Beyond "Justify"

From Discovering Geometry

Explain, Why?, Justify and Prove

Grab whatever geometry textbook happens to be nearby and scan the reasoning-and-proving exercises. (You can generally find two them at the end of the section after all the practice problems...) Take careful note of the language that's used in these questions. What exactly is the kid being asked to do when they're asked to defend their answers?

There's a variety of language that can be used for these exercises. Kids are asked to elaborate on their thinking using several different -- but apparently interchangeable -- prompts.  

In the exercises above we get a few common directives: "Explain." "Why?" "Justify your answer." If we poke around our nearby geometry text we'll pick up a few other phrases, like "How do you know?" and  "Explain your reasoning." 

Do all of these prompts sound the same to kids? Should they all? Do we want kids to think of an explanation as being roughly identical to a justification? Is answering "why?" the same thing as offering a justification? And how does all of this relate to that other core prompt, "prove"?

Reasoning Problem Makeover

A few weeks ago I wrote about something called the Hexagon of Proof, and that post was half-joke and half-serious. The half-joke part was the idea of making a catchy image that played off Bloom's Taxonomy. The half-serious part was the idea that we can teach proof more effectively if our classes have a healthy and varied diet of proof-like activities. There are natural bridges to be built between everyday discourse and the unnatural act of mathematical proof. 

We can do better than just asking kids to "justify" their thinking. There are lots of ways to provoke kids into expressing their reasoning, and there are some prompts that ought to see wider use. As an exercise, I rewrote one of the above problems in five different ways. As you read each problem, think about the different sort of student responses that each bit of prompting language might yield.

Exhibit A: Debating

Exhibit B: Disagreeing

Exhibit C: Convincing

Exhibit D: Explaining

Exhibit E: Teaching

Exhibit F: Proving

Bonus: Justifying

Does justifying have a different meaning to students than proving? I have no idea. Thoughts?

Exercises for the Reader

1. Which is your favored version of  the problem? Would you use different versions in different situations? Explain your answer.
2. Are there other versions of this problem that you can imagine? Construct an example.
3. "The language used in presenting reasoning problem significantly impacts the sorts of responses that a teacher can expect to receive." Do you agree with this claim? Disagree? Justify your response.
4. Challenge Problem! Samuel Otten (and colleagues) wrote a paper called "Reasoning-And-Proving in Geometry Textbooks." In it they analyze the types of reasoning-and-proving activities assigned in popular geometry texts. How does their analysis compare to the one given in this post? How would Otten respond to this post?

Questioning Wiggins' Definition of Feedback

Is this feedback or evaluation? From MathMistakes.org

Feedback vs. Evaluation

Imagine a baseball coach. The second basemen just struck out...again. After his at bat, the coach heads over to the player and puts her arm around his shoulder:

"Tommy, you haven't been hitting as well as you could've lately, amiright?"

Is this feedback? In a few places, Grant Wiggins argues that this is not feedback. Instead, this coach is offering an evaluation of the player's hitting. (See this, that, and the other thing.)

So what would feedback to the struggling player look like? Feedback would be judgement-free and informative. It would call attention to facts of the matter that the hitter herself likely didn't notice. It would look like this:

"Each time you swung and missed, you raised your head as you swung so you didn't really have your eye on the ball. On the one you hit hard, you kept your head down and saw the ball."

Wiggins thinks that this is judgement-free. And there certainly is no explicit language indicating evaluation of the hitter. The coach could've lead with "That wasn't a great at-bat," but she didn't.  It's tempting to say that the coach is only providing the hitter with plain, value-neutral information. So where's the evaluation in the coach's feedback?

I think the evaluation is there, lurking in the background. After all, imagine what the batter is thinking when the coach comes up. He just struck out, and he settles back on the bench. He knows his job is to get on base, and he knows that swinging and missing is not what you're supposed to do in the game of baseball. He knows how his teammates are hitting, he knows what sort of team they're playing. He knows whether he's the only one who's striking out and what people are expecting of him. And then the coach provides him with all this information on why he swung and missed. Here's probably how he's hearing this:

"Coach is saying my technique isn't good."

And is he wrong? Everybody involved knows that you're supposed to hit the ball in baseball. That's the point of the game! The kid and the coach both know what's expected of the hitter, and that shared context colors the feedback accordingly.

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Brief Linguistic Interlude

The point here is that you can say things without actually saying them. In philosophy of language or linguistics, this is called the pragmatics of speech. The context in which you say something contributes to its meaning. As a professor of mine put it, "I love cheese" can mean vastly different things depending on the conversational context.

• A: "Don't you love cheese?" B: "I love cheese." 
• A: "Do you love me?" B: "I love cheese."

In the first case, "I love cheese" means roughly that the respondent enjoys cheese. In the second case, B is saying something else: "No, I don't love you. I do love cheese."

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Can Evaluation Be Avoided?

Let's bring this back to math class. What your students hear when you give them information isn't just what you say. There's often a whole world of unspoken ideas communicated to a kid that depends on the classroom context.

Say that you give a kid the sort of feedback that Wiggins suggests is judgement-free:

On the first three problems you distributed the exponent to the terms inside the parentheses and ended up with a non-equivalent expression. On the fourth question you FOILed and ended up with an expression equivalent to the original.

If the task was to discover equivalent expressions, the kid is likely hearing something like an evaluation:

I screwed up the first three because I used a bad math-move. I should have FOILed for all of the problems. 

What Wiggins' calls "feedback" still contains plenty of evaluation, it's just happening silently as a result of the larger context. I think that it's much, much harder to give non-evaluative feedback than Wiggins suggests it is. (See his response here.)

What are the implications of all of this for giving responses to kids? I can imagine a few reactions:

• Evaluation is inevitable: "There's no way to give kids judgement-free feedback. This shows that really there's nothing wrong with evaluating kids, and actually it's often helpful, as long as its done in a respectful way."
• Purge evaluation from feedback: "Given how hard it is to give kids judgement-free feedback, we need to work especially hard to remove judgement from the feedback that we give kids."
• That's not what feedback is: "Feedback can't be conceived as judgement-free information, since context always infuses information with judgement. We need a new definition of feedback."
• Just do your best: "Yes, context can insert judgement into judgement-free language. You can't avoid it, but you can try to minimize how prominent that evaluation is in your feedback."

Part of this discussion needs to include research on the way evaluation can ruin feedback, and it'll also drive us to understand the how it is that feedback is supposed to help kids learn stuff. Wiggins sees feedback as helping learning by giving students information that they're missing. Maybe there are other purposes for feedback, though. 

Tell me, readers: what did I get wrong here? Did I get anything right?

Two Textbooks' Handshake Problems

Like so many classic math problems, the handshake problem is easy to state: "If everybody here has to shake everyone else's hand, how many handshakes do we need?" Because it's such a well-known problem, the way that a teacher or a textbook uses it can offer an interesting window into their perspective. The problem is common, so how you choose to use it speaks a great deal.

I'm going to share how my two favorite geometry texts -- CME Geometry and Discovering Geometry -- use the handshake problem. The question is: what do their approaches to the problem say about their takes on geometry?

Discovering Geometry's Handshake Problem:

CME Geometry's Handshake Problem:

Some Observations:

• Discovering Geometry breaks down the modeling process into eight steps, starting with completing a handshake table. CME doesn't offer any of those steps or supports.
• Discovering Geometry asks the student to model the handshake process with polygons and segments. CME presents the handshake problem and the diagonals problem separately, and then asks students to consider connections between the two questions.
• This isn't in the pictures, but the handshake problem is the first problem that appears in CME Geometry, while the handshake problem appears as subsection 2.4 in the Discovering text, titled "Mathematical Modeling."

I'm very curious to know what you folks out there think about these two presentations. My suspicion is that these two presentations speak to two different assumptions about how students learn to reason in mathematics. One assumption is that students need lots of informal experience that, with feedback and opportunity, will slowly shape their reasoning habits. The other assumption is that students need an explicit model of the reasoning process, either in addition to or at the beginning of their opportunities to reason on their own.

Your thoughts and criticisms are always appreciated, but I'll amplify the invitation for this post. I also know that authors of both of these texts hang out on the internet, and they could likely school us all a bit on their work.

Comments from the Bullpen:

Fawn and mrdardy want to see Discovering Geometry take away all that support for the kids, instead put it into a teacher's guide. They'd like to see CME get rid of the explicit connection between diagonals and handshakes, preferring texts that offer radically little upfront support for students.

l hodge finds CME's explicit call for a connection between diagonals and handshakes patronizing. Is there another way to push kids towards making that connection?

fivetwelvethirteen connects this with the CCSS standards of mathematical practice. How do you help kids get better at big skills like solving hard problems or mathematical modeling? My take: let's look to other fields, like literacy, because "problem solving" is as big a skill as "reading" or "writing."

A Tool For Exploring Transformation Rules

Using the fantastic Desmos graph and their API, I made a little environment to explore transformation rules. (Click here!) The app let's you type in a rule that determines the transformation and then visualizes the image for you.

Part of the fun of playing with this -- for me, at least -- is exploring a wider set of transformation rules than I'm used to visualizing. 

I love the idea of putting this in front of my geometry students next year before we study transformations. I think they'll have a lot of fun playing around with this. The other thing that I like about this app is that it provides a ton of Daily Desmos-style transformation problems.

Can you find a transformation rule that produces this image?

Can you find a transformation rule that produces this one? 

If you can find a rule for the rotation in that last picture, then give me ten minutes and I can explain what complex numbers are. Which is actually why I started to make this thing in the first place.

Enjoy! Let me know if you find bugs or think of improvements.

A huge thank you to the tremendous and brilliant Chris Lusto for helping me out throughout this project, and thank you to equally awesome Andrew Knauft for inspiring this project with his own work and for offering some crucial feedback and aid. These are good, generous people.

Taking Policy Personally

Lower East Side in the 90s. Source: G.Alessandrini

Schools have changed since Jose Vilson was a kid, and he’s not thrilled about it. More to the point, he argues that that these changes have taken away crucial social supports for Black and Hispanic students. In the opening chapters of This Is Not A Test we get the personal stories behind his concern and their relevance for education reform. All this gives us a chance to consider the relationship between incredibly personal memoir and national policy.

Vilson grew up on Manhattan’s Lower East Side in the 90s, a place where “no one wanted to live,” where “rats came as naturally as breathing and the phases of the moon.” This was a brutally unforgiving environment for a poor, Black, Hispanic kid to grow up in. You’d hope that school would help out a kid like Jose, but hey you’d hope for a lot of things. Too often, Vilson’s teachers taught him that school was a place where he was not welcome. There was (the super-heroically named) Mr. Missile, who punished young Vilson for the manner of his speech. See also: the English teacher who uttered the words “Well you don’t know anything, so I’ll move on.” See also: the guidance counselor who dismissed his grief, somehow ill-equipped to help a hurt student.

Fortunately, Vilson had help: from his mother, from his public elementary school, from Nativity Mission School, from Father Jack, Father C., Ms. Kittany and Mr. Wingate. Vilson’s schooling was a mixed bag, but he takes care to detail the kindnesses of the teachers and institutions that made him feel respectable and worthy. Ms. Kittany, for instance, dragged Vilson into the choir of his very white high school and did what the best teachers do: she patiently taught him.

During my last Mass before graduation, she just looked at me and signaled to the mic. I don’t remember much besides her crying as the bass took over my voice box…I never had her as one of my core subject teachers, but what she taught me about the power of my voice was one of the most important lessons I took away from my experience at Xavier High School.

These stories – the kindnesses and cruelties – have policy implications for Vilson. He makes the case that accountability and standards reform has reduced access to the kindnesses, offering little protection against systemic cruelty directed at Black and Hispanic children. He further argues that Black and Hispanic teachers are uniquely qualified to offer the sort of humanizing relationships that were so important in Vilson’s own childhood. He calls for the dismantling of the current high-stakes testing regime, a reinjection of the “warm-and-fuzzies” into schooling and the further recruitment of Black and Hispanic men into the teaching profession.

All these arguments hang on memoir. Are there limitations on how far the personal can take us in the realm of policy? In the early pages of Not A Test we see these limitations considered, and ultimately rejected. “I’ve been told that in order for my writing to be universal, it must turn away from things like race or nationality or the conditions of my upbringing,” he writes. “I have found that bringing my experiences into my teaching makes the lessons more profound.”

To be sure, the personal is crucial. And Vilson’s book is complex in its purposes. Fundamentally, this is a memoir, and Vilson is writing to tell stories the likes of which many in education have never heard. But he also wants to make a case about the direction of schooling in general, and here I felt myself wondering about how easily his stories generalized. To be clear, it wasn’t skepticism that I was experiencing. I found myself easily persuaded by Vilson’s stories and arguments. It’s precisely because I was so easily persuaded that I worry.

I’m left with a question: what is the proper relationship between memoir and policy?

Take, for example, Vilson’s case for the importance for increasing the number of Black and Latino male teachers. This is a position that he supports with stories, drawn both from his schooling and from his teaching. “The Black/Latino male students respond more readily to me,” he writes. “The girls in my class are more willing to share their experiences with me, and often look to me as a role model or father figure.”

Can such personal narratives serve as the bedrock of national decisions? Some will argue that they can’t. Who knows what factors are responsible for the special connection Vilson has with his students? Perhaps Vilson has such powerful relationships not because of his ancestry and gender, but because he is a remarkable and unusually empathetic teacher. There’s no way to know without expanding the survey. National decisions need to be considered from a national perspective, and this inevitably involves drifting up and away from the personal and taking in the big picture. Such a reader could glean questions for further study from This Is Not A Test, but never policy answers.

Standing on the other side are the defenders of narrative. These readers worry that the truth gets lost in national studies and large-scale surveys. Would a wider set of data discover that Black male teachers have a special role to play in the education of teens? Who cares! Vilson’s stories show and explain how Black male teachers can help, and if studies fail to capture that truth then all the worse for their rigor. Vilson’s stories show us that when a Black male student sees himself reflected in his teacher, that empowerment can carry the day. How do you propose to conduct a statistically relevant survey measuring empowerment?

These first chapters represent one end of a continuum – the intensely personal and specific – and most policy discussions represent the other. Should we read Vilson as arguing for shifting policy-talk away from abstractions towards the personal? What would such a debate look like? Are there also dangers in leaning heavily on the personal? Finally, what does the space between the abstracted and the personal in policy discussions look like?

I’m looking forward to hearing thoughts, disagreements, challenges and reflections in the comments.

This is the first entry in the Global Math Department's reading group on Jose Vilson's new book. Sharon Vestal will respond to this post with a post of her own, as will the other members of the reading group. We'll collect all the responses and put them on our reading group blog, so keep your eyes peeled for that.

James Baldwin's "A Talk To Teachers"

I began by saying that one of the paradoxes of education was that precisely at the point when you begin to develop a conscience, you must find yourself at war with your society.  It is your responsibility to change society if you think of yourself as an educated person.  And on the basis of the evidence – the moral and political evidence – one is compelled to say that this is a backward society. 

Now if I were a teacher in this school, or any Negro school, and I was dealing with Negro children, who were in my care only a few hours of every day and would then return to their homes and to the streets, children who have an apprehension of their future which with every hour grows grimmer and darker, I would try to teach them -  I would try to make them know – that those streets, those houses, those dangers, those agonies by which they are surrounded, are criminal.  I would try to make each child know that these things are the result of a criminal conspiracy to destroy him.  I would teach him that if he intends to get to be a man, he must at once decide that his is stronger than this conspiracy and they he must never make his peace with it.

This is James Baldwin talking to teachers in "A Talk To Teachers" from 1963. Pretty soon we're going to start reading and thinking about Jose Vilson's This Is Not A Test around this blog. If you're waiting around for your copy of Vilson's book, you could do worse than start with Baldwin.

Drop a comment if you've got thoughts!

John Adams on the Mathtwitterblogosphere

At the time of our revolution, there were those who advocated for a perfect democracy of the people. There would be one legislative body, there would be perfect and equal ("equal") representation of the nation. There would be no need for an executive branch or any sort of further central organization.

"To Adams this was patent nonsense. A simple, perfect democracy had never yet existed. The whole people were incapable of deciding much of anything, even on the small scale of a village. He had had enough experience with town meets at home to know that in order for anything to be done certain powers and responsibilities had to be delegated to a moderator, a town clerk, a constable, and, at times to special committees." (John Adams, David McCullough)

It seems to me that this tension -- between freedom, equality and centralization -- arises whenever a group of people attempts to coordinate at all. Coordination gets more and more difficult as the number of people involved in the community grows, and things can start to go bad almost unintentionally. Think unregulated markets, or the way crowds can be dominated by majority impulses.

Of course, Adams believed that many centralizing institutions were bad. He just wanted to build better, more perfect institutions.

We are living at a time when it feels natural to be skeptical of the structures of society. There's a longstanding decline in Americans' confidence and participation in our institutions. This cuts across the board: confidence in public schools, congress, organized religions, banks, professional organizations, community organizations, the courts, television news, unions, the presidency has never been lower.

I'm thinking of this because of Christopher Danielson and Anne Schwartz. Danielson recently published a piece titled "Not All White People" where he urges his white readers not to fight back when the conversation turns to race. He notices that when white teachers hear about racism they tend to worry "Am I racist?" instead of "Was what I did racist?". He'd like us realize that these are quite different, and that the relevant question is almost always the latter. He'd like to make a change in the way teachers online participate in and listen to discussions about race.

For her part, Anne Schwartz offered a critique of the selection of keynote speakers for an upcoming conference:

"If you know me you know that I love Dan Meyer, Eli Luberoff and Steve Leinwand. They are three really fantastic white men. But Twitter Math Camp is an outstanding conference run by amazing amazing women teachers, attended by a majority women and almost completely teachers I just wish the keynotes represented that fact. Max Ray said it last year and I echo it this year. More women, more diversity."

My argument is that both Danielson and Schwartz's critiques speak to the need for online communities of teachers to build strong institutions. Their critiques speak to systemic problems, and they call for improving our systems and organizations.

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There's a certain line of thinking that I hear in my discussions with friends and colleagues who are active on the teacher-internet. "What's great and different about twitter and blogs is that it's completely open and decentralized." Since the internet offers a level playing field good ideas end up flowering and being shared widely, allowing progress and creativity to burst forth and spread. Institutions are crusty and boring. Institutions are responsible for textbooks, lousy PD, edu-jargon, lazy teachers. We individuals? We're responsible for amazing blog posts, loose and informal PD and the most committed teachers on the planet.

If what you love about the internet is openness, you might be troubled that it favors white men. (Remember this?)

Like it or not, when you get a bunch of people together you end up with institutions and normative culture. Twitter has a culture. Blogs have a hierarchy. Twitter Math Camp is a gatekeeper. The question isn't whether we should have central institutions or not. The choice we have is whether to build strong institutions or weak institutions. Strong institutions are able to coordinate action, to move a community and make changes. Weak institutions are dominated by uncontrollable forces, emergent behavior, mere shouting and some kind of anarchy.

Twitter Math Camp is capable of turning this around. There are people in place who have responsibilities and accountability. These people can be asked to make diversity a priority. We will know if we succeed or if we don't. There are things that can be done.

But if you want to change the way business is done on twitter? Good luck with that. Enjoy shouting in the tornado of tweets and slowly watching your followers drop and follow you accordingly. Of course we should all speak up on twitter -- this isn't an argument against that. Instead it's an argument for the limitations of speaking up on a social media platform where there's precious little structure to the community.

Leslie Knope gets it:



To close: this is also an argument for the importance of the Global Math Department. Building strong institutions gives our community the chance to be more intentional, to know ourselves a little bit better. We can make our biases visible, and thereby have the chance to correct them. I don't just mean racial or gender biases here, by the way. How open are we to outsiders? Do we have tendencies to prefer certain learning experiences over others? Do we prefer certain forms of PD over others? I just finished reading a book about the ways that ideas ripple through the profession. Building strong, healthy institutions is how we gain control over those forces instead of merely being subject to them.

What I'm Working On This Summer

Yeah yeah manhattanhenge happens twice a year

Summer has arrived for me! Here's what I think I'll be thinking about over the next few months:

Complex Numbers - I've written a bunch about complex numbers over the past year or two, but I'd like to turn those ideas into something usable for other teachers. At TMC14 this summer I'll be running a session I'm calling "New Ideas For Teaching Complex Numbers," and I'll be working hard to have something that participants can take with them. Will this project take me into the Desmos API? I hope so. I also have Needham's "Visual Complex Analysis" on the shelf and I'd love to spend some time just reading and solving problems this summer.

Proof in Geometry - I'm slated to lead the morning sessions on geometry at TMC14, though there are parts of that plan that are still up in the air. Teaching geometry this past year for the third time, I became really interested in thinking about the role of proof in my classes. Proof is going to be the focus of our morning sessions in Jenks, and I'm looking forward to preparing for it. I'm going to be focusing mathmistakes.org on proof mistakes for a while, and I'm excited to dive into some research relating to proof in geometry. I'd love to take a closer look at how some of my favorite textbooks handle reasoning and proof, and I've been meaning to read Fostering Geometric Thinking much closer than I have yet.

"This Is Not A Test" - Through the Global Math Department, we've organized a book club for reading Jose Vilson's new book. There will be blogging being done by myself and others in the club as we read, and if you have the book you should dive in to our conversations.

On the backburner are a few other themes that I probably won't have time to attack with any sort of focus:

Exponents - I really do find exponents fascinating. You might hear a bit more from me on this if I find time to read about the history of exponent notation. Maybe I'll can pick this up in August after TMC?

Feedback - I think I've hit the edge of my knowledge on feedback, grading and assessment. I'm likely to pick this up again when school comes back around, since it's such a pressing teaching concern of mine.

History of Education - Over the past year I went on a bit of a history bender. I certainly feel like I have a better grasp on the world of education than I did a year ago, but it would probably do me well to sit on this all for a little bit. Summer's the time to dive back into some fiction.

The temptation for me is to just sprawl and sprawl and sprawl in my work, but I always end up happier when I pare down the list of projects to about three. Honestly I'm a bit nervous because there's at least one non-teaching thing I'm going to be also working on this summer and four seems like one too many. Shrug, we'll see how this goes.

You'll be able to find me in Uptown, NYC most of this summer, hopefully staying cool. If I'm not here, I'm likely in Portsmouth, NH or Jenks, OK or Philadelphia, PA.

Under The Influence Of A Theory I've Never Known

By all accounts, Merlin Wittrock was a highly influential educational researcher. He was highly cited. His generative theory of learning was the sort of thing that his colleagues called revolutionary. When he died people wrote articles like "M.C. Wittrock: A Giant of Educational Psychology." And on top of all that, Professor Wittrock had a bad-ass wizard name. Suffice to say, he was a big deal in educational psychology.

He was best known for his generative theory of learning. In short, Wittrock's theory states that deep learning has to do with the strength of the relationships between pieces of information instead of their mere storage. But as you can see from the fairly baroque diagram at the top of this post, that's a significant over-simplification.

If you know nothing about education, then you might think that the acclaim for Wittrock's theory was enough for his ideas to spread through teaching. You might imagine all three-and-a-half million of us teachers hanging out by the cafeteria and just shooting the breeze about Wittrock's theory and how it's changing how we teach fractions or something. I'd be all like "Hey guys so 'Problem-Solving Transfer' is the obvious choice, but I'd have to say my favorite Wittrock piece is 'The Cognitive Movement in Instruction'" and then we'd all argue boisterously but with obvious mutual professional admiration.

Anyway, that's not what happened:

"Generative learning never took root beyond the confines of its academic subfield, and certainly not in K-12 classrooms." (Schneider, 164)

In Chapter Five of From the Ivory Tower to the Schoolhouse, Schneider is interested in contrasting research that made it with research that didn't. You think that Gardner's Multiple Intelligences was destined for greatness? Check that against Steinberg's similar Triarchic Theory of Intelligence. So Bloom's Taxonomy was a winner, but what about Bloom's Second Taxonomy? Why didn't that stick?

Unlike research ideas that successfully made their way into the awareness of educators, Wittrock's theory umm fell like a rock. And it wasn't like he wasn't trying! He was leading professional development sessions for teachers. He tried to write articles distilling his work for practitioners. He spoke publicly and often.

What went wrong? Schneider lists the obstacles that Wittrock's theory faced:

• Wittrock was at UCLA and didn't have the prestige that a fancier professorship would've granted. 
• He did a lousy job turning his theory into bite-sized discussable, actionable principles for teachers. He tried, but he kept things subtle, abstract and detailed when they needed to be tweetable, down-to-earth and big-picture. Wittrock needed to go around saying something like "Project-based learning promotes lifelong learning" about his theory.
• He was too faithful to his own theory. In order for the theory to take-off, there would need to be easy applications for teachers to tackle. But third-party professional development salesfolk couldn't spin anything out of generative teaching.

Schneider sees Wittrock's theory as a failed bit of research. It penetrated educational psychology, but not teaching. And, indeed, I'd never heard of him or his theory until reading Schneider's book.

But here's what I can't shake: I had heard of ideas like his. Further, they challenged and provoked me. When I came into teaching I definitely saw understanding as a matter of information storage. I thought that I could teach skills by some sort of repetition, making sure that the explanations reached ever increasing circles of students. Somewhere along the line I ended up thinking that what matters more is the network of learning, the strength of the connections and the way new information can be invited into that intricate web. 

I wish that I knew exactly where my exposure to these ideas came from. It must have had to do with reading "How People Learn." It also probably had to do with my exposure to the CME Project and thinking through the ways that informal problem-solving can be helpful preparation for more formal learning. Certainly Dan Meyer played a role.

Here's what I'm thinking: if Merlin Wittrock influenced educational psychology profoundly, then he must have influenced me profoundly. All of my influences are themselves influenced heavily by educational psychology, after all. 

This doesn't contradict Schneider's point, but it does challenge his emphasis. Throughout his book, Schneider has been judging research against its widespread recognition and popularity. The theory goes like this: in order to become wildly popular a piece of research can't be too challenging, difficult or complex. Of course, this means that its fidelity will be up for grabs, but the exposure will inevitably send some educators back to the research to engage with the ideas more seriously.

But how many educators, in Schneider's vision, will end up engaging constructively with the research? How many will end up adopting the research in speech only, without really changing anything that they do?

Those are questions that are incredibly difficult to answer. But here are some more tough questions: Is there a path from Merlin Wittrock's writing to my understanding of learning? How many teachers were guided by Wittrock without knowing his name? How do we know that it's fewer than those that actually did something valuable with Bloom's Taxonomy? 

Still: it's easy to imagine a world where Wittrock was a more successful advocate for his ideas, and it's hard to imagine that world as a worse place than ours. Raymond: after five chapters and lots of hemming and hawing, I'm more or less convinced. What say you?

Raymond Johnson and I are reading Jack Schneider's new book, "From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education." This was Chapter Five, and Raymond will soon respond. Previously:

Chapter 1: Bloom's Taxonomy (Michael's post, Raymond's reply)
Chapter 2: Multiple Intelligences (Raymond's post, Michael's reply)
Chapter 3: The Project Method (Michael's post, Raymond's reply)

Chapter 4: Direct Instruction (Raymond's post, Michael's reply)

Direct Instruction Works, Sort Of

Direct Instruction works, it seems. See Raymond, recapping the latest, greatest chapter of From the Ivory Tower to the Schoolhouse:

Unlike Bloom's Taxonomy, multiple intelligences, and the project method, Englemann's Direct Instruction works (with the research to show it) when teachers are philosophically compatible with the method and they implement it with fidelity. 

Direct Instruction clashes directly with a lot of the teaching principles that I hold dear. It's a scripted curriculum. Everything that the teacher does follows the script, and that's a direct attack on the idea that classroom ought to involve much intellectual work. Englemann's program doles out rewards and punishments to reinforce behaviors in students, and not in the "aren't grades really a reward?" sense but in the real sense of using carrots and sticks to drive learning.

It shouldn't work. It works.

For a nice, balanced take on the implications of Direct Instruction for policy, for teaching, and for progressive teaching notions, I'll send you back to Jack Schneider for "Direct Instruction works. And I’d never send my own child to a school that uses it."

So maybe the effectiveness of Direct Instruction draws blood from progressive teaching principles and maybe it's compatible in some way or maybe it bla bla bla but Raymond cuts to the heart of things when he asks: what happens when there truly is a conflict? What happens when there is a major clash between the teaching principles that I believe most strongly and what research shows to be effective?

But where do we draw the line between philosophical compatibility and the need for teachers to be open minded? To be learners? As professionals, when should our philosophies give way to what we can gain from research, regardless of compatibility?

It seems to me that a major goal of Schneider's book is to teach researchers not to ask that question. The beliefs and values of teachers can't easily be changed, and to make an impact research needs to take this for granted. Should teachers be willing to change their core values in the face of compelling evidence? Absolutely. Will we? That doesn't seem likely.

Despite what the evidence may show, Direct Instruction can, at best, only sort of work because it's only sort of compatible with the profession's philosophical commitments. A truly effective idea that makes all involved want to throw up isn't "truly effective" in any interesting sense. (On the other hand, Direct Instruction only makes some people throw up, and it otherwise has a positive effect so that counts as progress.)

At the end of all this, we end up with a host of pragmatic questions for researchers to consider. Does my research idea attack core values of teachers? Do I think that I can change the core values of teaching? Do I think I can have a limited impact, even without becoming widely accepted? What's the path to impact? 

I feel compelled to close by sharing my excitement at better understanding an idea that I had only previously mocked. Direct Instruction comes out looking much better than Gardner's multiple intelligences in this story. DI has a strong research basis and can reasonably claim to have made a positive impact on schooling. The case for MI is much sketchier. If there's a moral here, it goes something like this: familiar ideas need the most scrutiny, but uncomfortable ideas need a fair shot. Scrutiny too, of course, but uncomfortable ideas are going to get that anyway.

Raymond Johnson and I are reading Jack Schneider's new book, "From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education." This was Chapter Four, in response to Raymond's post. Previously:

Chapter 1: Bloom's Taxonomy (Michael's post, Raymond's reply)
Chapter 2: Multiple Intelligences (Raymond's post, Michael's reply)
Chapter 3: The Project Method (Michael's post, Raymond's reply)

It's The Celebrities That We Need To Doubt

"I SAID NO SMILING DR. KILPATRICK. LET'S TRY IT AGAIN."

William Kilpatrick wanted to be famous, so he made himself famous. And he didn't just want to be academic famous, he wanted to be famous, period.

Already in his mid-forties, he sought--as he noted in his diary--to achieve "power and influnece" and to be remembered as an "original thinker," not merely an "acceptable teacher." (Schneider, 81)

Fortunately, Kilpatrick had a plan. As detailed by Schneider in the third chapter of From the Ivory Tower to the Schoolhouse, it involved selling himself to the country's teachers.

He set his sights not on gaining status among fellow academics, but rather on building his reputations among several hundred thousand K-12 teachers. Kilpatrick's strategy for raising his profile among teachers was, in his words, "to think of a small + popular book" that would "appeal constructively + so sell better." (81)

And that's basically what he did, though it turned out to be a wildly popular article ("The Project Method"), not a book, and, rather than selling it he ended up giving it away.

Kilpatrick tied his reputation to the notion that school should be structured around a projects (definition: "wholehearted purposeful act"). Teaching needs to be first and foremost concerned with the child's interests. Kids need to be engaged in their work in order to learn anything. School shouldn't be a preparation for life, it should be life itself.

Schneider maps how the idea caught on with teachers. Kilpatrick's appointment at Teachers College earned him the prestige that got his foot in the door. The world was ready for this idea -- it's not like teachers hadn't done projects before-- but Kilpatrick turned up the volume and expanded the meaning of "project" for wider application. Kilpatrick's idea was then transformed by teachers so that it could function as an "add-on" to what they were already doing, making the idea even more appealing. (In short, prestige + transportability + ease of application = profit.)

The California Mission Project

These days, William Kilpatrick (1871-1965) is dead. He's also very not famous. But his idea? You might say it's thriving, but that would very much depend on what you mean by "thriving." Kilpatrick wanted to restructure school around projects, making projects the central concept of schooling. Instead, projects are just part of the teacher toolkit, another activity that we might do with kids depending on the situation. The idea that projects should serve as the basis of schooling is seeing a recent revival through "project-based learning" but you'd struggle to draw a direct line of influence from today's advocates to Kilpatrick's ideas.

Schneider urges us to see today's widespread use of projects as a heritage of Kilpatrick's career. I'm capable of skepticism on this point. After all, part of Kilpatrick's success in spreading the notion of projects was that use of projects in education was already on the rise. The home project was an important part of rural vocational education. There was a journal devoted to the wider application of projects prior to Kilpatrick's article. I don't doubt that, in the short term, Kilpatrick made the idea more popular, but when the dust settled was the world much different than it would have been without the Great Man himself?

There's no way to know. So here's what we do know: Kilpatrick wanted to be famous and he made himself famous. His tool was the project, and it proved highly effective. His ideas were widely discussed, and eventually transformed and widely used by teachers. Some teachers used these transformed ideas well, but most didn't.

There's a lesson in this story about ideas in education and what it takes for them to spread. There's a lesson here about the path that a wildly popular idea will tend to take, and the sort of mutations it will accumulate along the way. But there's also a lesson here about fame and influence: Famous people become famous because they want to be famous, and we need to judge their ideas with the skepticism that sort of person deserves.

Raymond Johnson and I are reading Jack Schneider's new book, "From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education." Keep your eyes out for Raymond's response to this post. Previously: Chapter One and Chapter Two.