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.

But this is exactly why it’s wrong to declare, as you do, that anything that’s not direct instruction is discovery “under a different cloak”.

I absolutely agree that discovery can be as you describe. I agree that many kids are utterly uninterested in wondering or exploring. Moreover, I don’t think there’s anything wrong with their lack of curiosity, nor do I think that “the right teacher” could make them care.

And yet, I don’t do a lot of lecturing. I create tasks that don’t require wondering, but also aren’t instantly task-driven follow the numbers do what I’ve shown you. There is a lot of ground between open inquiry and direct instruction.

I was inspired by Shawn Cornally’s blog posts (Calculus: A Comedy). From this I stole the idea for allowing students to struggle with the problem of calculating instantaneous speed while wheeling around in the fresh air. For my group of high-achieving grade 12 students, this worked out remarkably well. They thought, designed, implemented, calculated, adjusted, re-calculated, and then struggled at the end when I stumped them with the request for speed at-that-exact-point. It led into an investigation, which led to some direct instruction, and culminated with my favorite part of the year: when I sit and read aloud to grade 12 students, from the book A Tour of the Calculus (by David Berlinski). Thus began our adventure in a story of mathematical beauty and genius.

After that first exciting lesson, I found that my students preferred to figure it out rather than receive direct instruction. I modified our lessons to give them more control, let them collaborate on discovering patterns for derivatives, and explain their reasoning to each other. The patterns are so important to understanding the “rules” of differentiation, and I had the impression that their grasp of integration was stronger (although it could also have been a stronger group of students).

At any rate, setting battles over educational theory aside, I tried this experiment in my classroom last year and have two big conclusions:
1) Assuming student understanding from both methods to be equal, reducing direct instruction made lessons much more enjoyable for everyone (which may increase understanding due to the happy amygdala)
2) As the teacher coach in these Calculus lessons, you need to know your audience. Some classes will respond well, some ideas are better adapted to this style of learning. Know your students and figure out the right dose of struggle vs explanation.

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.

So, as I detailed in this blog post, http://joyceh1.blogspot.com/2015/09/day-7-newtons-revenge-day-1-polygraph.html, I let students play Desmos Polygraph Parabola before knowing any vocab. (I also blogged about it previously with Kaplinsky’s Polygraph: points with the non-accelerated 8th grade class).

Students were frustrated with questions like “Is it negative or positive?” The more precise language was “Does it open up or open down?”

Halfway through the class, we took y=x2-1 and graphed it using an X-Y table as they suggested. After doing so, we graphed it. I asked them, what are some precise details about this graph that could help you describe it to a class mate?

They came up with Is it in all 4 quadrants? Open up? Open down? Does it have a y-intercept at (0,-1)? Does it have two x intercepts at this and that point (obvi being specific).

They also told me about the vertex. I asked them, how could you describe the vertex to someone? They said it’s where the graph doesn’t go below, or doesn’t go above. Oh ok, so that’s the minimum or maximum of the graph. Cool.

Students really were in tune to the interactive discussion we had halfway through class, and were excited to get back to the game, and succeed with their new vocabulary.

I know this isn’t calculus related, but definitely in the constructivist conversation. The “old way” of direct instruction lends to give you the vocab, then describe the graph. This is boring to many students.

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.

http://clearcalculus.okstate.edu/

I can imagine that via this type of work, yes, it would be quite possible to teach calc via IBL or guided-discovery. I think some of the project team might do it.

Similarly, Patrick Thompson has a calculus curriculum that’s pretty guided discovery:

http://www.patthompson.net/

The curriculum is described a bit here:
http://www.patthompson.net/PDFversions/2013CalcTech.pdf