Sometimes, I think we are making it all entirely too complicated. Sometimes I wonder if we have lost the forest through the trees. What do I mean by this? Well, in my experience, when we start with a rich math task, the rest falls into place. The engagement, the learning, the mathematical discourse, the standards for math practice all just happen as learners are drawn into problems. It really can be that simple.
I recently experienced this as I had an opportunity to work with some first graders. I found this task and many others here:
Pause and solve before continuing ....
After the students played and placed various numbers in the circles, they were asked to choose their own numbers and explain why they always ended up with the same number they began with.
The most fascinating conversations erupted.
Javon said, “I am going to use one million. I don’t know how to write one million. I will write 1M.”
Notice he didn’t ask me how to write one million. He also didn’t decide to use a smaller number because of this momentary stumbling block. He simply stated that he didn’t know and then solved his dilemma. He then proceeded to figure out the solution to one million minus one, and then that answer minus two. While solving, he was talking out loud to others and sharing his work and his thinking.
Joniel proudly worked with one thousand. He worked silently, but then announced, “Look, I did one thousand!”
All math friends were appropriately impressed as they jumped up to lean over his shoulder and look at his solution.
Makyna declared, “I am going to use an easy number so I can figure out the pattern.” She chose 20. Genuinely interested mathematicians looked over her work when she finished.
Without any prompting, the students spontaneously critiqued each other’s reasoning. They added onto each other’s thinking. They considered different solutions.
The standards for mathematical practice were alive and well. And they just happened.
The real magic took place when I asked, “Why do you always end up back at the same number you started with?”
“It’s because of the diagonals,” began Joniel.
I pushed further, “Tell me more. What do you mean by diagonals?”
He continued confidently pointing at the number ring, “See you have +2 and -2 and then +1 and -1.”
Javon jumped in, “Right so you just plus and then take it right off when you minus.”
Whoa - we were now talking about adding positive and negative numbers and ending with a net of zero in first grade.
Makyna added on using a mathematical model. “It’s like a number line. If you start on a number and then jump up, but then jump back the same, you’ll end up where you started.” She drew a picture of this. With no numbers just arrows indicating jumps up and then jumps back.
Every child understood what each other was saying. They built on each other’s statements and pushed each other’s thinking forward. Students did not need to be prompted to revise their thinking or to restate or add on to another’s. They were engaged in a natural flow of conversation that was all possible because of the math.
The Mathematics provided everything.
There was very little that I needed to do, except to step out of the way.
Since we were still learning and having fun, I kept it going. I posed the task, “Now create a circle ring of your own making up your own rules, but making sure you end up back where you began.”
The responses were exclamations of joy - “YES!” and “Wait, can I add like 1,000?!” And then all were in a big rush to get back to the math.
So what about that forest and those trees? There truly are so many trees to think about in any given lesson, but when we set students loose on some really interesting math tasks, they never fail to happily climb their way through that forest!
Where do you go to find rich math tasks? Please share.