(I will take up a bit of the question — perhaps not with force so much as curiosity.)

The ‘you’ is a general you, but it is (admittedly) with C. Draws in mind.

Before exploring a particular path, it is tough to know if it will lead somewhere worthwhile. If you have a sense of the type of path, or have walked similar paths, or however you wish to fill in this metaphor, then maybe you can make an educated guess at its potential worth.

As a mathematics instructor, you may know that, for certain functions, defining a new function in terms of slopes of tangent lines leads somewhere worthwhile. (More precisely: It can lead to the Calculus, which you may or may not feel is worthwhile!)

There is a way of teaching the Calculus in which you provide definitions, propositions, lemmas, theorems, and corollaries. There are good reasons, or at least reasonable reasons (I think), for teaching that way.

Here is another approach.

Draw a parabola; draw a tangent line; and let students sit with that or let them discuss it among themselves. (A good point at which to re-iterate from D. Meyer’s post: “I don’t think learning calculus without direct instruction is logistically possible over anything close to a school year…”)

How long do students sit with this parabola and the single tangent line drawn? I don’t know. Maybe close to a school year. Maybe 3 minutes. Maybe it’s the last thing students see before the weekend (or a break), and you ask them to think about it, then re-draw it when you all re-convene, and facilitate a discussion around student generated ideas. (Who thinks about such things once they’re outside of the classroom walls?)

These decisions depend on your students; depend on your course; depend on many issues related to classroom culture and climate; depend on administrations and parents; etc.

But mathematically speaking: You can draw a tangent line at any point on the parabola. Just one such line! And it is not obvious that there is just one tangent line for each given point on the parabola (it’s not obvious to me, at least) but if students can grasp that much, then we can think about what to do with these lines.

For each x-coordinate from our parabola, we have this unique tangent line; call it g(x) = mx+b. What should we do with mx+b? We could focus on m and think about how it changes. This leads to the notion of a derivative and, while I would rather not sketch it out now, one can scaffold (as a function of time, attention span, and so much more) towards the Calculus.

But we could also focus on b. What does this even lead to? (This is not rhetorical; what does it lead to?) Or maybe we want to focus on m and b, and think about a 3D plot where, instead of x-y-z axes, we have x-m-b axes corresponding to our parabola and its various tangent lines. What does this graph even look like? (Also: Not rhetorical!)

It turns out that pursuing ‘m’ as above is especially worthwhile (in some sense) but I don’t think even the brightest of students would/should be expected to guess as much from the outset.

When you are the teacher, there are so many considerations! Why start with a parabola? Why draw just one tangent line? At what x-coordinate should you draw the tangent line? (Should it be at the x-coordinate for the vertex? At least then you’d know its slope — i.e., 0 — and y-intercept… does that make it a better or worse choice?) How long do you leave students with such an illustration? Is it okay for students to pursue the 3D plot approach mentioned above? Is it okay for students to pursue totally different questions? And so on.

This is somewhat long. Should I have just given a direct answer? (If I had: Could a fair response to me have been, ‘If you support teaching without direct instruction, then why are you answering “directly” here?’)

Instead, please allow me to close by pasting the most interesting — to me — part of the quotation:

“I can’t even begin to imagine how I could teach the derivative through constructivist techniques.”

Direct instruction, constructivist teaching, a mixture of the two, or something else entirely different: Each of these, I think, can be an appropriate choice. But the tragedy in my reading is the phrase, “I can’t even begin to imagine.”

For my (perhaps idealistic) self, the most important choice in this context is not when to provide direct instruction and when to use methods from constructivist teaching; the most important choice is believe you can begin to imagine — whatever the topic may be, and to work hard to foster that same belief in imaginative potential for your students.

Students are usually not able to discover the solution to a problem. But they are well able to discover that there is a problem. The right track is: seeing – wondering – trying – failing – learning – mastering – modifying. I think you often demonstrated how the first part of this journey is a good example of learning by self guided discovery.

As a side remark: People, I often cannot follow your discussions because you use these American centered abbreviations. What is CPM, AP, DI? Americans seem to love acronyms. As a joke: PCMCIA = people can’t memorize computer industry acronyms.

What is CPM, AP, DI?

@Rene:

CPM: a particular curriculum out of a company in California. I’m afraid I don’t know much about it. They use the acronym for their own company.

AP: Advanced Placement. Classes in American high schools have a test that many colleges will take as college credit. (So AP Physics might give credit for Physics 101 in college, allowing the student to skip the class.) The acronym is super-common enough I’d call it fair use, since Advanced Placement in full would confuse non-US readers equally much.

DI: Direct instruction. Not sure why we need an acronym for this one.

Where are you from? Could you talk a little about how the different curriculum types differ from the US?

@Jason Dyer

DI: as Differentiated Instruction is the dominant meaning of the acronym “D.I.” in the education world, I think clarification is merited as to its meaning here.

In this discussion I supposed it to mean Direct Instruction, however I do think that the overabundance of acronyms(some American centric) detracts from the general discourse.

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 enamoured with one approach that they become ‘one trick ponies’. I fear it will not serve you or your kids well in the long run.

I’m curious: Is there a topic or concept that you can’t effectively teach using direct instruction?

Let’s be clear – I’m happy to go outside “traditional” teaching methods. I barely ever use textbooks and my classroom is frequently shambolic as I try to teach high and low levels simultaneously.

However, I refuse to go down the rabbit hole that is fake constructivism – what Sue Hellman explains above – where everyone knows that they are only “discovering’ what the teacher could tell them in a quarter the time.

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

Why would I expect my students to understand calculus? They’re sixteen and seventeen.

I’m much more interested in making sure they don’t learn any bad ideas.

Today one of my boys, not the worst, had to be explained that 2(x+3)^2 is not (2x + 6)^2. How much understanding of calculus is he likely to ever get?

My notes to Chester Draws were to emphasize there is conceptual underpinnings of calculus that must be developed with meaning for children – this is what is important in teaching calculus.

I strongly disagree.

Kids learn best by learning the process involved, so that they believe that they are moving forward – so that they think they are learning. They gain understanding over time for repeated application of process, helped by judicious pointing by the teacher.

Putting the requirement for “meaning” before process makes it harder for everyone.

Since the conceptual basis of calculus is particularly hard, I introduce it very late in the piece, and only to my top students. Since they know what the processes are, they have a framework on which to pin their understanding.

The right track is: seeing — wondering — trying — failing — learning — mastering — modifying.

This is where I really part company with the constructivists.

Learning this way is slow and annoying. The satisfaction a teacher feels when they see the “light bulb” go in a kids head is not necessarily shared by the student. If the process has been painful along the way, then the satisfaction they have is the one you get when a painful noise stops.

i taught myself CCS a year or so back. I could have worked it out by looking at other web sites until I figured it out – constructed it myself. That would have been mental, so I went and got a couple of books to help me. The whole time I was thinking that it would have been much better to have a decent teacher to direct me, and answer my questions.

The point to realise is that I didn’t want the satisfaction of learning CCS, I just needed the ability to do it. (I also never cared why CCS is the way it is, at any point in the process. I still don’t.)

Not a single one of my students will go on and do pure Maths at university. They do calculus because they need it as a skill. I’m doing them a disservice if I don’t teach that skill in the least painful way possible.

Yes I know that is not “educating” them. They don’t care, and I’m doubtful that education can be achieved from without at all.