!!! It’s the first week of the district wide blogging challenge that I created !!!
I’m really excited for this blogging challenge – to see teachers in our district who have wanted to blog have a reason to finally get started.
Anyways, the theme for week one is One Good Thing – to share something good that happened in your classroom in the past week and explain how it was celebrated.
It feels like on a daily basis, I’m carrying around my pride in my students within this bubble. I’m always proud of them – I teach a population of students who are in a daily fight with the low expectations the world has placed upon them, and every day that they show up in my classroom gives me pride that they haven’t given up yet. So the bubble always exists.
Sometimes it’s very fragile and small, and sometimes it inflates more and more.
And then sometimes it bursts, because it’s just too full of pride to be contained anymore.
My pride bubble burst on Friday.
My Algebra 1 students came into my class at the start of the year with little to no mathematical success in their histories. The challenge at the start of the year was to get them to even try. Throughout the year, we’ve slowly started doing some explorations/investigations at the start of new material in an attempt to expose them to the ‘real mathematical world’ where you aren’t just told a rule or formula; you discover it. When we first started doing these, most of what I would get were complaints like “how are we supposed to do this, we haven’t learned it?” and the like.
On Thursday, we began an algebra tile exploration on solving quadratics by completing the square. We’d already learned to solve quadratics by factoring and using inverse operations, and I’d alerted them to the fact that by the end of the year, we would have FIVE different methods for solving quadratics. I even warned them that this particular method would possibly be the least favorite for many of them, because that’s been my experience in the past with students.
We began looking at some problems together as a class – I explained that our goal was to make one side of the equation into a perfect square of algebra tiles, and we reminded ourselves that if we add extra tiles to one side, we must add those same extra tiles to the other side to keep the equation balanced. My pride bubble started swelling when we reached the point where we wrote the factored form of our first example as (x+2)(x+2) and one of my students offered, with no prompting, “couldn’t we write that as (x+2) squared?”
YES! WE CAN!
Then another student noticed that the problem suddenly looked like the ones we had been solving the previous week using inverse operations, and asked if we could solve it like those.
YES! WE CAN!
The next day, they were off, using their algebra tiles to complete the square and solve quadratics, on their own or in pairs. As I circulated, I kept hearing things that made my pride bubble swell more and more.
“No, remember, you have to split the x tiles evenly because we’re making a square”
“Wait, in this one the ones tiles are with the other tiles to start. Don’t we want them separate? Can we just subtract them to move them to the other side?”
“We’re always going to add positive ones tiles, right? It’s either negative times negative or positive times positive.”
“I don’t think we even need to use the tiles for this one. I know what’s going on.”
And these kids, who fought so hard against these investigations when we first started doing them in first quarter of this year, started asking me and each other extra questions that weren’t even part of the written instructions.
“Wouldn’t it be cool if there was a set of algebra tiles with an x cubed tile? What would that look like? It would have to be 3D, but how would you decide which side should be red and which side would be the other color, because there would be more than two sides but only two colors.”
“Hey, all of these have an even number of x’s.”
Me: “Would it be harder if there were an odd number?”
“Yeah, because you have to split them evenly”
Me: “We’re going to talk about that on Monday”
“Oh, man, that’ll be cool!”
(This was a student who just last quarter frequently sat in class mumbling under his breath about how pointless the class was and how much he wished it were lunchtime and failed over half his assignments)
“Are there problems that you can’t do like this? What would those look like?”
At this point, I was sitting at a table grading the previous class’ investigations because they were moving along so well without any prompting from me. The pride bubble was pretty huge at this point, and I was just sort of smiling to myself in the corner.
They started to get to the last two questions of the investigation, which asked them to look over all the problems they’d completed and try to find the relationship between the number of x tiles in the original problem and the number of extra tiles they’d had to add to complete the square. These types of questions have always defeated them in the past – I don’t think they’ve ever been asked to generalize before they got to me, and so they just fight against having to do it. They also hate to actually read instructions, so I was expecting all sorts of questions just because they didn’t want to read the fairly large block of text of the question.
Instead, they started to read the instructions aloud to each other. They started flipping through their packets to look at examples. They read the instructions line by line and paused to consider each piece.
All of them at least found the pattern that we were splitting the x tiles in half.
Many of them found the whole pattern and were able to use it to correctly solve one last problem without using the tiles.
One student, considering all his examples, asked, “Miss Mastalio, what’s the word for the answer to a division problem?”
This was his final answer:
This was when my pride bubble burst. I wanted to cry so many happy tears. These kids have fought and fought and fought thinking about how math works this year. Somehow, the dam has broken and they’ve worn down.
For some reason, this investigation wasn’t a fight. It was a triumph.
It was a great reminder to just keep trying. That they need practice grappling with new ideas, with finding patterns, with expecting math to have logical conclusions. That it will eventually pay off.
These kids are getting a school wide shoutout on Monday – these are read over our announcements and I individually named each student in the one I wrote after class ended. They were my #teach180 tweet for Friday and I’m so excited to do our formal notes on completing the square tomorrow and be able to say, “I know several of you already found this pattern; what was it?”
I love fourth quarter, when everything starts coming together.
(Here is the investigation I used, which is adapted from the exploration from section 9.4 of the Big Ideas Math Algebra 1 curriculum)