Engaging students in math has always been a goal for me. No… more than a goal…. a passion! And it’s not always easy to do. For example, I used to hate teaching students how to find the sum of an arithmetic series. I didn’t hate it because it was difficult to teach or because students had an overwhelming difficulty learning it. I hated it because I was the only one that saw the beauty in it. I was the only one who was passionate about it.
This lesson was “fun” (I use the quotes to denote that this was a fun lesson for me – not so much for my students). But this all changed when I allowed my students the opportunity to think for themselves.
The task was very simple in concept: Find the sum of the series of the numbers 1-20.
Before going any further, it may be useful to know about the
- Class norms:
- Estimate first,
- the answer is never enough,
- reasoning, explaining and looking for patterns are all expectations,
- if you found one way, look again, you may find a more efficient way,
- get out of your own head and talk about the math with your partner/group while you work
Several started adding 1 + 2 + 3 + 4 + . . .+ 19 + 20. I noticed this and asked those groups for one word to describe their strategy. Sample responses: boring, lame, tedious (actually proud of that one), calculator worthy…
My reply to each of their descriptions: If your strategy is [insert one: boring, lame, tedious, or just plain calculator worthy] why do you feel the need to use it?
Sometimes students get stuck in their own thinking and just need to be made aware of it. To help nudge students to think in other ways, I had bowls of tiles with the numbers 1-20 written on them available for groups to use.
It took several minutes before students began to grab tiles and began to notice things like:
- “Hey, Mr. W., we can make a bunch of 20s.”
- “We got a bunch of 21s. 10 of them. It can’t be that easy, right?”
- “We made 10s and 30s. How did you make 21s?”
- “We did the 20s too. That’s the easiest way for us.”
It was a bit chaotic, and I didn’t know it then, but there was a passion building. This wasn’t just engaging, these students were ALL IN. They were more than engaged and wanted to learn more about the strategies they came up with. They wanted to share. Needed to know. And the answer was almost irrelevant. The connections between all of their strategies became the focus.
From here, getting to the algebra made sense. How would you find the sum of the numbers 1-50? 1-90? 1-100? What about 5-50? Some saw their ideas with the tiles transfer easily to an algebraic expression and equation. Others not so much. So, more time to talk and share. More time to find a strategy that is more convenient to generalize for a series of numbers of any range. The success of the students’ mathematical ideas gave them power to reach further – to take another chance.
Teaching the lesson this way was a definite improvement on the original. In this version, the students’ ideas matter, so students matter. In this version, students think for themselves and collaborate with others, and in turn get validation of their thinking, so students matter. In this version, students built some passion. They fed off of each other. And the content mattered because of the students’ interaction with it.
Is this lesson the best it can be? I’m not sure. So, I’ll continue to try to improve on it.
Thoughts and comments welcome.