Two ten year olds stood at a whiteboard in room 118 with a group of parents seated in a semi-circle around them. There were other small clusters like this in 118, next door, and in a classroom down the hall. 4th grade was celebrating some math.
“I have 10 tickets. Half are lost. How many tickets do I have left?” asked one of the 4th graders.
“Think about it for a moment on your own, then turn and talk to a neighbor about your reasoning” said her partner to the 8 adults.
“Okay, let’s hear your strategies” — said the girls.
“Ten minus half of the tickets is five” — said one parent. “I’m going to write what I heard you say on the board” — said the other girl. “Tell me if you agree.”
Off to the side in a hushed tone one parent said, “Isn’t 10–1/2 just 9 1/2?” But the girls heard and said “Mr. Parent, could you say that to the rest of the group?” Ah ha! Now the girls knew that they had a math discussion on their hands.
Were the girls dismayed that their pupils had come up with an incorrect symbolic notation for their problem? Nope. In fact, they recognized this misstep for exactly what it was, an opportunity to learn. How? Through context and the Math Practices, they could explore with their pupils both the underlying concept coupled with the correct notation. They also knew that as long as everyone felt safe and was enjoying themselves, learning would prevail.
This is just one example of what a former teacher of mine, Dr. Tim McKeny, would call “mathy goodness” that permeated the Math Celebration on our second to last day of school.
To paraphrase Mark Twain (or was it Pascal or Franklin or Locke?), I didn’t have time to write a short blog post, so I wrote a long one. What allowed a group of 39 fourth graders with a wide range of abilities, talents, and curiosities all to have a successful Math Celebration? A glimpse into the culture of the Learning Laboratory in room 118 might shed some light.
Like most good ideas, I stole this one. This time from my teaching partner, Mrs. Matters, who teaches English Language Arts and science.
In English Language Arts each quarter, the students’ creativity and creation culminates with a Writers Celebration where students share their published works with family and friends who are invited to visit. The Writers Celebrations usually took place while I was teaching, but one so happened to align with a planning period. I attended. I was floored.
The students swelled with pride at their workmanship and enjoyed responding to questions about their writing. For families, the event communicated the classroom culture in a way I’d never seen before. For the students, this event helped solidify their identities as writers. Belonging, significance, fun.
Afterwards I ran to my math coach’s office and said “Nina, how do we have a Mathematicians’ Celebration!?”
Why was I so excited? Because communication is hard. Especially with families.
The families of these students deserved a better understanding of how their children were learning, and the more they knew, the better we could all work together. I didn’t yet know how to do that. With a Math Celebration, I could step to the side and let the caregivers see their kids in action.
So the question became, how do I translate a Writers Celebration into a Math Celebration?
My daily classroom structure was key. It’s where 9 and 10 year olds look, sound, and feel like mathematicians, and it’s what I wanted everyone to see, feel, and hear. Our structure was typically a Number Talk followed up by a problem of the day. The problem of the day has four components.
- Math by Myself: students first engage with a problem on their own. The students learn that I’m an observer/questioner, not a resource during this time. The students procure at their leisure whatever mathematical tools or they think will help them make sense of and solve the problem. Productive struggle ensues (also known as desirable difficulties).
- Plus Power: students work with a partner or small group to continue to make sense of the problem, solve it, compare strategies, or dig deeper into the problem.
- Knowledge Multiplier: we come together as a group and discuss.
- Reflection: students do individual written reflections. What they learned. Questions they still have. New questions that arose.
So how did this celebration unfold? I put the decision to the students. I started by asking them to describe their Writers Celebrations. What did they like about it? I then segued…
What if…we had a Mathematicians’ Celebration?
The room erupted. Energy and ideas turned into motion.
The students’ ideas were varied and highly individual. Some wanted to code a website that mimicked our classroom structure and let other kids around the world engage in problem’s of the day, make claims, and test each others claims. We used the word claim and conjecture interchangeably in our classroom. I don’t know if that’s a floggable offense.
Some students wanted to share claims that they had made and tested throughout the year. Others wanted to share the three models of fractions and illuminate how the models could help solve problems. Others wanted to get their audience involved. Number talks, problems of the day, and game shows were ideas that came to fruition.
We decided on a framework that could accommodate them all, left ourselves lots of wiggle room and ran with it.
So what was my role in all of this?
My role was project manager. I had to help the students execute our plan. This meant creating time to work on the projects, listening, questioning and giving feedback in tight loops that helped the students move forward.
We decided that “mathematicians celebration” was a mouthful so I said I’d come up with some other ideas for a name. The next day I recommended “math conference.” They thought that name stunk. They were right.
One of the girls showed particular interest in helping with the name so we settled on looking up all of the synonyms for celebration and seeing what made the most sense based on each synonyms’ definition. She methodically worked through the list and then informally polled her classmates about what they liked.
In the end, the students settled on “Math Celebration” as the best choice. Even though it was primarily one student who helped come up with the name, her peers had seen the latitude and ownership she was given over the process. We ended up not far from where we started, but now it belonged to the students.
I also provided perspective.
For example, several students wanted to share a claim they had developed about creating equivalent fractions. It was a beautiful thing that unfolded exactly as Van de Walle, the CGI folks, and other mathematics educators said it would. The students’ knowledge about multiplication and fractions had coalesced into the algorithm for creating equivalent fractions!
Initially, the parents would have just seen something like this, and probably thought “well, that’s how their teacher showed them how to do it” when in fact the students reasoned out the patterns on their own!
I can’t stress this enough. What you see in the picture to the left isn’t an algorithm I showed the students. It’s one they reasoned out on their own. What I did was help them to test what they had found and work it into a generalization. This seemed like a normal thing to the students. It was an extension of the mathematics they had been uncovering all year.
They didn’t realize that some people might not even be aware that this process could be uncovered without a teacher telling it to them.
I wanted them to to be able to make it clear to their audience that not only had they developed a tool that they could use to help solve problems but they completely understood it because they had built it, tested it, and found its strengths and limitations.
This preparation was not without its challenges. With 9 and 10 year olds, a confident mathematician one day, can be downtrodden the next. Personalities and conflicts arise. New students, less versed in our classroom culture, needed guidance. In spite of these challenges, we persevered and did our best to set everyone up for success.
On a positive note, it felt markedly different from the last two weeks of school during the previous year (my first). Instead of dragging to the finish line, we were racing towards a goal. It provided focus and motivation.
“Algebra? But that’s far too difficult for seven-year-olds!” “Yes but I didn’t tell them that, and so far they haven’t found out.”
This quote from the late, great Terry Pratchett exemplifies the process students undertook regularly. You’ll see evidence of it in some of the other contributions to our Math Celebration that you’ll read about below.
The “always, sometimes, never” framework was our workhorse for testing claims. One claim emerged from a student, let’s call him Clarence, after some fraction explorations that we got a lot of mileage out of. “If you multiply a number by a factor less than 1, then the product will be less than the first factor.” The kids tested this claim six ways from Sunday, experimenting with anything mathematical they could think of from negative numbers to decimals.
Just as exciting as this claim was a question that emerged from it.
“Would Clarence’s Claim work opposite with division?” asked Braddock.
This young mathematician posed the question, then went and found out. He extended his knowledge of the “other kind of division” (measurement) to make sense of what dividing by a fraction could mean. Division of fractions is no where in the 4th grade standards, but why would Braddock care about that? He was delighted to share his findings at our Math Celebration.
Braddock was not alone in using what he knew to figure out something he didn’t.
Alison and Edward decided they wanted to share about the conceptual understanding of multiplication. In particular the link between its symbolic representation and its models (equal groups, area).
Preparation for the Math Celebration served as a learning opportunity for these two. They extended their understanding of modeling multiplication with two factors to modeling it with three, uncovering volume and three-dimensions in the process!
Two other students Ashley and Josh wanted to divide a whole number by a fraction (9 ÷ 3/4) and explain why the quotient was larger than the dividend. They did so with an enormous ruler created on chart paper highlighting each of the twelve 3/4 increments. Preparation served as a learning experience for them as well. They not only extended whole number division concepts to fractions, but they played around with it so much that they began to see the tricks we use to divide by fractions. They noticed that if you multiply the dividend by the denominator, and then divide by the numerator, you get the quotient. They had uncovered the basis for invert and multiply.
Those in attendance for our first ever Math Celebration got it. The strongest sentiment was “I wish I’d learned math this way!” I don’t think it’s a stretch to say that the content wasn’t what impressed upon the attendees. It played a role, certainly, but it was the practices in action that exemplified what “this” kind of math environment can enable for students.
A few months after this blog post made the rounds Tracy Zager got a hold of me to learn more about the questions and conjectures that we explored in our Math Laboratory. I was ecstatic when she chose to include our story in her book Becoming the Math Teacher You Wish You’d Had.
Our class is a part of Chapter 11: Mathematician’s Prove. A chapter that features a number of math luminaries and begins with a quote from Paul Lockhart, author of “A Mathematician’s Lament.”
Growing up I hated math. It made me feel stupid, and I barely made it through my own K-12 math experience. With hindsight I know that I can understand and learn math. I also understand the deficiencies in my own K-12 experience. So to have our classroom included in a book with this title, which is is what I set out to do when I became a teacher, feels damn good.