Seems to me that constructivist education is already “in the water supply” of Less Wrong. My tweet-length definition of constructivism would be “teaching students so that they construct mental gears-level models of the subject, as opposed to just memorizing the teachers’ passwords”; and I assume most people here would agree that this is the right thing to do.
My question is how much this is a fair characterization—or perhaps it is abstracting away some very important stuff—and what are the best arguments against constructivism. Because, per Litany of Tarski, if constructivism is wrong, I want to believe it is wrong; although frankly I do not expect it to be fundamentally wrong, more like constructivists having some typical blind spots, in which case I want to know where they are.
The motivation for the question is to help a (constructivist) teacher do their job well, or to help a teacher unfamiliar with constructivism learn to do their job better. Therefore if the objection is e.g. that Piaget was historically wrong about some technical detail, the importance of this objection depends on how likely today someone is to make a mistake in designing their lessons because of that fact.
Here are my attempts to steelman the opposition to constructivism:
Excesses done in the name of constructivism, such as spending four years letting kids discover how addition of natural numbers works. (Either because according to the extreme interpretation, any guidance from the teacher would be a sin against the child’s independent construction of mental models; or because there is always one more “weird trick” that according to internet some natives used to add two numbers, and because having more models is always better, it is absolutely necessary to also teach kids this trick. As a result, 10 years old kids know dozen different ways how to add 15+8, each of those takes them five minutes, they will make a mistake in half of them, and they haven’t heard about multiplication yet.) These are rare extremes, and are not what a typical constructivist believes, but in a debate with an opponent of constructivism expect to get an example or two.
Trying to find a perfect model to teach may lead to unnecessary abstraction introduced too soon, and then having kids (and their parents) stumble over the things that were supposed to help them. For example, we all know that base-10 is arbitrary, but is it a good idea to introduce little kids to binary numbers before they have sufficiently mastered the decimal integers? Will it help them understand more deeply the logic behind the decimals, or will it just confuse them? Similarly, perhaps set theory is important at universities and some consider it a fundament of all math, but is it helpful to introduce the concept of a “set” at elementary school? (How specifically is “a set of two apples” better than “two apples”? Isn’t this also a teacher’s password on a meta level?)
Students are different. When you tell them facts, some of them will make the correct mental models immediately (or later, when they are reflecting on the lesson at home), others need more help. It is a tradeoff, where the time you spend helping the slower students form better mental models, is the time where the faster students could have been learning new things. The role of the school is not just to teach, but also to separate the better students from the worse, and providing credentials to the former. By teaching the facts and leaving the students to make the mental models on their own, you simultaneously teach more facts and separate the talented from the non-talented.
Constructivism is more fragile, because it requires that those who teach your students in previous grades become constructivists, too. If a class spent five years memorizing the teachers’ passwords, and you try to give them a constructivist approach in the sixth grade, it’s going to be difficult. However, if a class spent five years constructing mental models, giving them a list of passwords in sixth grade is not more difficult than giving a list of passwords to anyone else. Therefore, even if constructivism is superior in theory (which has to be proved yet), in practice it loses in an educational system with mixed teachers. You get a bunch of complaining constructivists, and a bunch of traditional teachers saying “guys, if you can’t teach in a real class, why don’t you go find a different job?” (Constructivism is also fragile in a system where all teachers try to be constructivists, but some of them do it wrong. Again, imagine teaching the sixth grade after five years of someone else doing constructivism seriously wrong.)
Constructivism goes against our evolved nature. Apes learn by copying, not by understanding. Often the things you copy work, even if you don’t understand why. (Washing your hands helps even if you don’t understand germ theory.) We may not even have a gears-level model of many things yet. Traditional education is compatible with our instincts, both for teachers and students. Perhaps we should be less confident about our ability to go against our insticts and achieve superior results.
Helping the students develop mental models might be actively harmful. Maybe they lose the ability to develop mental models without external help. Maybe the lose the ability to revise their mental models (if they believe they were taught the right ones). Or they may develop a habit of premature optimization, like whenever they hear a new fact, they immediately have to fit it into some model, because they were (implicitly) taught that facts not belonging to models are wrong. Teaching everyone the same mental model creates a mental monoculture, and the actual progress depends on some people randomly developing a superior model, therefore making all education constructivist may eliminate progress. (Also, if there is a disagreement about which mental model is better, which one should be taught?)
This might not apply the constructivism proper. But one thing that bothered me a bit about more progressive methods when I worked as a teacher was how they often became tools of manipulation. By creating the illusion of control and freedom I could get the students to reveal more of themselves, and that gave me more knowledge do use to figure out how to make them submit to the mandated curriculum. This might just be a problem if you are ethically oversensitive. But I prefer facilitating learning in environments where I do not have the power or any reason to force a certain outcome on the learner. And in those situations constructivism can be quite useful, as can drills.
In encouraging people to make models that fit their particular psychology me make them use models that are rare or seldom threaded. This might give them illusory sense of uniqueness, they might not benefit or expect to benefit from the knowledge of others.
Coming up with their own models they might have unique problems that if we encourage to take their own constructions seriously they will have problems that we don’t have answers to. A geology teacher teaching that earth is round in some capacity needs to be able to deal with flat-earther talking-pint and fumble points. taking a positivde spin on this is “they make their own reserach questions” but making a bad spin is “we spend more time of doing it wrong in X ways rather than the known working ways”
I’ve been thinking about this for a while. I spent the last 10 years teaching music lessons to children and a few adults. I haven’t studied education formally, but have read some of the literature that I thought would be helpful.
My feeling at this point is that constructivism is a useful thing to do, but that it takes a lot of investment on the part of the teacher to pull off at all. It also is slower than the “imitative” approach to education. It is obvious to me that it would overall harm, not help, a person to win a Nobel prize in physics by requiring them to reconstruct all of physics on their own first. But mixing in experiences of figuring out chunks of knowledge for oneself is more satisfying and useful than pure imitation.
Teaching still relies a lot on the teacher’s intuition and personal relationship with the student. It’s not enough to implement constructivism. To use it effectively, you need to implement it well, and on the correct problem area. Insufficient constructivism is not the cause of all learning challenges or experiences of student boredom. Figuring out the right blend is just one of several major problems in the overall challenge of optimizing a student’s education.
So to refine your original post, I recommend shifting more toward questions like:
When is constructivism the right approach for a particular student or subject, and why?
What proportion of constructivism vs. imitative approaches is ideal?
Can we know when constructivism is the key issue for a particular educational problem?
If one student is taught addition via constructivism, and the other by imitation, is there really a fundamental difference in their understanding of addition? Or is the difference in the amount of skill they gain in two different approaches to learning how to learn, rather than in the amount of skill they gain in the object-level topic?
How can we implement constructivist approaches for new topics where it’s not normally applied?
Maybe the constructivist approach works better for subjects with long inferential distances, such as math. Any individual fact is easier to memorize than to understand, in short term. The problem is, with memorization you are building a tower that will collapse under its own weight. Also, a misremembered fact feels exactly the same as a correctly remembered one, so there is no self-check.
I think you can’t use constructivism to learn what is the capital of France.
My first approximation for “when to use constructivist approach” would be like:
if there actually is a gears-level model;
if it is important to remember for more than one week (and writing it down is not an option);
especially if learning new skills depends on getting this one thing right.
It is hard for me to imagine someone not having a gears-level model of addition. But I guess someone who doesn’t, is at risk of making some stupid mistake in future (like, after returning from summer vacation, not having practiced addition for two months; or maybe a few years after finishing school), such as not aligning two numbers correctly, so that 111 + 22 = 331, or maybe with decimals 11.1 + 22 = 133 or 13.3, or something like that. Or would get confused when seeing an unusually written problem, such as 13½ + 24½.
My impression was that with constructivism, the question is not whether the student ultimately achieves a gears-level model, but whether they discover (“construct”) it for themselves.
I agree it’s hard to imagine addition without a gears level model.
Yes, but there can be a lot of nudging towards the discovery.
Like, if you want kids to find out that “a + b = b + a”, you give them hundred pairs of problems like “2 + 7 = ?; 7 + 2 = ?”. That achieves the goal more reliably and more quickly than merely giving them hundred random addition problems.
This assumes that educational paradigms matter a great deal. I would expect that skill around effectively projecting authority in front of children and classroom management are more important for the teacher to do their job well.
I’m sure constructivism works for some people some of the time but for a public mass education system it doesn’t seem scalable. One of my goto bloggers on education has written a lot against this idea, this is one of his posts. https://gregashman.wordpress.com/2020/05/23/whatever-happened-to-constructivism/comment-page-1/
Not sure if I am reading it correctly, but the main argument seems to be that “pure discovery” is inefficient compared to e.g. “guided discovery”… which is something that every constructivist teacher in my bubble would definitely agree with, and they would probably call the “pure discovery” a strawman of constructivism (the first bullet point in my question). Now the question is how much my bubble is typical among educators.
Making a note that whenever I mention constructivism, I should emphasize that I am talking about the “guided discovery” version.
In writing fiction, advice often given to new writers is “show, don’t tell.”
I’ve long thought that this advice is overly simplistic, or possibly incomplete. In the end, everything a writer writes is telling something.
“Mark was sad.”—Telling Mark’s emotion
“A tear fell down Mark’s cheek.” Showing Mark’s emotion, by telling what’s going on.
Without giving the matter an abundance of thought, Constructivism seems similar to me. Everything a teacher teaches is some kind of password. And even when the teacher is trying to get the student to construct mental models for themselves, the teacher is telling something.
So the question isn’t whether or not constructivism is superior to memorization, any more than showing is superior to telling in writing. Both are necessary; the question is at what level of abstraction each is best used.
My personal opinion is that routine skills—the kind of thing you have to do all the time, over and over again, like hammering a nail in carpentry or adding two number together in math—should be memorized, at least at first. Not because that’s inherently superior in any sense, but because it enables other, more complicated tasks to proceed faster. Those more complicated tasks are then good candidates for constructivism.
Another way to say this is that gears are made of atoms. Since education has to start somewhere, might as well start by memorizing gears so they can be constructed into complex machinery.
Then, later, you can go back and explain that gears are not the fundamental unit of matter, or ten is not a fundamental base, or nails aren’t the only way to secure two pieces of wood together.
Wait, wouldn’t that mean explaining difficult stuff to kids who never previously had the experience of understanding the simple stuff? And who were until now actively trained to memorize instead of understanding?
Also, what is the right moment to change the strategy? Every task is simple or complicated relative to something else. So when one teacher says “okay, this task is complicated, we should slow down”, another would say “nah, just memorize this too, the next chapter is the really complicated stuff”. Multiplication is complicated relative to addition, but simple relative to quadratic equations—should we memorize multiplication first or not?
If you change the strategy too early, like in the first or the second grade, did it actually save that much time in long run? If you change the strategy after four or more years, will the kids be flexible enough to overcome the old habits?
All good questions. My gut response is that these exact problems—which level to memorize vs. construct mental models—are exactly what I was attempting to communicate. You’re always telling something, even when you’re showing.
A more concrete response is that every level is constructed atop the one beneath it; multiplication is repeated addition, quadratic equations require multiplication, etc.
So for children, pick something near the bottom, something relevant to their actual, real, Newtonian lives (so addition, because adding things happens all the time, and not set theory, which is arguably more fundamental but further removed from the normal life of a child) and have them memorize it. Then construct the thing on top of it.
Once they’ve constructed the thing on top of whatever they’ve memorized, have them memorize that and construct the next layer. And then keep going.
A basic math course might then look like this:
Memorize addition, then
Construct multiplication on top of addition, then
Memorize multiplication, then
Construct exponentiation on top of multiplication, then
Memorize exponentiation, then
Construct quadratic equations on top of exponentiations, then
...
So the repeated pattern is:
Memorize the lower level skill, then
Construct the higher level skill, then
Repeat, but with the higher level skill taking the place of the lower level skill
This makes me think of climbing a ladder, where first you reach for a rung with your hands, then eventually step on the same rung as you climb higher.
This can also work going “down” as well to get to more fundamental levels:
Memorize Newtonian physics, then
Construct relativity (by exploring where Newtonian physics fails, or where the simplification/approximation reaches its limits), then
Memorize relativity, then
Construct quantum mechanics (similar to above, but with different limits/failures), then
Memorize quantum mechanics, then
Construct string theory or something...?
In the end, you’ve got to start somewhere, with something, knowing that you won’t be able to really explain it because doing so requires knowledge the student doesn’t yet possess (by definition). So pick something simple and get started up the ladder, knowing that at some point you’ll want to come back around and construct the original thing that was taught.
I find myself thinking “yes, obviously”, but at the same time there seems to be something we disagree about, so what it is exactly? I think it might be about what exactly the teacher is telling; to put it simply, at the extreme, there are two teaching styles:
build the models carefully, check that the students have the right models;
just say a bunch of true facts in random order without feedback, the smart kids will sort it out, and the rest of them… I guess they were not talented enough, “the camel has two humps”, etc.
And “constructivism” as I use it, kinda means: do more of the former and less of the latter. But there is of course more to it, like provide enough time for the kids to build and debug those models, show how the existing models relate to the new things, etc.
To get more specific, I believe that kids should definitely have a model of what addition is, on the level of “adding five and eight is like having five apples in one hand and eight apples in the other hand, how many total apples do you have?”. This model is enough to derive e.g. that 5+8 = 8+5; you just need to notice this first on a few specific cases, and then you can justify it using the model, like “if I am already holding the apples in my hands, and then I like switch my hands, the number of apples remains the same”.
Without a model, only having memorized lots of additions, if you ask whether addition is commutative, the answer will be like “well, I don’t remember any example to the contrary, so… maybe?”
Maybe what I am trying to say could be put like: of course you are always telling something, that is inevitable, but it is better to increase the parts where the kids themselves can connect the dots. And then we can go to specific techniques how to prepare the dots so that they are easy to connect.
100% agree. I suspect that set theory was just high-status at some moment and people couldn’t resist “hey, if we can make the kids use the word ‘set’, we will make them high-status mathematicians”. Of course it does not work that way; the things that kids do with “sets” at elementary school have nothing in common with the set theory as usual. So, drop the “sets” entirely, IMHO.
More general, we need to distinguish between something being simple in the… vulgar sense, such as addition being simpler (less work, easier to understand for a child) than multiplication, and being more fundamental from some perspective, like quarks are more simple (easier to define for a scientist) than apples. The former is a good heuristic for elementary schools, the latter is not.
Doesn’t this means that all skills—except for the first one—are explained first and memorized later? Then why make the exception for the first one?
Technically, addition of integers is also a sequence of skills. There is a difference between 2+3 (counting on fingers or memorizing), 8+5 (thinking how it wraps across ten: 8+2+3), and 7416+2872 (arranging vertically, then adding the digits starting from the last one).
(By coincidence, I recently made this tool to teach/train addition and subtraction; don’t mind the language, just click on the bullet points in order, it is self-explanatory. Needs JavaScript enabled.)
Maybe I should say it explicitly that I am not opposed to training/memorization, that I agree that it is super necessary, and I think maybe we should use computers at school and just spend the first 5 minutes of the lesson doing a quick “spaced repetition” exercise at the beginning of each lesson. I just think that the proper moment for memorization is after the things were properly understood.
...at some moment in future I would like to write a few articles on this, including screenshots from an actual constructivist math textbook that I already have at home, so that we don’t have to discuss this in abstract. The reason is that there were many shitty “constructivist” textbooks published in USA, and I want to make it clear that I am definitely not defending those ones, not even the ideas they were built upon, which to me feel like a strawman of the original idea. I suspect that if I showed you the actual textbook, you would see nothing wrong with it (at least compared to the usual textbooks). But I always procrastinate a lot with writing articles.
That “something” is the knowledge kids already have (hopefully) when they go to school. Mathematics is not a separate magisterium; it is an abstraction built upon things from everyday life. The primitive pre-mathematical knowledge is like “realizing that two apples are more than one”; you build on that.
As to the disagreement you mention, I think I’m starting from the position of the child being taught being a complete blank slate, which is obviously inaccurate when taken literally.
A blank slate would have to be told things first, before any models could be built, because you need a nonzero amount of knowledge about the world to build any models at all.
More realistically (and as you said), a child should be expected to be a functional human child by the time they get to a teacher (and math is not a separate magisterium), so starting with a model isn’t impossible (because the child already possesses facts to build the model with). I do believe that the memorize → model → memorize → model loop is how learning happens; the question is where in that loop the teacher meets the student.
Completely agree with you on Set Theory.
The exception is made because I assumed a human was starting with a complete blank slate, which is not literally true as I agreed above.
Perhaps a better example than addition would be the first time someone tries to learn a foreign language; I would argue that some vocabulary has to be memorized first, because that’s the foundation upon which everything else rests (you can’t start by trying to teach grammar, for instance).
I look forward to reading them!
I tried it out. Simple but nice! One thing I noticed was that for the answers that were “10”, you could just leave them as “1″ without it being marked as either right or wrong (green or red). Not sure if that’s a feature or a bug.
Wow, glad we came to an agreement, I actually didn’t expect that.
That’s on purpose, glad you noticed! Green = the correct answer. Yellow = not the correct answer, but a prefix of it (that includes an empty string). Red = neither the correct answer, nor a prefix of it.
Like, if the correct answer is “42”, then “4″ is yellow, because for all I know maybe you are halfway to writing the correct answer, so I don’t want to scare you needlessly. (Though maybe I should later update it to red when you leave the text field… and update back to yellow when you return? Nah, sounds like too much work.)
I like coming to agreements too!
Seems to happen to me here a lot more often than IRL.
The only problem with the box remaining yellow that I see is that it conveys partial information, because it turns red if the digit is wrong.
In other words, if a student wanted to fill out the boxes by brute force, without actually doing any math, just by trying numbers, they’d be able to get to multi-digit answers by trying out 1-9 until they found the number that didn’t cause the box to turn red, then moving on to the next digit.
Off the top of my head, the simple way to fix it would be to do the correctness check after focus leaves the box (triggered by leaving the box, as it were); that can apply to every box and ensures the student can’t brute-force the answer as above.
For some reason I procrastinate for months when trying to write articles, but can write an insanely long comment whenever I get angry about something. So here is a story about “constructivism” in education, as a Hacker News comment.
Give me a few more months, and I will probably rewrite it to a LW article, and then it will get like 5 karma total, heh.
I mean, what other response is possible when someone is wrong on the internet?
Either way, I’m looking forward to it.
Edit: after reading the comment, I feel like I have a better understanding of how we might’ve been talking past each other a bit. I do agree with your position.
Also, the history of education is a terrifying and depressing subject, in my experience.