I think these are both germane to how most educated non-professionals view math education as well.

In many ways I think that’s what we as teachers are trying to fix. We want students to be able to learn things without us saying. We are trying to create lifelong individual learners.

Profound and so true! Ashamedly, it took me several years of teaching before I realized any of them.

Wait, but in your original post you said that these were “The Two Lies of Teaching”. So is he saying this is true, or untrue? What are you saying?

(This may sound like sarcastic joke confusion, but it is real.)

Love it! I think that I will need to take those two lies into my next department meeting!

I think cbrown is right on. I was thinking about this as I read through the Khan Academy posts today. I teach some motivated, intelligent students in my AP Calculus course, and they take advantage of the recorded lectures we do. Of course, that doesn’t mean they work for everyone. i.e. one-size (or technique) doesn’t-fit all. as wrong as it is to assume a non-motivated student will simply learn by being told (or asked to watch a video) it is similarly wrong to dismis the entire concept if it works for certain students in certain situations (see Cheryl van Tilburg’s comment in the KA discussion.
I think some of us teachers are quick to criticize techniques that we don’t see working for OUR classroom or OUR students, but that doesn’t mean they won’t work for others. It depends on the context – right?
Don’t get me wrong. I think the debate is good, it makes me think, and I believe makes me a better teacher. However, our allegiance to methods we have used successfully doesn’t mean that everything else is terrible.

i think the spirit of the pronouncement that those two principles are lies is that it doesn’t really matter what you *say* in the classroom; rather, it’s the manner in which you say it and how you conduct yrself, how you treat the students and perhaps how you interact w/ knowledge, so to speak. the presumption is that there’s a kind of freudian man behind the curtain who’s inspiring the students, or not, and the man behind the curtain is the undefinable spirit of teacher (not spirit in the spiritual sense, but the person, the ethos of the person, his/her character). it’s that ethos that is essentially what gets transmitted over & above anything you say. the good teacher, according to this approach, essentially teaches students to want to learn, and then the learning begins to take care of itself; the student can then begin or continue carrying on the journey.

Last year I spent all Christmas break making lecture notes that were designed to show my Algebra 1 students why exponent rules worked they way they do (for integer exponents at least).

Low classroom assessment scores aside, the main reason I’ll never do it that way again is that when I would prod students as to why x^a*x^b=x^(a+b), they would say “because you told us so!”

This Khan stuff has been interesting. Silly me for not checking Google Reader for a few days and then finding myself 80 comments behind. Yikes!

Recently in second grade I had a girl who, at seven years old, had already been infected with the notion that math is a wolverine. I have never met someone so terrified of math, so much so that when asked a question like “So how did you know what to do with the pennies?” in small group setting, she would stammer, “4?” or some other hapless, nonsensical answer. The poor thing was paralyzed with fear.

I agree with a commenter on this post (or the previous one…I’m a little muddled now) who said that we live in a culture that wants to know “how” (maybe), but can’t be bothered with “why” or the “when” or any deeper inquisitive move toward understanding. This little girl is a product of this culture.

MBP had some interesting things to say above. Looking to these factors to reach students is key, rather than watering down the content to step-by-step procedures that, to a struggling student, may as well look like a well-stocked aisle of knick-knacks at walmart. I understand the possibilities for Khan as remediation. But for my terrified student? Over my dead body.

> x^a*x^b=x^(a+b)

When I teach my kids this, I don’t give them the formula. I have them expand the exponentiations, and combine them. It takes fractions of a second for some smart kid to wave their hands wildly to tell me they found a shortcut. When I go to check their work, and tell them not to share their observation with anyone else, it spreads through the room like wildfire.

You will note that I never ever told them what I wanted them to learn.

I’m not sure if this fits rule 1, or rule 2.

I feel like one of the best pieces of advice I ever received in teaching education fits with this concept.

Students learn by doing, and not by watching the teacher do anything.

I think it’s powerful to understand that every student, even the really good ones, doesn’t learn by being told something is true. Eventually every kid wants to find out if a statement works or not. Even if that statement is “can you do this?”

I’m thankful I finally have a legit reason to look up the word ‘roiling.’

Questions I now have, resulting from these two posts:

1. Where and why do teachers learn that the two questions in this post are (supposed to be) true? Surely they’re not taught in teacher prep courses, but it seems that many in the profession eventually fall into the trap. Is this a cultural snafu or an unfortunate consequence of the teaching process and some strange wrinkle tied to human nature?

2. When do teachers finally realize the truth of your post? What must take place for them to come to the understanding naturally?

3. How is it that Khan and others are able to successfully pull the wool over over so many eyes? Something’s gotta give.

Corollary 1: If a student can give a rule/answer/observation, it doesn’t mean they have learnt a concept.
Corollary 2: If a student can’t give a rule, it doesn’t mean they have learnt nothing about a concept.

This ties back to assessment – how do we determine what students can do? At what level do they understand a concept or in what contexts can they apply it?

Where and why do teachers learn that the two questions in this post are (supposed to be) true?

I am not claiming that the following are GOOD reasons, but here are the reasons why I spend most of my time in the classroom lecturing:

(1) My administration/parents/students think that lecturing and answering questions is how a teacher helps people learn. When I don’t lecture, the interested parties think that I’m goofing around.

(2) My school tries to cover lots and lots of material over a short (~130 day) year, and I’m held responsible if I don’t cover all the material that’s part of NY’s Regents exams. Lecturing is the fastest way to discharge my responsibilities.

(3) It’s not like what I’m doing isn’t leading to learning. I give skill tests, and my kids do OK on them.

(4) I teach three curricula and lecturing is what requires the least amount of preparation.

(5) It’s not like I walk into class with a script and just say things to people. I do what a lot of teacher-centered teachers do. I walk in, and try to hook them with an interesting question. No, my questions are not often the everyday, wolverine-declawing kind, but I do my best with mathematical context. I’m constantly assessing, both formally and informally. I stop every 10-20 minutes and let kids solve problems using our ideas, and they usually help each other. I cold-call kids. I demand questions. People have questions. I ask questions. We wait and think about them.

(6) Other methods of teaching require skills that I don’t yet have.

There are probably more reasons, but this is a start. As I get better/if I stick around, I’ll lean less on lecturing. But the point is that it’s not that I magically think that saying things is super-learning. It’s just the best that I can pull off, daily.

@MBP
You’re killing this thread with really honest, well-constructed comments. Number 31 in particular. It kind of took me off the holy mountain of ‘what math education is supposed to be.’ Well done.

My small qualm is that what you’re describing doesn’t sound like ‘lecture’ in the sense that I understand it. It sounds more like whole-class discussion (and really effective whole-class discussion). Lecture is talking at, not with. I haven’t seen you teach, but what you’ve described – cold-calling, pair work, thoughtful questioning, practice problems, feedback – sounds like a fluid, well-run classroom.

The discussion about the relative merits of ‘talking’ got me thinking. Imagine someone (say Khan 2.0) comes out in 2012 with videos of staggering quality. Just perfect. Multiple strategies, engaging hooks, critical reasoning, interactive, differentiated, etc.

As a student, you want to learn how to add polynomial expressions. Would you rather have access to Khan 4.0 (with pause, rewind, and assessment with immediate feedback) or a typical class with access to your teacher and peers?

And does your answer change if the topic changes?

Wow- good to see the heated and rolling debate here. I’ve been thinking about Kahn’s implications on education ever since the TedTalk he gave, and one thing all you critics need to do before you keep bashing is learn more about what “flipping the classroom” really means. I recommend checking out this site here, as these chemistry teachers have been doing it for some time: http://mast.unco.edu/programs/vodcasting/

Flipping the classroom when done well actually allows MORE time for inquiry-based investigations and project-based learning. So unless you’re pretty radical and avoid lecture/direct instruction all together, video lectures like Kahn could benefit you.

I also agree with the good point that videos lectures are “one-size-fits-all” but not “one-time-fits-all”. I’ve made many video lecture clips for my science students that get watched repeatedly by the same student- and that certainly has more impact than saying it one time in the classroom.

As a side note- unfortunately I think there may be a tendency for our media-saturated students to actually pay better attention to a video lecturing than a teacher lecturing. I have no data to back this up- but somebody should do a study of this!

I understand Dan’s critique that video lectures used blindly ignore a need for differentiation- but then since videos are easy to make and save, there can also open up some truly innovative ways to utilize them in a differentiated way. Again, refer to the link above for a good example- there learning is paced by the students.

I guess my overall takeaway is that video (and other technologies as well) are still hugely underutilized or improperly applied. So before we knee-jerk against something like Kahn we need to keep a more open mind.

If anyone has any more thoughts on what you see as the advantages of video presentations over textbook readings, feel free to let me know over at my blog (link above) to keep this thread clear of that tangential question.

My school tries to cover lots and lots of material over a short (~130 day) year, and I’m held responsible if I don’t cover all the material that’s part of NY’s Regents exams.

Practically every teacher of math at any level above basic algebra is in this position. Again, I teach college, so I probably don’t feel your pain quite so acutely (I am evaluated mostly on my popularity, rather than some measure, however deeply flawed, of how much my students have learned), but I am still pressured to cover most of the soul-crushing meaninglessness that is squeezed into some math classes. This is why I’ll be experimenting with “flipping” a couple of my courses in the fall, roughly following Eric Mazur’s lead. I feel that this is the best I can do for my students, given the circumstances — though I honestly don’t expect to be rewarded for it in the student evaluations.

This and the Khan thread have been fascinating to follow. I loved Sean’s brief pitch (comment 33) for a future technology enabled Khan 4.0 approach (though I don’t think of Khan as the inventor of this, so how about Math 4.0)

Add rich media, assessment and an organizer for all this and the result would be that skill and concept acquisition can be partially farmed out to technology so that:
a) kids have more options that suit their learning style and pace
b) teachers have tools to manage a more individualized approach to skills and concepts acquisition.
c) teachers and students have more time to apply mathematics which always gets short shrift by most of us and should be a rich and wonderfully engaging class activity.

MBP’s frank post (comment 31) makes it abundantly clear that getting from lecture based instruction to Math 4.0 is going to be tricky business considering the current emphasis on ‘getting through the curriculum’ and the lack of organized Math 4.0 tools and training to make it happen… but blogs like this certainly have us pointed down the right path.

Right. So “lecture” is about as precise a term as “love,” apparently. Right up there along with “say”.

And I’m not saying that this is the whole problem, except that this is the whole problem.

If ever we are to coalesce our thought about best practice into some kind of cohesive whole, and debate education realities meaningfully and efficiently within it, we have to apply the same rigorous process of creating definitions for these terms as readers did with “pseudocontext.”

R Wright asks:

Can you think of some specific classroom activity that does not clearly fall into, or out of, the “lecture” category?

Sure. I’ve got a whole mess of ’em. KWL. Think-Pair-Share. Orchestrating Mathematical Discussions, seatwork…

Lecture puts the teacher at the center of attention. Anything else isn’t that.

Oops.

Note to self: read the stuff between the commas. Sorry. I read the original as “…that does not clearly fall into…the lecture category”. I thought it seemed like too easy of a task!

Now that I have read more carefully, I would like to reframe your question. Dina isn’t arguing that the line is too fuzzy, but that we don’t all agree where it is.

So, R Wright, the challenge ought to be: Can you name a classroom activity that some would call lecture that others would not?

These I don’t have a whole mess of. But I’m optimistic that it exists.

When, for example, does Answering a student’s question turn into lecture? Is after a certain amount of time has elapsed? When the topic changes? Or was it lecture all along? Does it have to take place in a classroom (if so, then Khan is off the hook, right? Neither he nor his “students” are in the classroom).

Concrete example: Did I give my daughter a lecture on circles the other day?

Strong comment from R. Wright, the kind of comment that’s only generated by a math teacher who’s also taken the time to become familiar with the academic literature surrounding teaching. I think we need more of those in the k-12 arena, fwiw. Honest, instructional, and yet inviting.

Sure is a smart crowd around these parts.

Thanks for all your thoughtful responses surrounding my question.

First: I do believe “lecture” is both a pedagogy AND a technique to some teachers (as R. Wright insightfully points out). That being said, my goal was to clarify the exact meaning of the technique (as Christopher Danielson points out, equally insightfully.)

I still maintain, as the debate between Christopher and R. Wright amply and ironically suggests, that there is no shared concensus on what “lecture” actually means– particularly when we are still evolving our sense of how multimedia does and should play into lecture.

As a result, I think the debate suffers from circular thinking and missed connections. Easy fix, though, don’t you think? Do what is being done in this thread. Define your terms. No biggie. :)

This seems like the old Behaviorism vs Constructivism debate to me.

Why lectures are inefficient ?
Because students need to try before knowing.

The fact that a knowledge appears as a response to a question or a problem helps to build the understanding and the appropriation of this knowledge. It is why problem solving must hold an important place in any curriculum.

IMHO
In order to teach any notion the model should be :

1 – Discovery : evaluation of the initial conceptions of the students through a problem situation

2 – Familiarization : emergence of the characteristics of the notion, pooling of the strategies and procedures used or tried against the problem

3 – Training : structuring the notion in the student’s mind through simultaneous systematization (understanding the system of the notion, how it function and reorganize anterior conceptions) and automation (not having to think when reusing the notion).
At some point in this stage the students need some reflexivity on their process of learning : Why is this hard ? What is hard ?

4 – Reinvestment in a more complex situation, evaluation

first of all, i would think a proper definition of lecture would wrap it in the notion of an extended period of time. simply answering a question is not what we mean by lecturing. that’s silly. a student asks a question, you answer it in 15 or 30 seconds, check in w/ the student, and the question is answered. end of exchange. there’s been no lecturing. i would think a lecture is a prolonged monologue. okay, if it the answer takes longer we get into murky territory, but, still, as long as the answer focuses on a pointed answer to the question, it’s not a lecture. also, i would think a lecture is prolonged monologue primarily *generated* by the lecturer. when answering a question, in contrast, the answer is generated by the student. perhaps that’s an important distinction in the definition of lecture.

i like rook’s schema. reflection is essential, of course, as he/she says in #3. another essential ingredient that i try to include is history. i think it’s truly unfortunate that subjects are often taught (usually!) w/out a reference to the historical context of the knowledge that’s presented. i would even go so far as to say that knowledge w/out historical context is not knowledge–it’s something else, but it ain’t knowledge. of course we have to be talking about spectrum here. i can teach students how to write a personal essay, but w/out their knowing the circumstances surrounding the emergence of the modern personal essay in the 16th century, their knowledge of how to write one is … significantly limited. their understanding of how that form of expression came to be seen as important is crucial to their understanding of their own use of it.

Students need real experiences. Watching “Deadliest Catch” (as I am right now) does not give me even the remotest experience of the Alaskan fishing. I do not think I am qualified, even after 4 seasons. Watching someone else stack blocks/construct geometrically/scubadive/pace out a linear graph does not give me the experience of doing it myself. Anyone who thinks that has no memory of being a student.
My students’ biggest deprivation is the removal of actual experiences.
Despite their bad behavior, they need to make bicarb rockets/ barbie bungee/remote control car videos. Substituting a video does not do it.
I don’t even consider this to be constructivism, I consider it to be context. When I found out that a kid thought a question about cutting fudge into one-inch cubes was stupid and unanswerable because you can’t cut fudge ( it’s runny and you pour it on ice cream), I realized the problem they had was way beyond math.
While I like WCYDWT, I use them as cues for real experiences (which is why the mall with the escalators is hopping mad ).
I personally am grateful for every single thing that has been put up on the web for free to help students learn. Can’t hurt. Thanks to everyone.

(For clarity, that should be “unnecessarily.”)

I personally am grateful for every single thing that has been put up on the web for free to help students learn. Can’t hurt.

But this sentiment ignores opportunity costs (i.e. students’ time spent on something with no long-tem benefit, aside from helping them pass a test), and the hidden message I saw in the few Khan videos I’ve taken a look at (i.e. mathematics is an arbitrary set of definitions and procedures to be diligently memorized).

But lecture is bad :-) and telling might be pointless when students are ready to hear.

Arguing about a precise definition of lecture seems a bit pointless, I give my own after reading some comments nonetheless : lecturing is telling the discoveries of others.

I think that trying to define learning would be much meaningful (if not hectic)

In fact I postulate that there exist a definition of learning for which The Two Truth of Teaching are :
1 – If I say it they won’t learn it.
2 – If I don’t say it they’ll will learn it (when confronted to a proper situation and given enough time)

I still feel like we look at lecture as one thing. Lecture in and of itself is not inherently bad; however, it is a larger picture of our entire learning strategy. The key is looking at the two lies and recognizing something very obvious. They both talk about saying something, and it’s very obvious that students need to “do” in order to learn. Therefore, we have to look at regardless on if, how, when, and why we lecture, students still have to learn by doing.

It does depend on how you define learning : if it goes along the lines of discovering by yourself or as a group something, saying it will prevent them from learning.

If it refers to being able to reproduce, then lectures and electroshocks are very sensible means to teach.

I’m coming late to this party, but I do think that some working definitions would be useful to advance this discussion. I think the issue is that we’re discussing the efficacy of various learning methods, but lecturing is not a learning method–it’s a way of combining several learning/teaching methods.

So, here’s my attempt to find vocabulary for a more fine-grained vocabulary for learning methods.

If you reflect on what you know about the world, a lot of our knowledge comes because an individual “explained” an idea to us. And so instead of focusing on lecturing, I’d like to focus on explaining. People certainly sometimes learn when a matter is explained to them, so it’s natural that teachers will spend some time explaining matters to students.

A teacher has options when it comes to explaining ideas in a classroom. A teacher can explain things to a student, one-on-one, or a teacher can explain things to all of the students in the class at once. Another degree of freedom is that a teacher can either allow for questions from students during her explanation, or the teacher can not take any questions from students during her explanation.

Of course, the teacher can use props and video and other sorts of aids in her attempt to explain something to her student(s). It’s still an explanation.

So that’s the vocabulary. A teacher can choose to teach through explanation, and in doing so the teacher can explain to individuals or to the class, allow questions or not allow questions, use demos or not use demos. I’m sure we can add more to the degrees of freedom within explaining. For instance, we could distinguish between teachers who explain for an hour at a time (I’m looking at you, Sophomore year philosophy professor) and those that explain for just 4 minutes at a time.

I like this vocabulary for two reasons:
(1) Now I might know what positions some of y’all are defending.
(2) I think this way of talking about teaching gets at the heart of the confusion from students/parents/etc. We got so much of what we need to know from explanations. Why wouldn’t explanations work in the math classroom? Others will only see the light if we explain why explaining things, which works so often, fails so often in the math classroom.

I think we’re all missing the point. The major limit of lecture is language. Words. This is the medium of lecture. Children vary GREATLY in their ability to process is medium. Some children are expend less energy re-creating the meaning of words in their minds. That’s what has to occur, a complete re-creation of our thought process. We should focus much more on this BASIC fact. I’m not saying that we should dump down language. But we should be scientifically aware of the rate of information being disseminated in any given word or sentence and rate of speech. We need to focus on language as a perception

Lectures, because of their medium, many times muddle actities. Lecture should highlight, accentuate, a demonstration or interactive activity. They are quite convenient, and the results are many times qiute “average”. How many words out of every lecture are partially or completely discounted in a students “re-creation” of an idea. My guess, for many students is 85%. Do we have studies on this?

I teach children. I think sometimes many forget what the role accumulating knowledge as in done in school has in a child’s psyche. Math and any subject CAN be, and should be, a source of PRIDE for students. The lecture format has little is any room to nuture this very individual feeling of “I CAN”. That’s what students need. They crave it. Once again I believe the fault lies with the format of the lecture itself, NOT the teacher, although most certainly the lecturer. Every child participating in a lecture is covered by the same EXACT same BROADCAST. But children understand that they are all different YET they still can’t help judge their worth on their ability to understand and execute task X. So we have one broadcast, or word-stream, and many different children. Even when we don’t pick up on this unequalness, children do. It hurts them, plain and simple.

We need technology to augment a teacher’s ability to challenge individual students in an appropriate manner; that being any manner that leads to PRIDE, ACCOMPLISHMENT, and eventual success. Let your imagination run wild here. Its totally under-developed, practically non-existant arena for exploration. Its worth investing in.

sorry for the other post I should read what I type before submitting. I think we’re all missing the point. The major limit of lecture is language. Words. This is the medium of lecture. Math is very hierarchal in its language, it seems that we use WORDS to EXPLAIN more words. Look at numbers themselves, they are words. That’s a pretty high level of abstraction right there!

Lectures, because of their medium, many times muddle activities. Lecture should highlight, accentuate a demonstration or interactive activity. They are quite convenient, and the results are many times quite “average”. How many words out of every lecture are partially or completely discounted in a students “re-creation” of an idea. My guess, for many students, is 85%. Why do I guess this? Because you ask them to explain something back to you and they say, “this goes there.” They are sure “it” goes there. They’re right, and that’s what counts, they themselves need to feel right. Knowing and understanding WHY it goes there is too often secondary to “knowing” that it goes there.
.
Children vary GREATLY in their ability to process language. Some children expend less energy re-creating the meaning of words in their minds. That’s what has to occur, a complete re-creation of our thought process. We should focus much more on this BASIC fact. I’m not saying that we should dumb down language. But we should be scientifically aware of the rate of information being disseminated in any given word or sentence and rate of speech. We need to focus on language as a perceptible sense. Especially with younger children, they have to articulate what they learn if they’re to have a continuous “narrative”, and therefore a good memory, of their knowledge.

I believe that younger children rely heavily on their operative memory and working memory to recreate the PROCESS of a mathematical task. Instilling a more declarative memory “teaching experience” would feel more concrete and obvious to build off of in the long run. But I don’t think we insist on it. How many students actually have the idea of a number line present when they are adding and subtracting?

Math and any subject CAN be, and should be, a source of PRIDE for students. There’s a lot of knowledge to accumulate. The lecture format has little if any room to nurture this very individual feeling of “I CAN”. That’s what students need. They crave it. Once again I believe the fault lies with the format of the lecture itself, NOT the teacher, although most certainly the lecturer. Every child participating in a lecture is covered by the EXACT same BROADCAST. But children understand that they are all different YET they judge their worth on their ability to understand and execute task X (they can’t help it).

So we have one broadcast, or word-stream, and many different children. What kind of tension does this create? It works much better for some then others. Even when we don’t pick up on this unequalness, children do. It hurts them, plain and simple.

Lecturing has been a acceptable solution to unavailability of individual attention. But NOW, we can not accept it any longer because we have technology that COULD help us keep our “finger on the pulse” of the learning. A thousand forms of instant feedback and monitoring is what technology can bring, especially in a small relatively stable environment of the classroom.

Yes it is true that EVERY student must learn the same facts. But this will never happen if we do not AFFIRM the individual in a HEALTHY logical sequence and order. We need technology to augment a teacher’s ability to challenge individual students -in an appropriate manner; that being a manner that leads to PRIDE, ACCOMPLISHMENT, and eventual success. Let your imagination run wild here. Its totally under-developed, practically non-existant arena for exploration. Its worth investing in.