In the comments, Marilyn Burns distinguishes knowledge students should be told from knowledge they can reason through:

Explicit instruction (teaching by telling?) is appropriate, even necessary, when the knowledge is based in a social convention. Then I feel that I need to “cover” the curriculum. We celebrate Thanksgiving on a Thursday, and that knowledge isn’t something a person would have access to through reasoning without external input”•from another person or a media source. There’s no logic in the knowledge. But when we want students to develop understanding of mathematical relationships, then I feel I need to “uncover” the curriculum.

Chester Draws responds and asks a question which I’ll extend to anyone who takes a similar view of direct instruction:

You expect student to “uncover” calculus?

Do constructivist teachers quietly just directly instruct such topics? Do they teach them, but pretend the students found them out for themselves? I can’t even begin to imagine how I could teach the derivative through constructivist techniques.

Brian Lawler responds that, yes, it’s possible to teach calculus without direct instruction, and offers up his Interactive Mathematics Program Year 3 unit “Small World” as evidence. Pulling out my copy of IMP, though, I find pages in the Small World unit that directly instruct students in the calculation of slope, the calculation of average rate of change, and the definition of the derivative. This appears to answer Chester’s question, “Do constructivist teachers quietly just directly instruct for such topics?”

So I hope Marilyn and anybody else with similar ideas about direct instruction will take up Chester’s question with force. It’s an important one, and mandates like “uncover the curriculum” seem more descriptive of philosophy than practice.

It’s worth pointing out in closing that this direct instruction in IMP is preceded in each case by activities through which students develop informal and intuitive understandings of the formal ideas. This is in the neighborhood of pedagogy endorsed by How People Learn, which, again, you should all read. It just isn’t the example of direct instruction-less calculus Lawler seems to think it is.

BTW. Clarifying, because I’m frequently misinterpreted: I don’t think learning calculus without direct instruction is logistically possible over anything close to a school year, or that it’s philosophically desirable even if it were possible.

BTW. Elizabeth Statmore offers an excellent summary of the pedagogical recommendations in How People Learn.

BTW. Chester references “constructivist teachers.” Anybody who sniffs back at him that “constructivism is actually a theory of learning, not teaching” gets week-old sushi in their mailbox from me. I think his meaning is clear.

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Brett Gilland:

I consider CPM pretty strongly constructivist, and I am currently LOVING that calculus program. My kids are generating much better insights and dialogues around calculus than I have seen with other programs and what I am seeing suggests much better expected results on the AP exam. Don’t know if it is sufficiently pure to pass the “no instruction evar!” test that you seem to be using.

Also, it is probably worth noting that the VAST majority of AP workshops are centered around making instruction more constructivist, because the typical textbook presentation is so MASSIVELY DI. That makes this whole conversation feel like it is taking place in some bizarro universe. The question I tend to find myself asking is this:

“You expect students to understand calculus when taught using only DI?

Do traditionalist instructors just quietly slip in investigations, but pretend the students figured things out from their amazing lectures? I can’t even imagine how I would teach derivatives without heavy exploration of finite differences, secant and tangent lines, and distance/time vs velocity/time graphs.”

Sue Hellman:

No matter what anyone says, “uncovering curriculum” is just discovery learning & concept formation in a different cloak. I started teaching back in the 1970’s when this method was in full bloom. Once we’d set the stage for ‘aha moments’ of understanding to occur, we let the struggle ensue. We were limited to asking a learner nudging questions, redirecting his/her efforts in a more fruitful direction when the chosen path was a dead end, and simplifying the problem so he/she’d trip over the gem. As one after another student ‘got it’, we imagined we could hear a series of little light bulbs popping on over their heads until the light of understanding in the room was blinding!

The problem was that it’s impossible to be at every student’s side to ask just the right question at just the right moment all at the same time in a class of 25 or 30. First discoverers would end up telling their more baffled peers the secrets (hidden direct instruction). Some of the really lost got direct instruction from older siblings or parents.

The awful thing was that students who didn’t get it couldn’t turn to the teacher for help, because they’d only get more questions & more ‘lead up’ to the place where the leap of understanding had to be made. Direct questions were not to be met with direct answers. The teacher didn’t believe in telling.

And that’s the sort of dishonest thing about this method. The teacher has all the secrets. Everyone knows this is the case. The students’s job becomes finding and digging up treasure. For a some this is an act of learning. For many it’s like being in an elaborate guessing game with the prize of enlightenment denied those who are not capable players. The teacher had all the secrets but never tells.

But what many teachers don’t get is that taking students through a process of discovering Uncovering those secrets is not the same as constructing personal understanding. And they also don’t factor into the learning experience the fact that direct instruction is everywhere. If you won’t share the secrets & homework has to be done, kids will ‘uncover’ up what they need online. Kids can circumvent your process with the tap of a finger if it will get them what they (or their parents) believe they need.

Although deepening understanding is the goal, getting a toehold on being able to do some stuff can be a great place to start. How many of us who drive have a deep understanding of the physics, chemistry, and engineering that makes up our vehicles? Yet we can use them skillfully to solve problems. The same goes for cooking. If you can follow a recipe you can feed your family – which is a pretty fantastic result achieved without understanding how & why the recipe works. There’s nothing wrong with passing on knowledge and then getting on with helping kids learn how to apply it confidently and in a broad range of circumstances.

As teachers, it’s our job to pack our tool boxes with as broad a range of strategies as possible so we can help each kid forge the connection of skill development & growth of understanding. I urge colleagues not to become so enamored with one approach that they become ‘one trick ponies’. I fear it will not serve you or your kids well in the long run.

Greg Ashman, an advocate of “explicit instruction”:

In an influential book for the National Academies Press in the US [How People Learn], the constructivist position is explained in terms of the children’s book Fish is Fish. In the story, a frog visits the land, and then returns to the water to explain to his fish friend what the land is like. You can see the thought bubbles emanating from the fish as the frog talks. When the frog describes birds, the fish imagines fish with wings, and so on. The implication is that we cannot understand anything that we have not seen for ourselves; each individual has to discover the world anew.

Meanwhile, here is how the cognitive scientists who wrote How People Learn actually interpret Fish is Fish (pp. 10-11):

Fish Is Fish is relevant not only for young children, but for learners of all ages. For example, college students often have developed beliefs about physical and biological phenomena that fit their experiences but do not fit scientific accounts of these phenomena. These preconceptions must be addressed in order for them to change their beliefs (e.g., Confrey, 1990; Mestre, 1994; Minstrell, 1989; Redish, 1996).

To illustrate this phenomenon we need only look at Ashman’s essay itself. Ashman came to his essay with the common misconception that constructivists believe that “each individual has to discover the world anew.” Even though the How People Learn authors interpret Fish is Fish explicitly, that explicit interpretation wasn’t enough to dismantle Ashman’s misconception.

Perhaps the HPL authors should have taken their own advice, anticipated Ashman’s misconception, and addressed it explicitly. It turns out they did exactly that in the next paragraph:

A common misconception regarding “constructivist” theories of knowing (that existing knowledge is used to build new knowledge) is that teachers should never tell students anything directly but, instead, should always allow them to construct knowledge for themselves.

A book is nothing if not a medium for explicit instruction and Ashman illustrates the limits of that medium here. Explicit instruction is powerful, certainly, and I can’t think of any influential scholars, least of all the authors of How People Learn, who would deny it. But it often isn’t powerful enough on its own to remedy a student’s existing misconceptions. Luckily, How People Learn offers many more powerful prescriptions for teaching and it’s free.

BTW. Always relevant: The Two Lies of Teaching According to Tom Sallee.

I’ve thought about “modeling” more than I’ve thought about any other specialization in mathematics. I’m learning less and less about it each year so I’m hopping onto a different track for awhile, moving onto other questions. First, I wanted to collect these links in one location:

Print

• Missing the Promise of Mathematical Modeling. My Spring 2015 article in Mathematics Teacher analyzes the meager opportunities for modeling in school mathematics and recommends some solutions.
• The Checkout Line Is A Scam. (Or Is It? [Yes, It Is.]) My blog post illustrating modeling through the lens of grocery express lines, which are a scam.

Video

• Math Class Needs a Makeover. A 11-minute talk I gave for TEDxNYED in 2010. Couple million views.
• Fake-World Math: When Mathematical Modeling Goes Wrong and How to Get it Right My TEDxNYED talk + five more years of study + 50 extra minutes.

Summarizing all of the above in a single paragraph:

Modeling asks students a) to take the world and turn it into mathematical structures, then b) to operate on those mathematical structures, and then c) to take the results of those operations and turn them back into the world. That entire cycle is some of the most challenging, exhilarating, democratic work your students will ever do in mathematics, requiring the best from all of your students, even the ones who dislike mathematics. If traditional textbooks have failed modeling in any one way, it’s that they perform the first and last acts for students, leaving only the most mathematical, most abstract act behind.

Daniel Willingham, in his book Why Don’t Students Like School:

Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question.

Scott Farrar, on my last post on motivating proof:

I think this latches onto the structure of the geometry course: we develop tool (A) to study concept (B). But curriculum can get too wrapped up in tool (A), losing sight of the very reason for its development. So we lay a hook by presenting concept (B) first.

ISTE just wrapped. NCTM wrapped several months ago. What was accomplished? What can you remember of the sessions you attended? Will those sessions change your practice and in what ways?

Zak Champagne, Mike Flynn, and I are all NCTM conference presenters and we were all concerned about the possibility that a) none of our participants did much with our sessions once they ended, b) lots of people who might benefit from our sessions (and whose questions and ideas might benefit us) weren’t in the room.

The solution to (b) is easy. Put video of the sessions on the Internet. Our solution to (a) was complicated and only partial:

Build a conference session so that it prefaces and provokes work that will be ongoing and online.

To test out these solutions, we set up Shadow Con after hours at NCTM. We invited six presenters each to give a ten-minute talk. Their talk had to include a “call to action,” some kind of closing homework assignment that participants could accomplish when they went home. The speakers each committed to help participants with that homework on the session website we set up for that purpose.

Then we watched and collected data. There were two major surprises, which we shared along with other findings with the NCTM president, president-elect, and executive director.

Here is the five-page brief we shared with them. We’d all benefit from your feedback, I’m sure.

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Marilyn Burns on her reasons for attending conferences like NCTM:

I don’t expect an NCTM conference to provide in-depth professional development, but act more like a booster shot for my own learning.

Elham Kazemi, one of our Shadow Con speakers, tempers expectations for online professional development:

I have a different set of expectations about conferences and whether going to them with a team allows you to go back to your own contexts and continue to build connections there. Can we expect conferences and the internet to do that – to feed our local collaborations? I get a lot of ideas from #mtbos and from my various conversations and conferences. But really making sense of those ideas takes another level of experience.