Making It Stick…With Beanbags

The book Making It Stick – the Science of Succesful Learning has caused me to consider how I approach practice and assessment in my math classroom. The section “Mix Up Your Practice”, in particular, provides ideas for considering why spaced practice, rather than massed practice, should be considered in all courses.

But it was an anecdote which began the chapter on spaced practice which led to an interesting experiment for stats class.  The author presents a scenario where eight-year-olds practiced tossing bean bags at a bucket.  One group practiced by tossing from 3 feet away; in the other group, tosses were made at two buckets located two feet and four feet away.  Later, all students were tested on their ability to toss at a three-foot bucket.  Surprisingly, “the kids who did best by far were those who’d practiced on two and four-foot buckets, but never on three foot buckets.”


Let’s do it.

My colleague and I teach the same course, but on different floors of the building during different periods. Each class was given bean bags to toss, but with different practice targets to attempt to reach.

  • In my class, lines were taped on the floor 10 and 20 feet from the toss line.
  • For Mr. Kurek’s class, one target was placed 15 feet from the toss line.

Photo Oct 05, 9 33 54 AMAfter every student had a chance to practice (and some juggling of beanbags was demonstrated by the goofy….), I picked up my tape lines, and placed a new, single line 15 feet from the toss line.  Each student then took two tosses at the target, and distances were recorded (in cms).

We then analyzed the data, and compared the two groups (the green lines are the means):

bean bags

I love when a plan comes together!  The students, who did not know they were part of a secret experiment, were surprised by the results – and this led to a fun class discussion of mixed practice.  Here, the mixed practice group was associated with better performance on the tossing task. Totally a “wow” moment for the class, and a teachable moment on experimental design.

An Egg-Cellent Simulation

The scenario I used for a fun lesson with my 9th graders this week comes from a talk by James Bush from Waynesburg College, which I attended at the US Conference on Teaching Statistics in May.  James is a master of finding clips from TV and movies to use in his class to encourage discussion, and this clip from the Jimmy Fallon Late Show features a game called “Egg Russian Roulette”. I have embedded a clip here, but you can search for many times Jimmy played this game on his show.

For James’ college statistics courses, this clip is a helpful opener to the hypergeometric distribution, where we are interested in multiple successes from draws done without replacement.  While this setting could eventually be presented to my AP Stats students, it lives a bit outside of the scope of what we do at the college level. But there are some strong entry points for discussion with my 9th graders, including probability trees, conditional probability, and simulation.


Before showing the video, two volunteers were called to participate in a mystery game.  The two student volunteers became a bit nervous over their decision when a carton of eggs was produced, eventually shown to be filled with plastic eggs (awesome idea by James!). My first chance to try this with volunteers on my own was at Twitter Math Camp in June, and lots of fun tweets followed.

Thanks to Richard Villanueva, who recorded many of the My Favorites from Twitter Math Camp, we have coverage of the ganeplay.  Check out all of the videos of My Favorites from TMC15 on his YouTube Channel.

Next, I asked the class to think of questions they have about the game they saw, or in general about Egg Roulette.  A good starting list developed:

  • How likely is it that Tom Cruise would lose that quickly?
  • Once Tom picks a raw egg, how likely is it that Jimmy is safe on his draw?
  • Is it better to go first or second?

Photo Sep 18, 8 06 14 AM I then challenged my student groups to sketch out the first three rounds of egg roulette, and find the probability of Tom losing in 3 rounds. We had worked on trees the day before, and this game presented a good chance to apply what we had discussed earlier.

Photo Sep 18, 8 13 52 AM


After our analysis of the first three rounds, the conversation then moved to strategy: is it better to go first or second, or does it not matter?  Our “gut reaction” poll revealed that “it doesn’t matter” was the most common response, with “go first” was in second place.  The thought behind going first is that you could easily draw a hard-boiled egg, and thus put pressure on the other player.

To simulate the game, pairs of students were given one suit from a deck of cards.  The ace was moved aside, leaving 12 cards (representing the 12 eggs).  The 10, jack, queen and king then  represent raw eggs.  After shuffling the cards, students dealt cards into two piles, Tim and Jimmy. When a player was dealt two raw eggs, the game ends and the result recorded.  We were quickly able simulate over 50 plays of Egg Roulette, and the class results were recorded.

Egg Simulation

One student quickly identified, and then defended, that the game can NEVER go the full 12 rounds.  Also, some students noted that the second player (here, Tom) has one extra opportunity to lose the game.  A second straw poll revealed that student perceptions on the game had shifted – few thought it was a 50/50 game, and many saw that the first player held a disadvantage.


In teams, students are now challenged to explore a similar (yet shorter) Egg Roulette game, compute theoretical probabilities, conduct a simulation, and analyze the results.  I’m looking forward to some interesting-looking trees.  The document here shows guidelines for this activity, some assignment ideas, and a full tree for the first 3 rounds of Egg Roulette.

A Sneak Preview of My 2015-2016 Classroom

Today is the last day of summer vacation. In the past week, boxes have been unpacked, t-shirts and class decorations have been hung, and my awesome school custodians have provided me with even more whiteboard space – all the better for getting students up and moving


But beyond the physical layout for this year, here are some ideas I’ll focus on this coming year, many provided by my friends in the Math-Twitter-Blog-O-Sphere, the #mtbos for short.

GREETING STUDENTS WITH HIGH FIVES – Intertwined with all of the mathy goodness of Twitter Math Camp this past July was a simple and powerful device for student engagement from my friend Glenn Waddell – the High Five.

Each day last year, Glenn met his students at the door to give them a high five – a simple, caring gesture to establish a positive tone for class.  I often meet students at the door before class or linger in the hallway for informal chat, but I love the tradition and rapport Glenn establishes here and hope to emulate it.

ESTABLISH SEMI-REGULAR RANDOM GROUPINGS – this gem comes from Alex Overwijk, who is the king of Vertical Non-Permanent Surfaces and Visible Random Groupings. This year, I plan to randomly change my seating chart once each week, or at the start of a new unit – whichever seems to make the most sense at the time.  Traditionally, I’ll assign groups on my own and change them once or twice in a semester.  With some classes, I’ll allow students to choose their own groups.  But I have found that these practices often foster group-think, where a group will together develop the same bad habits through their work together.  I want more interaction, more sharing of ideas, especially in cases where students otherwise would not have encountered each other. I’m planning to assign each student a playing card on the first day, and set the new groups by dealing cards on the desks on days when it’s time to change.  I also confess here that a static seating chart was a huge crutch for me, as I would print out student names for me to glance down at when I needed.  Which leads into another goal for the new year…

better jobI MUST LEARN NAMES DAMMIT! – I confess this could be one of my weakest areas as a teacher. I could make all kinds of excuses for it, but it comes down to this – I drop the ball when it comes to learning and recalling my students’ names. We start school next Tuesday with a 4-day school week, and my goal is to know all names as they walk in the door by the first Friday.  I have already gone through my class rosters (which conveniently provide photos). How awesome would it be to know student names before they even walk in the door?

And beyond my current students, I am brushing up on names from students I taught last year. I’ve missed out on these connections for too long, and it’s my fault – time to work harder at it.

IMPROVING MY HOMEWORK PRACTICES – I don’t grade homework anymore, and in many cases have changed the nature of assignments. I’ve settled into the philosophy that I would rather have students think about a handful of meaningful, discussable problems rather than complete a laundry list. This year, I am looking to include more articles and video clips for students to observe and discuss in lieu of traditional assignments.

To go over homework, I often employ random methods to share works on my document camera, with mixed success. I’m finding that since I don’t directly look at assignments anymore, the completion is spotty at all levels. I may need to go back to a few minutes of checking and informal greeting at the start of a period to improve assignment fidelity.

grabUSING REFLECTOR TO ENCOURAGE PARTICIPATION – It can’t be the new school year without a new tech tool to try out. This year, I am looking forward to using the Reflector 2 program from the folks at Squirrels. This inexpensive software, loaded onto my laptop, allows me to relfect the screen from my ipad or iphone onto the laptop. I’m hoping this will allow me to be more hands-free for presentations, and hand over the ipad to students to take control – using Desmos or Deoceri to create works and share in front of the class. Also, I’m wondering what a class would look like where students could reflect their own phones onto the screen and share works. Day 1 of class could feature a “load test” – what happens when many, many students all try to reflect their graphs at the same time?

Now, out to the craft store to buy some last-minute stuff!

Desmos Lessons for AP Statistics

In the past year-plus, Desmos has added useful features to help those of us in the statistics world. The elegant addition of regressions (check out my tutorial video) has been a welcome new feature, and simple stats commands have also been added for lists.  Here are 3 Desmos creations which will become part of my classroom lessons for the 2015-16 AP Stats year.


That dreaded r-squared sentence…..yep, the kids need to memorize, but let’s add some meaning behind the “percent of variation due to the linear relationship….blah blah blah…” mantra.  Here’s an activity I do with my classes which has helped flesh out this regression idea.  To start, every student is handed a card face down with a prompt.  On my signal, the students turn over the card and respond to the prompt, with specific instructions not to discuss their response with classmates.  Here’s the prmopt:

An adult male enters the room. Estimate his weight.

After some nervous mumbling, I now hand out a second prompt card, and we will repeat the process.  But this time the card looks a little different.

An adult male who is {*see below} tall enters the room. Estimate his weight.

This time, I have 6 different versions of cards, and they are randomly scattered about the room.  Some cards say “5 feet, 6 inches” for the height, with other cards for 5’9″, 6’0″, 6’3″, 6’6″ and 6’9″.

After responses for both cards have been given, the responses are written on the board, along with the associated heights for the 2nd round of cards.  How did the background information given in the 2nd set of cards influence our responses?  Now the bait has been set to look at the Coefficient of Determination on Desmos.

rsqr1In this Desmos, heights and weights of adult males are given in a scatterplot. Activating the first folder – “using the mean of y1 for prediction” shows us the mean of all weights, and associated errors if the mean weight were used to predict for all men. The folder is activated by clicking the open circle to the left.

rsqr2Next, we can explore how the regression line helps improve predictions. Click the “LSRL and explained variation” folder and note the reduction of error.  The calculation for r-squared as the reduction of error is given, and can be compared to the calculated r-squared value from the regression.  Also, points in the scatterplot are draggable – so play away!


I have done this exploration of regression facts for many years, using worksheets from Daren Starnes along with Fathom. I find this Desmos version to be much easier for kids to handle, and it can be saved for future discussion.  And while in this demonstration I have all of the commands prepared for you, I would walk students through entering the commands themselves in class.

lsrl1First, we have a scatterplot with its LSRL included.  Activate the “mean of x and y” folder” and notice the intersection of these value lines. Here, the points are all draggable, so we can easily generalize that all LSRL’s pass through the point x-bar, y-bar.

lsrl2The second discovery is a bit more subtle.  Click the next folder, and now we have new lines 1 standard deviation in each direction for x and y.  Clearly, our intersection point is no longer on the LSRL, but what is its significance?  How far do we rise and run to get to this new point on the LSRL?  Some calculation and discussion may help students discovery this fact about the slope of an LSRL:


This is not a fact students need to memorize in AP Stats, but certainly the discussion builds understanding of regression beyond what our calculator provides.


binomialLists on Desmos have strong potential for investigating a distribution by using a formula repeatedly.  In this Desmos demonstration, students investigate the behavior of the binomial distribution, using sliders to define values for n and p in the distribution.  Activating the normal curve folder allows us to assess the “fit” of the binomial distribution against a normal curve.  I added the purple dots near the top to make it easier to investigate where the normal approximation is strong/weak in approximating its binomial cousin.

While Desmos has a while to go before it will replace graphing calculators in my AP Stats class, these activities will be part of my classroom this year.  Looking forward to creating and sharing more!

Twitter Math Camp – A Scalable Model for PD?

I’m finally gettick myself back to “real life” after about 3 weeks on the road, with stops at the Jersey shore, San Diego and Las Vegas. Sandwiched in the middle is the annual professional awesomeness of Twitter Math Camp. Now in its 4th year, TMC has evolved from a small group of online colleagues interested in discussing Exeter problems do a full-blown 4-day conference. Participants take part in the same morning session for each of the 3 days, a structure designed for digging deeper, encouraging conversation beyond the conference time, and developing ideas. In the afternoon, Keynotes by Ilana Horn, Chris Danielson and Fawn Nguyen inspire the crowd before afternoon sessions, which feel similar in structure to traditional conferences.  But with only 225 participants, the difference lies in the intimacy. Conversations easily move to meals and informal evening gatherings. The opportunity to extend the conversation with a speaker after the session hours is welcomed and embraced.

Compare this to the NCTM and ISTE conference models, even down to the regional and state-level conferences (full disclosure: I am programming co-chair of the upcoming Pennsylvania state conference, so I may wind up unintentionally yet, maybe I kinda-sorta mean it bashing myself here….let’s see).  There is a menu of sessions, some keynotes designed to draw folks in, and some planned sessions to wrangle folks together.  And vendors. Lots of vendors. No vendors at TMC…just straight-up PD, with the exception of sessions on Desmos and from folks at Mathalicious which begin to blur the lines between PD and self-promotion, but the mission is certinly not designed to support product. So, how is the TMC model different than the large-scale conferences? Here’s my non-exhaustive list:

  • Morning “themed” sessions at TMC encourage reflection through the week. Participants are expected to stick with their morning sessions and see it through.
  • The size of the conference provides laser-focus on math PD. No getting lost in the sea of 10,000 people in the convention hall.  The speaker you just saw in the last session may be sitting next to you learning along-side in the next session. Deeper conversation takes place at all hours.
  • Participants are encouraged to share out their experiences after the conference. Conversations continue via twitter, blogs and facebook long after the conference ends.
  • Teachers who cannot attend can participate and are welcomed into conversation. Global Math Department this week will feature a menu of speakers from TMC designed to summarize sessions and provide resources for those who missed the conference.  Presenters are encouraged to share resources for all on the conference wiki, and twitter conversations link teachers to teachers.

tmcThe morning session on Desmos I helped facilitate may have been the most powerful PD experience in my career. This is mostly due to the positive, team approach with enthusiasic colleagues who I admire greatly. Glenn Waddell from Reno and I have shared Stats ideas through twitter often, and see each other only now and then at conferences – his blog is a fountain of classroom resources.  Jed Butler has definitely become one of my go-to guys in the last year; his creativity and ability to build something new and meaningful quickly astounds me – check out the Desmos Bank he has developed, and share your works. And I was most excited to work with Michael Fenton. If you have never seen Michael’s Ignite talk – Technology and the Curious Mind – run there now….it’s only 5 minutes…we’ll wait for you… and visit the Reason and Wonder blog to get your feet wet with Desmos challenges. In the months leading up to TMC, we “met” a number of times via Google hangout to discuss what we wanted from our morning session – how do we structure the session for a large, diverse groups of learners. What themes do we want to develop through the conversations? How do we encourage learning to continue after the the conference has ended?  The team facilitation model has encouraged me to think this way as I consider other conference talks – hopefully starting with an ISTE session next summer with Jed.

What’s the future of the traditional “set and get” conference, in a connected world?  It seems that NCTM is starting to feel heat to change its model, as Matt Larson (President-elect of NCTM) attended TMC for a day with the NCTM executive director to soak in the experience, and presented a session in which NCTM’s Professional Learning Strategic Plan was outlined.  Some highlights:

  • NCTM will establish smaller, regional conferences based upon a theme, and replicate.  This sounded a lot to me like the Future Ready regional summit concept which is making the rounds this year – promoting a common message in more intimate gatherings,
  • Teams of professionals will be encouraged to attend and participate. How this works out financially is up in the air.
  • Reflective practice will become a bigger part of the NCTM message. This could mean promoting conversation after a conference through message boards (eh), allowing comments to published articles (I’d like to see this) or twitter/facebook/social media.

But in terms of PD, this exciting announcement leads me to believe NCTM is on the right track:

There are some promising developments here, though a problem of scalability will remain sticky.  TMC works because of its size and the zeal of its participants, and there is no desire to get much bigger. The math teacher twitter community is still small enough that conversations with colleagues from across the country are manageable.  What would happen is even 10% of the math teacher workforce became actively engaged?  It would be a great problem to have – but what gets lost?

Regional, focused conferences also sound great, but also present missed opportunity.  This year’s California TMC was amazing for me, as I had the chance to interact with west-coast math folks who I rarely see (or whom I have never met). Matt Vaudrey, Fawn Ngyuen, John Stevens, Michael Fenton, Peg Cagle….ok…..I’m stopping here….too many names to list. What connections are missed by regionalizing? Does it matter?

There’s a lot here to think about…check out the TMC wiki, find that 1 thing which fits in your classroom, and share it out.  The future of PD seems bright, but how do we manage it? I welcome your thoughts.

I Really LOVE These Math Tools, But…..

While the meat and potatoes of my recent ISTE session dealt with classroom use of the Desmos graphing calculator, a number of conversations with attendees after the session, both in person and via e-mail, moved in a different direction.  Specifically, the teachers I spoke with want to know where free apps fit in a handheld graphing calculator world?.  Some Q&A here….I’m hoping other will share their experiences and ideas.

Q: I’m not a 1-1 school.  How does this fit in my school?

Even though I work (and live) in a “nice” suburban district with decent financial support, I struggle to get technology working in my classroom.  A teacher across the hall from me has a cart, but the laptops there are slower than Cecil turtle and generally the educational payoff is not worth the technology aggravation.  We have a few computer labs available to use, but a lab scenario is often not what I am seeking for in my lessons.

cell phone

But all of my students have cell phones, and by the end of the first week with me we have used them a number of ways to explore and communicate.  Desmos works remarkably well on a cell phone, kids pick it up immediately, and many have it on throughout class time to use.  While the app is nice, our wifi is quite good so we prefer using the web version. For stats class, Stat Key was a welcome online addition, and allowed for many class investigations.

Q: But don’t your kids end up texting in class?

Yep, kids are generally weasels, when provided the opportunity.  But I recall my own 8th grade math class, where I passed notes constantly.  I’m relieved that none of my teachers told me I had lost paper and pencil privledges over my middle-school note-transit system.

I’m constantly reviewing my classroom management style, and make revisions based on readings and discussions.  I’m confident that handheld devices aren’t going anyplace soon, so I have two options: utilize the technology or bury my head deeper in the sand.  Sure, there are moments where I resort to silly tactics to focus my high schoolers – cell phones face down, corner of the desk, or away altogether.  But making sure students understand responsible use of technology should be build into our classroom mission; I’ll do my part to prepare them for these eventualities.

Q: Do you mandate your students purchase graphing calculators?

This question has many tentacles for me.  I teach honors freshmen, so my suggestion has usually been to consider purchasing a device, learn how to use it well in our courses, and this will put them in a good place for AP Calculus.  Also, I teach AP Statistics, where a graphing calculator is an indispensible tool and I do expect them to have one. (Yes, there are some great individual sites and apps for statistics. But the TI products are still ideal for what we do in AP Stats).  I also have a class set of Nspires, which helps with our non-AP students.

So, the short answer here is a conditional “yes”, but it is becoming much more difficult for me to stand in front of parents at Back to School night and justify the purchase, especially after I discuss the many tools we use in my class.  I also understand that while I am comfortable with many new, free tools, many of my colleagues are not.  I need to consider where I reside in my department’s tech eco-system.

I’m expecting that my answer to this question will shift to a definitive “no” in the next few years.  Until then, some creative solutions, like graphing calculator loan-out programs, may be a way to go.

Q: What about standardized tests?

In AP Stats, students are expected to bring a device to use to the exam, and know how to use it.  So, there is responsibility on my end to ensure that my students have meaningful problems and practice.  There has been chatter of AP eventually moving to an online administration, but I didn’t hear anything concrete about this at last month’s AP reading.  SAT and ACT exams still expect students to bring their own approved calculator devices.  With many of the recent bad press there has been over exam exposure and cheating, I have trouble seeing a scenario anytime soon where any communication devices would be allowed.  Put another notch in the “I still need my students to have a graphing calculator” column.

But if you take a look at some online versions of state and national assessments, you’ll see students provided tools within the test.  And there are some exciting things happening regarding ipads and other non-traditional devices. Texas recently approved the Desmos test-mode app for use on state assessments, where the first attempts at implentation occured recently. Cathy Yenca chronicles her experiences with this on her blog, and you can read more about implentation issues on the Hooked on Innovation blog.

It’s an exciting time to be a math teacher, but also one where some technology growing pains will occur.  Looking forward to hearing what other schools and districts are thinking.

ISTE 2015 – Keep the Learning in Focus

Anytime I do a blog post which is a list, my traffic shoots up.

– A friend / tech-blogger

cuethinkThis post has been rattling around in my head since the end of the ISTE (International Society for Technology in Education) Annual Conference last week. I most appreciated the chance to meet and discuss technology war stories with folks I had only “met” before through twitter, make new connections in the math world, and think about how new ideas and products will change my classroom culture.  But trying to summarize the experience in a blog post is difficult.  There’s just too much stuff – new tools, inspiring people, great school action – how can I fit it all in?

I got it — I’ll make a list!!!

Often, the most popular presentations are those which feature lists – it’s a great device for getting foot traffic to your session.

  • Amazing Chrome Apps and Extensions to Enhance Teaching and Learning!
  • The Magic Bag of New Presentation Tools for Teachers
  • 60 in 60 – App Attack

I confess I enjoyed Steve Dembo’s list session – “Something Old, Something New” – which challenged participants to share tools on Twitter and think about how “old” tools still could be thoroughly effective in the classroom, and not to toss them for new things, while also embracing the new.  Sessions featuring lists can be intoxicating hours of “wow”.

After my session featuring Desmos on day 1, I was energized to think about a session for next year.  There were few math-specific sessions at ISTE, and the group I worked with seemed appreciatice and eager for more.  There are many, many math tools I use in my classroom, and I’d love to share them…I even have a title:

The Math Tech Tool SmackDown!

60 minutes – 10 to 15 math tools, lots of oohs and aahs over their wonderfulness, some quick examples, a few cute anecdotes, and everyone leaves happy.

Teachers LOVE lists!

The list is also a cop-out.

List sessions are often one-sided affairs.  The presenter moves rapid-fire through tools with examples, and the time crunch to get to everything means little time for discussion.  The application and personalization are left for the user to figure out later.  They aren’t BAD sessions at all (heck, my last post on this blog is a list…and you’ll find many other lists buried here on the blog), just know going in that discussion of pedagogy will not be the order of the day.  Follow up that list session with a smaller group opportunity and syntthesize your new learning immediately.

I’m suddenly feeling less excited (and a little guilty) about my Math Tool Smack-Down.  Some twitter sharing from a colleague helps lend some clarity to my thoughts:

Yes!  It’s about best teaching practice – not the tool (duh!).  It’s easy to forget that in the tsunami of stuff (and swag) at a big tech conference.

Jed Butler is such a great math resource, and an awesome friend.  He came as a participant to my Desmos session, and ended up being a vital resource when the tech went south.  He also acted as my button-pusher, and general problem-solver.  On the last day of the conference, a lunch conversation of math tools developed into a potential ISTE talk for next year, featuring problem-posing as a framework for making use of apps and tools.  Such exciting conversation, and there will be a lot more to come this month when Jed and I (along with Mike Fenton and Glenn Waddell) share Desmos morning sessions at Twitter Math Camp.

Extending conversations beyond conferences – one of the most powerful aspects of my participation in the Math-Twitter-Blog-OSpehere.  Keep a lookout here on the blog as we get deeper into July as the group shares out classroom ideas.

Thanks to Priness Choi for sharing out her experience in my session.  Yes, I move around a lot!