Bingo Lingo


This time of year, the standards used to measure the success of a lesson may look different than they do at other times of the year.  For example, some teachers might consider “Students not using worksheets to have paper airplane throwing contest” to consistute a lesson well executed.  To a certain extent, I am joking, but there’s a thread of reality there.   Think back to a time when your excitement for a future event prevented you from doing anything productive.  Now imagine leading a room full of 32 people with that same excitement and handing them a manual for their new scanner/copier.  You get the idea.

I can usually distinguish between my being pleased with a lesson based on lowered expectations and my being pleased with a lesson because of a high level of learning and collaboration.  Today was the latter, with my 9th grade probability and statistics class again.

I found this on Don Steward’s website.  If you have seen his blog and are not fascinated, or at least intrigued, we cannot be friends.  He comes up with some amazingly simple, yet elegant classroom problems.

Picture1  Picture2 Picture3


We started this yesterday.  They are in groups of 4; the oldest student in the group got to choose first and so on.  Then they played three “games” using a pair of dice and a whiteboard with their numbers on it.  Today, they worked on figuring out why the “6,7” card was the best and determining how to rearrange the numbers  on the cards to make them all equally likely to win.

I’ve had this glossy paper in my room forever, so I decided to have them make a mini-poster with their solution and some reasoning.  Here are my two favorites:

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Talking Pizza and Pennies


Today was a banner day in my ninth grade probability and statistics class.

First, our number talk was a bite out of the real-world and not the “you and 5 friends share 8 pizzas” kind of real-world.  When my daughter has a babysitter, as she did last night, I usually spring for pizza.  (Yes, our vegan lifestyle maintains a real iron-grip on nutrition when mom and dad are gone.)  I even splurge on the good stuff:  $5 Pizza.

With tax, my vegan-less vice cost $5.36.  I gave the cashier $20.11.  How much change did I receive?

Lots of great strategies:  counting up, counting down, counting to the middle even.  It’s worth noting that the two students in each class that insisted on stacking the numbers and borrowing were not able to do so correctly.  I say this not to discount the standard algorithm.  Rather I wish to point out that in this case, when it’s necessary to borrow three times, the standard algorithm is blatantly inefficient.

The students had to know why on earth I would give the cashier $20.11 rather than just $20.  The answer: Quarters.  Because if you’re at the store with a 4 year old and you do not have a quarter for a gumball machine, god help you.

The main portion of the lesson was the real magic. This problem is from Strength in Numbers by Ilana Horn:

Imagine that you have two pockets and that each pocket contains a penny, a nickel and a dime.  You reach in and remove one coin from each pocket.  Assume that for each pocket, the penny, the nickel, and the dime are equally likely to be removed.  What is the probability that your two coins will total exactly two cents?

They sit in groups of three or four.  I gave each group a large piece of paper, had them put a circle in the middle for their final solution and then divide the paper into 4 sections for their individual work.  When looking through my pictures of student work, I noticed that I have a tendency to capture correct work (but differing methods), but I do not take photos very often of incorrect work.  Today, I changed that.

Here is a sample of their strategies for determining the number of outcomes:

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The level of discussion was exquisite.    But what’s more important was that they were able to work together to organize their thinking and to make sense of their solution.  They built on what they knew an gained conceptual understanding as a result.  In addition, they were able to focus on understanding their path to the solution rather than simply being satisfied with the solution itself.  I’m very proud of them.

Making Math Talks a Habit

How many dots are there?  
How many dots are there?  

One of the best experiences about being a teacher is the opportunity to bear witness to student sense-making.  I enjoy hearing learners help one another develop different ways of approaching problems because I know this is a skill that will transcend mathematics class into when-are-we-ever-going-to-use-this land.

I was first introduced to the idea of a Math Talk when I was taking Jo Boaler’s online course How to Learn Math.  This one is simple enough that anyone able to count can do it.  Seriously, take a second and give this one a go:

How many dots are on the card?  How did you determine your answer?  

How many dots are on the card?  How did you determine your answer?

The answer of ten is hopefully quite obvious to your students.  But it’s the incredible number of ways in which they determined that answer that blows me away.  Is it two rows of 3 and two rows of 2?  Or is it 4 diagonals of 1, 2, 3, and 4?  Maybe 5 in the top 2 rows and 5 in the bottom 2 rows?  Perhaps 5 pairs of vertical dots catches their eye?  THESE ARE JUST DOTS, PEOPLE!  All of this awesome thinking over dots arranged strategically on a piece of paper.  But these dots opened the door to my getting my students to explain their thinking to one another.

Fast forward to MCTM this past weekend.  I was reminded of the power of the Math Talk at a session hosted by Christy Pettis and Terry Wyberg.  I knew Fawn Nguyen had some wonderful examples on her website, so I jumped in.

The results have been lovely.

Monday:  Which is greater 79×25 or 75×29?

Tuesday: Visual Pattern #10

How would you have determined that there were 85 puppies in step 43?

How would you have determined that there were 85 puppies in step 43?

Wednesday:  Which is greater 12/17 or 5/8?

There were many lovely responses to all of these questions in each of my classes. But the one that stands out as my favorite was Caytlin in my 5th period Algebra 2 class.  For Wednesday’s problem, Caytlin says that it’s easier to compare the reciprocals of those fractions, so she flipped them over to compare 17/12 and 8/5.  When converted into a mixed number, 1 and 5/12 is smaller than 1 and 3/5.  The opposite would be true for the reciprocals of the numbers.  Therefore, 12/17 is larger than 5/8 since its reciprocal is smaller.

Honestly, isn’t that golden!?  What I love about math talks is that students are asked to make sense of the problem themselves.  They aren’t shown an example or taught a rule.  They develop their own method and then help their classmates by sharing it.  There have been a lot of good experiences in my classroom this year, and math talks rank up there near the top.

(For additional information on math talks, I recommend the book Making Number Talks Matter by Cathy Humphreys and Ruth Parker)

Secret Teacher Problems

Colorful chalks on the paver street. Some of them broken some of them not.

I came clean recently about my addiction to alcohol, and this past Tuesday, March 31st, I celebrated 3 years of sobriety.  As problems have arisen (a broken thumb is a fine example), I know that no problem is so big that alcohol won’t make much worse.  Trust me, I’ve tried alcohol as the solution more times than I can remember.  It failed ten times out of ten.

Rewind about 4 years, and you’ll see a teacher who’s put together on the outside, but falling apart within.  I kept beer hidden in the garage, wine stashed in the basement, and vodka in my purse.  All because of fear.  Fear that people would discover the unhappy person that loomed underneath my smile.  After a drink (or five), I could keep up that smile that I believed others wanted to see.  I had a happy marriage, a beautiful daughter, a great job, and financial security.  There wasn’t a reason to not be smiling, so instead, I drank.  And then I constantly lived in fear that my secret would be discovered and that i wasn’t truly as happy as I outwardly appeared.

As teachers, we are all control freaks. Every. Single. One of us.  I’m even exhibiting controlling behavior with my insistence of putting on a happy face outwardly.  It’s a more subtle type of control, but it keeps others at bay from what’s going on beneath the surface.  It’s similar to those who use humor to deflect serious conversation.  If we keep people laughing, we deflect their attention from the pain that lurks beneath the surface.

Yes, the degree at which we exhibit control varies from teacher to teacher, but can you think of anything more terrifying than a chaotic classroom? I know I’ve been awakened from that nightmare more than once.

Even if you’ve had no brush with addiction I’d venture to guess that the Serenity Prayer is something familiar to you:

God, grant me the serenity to accept the things I cannot change, the courage to change the things I can, and the wisdom to know the difference.

Putting any religious context  aside, this prayer has powerful implications to the classroom . what is it in my classroom that I have the power to change? Do I have the necessary courage to face the challenges head on? Can I discern the difference between what is teachable and what is not?

More important recently, am I able to take what I preach in my classroom and apply it to my personal life? Do I have the courage to say “no, I cannot,” but more importantly be at peace with a decision to step away?

Because as control freaks, it’s one thing to say no and quite another to accept that decision. Relating this back to the Serenity Prayer, I’m very good at trying to change the things I can. I’d also say I do a decent job at knowing the difference between what I can change and what I cannot. It’s the serenity to accept what cannot be changed that gets me every time.

What a Difference 12 Kids Make


We’ve entered Spring Trimester and the volatile Minnesota weather is cooperating thus far.  If there’s a silver lining to last year’s Spring suckfest, the lack of warmer weather put off the end-of-the-year slide until closer to May.  I’m not sure we’ll have the same luxury this year.

I teach the same level of Algebra 2 that I did last year but my class sizes are a more manageable 22-24 rather than the monstrosity of 36 I had last year.  I know class size isn’t high on Hattie’s list of influences on student achievement, but providing formative evaluation (something VERY influential, according to Hattie) is much more doable with 20-something rather than 30+.

I’ve left the desks in pods because I’m convinced students interact and collaborate mathematically more often when they have multiple classmates within conversation distance.  I want to switch their groups periodically, if only I could get them to sit in their assigned seat!

One of my go-to resources is the Mathematics Assessment Project. Their lessons are robust, and provide good opportunities for students to have great conversations around the mathematics.  This lesson on investments is no exception.  The main activity is a card sort where students match a principal and interest rate of an investment with a formula, graph, table, and description.  But the everything from the pre-assessment to the closing slide makes students think and share.

Here are the openers of the main lesson:





My assumption, not being familiar with this group of kids, was that they’d go right for the obvious – Investment 3 is the odd one out because it has a 10% interest rate and the others have a 5% interest rate.  I underestimated them.  They came up with very creative, thoughtful reasons why each investment could be considered the odd one out.  I really like these questions because all three can be correct, and students have an opportunity to defend multiple answers.

The card sort was also spectacular.  I was able to have great conversations with each group about their thinking. (Yes, that’s much easier to do with 24 rather than 36).  What a difference 12 kids makes.  There is so much to this lesson to love.  If you have a unit on exponential functions, give it a try.  I’d love to hear how it went in your classroom.

Emotional Baggage Around My Wrist


This blog post is more personal therapy than it is educational reflection, but I hope that others dealing with similar strife can relate and find solidarity.

Two weeks ago, I was in a car accident on the way to school on a clear, dry, sunny Monday morning.  I was heading north on a county highway when a car pulled out from my right to head south.  My SUV slammed into him at about 50 miles per hour deploying all of my airbags and sending my car into oncoming traffic.  When my car came to a stop on the opposite side of the road, I was relieved that my Sync system was dialing 911 on my behalf.  Who knows where my cell phone was at that point.  Fifteen minutes later, a police officer arrived.  The driver of the other car was issued a citation for distracted driving and my mangled Escape was hauled away.


My Ford Escape immediately following the accident.

The pain that remained in my hand kept intensifying as the morning went on, so instead of waiting for a doctor appointment to get myself checked out, I headed for the ER.  Sure enough, my left thumb was fractured.  The airbag that saved my face instead broke my thumb when I gripped the wheel, bracing for impact.

I was angry.  Thumbs are important.  Two working thumbs are a lot more productive than one.  I didn’t cause this accident, yet I’m the one left with a broken hand, smashed car, and sore neck. To add more gas to my resentment-fueled fire, the Doogie Howser-esque orthopedic surgeon informed me that my busted digit would require pins to heal properly.

This past week, I’ve thankfully moved past anger but haven’t seemed to be able to rise above the emotional grip this hand cast has had on my day-to-day functioning.  I couldn’t imagine that having difficulty taking the cap off of a dry-erase marker would have such a strong emotional impact.

This week we began a new trimester, so I struggled with the decision to take a few days off of work so I could wrap my brain around this injury and release myself from the emotional handcuff surrounding my thumb.  Our culture accepts physical injury, but the unseen toll on our mental well-being is what really needs the most care.

The part of the first step in Alcoholics Anonymous is accepting powerlessness over alcohol.  While this is true for addictions in general, that thinking is also applicable here.  I am powerless over my broken thumb, the 6-week casting period and my physical disabilities resulting.  My emotional interpretation of the event, however, is completely in my control, and I’m determined to make the best of it and be a better person because of it.  Life is full of lessons, and this one is teaching me humility.  I am determined to be humble enough to accept it.

Dear Students: No. I’m Not Sorry.

Picture from:
Picture from:

I read a very moving blog post in which a teacher apologizes to her students for the problems plaguing our education system and hold students back.  While I agree with Lizanne Foster’s view on the way schools are structured, I felt she was taking blame for elements of students’ experiences over which she has no control.  I wanted to follow her blog up with a reminder that students don’t need teachers to feel sorry for them; they need teachers to empower, inspire, and motivate them to do better when they leave the classroom than they did when they entered.

Although you have to be at school so early each morning, I promise to make those early morning minutes worth getting out of bed for.

Even though you have to ask permission to use the restroom, I promise to respect your good judgement for appropriate times to leave the classroom.

I promise to create opportunities for you to get out of your seat and move about.

I promise that even though you are pre-grouped by age, I will provide you with problems that engage all levels of intellect:  problems that stretch you as well as provide scaffolding as needed.

I promise to create an environment where you feel safe in making mistakes.

I promise to allow you to solve problems collaboratively because I know “together” is where the best solutions come from.

I promise to work hard to provide the support you need to further your learning.

I promise to do my best to help open your mind to subjects and ideas that may have once seemed boring.

I promise that you will have my respect at all times and do not have to earn it.

I promise to never make you compete for grades.

I promise to give you opportunities to apply mathematics to solving our world’s economic, environmental, and political problems.

I promise to encourage curiosity throughout your learning experiences.

I promise to always let you examine, explore, experiment and experience.

I will try every day to re-ignite your passion for learning you had when you were young.

I will attempt to bring out your inner-scientist/writer/architect/artist whenever I am able.

I accept and understand that you were born to learn and that memorizing is not learning.

I promise to never make you feel that the only learning that matters is learning happening in a classroom and I promise to never focus your learning on just what will be covered on the test.

I promise to facilitate as much “out of the box” thinking as I can and will always present problems that allow for multiple solution paths.

I am mostly powerless over these powers-that-be that determine funding for your education, but I will do anything I can within my control to make learning in my classroom a positive experience for you and your classmates.


Your Teacher