I’ve been reading through some of my older ideas on inquiry. Early in my teaching, I took inquiry as a kind of golden ideal — it was hard, but it was the best way to teach math, and I needed to work my way in that direction. I didn’t have a very specific idea of what I meant, but it more or less meant kids figuring out mathematical concepts themselves.

I’ve come to dislike the term inquiry. There’s plenty of good in what is often termed inquiry, but my dogmatic approach led me to some ineffective teaching strategies. Most of all, inquiry is a pretty divisive term, and I think a specific conception of exactly what it is that helps students learn and why is worth figuring out. Here are the issues with an inquiry approach that I’ve had in my teaching.

**Inquiry Assumes Kids Remember What They Figure Out**

It might be true that figuring something out for yourself leads to better retention, but that’s not the whole puzzle. If I want a student to be able to apply a mathematical idea six months after he or she leaves my class, there are a ton of variables in play. They need to have applied the idea in a number of different ways, spaced over a period of time, tied to what they already know, and in a way that emphasizes essential elements of the concept. Inquiry is one approach here, and it’s not invalid, but I think my assumption that kids figuring things out for themselves is inherently good led to some questionable decisions. I would hypothesize that learning depends less on how an idea is first introduced, and more on the thinking that students do with that idea down the road.

**Are They Really Just Playing “Guess What’s In My Head”?**

I understand different mathematical ideas in my own idiosyncratic way. Too often when I went into a lesson wanting students to figure something out, I wasn’t actually asking them to figure something out about the math, I was asking them to figure out how I understood the math. It’s a subtle distinction, but it led to long worksheets where students answered questions they didn’t know the meaning of in an attempt to lead them to a very specific understanding that I had envisioned. Even if I do have some superior conception of how a mathematical idea operates, students aren’t doing the hard intellectual work here. Instead, they’re being conditioned that learning math is about guessing what the teacher wants students to say, and being led by the nose until they get to some magical formula that I could’ve just told them to begin with. Students learn by making their own sense of the math they’re encountering, but a classroom where they’re constantly guessing what’s in the teacher’s brain builds beliefs in students that make this learning impossible. I’ve given classes way too many worksheets where they solve what are (to them) a series of simple but unrelated problems. Then I ask them to make an inference about a new problem. A few students offer incorrect ideas, then a student “gets it”. I repeat that student’s answer, have everyone write it down, and I delude myself into thinking that the entire class just learned something.

**There Isn’t Just One Big Idea
**When I was focused on teaching so that students figured things out for themselves, there was usually a big, magical “key understanding”. A classroom where students are just trying to guess this mysterious “key understanding” in my head seems likely to be unproductive, but I think the illusion of one big idea is damaging as well. Mathematical ideas are messy, and coming to a deep and flexible understanding of math involves dealing with ambiguities, special cases, and unusual applications. If I have some illusion that, just because students found the rule for the Pythagorean Theorem instead of having it told to them, they will be able to apply it in new ways, I’m taking away opportunities for learning. Students need to know the formula — but they also need to understand non-examples, relate right triangles to the coordinate plane, find right triangles in complicated figures, drop auxiliary lines, recognize patterns in Pythagorean triples, relate the Pythagorean Theorem to acute and obtuse triangles, and much more. There isn’t a finite set of key understandings that will get students there — all of those ideas are interconnected, and build a broadly applicable schema, but only if students are wrestling with these interconnections. Inquiry doesn’t make any of this impossible, but oversimplifying inquiry to students figuring out one new big idea obscures the complexity of the relevant mathematical ideas.

**Inquiry definitely does some things right. **But I think an oversimplified understanding of inquiry chasing the idea that students should figure everything out for themselves leads to a classroom that is centered on students guessing what the teacher is thinking, at the expense of doing original mathematical thinking of their own.

I don’t mean to attack anyone else’s practice, or declare that inquiry is useless, or take a side in the math wars, just to illuminate the issues I’ve encountered. I’ll put some together some thoughts on the good in inquiry in my next post.

BrianI like the idea of what spiraling brings to activity based inquiry…been reading some on interleaving learning and see a real connection to what I used to do with inquiry in the classroom.

dkane47Post authorI think that’s one pairing that makes a ton of sense — take away the pressure of “they have to know x by the end of the period so we can do y tomorrow” and an inquiry approach becomes much more meaningful.

blaw0013Much appreciated – thank you for sharing your thinking / evolution with regards to inquiry-based education. Resonated well with mine!

dkane47Post authorThanks, Brian!

NicoleI think that “inquiry assumes that kids remember what they figure out” has also been my primary motivation for teaching that I have done that would fall under the category of inquiry-based. But when I think back to those lessons (“discovering” the Pythagorean Theorem and exponent laws spring to mind), I don’t think that students have done actual inquiry. And this is primarily for two of the reasons that you have brought up– I wanted them to notice one specific thing that was already in my head and I expected that they could discover and deeply understand it in a day. For me, I think the major issues in my own inquiry-based teaching is that I have pushed kids to “fluency” too quickly. To get the benefits of the productive struggle of inquiry I think I need to have students spend more time on it with less direction from me. That way as they apply the idea(s) in the future, they can do the recreation and remembering necessary because they owned the process in the beginning.

dkane47Post authorInteresting you bring up fluency — it’s mentioned in both the CCSS and the Principles to Actions Practices. I’m not sure I have a great understanding of what fluency means in this context, in particular across a range of content areas. I totally agree about the amount of time spent on a topic — there’s some interesting brain research that suggests it’s more or less impossible to get something into long term memory in the span of a day or two. That seems to have a ton of implications that go way beyond the possible benefits of inquiry.

“they owned the process in the beginning”. I intuitively agree with this statement, but I’m not sure why. What is happening here? Is it just a questions of engagement or motivation, or is there something else going on?

NicoleThe Strands of Mathematical Proficiency chapter in Adding it Up has had the biggest influence in how I view fluency. They define procedural fluency as “knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.” I think that the fluency is often misunderstood as a being quick and accurate with procedures, but that misses the flexible and efficient part which is where I think the real thinking comes in. At it’s best, I believe that inquiry, as you’ve defined it, should be part of the process of building procedural fluency from conceptual understanding.

LisaDarn you for pulling me out of my “I’m not going to stay current on blogs and Twitter because I’m on vacation” mode… 😉

You expressed very well how I feel about inquiry. For years, it has seemed like the “ideal” – especially in Geometry. I could never figure out why I couldn’t get it to work out the way it was “supposed to” in my classes. I think, for me, it ended up being a big “guess what’s in Mrs. Henry’s head” whenever I tried to structure lessons that way. On some occasions, for some students, it worked out. But for the vast majority of my students and the number of times I did it, it didn’t work and I could never peg why it didn’t. I mean, after all, I structured the inquiry so it should have lead to EXACTLY what I was thinking. How could my students not get it? Can’t they read my mind?? When you wrote that section of the blog, the bell just went off in my head as to why it didn’t work.

As we have moved to Common Core and I have examined the standards, one of the questions that has hovered in the back of my mind is whether it matters /how/ a student gets to an answer. If they do something that is mathematically sound, will the student receive full credit on the standardized test? If it’s a multiple choice question, the grader won’t have any idea how the student got to the answer. If it’s a question where the student has to show his or her reasoning, then the grader will know. Where it concerns me is if I allow any mathematically sound approach in Algebra, what happens when the student is scored by someone who has been instructed to only allow approaches that use equations? If the directions clearly state that the student must use equations, it’s a moot point. But if the directions ask the student to solve the problem and no qualifying directions are given, in my mind, the student should be able to use any mathematically sound approach. Otherwise, once again, we’re asking the students to read our minds.

My natural follow up to this post would be how do we best include inquiry in our approach? How do we make it authentic, where the student is actually working through their own thinking rather than trying to figure out what we’re thinking? Based on your closing lines, I am hoping that is what comes next. I look forward to it. Thanks for making me think about this!

–Lisa

dkane47Post authorThanks for this, Lisa. It reminds me of something I was reading in Embedded Formative Assessment recently, by Dylan Wiliam. He asked students to count the number of rectangles formed by a set of horizontal and vertical lines. He had articulate what he called “process criteria” — students should start with simpler cases, and use these simpler cases to build up to the question he asked.

He was surprised when, after just a few minutes, a student presented a general expression for any number of m and n horizontal and vertical lines, with no scratch work. He was initially upset and asked the student to show him his scratch work — but the student had actually had used a really insightful piece of combinatorial reasoning to find the general form.

I’m getting a bit off-topic — but Wiliam’s point was that, by coming into a class with “process criteria”, he constricted student thinking, and that what mattered more was the quality of that thinking, no matter what pathway students take. That seems pretty powerful to me — but also difficult to apply in the classroom.

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fawnpnguyenI hear the terms “inquiry-based” and “project-based” being tossed around the internet and never really understood what they mean. Or what they look like in the classroom. After all, if I don’t see it in action — with real students in real time (and over a period of time) — then I remain skeptical. I don’t think anyone who advocates for inquiry would claim that it’s about guessing what’s in the teacher’s brain. But without the teacher’s intentional problem-posing, structuring, scaffolding, connecting, and more connecting, then it becomes just that [guessing what’s in the teacher’s brain] — and the student is doomed. I hope inquiry does not mean walking a kid to the edge of the forest and say, “Go on. Find your way and I’ll meet you on the other side.” Anyway, I think the biggest enemy/challenge of inquiry-based learning is TIME. We say we want kids to learn at their own pace. This is a big fat lie as long as we continue to have due dates and grades.

I think of the slogan “inquiring minds wants to /know/.” Says nothing about wanting to figure it out on their own. 🙂

Thank you, Dylan!

dkane47Post author“We say we want kids to learn at their own pace. This is a big fat lie as long as we continue to have due dates and grades.”

Love this. I’ve been thinking more about some of these principles as ideals — but ideals that often don’t work out in the realities of the institutions and classrooms that real teachers work in. That tension is a tough one.

Thanks for pushing my thoughts!

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Hannah A.I am definitely one that advocates for the fact that if you figure something out for yourself you learn it better. There is that “ah-ha!” moment that I think has value to students and their learning. However, I completely agree with you that there are many other factors that come in to play in order for students to actually retain this information.

Is it really true inquiry for the students if it’s a guessing game of my thought process of the concept? The students are working through the concept how I thought it could be discovered but in reality, mathematics has so many different avenues students could take and come to the same conclusions. These possible different avenues could be a really important discussion to have in the classroom and we could be missing out on these valuable conversations if we continue with inquiry how we are. You also mentioned that we use inquiry for the main concept but there are so many other aspects that go into fully learning and understanding the concept and having the ability to apply it to new situations specifically (your example of Pythagorean Theorem). These other “sub-concepts” are really important to a student’s understanding of the material but aren’t discussed in an inquiry approach. I think I have come to the conclusion that inquiry is a useful tool in the classroom but maybe this isn’t how students should learn right away. I wonder if there is still value in inquiry if we used it to discover the sub-concepts after instructing students on the overall concepts.

dkane47Post authorThis is interesting. One idea I’ve been playing with for a while is that students should spend time doing “inquiry” type thinking for a similar type of problem — call them sub-concepts, or transfer, or whatever you like. One example — I just finished teaching conics through a variety of methods, mixing some inquiry with some explicit instruction, with lots of practice in different forms. One lesson I took a pure “inquiry” approach on was asking them to write a guide to graphing equations with the general form (x-k)^3+(y-h)^2=1. It was a fascinating opportunity for student discovery and transfer — many of its characteristics are similar to those of conic sections, but some are different. Students had the tools to do great thinking, it was low stakes so if some students weren’t thinking at the level I was looking for it wouldn’t set them back learning other material, and they had the opportunity to make sense of the material on their own terms, similar to what you alluded to.

I wonder if more tasks like this are a viable solution? I often think that the big ideas students need are, in most cases, remarkably simple — it’s applying them in different contexts that is complex. We can try to have them memorize every possible context — or we can use this approach to build deep, flexible understanding that is more likely to transfer.

Thanks for your ideas!