Monday, June 12, 2017

Lessons from the Math Team Kids

Math Team Awards Night 2017
For the past three years I have been filling my free time with a fun little extra-curricular project: Math Team for third, fourth, and fifth graders. Being a high school math teacher, but also a mom, this was a way for me to enjoy some mathematical time with my own children and their friends. I remember my own mother doing something similar when I was in elementary school.

So I sent home a flyer. I thought maybe I'd get a dozen kids who were interested in after school math with me. In the first week, I had 36... and they kept coming back - week after week for an hour and a half of additional math after the school day was over. This year, our third year, we had 70 students coming once a week all year long in two separate after school programs and another dozen coming for math club in the middle school. Hooray! I am very proud of these kids and so grateful for the abundant support of staff, administration, volunteers, parents, community, etc. Go math team!

Now that awards night has ended and there is a moment of calm, I can pause to reflect on a couple lessons I have come to appreciate and don't want to forget:

Mathematical Ability is a Many Colored Beast 

Some kids joined Math Team because they felt 'good at math', others because they liked puzzles and other mathematical amusements. Some joined because their friends were joining. Some joined because their parents wanted them to improve their math skills. Some joined because they had no where else to go. Ultimately, despite our differing motivations, ages and abilities, we spent time together engaging in and talking about math every week from October through May.

There were some kids who were fast with tricky calculations in their heads. They made our jaws drop. There were some kids who consistently saw a way through a problem that was beautiful and efficient and different from anything else we thought of. They made our eyes open wide with wonder. There were some kids that could explain their thinking so that it felt clearer than our own. They made us want to listen. There were some kids who could listen to a forming idea and help nudge it in a productive direction. They made us want to share our thoughts. There were some kids who jumped up and down and shared every idea, productive or not. They made us feel excited and happy to be together. There were some kids who persisted in asking question after question. They helped us overcome our own feelings of uncertainty and self-doubt. There were some kids who could sit and struggle with a single problem long after their peers had given up hope. They helped us to remember to take deep breaths and let the rest of it go.

I see that any one of these traits can grow to become the foundations of a successful mathematician. I do not need to be fast with mental calculations to be a successful mathematician, but at the same time, those who ARE fast are amazing. It's important for me to own this: each strength is beautiful in it's own right. Of course, the ultimate goal is to recognize and cultivate habits in each area, celebrating our personal strengths and learning from the strengths of others. In this way we find that we continue to grow as mathematicians far beyond our original ideas.

Heterogeneous Groups are Not the Best Groups for Mutual Growth

I know this heading seems a little contradictory to my poetic "we all learn from each other" musings in the previous paragraphs. I truly loved the mixture of 3rd graders who just learned to multiply with 5th graders who are dabbling in algebra in their 'other' free time. There is so much to celebrate in one another and we did enjoy time together as a whole group every week.

But when it came to small group explorations, my experience told me this:

There's only so much waiting time you can expect from a kid who solves a problem first. She wants to share her ideas and her pride and will too soon tell the rest of them how to solve it, often to unappreciative ears. The ability to foster productive struggle, encourage diverse ideas, and grow together as a group is a skill that takes years for a teacher to develop. It does not come easily to kids and often results in frustraton when required.

Low floor, high ceiling tasks were the hallmark of our sessions - a perfect tool for our mutual growth, but they too had a down side. There are kids who are perfectly happy sitting on the floor and discovering for themselves all that is there. And there are also kids who see the stuff on the floor and also the stuff on the ceiling and want it all. These two groups of kids are a mutual frustration to each other. One's desire to climb to new heights causes the other to be ashamed of staying on the floor. They know it's awkward and try to remedy it.  They 'tell' eachother what to do. They pretend they understand. They are all trying hard, but no one feels good.

As a group, we did our best growing when we were with our friends. Some friends met for the first time at math team. Some friends needed more frequent reminders to focus. Some needed to be introduced to new 'math' friends. Some needed more adult 'support'. But people are friends because they complement each other: "I like the way you think and I think your quirks are funny."

We did not need to all work on the same task. We did not need to take tasks to the same levels. We did not need to answer the same questions. We needed our friends. It was loud, but it helped each one feel good about his own growth, and want to keep coming back for more.

I feel like I could go on forever:

  • about the unexpected rewards of struggle, failure, and challenges that are not easily mastered; 
  • about my own struggle to find the right balance of presenting a problem, but not the solution, while sustaining interest, guiding growth, and nuturing each child's needs;
  • about the difficulty of connecting with every kid when there are so many;
  • and about the wonder and glory of a community of support that shows up to help, sends treats to eat, and simply says, "Hey, I noticed what you're doing here. Great job." It's this kind of support that helps us teachers move on to the next day. Thank you.

Sunday, April 30, 2017

My Experience Writing a Conference Proposal for NCTM

This year at NCTM in San Antonio I realized that I have reached my 25th anniversary of NCTM membership. For twenty-five years I've attended inspirational talks, read well-informed ideas, and engaged in important conversations. And each year I think to myself... "maybe I could contribute something." 

Well, this is it. I've officially submitted my first conference proposal for NCTM 2018 in Washington DC. The process of writing the proposal was grueling but undeniably useful. A couple things I've learned? It is SO difficult to:
  1. fully develop an idea now for a potential workshop one year away... so that it feels fresh and current both now and then.
  2. write 'explicit' and 'specific' descriptions that are engaging and thorough, and within the character limits defined by NCTM.
  3. appreciate just how important it is to focus on struggling and under-represented learners.
Thank you to Robert Kaplinsky and Dan Meyer for your professional leadership and to my friends, family and colleagues for your motivation, support, editorial advice, and encouragement. We'll see what happens, but the proposal has been submitted and here it is:

"From Number Lines to Logarithms: How Forgotten Instincts Can Spark Deeper Understanding"
Research suggests that logarithmic thinking is innate. We are born with a number line in our heads, but the numbers are spaced in equal RATIOS instead of equal distances. We will explore how reigniting this instinct can pave an enlightened pathway from number properties and fractions, through ratios and means, to rational exponents and logarithms.

Write the participant learning outcomes of your presentation, including an explicit description of what participants will learn. Please also provide an overview describing how time will be allocated during this presentation.
In four parts, this session seeks to answer these guiding questions:
  1. Introduction - Given a number line from 1 to one billion, where is 1 million? Where would your students place it? What research-based implications can we make? How can we turn our confusion into flexibility and sense making around additive and multiplicative mindsets?
  2. Foundations - How does this flexibility illustrate basic number properties? How does it change our understanding of fractions? How does your vision of later connections impact the way you teach foundations?
  3. Connections - How do ratios compare to fractions? How does our new flexibility change the way we think about "the middle" and "equal sharing"? How do factors and terms help us recognize what kind of thinking is required?
  4. Breaking Barriers - Can we use our new flexibility to break down the confusion surrounding exponential models, logarithms, rational exponents, inverse functions, geometric vs arithmetic sequences and means... and more?
What is the key mathematics content that is a focus of this presentation?
This presentation is about how exponential models/patterns have many connections and similarities to linear models/patterns. Noticing and celebrating these similarities helps make sense of: 
  • Additive and Multiplicative Number Properties
  • Fractions and Ratios
  • Linear vs Exponential Models
  • Logarithms as Inverses and Related Facts
  • Integer and Rational Exponents
  • Arithmetic and Geometric Sequences
  • Arithmetic and Geometric Means

How does your presentation align with NCTM’s dedication to equity and access?
I attribute part of my success in mathematics to an occasional willingness to accept that “it will make sense later.” This kind of persistence through adversity in mathematics is admirable, but it also pushes the limits of student engagement and promotes boredom and withdrawal. My presentation is rooted in the idea that logarithmic thinking is innate. Embracing this instinct makes sense-making and flexibility with multiplicative and additive mindsets accessible to all... now, not later.