make something math-y?  A few ideas started to take shape.  I would make a math puzzle.  The pieces would stand for the essential components required to make a complete picture.  But what was the complete picture?  What did it represent?  A well composed curriculum design plan?  A successful math classroom?  A well composed curriculum design plan for a successful math classroom? Something else?  

While I wasn’t sure about all the specifics, the puzzle idea felt right. If all the necessary components fit nicely together, you have a sum that is greater than the parts.  Puzzlers often use a variety of strategies and methods to reach the desired end point.  And of course, with a puzzle, the process can be collaborative and fun and just as satisfying as the finished product.  So yes…a puzzle.  But not a typical puzzle with only one solution.  That would not have the parallels I needed. Depending on the community of learners, the plan for a successful learning experience changes…sometimes drastically.   The puzzle had to have multiple correct and creative solutions that include all the pieces. It had to be fun and appealing and invite play without being intimidating.  Puzzlers should want to interact with it in their own way and figure out an end result that would be satisfying to them, but I should also offer prompts for users to examine, challenge, and test.

A few puzzle options seemed to fit nicely with my plan:  3D Pentominoes and Soma Cubes.  I’ve been wanting to create both for a while after following the adventures of Mark Kaercher and others on Twitter.  Perhaps this assignment will give me that needed push to get going.

The Product
I tried to visualize my ideal finished product.  It would be a beautiful (fingers crossed) 3D pentomino puzzle.  Pentominoes are the 12 unique ways that 5 squares can be arranged edge to edge to create a polygon.  Then they can all fit together to form different rectangular solutions.  3D pentominoes just take this idea and make the pieces 3D – cubes instead of squares. Having twelve little structures to work with meant that I could have each one represent one of the lenses discussed this week and still allow room for other factors that might be specific to the math classroom.  The pieces resemble and are often named for letters of the alphabet.  Should I assign a lens to each?  There’s a “P”…that could be postmodernism or maybe persona?  Or could it represent product models versus process models?  If the “P” represents Persona, maybe I could find mirror tiles or stickers to use on the faces to represent students seeing themselves in the design.  Or would reflective or sparkly paint work? The “U” could be UDL…my mind was overrun with ideas and possibilities.  I was getting bogged down with the details and the bling before I had the structure.  Maybe I’ll think about those specifics later…

The super cool thing about pentominoes is that they can fit together in many different ways.  One solution is a 6 x 10 rectangle, another is a 15 x 4.  What other combinations exist?  I remember the great math-y conversations I have had with students when playing with pentominoes in the past.  3D pentominoes take it up a notch, opening up the possibilities for discussion and investigation even further.  Beyond my given prompts, students playing with these pieces might make a completely different creation that just looks or feels cool by interacting with the parts.  The more we play, and test, and challenge, the more we can understand how the parts relate to each other and a more complete view of the possibilities can be realized.  Yes.  This fits nicely with my math ideals and my learning this week.  With my finished product visualized, I was ready to plan out the process.

The Process
The first thing I needed were cubes.  Lots of cubes. The ones I purchased at the local craft store were not going to work.  A quick measure confirmed what my eyes had suspected.  These cubes were not quite cubes…fine for other crafting but a disaster for this project.  I needed to start from scratch. While my awesome husband, Kyle, cut some 1 inch cubes from a 2 by 4, I assembled my other materials and made a schedule of what needed to be done each day so I could meet the deadline. I was ready to work.

Kyle piled 70 cubes in front me.  Perfect.  Enough for my twelve little 5-cube structures and some extras in case I had to redo a couple.  Once I had my cubes sanded, I assembled my pieces and glued them in place.  They were now 3D pentominoes.  I lined them up assembly style and set to work with my schedule in mind.  Today they had to be primed and painted.  That’s when I started to notice some interesting things…

Some structures were easier to paint than others. Some structures required a different brush. I started with the simplest piece and attempted to apply that process to the others – nope my skills didn’t translate.  Figuring out what worked best for my complicated structures, and handling them first would have saved me time and built my skill set faster.  I made assumptions about what paint colors would work best and which might require additional coats.  I was wrong.  Trying to stay on schedule, I attempted to get a second coat of paint on too soon…it didn’t stick.  In fact, I had to sand and start fresh when I tried to do too much too soon.  I started out with the same plan for each piece but quickly had to adjust.  The tools, process, number of coats, and wait time between coats was different for each structure.  I didn’t give up (even though I had a cool idea involving a Jenga game in my back pocket in case of disaster).  I examined the progress carefully for each piece during each stage to make sure my efforts were successful.  I made sure they had what they needed before I tried to move them on to the next step.  I continued to adjust my timeline and rework my plan.  I talked about it with my husband and I checked in with my girls.  How do you think I should handle this piece?  How long between coats?  The others had an extra coat – does this piece need one too?  In the end, the pieces didn’t all go through the same steps at the same time…but they all got there.  When I really tuned in and allowed some flexibility with my plan, I could  figure out what each piece needed and when they needed it. 

In the end…
I did it!  Process and project complete.

My 3D Pentomino Puzzle!

I have come to realize that these sweet puzzle pieces are all the little personalities that I consider when designing learning experiences.  Some soak up the learning quickly, while others need more time or alternate tools and techniques to have the same result. Some are so complex that I need to expand my skill set and try something different and new to have the desired result.  I have to choose my materials and methods with this in mind. 

The complete picture is my ideal learning community.  Everyone is beautiful and unique.  Everyone has a space and we are all connected, considered and valued.  Interacting with this puzzle will remind me that there are many ways we can all fit. The typical 6 x 10 x 1 rectangular prism is the solution I know well, but what others are out there?  What learning and experiences do I need so that I can reimagine how we can all fit together?

You could argue that my 3D Pentomino Set is not really connected to this course at all.  This was something I’ve been wanting to create for some time – this assignment just gave me an excuse to do it.  

And, well – that’s true…or it was true.  Pentomino puzzles represent a lot of what I value in math instruction:  Have stuff to hold in your hands. Play.  Play your own way then try out a method your friend suggests.  Invite everyone in. Try given challenges and stick with it until you can be proud of your endpoint. Talk about your creations with a friend and ask questions about their work.  Reflect on the process.  This is how I operate…how I already operate.  So how is this course reflected in my work?

The act of creating these pieces and reflecting on the process represents the act of consolidating my new learning and will guide my practice moving forward…

Decide on the desired end result.  Plan out the steps you may need to reach that goal but be flexible.  Know when to ask for help.  Make sure everyone is primed and ready for the learning to come.  Consider the needs of the complicated or exceptional first in designing the necessary steps to reaching your goal. What are the different tools, techniques, and methods that might be needed? Have them ready. Remember that taking small meaningful steps in the right direction is faster in the long run than having to backtrack or begin again.  All these ideas have become solid.  They are in the brushes I used, the paint that I chose and the glue that holds everything together.

Over time the paint may fade.  The wood may warp and the pieces may no longer fit together the way I want them to or think they should.  This will be the time for a new puzzle…a chance to reconsider the make-up of my new classroom community. Who are all these little people?  What do they need to be successful?  What does success look like now?  Our learners change and their needs change. Reimagining and redesign is a necessary component in all aspects of education.  We need to continually respond to the changing needs of our learners and dream up new ideas of what education could and should be. 

This puzzle was a challenge and a delight – just like each and every learning community I have had the pleasure to lead.  I’m proud of my beautiful finished product.  But for me, it was the reflecting I did throughout the process that was the most meaningful.  I am hoping that I continue to examine curriculum design from the eight lenses we discussed these last two weeks.  But, like all things, some lessons will stick and others will not.  As long as I keep students at the center of everything I design, the pieces of the puzzle will fall into place.  

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It’s a Wrap!

Ahhhhhhh March Break.
Time to relax…time to regroup…time to reflect with a blog post?

My Dad always said, “The meal is not over until the dishes are done”.  I’m realizing that the apple doesn’t fall far from the tree.  I can only quiet my mind when I’ve taken a few moments to record what I’ve noticed and made the adjustments to my activities and practice that an honest reflection often demands.  What about students?  How do they process the lessons they take in, or decide what activities have value and which are the fillers between the ones that really count?  What messages, intended or otherwise, are we sending to kids with the tasks we assign and how we serve ‘em up?

This last week before break was incredibly hectic.  Everyone was exhausted and just trying to hold it all together.  Some teachers were organizing fun extras for students and I started thinking about what that could look like in math class.  What guiding principles should I be mindful of so that activities created were both fun and valuable?  How could that value be communicated to students with my delivery and wrap-up?

Math games are an obvious choice before a break but math games can be tricky.  While some students love competing it can be incredibly stressful for others.  When the focus is on speed and answer-getting, is the game really fun?  For who?  I remember realizing one day, part way through a Kahoot, that the fragile confidence I had painstakingly built in a few of my anxious math learners was completely shattered in a span of 5 minutes when it was made plain that they could not match their peers in speed and accuracy.  As someone who tries to send the message daily that math awesomeness is not just quick calculations, certain math games just don’t make the cut.  Mental note:  if the game is just a who-can-get-this-the-fastest…I’m out.  First principle:  do no harm. 

My second guiding principle is purpose.  Why are we doing this activity?  Is it just for fun?  Is it a review?  Is it a mental health break or a way to enjoy the areas of math not often covered in our curriculum?  With report cards looming students want to know if the activity matters.  Will it count on their mark?  Why participate?  I have to know why and what it is myself so I can be honest and upfront with the students. If the activity can be both fun and relevant, great! Focusing on the delivery and wrap-up will be key in communicating that message.

So, my list of criteria was made.  Relevant, social, with some embedded strategy and competitive fun but without too much pressure, spotlight and speed.  Did such an activity even exist?

Knowing some teachers would like to be outside and others in need of a quiet relaxing period, I created a Station Activity.  Set up scavenger hunt style, with 8 pit-stops around the school, or assigned digitally in google classroom, it was adaptable to one’s desired delivery needs.  Completed by students during an event-filled Winter Carnival Day, teams could travel the loop in any order they chose, working together to complete the problems.  As for the wrap-up, I supplied answers and notes – so teachers could have some guiding questions to support their discussions along with possible answers to review with their classes.  Answer sheets had helpful tools like hundred charts, coordinate grids, and number lines, and students were seen having great conversations on a variety of topics.  It was social, relaxed, and I dare-say…fun? The next day I found myself in different classes in different roles as I filled in where needed.  I asked the students how they enjoyed the Station Activity.  I was glad I did.  When you ask for feedback from junior high students – you get it!  Some loved the word search and some hated it.  Some thought the mystery number question was tricky but still thought they got it.  Wait…what?  You didn’t correct it together?  The more I talked to students, the more I noticed that common thread.  Many students didn’t know how they did, if they were on the right track or way off track.  The consolidation piece was missing.  Many teachers supervised the activity, but did not do any wrap-up.  So in spite of a reasonably fun and engaging social activity, the message that was received loud and clear by kids was – who cares?  Your effort does not count.  Your reasoning and conversations are not worth repeating.   Don’t get me wrong…I get it.  Sometimes just running around outside is what everyone needs to get through the day – especially during the week before a break.  I guess I just would have made something different if I suspected the wrap-up wouldn’t happen.  Something that gave students some kind of feedback along the way.  This wrap-up…or lack thereof…was wildly unsatisfying.

My tired brain struggled.  Was there another activity with some of those same qualities but perhaps a more embedded feedback system?  My colleague had a suggestion at exactly the right moment:  “I’d love to do a Math Market on the Friday before the break!”  Yassssss!  Math Market! Of course!

Math Market is an activity where students work in teams to answer questions they choose.  Teams purchase math problems with super-fun fake money and sell back the solutions at a profit.  As the difficulty increases…so does the cost per question…but the profits increase as well.   This could be just the right activity before break.  It ticks a lot of boxes.  Math Market:

  • promotes teamwork
  • offers choices
  • encourages revision and precision 
  • suggests some urgency, but on your own terms
  • is intensely satisfying

I went for it. I put together a Math Market for middle schoolers working on problems involving integers, fractions, decimals, and percentages.

While I didn’t get to see this one in action, yet, I am told that the students loved it.  Lots of the same great conversations from earlier in the week, but regular feedback and consequences as answers are accepted or rejected, so adjustments in strategy can be made along the way.  In the end, money was counted and a team was declared the winner.  Success!  Success?  I still wasn’t quite satisfied. 

I wondered how students felt when the market was over.  Did they wonder if they could handle more difficult questions than the ones they chose?  Did they think about a question they struggled with and wonder if there was a more efficient strategy?  Were they happy to walk away with their third or fourth place ranking or did that activity solidify a false idea that the answer was the only important part?  

I remembered back to a course I had taken where groups were frequently asked to summarize their ideas, conversations and solutions on a slide.  Each group’s slide was part of a class slide deck and students had the opportunity to do a gallery walk to check out the solutions of their peers.  It was always the most satisfying part to me – the wrap-up.  The summarizing and vocalizing helped me consolidate my own ideas.  Seeing another persons’ solution was so interesting – especially when they took a completely different approach to the same problem or were working on entirely different tasks. Was there a way to incorporate these ideas into my activities?

I decided to include a class slide deck in The Math Market Activity I shared. Perhaps at the end of the market, students could choose their own slide and answer the question that they liked best.  Students could see the problems they didn’t pick and check out their peers’ solutions. Perhaps this is a way to emphasize processes as well as solutions.  Maybe I could ask students to create a question that could be used in a future Math Market and include a possible solution.  What level of question is it?  Why?  All skills and noticings that might promote new growth in interesting ways.

How else could we encourage students to reflect, wonder, and question? 
How do we keep the curiosity going and track the process?
Not all students have a “memorable math moment” during the same activity.  
If we, as teachers, stay tuned in, we can notice when it happens.  Will shining a spotlight on those that do have a big moment help others recognize it when it happens to them?

I’m reminded of a class earlier in the week when I arrived a little early for math time.  Students were doing book talks.  Standing up in front of their peers sharing what they had read and connections they had made…building interest without spoiling the ending. The enthusiasm in students’ voices when they speak about a book that had a powerful personal impact is contagious, as is the plug where they encourage others to read it.  Could this idea work in math class? 

I’m imagining a guided reflection where students choose a problem, activity, puzzle or math moment to write about or present.  Something that made a personal impact.  Maybe it was from the Station Activity, Math Market or something else. Maybe it stuck out because they could solve it so easily that day and couldn’t a few days before.  Maybe it surprised them that they could solve a percent problem mentally or reason through a fraction operation without resorting to decimal conversions.  Evidence of growth. Their evidence…their reflection…what they value.  Perhaps it starts with a weekly reflection that grows into a book talk format.  Maybe those proud moments are captured in a digital portfolio in a form that makes sense for the student…picture of work, audio recording, video of their presentation for a small group or to the class.  A collection and record of math growth over time…the wrap-up of one problem that leads to the curiosity for another.  Hmmmmm… 

I can imagine that as a teacher, seeing what strikes a chord in students would guide my choices in the activities I select and create.  Which are the intensely satisfying ones?  Where can I find more like these?  Who hasn’t had a math moment yet?  Why not?  What could work for them?  Classroom libraries have their novels arranged in categories like action adventure, biography, romance…what are the leisurely math equivalents?  I get giddy at the thought of students pursuing and selecting their own fun math activities to explore, challenge and delight and then sharing that journey with their class communities.  I’m sure it could work.

When we place emphasis on the wrap-up and reflection (as well as on completing the task) we send a clear message to students.  I care about you and what you like.  Your learning is significant.  This conversation is important.  Your opinion on your problem solving matters.  We can learn from each other. I care about what you have noticed in your personal learning journey.  And yes – running around outside to find problems and counting up your earned play money is fun.  And math activities can and should just be fun sometimes. But, put fun, learning, communication, and personal reflection together?  Well, then you have found the sweet spot.  I have rarely had the question, “Does this count?” in those moments.  The value is felt.  Then I know…it’s a wrap.

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Aha! Moments

Ohhhhhh the Aha! moments.  As an educator I’ve witnessed more than a few.  It’s actually the most important part of my job:  setting up the conditions that will allow Aha! moments to happen.  When they happen to me, as opposed to in front of me, I really take notice.  I write about them, plan around them, and embark on new journeys because of them.  

The Aha! moment that started this latest adventure, my pursuit of a Connected Classroom, began in the spring of 2020.  You remember it:  the exhaustion, the fear and dread, the not knowing how to connect and learn and lead from a distance.  Or maybe you were further along in your technology integration journey and weathered the storm in a more productive way.  For me, it went something like this…

As soon as the pivot to at-home learning was announced I got right down to business.  I created slide decks containing links to video lessons that I painstakingly recorded using Screencastify with Google Slides.  I included levelled questions to engage all learners at their current stage of understanding and had fun add-ons like math art and puzzles.  I would share these through google classroom and monitor the progress.  I’d create, share and correct, create and share and correct.  I had back issues and carpal tunnel syndrome from the effort and I missed my students immensely.  While we met in Google Meets to check in, it wasn’t the same as the in-person learning experience.  I struggled to engage students in real time and the collaboration and building on each other’s ideas that is such a vital part of math learning was completely lost.

On my mind too, were the two students in particular that I hadn’t managed to engage much even in person.  While I worked hard and made strides in building a relationship with them, that hadn’t yet translated into any real learning or effort on their part. Once we were learning remotely – those two students were completely silent.  I was worried…but also exhausted, and out of ideas.  Something had to change.

I met with another teacher,  my trusted friend, Valerie Targett, and we tried to problem solve together.  What we were doing was not sustainable.  Even worse, after our hours and hours of attention and effort – what we were doing was not effective for many students.  Luckily, I was becoming more active on Twitter.  I had made the move from silent observer to casual and occasional questioner.  I heard mention of an ed tech tool that seemed interesting…Flipgrid.  While nervous to attempt something completely new from afar, I had to make a switch.  Feeling so disconnected from my class and so ineffective as an educator was zapping my strength. I needed to hear and see each member of my class.  Val and I made an agreement.  We would both try something new and report back in a week.  For me, it was time to give Flipgrid a go.

That experiment resulted in the tweet you see here:

Success!  I finally connected.  I heard voices.  I saw faces.  And they saw me and each other.  We laughed and joked and eventually collaborated on some math too.  It was magic.  But the best part – the Aha! part was when I noticed who had responded first.  Yup – you guessed it – my silent duo.  I literally had tears in my eyes as I watched and re-watched their thoughtful videos and rehearsed content.  They cared.  They finally engaged.  I just needed the right medium.  For them – it was Flipgrid.  

I realized too, the mistakes in my initial efforts.  I was trying to duplicate my classroom environment online.  Instead of looking for a tech substitute for my in-person activities,  I should be thinking about what I want my students to be able to learn and do, and look for the appropriate ed tech match.  Over the next year I have done just that.  I have extended my ed tech repertoire and in doing so realize how much there is still to learn.  Every time I try something new, I see a new set of kids light up.  While some students will learn in spite of what you do, (yes…I can admit that) I take special notice of what activities and tools allow my reluctant or otherwise exceptional learners to shine.  And like always, I try to offer choices in how students engage in and showcase their learning; only now the choices I offer include a few tied to technology of some sort.

I have learned how to create self-correcting activities like pixel art (Thanks Souad El Achkar!), and activities in Desmos (Thanks Erick Lee and Erika Swinemer!).  I’ve seen the limitless potential of Mathigon Polypad (Thanks David Poras!)  and watched ideas and proofs come to life with Geogebra (Thanks Shelby Strong and Tim Brzezinski!)  I’ve gotten nervous and excited along the way as I tested out my creations in classes and crossed fingers that they worked. While learning remotely kick started the process, it was the results that kept the technology integration going long after the at-home learning phase was over.

About 20 years ago I felt the same electric excitement for trying new things in the classroom.  It came when I was first introduced to using concrete materials and manipulatives for learning in mathematics.  The Aha! moments that arose, and continue to arise, from that enterprise have kept me glowing for two decades!  And now this – technology integration:  another exciting source of energy with endless possibilities.  My new adventure has begun.

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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?
HOW AM I GOING TO MAKE IT POSSIBLE FOR THEM TO FIGURE IT OUT FOR THEMSELVES?!?

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
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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 https://www.tangram-channel.com/ 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 https://arbitrarilyclose.com/home/

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

Probability

Adding & Subtracting Fractions

Area and Perimeter Basics

Ratios

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!