## Learning Calculus Without Direct Instruction

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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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.”

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.

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.

1. #### MQ

September 14, 2015 - 2:28 am -

(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.

2. #### Evan Weinberg

September 14, 2015 - 6:42 am -

I don’t see it as a violation of constructivist philosophy to give suggestions of what knowledge might be useful in a conceptual development of Calculus ideas. A teachers, we frequently give/elicit background knowledge of this type both for assessment and conceptual development purposes. This isn’t DI necessarily, but it might skip over the part where students are rediscovering the slope formula again so that they can instead see what is different about the Calculus perspective themselves. Providing opportunities to figure a way through the concepts with gentle nudges in productive directions puts discovering these concepts within reach.

I see it like this: if someone drove me up a mountain blindfolded, threw me in the woods, and said ‘find your way back to base camp’, I’d have a hard time. I’d construct a path in some direction for sure, but there’s no promise I’d get to the intended destination, or even in the same direction as anybody else.

If, on the other hand, I was given a compass, a map, and maybe a radio to talk to others that were also working toward the same goal, we’d at least help each other along in a similar path. We wouldn’t be told exactly how to get down by someone that knows the answer, but we wouldn’t be crafting a compass from scratch either.

A happy medium is always there, especially in areas of math like Calculus that, at one time, mathematicians didn’t agree should be able to exist at all.

3. #### Joshua

September 14, 2015 - 7:45 am -

Unintentionally, I’m seeing several young children develop calculus (or important concepts in it) without direct instruction. Here’s the scenario:

I teach a computer programming class to 6th graders and my own, younger children. They like creating games, which have moving characters, and some have started making car games. How should a car move? Well, at first they program so that it goes forward when they are holding a key down, but that doesn’t look or feel to them like a real car. So, they get in and start modifying the equations of motion of their virtual objects. Before long, we’ve suddenly got position, velocity, and acceleration.

As I said, this is an unintended result, so I can easily imagine what someone else could accomplish with an intentional selection of tools and guided experiences.

4. #### Dan Meyer

September 14, 2015 - 7:46 am -

Evan:

I don’t see it as a violation of constructivist philosophy to give suggestions of what knowledge might be useful in a conceptual development of Calculus ideas.

Neither do the authors of HPL, by my reading. My questions are directed at people who believe students can and should learn calculus without direct instruction. Why and how?

5. #### Dan Meyer

September 14, 2015 - 7:50 am -

Joshua:

So, they get in and start modifying the equations of motion of their virtual objects. Before long, we’ve suddenly got position, velocity, and acceleration.

Can you describe how they managed this without any direct instruction? It seems there’s a lot of cognitive work tied up in “before long.”

6. #### Jason Dyer

September 14, 2015 - 11:18 am -

If you really want to see a calculus-from-scratch-with-everything-discovered check out Creative Mathematics, by H.S. Wall. (I have a review here.) It’s written via the Moore method which is about as hardcore constructivist as you can get. Keep in mind Moore’s audience is other aspiring mathematicians.

Noteworthy: there are some major edits from the standard curriculum, plus there are *still* bits where the book just hands over the keys.

7. #### josh g.

September 14, 2015 - 11:19 am -

Does “googling it” or “finding examples online” count as direct instruction in Joshua’s case? Because that’s usually how people figure out how to program stuff.

8. #### blaw0013

September 14, 2015 - 11:26 am -

Dan, I am not sure what you are shooting for with this post. It seems you have set out a straw-man debate, the notion that there may be an argument for no direct instruction ever. I think Marilyn Burns’ summary is PRECISELY on point.

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. Standards algorithms for procedures/skills must be taught explicitly, as Burn’s says “covered.” But, what is the impetus to learn standard algorithms, other than to fulfill the needs of a governing, disciplining body – such as CCSS or NCTM, or most educational institutions I know.

But for the child to learn calculus, free from social conventions, free from it necessarily being Newton’s calculus, can most certainly be taught free from direct instruction.

That last point distinguishes learning from being educated.

9. #### josh g.

September 14, 2015 - 11:40 am -

“But, what is the impetus to learn standard algorithms, other than to fulfill the needs of a governing, disciplining body – such as CCSS or NCTM, or most educational institutions I know.”

10. #### Brett Gilland

September 14, 2015 - 12:38 pm -

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.”

11. #### educationrealist

September 14, 2015 - 2:42 pm -

I’m not a constructivist, but I do find that it’s usually more effective to let kids figure out something with a directed activity than it is to just lecture through it. And the trick is finding that directed activity.

That’s where “beginning to imagine” comes into play.

For example, I’ve long been unhappy with my introduction to quadratics because unlike lines and exponential functions, there isn’t an obvious, intuitive application to give the students that leads them to the parabola. Then I realized a year or so ago that the product of two lines was a parabola. So after showing linear addition and subtraction, I moved into linear multiplication. This not only leads to the parabola, but leads to it in the factored form. It then leads from the factored form, via binomial multiplication, to standard form, and so on. Also helps me reinforce functions, function notation, etc.

I don’t teach calculus, but I’m sure that many concepts in calculus lead themselves to this sort of introduction.

I can’t *prove* that it’s better for kids to see where a concept originates, but I have found this method leads to better engagement. That is, when you teach kids who tune out a lecture and wait for you to come around and explain it personally, it’s nice to make them do the work first.

• #### Clara

September 14, 2015 - 4:04 pm -

We are graphing quadratics I co-teach with a person who does almost everything via direct instruction (how she was taught to teach. I teach in the way described above, by multiplying linear expressions, creating tables, and graphing the results. As Educationrealist notes, it leads to all those other great ideas that are often taught with direct instruction. I’ll be comparing the results: student engagement, understanding the content, modeling their understanding…..

12. #### Dan Meyer

September 14, 2015 - 2:52 pm -

Brian:

Dan, I am not sure what you are shooting for with this post. It seems you have set out a straw-man debate, the notion that there may be an argument for no direct instruction ever. I think Marilyn Burns’ summary is PRECISELY on point.

I read Marilyn as saying that social knowledge requires direct instruction but logical knowledge should be “uncovered.” So I realize that no one is arguing “no direct instruction ever.”

But Chester asked how students could learn the entire logic of calculus without direct instruction, without also being Newton or Leibniz. To which you referred him to IMP. At which point I had to say, hold on, IMP does directly instruct.

Perhaps you’d like to clarify your response to Chester.

13. #### Dan Meyer

September 14, 2015 - 2:55 pm -

@Brett, would you do us all a favor and get a blog and a Twitter account running and update us when you do? Like your vibe, insight, etc. You’re always welcome here until then.

14. #### Brett Gilland

September 14, 2015 - 3:08 pm -

Heh. I keep trying to restart my blog and then get overwhelmed with life and it dies a slow horrible death. Actually told you about it at NCTM Denver because I had just been picking fights with you via my blog at the time. Don’t know that I have posted since. Maybe this year… Maybe.

Still, thanks for the compliment. It is truly appreciated.

15. #### Dan Meyer

September 14, 2015 - 3:09 pm -

josh g.:

Does “googling it” or “finding examples online” count as direct instruction in Joshua’s case? Because that’s usually how people figure out how to program stuff.

Probably the quintessential case of needs-based direct instruction. It’s so much harder, by comparison, to appreciate and make sense of Stack Overflow postings for programming problems you haven’t had or made sense of yet.

16. #### Simon

September 14, 2015 - 6:47 pm -

In general, based on my experience with authors like Constance Kamii, and the current Japanese math teaching system, Marilyn Burns is right on. Things that students wouldn’t know, like naming conventions, ways for writing expressions, or things that students couldn’t be expected to figure out for themselves, are directly told to students. Otherwise, students should have the opportunity to work with problems that they haven’t seen before but could use prior knowledge to get at. Consider Jo Boaler’s pattern finding exercises, pretty much all students can talk about how they see change happening but the teacher can help them have a language and mathematical way of describing what they are seeing.

17. #### Rene Grothmann

September 15, 2015 - 2:31 am -

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.

18. #### Joshua

September 15, 2015 - 2:39 am -

@Dan and @josh g.

I think the steps and insights required are reduced because of the environment in which the kids are programming. We are using a turtle graphics variant, so you get a couple of things for free:
– position: all programs are visual, so you literally see the position of your turtle. Even better, there is a little box that shows x, y coordinates if you want to look at that.
– motion: the fundamental commands are commands of motion from the perspective of the turtle. That means we are working with velocity as a basic building block.
– incrementing variables: this is a super common programming idiom. I’m sure I showed them an example of “x = x+1” at some point. Yes, that’s a bit of direct instruction (if I understand the term) and a phrase that is fun to type on a mathematics blog.

So, the relationship between position and velocity is basically embedded and they just need to difference again to get acceleration. That requires a mild combination of 2 programming concepts.

19. #### Jason Dyer

September 15, 2015 - 8:09 am -

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?

20. #### timfc

September 15, 2015 - 11:59 am -

While not a full curriculum, you might check out:

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

21. #### mike

September 15, 2015 - 3:41 pm -

@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.

22. #### Martin Joyce

September 15, 2015 - 5:21 pm -

At the CPM academy of best practices we got the tour of Desmos.com by the creator Eli. We focused on Polygraph: Hexagons and the frustration (headache) caused by students inprecise language to each other created a need for the vocabulary (aspirin). Eli cited the research of the intellectual need, where students don’t see a need for the vocabulary until they need to describe it.

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.

23. #### suehellman

September 15, 2015 - 5:44 pm -

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.

24. #### R Eichholtz

September 16, 2015 - 11:33 am -

This is my first post on your comment wall. Last year I tried teaching Calculus with reduced direct instruction, and to my pleasant surprise, it worked out quite well!

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.

25. #### Zach

September 16, 2015 - 5:15 pm -

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

26. #### education realist

September 16, 2015 - 8:52 pm -

“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 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.

27. #### Chester Draws

September 16, 2015 - 11:03 pm -

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.