Problem-Based Learning Needs A Different Crux

Geoff Krall:

The crux of Problem-Based Learning is to elicit the right question from students that you, the teacher, are equipped to answer. This requires the teacher posing just the right problem to elicit just the right question that points to the right standard.

Our existing knowledge and schema determine what we wonder so kids wonder kid questions and math teachers wonder math teacher questions. Sometimes those sets of questions intersect, but they’re often dramatically disjoint.

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.

If PBL is to survive, it needs a different crux. Here are two possibilities, one bloggy and one researchy.

First, Brett Gilland:

[The point of math class is to] generate critical thought and discussion about mathematical schema that exist in the students minds. Draw out the contradictions, draw attention to the gaps in the structures, and you will help students to build sturdier, creatively connected, anti-fragile conceptual schema.

Second, Schwartz & Martin:

Production seems to help people let go of old interpretations and see new structures. We believe this early appreciation of new structure helps set the stage for understanding the explanations of experts and teachers – explanations that often presuppose the learner will transfer in the right interpretations to make sense of what they have to say. Of course, not just any productive experience will achieve this goal. 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).

Notice all the teacher moves in those last two quotes. They’re possible, desirable, and, importantly, replicable.

2016 Jan 12. Logan Mannix asks if I’m contradicting myself:

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”?

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.

2016 Jan 13. Geoff Krall writes a post in response, throwing my beloved Harel back at me. (My Kryptonite!) It’s helpful.

About 

I’m Dan and this is my blog. I’m a former high school math teacher and current head of teaching at Desmos. More here.

27 Comments

  1. 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.

  2. Thanks for the question, Logan. I added it and this response to the post:

    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.

  3. 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.

  4. I feel like Geoff and Logan are essentially taking the intermediary step of a three act task out of the equation.

    Sort of like this: http://www.speed-light.info/speed_of_light/speed_of_light_fold2.gif

    I’ve done a handful of three act tasks in my classrooms over the years, and Geoff/Logan are correct to some degree. But I also have usually had students suggest questions that are very similar to what I want to hear – that is, what the ‘act 2’ responses are.

    I have usually been able to take questions that students ask and massage them into questions worth answering through questions of my own. I once tried the ‘predictive questioning/responses’ protocol talked about in 5 Practices for orchestrating Mathematics Discussions. (http://www.amazon.com/Practices-Orchestrating-Productive-Mathematics-Discussions/dp/0873536770)

    In the end, I almost think the next time I get to do a 3 act task I should give more of just a 101qs.com thing and let the kids go from there/design multiple 2nd and 3rd acts perhaps too if needed.

  5. 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)

  6. Geoff:

    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?

    No, please see my response to Logan directly above.

    Greg, cognitive conflict isn’t an explicit feature of the Preparation for Learning research. If you haven’t read it, you should. I’ve only seen you critique Kapur’s productive failure pieces, which are less rigorously controlled & designed.

    And I’d already caught wind of that piece (and emailed the author for a PDF) from my trad twitter lurking. Appreciate you keeping me in the loop just in case, though. Looks interesting.

  7. 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.

  8. Dan – My bad! The terminology is so confusing sometimes. You are quite right, they are preparation for learning studies as in the link I just posted.

  9. 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.

    At some point if you don’t stress building skills as equally important as getting them thinking about the subject, then you get students who are passionate about Maths, but not much good at actually doing any of it.
  10. Chester:

    At some point if you don’t stress building skills as equally important as getting them thinking about the subject, then you get students who are passionate about Maths, but not much good at actually doing any of it.

    False choice, IMO.

  11. 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.

    It is probably worth noting that CL theory and its discussion of schema construction is often considered a fairly conservative theory that overwhelmingly supports traditional teaching methods. I think that this is horribly incorrect working on a post to that effect right now). I just add that to allay any concerns that schema construction is hippy dippy head-in-the-clouds stuff.
  12. 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. It is an incredible teachable moment when I have to investigate and wonder along with them in a mathematical sense. The Three-Act Model certainly promotes that possibility and having answers ready is in a procedural sense, anticipated, but conceptually….I am not convinced. I can’t be expected nor do I wish to pretend to know everything, hence the purpose of a growth mindset. Great, thoughtful, and motivational entry today Dan. Thank you! Now, back to prepping today’s lesson…

  13. “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.”

    This seems pretty disingenuous when when you have an Act 2 pre-prepared and pre-packaged. Why bother organizing tasks by standard if you don’t anticipate certain content-oriented outcomes?

    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/

  14. Geoff:

    This seems pretty disingenuous when when you have an Act 2 pre-prepared and pre-packaged. Why bother organizing tasks by standard if you don’t anticipate certain content-oriented outcomes?

    Fair question.

    If a first act (or visual pattern, to generalize just a bit) depicts a linear model in action then it’s likely students will find it productive to apply a linear model. That’s why I tag my work with standards.

    But what I don’t think is possible, desirable, or replicable is to get kids to ask a question about linear modeling, and to not ask questions that aren’t about linear modeling.

    It’s the right question-eliciting move in your chain up above that I think PBL advocates should reconsider.

  15. Belinda Thompson

    January 13, 2016 - 7:40 am -

    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.

  16. The longer I watch this conversation, the more it seems like it is more a case of reading each other’s statements as being much stronger than they are than it is a case of genuine disagreement. I am going to offer a few benevolent interpretations of the progression of this argument below. If I get something wrong, let me know. Benevolent =/= infallible.

    In his original post, I think Geoff is pushing on the idea that we want to have kids explore math, but that we want them to explore a particular topic on any given day, so we need to design our tasks to guide our discussions to that topic.

    Dan read that as the stronger claim that we are trying to manipulate kids into asking a very particular question so that they can discover just the right answer. This feels like massive coercion. What’s worse, it is disguised as free exploration (disguised, because we were funneling them to this question all along). While I think that reading is stronger than Geoff intended (given his follow up responses), it also seems an understandable reading of the excerpted paragraph, which reads as more than a little Machiavellian to my eyes.

    And here is where Pershan’s Maxim may come into play: The more specific you get, the less disagreement there is. My lessons are significantly more structured than 3 Act Tasks. I am very explicit about what I want them to explore. I simply emphasize that we are _exploring_ that area, not marching forward to the Truth. I am not a fan of unstructured Discovery learning. (Nor is Dan, who opined on twitter about being lumped in with them in the infamous 2006 Cognitive Load article from Kirschner, Sweller and Clark.) I want kids to focus their explorations on a given topic in a given area. Ideally, they will _eventually_ make some practical discoveries related to a given standard.

    My description that Dan quoted above has more to do with how I view the process of learning and concept acquisition- lots of exploration, rarely linear, and activating/modifying existing schema during exploration of a topic, _even when performance is not actively improving_- than it is to do with much of anything else. Thing is, I don’t think Geoff is that far off from where I am (if there is a difference at all). I just think he stated his case with an unfortunate word choice that suggested strong external control of student thought patterns, something which I am pretty sure Dan finds reflexively objectionable.

  17. 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.

  18. Another difficulty with fishing for/forcing the correct question is that the students are being driven to a task which they are not supposed to fail. Lack of opportunity to practice failing (and then either finding a new approach, or asking what they can figure out) is a major problem in both Mathematics and Science education. Real problem solving skills involve failing and trying something else, not only asking questions that you can easily answer. Often times finding out what you can’t figure out will be more inspirational than whatever you were trying to do in the first place (e.g. Newton inventing Calculus to solve Physics problems)

  19. 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.

  20. “It’s the right question-eliciting move in your chain up above that I think PBL advocates should reconsider.”

    Ah, now I see and understand the criticism a bit better. Perhaps ineloquent, perhaps semantic, the phrase “right question” understandably rubs folks here the wrong way.

    That said, in my practice I am looking for a student question that relates to the intended content on which I can pounce, to ask more probing questions and dig deeper. To me that’s the whole point of sense-making. While (of course) honoring and exploring unanticipated questions.

    I appreciate Brett taking a charitable reading of the commentary. And Dan for challenging me on imprecise language.

    Also, don’t lump all PBL advocates in the same boat. Many are much more eloquent and precise than me.

  21. “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.

  22. Michael Paul Goldenberg

    January 13, 2016 - 6:21 pm -

    Just wanted to see if anyone else gets uncomfortable with “just the right question that points to the right standard” not only with “the right question” (which I think has been discussed productively already), but also the very idea of “the right standard” looming above everything.

    Does that mean that if the teacher asks a provocative question that stimulates student thinking in mathematically meaningful and productive ways (or a student asks such a question or answers a teacher or student question in a way that leads to meaningful mathematics, etc.), that the Standardista in the corner gets to blow a whistle at the end of the lesson, should not only the “right standard” not get pointed to, but perhaps no immediately identifiable standard (as in, could there be important, powerful mathematics that arises in problem-based math lessons that is outside of the rather narrow confines of our current national standards or some other set of standards in the future?)

    Maybe this is a trivial point, or “too political,” or just not getting at anything folks here want to discuss. If so, my apologies. It’s just the rebel in me, I suppose.

  23. Geoff:

    I appreciate Brett taking a charitable reading of the commentary. And Dan for challenging me on imprecise language.

    Same. Brett’s mediation has made our marriage 10x stronger. Appreciate your follow-up post too, Geoff, which I’ve added to the body of the post.

    Ethan:

    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.

    Probably. Except I run back to their questions at the end of the period, many of which have been answered in the course of answering our focus question. I say, “I love these questions and I hope we get to all of them by the end of our time today,” and they know I mean it because we do.

  24. False choice, IMO.

    Well, yes, it is a false choice. That’s my point.

    I believe we need to drill, and we need to open minds to what that drill allows us to do.

    But that’s not what those Krall and Gillard quotes say. A person could quite reasonably take from them that practice of skills was unimportant. After all, in their discussion of what is important in a Maths class they do not mention anything about teaching of skills.

  25. 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.