Making Math Talks a Habit

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)

Number Talks – Bumps in the Road

Today, I presented, with my colleague, Denise Anderson, a session on Number Talks at the Minnesota Department of Education.  The topic was ‘Where do we go from here:  Managing bumps in the road.’

Here are the slides if you are interested:

I’ll be presenting this session as a webinar on Tuesday, May 24th.  Details here:

My favorite slide:  0513161500



Curiosity Driven Mathematics

In my very first years of teaching, I used to have students ask me, in that age-old, cliche teenage fashion, “When are we ever going to use this?”  I vividly remember my response being, “Maybe never.  But there are plenty of other things we do in life, like play video games, that have no real-world application. That doesn’t seem to bother us too much.”

In fact, if every moment of our lives needed to apply to the bigger picture, the REAL-world, when would we do anything for pure enjoyment? or challenge?  or even spite?  I know kids are capable of this because some of them spend hours upon hours a day engaging not only with a video game but also collaborating with other people through their game system.

And furthermore, where do we think this resentment for learning math really comes from?  I have a guess…probably adults who have realized that through the course of their lives, being able to solve a polynomial equation algebraically is not all that useful! News flash, math teachers:  Our secret is out! 

There are many kids across all levels of achievement that will not engage in the learning process simply because the state mandates it or the teacher swears by its real-world relevance.  Students (and arguably people in general) are motivated by immediate consequences and results and cannot easily connect that the algebra they are learning today will be the key to success in the future.  They do not care that if they don’t nail down lines, they’ll never have a prayer understanding quadratics.  If they are bored to death by linear functions, I can’t imagine that they have even an inkling of desire to comprehend the inner workings of a parabola.  

What does resonate with learners is the satisfaction of completing a difficult task, puzzling through a complicated scenario, or engaging in something for pure enjoyment.  Kids are naturally problem-solving balls of curiosity.   There are ways to provoke curiosity and interest while simultaneously engaging in rich mathematics.  I think many teachers assume that in mathematics, especially Algebra, curiosity and deep understanding need to be mutually exclusive, and I’m positive that mindset is dead wrong.  For example, show this card trick to any group of kids, and you’d be hard-pressed to find a group who isn’t trying to figure out how it works.  I also think you’d be hard-pressed to find the real-world relevance to a card trick.  It’s still no less amazing, as well as algebraic.  



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.

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.





Safe Spaces at #TMC15

We talk a lot about creating safe spaces for students, but it’s important to acknowledge that teachers need those safe spaces as well.  In many ways Twitter Math Camp IS that safe space where math educators can explore their ideas without judgement and overall be themselves.  But I realized this time around that as inviting as #MTBoS is, it’s ok to seek out further safety nets.

Based on my own experience and what I’m reading on Twitter today, it seems as though there are many of us that need a safety net from the safety.  An example:  everything from Twitter Math Camp ends up on Twitter. There was a presenter who purposefully refrained from using pithy comments simply so they couldn’t be taken out of context on social media.  I’m sure we’ve all experienced a tweet interpreted differently than we intended.

It’s not a secret that I deal with alcoholism.  But this post isn’t about my issue in particular.  It’s about our need, no matter the issue, for a safe space.  And there needs to be safety within the safe places.  Anne Schwartz talks about this in her recent blog post where she talks about surrounding herself with the people she needed after being apprehensive about attending TMC this year.   For Julie, the safe place was the Piano Bar where she could break away and be free to dance with close friends like she loves.

It took me until Saturday to recognize what that safe space looked like for me.  And once I realized it, it was so crystal clear, I can’t believe it took me 2 TMC’s to figure it out.  I don’t mind being around the alcohol one bit, and I don’t want people to feel uncomfortable drinking around me.  In fact, the silliness at the end of the night is usually something I enjoy (sorry not sorry, you guys are hilarious after a few drinks).   BUT at some point in the night, I need a safe place a deep conversation with someone who isn’t drunk.    At TMC14, that person was Justin Aion who walked 1.5 miles with me to get club soda and spent a good deal of Saturday night listening to me.  In Claremont, the sobriety of my pregnant roommate, Teresa, was more important than I realized pre-TMC.  Thank you, Teresa, for just being in the right place at the perfect time.

The patio on Saturday was delightful.  So many people, so much joy, so much community.

But all I could see was the alcohol.  It literally was suffocating me.

I know there were probably lots of you that weren’t drinking.  But the addicted mind sees what it wants, and my imagination had created a courtyard drowning in liquor.  So I returned to my room, texted a friend and called my husband.  This was a powerful realization for me because although those safe places presented themselves organically in previous TMC gatherings, it’s vital that I proactively ensure that safety exists from the get-go.  And that’s what I will do from now on before heading in unprepared.

I know a similar story can be told for a lot of us regarding our interactions in large groups of semi-familiar people.  I would encourage you to look deeply inside and identify the source of your discomfort and examine what can be done to alleviate it.

Yesterday:  As much as I get frustrated by the attitudes and actions of my 5th hour, much of my resentment stems from the fact that I believe the situation in my class is my fault.  I feel like I’ve conditioned them by accepting disrespectful behavior in order to keep kids in the classroom.  As a result, the entire learning environment has suffered.

Today: So that was the beginning of yesterday’s post. I was concerned going into today’s class. Last Friday of the year and the fact that the school has been a circus compounds the issue. I was expecting chaos, but what I got was mathematical success. The difference was I demanded their attention in a more respectful way. I was firm, but polite, and it payed it’s dividends in student engagement.
We began with a simple math talk that I modeled from Fawn Nguyen’s March 21st math talk:

Today is the 30th day of the month.  Write as many equations you can that equal 30.

I gave them about 5 silent minutes. Then I let them use their calculators to come up with more gems.  At the end, I had them share their favorite or most complicated equation on the whiteboard.

IMG_5388 IMG_5389 IMG_5390


Here’s where the real magic happened.

Me:  Look up here and see if there are any equations you disagree with

Lots of discussions ensued about order of operations, square roots, rounding, parentheses, etc.  Overall, the activity lasted 30 minutes, which was about 29 more minutes of math than we did yesterday.

But the fun doesn’t stop there.  To boot, I introduced the Mathalicious Decoder Ring Lesson.  We watched the Christmas Story clip and talked about what a decoder ring does.  What I liked is that most of them were trying to figure out how the decoding worked, rather than just “get the worksheet done.”

Justin Aion stated poetically on his blog today the exact way I feel about this class:

We teach students long-term strategies to accomplish short-term goals and often don’t see any progress.  If we are very lucky, we’ll see the kind of growth we want by the end of the school year, but the growing season on students isn’t as regular as it is for other crops.  Each seed needs its own time to grow.We desperately need to get away from the notion that if it hasn’t sprouted by the beginning of June, then it must be a defective seed.