I’m about to finish my 4th year of teaching. This was my first year as a middle school math teacher and my first year really trying to create an inquiry-based math classroom. It’s hard.
What is an inquiry based math classroom?
I generally see it as getting students curious about mathematics and driven to understand and explore and willing to fail.
How do you generate curiosity and inquiry about mathematics?
I did several things this year. At the root of a lot of it, I made space for discussion and dialogue in small and large groups. When presented with most problems in class, I had students:
- attempt problems on their own
- discuss it with their partner to see if they agree
- share with the class
This teacher does a great job of giving think time, time to discuss in small groups, and finally discussion in the large group.
How do the discussions go?
With every lesson I did, I would have a few opening questions that were hopefully just a little out of reach for them; usually something they have some background knowledge or intuition that pertains to it, but not something I’ve necessarily taught them how to do. I’d always tell them to try something, even if they’re not sure; that we’re doing this to get them thinking about these concepts before we learn them which helps our brains process the new information.
For example, in a lesson on using percents to solve tip/tax/discount types of problems, I would give the following opening:
“If you bought a diamond ring for $300 but got 50% off, how much does it cost now?”
Most students know 50% means half. They’d probably be able to solve this one withoutout much help from me. My second opening question may be something like:
“You buy $40 worth of clothes and have to pay 10% tax. How much is the tax?”
Again, students may intuitively divide by 10. My third question may be:
“You have a $60 bill and want to leave a 15% tip. How much of a tip would that be.”
This is where (hopefully) some magic happens; it’s not as intuitive as the previous two but doesn’t seem impossible either. Students try all sorts of methods. I then have students turn and share with their neighbor and you can hear the background knowledge coming to the surface:
- “percents mean like, out of 100…’
- “I tried to find 1% and then just multiply that by 15…”
- “…why did you multiply by 15?”
When I’ve created questions in that sweet spot of not too obvious but not too far out of reach, these conversations can go on for several minutes. When the conversations start to die down, I ask some people to share their thinking at the board. People give arguments, others disagree with their arguments. There are aha moments, and periods of confusion, all while I never really tell them which method is right or wrong, I just ask clarifying questions. That in itself this takes tact because when a student is doing it wrong, our teacher voice of asking ‘so you’re saying you’re supposed to multiply?’ can give off the slightest inflection that the student is off track.
In my 6th Grade math class, I asked students to convert 35.8% to a decimal. After working on it, they began sharing ideas. In the middle of the discussion, I started recording the audio on my iPad and I’ve included some clips of the conversations below:
Here, one student defends why he felt 35.8% did NOT equal 358/1000—, but instead was 35 and 8/10. He then wonders if it could be 35.8/100
Another student begins to defend why it IS 358/1000, by explaining how 25/100 is the same as 250/1000. Another student shared an ‘aha’ moment.
This is sometimes the meat of my lesson; the conversations lead to the students basically teaching the concept as they work through one misconception after another and build the connections through discourse. I will then go through and teach the lesson, usually referring back to what many of the students had already said.
When this is done well…
These opening questions can really get the students invested in the new learning. They’re usually wondering ‘was I on the right track or not?’ Afterwards, students often share misconceptions they had coming into the lesson and why their thinking was misaligned.
The challenges in this practice…
The above discussion clips were not demonstrative of the discussions I had everyday; this one was a bit exceptional. The challenge is being able to maintain that quality of discussion and have them consistently.
When poor questions are asked (the questions are too obvious or too difficult), students are bored or frustrated and I feel like a traditional old school lesson of ‘watch me do this and now you try.’ It’s hard to create just the right questions, and on top of that, one set of questions can spark such great conversation in one class while the class right after them turns into most of them just getting the same thing and moving on.
It’s also difficult when you anticipate them to go in one direction, but then they sort of just make seemingly insignificant observations and no one really seems to take a real foothold into the concept. Sometimes when students are sharing, ideas are just all over the map it creates a less productive sense of confusion that has no real direction.
Additionally, maintaining an attitude of respect can be challenging. One day in my 6th grade math class, a student was a bit off base with his thinking and the students began to ridicule him; comments like ‘man, in his world, 1+1 must equal 5 or something.’ He was fine and laughing it off, but I had to have a discussion about how when some ideas begin to get mocked (even if done in friendly jesting), we begin to lose the safe space we’ve created. Students who may be unsure of themselves may not want to risk going up and being wrong.
How else to you develop an inquiry based classroom?
Another way I’ve tried to develop a little bit more inquiry and curiosity is through Dan Meyer’s 3-Act-Problems, which I discuss in PART 2 of this blog series..
This sort of leaning is more enjoyable, and works well.
It is also really, really, slow. You don’t get as much done in a year. You then push the next year’s teacher into catching up all the stuff that you didn’t quite get round to, which isn’t fair on them. If everyone does inquiry based learning then children leave school being able to do simple maths better, but never quite getting round to the difficult stuff that they need for university.
In my 6th Grade math class, I asked students to convert 35.8% to a decimal. You’re taking lessons to learn what can be taught in 10 minutes. Fine, but there are consequences to that.
It is also interesting only for a while. If you teach like this all the time you really need to deal with the social loafers. The slow kids who once the excitement of the new method has worn off, just get to hide. At least with “ordinary” teaching you can at least see if they aren’t working. The fast kids who get ground down into boredom as the pace of lessons never picks up. I hated such lessons with a passion when I was a student because I had the answer 10 minutes in, then spent the rest of lesson totally bored.
My position is that just because it works for you doesn’t mean it works.
Thanks so much for reading Chester 🙂
I totally get what you’re saying. It’s been the rub as I’ve explored this type of teaching. But I would say that ‘covering’ a whole year’s worth of material has no guarantee that all the students have learned those concepts by any means (which is why high schoolers still end up struggling with integers and exponents that they ‘learned’ about in middle school).
My hope is that developing deeper understanding of big mathematical concepts allows students to making larger connections in mathematics later because they’re more aware of the ‘whys’ behind the math (that hopefully they discovered through carefully planned problems and discussions) and they’re not just trying to remember the algorithm their teacher told them about.
I try to vary it up and will have a little bit more lecture for concepts and ideas that lend themselves more to that. Just trying to find the right balance I guess.