I really wanted to see if a group of 3rd graders could conceptually understand a visual pattern that is quadratic. Yep, I wanted to teach a high school concept to 8 and 9 year-olds. Lord, please give me the strength to survive.
The kids had fantastic conversations, immediately turning toward their table groups and engaging in discussion. They noticed baseballs, going from least to greatest, they noticed rows, they noticed that it was growing by 2 rows, and then a kid said something else:
"They are arrays."
I... wasn't expecting that from a 3rd grader. Maybe I should've been, but it was certainly a pleasant surprise, and a nod to the teacher for encouraging strong academic vocabulary. Remember, high school teachers, that kids are learning about arrays in 3rd grade, and doing well with them. We don't have to "dumb it down" for our older kids!
I wrote down their noticings on the board, just so I could keep track.
Me: "OK, I'm going to count these ones. One, two, three, four,...., eighteen, nineteen, TWENTY! Ah, that was the only way to do that. Right?"
Me": "Wait. How did you do it?" <points to squirrely 3rd grader>
Squirrely 3rd Grader, eager to show off: "I multiplied the 4 times the 5 and I got 20." <smiles>
Me: "Oh. Yeah, your way was much more efficient than mine."
Their earlier work with arrays had paid off. Knowing that they had done this, based on work that my son had brought home, I was hoping that it would be the natural next step, but I wasn't certain. From there, I asked them about step 4 in the pattern. They were to draw out what they thought it should look like, then figure out how many baseballs would be in the array. Some students struggled, but with a little help in noticing how the rows and columns were growing each time, they were able to sketch it out and either count or multiply to find the total number of baseballs.
There was one student who figured out the 4th step as 7+7+7+7+7 and I got giddy. Yes! The array helped her notice that she could keep adding the rows up to determine the total! After coming to the consensus that the 4th array would be a 7x5 and that there were 35 baseballs, I asked for the next step after that. I knew that we were reaching further into the abstract, with visuals a thing of the past as the class was to now rely on the pattern that had been built. To my surprise, many of the students were already working on step 5 in the pattern, and had answers to share without needing to draw out the baseballs.
"As we get further and further along the pattern, it is going to be harder and harder to keep drawing out the baseballs. This is why the numbers are going to help us a lot more. Could we represent the next step in the pattern without drawing the array?"
And this, this right here, is where empowerment happened. Kids were already ahead of me, writing 9x6 for step 5 of the array. I could see it on a few kids' faces that they were proud of what they had found and could get a solution. I say a solution, because some kids did not multiply correctly, but that's not what I was there for. I wanted to see if kids could identify a pattern and build it out.
Then... as if it were magic... I dropped the truth on them that I was waiting for.
How did they react? They were excited. They were empowered. It was so cool to see the look on their faces as they were told that they were doing math that was designed for kids 6 years older than them, and they understood what was happening. I gave them the expression (n+1)(2n-1) as we discussed how the columns went up by one, and the rows were doubled minus one (that part was a little bit fuzzy for some, but they played along with it). We don't do two-digit-by-two-digit multiplication in third grade, so I didn't continue with the pattern toooo far, but some of the kids did. In high school, per Fawn Nguyen's visual patterns prompt, I would ask the class to determine the 43rd step, necessitating some sort of algorithm. But here? No way.
When I picked my son up from school, I asked him if he liked the lesson. "Yeah," he said, "it was fun to do high school math. I felt really smart."
So here's my sales pitch to elementary and secondary teachers alike: work together. Find some common threads that elementary introduces and high school expounds upon. Patterns and linear/quadratic functions are a good one, but there are many more. More importantly than that, though, is the idea of vertical articulation and that it doesn't need to be with a single grade level lower/higher than the one you currently teach. For the kids who will spend their entire K-12 career in the same school district, or area, it is a shame that they receive 13 isolated years of education. We need to be doing a better job of helping students see the connection from what they learned in years prior as well as the learning that they will be doing in years to come.
I have an up-and-coming 1st grader and 4th grader next year, so I can only hope that I get the chance to keep doing a lesson like this one to help others feel "really smart" for doing math.
Happy "Every Kid Deserves To Feel Really Smart" Fishing