Check out ‘Challenges Of An Inquiry Based Math Classroom Part 1‘ to see how I develop inquiry through dialogue & discussions in my classroom.
Another way I’ve tried to develop a little bit more inquiry and curiosity in mathematics is through Dan Meyer’s 3-Act-Problems (check out Meyer’s TED talk on math education in the US if you’re interested). I had heard of these 3-Act-Problems for a long while but didn’t actually try one out until later in the year. The 3-Act comes from the idea of how a story unfolds; Act 1 is where you grab your audiences attention, Act 2 is where the plot unfolds, and Act 3 is the resolve.
The first one I did was called ‘Finals Week.’ It starts with a video below:
Act 1 is where you show the students the video and they generate questions about what they saw. Allow for all types of questions, not just ‘mathy’ ones (one student asked ‘what does this even have to do with math’ after we watched it). You do have a question in mind that they hopefully go to (in this case, which one has the greatest caffeine concentration). After they ask the questions, you tell them that together you’ll try to answer all of them but you really want them to help you out with a few specific questions that were asked. You then ask students to give an estimate or guess for their answer and justify (again, I have them do this independently, then discuss with a neighbor, then have a few share out loud). In this example, I had them rate them from least to most effective. You can write down the guesses to come back to later as well.
This act breaks down the barrier between ‘math folks’ and ‘non-math folks.’ Anyone can take a guess. Anyone can ask a question about something they saw.
Act 2 is where you ask the students what information they need from you to answer the question. Make a list of what they share. I usually ask ‘is there anything on this list that you wouldn’t need to answer the question?’ Students discuss and share why certain pieces are (or aren’t) needed. In our example, some students said you would need to know how he felt after he took the drink, but others argued that the effect the drink had on him has nothing to do with the concentration of caffeine in the drink.
Once the list is dwindled down, you then give them that information and let them try to answer it. I haven’t come across a situation where they request information they need that you don’t have. Not sure how you would go about that w/out saying ‘well I guess you don’t need it’ or maybe ‘lets see what you can do if I just give you THIS information.’
By this point, they’re really invested. Problems that have multiple solutions are the best. I’ve heard Meyer share that this is a great place to teach the new concepts as it has an immediate application. When I did this first one, it was right after our lesson on creating ratios and unit rates. Still, students used different methods to solve it. Students either try on their own or collaborate with a partner to try and solve it. We spend time sharing methods and techniques.
Act 3 is where the answer is revealed. Verify who’s guess was the closest. Having students share why their may have been incorrect has a ton of great value as well. Some 3-Act-Problems have a ‘sequel’ portion where follow up questions are asked. I imagine many ‘sequels’ could come out of student-generated questions.
I’ve only done a few of these 3-Act problems. Students who aren’t super into math request to do these. One young lady said ‘You know, I didn’t hate this.’ Total win.
What I’m working through is the time these take. Some are longer than others and there’s the art of knowing when to go ahead and move on from hearing students explaining their thinking. You get excited that so many of them want to but then you start getting kids check out because it starts getting redundant or they start getting more confused. With ‘Finals Week’, a part of me felt that they weren’t getting enough practice or exposure to different types of unit rate problems when we’re spending at least half of a class or more on one problem. But then the question of quality vs. quantity arises.
How much do you let them struggle in Act 2 before teaching it, and when you do teach it, what does it look like? Traditional lecture? Do you just tell them how to solve it, or do you give them a different example to explain the concept, then letting them apply it and contextualize it to the current problem? Do you have a variety of student methods and then show Act 3 to see who was right? What if no one was right? Do you swoop in with the instruction?
I’ve had students do well on a 3-Act-Problem but then have no idea what to do on a word problem presented to them addressing the same concept. The knee-jerk reaction is to just move back to paper-pencil word problems but I want them to be able to do both. This method not only allows them to think more conceptually and become problem creators (instead of just problem solvers), it’s getting them excited about mathematics which in itself is crazy valuable.
There also isn’t a 3-Act problem for all the concepts I’m doing. Of course, nothing is stopping me from creating my own except the time that goes into it. These also aren’t an ALL-OR-NOTHING sort of resource so having them sprinkled in throughout different units can be equally effective.
thanks for sharing this. One of the parts of the “3-Act” sequence that I would push back on (and always have) is this quote: “this is a great place to teach the new concepts as it has an immediate application” – yes, it’s great that there’s an “immediate application” but the teacher is still teaching it as opposed to the kids coming up with ideas of the actual mathematics themselves. I believe that as much as possible students can and should do this (yes, the actual construction of the mathematical knowledge) themselves – not just motivate it with a cool problem, but actually come up with the mathematics, themselves. Because if not, are we really increasing their agency at all? Aren’t we really just giving them a cool show about how to use the math that we (the ultimate holders of the math knowledge) will then show them how to use?
To me PBL and IBL are more than that – it’s actually honoring the students’ voices in the *mathematical* process and the sense-making process as well.
To your point about time -yes this does take time, but theoretically and from my own experience, it does not need as much “practice” – and it also spirals. According to Dewey, a huge part of the learning process that is often not giving its due is reflection and discussion (especially in math class, so what might feel like a lot of time for you might be actually just the right amount of time!
hope that makes sense,
Thanks so much for reading and sharing. I guess I struggle with knowing how to appropriately scaffold students in those conversations and knowing what questions to ask to lead them to discover the math on their own. The times I did the 3-Act’s this year, they were usually able to make their way through the problem with the help of one another and I had them sharing their thinking. It comes back to knowing the types of problems to present; knowing what they’re ready for.
I’ve had a few times I posed a problem that sort of just led to students sharing all sorts of methods, none of which were hitting the mark, and me not really knowing how to proceed. Usually I’d try to ask questions to help them see their own misconceptions and redirect their thinking in the correct direction, but then has it become less inquiry based at that point?
How do you support them when they aren’t moving towards understanding of the concept?
As far as time, makes me think of this NYT article on how math ed in Japan is so much more inquiry based and how they spend 30-45 min deconstructing one problem – http://nyti.ms/1nTFs4i
This discussion also reminds me of an article I read called ‘Never Say Anything A Kid Could Say.’ : http://roosevelt.4j.lane.edu/wp-content/uploads/2013/08/Never_Say_Anything_A_kid_say.pdf
Thanks for engaging me in this conversation!