Stringing Students Along


If I’ve done one thing consistently this year, it has been Number Talks in my Probability and Statistics classes.  I have seen students who, at the beginning of the trimester, told me flat out, “I can’t do math in my head.” Now that Trimester 1 is coming to an end, those same kids are volunteering multiple strategies in these mental math challenges.

During the trimester, we started with the dot image below and have moved through the four operations, onto decimals, and even dabbled in fractions and percents.


How many dots are there?  How did you count them?


What’s important to me with these number talks is the visible improvement I saw in my students’ confidence and flexibility with numbers.

I’ve shared before about my experience with number talks and I plan to continue these throughout the rest of the school year.  But at the NCTM Regional conference in Minneapolis a couple of weeks ago, I had the pleasure of attending Pam Harris’s session on Problem Strings.  I found that problem strings are very useful when wanting to elicit certain strategies or move toward generalization of a strategy.

Here are my notes from a problem string I did recently with the same group of students I have been doing number talks with.


I noticed:

  • Many students did not use “17 sticks in a pack” to figure out sticks in 10 packs
  • Many more strategies than expected were shared to find the number of sticks in 6 packs of gum.
  • Most students were able to generalize about number of sticks in n packs.
  • Participation increased with the multiple opportunities to volunteer their strategies.
  • Students could see relationships between the numbers and find the solution in multiple ways because of that relationship.
  • There are many implications of these problem strings in secondary mathematics. In this example, the slope formula can be easily elicited through further exploration of the table we made.

I’ve read all of Pam’s books, but getting to see her present problem strings in person really illuminated how these can be useful in my classroom. Thanks, Pam, for opening my mind to this and letting me fangirl you.  I’m looking forward to doing more of these, including recording them.  Stay tuned.

Regional Reflection – Releasing my Grip


As humans, our complex brains are able to create such detailed visions of the future.  We build things up (or down) in our minds that reality can’t possibly compete with.  Until we let go of what we believe should happen, we are unable to fully experience the beauty of what actually is.

Proposals for the NCTM Regional Conference here in Minneapolis were due in September of 2014.  This means I have had over a year to continue to wind the anxiety yarn into one giant ball of stress.  But sweet relief occurred when I released my iron grip on my expectations and began to appreciate the phenomenal power of educators coming together.

First off, thank you, from the bottom of my heart, NCTM, for  your support of the MathTwitterBlogosphere at the NCTM conferences. I spent much of my time at the #MTBOS booth in the exhibit hall.  Sharing this wonderful, supportive, organic community with other math educators has been as fulfilling as it has been fun.


East Coast meets Minnesota Nice


You guys have something called the “Trap Team?”


Woman: You didn’t say there would be math. Christopher: Actually, I said there would be nothing BUT math.


When Nicole Bridge gets fired up, the magic happens.

When asking people in the Exhibit Hall “are you on Twitter?” the most common response was “yes, but I don’t tweet.  Think of the student in your class that thinks very deeply, submits very thoughtful work,but doesn’t raise his/her hand in class to volunteer his/her thinking.  I’d hope that most teachers would agree that these students are still valuable members of the classroom community.  It works the same with the online edu-community.  Plus, I’d venture to guess that many people who actively tweet with other math educators started by diving down the rabbit hole of math blogs.

Max Ray-Riek led a panel where we discussed this problem and blog post of mine.  Next week we venture into rational functions in college algebra and I anticipate good times to be had once again.

An hour later, Carl Oliver and I spoke on statistics, social justice, and how to have safe, productive conversations with students around the issue of race and equity. Here is the link to the slides.  The discussion centered around these data sets:

Using Local Data to Teach Statistics

Using Local Data to Teach Statistics (1)

I really enjoyed giving our presentation and a lot of great discussion ensued.  But ultimately, I’m thankful to the MathTwitterBlogosphere for being the catalyst of the great discussion we get to take part in, day in and day out.  I had never met Carl Oliver in person before Wednesday.  But the powerful connections we (all of us) have made with one another, make it possible for an algebra teacher in New York and a stats teacher in Minnesota to get together and share their passions with fellow educators. It allowed a teacher in Massachusetts to spread the fire she started in Boston on to Atlantic City, Minneapolis, and Nashville.  And that fire is continually kindled as we welcome, share, engage, and support over and over and over again.  Thank you, #MTBoS for being the genuine, authentic community that has naturally produced so much awesome for so many teachers.

But Would You Put Money On It?

Doubles Dilemma (1)

I have felt one of two extremes every day this school year:

  1. My students aren’t learning anything meaningful, it’s impossible to do everything I need to do well, and my brain is on fire.
  2. Cheers!  My students had fun while making meaningful mathematical connections.

Today was the latter kind of day so I thought I’d take a few moments to embrace it.

I proposed this scenario to my non-AP probability and statistics class:

Doubles Dilemma (1)

I had students discuss their initial reactions.  Many of them mentioned specifics like “1 out of 6” and “36 possibilities” but for the most part, the students were willing to put their hard earned money on the line for a chance at avoiding doubles.  (To be clear, no actual betting went on in my classroom)

Then we rolled until we got doubles.  And rolled again and again and again.  I have one computer and a class set of TI-84s.  So, naturally, we made a class dot plot of our average number of rolls to get doubles.



Now that our data was collected, I asked them again if they would take the bet.  Since $5 didn’t seem to be enough money for them to really consider the probability, I upped the wager to $100.  That seemed to be enough money for them to consider the results of the experiment and think twice about putting up $100 because they feel lucky.

Thanks to Chris True, Mathematics Professor at the University of Nebraska, who proposed this scenario at an AP Statistics training I attended this summer.


We are Better Together


We are better together.  Say that out loud.  Go ahead.  I’ll wait.

We are better together. 

I just returned home from Edcamp Math and Science at Eden Prairie High School.  (Beautiful campus, by the way.  Thank you for hosting us.)  I’ve made a conscious decision over the last year or so to only attend conference sessions on topics I’m already using so that I can refine and improve.  It’s too easy for me to get swept away in the glitz of new classroom tools that draw me in with edu-buzz-agogy like “classroom engagement” and “streamlined feedback.”  Instead, I focused on two things:  Number Talks and Desmos.

I attended Christy Pettis and Terry Wyberg’s session on Number Talks at the state math teacher conference last May and learned a lot, so you didn’t have to twist my arm to get me to listen to them again.  A quick survey of the room revealed that the group ran the gamut of novice to expert when it came to experience with this transformative classroom routine.  I’ve used these in my classroom regularly and was still able to gain many useful strategies to make this process even better.  I loved how Christy was able to turn the strategies into area models so that students make that connection.  That was something I had not thought of but will definitely be implementing starting Monday.    Again, it’s worth repeating:  We are better together.   Here are my notes:


Next up:  Desmos.  The program speaks for itself but it was lovely to have someone on their payroll available to demonstrate its flexibility.  Thanks, Christopher.  Who knew projector mode was so amazing! And I never knew how to create a dragable point.  Child’s play, I know, but new to me.


Right before lunch, I joined Seth Leavitt for a conversation on race in math and science.  An overarching theme was that students of color are over-represented in remedial math classes.  Seth encouraged a continued conversation with leaders from our school districts on equity and access in mathematics and science.  I’m committed to this ongoing discussion in St. Francis and to ensuring our students of color have opportunities to take high level mathematics.

Thanks, Casey Rutherford, for organizing this again this year and allowing us to get together and get better.  Teaching is hard, but we are better together.





Conceptual Function Foundation

I’ve taught College Algebra for a number of years.  This course and College Trigonometry replace Pre-Calculus at our school.  I’ve struggled with helping kids with functions because of the variety of background knowledge they have on the topic.  I have tons of good activities, but never one that really built a conceptual foundation of the important features of functions in general.  It’s not that it’s impossible to create a conceptual foundation after procedures have been introduced.  It’s just really difficult to do.  (Remember this post from Christopher Danielson?)Enter New Visions for Public Schools.  Unit One of their Algebra 2 course allows kids to make sense of families of functions in their own way.

You can see the details yourself on the link, but in summary, kids sort graphs according to their own criteria and then build a definition of a key graph feature and re-sort accordingly.  Students then form new groups and share their key feature with their classmates.  Finally, the group as a whole creates statements that link the key features together.

Dan Meyer states that math is the process of confusing and unconfusing.  This progression does that perfectly.  Conceptual Understanding Achievement unlocked!


  • Students are asked to make sense of graphs based on their prior knowledge.
  • They develop a need for certain vocabulary such as “turning points” as they discuss key features of the graph. For example:
  • They need to take responsibility for their learning because they need to teach it to other students during the “jigsaw” portion.
  • They have to ask clarifying questions of each other rather than of the teacher creating student-centered discussion.

The real beauty was watching three days of making sense of graphs come together with the vocabulary.  The students are asked, with their groups, to find how the key features are related and how they are not related.   I wish I had an audio recording because it was some of the most beautiful student discussion I may have ever heard.  I captured this moment just so I could remember it:

Students discussing how function features are related and how they are not related.

Students discussing how function features are related and how they are not related.

Also, here is one of the reference graphs a student made:


Where Do We Go From Here?


Hundreds of students come in and out of my classroom every year.  And after four short years in high school, they are onto the next stage of their lives, whatever that might be.  I get a few friend requests on Facebook from former students, but very few relative to the number of students I’ve taught over the last eleven years.  Seeing them grow into adults with spouses and jobs and families always brings me joy. But so many of them I never hear from again, and there’s nothing wrong with that.  They go out into the world and grow up.  We have to assume we did the best we could to make a positive impact.

I have a folder in my file cabinet where I keep special mementos from students:  thank you notes, drawings, and other delights I’ve collected over the years.  Andrew Stadel recently requested memorable moments from our teaching careers and so I went digging through this file folder to find Algebra version of M.C. Hammer’s Can’t Touch This that I adapted for my class made up of mostly choir and band 9th graders. [No, I’m not sharing it, and No, there was no cell phone video back then].

As I dug into that folder, I also found this:


It didn’t seem to have the same pick-me-up tone as the other papers in the folder, but I know exactly why I kept it – to remind myself of my privilege.  To make sure I am always cognizant of the struggles my students endure when they aren’t in my classroom. And to make it clear to myself that I teach people first, not mathematics.

Never for one second did I believe that this kid ended up in jail as he was convinced back then.  I reached out to him using my old stand-by:  Facebook.  Not only is this student not in jail, but he is thriving as an entrepreneur in IT, has a child on the way, and is living happily with his beautiful fiance.  With his permission, I am telling his story of triumph over his adolescent years where happiness seemed out of reach and success seemed hopeless.  His story of resilience has made a positive difference for me as an educator and will continue to help the future students that step into my classroom year after year.

Appreciating the Larvae and the Butterflies


Last month, I was attending a meeting with our district’s curriculum specialists.  A science specialist said this:

When we order our larvae this year, can they be sent directly to the elementary school or do they have to go through the office of curriculum and instruction?

The question made me laugh out loud simply because of its somewhat unexpected nature.  But then I wrote it down, and I stared at it and thought why on earth would the larvae used in an elementary classrooms need to go through the office of curriculum and then distributed to the teachers?  But this is how it always was done.  The teacher went on to explain how the larvae ended up being very small and many didn’t hatch as expected leaving some students to be very disappointed and even sad.  Because every child wants his or her larva to turn into a beautiful butterfly.


There was a simple fix.  Send the larvae directly to the schools.  And that’s what happened.  But the question made me think about our jobs as educators.  Criticism of the work others do is a tempting thing to partake in.  But do we really have any idea what our fellow teachers deal with?  It would be easier to criticize this science teacher’s end result:  disappointed students. But the real culprit: a supply-chain management issue.    Another example, I get frustrated when kids don’t have pencils. But I would have no idea how to explain to a child that his or her larva didn’t survive and become a butterfly.

School is well underway, and here is my challenge to myself (and anyone else who would like to join me).  Let’s take time to appreciate that each of us is doing the best we can.  Blaming is easy.  If kids can’t factor, it must be their previous algebra teacher.  If kids can’t multiply, it must have been their elementary teacher.  In reality, learning is a very complex, non-linear process that does not necessarily bear fruit on a regular, seasonal basis.  Sometimes, we only get to see the larvae-portion of another colleagues job.  Larvae are easier to criticize than the beautiful butterfly that results.