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!

8 Take-Aways From USCOTS

This weekend, I spend two days at USCOTS, the United States Conference on Teaching Statistics at Penn State University. The opportunity to connect with old friends, share ideas, and reflect my own practices was exciting. Here are just a sampling of my experiences, many of which could be their own blog post.  You can find many speaker presentations and more resources on the CauseWeb site. Hope you enjoy!

MOVING FROM SANITIZED DATA SETS – a Keynote by Shonda Kuiper of Grinnell College noted that while interest in the study of statistics is at an all-time high, are we really preparing students to apply statistical concepts in a realistic manner? Shonda challenged the group to move from canned textbook data sets and let large, real data sets drive coursework.  Her Stats2Labs website is a treasure chest of activities and data sets, from which she shared a rich set of NY Stops and Arrests, and Shonda shared her methods for using the set to faciltate discussions.  For my high school classes, I am most looking forward to using the Tangram Game applet, which collects many variables on gameplay.

I often tell my students that statistics is often about telling a story, and was thrilled to hear a college professor share this theme as well!

Photo May 30, 9 40 44 AMSTUDENT POSTERS – the poster sessions were interesting to me as a high school teacher, as my own students are preparing for “Stats Fair” next week.  How awesome to show my students that the presentations they are about to share are not too dissimilar from those they may encounter later in their academic careers, just in the sophistication of the studies and methods. You can’t beat having a small-group discussion with Beth Chance regarding the Rossman-Chance stats applets, which should be a part of every Stats class.  Posters on HS integrated math programs, flipped learning, and formative assessments provided info to think about for next year.

EGG ROULETTE – a session by James Bush and Jen Bready led to a fun “hook” I hope to try with my 9th graders next year.  James and Jen are masters of using video and pictures from the media to engage learners and grow discussion. Here, James chose Doug Tyson and I as “volunteers” to participate in a game. I quickly became worried when it was advertised as “Egg Russian Roulette”, and a clip from the Jimmy Fallon show was played:

me eggsWhat have I gotten myself into here?

The plot thickens when an egg carton makes an appearance…but filled with plastic eggs, some containing packing peanuts. I lost after 3 picks, and a simulation of the activity ensued from the group. Is there an advantage to being first? James alleges the person playing first loses 5/9 of the time. Try a simulation with your classes and find out!

CATCHING UP WITH OLD FRIENDS – Ruth Carver teaches high school about 20 minutes from me, and I cherish the times we find to trade stories.  By the time arrived at the conference, Ruth was already gushing over the many great sessions she had attended, and shared a quote from Dick DeVeaux which applies nicely to all classrooms:

Students like uncomfortable learning less and less.  They like things clear as a bell with no sweat, no thinking, no neurons firing.  They are confusing easy and comfortable with learning. To use a sports analogy, “Is that what you want from your sports practices – easy, comfortable, they didn’t break a sweat?  Well it should be the same with your classes; they should be sweating afterwards.  It’s hard stuff; they should be thinking hard.

What are we all doing to make sure our students sweat in math class?

Photo May 29, 6 04 25 PMIn return, I shared one of my new favorite ways to collect fun data: the website How well does it predict your age? What if you smile? What it I wear a hat? You will be toying with this and your friends at your next gathering.

SIMULATION-BASED INFERENCE –  This has become a hot topic in the stats world – I have come to use the StatKey site more often in my classes to have students simulate distributions – and was eager to learn how to leverage simulations with traditional hypothesis testing methods. Robin Lock and Kari Lock Morgan  shared examples where simulations allowed us to compute simulation distributions, but then move those results into traditional distributions and test statistics. My AP classes have generally been “successful” in that my AP passing rate is quite good, so it becomes tricky to want to ditch old methods. But the experiences and communication gained from simulation methods are too rich to be ignored. Infusing my classroom with more simulation-based inference could dominate much of my planning for next year.

CONNECTING – Connection was the theme of the conference, and a part of all of the talks. I strengthed bonds with old friends (many of whom I will see in 2 weeks for the AP Stats reading), and appreciate the many new folks I met for the first time. Doug Tyson’s silly selfie challenge gave me the courage to say hello to many people I wouldn’t normally have approched.  And though Doug won the challenge with a late-Friday “get” of Jessica Utts, the new AP Stats Cheif Reader – which I came so close to photo-bombing, I’ll take my photos with Allan Rossman and Roxy Peck as a well-deserved second-place.

Photo May 29, 6 17 19 PMPhoto May 29, 6 48 08 PM

DISCONNECTING – The saturday lunch-time talk by Michael Posner of Villanova University inspired the group by sharing the many connections he has made with the Stats community over his career. Michael often shares at our local PASTA (Philly-Area Stats Teachers Association) meetings, and I appreciate his desire to connect with high school teachers.

While explaining the power of connections he has made, Michael also challenged the group to disconnect, and reflect upon their teaching.  In particular, are we using our Stats expertise to clearly measure the efficacy of our teaching methods?  And while sharing ideas at conferences can be energizing, how do we personalize what we have gathered to work for our classroom?  Such great themes to consider at the close of a conference.

AFFIRMATION AND REFLECTION – When I first started teaching AP Stats, I was cautioned that stats teachers are often the lonliest people in their departments. Walk into a high school math planning room, bring up methods for solving quadratic functions, and you may soon have a full group conversation.  But try to start a discussion of two-sample t-tests?  Crickets…. This conference was attended by about 450 passionate stats people, with only about 10% being high school folks. But the college crowd could not have been nicer or more accomodating in wanting to share their ideas.  The entire experience left me energized that I am headed in (mostly) the right direction in what I do to encourage stats study, and with plenty of resources and connections for improving my practice.  Looking forward to USCOTS 2017!

Put(t)ing Rational Numbers in Order

Many of my friends and followers have caught onto one of my guilty pleasures: my wierd fascination with The Price is Right (read about Price is Right and counting principles in this old post).  Here’s how a pricing game made for a fun review activity, and also made my life flash before my eyes (read to the end for that).

Here in Pennsylvania, we use the PA Core Standards.  For Algebra 1, here is a standard under “Anchor 1″:

A1. Compare and/or order any real numbers.  Note: Rational and irrational may be mixed.

Seems innocent enough.  Here is a sample “open-ended” task used to assess understanding on our state’s Keystone Algebra 1 exam:

image001 (1)

Exciting….now let me go over here and watch the paint dry….

But during the NCTM conference, a lightning bolt hit. I was checking out a putting game at one of the booths, and I suppose rational numbers were on my brain….Hey – Golf + ordering rationals = feels like Hole in One to me!

In the Price is Right Hole in One game, contestants place groceries in order from least to greatest by price.  The number of items they can order until they are incorrect determines where they putt from. After a quick trip to the sporting goods store to find a putting cup, and some time with a Google Doc, we’re all set!

To start, I created a Google Slides presentation with 6 games.  Each game has 6 numbers for students to put in order:

During the game, all students in the class had about 2 minutes to place the numbers in order.  They, we randomly drew our “contestant”, who came to the board to fill in the 6 boxes on the board.


Next, we went through the numbers from left to right, and determined how far the contestant had gone in successful ordering.

puttOn the floor, 6 lines were taped.  Line 1 was on the other side of the room, and the lines were closer and closer to the hole. If a student had 4 numbers correctly ordered, they were allowed to putt from line 4.  Two students were able to order all of the numbers and tried their putt from about 2 feet away.

Those who made their putts earned candy to share with their group.  In about 20 minutes, we got through 4 games – not bad for ending a Friday on a fun note.

But be careful! My last “contestant” – one of my less cooperative students and a sometimes hot-head – was able to putt from line 6 with the help of his group.  After missing the first putt, I reminded him that the game is really Hole in One – OR TWO, and had a second chance. Lining up the putt…he took it easy…and missed again.  This is when he raised the putter up and, for a brief second, it looked like the putter could end up flying in my direction.

“Sean, just pick up the ball and put it in the hole….here’s some candy…”

It Took Me 2 Years to Get This Approach to Imaginary Numbers

This past week the NCTM annual conference was held in Boston, and what an enriching epxerience! What made it so special this time around was meeting and hearing from my PLC of Twitter friends, many of whom I had admired from afar for some time. I’ll discuss the power of the MTBoS (Math-Twitter-Blog O’Spehere) in a later post.  Today I want to focus on a powerful session I attended in Boston, and how a new persepective developed – even after a 2 year delay.

The story starts 2 years ago at Twitter Math Camp in Philadelphia.  At that conference, I participated in an Algebra 2 small group, facilitated by the super-creative Max Ray, from the Math Forum. Splitting into smaller groups, I worked with a team to think about rational expressions – a unit which is often dry as sand in Alg 2 courses, and where I thought we could make some head-way. While we worked on our slightly-less dry, yet safe lessons, Max and a small group were discussing complex numbers on the board. There were mysterious circles, transformations, and discussions I didn’t understand.  I suppose I was taught about complex numbers the “traditional” way – we need them to solve certain quadratics and memrize some wierd rules about their behavior. We perform strange operations on them, and we definitely don’t ask why. I suppose I could have simply wandered over to the group and found out more, but the mathematical intimidation factor was high – I’m sometimes too proud to admit what I don’t know.

Fast forward 2 years, and I see Max is presenting a session with Michael Pershan. This is a must-attend. Two engaging speakers whom I appreciate for their ability to use students’ natural curiosity to facilitate math conversations.

Here’s the set-up: Michael finds a handful of volunteers to stand at the front of the room, standing on a hypothetical number line (Max stands at zero). The participants are then asked to consider the following transformations to their value, and move accordingly, returning after each move to their original position.

  • Add 2 to your value – participants all move to the right 2 spaces.
  • Multiply your value by 3 – participants all move to the left or right accordingly, depending on whether their original value is positive or negative.
  • Multiply your value by -1. OK, now the plot thickens.  While we can find our new position, Michael does a materful job in having participatins reflect upon the transformation. The first two moves required left and right shifts; here we need to consider a rotation about the origin. This rotation provides a rule for multiplication by a negative.

Photo Apr 16, 12 36 08 PM

The table has been set, the silverware polished. and now we need some new volunteers. We have a new number line, and some new transformations to think about.  BUT this time around we want to complete our movement by using the same transformation twice.  Let’s roll!

  • First, add 4 by using the same transformation twice.  This is a nice appetizer – let’s move 2, then 2 more.
  • Next, multiply by 9. This is a little trickier, as some folks almost crashed into the next presentation room. But two multiply by 3 moves do the job.
  • Now, multiply by 5. Oooh….we have an entry point into radicals. Some quick discussion, and we two moves – multiplying by a little mroe than 2 each time.
  • Finally, multiply by -1…in two moves…..

WAIT!  This is the stuff Max was talking about 2 years ago that I didn’t get.  The bulbs have gone off.  I GET this now!  We do a 180 degree rotation do perform a multiplication by -1, so now we need two 90 degree rotations.  And now we have an entry point into imaginary numbers, without the scary-sounding term.

What I appreciate most here is that we don’t need to wait until deep into algebra 2 to think about the imaginary unit.  These concepts are accessible to younger students, and we have a responsibility to achieve some conceptual buy-in before just thrusting abstract ideas in front of our students. You can find Michael and Max’s shared files here on their Teaching Complex Numbers page.

I get it now…I think….and I’m not ashamed to say it took me 2 years.

UPDATE: You need to immediately run to check out the fun summary Ashli has provided of this session. Her notebook sketches are unreal (in the non-numbr sense)!

Statistics Arts and Crafts

The Chi-Squared chapter in AP Statistics provides a welcome diversion from the means and proportions tests which dominate hypothesis test conversations. After a few tweets last week about a clay die activity I use, there were many requests for this post – and I don’t like to disappoint my stats friends! I first heard of this activity from Beth Benzing, who is part of our local PASTA (Philly Area Stats Teachers) group, and who shares her many professional development sessions on her school website. I’ve added a few wrinkles, but the concept is all Beth’s.

ACTIVITY SUMMARY: students make their own clay dice, then roll their dice to assess the “fairness” of the die. The chi-squared statistic is introduced and used to assess fairness.

clayYou’ll need to go out to your local arts and crafts store and buy a tub of air-dry clay. The day before this activity, my students took their two-sample hypothesis tests.  As they completed the test, I gave each a hunk of clay and instructions to make a die – reminding them that opposite sides of a die sum to 7. Completed dice are placed on index cards with the students names and left to dry. Overnight is sufficient drying time for nice, solid dice, and the die farm was shared in a tweet, which led to some stats jealousy:

The next day, students were handed this Clay Dice worksheet to record data in our die rolling experiment.

In part 1, students rolled their die 60 times (ideal for computing expected counts), recorded their rolls and computed the chi-squared statistic by hand / formula. This was our first experience with this new statistic, and it was easy to see how larger deviations from the expected cause this statistic to grow, and also the property that chi-squared must always be postivie (or, in rare instances, zero).

Students then contributed their chi-squared statistic to a class graph. I keep bingo daubers around my classroom to make these quick graphs. After all students shared their point, I asked students to think about how much evidence would cause one to think a die was NOT fair – just how big does that chi-squared number need to be? I was thrilled that students volunteered numbers like 11,12,13….they have generated a “feel” for significance. With 5 degrees of freedom, the critical value is 11.07, which I did not share on the graph here until afterwards.


In part 2, I wanted students to experience the same statistic through a truly “random” die. Using the RandInt feature on our calculators, students generated 60 random rolls, computed the chi-squared statistic, and shared their findings on a new dotplot.  The results were striking:

FullSizeRender - 3

In stats, variability is everywhere, and activities don’t often provide the results we hope will occur. This is one of those rare occasions where things fell nicely into place. None of the RandInt dice exceeded the critical value, and we had a number of clay dice which clearly need to go back to the die factory.

Introducing Discovery Hour with Codebreaking

Our school has been on a semester block schedule for over 20 years, with some tweaks made to accomodate building size, AP courses and electives. But this year brought a major schedule change, and an opportunity to think about how we use time to engage students. After periods 1 and 2 (each 75 minutes), all students move into something called HATS period. The acronym stands for Hatters Achieving Targeted Success, and during the period students have a lunch period, along with assigned time with teachers. It’s a great mid-day block for students to touch base with activities and clubs, seek help, make up work, and our RTII team has utilized the time to meet formally with students and facilitate individual help sessions.

I saw an opportunity to engage students in meaningful activities during this time, and have started Hatters Discovery Hour – modeled after the Genius Hour concept many elementary and middle schools offer. My thought is that so many of our teachers have awesome ideas to share which don’t quite fit class time. Also, it’s an opportunity for students to experience teachers they may not cross paths with during their high school career. Let’s build more connections!

The past 2 months have seen some fascinating offerings.  Our No Place for Hate Team has used Discovery Hour to facilitate open discussions on race relations.  Meanwhile, a science teacher shared his experiences working as an EMT in a medical diagnosis session. Juggling was the fun focus of one session, and Discovery Hours on memory systems, photography and meditation are in the works.


For my Discovery Hour session, I shared many of my collected activities on codebreaking. With Oscar season just passed and some simmering interest in the Imitation Game, it was a perfect time to talk about the role of codebreaking through history. Even better, my principal and district curriculum director (and my former boss) were on hand to join in the fun:

I was ambitious, trying to fit 4 codebreaking challenges into the hour. In the end, we had just enough time to keep things moving and hold some fun discussions in these 4 areas. Scroll below to download the handouts.

CRYPTOGRAMS – We started with a basic letter-to-letter cipher. I used a long quote from Bill Gates, which almost turned out to be too long – as I felt a time crunch hitting early. But longer quotes allow more entry points, and I couldn’t pull my principla away from the challenge!

CAESAR SHIFTS – Here we used an online applet to explore shifts, and this provided an entry point for modular arithmetic, which few of the students had encountered before.

HILL CIPHER – By now we had established that the first two coding procedures did not seem too secure. I have shared Hill Cipher with students in my classes before during matrix units, and again a cryptography website was helpful in providing some easy codebreaking trials. When I have done these in class, I often develop problems which get around the modular arithmetic issue (it takes longer to discuss than I often have time for) but we were able to squeeze in a 5-minute mod primer.  See below for other Hill Cipher problems I have used.

THE ENIGMA – The cherry on the sundae, and where many students were stunned by the complexity. This online Enigma simulator is one of my favorites – I love the visual of the wiring. So many good questions concerning inverses, how codebooks were traded and how the British broke the code. I left enough time to show Numberphile’s Enigma video, which capped off the hour nicely.

Looking forward to sharing more of what I know in later Discovery Hour sessions, and thrilled so many of my colleagues are buying into the idea.

Inverse Function Partner Share

We’re working through functions in my college-prep pre-calculus class; meaning a more rigorous treatment of domain, range, and composition  ideas than what students experienced in earlier courses. As I was about to start inverses last week, I sought an activity which would provide some discovery, some personalization, and less of me rambling on.

These are the times when searching the MTBoS (math-twitter-blog o’sphere) leads to some exciting leads, and the search for inverse functions ideas didn’t disappoint – leading me to Sam Shah’s blog, and an awesome discussion of inverse functions which I turned into a sharing activity. A great list of blogs and MTBoS folks appears on this Weebly site.

To start, I wrote a function on the board, and asked students to think about the sequence of steps needed to evaluate the function:

The class was easily able to generate, and agree upon a list of steps:

  1. Square the input
  2. Multiply by three
  3. Add 1

From here, I asked the class to divide into teams of 2. Each partnership was then given two functions on printed slips (shown below) to examine: list the steps of the function, and provide 3 ordered pairs which satisfy the function.



Notice that the functions are arranged so that A and B in each set are inverses.  Partners were given two different functions, but never an inverse pair. So a team could get 2A and 4B, but not 3A and 3B.

My plan was to complete this entire activity in one class period, BUT weather took hold. They day we started we had a two-hour delay, and the next two days were lost due to snow, then a weekend. SO, the best-laid intentions of activity, sharing and resolution became activity…..then 5 days later.

As we started the next class day, I asked students to review their given functions (and re-familiarize themselves), then seek out the teams who had the other half of the function pair and share information. So a team which had 2B sought out 2A, and so on.

After the sharing, a classwide discussion of the pairs was then seamless. Students clearly saw the relationships beteen the inverse pairs and the idea of “undoing” steps, and we could now apply formal definitions and procedures with an enhanced understanding. Also, by sharing ordered pairs, students saw the domain-range relationship between functions and their inverses, and this made graphing tasks much easier. I’m definitely doing this again!

Finally, notice that pair 2A / 2B features a quadratic / square root. While we didn’t dive right in at the time, this set the trap for a discussion of one-to-one fucntions and the horizontal line test the next day.