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. As president of Generation YES, 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.
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.
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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.