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Why Minimal Guidance During Instruction Does Not Work (2006) [pdf] (usc.edu)
122 points by eBombzor 5 months ago | hide | past | web | favorite | 50 comments

After skimming the paper, I get the sense that there's this underlying assumption that the goal of education is to get students to learn specific facts, i.e. that a physics student should learn the formula that force equals mass times acceleration.

While there is of course lots of value in ensuring that students can learn specific facts and figures, the criticism (if I recall correctly) is that an approach that dictates specificity of recall as the end goal is not an approach that induces a lifelong love of learning. Most of the value in an experiential-based approach is not limited to the specificity of facts learned during the experiences, but in the cognitive linking of learning with the dynamism of the experience, which encourages independent growth.

This leads back to the philosophical question of whether our school systems should focus on graduating students who know specific knowledge (an approach that leads to standardized testing), or students who have become independently capable lifelong learners (a more holistic approach that tries to adapt to students' individual strengths). Which approach better prepares students for life after school, and equally important, which approach can prove to scale to the levels required for public education?

I've increasingly come to value mastering the basics.[0]

A quote from Michael Nielsen in his great article Augmenting Long-Term Memory (http://augmentingcognition.com/ltm.html) puts it very well:

"Over the years, I've often helped people learn technical subjects such as quantum mechanics. Over time you come to see patterns in how people get stuck. One common pattern is that people think they're getting stuck on esoteric, complex issues. But when you dig down it turns out they're having a hard time with basic notation and terminology."

[0] I suspect individuals who learn things relatively easily may undervalue that skill, because they typically were ahead in school.

It isn't just the basics that need mastering, but all of it [1]

Adding knowledge to your own knowledge graph requires existing context, without context, there is no place for new knowledge to attach.

Without master, eventually the new knowledge is simply remembered and not fully understood. The knowledge front is a fixed distance from the last fully mastered concept.

It is ok to go slow, who cares how fast shaky knowledge is consumed. You want a building that will stand, not one that was built on time.

[1] https://en.wikipedia.org/wiki/Mastery_learning

I also tutor/educate. I've fixed people in 1 week, what the education system couldn't accomplish in x years.

I just reach in, and remove the blocks. Once they get it, they are never the same again.

True story, somebody came up to me, who was having severe difficulty. They were studying for their GED (700+ hours in) and couldn't crack it. He asked for my help and I told him I was busy at the moment, but that I could give him 30 minutes in a couple of days - but that I would be happy to talk to him right now for 5-10 minutes. So I basically grilled him for a couple of minutes, then gave him a complete dump on what he needed to do in a highly-abbreviated manner. About a week later, I see him and tell him I'm free to sit with him again if he likes, to which he replies "No, thats not necessary - I passed. I followed your advice." :)

I get people calling me up after decades (for whom I only sat with for 1-3 hours) telling me about how much I changed their lives.


I also do "IQ" raising. Just being around me gives you about a 10-12 point jump. The longer the period, the more permanent the results. Takes about a couple of weeks to kick in. Takes about about 1-2 years for full raise (permanent, for life).

And shit, if you give me somebody smart-smart whose open & ethical? :) Like gas, on a fire.

My only real constraints are:

a) I don't put out anything dual-use. b) I don't hack people for harm or personal gain. c) I don't enable crimes, criminals, or unethical people.

It's not a either-or choice. You can certainly know a lot of facts while at the same time developing those skills. In fact, my guess would be that the two are related: learning a lot of facts teaches one to learn arbitrary things, even things that aren't that interesting. This, learning things not that interesting, is the skill.

My money says an education is about developing a certain set of habits. The base one is probably being able to sit down by yourself and memorize a bunch of arbitrary detail that you can recall on command, but that's just the base, maybe 1% of the habits you should get.

I don't know, though. We may just suck at managing public education, not implementing it. In that case, no discussion of underlying theory is ever going to amount to much more than noise.

Sherlock Holmes famously refused to learn things that would be useless for his job, to keep his mind tidily organized. He is a fictional character, but the idea that learning facts can be detrimental to someone's ability is interesting.

Look at yourself: browsing Hacker News and Twitter in the morning to achieve little by the end of the day.


He was also famously interested in all sorts of trivia (such as types of London clays, and types of cigar ashes) because it helped him with his job.

I think "neo-generalist" would be a good way to describe Holmes: https://hackernoon.com/the-neo-generalist-and-why-we-should-...

For engineers and developers, many of whom come from a STEM background, learning about the humanities can be an enriching experience and even help them design better systems. I'd highly recommend it to anyone who has the time -- and it doesn't have to mean learning random facts. There are a lot of good reading lists, podcasts, etc, that one can go through in their spare time.

It's an extremely relevant point. I don't know. You may be right!

There is another school of thought that truly interesting and transformative work comes from joining up things that naturally don't go together. In this school of thought, people who focus too narrowly only ever do "normal" science, as Kuhn taught. They never achieve breakthrough.

I suspect that both are valuable: skimming and remembering/assimilating random surface-level material from all over, and deep-diving into only those things that you feel matter.

As an official Old Programming Dude, and a self-taught coder/consultant, what I see in the dang kids of today is a lack of being able to manage both needs. They're either great at one or the other. Neither bunch amounts to much intellectually in terms of being interesting, but at least with the focused group they are able to do some really cool stuff.

Personally, I need more focus. I know this. (And I plan on finishing up 4 hours of coding and two essays by the end of the day, kid!)

ADD: One of the things I've noticed about focus is that when solving problems, I work best when I manage my focus. I'll put the problem on the monitor or on the wall where it's always visible. Then I'll go do something else. Anything. As long as I can see the problem, somehow-or-another my brain keeps working on it, even if I'm not aware of it. So for many, the problem may not be HN/Twitter, but the inability to control the subtle parts of their work environment that might be preventing them from working at a higher level.

>but the idea that learning facts can be detrimental to someone's ability is interesting.

It's interesting. I would say the issue with this idea is, is you can't predict how learning random information and combining that into new ideas will benefit you. The only reason this worked for Holmes was because his character was written that way, not because it necessarily has any merit. Maybe it reflects the thinking of the time, or Doyle's own beliefs on the matter.

Although I don't think there's a problem with being choosey, like maybe you don't care about the WW2 Pacific Theater, so why learn facts about that when your interest is the history of computers?

Just as an aside, the BBC version of the quote: https://youtu.be/OVXt8jBAGkc?t=87

For the last few years I took some time off to recover from a mild heavy metal poisoning, and used the time to learn learn learn. I was pretty lazy in childhood and now had a desire to learn, but also to prove to myself that I can study diligently for an extended period, which I never did (had to do) before.

So I took some 60 courses on edX and Coursera mostly (but also Youtube channels and some Khan academy for refreshing some basics, plus some reading) - at the time everything still was free. Most of the courses were "life-sciency", stuff I never had to deal with since school and never thought I would ever touch. More than one course in each chemistry, org. chem., bio-chem., physiology, anatomy, neuroscience, lots of statistics (again - had forgotten everything, this time I actually cared).

The more I learned the more I can relate to the Sherlock Holmes quote. It's no use. The deeper I went, the more courses I took, the more I forgot. Also, the more I felt that I don't have any practical use for any of it. It made me feel good, but society won't benefit one bit from my learning. Humans specialize, and most people who do good work in their place in society really don't benefit from cramming ever more stuff into their heads. At this point I would not even care if I met somebody who really didn't know about earth and sun, as long as they perform well in their actual job.

Part of it is that the more I learned the more I also learned that I still know nothing at all. I can take all the basic courses about medical topics, it does not help one bit to make any practical decisions- There is a high threshold before the knowledge is practically useful (you can't be 1/3rd of a doctor).

I have switched to mostly learning things related to my actual work now and stopped taking courses outside my field. It just felt increasingly futile. Sure, I now have quite good basics but it was a huge investment, if I hadn't had anything else to do during the time anyway it would have been quite a waste. I would not expect anything similar from larger society, let them specialize.

I now think trust is far more important. It's okay if people don't know much or anything about a lot of subjects - as long as they trust others who do. Things like for-profit health care lead to things like skepticism of the specialists. I too had plenty of doctors who tried to sell me useless stuff (network marketing!) and useless treatments when I tried to find someone for the problem I had. The missing trust is something rarely if ever mentioned in discussions about "anti-vaxxers". Even those who think they are stupid know nothing at all about the immune system and about how immunization actually works (edX has some good and detailed courses, and it's a lot and can be quite boring unless one really is into the subject) - the difference is not that they know anything but that they trust official authority. So the question is not how do you teach immune system fundamentals to skeptics, but how do you create trust that they don't have to? There are enough headlines to show that mistrust of the profit-driven system is not exactly completely stupid. That it shows up as anti-vaccination attitude is, I think actually secondary and incidental and not the actual root problem.

In a highly specialized society we need trust, and we don't need to want to teach everybody lots of everything. Food for thought - ignore the product this is about, it's just an example: https://medium.com/@kevin_ashton/what-coke-contains-221d4499... (see the reader-highlighted text at the end for the summary).

The issue I heard people saying was that not just love for learning was issue, but also recall in long term, lack of understanding and students freezing when they encountered just slightly different problem then the one they have been taught directly. E.g. the act of actively solving problem, having to try out different options and think makes you better able to derive formula again when you forget it or to use it in different contexts.

Also that students who are taught via direct instruction only (everything on the platter) freeze when they encounter non straightforward exercises. They are unused to initial uncertainty, doubt, assume that if first attempt failed then it means something bad and there is no point in continuing etc.

Obviously, in practice it is not and should not be exclusive dichotomy between the two. The teacher should be able to teach some parts of curriculum via direct instruction and others via minimal guidance depending on what is taught, where students struggle and looking for reasonable trade-off between the two.

    > [Direct instruction students] freeze when they 
    > encounter non straightforward exercises. They
    > are unused to initial uncertainty, doubt,
    > assume that if first attempt failed then it
    > means something bad and there is no point in
    > continuing etc.
I would argue that students "freezing up" when presented with new problems is _also_ a part of a solid educational approach. Careful guided instruction when introducing new topics is absolutely necessary. But at some point the student is going to need to practice "open-ended" problems.

It is sort of like weight lifting, to really develop one needs to reach the point of failure in a controlled way, regularly and repeatedly, BUT NOT ALL THE TIME.

What gives you that impression? I don't think any physics teacher would consider that their goal; rather, they would want the students to gain an understanding of the concepts of force, mass, acceleration and how they interact. Being able to easily recall the formula describing that relationship would be a side-effect.

I also read the abstract of this paper, and I didn't see any evidence of that underlying assumption.

We need both.

I as a tax payer am paying for kids to attend school because food is not free for everybody - somebody needs to work no matter what system you come up with. It is better a better investment for me to get you educated so you can support yourself, than to remove your staved to death body from the street.

There are some facts that almost everybody universally needs to know to support themselves: you need to know how to read, write and some basic arithmetic. F=MA is useful to enough jobs that it is best for everybody to know it even if whatever job you end up with doesn't need it.

The above is intentionally overblown to make the point.

Specificity of recall is not an end, but it is a necessary means to the end goal of education. We can only know that someone is an independent lifelong learner if we can tell that they have learned something; not only that, but that they have learned a lot of related, non-obvious facts, and can work with them. This dichotomy, between learning facts and learning concepts, is not an innate one and I wish that notion would go away.

And this leads to savvy kids at 14 asking the teacher " miss what's the learning experience".

It also leads to say Eng Lit students only reading the text's (or parts of the texts) on the test and not actively reading other related works in the cannon.

My knee-jerk instinct was to read this paper and take it as truth in the sense that it's "debunking a popular belief," much like the "types of learning" belief. However, this paper is 12 years old, so I was curious about whether there had been any responses. I found this blog entry, which seems to be relatively well-researched: http://bdld.blogspot.com/2011/06/five-years-later-review-of-...

Especially of note is the section that states:

> The paper relies heavily on Cognitive Load Theory, yet we have to realize that it is still a theory rather than a law.

The author also quotes Joel Michael to unpack some of the difficulties in studying educational methodology in general, which gave me some perspective. It's something I've thought about a lot since many Japanese people mourn the "yutori kyoiku" (https://en.wikipedia.org/wiki/Yutori_education) philosophy and I would often wonder which generations of students were a result of that system and which were simply molded by other social pressures.

I felt the paper's reliance on cognitive load theory was a bit problematic.

I'd have a different theory about the problems of "experiential learning" in a class room setting. I've taught myself a lot on my own and so the idea of exploratory learning always appealed to me. But all the experiential learning circumstances I've been exposed to just felt like tricks. The teacher has something in mind they want the student to learn, doesn't tell them exactly what that is, and still expects them to learn to exactly that thing, rather than some other things that the circumstances presented might teach the student.

Actually, this relates to a lot of contexts. One example is those terrible job interviews where the interviewer says something vague like "tell me about yourself" but expects to learn specific thing. Another is Role Playing Games (I'm an avid player of these). There's a classic problem where a dungeon master expects their players to explore a city and make a specific discovery with the clues given. That's typically much harder than the DM imagines and experienced DMs learn to just set-up situation and let whatever the players discover be what happens.

In general, I think a teacher or authority should say what they want in a learning or examination process, if they want a specific thing. Because if they don't say that, their idea of what a person would "naturally" discover through exploration might actually be wrong, indeed given the natural variation in people's cognition, it probably is going to be wrong some large portion of the time for many students.

Cognitive load theory is fairly straightforward. People are able to handle 7±2 items in their working (short-term) memory at any one time. This means reliably 5 items. Subtract one for standard normal distractors. Thus students should not be expected to have to juggle more than 4 items at any one time. The problem here is that what an item is for a student is different than what an item is for an instructor. This is due to "chunking" (technical term from psychology). For the instructor, discovering what an item is especially difficult, as they may not be aware of what they had to learn to know what they now know. Personally I'm all for minimal guidance, but it has to be closely aligned to learning outcomes, and the instructor needs to understand what an item means for the student, if they are to have a hope of introducing less than four things at a time.

CLT is a solid theory with an extremely good basis. However understanding how to apply it is hard given the knowledge difference between student and instructor.

Cognitive load theory is unfalsifiable and has numerous conceptual and methodological problems: https://edtechdev.wordpress.com/2009/11/16/cognitive-load-th...

There have been numerous refutations of this opinion piece: https://edtechdev.wordpress.com/2007/07/25/problem-based-lea...

As well as criticisms of their central underlying theory, cognitive load theory, which is unfalsifiable: https://edtechdev.wordpress.com/2009/11/16/cognitive-load-th...

And there is a wealth of counter-evidence on how constructivist inspired teaching techniques (like active learning, inquiry learning, problem-based learning) and technologies (like simulations, modeling tools, games) are more effective for student learning: https://edtechdev.wordpress.com/2015/04/03/evidence-for-vari...

Unfortunately, fans of the neo-traditional perspective presented in this article plastered it all over Wikipedia 10 years ago, overwriting and ignoring any contrary evidence and opinions, leading to today: https://edtechdev.wordpress.com/2007/12/26/an-argument-for-k...

Excellent. Thank you.

Do you know of a handy (ELI5) layperson's explanation for state of the art learning techniques?

As a parent of K-12 students, what should I be advocating?

As a life-long learner, what style of courseware should I seek out?

As a person who has taught & mentored 100s of peers how to code, informally, what's are some strategies and techniques I should consider?


A good friend teaches highschool physics using ASU's Modeling Instruction Program. He claims terrific results.


As a layperson, I can't make heads or tails of the online resources. What he describes, using the Socrates method in the classroom, sounds exciting. But I don't know enough to act on that tidbit.

By far the best summary of current learning research is available at the Bjork lab's website: https://bjorklab.psych.ucla.edu/research/

This blog post makes a strong case for Direct Instruction[1], with links to a recent meta-analysis: https://marginalrevolution.com/marginalrevolution/2018/02/di...

[1] https://en.wikipedia.org/wiki/Direct_instruction

Not OP (and I don't have much time), but as of about 5 years ago (last time I was spending time reading papers), the biggest thing I ran across was the spacing effect (which is different than spaced repetition). You may have heard of spaced repetition. Very quickly whether or not you remember something is a random variable with a vaguely gamma distribution. Your chance of remembering is a function of how long it was since you last looked at it. So if you wait 5 minutes, you might have a 90% recall chance, whereas if you wait 5 days, you might have 20% recall chance. The speed that the recall chance goes down is a function of how many times you have reviewed something and how much time has gone on between reviews. Better than trying to understand my terrible description, look up "forgetting curve".

Anyway, in spaced repetition, you usually try to time your reviews so that you have a 90% recall chance. This minimises the amount of time you have to do review. Also, initial remembering of something takes a lot of time, while reviewing something you already remember takes very little time, so you try to make sure that you review it before you have forgotten it. Usually at around 90% recall rate, after 12 reviews, you're many, many months between reviews, so it's insanely efficient.

However, there is a completely different thing called the "spacing effect". What that says is that if you forget something, and then relearn it, the forgetting curve will be much, much, much shallower than if you had done the spaced repetition (papers on spacing effect don't put it that way at all, so you're getting a big dose of my bias -- better to do some research on it. But I'll continue anyway since I think this is the easiest way to understand it).

What this means is that you can remember things much, much more easily if you forget them first and then relearn them. So in this school of thought (literally -- ha ha) you learn something, then you wait until you forget it, then you learn it again. In studies, this has been called the most profound learning effect every seen (pretty powerful words!)

So there are some other people who like the spacing effect thing and thought there must be a better way than randomly waiting until people forget stuff. So they came up with an idea called "interleaving". In interleaving, instead of learning one subject for and hour or two and concentrating on it until you understand it, you study something for 10 minutes, then study something else, and then something else, etc, etc, etc. After 6 or 7 goes of this, you circle back to the original thing and study it again. The idea is that by keeping distracting yourself, you have no way of remembering anything at all -- which speeds up the spacing effect. (Though, one might note that this starts to get into the realm of spaced repetition -- and I think it's not a coincidence).

Basically, in interleaving you try to mix everything up and make sure that you are constantly looking at something new. But, of course, this sucks. I mean it sucks enormously. You never feel comfortable. You can't remember anything. As soon as you think you are getting the hang of something, the topic changes and you freaking forget it all. It's hell.

Which has led this kind of thing to be referred to as "desirable difficulty". If the studies are to be believed, the potential is incredible. I'm not sure I believe the studies completely, but I've experimented on my own students (when I was teaching) and I was pretty impressed with the results I was getting. Still, really early days with this research, though and I wouldn't be surprised if someone replies to this message with something like, "Yeah, somebody found out in the last few years that it's a load of BS". Education and psychology are pretty damn hard to do studies on ;-)

Interesting! The authors state their findings in the conclusions:

> After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly sup- ports direct, strong instructional guidance rather than constructivist-based minimal guidance during the instructions of novice to intermediate learners.

I’m disappointed, though, that having a teacher who teaches physics by standing and talking in front of a class appears to be optimal. I wish there would be a more engaging alternative.

I don't know what level of physics you're taking, but the lecture portion is mostly motivational introduction (literally motivating particular types of solutions or insights), but it's not where you learn the details of models or techniques for solution. That is done in some combination of reading the text (before class), previous or concurrent math classes (algebra, calculus, analysis, linear algebra, ODE, complex analysis, PDE, abstract algebra each go really well with particular topics), recitation with a TA (where you can ask for aditional detailed questions), and homework sets (that you'll want to discuss with the TA as well). If you have good labs, you'll actually prove many of the relationships by experiment... and you'll see how hard it is to actually do science (there are so damn many ways to do physical things wrong!).

Like programming the details involve sitting down and cranking out the solutions and having the aha moments by yourself. It's easy to be wowed by a lecture, but if you can't teach it to someone else you don't really understand it.

Nothing teaches like experience :-) as I am sure one of my class mates in GCSE physics who connected up a capacitor the wrong way round and experienced a small explosion :-)

You are right, our teacher / school could have combined upfront teaching with all the activities you've described!

Yeah, it can be frustrating. Try to develop friends with the same interest at multiple levels, and you can fill in some of the gaps yourself. Asking the professor during office hours what other learning opportunities they know of which will help the depth of the course may get you an advocate and a few suggestions. They may be frustrated and burned out too.

The most frustrated I ever saw professors were teaching pre-med, where the students literally only cared about the grades.

The Flipped Classroom[1] is one alternative that has gained some momentum. Basically, watch the lecture on your own before class, so classroom time can be interactive.

1. https://en.wikipedia.org/wiki/Flipped_classroom

This was implemented in some of my classwork when I was attending university - for my language and intro physics classes. Unfortunately, the classes that I took that did this had outlandishly bad commercial implementations of this that we were required to use. Basically, we would watch short video lectures on the topic and then have some homework on the video's topic. Unfortunately, there seems to be no pressure for these companies to make a nice-to-use service, since every website I had to use for this was implemented quite horribly.

One example of how terrible these services are was from my foreign language class. Pearson (fuck Pearson) has a service for that which strictly disallows any international keyboard input, and to write non a-z characters you would have to use their very clunky ui. And for their video player would not allow you to rewind or skip forward in any useful way.

I ended up writing some greasemonkey scripts to re render all of their web content. While doing so, I figured out that they also send all of their answers to homework in the html, so I also wrote a script that could allow you to see the answers - which I ended up using sometimes when questions had multiple correct answers but I couldn't figure out which one the homework system wanted(i.e. different words possible, multiple valid verb conjugations valid, etc.).

It's a shame that online textbook/lecture/homework systems for college coursework tend to be so terribly designed - the most difficult part of coursework shouldn't have to be figuring out how to interface with a clunky web ui. I think there is serious potential for a company to make a well-designed system for coursework if they could market it to colleges correctly - maybe survey students on what they like/don't like about existing products and show to colleges that these systems are seriously lacking. Unfortunately, the big textbook companies seem to have a monopoly on curriculum design, so getting these systems to mesh well with the curriculum could be difficult.

You should publish this as a chrome add-on.

Yeah I should. Problem is that I need to put some finishing touches on it before it would be usable for other people and I don't have access to the system anymore because I graduated so I need to find someone with access to test it out. If anyone has access to myspanishlab by Pearson let me know!

Guided instruction doesn't have to mean students don't participate.

Good point! I understood the article that this was excluded, but you are right, I was mistaken: you can guide upfront and still have students participate (eg., afterwards).

I meant something more than that. Maximally guided instruction doesn't rule out checking for understanding every 10 seconds for instance. It doesn't rule out choral responses. IMO the main benefit of explicit instruction is that it provides a framework for constructing the prompts that check if each atomic concept has been correctly generalised. It's extremely hard to do as a teacher and requires deep subject expertise and very good technical abilities regarding how to communicate.

> Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although un- guided or minimally guided instructional approaches are very popular and intuitively appeal- ing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that con- sistently indicate that minimally guided instruction is less effective and less efficient than in- structional approaches that place a strong emphasis on guidance of the student learning pro- cess. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide “internal” guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described.

This guy promotes evidence based education, meaning you can't "guess" or "hope" that an educational technique works, you have to be able to prove, demonstrate and repeatedly get effective results with your education. Direct instruction (from the little I know) works, and has been rigorously tested scientifically.


The quote on the home page:

>"Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.

--J.Tukey, 1962"

Coincidentally I just finished reading the section on minimal guidance in the great book, "How I Wish I'd Taught Maths" [1]

Here's the relevant bit:


Guided discovery lesson plans can be found on most topics in maths. Geometry is a particularly fertile breeding ground. Take something like circle theorems. Instead of simply explaining to students the Angle at the Centre relationship, why not have them discover it for themselves? Give them a set of blank circles, instructions to construct several formulations of the theorem, each time giving them complete freedom as to where they place their three points on the circumference, challenge them to measure the two relevant angles and then see what they notice. Students get important practice of measuring angles, a feeling of involvement in their own learning, and may even teach themselves a key GCSE topic without me needing to say a word. What could possibly go wrong?

I was particularly proud of a guided discovery task I came up with for introducing some of the more complex laws of indices to my Year 11 class two years ago. The worksheet looked like this:


Nice, eh? Again, I ask the question: what could possibly go wrong? Well, quite a lot, as it turns out.

Take the laws of indices lesson. The best that can happen is that all students discover the laws of indices for themselves, leaving no gaps in their knowledge, nor developing any misconceptions, in a reasonable time frame. We can then proceed with the rest of the lesson, maybe moving on to application questions, or interleaving other topics into the examples (see Chapter 12), such as indices involving surds or fractions. How often does that actually happen? In my experience, literally never. What actually happens is that one or two students discover exactly what I wanted them to discover. They are feeling great about themselves, and rightly so –as we have seen in Chapter 2, success is motivating. A handful of students have some kind of idea what is going on, but with an eclectic mix of gaps in their knowledge and newly formed misconceptions. Some of these students are aware they have gaps and misconceptions, others are blissfully ignorant. And the rest of the students do not have a flipping clue what is going on. They are feeling confused and pretty down about themselves when they see their fellow classmates have figured it out. Any form of decent formative assessment strategy (Chapter 11) quickly reveals this disparity between levels of understanding, and as such I cannot move on with the lesson. So what do I inevitably end up doing? Teaching the laws of indices, of course! Maybe I will set those students who seem to have understood it off on the work I hoped everyone else would be moving on to –mind you, I would really like them to hear my explanation and do the worked examples, but how can I justify doing so when they have demonstrated their understanding? Hmmm…

Anyway, back to the rest of the class. By this stage, I am 30 minutes into a 50-minute lesson, rattling through a series of worked examples on the laws of indices far quicker and with much less care than I should. There is zero time for the students to practise their newly acquired skills and hence consolidate their knowledge, nor sufficient time for me to do any kind of application questions which would show them the full breadth of the topic.

But it is even worse than that. Even if I could somehow freeze time and spend those lost 30 minutes going through carefully structured and well-chosen worked examples, I am not back at square one. I am behind square one, because my students are no longer coming at the topic with fresh eyes. Many of those who failed to ‘discover’ the key relationships have already decided that indices are difficult, and yet another area of maths that they don’t understand. It’s going to take more than my magically retrieved 30 minutes to turn that one around.


Later the author describes how his approach has now changed to direct guidance. Of this approach he writes:

A common complaint I hear from teachers when I describe this approach is that students are not as actively involved as they would be during guided discovery. My response is that it depends on what you mean by active, and its anthesis, passive. If active students are ones making noise, working in groups, moving around the classroom, going about the task several different ways, getting some things right but plenty of things wrong, whereas passive students are sitting there quietly, thinking hard about the mathematics I am presenting, then I know which one I would prefer, especially at this early knowledge acquisition stage. For me, such ‘activity’ is exactly the poor proxy for learning that Coe (2013) warns us about. Students may well be active, but active doing what? What are they thinking about? What are they expanding their precious, limited working memory reserves on? During these demonstrations, my students are active in another sense. They are actively thinking hard about the matter in hand – or at least I am creating conditions to give them the very best chance of thinking hard about the matter in hand, and nothing else. Such activity is impossible to see, hence it is often dismissed as passivity and a lack of engagement. But periods of quiet contemplation like this are the key to learning, especially when we consider in greater depth the limits of working memory

1] https://www.amazon.com/How-Wish-Taught-Maths-conversations/d...

Interesting. But it seems to me that the teacher is in a bind because of the assumption that it is necessary that all students learn the same things. If we really want everyone to have an interest in learning then we have to accommodate all the different abilities and interests that the students have. What the teaching profession has to be able to do is not to teach specific subjects but to enable the students to learn and somehow enthuse them to do it.

The problem is that then it is hard to measure the result. And in these days when quantifiability is regarded as the only necessary attribute of pretty much anything we end up falling back on measuring what we can rather than producing students who want to learn and are able to learn.

And when it comes to employment what most employers need is not someone with specific skills in mathematics or computer programming but someone who is willing to work, to improve, and to work with the rest of the department, someone who sees beyond narrow self interest.

> it seems to me that the teacher is in a bind because of the assumption that it is necessary that all students learn the same things.

That's not the starting point. The starting point is that there is one teacher and many students. The necessity for student to learn the same thing derives from the impossibility for the teacher of teaching more than one thing at the same time.

> we have to accommodate all the different abilities and interests that the students have

The craft of teaching is actually the opposite: it is to boil down the necessary priors for some specific learning to occur, and ruthlessly disregarding all other factors. If those priors are not in place you go back and teach them instead, recursively. It helps a lot obviously if students are grouped by current capabilities.

> What the teaching profession has to be able to do is not to teach specific subjects but to enable the students to learn and somehow enthuse them to do it.

Failure is demotivating; success is motivating. Motivation is baked into the pie.

> And in these days when quantifiability is regarded as the only necessary attribute of pretty much anything we end up falling back on measuring what we can rather than producing students who want to learn and are able to learn.

In the world of education, quantifiability is absolutely 100% not a priority. If it was, successful approaches to instruction would have propagated long ago, instead of the typical aspirational mush that passes for analysis these days.

> And when it comes to employment what most employers need is not someone with specific skills in mathematics or computer programming but someone who is willing to work, to improve, and to work with the rest of the department, someone who sees beyond narrow self interest.

They need both actually. Neither competence nor motivation alone will do the job. And frankly, it's somewhat going beyond your remit as an educator to decide what students should be motivated by. That's on them. The teacher's job is to teach.

Also relevant- (2004 pdf)


But, yeah, I think anybody who has taught anything beyond the experiential discovers this. Mayers summarizes it as people have confused constructivist theories of learning with effective ways of teaching. We might learn through constructivism, but learning is more efficient if we guide that construction.

I'm a former teacher and I disagree pretty strongly with the paper based on both my experience and on the fact that this paper has been pretty thoroughly gutted by the literature. The more specific you want the knowledge outputs to be, the more guidance you have to give, as a general rule, but inquiry-based approaches lead to something I call "white swan" learning (with apologies/thanks to Taleb). People often find something through the inquiry that they're enthusiastic about learning, and then they continue learning about that long after they've completed whatever was required. It's not something you can plan, and you don't know which or how many kids will find something with which project, but it happens reliably and for each of those kids finding that topic (and the joy and practice of learning something they are interested in in-depth) is far more effective than semesters of "normal" learning.

If you want your students to discover the Newton's second law you are going to have a hard time getting them to do so in a two-hour lab. It took Newton himself considerably longer, and he was actually deeply interested in the problem.

Inquiry does work well, but not without guidance, unless your sole outcome goal is the inquiry process itself. Even at the other end of the spectrum, the Ph.D. candidate or post-doc, things can get pretty miserable if they are poorly mentored.

So while I applaud you for helping develop natural curiosity (but does it stick? That's another question. Do those kids go on to ask more questions like you've started an engine in their minds, or two years later are they just like everybody else...), if you have curricular goals to meet such that their future classes depend on them knowing the content you are supposed to teach, guiding them is much more efficient.

Proof is watching average machinists struggle to get decent postable CNC machine instructions from the Autodesk 360 “free” CAM system. Well-trained machinists and college guys have no problems. But most would be better off manually programming.

This paper is really fascinating (because one of my flatmates is a huge advocate for unguided learning). Do any newer papers like this exist? Maybe even one that shows that minimal guidance could be positive?

I'd love to read more.

You may have already seen this but @sdrothrock posted this interesting counter-article that defends constructionism (minimal guidance).


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