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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.

August Remainders

I’m bringing this feature back on the encouragement of Tracy Zager and because I ought to have more to show for the truly inappropriate sums of time I spend trawling the mathtwitterblogosphere.

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  • Patty Stephens is an instructional leader in Washington state. I’m hoping she writes more about her Teacher Fellows program, which attempts to build teaching capacity throughout the state. (Ditto Bryan Meyer about his Teacher Partnership Program, while I’m here.)
  • Bridget Dunbar has been blogging and tweeting for years but left her first comment on my blog last week, pointing me to her exceptional post comparing the pros and cons of three representations of the same problem.
  • Ryan Muller is a software developer who writes about education research at his Learnstream blog. He seems curiously unaffected by education research’s typical turf wars, just happy to read and write about what he reads. Refreshing.
  • Timothy McEvoy writes thoughtfully and critically about math edtech, a genre of writing that is in short supply.
  • Tom Bennison runs the #mathsjournalclub chat on Twitter – a discussion group for math education research, which is the kind of social unit I’m already missing from grad school.
  • Julie Reulbach offers us all a daily photo and caption from her innovative algebra classes. Yes, please.
  • Kris Boulton applies our headache metaphor to a question about slope. Watch how his subtle alterations to the same task make its mental controversy more acute. More like this, please.
  • John A. Pelesko and Michelle Cirillo are a university-level mathematician and math education researcher, respectively, and have paired up for a blog dedicated exclusively to mathematical modeling! Their post “Is this mathematical modeling?gets it.
  • Sam Shah’s new group blog around good questioning strategies had me at “Sam Shah’s new group blog.”
  • Nathan Kontny writes breezy narratives about entrepreneurship, at least one of which (on audience) is still rattling around my head one month later.

This is from a worksheet I assigned during my last year in the classroom:


There are lots of good reasons to ask students for multiple representations of relationships. But I worry that a consistent regiment of turning tables into equations into graphs and back and forth can conceal the fact that each one of these representations were invented for a purpose. Graphs serve a purpose that tables do not. And the equation serves a purpose that stymies the graph.

By asking for all three representations time after time, my students may have gained a certain conceptual fluency promised us by researchers like Brenner et al. But I’m not sure that knowledge was ever effectively conditionalized. I’m not sure those students knew when they could pick up one of those representations and leave the others on the table, except when the problem told them.

Otherwise, it’s possible they thought each problem required each of them.

The same goes for representations of one-dimensional data. We can take the same set of numbers and represent its mean, median, minimum, maximum, deviation, bar graph, column graph, histogram, pie chart, etc.

So here is the exercise. Take one representation. Now take another. Why did we invent that other representation? Now how do you put your students in a place to experience the limitations of the first representation such that the other one seems necessary, like aspirin to a headache?

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Howard Phillips:

Ok. First is bar chart, second is box plot.

All situations in statistics require some data, and the best data is that which students compile themselves. For this comparison a single set of data is best presented as a bar chart, but compare the data from five or more distinct groups of subjects, same measure, and the multiple strip bar chat is a bloody mess. Five box plots above the same numberline, and so much more is revealed, at a small cost of loss of detail.

I used to think that box plots were a waste of time until I saw the above usage.

Kim Morrow-Leong:

The same is true of physical representations. I am thinking of many algebra growth problems that involve squares and growing patterns. It is valuable to ask students to go through the actions of adding squares to watch a pattern grow through the addition of tiles. This action can help them have the physical experience of a rate of change. But this representation also has its drawbacks. It is clearly cumbersome and not efficient.

Daniel Schneider:

I think of them all as connected to making predictions about data – certain representations lend themselves to different ways in which data is presented, and certain representations help make predictions about that data.

Tables are great when you need to generate data from a scenario – you have a situation that has been given to you and you need a place to start. Creating a table for some initial data helps you see the patterns in whats happening and helps you make littler predictions about where the data is going. If I want students to appreciate tables, I give them a visual pattern or a scenario problem with a starting condition and a rate of change, then ask them some questions about what will happen.

Graphs are great when you’re given several random data points that, even when arranged as a table, don’t indicate a clear pattern. Sometimes plotting these visually helps you predict what other points could be missing, or what other points exist as the pattern continues. This is especially true for situations whose solutions depend on two variables, such as only having 30 dollars to spend on item A that costs 2 dollars and item B that costs 1 dollar. When I want students to appreciate graphs, I give them one of these situations (which usually lends itself to standard form of an equation, but they don’t know that) or I give them several data points and ask them what’s missing in the pattern. This is easier to see when organized visually and you you a particular shape to your points rather than a random collection.

Equations are the most efficient way to make predictions about patterns – if you’re given an equation, there’s no reason to have any other representation. Equations are useful for predicting far into the future for your data – maybe you can figure out the first few terms of your pattern, but trying to generate the 100th term is inefficient. Using an equation is like being omnipotent with a set of data. When I want students to appreciate equations, I give them a scenario but ask for a data point in the absurd future where the table or graph necessary to find the point would be too large and unwieldy to use.

The order I’ve presented these in this comment is also my typical order for presenting these representations to students: tables are useful at the beginning to generate data; graphs are useful once you have lots of it that may or may not be organized and may be missing some points, and equations are good for predicting the future.

A curious consequence might be: it’s not particular situations that necessitate one representation versus the other; rather, its what data you choose to give them at the beginning and what you ask them to do with it that makes one representation more valuable than another.


So very true. This skill seems to be neglected in our classrooms. Computers can take one representation and switch to others over and over again, much faster than humans can. If switching back and forth is your only skill, I can easily replace you with a $100 calculator from Target. And the calculator will be faster and more accurate.

But if I’m training students to be problem solvers who are smarter than computers, the “which representation is needed here” is a much more important question. I’m not aware of a computer that can answer that question.

Chester Draws:

Draw a simple line on a graph.

Now what is the value at x = 1.37?

Now they see that the equation is quicker and more accurate than the graph — even when inside the graphed region.

Or draw two lines that do not meet at integer values. Where do they meet, exactly? Hence that simultaneous equations are better in some situations than graphs.

But again, we can draw y = log x crossing y = x2 quicker on our graphics calculator than we can solve it.

(Of course y = x2 doesn’t cross y = log x, but they only know that if they graph it!)

BTW: Essential reading from Bridget Dunbar also: Effective v.Efficient.

A Response To Critics

Let me wrap up this summer’s series by offering some time at the microphone to two groups of critics.

You can’t be in the business of creating headaches and offering the aspirin.

That’s a conflict of interest and a moral hazard, claims Maya Quinn, one of the most interesting commenters to stop by my blog this summer. You can choose one or the other but choosing both seems a bit like a fireman starting fires just to give the fire department something to do.

But “creating headaches” was perhaps always a misnomer because the headaches exist whether or not we create them. New mathematical techniques were developed to resolve the limitations of old ones. Putting students in the way of those limitations, even briefly, results in those headaches. The teacher’s job isn’t to create the headaches, exactly, but to make sure students don’t miss them.

To briefly review, those headaches serve two purposes.

One, they satisfy cognitive psychologist Daniel Willingham’s observation that interesting lessons are often organized around conflict, specifically conflicts that are central to the discipline itself. (Harel identified those conflicts as needs for certainty, causality, computation, communication, and connection.)

Two, by tying our lessons to those five headaches we create several strong schemas for new learning. For example, many skills of secondary math were developed for the sake of efficiency in computation and communication. That is a theme that can be emphasized and strengthened by repeatedly putting students in a position to experience inefficiency, however briefly. If we instead begin every day by simply stating the new skill we intend to teach students, we will create lots and lots of weak schemas.

So creating these headaches is both useful for motivation and useful for learning.

Which brings me to my other critics.

This One Weird Trick To Motivate All Of Your Students That THEY Don’t Want You To Know About

There is a particular crowd on the internet who think the problem of motivation is overblown and my solutions are incorrect.

Some of them would like to dismiss concerns of motivation altogether. They are visibly and oddly celebratory when PISA revealed that students in many high-performing countries don’t look forward to their math lessons. They hypothesize that learning and motivation trade against each other, that we can choose one or the other but not both. Others even suggest that motivation accelerates inequity. They argue that we shouldn’t motivate students because their professors in college won’t be motivating.

I don’t doubt their sincerity. I believe they sincerely see motivation as a slippery slope to confusing group projects in which students spend too much time learning too much about birdhouses and not enough about the math behind the birdhouses. I share those concerns. Motivation, interest, and curiosity may assist learning but they don’t cause it. In the name of motivation, we have seen some of the worst innovations in education. (Though also some of the best.)

But there are also those who do care about motivation. They just think my solutions are overcomplicated and wrong. They have a competing theory that I don’t understand at all: just get students good at math. It’s that easy, they say, and anybody who tells you it’s any harder is selling something.

Success in a skill is self-motivating.”

Many forget that there’s intrinsic motivation to simply perform well in a subject.”

And, yeah, I’m sorry, friends, but I do have a hard time accepting such a simple premise. And I’m not alone. 62% of our nation’s Algebra teachers told the National Mathematics Advisory Panel that their biggest problem was “working with unmotivated students.”

I see two possibilities here. Either the majority of the nation’s Algebra teachers have never considered the option of simply speaking clearly about mathematics and assigning spiraled practice sets, or they’ve tried that pedagogy (perhaps even twice!) and they and their students have found it wanting.

Tell me that first possibility isn’t as crazy as it sounds to me. Tell me there’s another possibility I’m missing. If you can’t, I think we’re dealing with a failure of empathy.

I mean imagine it.

Imagine that an alien culture scrambles your brain and abducts you. You wake from your stupor and you’re sitting in a room where the aliens introduce you to their cryptic alphabet and symbology. They tell you the names they have for those symbols and show you lots of different ways to manipulate those symbols and how several symbols can be written more compactly as a single symbol. They ask you questions about all of this and you’re lousy at their manipulations at first but they give you feedback and you eventually understand those symbols and their basic manipulations. You’re competent!

I agree that in this situation competence is preferable to incompetence but how is competence preferable to not being abducted in the first place?

If that exercise in empathy strikes you as nonsensical or irrelevant then I don’t think you’ve spent enough time with students who have failed math repeatedly and are still required to take it. If you have put in that time and still disagree, then at least we’ve identified the bedrock of our disagreement.

But just imagine how well these competing theories of motivation would hold up if math were an elective. Imagine what would happen if every student everywhere could suddenly opt out of their math education. If your theory of motivation suddenly starts to shrink and pale in your imagination, then you were never really thinking about motivation at all. You were thinking about coercion.


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