My recent post about the differences between Salman Khan and Conrad Wolfram’s TED Talks (**Compare and contrast: using computers to improve math education**) brought a lot of traffic to the blog, some great comments, and more than a few Twitter conversations about how to teach math.

So I’d like to get more specific about what I think is wrong about the Khan Academy approach by writing about things I see as wrong with the way we teach math in the US.

No matter if we agree or not about Khan Academy, I’m fairly certain we can agree math learning is not going as well as we’d like (to say the least.) Too many people are convinced by the system that they “hate math”, and even students who do well (meaning, can get decent test scores) are often just regurgitating stuff for the test, knowing they can safely forget it shortly afterward.

There is plenty of blame to go around… locked-in mile-wide inch-deep curriculum, focus on paper and pencil skills, lack of real world connections, assessments that are the tail that wag the dog of instruction, a culture that accepts “bad at math” as normal, teacher education programs that have don’t have enough content area specialization, … you can probably add to this list.

I can’t tackle all of these. But if you are interested, I’d like to share my thoughts about Khan Academy and a few epic math myths that are relevant to a discussion of the Khan Academy. In America, these myths are so pervasive that even people who were damaged by the way they were taught themselves accept them and insist that their children be taught using exactly the same methods.

I think these myths explain both the widespread acceptance of Khan Academy as a “revolution” and also why in reality it’s not going to change anything.

**Myth: Learning math is about acquiring a sequential set of skills (and we know the sequence)**

** **I think people have a mental image of math that looks something like a ladder. You learn how to add single digit numbers – rung one. You learn 2 digit addition – rung 2. You learn 3 digit addition – rung 3. In this model, you get to rung 3 by throughly learning rung 1 and then rung 2.

The myth continues with the idea that the march up the ladder goes faster if we tell children exactly how to do the problems step-by-step. In the language of math instruction, these step-by-step processes are called algorithms. Some kids “get it”, some don’t, but we accept that as a normal way that learning happens, and “help” the ones who don’t get it by drilling them harder in the step-by-step process, or devising additional tricks and supports to help them “remember” how to solve the problem.

If they don’t learn (meaning pass tests), we take this as evidence that they haven’t practiced the steps well enough, and prescribe more of the same.

Khan Academy plays perfectly into this myth. Here are a convenient set of videos – you just find the one you need, push play and the missing rung in your mental math ladder is filled in.

A corollary to this myth is that we can test students for these discrete math skills, see which “rungs” are missing, and then fix that problem with more instruction and practice on that specific skill.

**Let’s diagnose how we think about learning a simple math skill**

When we teach 2-digit addition, we immediately introduce the algorithm of “carrying”. You should know, though, that the U.S. form of carrying is just one of many addition shortcuts, not handed down on stone tablets. It’s not used world-wide, nor is it something that people naturally do when adding numbers. But it’s cast in concrete here, so we teach it, then we practice that “skill”. With our ladder model in mind, if a child can’t answer the 2-digit problems correctly you do two things: 1) Do more practice on the rung under it, and 2) do more practice in the algorithm, in this case, carrying.

The problem is that if a student has simply memorized the right answers to rung 1 without real numeracy, reviewing carrying will not increase that understanding. In fact, it will reinforce the memorization – because at least they are getting SOMETHING right. They are like the broken watch that’s right twice a day. This issue gets worse as the math gets more complex – the memorization will not be generalizable enough to solve more complex problems.

**A different vision of learning**

*“Some of the most crucial steps in mental growth are based not simply on acquiring new skills, but on acquiring new administrative ways to use what one already knows”. “ Papert’s principle” described in Marvin Minsky’s Society of the Mind.*

If this is true, and since these administrative skills are not sequential, it makes it less likely that we really learn math in a sequential way. I think we’ve all had similar experiences, where a whole bunch of stuff suddenly makes sense.

This different vision of how people learn is called “constructivism“. It’s a theory of learning that says that people actively construct new knowledge by combining their experiences with what they already know. The “rungs” are completely different for each learner, and not in a specific order. In fact, rungs aren’t a very good metaphor at all.

*“…constructivism focuses our attention on how people learn. It suggests that math knowledge results from people forming models in response to the questions and challenges that come from actively engaging math problems and environments – not from simply taking in information, nor as merely the blossoming of an innate gift. The challenge in teaching is to create experiences that engage the student and support his or her own explanation, evaluation, communication, and application of the mathematical models needed to make sense of these experiences.”* – Math Forum

**Learning theory? What’s the point?**

We need to talk about learning theory because there are different ones at play here. And to be complete, we are also going to need to talk about teaching theory, or pedagogy, along the way. Constructivism doesn’t mandate a specific method of teaching, but is most often associated with open-ended teaching, constructionism, project-based learning, inquiry learning, and many other models. Most of these teaching models have at the heart an active, social view of learning, with the teacher’s main role as that of a facilitator.

However, the teaching theory underlying most of American math education is instructionism, or direct instruction – the idea that math is best taught by explicitly showing students how to solve math problems, then having students practice similar problems. Direct instruction follows when you believe that math is made up of sequential skills. Most American textbooks use this model, and most American teachers follow a textbook.

This is important distinction when talking about Khan Academy. Khan Academy supports teaching by direct instruction with clear (and free!) videos. If that’s your goal, you’ve found the answer…. but wait…

**Is clarity enough?
**Well, maybe not. Even if you believe in the power of direct instruction, watch this video from Derek Muller, who wrote his PhD thesis on designing effective multimedia for physics education. Really, if you are pondering the Khan Academy question, you must watch this video.

*“It is a common view that “if only someone could break this down and explain it clearly enough, more students would understand.” Khan Academy is a great example of this approach with its clear, concise videos on science. However it is debatable whether they really work. Research has shown that these types of videos may be positively received by students. They feel like they are learning and become more confident in their answers, but tests reveal they haven’t learned anything. The apparent reason for the discrepancy is misconceptions. Students have existing ideas about scientific phenomena before viewing a video. If the video presents scientific concepts in a clear, well illustrated way, students believe they are learning but they do not engage with the media on a deep enough level to realize that what was is presented differs from their prior knowledge. There is hope, however. Presenting students’ common misconceptions in a video alongside the scientific concepts has been shown to increase learning by increasing the amount of mental effort students expend while watching it.” *– Derek Muller, Khan Academy and the Effectiveness of Science Videos

Derek makes an interesting point – clarity may actually work against student understanding. Videos that slide too smoothly into an explanation do not give a student a way to process their misconceptions and integrate prior knowledge. The very thing that makes the videos so appealing – Khan’s charisma, sureness, and clarity may lull the viewer into comfortable agreement with the presentation without really absorbing anything (Research references and Dr. Muller’s PhD thesis on this subject)

**Hooks, not ladders**

This goes back to my original point. People learn by reorganizing what they already have in their head and adding new information that makes sense to them. If they don’t have a “hook” for new knowledge, it won’t stick. The tricky part is, though, that these hooks have to be constructed by the learner themselves.

Wishful thinking about downloading new information to kids is just that – wishful thinking.

There is no doubt that Khan Academy fills a perceived need that something needs to be fixed about math instruction. But at some point, when you talk about learning math, you have to define your terms. If you are a strict instructionist – you are going to love Khan Academy. If you are a constructivist, you are going to find fault with a solution that is all about instruction. So any discussion of Khan Academy in the classroom has to start with the question, **how do YOU believe people learn?**

I have more to say about Khan Academy and math education in the US — this post turned into 4 parts!

Part 1 – Khan Academy and the mythical math cure (this post)

Part 2 – Khan Academy – algorithms and autonomy

Part 3 – *Don’t we need balance?* and other questions

Part 4 – Monday… Someday

Stay tuned – Sylvia

**My context for these posts: **I fully admit I’m not an expert in math or math teaching, just an interested observer of K-12 education in the U.S. In my work, I have unique opportunities to see lots of classrooms in action and talk to lots of teachers. It means I get to see patterns and similarities in classrooms all over the country. I don’t intend to do a literature review or extensive research summary in these posts. This comes from my personal experience, my master’s degree in educational technology and draws from a subjective selection of research and sources that have had a deep impact on my thinking about learning. Finally, I am NOT trying to tell teachers what to do. I’m not in your classroom — that would be silly.

You just saved me a lot of writing. I was posting about the emporium model of course redesign, and you made several of my points for me. So I just linked to this post.

I agree with everything that you say in this post. I don’t think that the problem is with Salman Khan, who is doing the best he can with a limited understanding of education, but with Bill Gates. Gates, through his foundation, is really pushing for this model of education in America. The Gates Foundation has been offering money to colleges and high schools to change how they teach, and faculty are getting administrative pressure to change.

There is one faculty member at our college who has been teaching for forty years. He has a list of all the technologies that were supposed to replace him. As the Khan Academy matures, it will become a tool in the educator’s toolbox, but nothing more.

I see math education as two strands: computation (yes, a ladder like approach) and investigation-content, skills, knowledge.

Similar to any skill learning, I still think it’s important for students to learn the facts and algorithms by practicing a few times a week.

I also think that the love of math and true math learning comes from the investigations that allows students to “hook” into the math. I really like Arthur Hyde’s books which promotes teaching math using cognitive strategies similar to those used in reading and outlined by Ellin Keene in Mosaic of Thought.

I think the classroom model should include hands-on investigation/real-world problem solving, direct instruction, practice, discussion, tech and more. A varied approach with continual assessment (informal, observation, and some tests) to see who has mastered what, and what questions remain.

I’m open to discussion and want to learn more about math instruction. My biggest issue with math teaching is time — at my grade, 4th grade, we have so many goals and so little time. I’m not a proponent of more time right now as students and teachers are working really, really hard and too much time on task would make everyone too exhausted and dull.

Wow! Great post. I particularly enjoyed the video about relating the research done on science instructional videos to the Khan Academy.

Do you think the same problem discussed in the video with regard to science applies to math instructional videos? Do students tend to have misconceptions about how to factor or reduce fractions? I’m not a math instructor (I’ve always taught science), so I’m just not sure how analogous the problems would be for math and science. In science, especially when it comes to teaching about forces, students have all of this experience with the world that leads to folk understandings of how things work. Does experience with the world lead to similar misconceptions in math? As I said, I’m just not sure.

I’ve tended to think that there is potential with the Khan Academy, but not as a curriculum unto itself precisely for the reasons that you relate early on in this post: Math is not just a series of perfectly ordered skills like levels in a game that you must achieve and move on. In my mind math exists because of it’s utility in commerce, science, and construction. So I’ve always liked curriculum that spends more time using math to solve these types of problems as Dan Myers does. But I can see the videos working well as supplemental material so that students can quickly and easily refresh the skills. But as I said, I am not a math teacher, so this may all relate to my folk understanding of math instruction.

Helen, I think it’s exactly when we disconnect math from the real world that we get into a lot of trouble. Math is a way to describe your world with precision, to convey information to others, and to predict the future. Students do have a lot of experience in the world about math that gets ignored because we are busy teaching something we call math, but it’s more about abstract arithmetic.

Was that “science video” clip actually supposed to answer the question asked in the pre-test? The part you presented here did not. So, it makes sense that revising it to actually address the question would mean students learned more.

In my experience,Those misconceptions are *huge.* However, many math instructors infuse their instruction with “of course you remember this from…” — when, actually, if they do remember, they remember the simplified model. I’ve had a lot more success when I directly address the idea that, for instance, we *want* adding to make things bigger, but if we’ve got negative numbers involved, the pattern changes.

… and the disconnect of math from the real world… starts really, really early. 90% of the brilliant ideas to “make it real” don’t. The “make it real” part just sticks another Layer Of Procedures to Memorize into the mix.

I would venture that *most* adults don’t know what fractions mean unless they’re quilters, cooks or carpenters.

Wonderful post! I heard on a talk on this subject just last Friday by the wonderful speaker Eric Mazur. For anyone who hasn’t seen it, I recommend his lecture ‘Confessions of a converted lecturer’ (which is on Youtube) which is all about how we can use Peer Instruction to help students learn things for themselves, rather than being told information by teachers/lecturers.

At the University of Edinburgh we’re keen to try out some of these methods, but the difficulty is in learning exactly where the misconceptions are and what questions we can ask students which will allow them to think about the concepts and change their minds. It somehow seems easier in Physics than in Math but that shouldn’t stop us trying! I’ll be blogging about this later in the week and hope to get lots of suggestions.

Wow, I’ve never gotten a comment from a sheep before 🙂

I would guess that Physics is easier than math because introductory physics has a lot of “hooks” for students. They have direct experience with gravity, friction, force, acceleration, etc.

It’s hard to connect abstract computational processes with real experiences, and a lot of the math we teach has no real need except that it was invented to help people solve problems of dealing with big numbers before computers were invented.

Is it possible for you to change the curriculum and get rid of some of the math for maths sake?

I’ll bite on Sue J’s comment: “Was that “science video” clip actually supposed to answer the question asked in the pre-test? The part you presented here did not. So, it makes sense that revising it to actually address the question would mean students learned more.”

The question he asked was:

Consider a basketball player shooting from a free-throw line. After the ball leaves his hand, the force on the ball is

a. upwards and constant

b. upwards and decreasing

c. downwards and constant

d. downwards and decreasing

e. tangent to the path of the ball

The video lesson says,

“While a juggling ball is in the air–we’ll ignore air resistance because it’s so small–only one force acts on he ball throughout its flight. This is the force of gravity, which is constant and downwards.”

I don’t understand how this is not answering the question. Sure, the ball in the problem is a basketball and the ball in the explanation is a, er, “juggling ball,” but I don’t see that making any difference.

My question is–and I’m going to skim his thesis to find out–could students watch the video as many times as they wanted, or stop the video, or use it during answering the question? The assumption seems to be that videos are too passive, but not giving a student control over the video in the experiment eliminates engagement options, which would make the conclusions it makes about Khan academy videos invalid (or at least, only valid when a student cannot interact with a KA video).

Hi Stan,

I think the point of his thesis is he wants to answer this question, “Is there a difference in science learning when you show videos that do not address misconceptions and show only clear explanation of concepts versus videos that begin with common misconceptions and then present the correct science to refute those misconceptions? ” I assume that in his two groups the variables that you suggest (such as learner control, repeated viewing, etc.) were controlled. They simply weren’t part of what his research was designed to test.

It looks like the data he collected showed a statistically significant difference between the learning of the two groups. So in no way does it make sense to say that his conclusion about the Khan Academy is invalid because of a lack of learner control. That is unrelated to his question. They may or may not have had control over the video, but as long as the two groups were controlled for this variable, it doesn’t matter. His conclusion regarding the need for addressing misconceptions in science seems to be a pretty reasonable critique of scientific videos that only show clear and correct explanations.

I think a more interesting question would be whether or not you can extend his conclusion to math instruction at the Khan Academy. Would learners learn math more effectively if those videos began with misconceptions and then explained why those misconceptions are incorrect? Certainly students make common mistakes in math, but are they misconceptions the way that students develop scientific misconceptions based on their experience of the world. My hunch is it wouldn’t be as useful. You couldn’t go up to people on the street and ask them what are the steps to factoring quadratic equations the same way you can ask why there are seasons. One is far too foreign and abstract a concept for the everyday person to have a “folk” or mistaken understanding of.

Heidi,

I found an answers (well, not really, but it’s something). From Derek Muller’s thesis (pg 162), which is linked, or linked through another link above:

“Attempting to use all ﬁrst year students as participants meant multimedia treatments could not be tested in a lecture environment. Instead, students watched multimedia streamed through the Internet on computers. Consequently, many variables were uncontrolled. Participants accessed the experiment wherever and whenever they liked. They may have used resources or consulted with peers when answering the pre- or post-test questions. Although the time between the start and end of the multimedia was calculated, an appropriate length of time was no guarantee that a student actually watched the treatment. These were features of the methodology. The ability of students to participate as they saw ﬁt ensured the results could be generalized to authentic learning environments, a central requirement of the design experiment methodology.”

So, it’s still not clear if students were able to pause, rewind, and skip around on the video during answering the question. The last sentence in the quote above–about “ensuring the results could be generalized” is not convincing, including when the results are generalized to use of KA. In other words, it’s unreasonable to assume that results could be generalized to the set of all “authentic learning environments,” including one where KA videos are used.

As for your question about math vs physics and folk notions of the real world, I have a similar hunch. However, that interesting question reveals part of the problem with the original blog post: it confuses physics videos with math videos; KA videos with the KA program, which contains practice problems and community responses; one different learning situation with another.

I purposely didn’t think hard about what “force” actually means when I saw the question and the video. Technically, the question is “answered,” but — partly because yes, the juggling balls aren’t basketballs — a person isn’t necessarily going to realize that, even though the same word is used in all three settings, that force doesn’t mean speed the third time, either.

It’s almost impossible to explain to somebody who understands forces well. That “gravity is the only force” just didn’t pierce my process of “what are the forces on the ball,” so the idea that the pressure of my hand was still, somehow, lingering and guiding the basketball forward, remained.

Essentially, if I hadn’t already confronted the fact that as an uninstructed person, you were really asking about speed, not force, then you talking about the forces on the ball in the air doesn’t connect. Talking about the only force being gravity doesn’t teach it. It gets recorded in the brain as “yea, gravity is what makes it slow down. Science is cool!” The “only,” because it contradicts the person’s model of the working universe, gets passed over and ignored.

I’ve found that for many learners, paying more attention to the language is a big help. Clarifying that force is force and speed is speed (and if I’m trying to teach you what force is, I’m going to say “speed,” not “velocity,” because I want to reduce the number of times I say “velocity” and the learner thinks “speed” and I’m teaching something else so i can’t tweak *that* misconception.)

Sue J-

Good points. I just wanted to make sure that we both agreed it did “technically answer” the question, even if it was a fruitless answering.

May I suggest that what we’re seeing is what we’d generally expect if we consider dual-process models of cognition that propose a distinction between automatic/intuitive thinking (fast, easy on attention, associative) and deliberative thinking (slow, taxing on attention, logical). Basically, Daniel Kahneman’s System 1 vs System 2 (Kahneman’s 2002 Nobel Prize lecture has a brief synopsis of this).

Looking at the experiment in these terms, we should expect people to process information quickly and automatically until something causes them to notice an error in their expectations. Recognizing the error, and being interested in the context surrounding the error, can then lead to deliberation.

The first answer video contains nothing to draw attention to errors in expected answers. Minds glide along in System 1.

The second answer video is more likely to trigger a viewer to notice an error in their expected answer to the original question. A few minds think “Wait, that isn’t right?” and engage some System 2.

I glanced through Muller’s thesis’ list of references, and I saw mostly “education” sources, and no “cognitive science” sources, but I didn’t go through it with a fine-tooth comb. Muller doesn’t seem aware of this particular area of cognitive science at all, so I thought I’d mention it here.

Thank you — I was also unaware of that model, but was simply reconstructing my processing and my experience tutoring, which goes infinitely better when I target student misconceptions, go a step or three below that and try to rebuild. This will be useful for my little chat about negative and positive numbers Tuesday here at the Center for Academic Success…

Stan and Sue – I have to congratulate you both on such a wonderful example of how educators can have an intelligent conversation about learning! It’s so much better than just assuming you can push the play button and students will learn.

Sylvia, that’s a very smug quip about KA that is neither good-natured, nor deserved.

Who assumes “that you can push the play button and students will learn?” Is it Khan? Is it the teachers in the pilot project at the Los Altos schools? Who is it?

Hmmm… it didn’t strike me as smug, because I interpreted it as a metaphor (hey, I’ve got models and metaphors on the brain right now). It is entirely too easy for me to think “If I present it, they will absorb it in the framework I’m thinking and learn.” It’s something we *all* have to work at. (Perhaps this is a perfect example of that… Hmmm… *perhaps* online communication has helped us realize how easy it is for what one person thinks s/he’s saying to be interpreted wildly differently?)

Right now, I’m planning an hour-long presentation of “why all the negativity about integers?”

One reason the cognitive science model is very useful is that it answers my hugely nagging doubt: how in the world, in one session, can I help students who have struggled with this all semester/ for several semesters? Several of the students attending are folks I’ve worked with extensively, 1:1 and… yes, they still make the Same Mistakes.

It is very helpful for me to consider that — away from Math HOmework — *just* building a novel concept could help… but that “pushing play” is less useful than building in genuine interaction — and would be even less so if I were dealing with some of the very hands-on “don’t make me think in symbols!” folks that I’d really like to reach.

Now, this challenges some “we must make it real! we must make it real!” lines of thinking IMO. Sometimes, I think, it helps to focus on the idea; to identify the abstraction and work with it and wrestle it to the ground… but with the grip of understanding, not the distant “am I sticking the symbols in the right pattern?” procedural mastery that these guys are finally realizing isn’t working well at all.

Sue J,

Here is the quote from the blog post, which is above somewhere:

“Khan Academy plays perfectly into this myth. Here are a convenient set of videos – you just find the one you need, push play and the missing rung in your mental math ladder is filled in.”

Perhaps its not smug, it just struck me as such. Just as one must be precise when discussing speed versus velocity, one must be precise about the subject of criticism (KA? implementation of KA?).

Granted, I don’t know who the “you” is in the quote above. That’s one of the problems in the series of blogs that are supposed to be about the Khan Academy (I think, at least). I don’t know who is being criticized, so it’s impossible to see if the criticism is accurate.

I addressed Muller’s study at first because, at least in his case, I could tell precisely what he was criticizing and why, and I could look at his methodology. I bumped into this blog because I was looking for critiques of the KA program, and I’ve both been left wanting, and feeling that Khan and the Khan Academy have been unfairly characterized.

Stan- I meant that sincerely as a compliment to the interesting and civil discussion between you and Sue.

Hey, once a thread gets past three posts, it gets unraveled. Your response makes sense in that context.

I, too, appreciated an analytical discussion of Khan; somebody linked to this blog to support his statement that KA was being wrongly touted as the cure for the ills of education, and I had shared a near certainty that no, it wouldn’t work for my students, but I wanted to be able to say why in ways that didn’t sound like “our students just don’t have the independent learning skills to learn from that.”

Sylvia-

I apologize for my misinterpretation. Perhaps at some point I will place a critique of the series in the comments to the last part, but as stated, I’m not always sure who or what is being criticized in the posts.

Sue J-

Curious: have you tried using KA to relearn something you’ve forgotten, or learn something new? (Assuming you can find something that fits the bill in the math topics that have exercises.)

I think one of the reasons that many educators are pushing back on the Khan Academy as a revolutionary idea is that in some ways it is nothing new, but yet is being touted by some like Bill Gates as a the solution to educations problems.

If you think about it, how are the videos any different that reading a chapter from a text book? The content in either case can be structured well or poorly. I liked the video above because I hadn’t seen as much criticism of the structure of the videos themselves, and that raises an interesting point. Both these videos and the textbooks are available potentially 24/7. The videos are available as long as you have a network connection and the book would be available as long as you lug it around. In both mediums you can “stop,” “rewind,” and “move at your own speed.” What makes the video perhaps superior is that you have the opportunity to dual encode (in a difference sense than dual processing above). That the information allows for visual and auditory processing which has been shown to have greater effect that one single processing modality.

Another way that the videos are nothing new is that video has been used in education for quite awhile. I remember showing Bill Nye the Science Guy and Nova on occasion when I taught science. Now these cases lacked the learner control that the Khan videos make possible and weren’t available on demand for students to access 24/7, so in this way Khan makes an improvement.

What makes Khan unique is that it has a diagnostic system of “quizzing” students to see if they’ve learned the material that is much more responsive than a teacher could be on his or her own without the help of a pre-set system.

The question is whether learn through a passive model – > quiz, learn – > quiz, etc. is a complete curriculum? I expect not. I think what makes math exciting is its application to solve problems. So if students never work together and “use” math, it would be a shame. Students at the end of the day may be able to complete the steps to solving a problem but it won’t be of any real-world and enduring value since the application won’t have been a part of the instruction.

Another reason that many educators object is that for years there has been a shift to believing that the “sage on the stage” approach to learning is not individualized enough and is based on the idea that learning is a simple transfer of information from the expert to the novice. The Khan Academy, if used by itself, simply replaces one “sage” with another.

Here’s my response to Khan: I think that Sal Khan is a well-intentioned, altruistic, and talented individual who wants to solve a problem and wants to make that solution free and available to educators. His videos and system are a good idea but could use tweaking and revising. Rather than immediately building out his library to make as many videos as possible, some work should be done to assess whether the content and structure of the video are presented in as pedagogically sound way as possible (the video above makes such a good point about science instruction. A similar investigation into what video structures are effective in math would be interesting – are real-world examples helpful? if so, should you begin with a real-world problem and teach the math as a solution? should you teach the math first and then show its application? is the black background with neon as effective as other contrasts and capture formats? Finally, those with power such as Bill Gates should dial back their praise. Khan deserves praise, but I think even he shies away from the idea that this is the panacea solution to education’s woes.

I am going to head over there …. when I hve a little more time. I’ll have to see about that “visual and auditory” aspect — for students who “need to see it to understand it,” often seeing scrawling the symbols out doesn’t do the trick… they need to see the why part. If they still don’t connect the symbols of fractions with the concept of parts and wholes, then seeing the procedures doesn’t even bring up those cognitive conflicts (especially if they’re doing more advanced algebra, because they’ve passed enough quizzes because they’ve learned procedures).

I don’t know why I never thought of going to Khan and pulling up something that I’ve forgotten — which is quite a bit 😉 Back when I was taking Calculus in 1977, we coined “physics teacher syndrome” as the problem when a nice teacher knew the topic so well that s/he couldn’t grasp how anybody couldn’t already understand it… when a prof substituting in a basic class here explained that, oh, it was either absolute value or an expression as numerators of fraction, was a “vinculum,” I was grateful for the regular teacher’s use of more mundane vocabulary…

Just to answer that question – students were allowed to do whatever they liked with the video: stop, rewind, play – and watch it as many times as they liked (or go make a cup of tea instead of watching it). They were only allowed to proceed to the next page after the video had ended.

I teach physics and believe me, the misconceptions of students are exceedingly difficult to overcome. When I started teaching physics I lectured well and showed my students how to solve problems step by excruciating step. I assigned more problems, more homework, pulled out the best videos on the subject; I did everything in my limited arsenal to destroy those pesky barriers to learning. Nothing really helped until I started teaching using Physics Modeling Instruction.

Now my students construct models of physical phenomena and apply ideas they have come up with on their own that are based on experimentation rather then previously held (and often incorrect) ideas. They solve problems with little or no assistance using equations as well as graphs, force diagrams, and charts. I lecture infrequently and my students’ scores on the required district tests are far and above those sitting in a more traditional physics setting. There’s more information at modeling.asu.edu

Derek’s video is enlightening and at the same time presents information that is expected. The only difference between a video and a lecture is that the video can have higher entertainment value and can be replayed. It is still, however, a lecture at its core. Derek identified a small addition to videos that can improve student learning. He’s to be commended for his work and thanked for bringing it to our attention. I found it to be extremely interesting as well as somewhat disturbing.

So… what’s disturbing?

Derek – thanks for that clarification.

Redpony – Every dissertation has to be narrowly focused around one question – they are supposed to advance and contribute to the field in one specific area. I’m sure Derek has a lot of material that didn’t make it into the dissertation (or the video.)

I really appreciate your comment since you are out there teaching physics trying to figure this out. Some of the most ardent constructivist teachers I know are reformed instructionists. I’ve heard the most interesting confessions about midnight epiphanies when they realize that perfecting the ninth verse of their long division song won’t help kids learn. (By the way, I added HTML to your comment so the ASU Modeling Instruction Program link is live. Hope that’s OK.)

And Sue – sorry I haven’t been responding to you, but I do appreciate you keeping the conversation going! Thanks.

@SueJ what you call “physics teacher syndrome” I know as the ‘paradox of expertise’.

Whatever you call it, it’s my belief that being aware of this problem is enough to allow one to adopt a ‘sponge learning mode’, by which I mean a deliberate attempt to ignore preconceptions when being taught something new. For this to be effective, however, one does have to recognise that one is in a potential learning scenario so as to ‘trigger the sponge’, as it were.

Hmmm…. how does one “ignore” a “preconception?”

How does one discern between background knowledge and preconception?

“Knowing” that adding makes things bigger is often what’s behind some of my students’ struggles with understanding negative numbers. Unless I challenge that preconception directly, they will “ignore” their preconceptions, and follow directions, and subtract instead of adding. However, since it makes no sense in that model, they tend to forget it. I also suspect that this “the rules can change” experience means that when it’s time to learn subtracting integers, and they’re told “change subtraction to addition” (of the opposite, but that gets dropped), they figure ‘okay! that’s consistent! When I thought we were going to add, we subtracted. Now … I’ll add.” Confusion ensues and even if they manage to structure a way of getting things right on *that* test, when they’re supposed to integrate that into further algebra such as 3 – (x – 7). However, they accept this confusion as “what math is all about until you’re not forced to take it any more.”

(take out the “and” in the penultimate sentence above… Confusion ensues even if htey manage…)

I had a long chat with a biking buddy yesterday that reminded me that educated people have absolutely *no* idea just how little the regular folks in the world think about the numbers in their lives.

This post shows what has been wrong, and continues to be wrong about academic thought in education. This “it has to always be done this way and never the other way” is so old and so wrong. Whole Language vs Phonics – it can’t be both right? Memorizing math facts, not memorizing any math facts – no middle ground right?

I think the Khan videos are a wonderful supplement to a curriculum and provides opportunities for students to get extra/different help with various concepts. I provide my students with classroom instruction, math DVD’s, math websites, the Khan videos, tutoring, textbooks, manipulatives, etc hoping that one of these resources will be helpful. I refuse to just try one system. I just plain old refuse.

This article is the worst kind of useless. The fact is that Khan academy *works* where classroom instruction alone doesn’t. When I left high school I had something like a pre-algebra education in math. I had *passed* algebra and trigonometry classes in high school, but only by rote memorization of the mechanics of problems for that particular test, and I had *no* conceptual understanding of the why or how I was solving these problems. When I left high school I retaught myself everything through calculus using exclusively khan academy. Suddenly, I could pause and replay the teacher, I didn’t have to be introduced to confusing new notation in text form (which was one of the bigger stumbling blocks for me in traditional instruction) and I didn’t have to worry about holding up the rest of the class if I had a question. Khan academy allowed me to get the background i needed to become an engineering undergraduate student, and it was orders of magnitude more effective then any instruction I’ve ever gotten. This article reads to me like someone who is worried that sitting in a classroom and lecturing to 20 or 30 vacant stares is no longer going to be an acceptable way to teach upcoming generations. Khan is living proof that in many cases being educated on how to educate makes you a worse teacher, not a better one. Teachers should be experts in the sub

The Khan Academy worked for you. If that helps you become a good engineer, excellent!

Do you think that it works for everybody? If you believe that it *could* work for anybody, if they were simply focused and determined as you were, is exactly the problem I have with portraying Khan’s videos as *the* cure for the problems we have teaching people math. It labels those who don’t learn the math as lacking in character and justifies writing scads of people off as deserving the educational and economic limitations this lack of numeracy & education places on their lives.

Sylvia–thanks for fixing the tag on the link. I didn’t think about that.

Sue–What I find ‘disturbing’ about Muller’s research is that students seemed to be more confident in their WRONG answers after viewing the videos. That makes me wonder how many of MY students left my class quite confident in something that was not correct.

I realize that Muller’s research isn’t a panacea but it sure makes me hope others follow up on it with more.

As far as Khan Academy goes, I’m sure it has a place in education. I just hope administrators don’t start thinking it’s the answer to all education woes. No doubt it’s very useful for some learners and not for others. I do wonder, though, how many students watch those videos and walk away more confident in their own incorrect knowledge.

The problem is not the concept of “flipping” the classroom, but rather the nature and quality of the recorded instruction (and, I would argue, of ANY instruction). If a teacher can create instructional videos that highlight and then challenge misconceptions, and provide them for students anytime/anywhere, this will indeed free up more face-to-face time for differentiation, clarification, and deeper investigation. We shouldn’t be afraid of flipping classrooms. We should be afraid of this kind of direct instruction wherever it is — in a video or “live and in person every day, period 3.”

@Jeremy – I think actually your situation proves the opposite of what you are saying. You learned math when you had a reason to learn math. It just proves the old saying — when the student is ready, a teacher appears.

But to assume that your experience indicates anything about anyone else’s learning or about all teachers seems unlikely.

I have an odd question that’s not totally related to Kahn but which I have frequently wondered about… I understand that most people don’t learn very well by reading/listenng, but some do. I’m one of the ones that do and now that a lot of the instructional material on the net is in video form (as Kahn is) I am considerably frustrated by being forced to watch repudiative slow videos because the site offers no textbook-style alternative (it’s hard to speed-read a video!).

I know that those who have significant trouble with the material get most attention (and I understand why it will always be that way), which I guess is why I don’t see this discussed, but is your ‘perfect world’ one where all instruction is available in multiple formats and multiple styles (including boring ones!) or one where everything is multimedia and interactive? It seems very ,uch from discussions that the multimedia/interactive is what people are aiming for and what’s happening in the real world.

As your classrooms and teaching styles change, how do you deal with the few students who do worse with the new styles? Perhaps not worse in terms of test scores, but kids who disengage and become insanely frustrated with the different style.

(I know I am doing a crappy job here of owning my privilege in being a learner who excels with textbook/lecture learning and I apologize. I know there are always a tiny proportion of people who are worse off after any change and I am not meaning to imply this is a reason not to change. I fully understand and agree that textbook/lecture learning is not ideal for the vast majority of people and therefore shouldn’t be the dominant school system method of teaching, I just happen to be in that small minority where it is best for me. It gets frustrating and I’ve never really had anybody interested in discussing it).

I’m tempted to speculate on the nature of the ‘privilege’ of learning well from lecture & text — while different people are wired differently, I believe it’s possible to figure out how to learn from texts and lectures, if you understand what the symbols stand for. WHether we should make teaching stuff that grabs attention and entertains vs. teaching students to distill knowledge themselves is another matter for discussion.

I don’t think anybody is out to eliminate lectures or texts; the discussion is about what kinds of videos are effective at teaching… I have seen many discussions about the disadvantages and frustrations of students being held back as teachers teach to the “lowest common denominator,” but it just doesn’t happen to be what’s being talked about here.

If I were forced to watch videos, especially about stuff I already understood, I’d smuggle in a text to ponder — or find some more challenging way to engage in the content as the video rolled (generate better examples than they had, for instance, or experiment with exactly how much more efficiently the concept could be stated with a few symbols).

And if those videos were repudiative then I’d want to explore what they were repudiating. If you meant “repetitive” — I’d smuggle in a dictionary and learn words like repudiative and sprinkle my comments with them.

While I certainly appreciate the discourse and the intellectual exercise granted by Derek’s and Sylvia’s articles, it feels a little like philosophizing about them number of teeth in a horses mouth. Put more simply, the proof of the pudding in the eating. I didn’t see either address whether or not students who participate in the Khan Academy or similar programs actually learn? Since they go through the mastery process it seems to me that something must be going on. I would still like to see the data. If the flipped classroom helps students learn the the model whether constructivist or instructionalist is immaterial.

This is much like the argument often found in gyms. Hard core bodybuilders will swear by free weights as the best way to build muscle and will eschew the machines all together. The reality is that people do get in shape, build muscle and increase strength using the machines as well. Whether one is faster or “better” I’m sure depends on a number of factors including the client (student). So likewise, if students and possibly more students can learn using the Khan Academy, flipped classroom approach, then that’s one more arrow in our quiver to use to nail down this process we call education. For now let’s gather the data and look at the results. Let’s open the horse’s mouth and count.

I have not seen any mention in this thread whether there is something to the KA form factor–short targeted, granularly accessible, reviewable vidoes–that is different from the traditional learning paradigm. Mostly, the issue is avoided by discussing preferred alternative teaching methods such as “modeling” which dispense–in whole or part–with the instructor paradigm. There is an assumption–that a good video, a good text, and a good class intructor are essentially equivalant. But anyone who has tried to “read into” a new math area already suspects that math textbooks–even good ones–are difficult to work through. It seems the geometric and conceptually entwined nature of math makes it harder to read about than History or Literature. I suspect KA type instruction will prove to be revolutionary, and modeling will be a fad. What is interesting is that the question–unlike phonics v whole language–will not be decided within academe–as students will have outside choices online.

Welp, it’s not avoiding a topic to happen to be talking about something else…

Wouldn’t the geometric and conceptually entwined nature of math make fragmented, granular videos *less* appropriate for instruction? Taking grains here and there helps reinforce “math is a mess of procedures to memorize and hope you pass the test so yo8u can forget them again” paradigm.

I just started looking at the video for figuring out averages, since I have a student looking for resources to review that.

It starts out with a black screen and a guy doin’ the lecture thing… telling me that hey, I might use the word average non-mathematically, saying “the average voter wants…” or “the average student would like to leave early.” He doesn’t go into what teh words mean there, perhaps because he really should have said “typical,” not average… that meaning has nothing to do with the concept of “average.”

How does he explain what average means? Well he at least mentions once, in passing, that the average is between those numbers. (Not between the highest and the lowest, just ‘between the numbers,” even though it’s *not* between most of the pairs of numbers.) Then he says that we can “kind of” use 7.25 to *represent* that set of numbers.

Huh?

What does that mean?

Then, when presenting what he calls a harder problem — you’ve got four scores, you figured out the average, now what do you have to do get to have a higher average — he tosses in things like “you know that that the sum of the first four scores is 4 x 84” — I do? And just how do I know that?

Oh, because of course, I really already know this. Then why do I need this video? I mean, he could have said “the sum of the first four scores is 336 — because you just added them!” … or, perhaps, given that missing explanation of what algebra means. NOpe.

“You sum up the first four exams here” — oh. Except he’s pointing at multiplication. I thought sum meant add!

He solves for x by multiplying by five… well, I hope you’ve already learned that. He’ll tell me that the dot means times, though.

The current comments indicate a *lot* of still-confused people.

Modeling is not just a fad, Paul. It is the way scientists learn science. You might want to read the research before making such sweeping remarks. I teach physics using Modeling instruction and it is amazing. Much better than KA, in my opinion. KA may be a good resource that students can use for review but in my experience videos do not help students comprehend physics any more than do lectures. I watched some of the physics videos and was sorely disappointed with not only the content, but the presentation, as well. The videos are not what I would consider top quality instruction.

@REdpony, You make the judgment that KA videos are not “top quality instruction”. interesting opinion but the the proof of the pudding is in the eating. Let’s look at the results. Are students learning and retaining the information? Are they demonstrating knowledge? Are they passing tests (although I don’t think it’s the best measure if it’s one we are currently using then let’s measure)?

So far all of the negative response to KA is a lot of hand wringing and supposing. I say let’s do the research and look at the results.

Look at the comments and the number of people saying “please explain.”

Look at teh videos and compare that to what we already *know* about effective teaching. Does it match? Not particularly.

Are students learning and retaining information? What information is in the videos to retain? Is it what we want to be teaching? (See the previous post for the details.)

Whenever some ones says, “we already know…” That sends up a red flag; especially when it comes to education. The one thing we know it that a variety of things, “work” and students learn in a variety of ways. KA is probably simply one other way. So I still propose the scientific method. Let’s see how students (like the ones in Cali and Ari) do when using KA and compare to traditional methods. Let’s see what the data says. Who knows, maybe there is still room to add to “what we know is effective teaching” and more importantly, effective learning.

Welp, yes, we’ve covered that — that yes, this will work for some students.

I’m honestly not just being an impulsive dismisser of this. However, I can state with some confidence that there’s a few tons of research that says that when people learn anything, including mathematical procedures, it works better when there are connections to things they already understand.

There are an *awful* lot of people who have bad experiences with math. I would be hesitant to inflict “research” on them with instructional materials that “introduce” averages by calling multiplication “summing up” and saying that 7.25 “represents” 2,3,5 and 20.

… HOWEVER.

I do absolutely think it is worth researching the use of video instruction. We *do,* too easily, throw babies out with bath water; it would be a mistake to dismiss this approach. If we want to see if video instruction works, let’s make some really good, well-edited videos. Camtasia is callingme!

Research has already proven that Modeling instruction and other student-centered constructivist teaching methods are much better at teaching physics than traditional methods (i.e. Lectures). I do not understand why others dismiss so easily 20+ years of research on learning physics by top universities such as Harvard, U Washington, Arizona State, and North Carolina (there are many others, too). I am not saying that KA doesn’t have a place in education, but I do not see it as the panacea that others seem to. I want my students to learn by doing physics, not by watching equations appear on a black screen while a disembodied voice explains. Very few students can actually learn and retain information presented in this manner.

From my experience as mom of 3 kids – ages 9, 11 and 15 – and knowledge of math instruction in wide variety of schools, both public and private, in the greater Boston area (city of Boston and multiple suburbs), I completely disagree with the statement that “… the teaching theory underlying most of American math education is instructionism, or direct instruction… Most American textbooks use this model, and most American teachers follow a textbook.”

Almost every school district and private school that I’m aware of in this area use a math curriculum/textbook that is constructivist (and decidedly NOT traditional or “direct instruction”). I’ve seen a LOT of dissatisfaction with this type of teaching from both parents and students — both from students who dislike math and feel that they’re “bad” at it, as well as those who like math and feel they’re “good” at it (and who already have a good sense of numeracy so commonly feel bored/completely unchallenged with constructivist math instruction in elementary school). I feel there needs to be a better balance between the two types of teaching and sense that Khan Academy fills a void of direct math instruction that’s missing in the current state of American math education.