As a science teacher follower of your blog, I’m not sure I follow. Isn’t that what you are trying to do with many of your 3 act problems? Get a kid to ask questions like “is there an easier way to do this” or “what information do I need to know to solve this”? That question then sets that student up to be engaged enough to receive a little instruction?

I certainly feel like that is what I’m trying to do in science often. One of my teaching mentors always said, “start with a question.” “Asking questions and defining problems” is one of our 8 science practices in the NGSS. Trying to find ways to get students to ask “why does that happen” or “how do we change that” seems like the one of the most important steps I can take in getting students to engage with material.

That is what I see as valuable in PBL. It is messy. It takes forever, but it helps provide the “why” that I think so much of our education is missing.

I prefer a Jedi mind tricks over mind control.

I’m not sure what I wrote is appreciably different then a three act math task. Isn’t the point of the three at math task to get kids to ask the question you want them to? Isn’t the point of showing a video of a water tank filling up to get kids to ask how to find the volume of the object? Isn’t the point of the file cabinet task to have students calculate the surface area of the file cabinet?

I actually particularly like the last sentence of the research you cited:

“It is important to shape children’s activities to help them discern the relevant mathematical features and to attempt to account these features (2004, p. 134).”

The goal of my post was to do just that. To help teachers design mathematical experiences and activities such that they can discern that relevant mathematical features. Or design tasks that rely upon the teacher to help students find those relevant mathematical features.

Clearly and perhaps that should’ve been stated more clearly we want to give students the freedom and agency to ask new and interesting mathematical questions that perhaps we haven’t even thought of. Those are great experiences too! In addition to more exploratory tasks I also want to design tasks that allow for targeted learning to ensure that I’m covering the standards. And that’s where making sure kids are asking the right questions can be helpful.

Interesting post, Dan. The statement from Schwartz & Martin seems to draw on the idea of cognitive conflict, at least in part, in that it is asking students to move away from old interpretations.

As a coincidence, an interesting paper has just been released on cognitive conflict. I haven’t studied it fully yet but I thought you might be interested:

http://psycnet.apa.org/journals/edu/108/1/98?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed:%20apa-journals-edu%20(Journal%20of%20Educational%20Psychology)

Dan – I was referring to the specific statement in your blog post which *does* seem to have at least a passing relationship to cognitive conflict. You may disagree.

I am aware of the PF studies of Schwartz and others.

I fully expect you to be all over this stuff but the link might perhaps be useful to others who are following this thread and were unaware of the study.

Dan – On the Schwartz and Martin study, your readers might be interested in the following, particularly the discussion at the end (it’s not solely about this study but it is interesting):

http://aaalab.stanford.edu/papers/Constructivist_Assessments_Final.pdf

I have focused on Kapur because, unlike you, I tend to think that these experiments are better controlled (see Sweller’s questions).

I look at the Gilland quote and I just can’t see this being the point of Maths class, any more than generating correct answers is. Surely the point is to build a student with more Maths understanding that can be used by them to solve problems.

Focusing on building mental schemas is all very well, but if it isn’t followed up by building of skills then it is just as worthless as skills without understanding.

If your main aim while teaching is to get interesting questions asked, then you will get interesting questions asked. But that won’t make them any better at answering them. This is not to say that interesting questions and discussing contradictions aren’t good, just that they should not be the aim.

Chester,

I should be clear that I am using the term ‘mental schemas’ as it is used in the literature on Cognitive Load (and I assume other ed research, but that is where I first encountered the term). In this usage, building mental schemas isn’t distinct from building skills. In fact, skills are embedded within mental schemas via working memory and combining of interactive information pieces. I am not sure CL theorists would actually recognize a significant distinction between skills knowledge and fact knowledge (on storage- accessing them is somewhat distinct and has slightly different impacts on cognitive load).

All of that is a long winded way of echoing Dan’s “false choice” with a bit more detail and clarification of terms.

“Which makes Geoff’s “crux” a form of mind control, or maybe inception, which is impossible. Kids wonder so many wonderful and weird things. And even if that practice were possible, I don’t think it’s desirable, since it seems to deny student agency while pretending to grant it. And even if it were desirable, I wouldn’t have the first idea how to help myself or other teachers replicate it.”

“I am interested in question-rich material that elicits lots of unstructured, informal mathematics that I can help students structure and formalize. But I never go into a classroom hoping that students will ask a certain question. I often ask them for their questions and at the end of lesson we’ll try to answer them, but there will come a moment when I pose a productive question.”

Not sure if it’s worthy of a standalone follow up post, but,
1) I like kids asking questions about math
2) It’s lovely when kids are asking about the math content I’m trying to teach

FWIW, I’ve blogged about various problem implementation types in the past, including problems that might be considered more exploratory and ones in which the teacher may or may not know the solution or most efficient solution path.

http://emergentmath.com/2013/02/24/developing-a-taxonomy-of-problems-not-all-problems-are-implemented-equally/

So thought-provoking!

Two statements will stick with me. One from Dan: “…it seems to deny student agency while pretending to grant it.” Kids have been ‘playing school’ in math class for years and they’ll quickly see right through this variation of it. They are more than willing to take the path of least resistance and fall back into a role of trying to guess what it is the teacher wants them to say or waiting the teacher out.

The second is from Meryl: “Personally, one of my greatest hopes during every lesson is that the students will ask me something for which I do not have an answer.” What a great job we have as teachers that we get to keep learning along with students!

Of course part of that job is knowing the tasks and problems we present to students inside and out so that we can make our best predictions about how students might approach them in order prepare questions and prompts we might pose about their approaches. Along with that is knowing the relevance of those tasks and problems to the standards or concepts or whatever it is we’re trying to teach. I love when a kid successfully avoids the approach I was really hoping for. It makes me re-examine my assumptions about the task and my students’ understanding and flexibility. Good stuff.

So I do generally use a problem based format. But the structural issue I see with it is that even when student thinking goes in the direction that you expect its not the same experience for each kid. What happens instead especially when working in groups and talking with each other, is that insights and intuitive leaps are very unevenly distributed. Every time most kids are essentially learning via their classmates thinking. That’s not really very different from if I structured an explanation (well I think I could probably be clearer and bit more concise in many cases)

This doesn’t negate the value I see with the format but it does provide some comfort when I go around and try to facilitate students who are stuck.

Many times teachers abandon good tasks or PBL because they do not get their students to a certain point fast enough or the right questions come too late for the time constraints of the class. A couple thoughts …

PBL takes time to perfect if it ever can be perfected.

We can’t neglect the idea of lesson study, action research, and growth mindset to help us shape the “right” problem and equip us to answer the questions (right/appropriate or not appropriate) students will generate.

“I am interested in question-rich material that elicits lots of unstructured, informal mathematics that I can help students structure and formalize. But I never go into a classroom hoping that students will ask a certain question. I often ask them for their questions and at the end of lesson we’ll try to answer them, but there will come a moment when I pose a productive question.

The possibility of student learning needs to rely on something sturdier than “hope,” is what I’m saying.”

I really appreciate your and Geoff’s posts on this. I wonder in terms of formalizing 3-Act Math as a routine if the shift to the teacher posing the “productive question” might eventually lead to students losing motivation for questioning since they know that Act One is “rigged,” so to speak. I definitely agree with needing more than “hope” when designing a lesson, but I wonder the extent to which flexibility can come into play in putting more student control on the “productive question.”

I’m not sure IF there is a way to resolve this with your valid critique of needing more than “hope,” but that’s one thing spinning around my mind after reading your and Geoff’s posts. I think he’s also circling around this idea in his reply to this post.

No doubt this has been written about many times before, but does it not deserve to be repeated that many kids actually like problem-based learning environments, even if that might be less efficient and growth-inducing? There can be resistance to inquiry-based math if students only get it once in a while or never. If the environment isn’t set up for students to be inquiry-based at least 20-25% of the time, in my opinion the 3 acts won’t really sustain themselves with a majority of students when you whip out inquiry questions once in a blue moon.

Further, there’s still just going to be a ton of teacher resistance to the amount of time it takes to do inquiry right. If there’s a sense you can’t cover enough ground, teachers are going to panic and go with what moves students through the material at a brisker clip. Again, mastery through inquiry will only work well if happens once a week or more. But on the other hand, there’s the possibility of going too far: one truly masterful teacher that I know runs inquiry 50-60% of the time and thinks this has hurt her scores because she didn’t push fluency enough.

So, did the kids learn/grow more (in the big picture) because of 60% inquiry, or was that too much and they really do need more fluency? No really easy way to answer that question in the current environment, much less any easy consolation if everything has to be done based on scores and a strict curricular schedule that fits testing windows in February and May.

I love inquiry, and I think it is really great for kids’ minds. But I don’t think we should kid our selves into thinking that there’s not a pretty big role for problem-based learning that still must exist.