Shining a light on how we see numbers

fullsizerenderI love numbers–especially their visual representations.  So when I read this week’s Table Talk Math newsletter that was all about different ways to “see” numbers my mind started thinking about all the great conversations I get to have with students about numbers, and their representations with dice and ten frames and square numbers, cubic numbers and triangular numbers and Pascal’s Triangle and… well you get the picture.

So you can imagine my dismay when I was lying on my back, staring blankly at the ceiling, brain filled with mathy thoughts, and I notice the new LED light bulb I recently installed. And I had to count the number of LEDs that were in it.

I mean, I could see five, but knew there were six.  This got me thinking even more about my bias to visualise numbers like they’re on the square faces of dice. That’s where the five came from, yet the additional one in the middle makes for six.


This led me to wonder what numbers would look like on circular faces.  And if that representations would even be helpful. What do you think?

Circular Numbers (1).png

As I started drawing numbers, one  through five made sense to me.  I really don’t like the lack of rotational symmetry in how I drew two, but I don’t think there’s anything that can be done about that.  After five, I was puzzled at what six should look like.  Should I add a center dot to five like in my LED light, or should  I double three, then add a dot for seven, double four for eight, then add a dot for nine, etc.? If I went with the LED model of six, then there’s a problem in imagining seven.

I’m confused at where the pattern should start, but mathematically that’s a fun place to be.  This has also led me to wonder what dice with round faces would look like too.  The number of faces on such a thing might have implications on the significance of this whole exercise, but that’s a puzzle for another day.

Tell me, where do you see the pattern repeating? Would you do anything differently? Feel free to make a copy of my Google Drawing to play with the numbers!


The Great Paper Roller Coaster Challenge

IMG_3284.JPGI can clearly remember many examples of the opportunities my teachers gave me for hands on experiential learning.  And I know that many of these experiences directly led me to academic and personal successes.  So I try to make sure that I have similar opportunities for my class.

One of my favourite examples, is what I call the Great Paper Roller Coaster Challenge, as adapted from Andrew Gatt on his website.

Essentially, I tell the students that the city is looking to add a roller coaster attraction to a local park and is looking for mock ups of potential designs.  Students score points for the length of time their coaster takes, the excitement value, the sturdiness of their build, and the use of a theme.  They then subtract points for the cost of materials.  Bigger isn’t always better! They also have a limited amount of tape, so they have to conserve it or buy more themselves.

You can see it’s very easy to frame this project as science or STEM, and outwardly, I do call it science and have students learn about the physics and math, but there’s a hidden curriculum here too.  What makes this project really great is what’s not taught.

As students work in small groups, I know they’re going to have interpersonal challenges.  Sometimes these are small, and quickly resolved, and other times they require mediation. These bigger ones give me a chance to sit down with students and have a frank discussion. Kids get a healthy dose of empathy and understand each other on a deeper level.  I actually look forward to when the kids say, “Mr. Inscho, we just can’t work together!” In the end, these often become the most effective groups. The approaches to collaboration are fascinating to watch, too.  Some groups delegate, some work as a unit, and others, they find a way that defies my description but still get to a finished project.

Individually, students learn to work carefully, patiently, creatively and think critically.  If students are hasty or careless, their coasters fall apart and simply don’t work.  It’s not uncommon for components to be disassembled and reassembled multiple times in order to get it right at the beginning of the project.  By the end they’re much more patient and careful in their building.  Simply by the nature of this project, I know that students are going to have challenges in bringing their creativity and design to functional reality.  As I circulate, I offer some suggestions but generally let kids figure it out.  It’s not beyond them and they’re proud of the solutions they come up with.

Finally we showcase the coasters and have other classes come through the room for an hour or two.  Younger kids are in awe and excited to try them out while mine are pleased to share their creations.  They’re also proud to explain to their peers what problems they overcame, and how they did it. Occasionally they use some scientific language to do it, but I’m satisfied with their applied knowledge.  Yet often they are unaware of their most significant learning: the perseverance, critical thinking, and collaboration.  Perhaps I should make that more explicit.  Perhaps not. Would you?


Engaging Students from the First Minutes of the Day.


About this time last year, our K-7 school was having difficult discussions about ways to engage students better, and sooner in the day.  We were particularly concerned about those students who were so disengaged, that they wouldn’t arrive on time. Too many were arriving 15, 30, even 60 minutes late almost every day.  Our solution to the problem was to make the first minutes of the day some of the most fun.

As a staff we committed to doing 15-30 minutes of Daily Physical Activity (DPA) first thing in the morning.  But we also changed the look of DPA.  Too often we sent kids for a run and called it DPA, and many kids hated it.  They walked, they hid, they complained and there was little to no value for most.  It was one more reason not to come to school.  In the reimagined version, our administrators paired up similar grade classes and scheduled them one of five locations in the school that we would rotate through each day of the week. Teachers committed to adding variety to the activities and making them fun–to great success!

In the end, there are still some students who continue to arrive late, and we may never reach all of them, but we have noticed better attendance in general.  Additional benefits we’ve seen include a greater connection with students as this time allows for us to have a brief conversation with many students outside the academic setting.  General fitness levels are increasing, as is the sense of community because kids spend the time playing together, laughing together and talking.  All too often that is lacking in their lives.  Finally, having that chance to decompress from the stresses that they may be leaving at home, connect with an adult and expend some physical energy is translating to greater self regulation in the classroom.  Kids return from DPA ready to apply their minds, now that they have satisfied their bodies.

Here’s a look at what my grade partner and I do each week:

Day: Monday Location: Upper Covered Patio and Playground
We use the playground equipment to do timed fitness circuits.  Stations often include running low hurdles, step ups, pull ups, planking, tricep dips, push ups and running a lap of the building. Over time, enthusiasm for this has waned and we need to consider ways to remotivate.  One successful strategy is to make it an AMRAP (as many reps as possible) workout.  Last week I had students run one lap, do 20 jumping jacks, 20 step ups, 20 tricep dips, and 20 pull ups.  They had 15 minutes to do as many as possible and count their reps for points.  The goal is to improve on the number for next week.  I’ve never seen them sweat so much.  This strategy is a keeper.

Day: Tuesday Location Cement Basketball Court
We use the court to run relays that build coordination. The classes are divided into four groups to get as many kids moving at once in this small space. The relays might include:

  • sprints
  • jogging backward
  • forward lines (run out to the second line, back to the first one, out to the end line, and back to the beginning.
  • backward lines (as above but students run backward as they run out, forward as they return.
  • crossovers (aka grapevines)
  • side gallops
  • hopping, on one foot or two
  • walking lunges

To step up the level of fitness, sometimes we’ll have the next person in line hold a plank or do pushups while they wait.

Day: Wednesday Location: Lower Covered Patio
This is a bit of a games day.  Four square is popular with our students and this location has two courts that students use for four square. We also bring out tennis balls to play pass, skipping ropes, and crank up some music.  Some of the students are really into volleyball and will pass a set or pass a volleyball around in a circle during this time too.  Wednesday’s DPA is a class favorite because it offers the most free play with a little bit of structure to get it going.  It also offers the most opportunity to chat with students while just passing a ball back and forth, or engaging in a four square game with them.  This is the day that DPA often extends well beyond 30 minutes because we see so much value in kids being kids and playing.

Day: Thursday Location: Lower Playground/Turf Fields
The schedule says provides an option on Thursday to use the lower playground, or make the 10 minute walk to the the local turf fields.  We’ve been using the turf fields for more sustained running.  For about 12 minutes of running, I call out lap times as students make their way around the fields.  Some other classes use the fields for more relays.

Day: Friday Location: Gym
Friday is an extra special fun day because we’re in the gym with music pumping again.  We do extra silly relays, and relays using gym equipment here.  In addition to many of the same relays we do on Tuesdays, we spice things up by having students do wheelbarrow races, run in partners inside a hula hoop, have to hold beanbags head-to-head with a partner, or do a somersault on a mat as they run to the other end of the gym.  Crab walks, leap frog, and bear walks are easier on the gym floor than the cement outside.  One of the class’ favourites is to roll a hula hoop to the other end of the gym and back, but if a student can dive through it without knocking it over, no other members have to run.  As my grade partner and I find new ways to get creative and come up with sillier and sillier relays, the students respond in kind with good sportsmanship and lots of laughs.  Students are able to laugh with each other instead of at each other and the community really grows in a positive way.

A note on rainy days:
If you know anything about the west coast of British Columbia, you know the the weather is predictably unpredictable and we get a lot of rain.  For this reason, the school field was never a scheduled location as it can get quite muddy, though many of our classes chose to do activities there during better weather. 500 Up is one of my favorites requiring little equipment to have lots of fun on the field.  Track and field activities might also happen on the field.  Our school is also fortunate to have a number of covered patios and walkways.  On rainy days, we can still go outside under a patio and play a game of Simon Says that involves lots of fitness and yoga poses.  Students getting caught not doing what Simon says have to run out in the rain around a nearby tree.  The covered walkways are supported by poles that we use too.  Running around the building in small groups at the sound of a whistle, students run under the walkway in one direction and then slalom through the poles on the way back to stay out of the way of the next group.  Blowing a whistle helps to spread out the groups and keep a safe distance.

Making a school-wide effort to commit to DPA has had noticeable results in our school this year and has addressed some of the concerns that we set out to.  To our surprise the rewards have been even greater than expected from that conversation a year ago.  Is it time for your school to have the same discussion? Maybe it’s just you and one or two other teachers that commit to trying it. What could you lose? We gained attendance, engagement, fitness, fun and community.

What is a Bully’s Mindset?

I read an article last week, and watched its accompanying video that discussed bullying behaviour from the perspective of evolutionary psychology.  It suggested that bullies act the way they do not out of a lack of self esteem, but because they have a high one that they seek to maintain, and that bullying builds popularity and social status that eventually lead to greater mating potential.  Furthermore, they’re genetically predisposed to the behaviour and there’s not much that can be done about it, except perhaps to provide alternate pathways for status and achievement to kids hardwired this way so that they they’ll not need to bully others in their pursuit of status and popularity.  To me, this is akin to buying off the gangster with “protection money”. Some of this may make sense in isolation, but it seems to me that there has to be more to the story.  Especially because there doesn’t appear to be any evolutionary advantage to being hardwired to be meek and gentle, and very little to being empathetic, yet all these qualities exist (though I do recognize there are theories to explain each of these qualities too).

This has been bothering me for a week and it wasn’t until I started reading Carol Dweck’s Mindset yesterday that I started to put the pieces together.  Admittedly, I’m only 80 pages in, but as I read, Dweck’s description of those with a fixed mindset continually reminded my of my most difficult, bullying students. These students constantly seek to reaffirm their superiority, they are afraid to take risks for fear of demonstrating a deficiency, and seek the easiest paths to success.

Is it possible then that bullies aren’t genetically hardwired, but come to school with a fixed mindset for social prominence, or are even put in a fixed mindset by our school systems?  Dweck does a great job to explain how people can be inadvertently, and intentionally put into a fixed or growth mindset, yet; she makes it clear that whatever your initial abilities or talents may be, your qualities are not fixed.

“Mindsets are an important part of your personality, but you can change them.  Just by knowing about the two mindsets, you can start thinking and reacting in new ways.” p.46

And so it may be that bullies act from a fixed (but changeable) mindset that drives them to be socially dominant at whatever the cost, and the reason that buying them off appears to work is because they are also prone to take the easiest path to success–and there’s no risk involved with taking a buyout.  As effective as this appeasement model may be (as long as the buyouts continue), it isn’t morally right.  It would appear to me that if we can change the mindset of bullies, we would be off toward a much more successful path.  I am excited to finish reading Mindset to learn more on how I can go about doing that.

Do you think traditional approaches to bullying are effective, or do either of these two perspectives change anything for you?  I’d like to hear and discuss it with you!