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.

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.

Magic Doesn't Always Happen in my Classroom, but When it Does...

I used an activity this week where I introduce the method of using Lowest Common Denominators to eliminate fractions and decimals in 'tricky' equations. On the activity page, I refer to this trick as a 'good idea,' but in class I said, "It's like a magic trick: I wave my LCD wand and presto, no more fractions!" They laughed, so I kept it up.

"Ack. This one looks yucky. I need some magic." Snicker.
"See, here's where the magic happened." Chuckle.

At one point, a student raises his hand and says "You know Mrs. Allman, it's not really magic, it's just logic."

Nice. But aloud I said, "You know what you are? A muggle."

Jaws dropped. Did she really just call him a muggle? Putty in my hands now.

The next equation had some decimal coefficients: 

1.2x + 0.4 = 7.6

I asked, "What's the LCD?"

Someone said, "There aren't even any fractions." (I swear, he wasn't a plant.)

I spun around and looked again, "There AREN'T?"
So I waved my hands and said, "What's this number?"
"One point two."
"Didn't your 5th grade teacher tell you never to say the word 'point' in math class? How are you supposed to say it?"
"One and two tenths."

There was an audible gasp from the class. I cannot make this up.

So now they're fighting over whether 5 or 10 is the LCD of this group (cool) and whether it even matters, and which is more magical, and I could NOT have planned this better. At one point, a student shows me his work "Mrs. Allman, I used a different number to multiply both sides and I still got the right answer. It doesn't matter what you use."

AWESOME!
"Ah," I said, "And that's a beautiful thing. You've confirmed one of the properties of equality: that you can always multiply both sides of an equation by ANYTHING, and it will not change the solution. The magic comes from knowing which numbers will make the decimals (or fractions) disappear, and (hopefully) make your life a little easier."

There might still be pixie dust on the floor.

Powers, Roots, and Logs are Related Facts

This year, in my Algebra 2 class, I prefaced our individual function units with a overview of functions in general. One thing that happened is that before we studied exponential functions, we had a decent understanding of inverses and how several functions are related in this way. Several weeks before I ever needed to hint at the existence of logarithms, the students saw the need for an inverse to an exponential function and also were stymied by the relationships that are already comfortable to them: namely, the existing inverse relationships between powers and roots.

So this month, when we got to the middle of our exponential function unit, I decided to present logarithms as a group of THREE related facts in a fact family.

We listened to a fabulous Radiolab program that presents numbers and logarithmic thinking as a human interest story. I am so grateful for programs like this that do my hard work for me!

Then we talked about how powers, roots, and logarithms are all different ways to say equivalent things, while each highlighting a different feature.

Fact families are something that students are familiar with. No one batted an eye.
When I asked them what is meant by logarithmic thinking, I was happy with how their explanations centered around exponents and thinking about 'how many times a number is doubled or tripled,' etc.


We concluded with some fact practice by using fact triangles and naming the three related facts. I made these awesome octahedral dice with fact triangles in base 2, base 3, base 4, and base 5. Their job? Roll a die and record the three facts that can be written from the trio of values.

The students whipped through it, never complained, and had 100% accuracy on our quiz the following day. I'll count it as a worthy addition to my filing cabinet.

Algebra 2 is All About Exponents

I don't know if this makes function families easier or more complicated, but I realized this week that everything we cover in Secondary Algebra (1 and 2) can be reduced to two basic function families: f(x) = n^x and f(x) = x^n (trig functions excluded, but we don't cover those until precalc at my school). I don't remember this ever getting pointed out to me when I studied functions, and seriously, how many years have I been teaching this? I just find it very interesting.

We made these organizers in class today (and yesterday). We've already got our brains wrapped around transformations and compositions, although we have thus far stealthily avoided talking about operations on functions (adding/ subtracting and multiplying/dividing)... other than to say, "Ugh, two x's, that looks messy." Which of course, it IS, right? And isn't that the point?  At least, I think that's my point this time through. Basic functions that involve these families, simple transformations, and compositions where one step follows another in a specified order... this kind of function is not hard to work with. They are logical and orderly. It's only when we start multiplying and dividing or adding/subtracting functions that stuff gets tricky and we need to start pulling out new tricks like factoring and zero product rule, and complex numbers, and extraneous solutions, and reducing rational expressions, and limits, oh my.

Why aren't exponents and roots always inverses?

We've been studying functions in my Algebra 2 class. I'm taking an entirely new approach this year. It's going well, but the jury's still out. So far, one thing I'm really happy about is their excellent grasp of inverse functions as a process of 'undoing' whatever the original function did. Today in class we were organizing our thoughts about the different function families. We'd drawn out some nice examples of expoential growth and decay and I asked if we could figure out the inverse functions. I EXPECTED them to see that they could not write a function (we haven't done logs yet), but that they could use the tables and graphs that they just made to draw an inverse graph. Our function was a simple "2 to the power of x."

But they surprised me. "The inverse would be the xth root of 2," someone said.

Several agreed. I was dumbfounded. Why not?

After all, isn't it TRUE that:

for all real values of n?

That is, I know there are some issues with this as a blanket statement. For example, even values of n only work if we restrict the function domain. But for the most part, this is entirely true and logical.

So why is it FALSE that:

Maybe you, like me in class today, are scratching your head now and wondering... could that be right? It isn't. We checked. Choosing an input for the original function, applying the function and then applying the inverse function will not return us to our original value. But WHY?

Help me out here. Can anyone provide a purely sensible argument for why this will not work? Not just a demonstration of HOW it doesn't work. I can supply several of these now. I want to know why.