Does anyone else hate “discovery methods?”

[Matryoshka]

20 minutes ago, Mrs. Tharp said:

Yeah, my issue with it from a ps perspective is that the method will leave behind everyone with difficulties inferring, which includes many neurodiverse folks, ESL kids and people who just need a concrete presentation. In effect, it is biased in favor of those who excel at abstract thinking and who have outstanding EF skills. The method is terrible news for educators who try to narrow the achievement gap, since minorities, the poor, and the neurodiverse traditionally struggle more with these skills.

Even with guided discovery, I think it's most appropriately done 1:1 or in very small groups.  Discovery also has to include sustained interest, and many kids in a large group where someone noticing and redirecting or putting them on the right path isn't really feasible are going to get lost and give up.  Discovery is only fun when you are discovering something, not feeling like you're wandering around in the forest or beating your head against a wall.

Discovery can also work great for self-teaching, but only if a person has inherent interest in discovering something about that subject.  Not all people are interested in 'discovering' math algorithms, how chemistry works, or the intricacies of foreign grammars.

[Not_a_Number]

Just now, Matryoshka said:

Not all people are interested in 'discovering' math algorithms.

I’ve never made a kid discover a math algorithm. However, you do not need math algorithms to add, subtract, multiply or divide.

[Matryoshka]

2 minutes ago, Not_a_Number said:

I’ve never made a kid discover a math algorithm. However, you do not need math algorithms to add, subtract, multiply or divide.

LOL, no, I'm just generalizing.

[Zoo Keeper]

6 minutes ago, Matryoshka said:

...

Discovery can also work great for self-teaching, but only if a person has inherent interest in discovering something about that subject.  Not all people are interested in 'discovering' math algorithms, how chemistry works, or the intricacies of foreign grammars.

You obviously must have met my children.  😉

 

[Not_a_Number]

3 minutes ago, Matryoshka said:

LOL, no, I'm just generalizing.

But I think this is a crucial issue that was brought up earlier. Your learning progression CANNOT look the same using discovery-based learning. I taught DD8 the algorithms in about a day when we were about done learning the operations. The algorithms are computationally efficient but cognitively extremely taxing to UNDERSTAND. If you want kids to learn the algorithm within a few months, you’d never use the discovery method, because the algorithm only makes sense when lots else is automated and requires no working memory.

Edited November 15, 2020 by Not_a_Number

[Mrs. Tharp]

Exactly!

[Matryoshka]

7 minutes ago, Not_a_Number said:

But I think this is a crucial issue that was brought up earlier. Your learning progression CANNOT look the same using discovery-based learning. I taught DD8 the algorithms in about a day when we were about done learning the operations. The algorithms are computationally efficient but cognitively extremely taxing to UNDERSTAND. If you want kids to learn the algorithm within a few months, you’d never use the discovery method, because the algorithm only makes sense when lots else is automated and requires no working memory.

Yes.  That's why I never have a kid memorize, say, a list of 'prepositions' (although I do give a definition - but that can also be hard to grok before deeper understanding). Because virtually every 'preposition' can also function as at least one other part of speech.  It only ends up leading to confusion, not understanding.  So I teach them how to figure out what every word is doing, and then we know what part of speech it is in that sentence.

[Not_a_Number]

Just now, Matryoshka said:

Yes.  That's why I never have a kid memorize, say, a list of 'prepositions' (although I do give a definition - but that can also be hard to grok before deeper understanding). Because virtually every 'preposition' can also function as at least one other part of speech.  It only ends up leading to confusion, not understanding.  So I teach them how to figure out what every word is doing, and then we know what part of speech it is in that sentence.

Exactly. It sets the stage for later learning to do some “guided tinkering.”  But sometimes, people just seem to dump things on kids and expect them to figure it out. That doesn’t work for most kids.

[Porridge]

17 hours ago, Matryoshka said:

I'd never heard that word before, so I looked it up - the only definition I found (in multiple places) was:  A public lecture or reading, especially delivered at a college or university.

I'm guessing that's not what you're referring to here, lol?  Did 8 adopt and redefine this term, or is this from some other pedagogy?  What would be a proper definition for what you're meaning by that term?  (I mean, I think I get the gist, but I'd rather not start to bandy about terms with only a "I think I sorta know what you're getting at" level of understanding 😉 )

@8filltheheart explains it best. See post #1 and #18: Maybe @8filltheheart will jump in, as I might inadvertantly butcher the topic 🙂

To be clear, I don't think that *prelection* the way 8 or the Jesuits might define the word precludes what some might consider "discovery methods."  But, as far as I understand it, it's about giving the learner a rough framework  - some guideposts and signage, a sense of place (where the knowledge might fit into prior knowledge or the "bigger picture," a sense of purpose, and a sense of connection.

[Mrs. Tharp]

I guess I always took the concept of prelection for granted, as one of the foundations of good teaching. I don't think it would necessarily preclude a discovery approach in the abstract.

[Janeway]

On 11/11/2020 at 1:59 PM, drjuliadc said:

I also hate Socratic questioning as a teaching method.  If I’m learning something I just want to be given the information, not figure it out myself.

It did occur to me that I might just be intellectually lazy. I wouldn’t be mad at you if you told me that. I have really thick skin. I just want to know the truth, even if it hurts.

It could also be that I was never taught that way and I am just not used to it. I tend to be very good at academics, but I’m really a boss at rote memorization and not as stellar at problem solving. Maybe it uses the parts of my brain that aren’t as optimal.

I wouldn’t mind hearing from people who love it either. You don’t have to agree with me.

I think discovery method has limited use and is actually the teacher being lazy. It can also lead to learning things wrong. It is like when they tell the students to creatively spell things, then years later, the kids cannot learn the correct way to spell because they already "learned" the creative ways.

[Matryoshka]

1 hour ago, JHLWTM said:

@8filltheheart explains it best. See post #1 and #18: Maybe @8filltheheart will jump in, as I might inadvertantly butcher the topic 🙂

To be clear, I don't think that *prelection* the way 8 or the Jesuits might define the word precludes what some might consider "discovery methods."  But, as far as I understand it, it's about giving the learner a rough framework  - some guideposts and signage, a sense of place (where the knowledge might fit into prior knowledge or the "bigger picture," a sense of purpose, and a sense of connection.

Thanks - I had gone and looked for a post from 8 upthread and hadn't found one!  I'll go and look further.  But, yeah, what you say here makes perfect sense.  I honestly think whole-to-parts vs parts-to-whole is a false dichotomy.  I like to give a sense of the big picture (whole) but then break it down into parts and have them see how it fits together, so that sounds pretty much like what you're talking about.

With 'discovery' in schools, I feel like they're given a bunch of legos and told to build the Taj Mahal with no picture to guide them and perhaps not even an explanation of how legos fit together...  Show them a picture and give them some pointers on different ways things might best fit together and which bricks might be most useful in tricky situations, and help them when they get stuck, for heaven's sake!

And the 'traditional' method seems often also to skip the big picture - you get the pile of blocks and are told painstakingly which one to use in what order, but are still never shown what it is you're building or why...

1 hour ago, Mrs. Tharp said:

I guess I always took the concept of prelection for granted, as one of the foundations of good teaching. I don't think it would necessarily preclude a discovery approach in the abstract.

This.  But it sounds like the pre-part is getting skipped a lot...

[Not_a_Number]

1 hour ago, Matryoshka said:

Thanks - I had gone and looked for a post from 8 upthread and hadn't found one!  I'll go and look further.  But, yeah, what you say here makes perfect sense.  I honestly think whole-to-parts vs parts-to-whole is a false dichotomy.  I like to give a sense of the big picture (whole) but then break it down into parts and have them see how it fits together, so that sounds pretty much like what you're talking about.

I agree with you there. I'm going to start a new grammar thread in a bit, I think, because it sounds like we've very in sync, teaching-wise, and I'd love to hear what you did.

 

Quote

With 'discovery' in schools, I feel like they're given a bunch of legos and told to build the Taj Mahal with no picture to guide them and perhaps not even an explanation of how legos fit together...  Show them a picture and give them some pointers on different ways things might best fit together and which bricks might be most useful in tricky situations, and help them when they get stuck, for heaven's sake!

And the 'traditional' method seems often also to skip the big picture - you get the pile of blocks and are told painstakingly which one to use in what order, but are still never shown what it is you're building or why...

That's a brilliant analogy. I totally agree with you. 

OK, here's one of my favorite examples of a teaching "aha!" moment. It's a little abstract for people who haven't seen it, but I think it illustrates that "discovery" vs "not discovery" isn't really a sensible dichotomy, either. 

I've been doing a lot of experimenting in AoPS precalc. One of the things we do in this class is graph parametric equations: graphs where both coordinates depend on a single parameter, which you often think of as time. Anyway, one thing you can do with some simple parametric equations is to calculate their speed. 

Well, when kids see a line and see the question "What is the speed?", they immediately jump to the technique they've seen before for questions like this, and they give me the slope. Except for parametrizations, that's wrong. We do NOT have a graph of the time vs. distance: we have the graph that the particle is actually traveling along. And the question was how to get kids to compute this speed effectively. 

 

Here's are some things I tried: 

1) Tell them to compute the speed without any other hints (pure discovery.) Tell them that the answer is NOT the slope and to think again themselves. 

 -- This was a terrible failure. None of them understood. 

2) Tell them HOW to compute the speed. I told them to look at two points that were apart 1 unit of time, then to calculate the length of the line segment between them.

 -- This would work for a single example, but then when we'd get to a DIFFERENT parametrization, the kids couldn't do anything at all. For example, if we were parametrizing the same line, the kids would assume the speed was the same. If we were parametrizing a circle, they had no clue what to do. I could then tell them how to compute it again, but they obviously hadn't internalized the essential idea. So, ultimately, this was a complete bust -- they could only do it if I gave them a formula. 

3) Tell them that the particle was moving along the line, and remind them that speed was distance traveled per unit of time and then... wait. Wait without helping them to calculate, except for sending messages to kids who were confused. 

-- This was very successful. After doing one example like this, they could do all the examples for the rest of the class. They figured out the essential idea you needed to figure out the speed of a parametrization. 

 

So, you can see the limits of the discovery vs. not discovery model here. Was the thing that worked "discovery" learning? I didn't tell them a formula, and I made them compute it themselves. So that's like discovery. But I also directed their attention to an essential feature of the question, which is very much direct instruction. The point was there was an essential PRINCIPLE at work, and that principle needed to be communicated. After it was communicated, they could take over themselves. And both trying to get them to figure out the principle themselves or to try to intuit it from formulaic instructions was a flop. 

Edited November 15, 2020 by Not_a_Number

[wendyroo]

3 hours ago, Janeway said:

I think discovery method has limited use and is actually the teacher being lazy. It can also lead to learning things wrong. It is like when they tell the students to creatively spell things, then years later, the kids cannot learn the correct way to spell because they already "learned" the creative ways.

In my experience, discovery teaching done well actually requires a very knowledgeable, skilled, dedicated and patient teacher...not a lazy one.

For example, say my goal was for a child to discover how to change fractions into infinitely many equivalent fractions.

First, I need to be tuned in enough to the student to know that they are conceptually ready to discover that concept. I think just telling kids something can gloss over all sorts of cracks in their foundational understanding. If I tell a child that the algorithm to make equivalent fractions is multiplying the top and the bottom by the same number, I can quickly have them churning out correct equivalent fractions...even if they have no clue why that works or even precisely what a fraction is or means. But it will be much more difficult for them to discover the concept of making equivalent fractions if they don't firmly understand fractions.

Second, I need to set them up for success. I normally open with a situation or word problem that will naturally lead to needing an equivalent fraction. I carefully choose what manipulatives will best help the child experiment with the concept. I carefully word my question or problem statement so that it is precisely and correctly asking for the information I am looking for.

Third, I have to be ready as their guide to recognize and cheer on any baby step in the right direction and ask meaningful, respectful questions to help them realize and redirect their own mistakes...and most importantly, to correctly judge when what they need is for me to give them time and space. In my experience, this is WAY harder than just telling them the right answer. There are so many paths they could take toward discovering equivalent fractions, and all of them, even the meandering ones with lots of backtracking, are equally valid and valuable. I need to understand fractions so thoroughly that I can see the progress and hints of understanding no matter how divergent or nonstandard their thinking.

Lastly, as they piece together a conceptual model, I need to gently push back and force them to explain and prove their method. In my opinion, that is the strength of the discovery method. When done well, a student should come out of it not just able to parrot back the method and rotely employee it on standard, predictable problems, but also fully understand why it works, other methods that work equally well, and how various methods could be applied to never-before-seen problem types.

[Shoeless]

3 hours ago, Janeway said:

I think discovery method has limited use and is actually the teacher being lazy. It can also lead to learning things wrong. It is like when they tell the students to creatively spell things, then years later, the kids cannot learn the correct way to spell because they already "learned" the creative ways.

It sounds like you are confusing discovery method with some sort of laissez-faire free for all.  Discovery method isn't completely devoid of guidance or anyone overseeing what is being learned. There's a goal in mind, and I ask questions of my child that help advance him to that goal. 

Discovery method isn't devoid of content or follow up. If the kids are "discovering" wildly incorrect information, it's up to the teacher to redirect them and ask "Explain to me why you believe the sky is green" and ask enough of the right types of questions so that the student will eventually say "No, that can't be true. The sky is really blue". 

[Matryoshka]

59 minutes ago, wendyroo said:

In my experience, discovery teaching done well actually requires a very knowledgeable, skilled, dedicated and patient teacher...not a lazy one.

For example, say my goal was for a child to discover how to change fractions into infinitely many equivalent fractions.

First, I need to be tuned in enough to the student to know that they are conceptually ready to discover that concept. I think just telling kids something can gloss over all sorts of cracks in their foundational understanding. If I tell a child that the algorithm to make equivalent fractions is multiplying the top and the bottom by the same number, I can quickly have them churning out correct equivalent fractions...even if they have no clue why that works or even precisely what a fraction is or means. But it will be much more difficult for them to discover the concept of making equivalent fractions if they don't firmly understand fractions.

Second, I need to set them up for success. I normally open with a situation or word problem that will naturally lead to needing an equivalent fraction. I carefully choose what manipulatives will best help the child experiment with the concept. I carefully word my question or problem statement so that it is precisely and correctly asking for the information I am looking for.

Third, I have to be ready as their guide to recognize and cheer on any baby step in the right direction and ask meaningful, respectful questions to help them realize and redirect their own mistakes...and most importantly, to correctly judge when what they need is for me to give them time and space. In my experience, this is WAY harder than just telling them the right answer. There are so many paths they could take toward discovering equivalent fractions, and all of them, even the meandering ones with lots of backtracking, are equally valid and valuable. I need to understand fractions so thoroughly that I can see the progress and hints of understanding no matter how divergent or nonstandard their thinking.

Lastly, as they piece together a conceptual model, I need to gently push back and force them to explain and prove their method. In my opinion, that is the strength of the discovery method. When done well, a student should come out of it not just able to parrot back the method and rotely employee it on standard, predictable problems, but also fully understand why it works, other methods that work equally well, and how various methods could be applied to never-before-seen problem types.

Yes. This is discovery done right.

This is, though really hard to do with a group of 30 kids, and I also have almost never seen it done this way in a ps. To manage this with a large group takes not just a not-lazy teacher, but an amazingly skilled one.

Edited November 16, 2020 by Matryoshka

[Gil]

I don't think Socratic Questions and "Discovery Methods" should ever be used for the fundamentals/basics of any field. It's just a huge waste of time and very inefficient. You risk fatiguing the students during the "learning the basics" phase and burning them out of the area/field of study. You also create a huge risk of missing some basics and creating a bunch of misunderstandings that get over looked and go unchecked and unchallenged for who-knows-how-long.

My own style of teaching is very direct and explicit. I am a firm believer in using intelligently designed drill and copious amounts of deliberate practice. I excel in teaching the fundamentals/basics of a subject to mastery. It's very easy to come back around to the basics and ask questions/assign tasks that elevate their understanding of "the basics".

"Discovery methods" are very inefficient. I dislike them because they waste so much time and can be hugely frustrating for me as a teacher and my kids as the students. Socratic Questioning works when you have a Questioner who is

1) A Master of the subject
2) A Master of the technique itself
and a Learner who is
3) receptive to and progresses from round-a-bout Responding to Questions with Questions method (it would've driven me insane)
4) prepared to benefit from the Socratic Journey.

 

[Not_a_Number]

4 minutes ago, Gil2.0 said:

"Discovery methods" are very inefficient. I dislike them because they waste so much time and can be hugely frustrating for me as a teacher and my kids as the students.

What's your definition of "discovery methods"? 

[wendyroo]

40 minutes ago, Matryoshka said:

Yes. This is discovery done right.

This is, though really hard to do with a group of 30 kids, and I also have almost never seen it done this way in a ps. To manage this with a large group takes not just a not-lazy teacher, but an amazingly skilled one.

I think it can be done in groups, but, as you said, is very difficult. I also think some concepts are particularly well suited to group discovery.

I once read about an elementary teacher trying to have her class reinforce perimeter versus area and discover that various rectangles can have the same areas and different perimeters. She had cut a bunch of rectangles, all of which had the same (integer) area, out of construction paper. She split the class into groups, each with one rectangle. Using a unit square, the groups measured their rectangle's area and perimeter.

The teacher then had the groups mingle and confer with each other in order to line themselves up across the room by their rectangle's area. Lo and behold, they soon realized that all the areas were the same! The teacher led a Socratic discussion until the students clarified that that meant the rectangles would all "hold" the same amount, that they all had the same amount of inside area, that even though they all looked different, that it took the same amount of unit squares to cover them.  

Then the teacher had the groups line up in order of perimeter. Hark! Far from being all the same, now they were all wildly different. More discussion clarified the idea that these rectangles, that all had the same inside area, all had different amounts of edge length. To probe their understanding, the teacher challenged the class to choose (and defend) the best rectangle if a farmer was trying to pen in animals using as little fence as possible, and the best rectangle if you were baking a cake and all wanted to get the most frosting on your piece.

I think discovery learning like this can work quite well. I think one defining feature might be that you are trying to get the class to discover an intriguing idea rather than a particular, foundational skill. If at the end of the session, some kids still don't truly grasp how or why rectangles can have the same areas and different perimeters, that's okay. It's not something that mastery is absolutely required right then in order to progress to the next topic. And even for those kids, I think in most cases the discovery session was still worthwhile. They might not have fully grasped the new concept, but they still got to review previous concepts and get introduced to the new idea.

[Gil]

34 minutes ago, Not_a_Number said:

What's your definition of "discovery methods"? 

Just looking on Dictionary.com one will find the following for "discovery method"

Quote

noun Education.

a largely unstructured, situational method or philosophy of teaching whereby students are permitted to find solutions to problems on their own or at their own pace, often jointly in group activities, either independent of or under the guidance of a teacher.

Which is more-or-less what I understood "discovery methods" to mean, but I'm not sure that I have my own definition of it.

[Not_a_Number]

2 minutes ago, Gil2.0 said:

Just looking on Dictionary.com one will find the following for "discovery method"

Which is more-or-less what I understood "discovery methods" to mean, but I'm not sure that I have my own definition of it.

Ah, yeah, I definitely don't do the "unstructured" part of that. Hmmmm. For me, "discovery" just means giving the kids ownership... but I still carefully select the problems. 

In the math department at UT Austin, they had "Moore method" classes where the kids would learn the material by going through problem sets that sort of guided them through the material, then taking turns presenting them to the class. The problem sets were very carefully chosen, so they weren't unstructured at all. However, the idea was that the kids discovered things themselves. DH taught one of these and said that it worked extremely well, although one drawback was that the stronger kids had to wait for the weaker kids to catch up. 

[Gil]

But Moore Method is very systematic. It's more like a Programmed Course than just some willy-nilly Discovery Method.
In a Moore Method math class, the students as well as the problem sets are supposed to be very carefully-selected in advance.

[Not_a_Number]

5 minutes ago, Gil2.0 said:

In a Moore Method math class, the students as well as the problem sets are supposed to be very carefully-selected in advance.

DH, who is sitting next to me and has taught several such classes, does not agree that the STUDENTS are carefully selected. They simply signed up. 

(Oh, he instructs me to tell you that the phrase "Moore Method" is not supposed to be used anymore, because Moore was a huge jerk, and even the math building at UT Austin is no longer named after him. Apparently, now it's "inquiry-based learning.") 

Edited November 16, 2020 by Not_a_Number

[kiwik]

On 11/15/2020 at 10:38 AM, Mrs. Tharp said:

A method in which kids are frequently asked to explain why they are doing what they are doing, there are lots of open ended activities, and learning through play is emphasized. Practice is deemphasized. I remember one such program regularly asking my six year old to explain, in writing, why 2 and 3 made 5.

Is that enough, or do you want something more specific?

I don't hink I would class that as discovery.  I would probably count groups doing more complex sums and explaining verbally the different things they tried.  As an adult though I prefer to be told how to do a new job than left to struggle by myself.

[kiwik]

8 hours ago, Mrs. Tharp said:

 

Yeah, my issue with it from a ps perspective is that the method will leave behind everyone with difficulties inferring, which includes many neurodiverse folks, ESL kids and people who just need a concrete presentation. In effect, it is biased in favor of those who excel at abstract thinking and who have outstanding EF skills. The method is terrible news for educators who try to narrow the achievement gap, since minorities, the poor, and the neurodiverse traditionally struggle more with these skills.

I have watched modern leaning environments (aka barn schooling) and project based work take hold in NZ.  I think in a few years the ONLY kids with good literacy and numeracy at the end of primary will be those who after-school or who can pay for a tutor.  Which is odd since they are always talking about improving education for Lowe socio economic and struggling learners.

 

I think both discovery and unschooling methods require excellent teachers and low child to teacher relationship.

Edited November 16, 2020 by kiwik

[Shoeless]

What are y'all seeing out there in the world of learning that makes you think discovery method is "willy nilly" and "lazy"?  Are y'all really seeing teachers dumping kids in a room with with a periodic table and a solar system model, and saying "Lemme know what you come up with!" and then rubber stamping it as a success when the kids decide the moon is made of green cheese? lol

Because of course that won't work!  There's an assumption of foundational knowledge and a knowledgeable facilitator guiding the process and encouraging the kids to make connections between concepts. I find this far less tedious than rote memorization, and it also results in increased retention of the material.  There was a lot of material I memorized in high school and college, and it all went poof as soon as the last test was completed. If I'd had more time to engage with the material in a more meaningful way, at my own pace, I would have likely remembered a lot more of it.  

[Clemsondana]

I think that what can make this hard is that people respond so differently.  The example that Wendyroo gave above looks well done and is probably similar to what I would do if I were asked to teach an interactive lesson in a group setting.  But, I look at my particular kids and am not sure that it would have actually worked.  If given multiple rectangles, my older would have quickly seen that you could have the same perimeter give multiple areas, and would have actually preferred to just try drawing it on graph paper so that kid could see what the biggest area that you could get was.  My younger would have loved the interactiveness of it but probably wouldn't have generalized it.  We're doing the Arbor Math pre-A program right now which is written to be discovery-ish.  What I'm seeing is that kid quickly grasps concepts and can figure things out but doesn't see the rule without a lot of guidance.  Kid is pretty good at pattern recognition in other contexts, but I don't think that kid cares about math enough to want to think enough to figure it out.  

And, now that I think about it...I think that the way that I talked about area and perimeter in the beginning was even more straightforward.  I defined the terms and then gave the kids a loop of yarn.  I asked them to make a shape that would hold a lot and then a shape that wouldn't hold much, they quickly saw that the same perimeter could yield different areas, and then we moved on to calculating them using unit squares.  We used unit cube blocks to see that a 3x2 square really did have 6 squares in it or to move on to volume.  I think for me I'm bigger into having them visualize or verify than discover - sometimes they do discover it on their own, and seeing that the math really does represent concrete, real world things is helpful, but it isn't discovery as much as reinforcement.  

[wendyroo]

After thinking about it for a while, I think the most common kind of "discovery" learning at my house is me simply not teaching a student a short cut until they have mastered the concept enough to figure it out on their own.

My 5 year old is building her understanding of place value, but I know it is not rock solid. How do I know? Because every gosh darn time I "build" a two digit number on the abacus or out of base 10 blocks, she still counts it out: 10, 20, 30, 40, 41, 42, 43! She isn't at the point that she can just look at 4 ten rods and 3 unit blocks and name it as 43. Even if she knows she has one ten rod and 8 unit blocks, she still has to individually count them out...as the monotonous boredom slowly kills me. 😉 And yet, I am not saying one word to her about it; I fully intend to sit tight until she figures it out on her own. I have no doubt that I could teach her to parrot back that 4 tens and 3 ones is 43, but I think she will have a much more robust understanding of the concept if I let her laboriously count out every number until she realizes for herself exactly why she doesn't have to.

My 7 year old is so close to figuring out cross multiplying for himself. Soon my 9 year old will discover the product rule of exponents which will make his life easier. My 11 year old will realize that sometimes the easiest way to figure out the probability of something happening is to subtract from 1 the probability of it not happening. And when they figure those things out, they won't just be parlor tricks that someone else promises work, but completely understood methods that they can prove to and for themselves are reliable.

[Not_a_Number]

23 minutes ago, wendyroo said:

After thinking about it for a while, I think the most common kind of "discovery" learning at my house is me simply not teaching a student a short cut until they have mastered the concept enough to figure it out on their own.

My 5 year old is building her understanding of place value, but I know it is not rock solid. How do I know? Because every gosh darn time I "build" a two digit number on the abacus or out of base 10 blocks, she still counts it out: 10, 20, 30, 40, 41, 42, 43! She isn't at the point that she can just look at 4 ten rods and 3 unit blocks and name it as 43. Even if she knows she has one ten rod and 8 unit blocks, she still has to individually count them out...as the monotonous boredom slowly kills me. 😉 And yet, I am not saying one word to her about it; I fully intend to sit tight until she figures it out on her own. I have no doubt that I could teach her to parrot back that 4 tens and 3 ones is 43, but I think she will have a much more robust understanding of the concept if I let her laboriously count out every number until she realizes for herself exactly why she doesn't have to.

My 7 year old is so close to figuring out cross multiplying for himself. Soon my 9 year old will discover the product rule of exponents which will make his life easier. My 11 year old will realize that sometimes the easiest way to figure out the probability of something happening is to subtract from 1 the probability of it not happening. And when they figure those things out, they won't just be parlor tricks that someone else promises work, but completely understood methods that they can prove to and for themselves are reliable.

Thank you, Wendy. You put it exactly right, just like you usually do 😉 . I do NOT give shortcuts that my kids aren't ready for. I give them a definition and I watch them use it until the shortcut becomes relatively clear to them. Only then do I talk to them about the shortcut. I may nudge them towards the shortcut, but I don't push them if they aren't ready. 

The amazing thing about this approach is that even if the kid forgets the shortcut, they can always go back to the DEFINITION to figure out the shortcut again if they need to. If they never understood the definition in the first place, if they ever forget the shortcut, all that's left is a yawning chasm. 

So if if my kiddo ever forgets how to add fractions, she'll benefit from her robust understanding of what a fraction is, what addition is, and what division is. She'll be able to figure it out with or without a formula.

[daijobu]

When possible I refrain from giving away the solution to a problem because I want the student to be able to discover it on their own.  Because like it or not, they will need to be able to do this on an exam, and I want them to be prepared for that situation.  

For example, a problem had defined the operation "#" to mean:  a # b = a.  My student could use this for specific problems, but when asked to prove this operation is associative and he couldn't tell me the answer to x # y = ?  I didn't really want to tell him, so I started with what I knew he knew:

Me: "What is a # b?"  

Student:  pause.  "a."

Me:  "What is 1000 # b?"

Student: long pause.  "1000?"  

Me:  "What is "x # y" and so on.  

Now I need to coach him to understand what it means for an operation to be associative.  When he gives me a specific example, like 2#(3#4) = (2#3)#4, that's all well and good... for that specific example.  Again, I don't want to do all the legwork, because I think forcing him to figure it out on his own will serve him well in the long run when he'll be asked to solve this without me around to coach him.  

Me:  "What is a#(b#c)?

Student: long pause.  "???"

Me:  "What do the parentheses mean?"

And so on.  He'll learn so much more if he's forced to discover this himself or at least when I force him to return to first principles and derive what he needs to know.   Yeah, it's a headache, it would have been easier to just show him, but I figure that's what I get paid to do.  

[drjuliadc]

What an amazing explanation Wendyroo, revealing your status as Homeschool Mom Extraordinaire, or perhaps an actual homeschool mom ninja, without sharp weapons.

[Not_a_Number]

On 11/16/2020 at 8:14 PM, daijobu said:

When possible I refrain from giving away the solution to a problem because I want the student to be able to discover it on their own.  Because like it or not, they will need to be able to do this on an exam, and I want them to be prepared for that situation.  

For example, a problem had defined the operation "#" to mean:  a # b = a.  My student could use this for specific problems, but when asked to prove this operation is associative and he couldn't tell me the answer to x # y = ?  I didn't really want to tell him, so I started with what I knew he knew:

Me: "What is a # b?"  

Student:  pause.  "a."

Me:  "What is 1000 # b?"

Student: long pause.  "1000?"  

Me:  "What is "x # y" and so on.  

Now I need to coach him to understand what it means for an operation to be associative.  When he gives me a specific example, like 2#(3#4) = (2#3)#4, that's all well and good... for that specific example.  Again, I don't want to do all the legwork, because I think forcing him to figure it out on his own will serve him well in the long run when he'll be asked to solve this without me around to coach him.  

Me:  "What is a#(b#c)?

Student: long pause.  "???"

Me:  "What do the parentheses mean?"

And so on.  He'll learn so much more if he's forced to discover this himself or at least when I force him to return to first principles and derive what he needs to know.   Yeah, it's a headache, it would have been easier to just show him, but I figure that's what I get paid to do.  

Your example is interesting, because I often find myself doing things like this with my AoPS kids. And I will say... I don't on average find it all that effective. Usually, what's happening in situations like this is that they genuinely don't have a great mental model for something (in your example, I would guess they don't have a great model for a variable.) So, while asking them to think about it is indubitably better than just giving them the answer, I find that I have to do this a lot and I don't get great outcomes out of it. We do MUCH better if we get to the root of the problem and build a good mental model in the first place. 

 

[drjuliadc]

Not a number gets a prize too. I think you may have been responsible for reinvigorating this board.  Since you weren’t here before you were here, you might not know that posting had died down a lot. It seems like you started posting interesting things, and interesting people started posting again too.  Just thought you should know that. I’m sure you are a mom ninja too.

I probably don’t have the greatest working memory. I am pretty disorganized. Nothing that stopped me from being a very good student though. Both of my parents were absent minded professor types.  Maybe that is a clue to my discovery method deficiency syndrome.

[Not_a_Number]

Just now, drjuliadc said:

Not a number gets a prize too. I think you may have been responsible for reinvigorating this board.  Since you weren’t here before you were here, you might not know that posting had died down a lot. It seems like you started posting interesting things, and interesting people started posting again too.  Just thought you should know that. I’m sure you are a mom ninja too.

I probably don’t have the greatest working memory. I am pretty disorganized. Nothing that stopped me from being a very good student though. Both of my parents were absent minded professor types.  Maybe that is a clue to my discovery method deficiency syndrome.

That's sweet of you! 

I've been here for 2 years, though. I just had a different name for most of that time... there was a privacy breach and I had to delete the old account. 

[drjuliadc]

You didn’t disguise your identity very well. I AM talking about when you WERE a number. Haha.  It doesn’t seem as long as two years ago, but it wasn’t recently.  

I acquired 4 children starting when I was 47 and ending at age 50.  That has done weird things to my sense of time.

[Not_a_Number]

33 minutes ago, drjuliadc said:

You didn’t disguise your identity very well. I AM talking about when you WERE a number. Haha.  It doesn’t seem as long as two years ago, but it wasn’t recently.  

Hahahah, I was a number and a shape, lol. And yes, I wasn't really trying to disguise my identity! Just to delete old threads in which I wasn't discreet enough. (Someone complained to my work, ugh.) 

 

33 minutes ago, drjuliadc said:

I acquired 4 children starting when I was 47 and ending at age 50.  That has done weird things to my sense of time.

I can imagine! 

[kiwik]

On 11/16/2020 at 7:28 PM, MissLemon said:

What are y'all seeing out there in the world of learning that makes you think discovery method is "willy nilly" and "lazy"?  Are y'all really seeing teachers dumping kids in a room with with a periodic table and a solar system model, and saying "Lemme know what you come up with!" and then rubber stamping it as a success when the kids decide the moon is made of green cheese? lol

Because of course that won't work!  There's an assumption of foundational knowledge and a knowledgeable facilitator guiding the process and encouraging the kids to make connections between concepts. I find this far less tedious than rote memorization, and it also results in increased retention of the material.  There was a lot of material I memorized in high school and college, and it all went poof as soon as the last test was completed. If I'd had more time to engage with the material in a more meaningful way, at my own pace, I would have likely remembered a lot more of it.  

What happens is schools get these wonderful ideas but then they interpret them oddly and alter them to allow for having 30 kids.  Then you end up with teachers using independent learning as an excuse not to teach.  My feeling is that discovery act are great with a great teacher but mediocre direct instruction beats mediocre discovery.

[bookbard]

Yeah, I think the problem with the term 'discovery learning' is that it means something different to everyone. I took it as child-directed, others as guided teaching, others as socratic . . . and still others as mere exposure to materials. A bit tricky to debate whether 'discovery learning' is effective when there are so many different definitions involved. 

My eight year old just designed a device which incorporates a dyson sphere and an earth shield, to protect the world from global warming while providing unlimited energy. I'd never heard of a dyson sphere before - I still don't really understand it. But he'd come across it online and when I was chatting in the car about global warming solutions (after he mentioned how hot it was getting), he put the two things together and drew up a design. 

That, to me, is child-led and discovery-based learning; it's flexible thinking, making new connections and new ideas from information drawn from different places. And it's far more than filling up a pot with pre-selected knowledge. 

[drjuliadc]

29 minutes ago, bookbard said:

Yeah, I think the problem with the term 'discovery learning' is that it means something different to everyone. I took it as child-directed, others as guided teaching, others as socratic . . . and still others as mere exposure to materials. A bit tricky to debate whether 'discovery learning' is effective when there are so many different definitions involved. 

My eight year old just designed a device which incorporates a dyson sphere and an earth shield, to protect the world from global warming while providing unlimited energy. I'd never heard of a dyson sphere before - I still don't really understand it. But he'd come across it online and when I was chatting in the car about global warming solutions (after he mentioned how hot it was getting), he put the two things together and drew up a design. 

That, to me, is child-led and discovery-based learning; it's flexible thinking, making new connections and new ideas from information drawn from different places. And it's far more than filling up a pot with pre-selected knowledge. 

That is so awesome.  I am impressed.

This summer, my four year old used the word suppository.  I asked him where he heard that word.  He said, The Grammarly commercial.” That is the level of discovery learning around here. Haha.

He and his sister are 5 now. She told me last night her throat was malfunctioning. That has nothing to do with the discovery method but I’ve got to write these hilarious things down or I will forget them.

[Not_a_Number]

On 11/18/2020 at 1:35 AM, kiwik said:

What happens is schools get these wonderful ideas but then they interpret them oddly and alter them to allow for having 30 kids.  Then you end up with teachers using independent learning as an excuse not to teach.  My feeling is that discovery act are great with a great teacher but mediocre direct instruction beats mediocre discovery.

Hmmmm, mediocre direct instruction... I actually don't know. My experience with kids who have had direct instruction from people who know very little math is that they come into my classes not understanding ANYTHING. They wind up thinking math makes no sense. They can't estimate something like 300/0.0145 for me without doing long division, because they have no picture of what division is AT ALL. They can do the algorithms, but they aren't sure what the meaning of what they are calculating is. 

I'd rather have zero learning and board games that this kind of instruction. It's easier to take people from 0 to 100 than it is to unteach them!!

Edited November 22, 2020 by Not_a_Number

[Xahm]

I think discovery method is one of those trendy terms that gets slapped on things and so abused. To my mind, using a discovery method means allowing students to try different things to see what works best and what doesn't work at all, as well as things in between. It starts with itty bitty babies as they learn to grab a toy then later learn to feed themselves. It is inefficient and often messy, especially then, but it's necessary. As kids' worlds get bigger, a combination becomes important. I tell them directly that wild animals can be dangerous; I don't let them discover it. We discover that dogs are soft and furry, but I teach them that bears are, too, and that having hair is one part of being a mammal. When they are learning how to cook, we directly go over some things, but I intentionally let them discover others. They've tried lots of ways to crack eggs, to mix ingredients, and to measure carefully, but we drilled how to properly turn on the gas stove without poisoning or blowing up the family. We've discussed and experimented with various substitutions. It's horribly messy and not an efficient way to put food in the table short term, but I have too many friends (especially a few years back. Most seem to be learning more as they gain this experience) who simply stop cooking in the middle of a recipe and run to the store or throw everything out because they run out of one ingredient and have no idea what to do if they can't follow the exact written directions. They haven't discovered anything about cooking, and their explicit instruction didn't include every situation they run into. 

This is rambling and perhaps doesn't make sense, but I think some methods are much better at teaching something quickly and others are better at teaching for long term success and understanding. In a standard educational situation, we must make choices about which things merit the extra time needed to allow for discover and which are better taught directly. Ideally there should be lots of opportunities to get messy with ideas when very small because it builds a pattern of thinking and lets one go forth with an outlook that assumes that things are understandable, not some opaque mystery that must simply be accepted and memorized.

[kiwik]

On 11/19/2020 at 8:17 AM, Not_a_Number said:

Hmmmm, mediocre direction instruction... I actually don't know. My experience with kids who have had direct instruction from people who know very little math is that they come into my classes not understanding ANYTHING. They wind up thinking math makes no sense. They can't estimate something like 300/0.0145 for me without doing long division, because they have no picture of what division is AT ALL. They can do the algorithms, but they aren't sure what the meaning of what they are calculating is. 

I'd rather have zero learning and board games that this kind of instruction. It's easier to take people from 0 to 100 than it is to unteach them!!

But it isn't usually zero instruction and board games.  It is confused kids who don't understand being told to ask their classmates or use khan academy and still not understanding or being deliberately mislead.  It is the kid with dyslexia and auditory processing sitting staring into space or doing a bunch of irrelevant stuff on a Chromebook.  We had teachers who really didn't understand maths and who weren't great writers but they gave direct instruction with textbooks.  We at least learnt enough for everyday life.  Most of those a generation below me (some of my much younger siblings) can't manipulate numbers in their head at all. They did adjust the syllabus after that but I have met people who just left school or couldn't divide by 10 without a calculator.  If the kids aren't being taught and aren't learning why pay the teachers more than babysitters?

[Not_a_Number]

5 hours ago, kiwik said:

But it isn't usually zero instruction and board games.  It is confused kids who don't understand being told to ask their classmates or use khan academy and still not understanding or being deliberately mislead.  It is the kid with dyslexia and auditory processing sitting staring into space or doing a bunch of irrelevant stuff on a Chromebook.  We had teachers who really didn't understand maths and who weren't great writers but they gave direct instruction with textbooks.  We at least learnt enough for everyday life.  Most of those a generation below me (some of my much younger siblings) can't manipulate numbers in their head at all. They did adjust the syllabus after that but I have met people who just left school or couldn't divide by 10 without a calculator.  If the kids aren't being taught and aren't learning why pay the teachers more than babysitters?

Yeah, it really depends how you're trying to use the discovery method. I wonder if just not teaching math and doing board games is worse than doing bad direct instruction or not? To be clear, neither of those are what I would use to teach, I just have no idea which of those is worse... practically all the kids I taught calculus didn't think math made any sense, so I wonder if NO teaching and just natural interaction with numbers/shapes/logic is worse than that. 

But if you're trying to get kids to discover standard algorithms and not immersing them in numerical reasoning, then that's probably even worse than mediocre direct instruction. 

What did your much younger siblings' math education look like? How were they "taught"?  

[daijobu]

15 hours ago, kiwik said:

But it isn't usually zero instruction and board games.  It is confused kids who don't understand being told to ask their classmates or use khan academy and still not understanding or being deliberately mislead.  

This is some current pedagogy that I'm finding a bit mystifying .  Students are often asked to  join into small groups and discuss things together, often for minutes at a time, and then everyone will reconvene to the whole classroom.  It seems like a such a waste of time.  My dd reported a college professor would do this, but allot about 15 seconds to the task.  She speculated he also thought it was a waste of time but since he was required to use this technique, he minimized the lost instruction time.  

[Not_a_Number]

22 minutes ago, daijobu said:

This is some current pedagogy that I'm finding a bit mystifying .  Students are often asked to  join into small groups and discuss things together, often for minutes at a time, and then everyone will reconvene to the whole classroom.  It seems like a such a waste of time.  My dd reported a college professor would do this, but allot about 15 seconds to the task.  She speculated he also thought it was a waste of time but since he was required to use this technique, he minimized the lost instruction time.  

Huh, weird. I like flipped classrooms, where kids work on problems for some of the class... but not so much discussion groups. 

[Clemsondana]

I think that it's a version of the 'Think, pair, share' method that was (is?) popular.  I had a chemistry professor who used it a lot and I hated it.  If you understood, then it probably worked OK.  If your group didn't, then it was wasted time.  I think that working problems in class instead of as homework can be highly beneficial if the instructor is available to help, though.  When we do genetics problems I give time to work, individually or in groups, but I circulate constantly so that I can catch mistakes before they're repeated them through the whole assignnment.  I think that's one of the biggest issues with discovery and peer teaching methods - the potential for learning it incorrectly.  With oversight, intervention, and guidance it can work well, but I think that many people's experience of it is...not that.  

[Not_a_Number]

6 minutes ago, Clemsondana said:

I think that it's a version of the 'Think, pair, share' method that was (is?) popular.  I had a chemistry professor who used it a lot and I hated it.  If you understood, then it probably worked OK.  If your group didn't, then it was wasted time.  I think that working problems in class instead of as homework can be highly beneficial if the instructor is available to help, though.  When we do genetics problems I give time to work, individually or in groups, but I circulate constantly so that I can catch mistakes before they're repeated them through the whole assignnment.  I think that's one of the biggest issues with discovery and peer teaching methods - the potential for learning it incorrectly.  With oversight, intervention, and guidance it can work well, but I think that many people's experience of it is...not that.  

Yeah, that's what I did, too. I did half a class's worth of lecture, then the kids would work problems and I'd circulate. 

Edited November 23, 2020 by Not_a_Number

[daijobu]

9 hours ago, Clemsondana said:

I think that it's a version of the 'Think, pair, share' method that was (is?) popular.  I had a chemistry professor who used it a lot and I hated it.  If you understood, then it probably worked OK.  If your group didn't, then it was wasted time.  I think that working problems in class instead of as homework can be highly beneficial if the instructor is available to help, though.  When we do genetics problems I give time to work, individually or in groups, but I circulate constantly so that I can catch mistakes before they're repeated them through the whole assignnment.  I think that's one of the biggest issues with discovery and peer teaching methods - the potential for learning it incorrectly.  With oversight, intervention, and guidance it can work well, but I think that many people's experience of it is...not that.  

I like having students tell me how to solve a math problem while I'm at the board, because:  The other students will only see the correct way to solve the problem.  The student dictating to me is forced to use their newly learned math vocabulary, so when they say "that corner in the upper left" I can gently interject, "oh you mean vertex A?"  When students see that their peers are solving the problem, I think it seems more do-able than if I am.  I allow other students to volunteer their different solutions, or I will show a more efficient solution than one proposed by the student.  And sometimes none of the students were able to solve it, so I teach it directly.  

All this is after they have all attempted the problem individually, never in groups.  

[wendyroo]

On 11/22/2020 at 8:36 PM, Not_a_Number said:

Huh, weird. I like flipped classrooms, where kids work on problems for some of the class... but not so much discussion groups. 

I like the idea of flipped classrooms, but all of my experience as a student makes me question how well it actually works in most schools.

In my experience, the vast majority of students do the absolute minimum of work (especially outside of school hours) that they can possibly get away with. When I was in high school, probably 75%ish of students did the homework, but only 10-15% of students did any reading assignments, especially pre-reading of material that would be then covered in class.

Unfortunately, these percentages did not seem to change much when I went to community college and then the local university for dual enrollment. The class would be assigned reading in either a novel or textbook, but during class discussion it quickly became crystal clear that most students had not even cracked the book in preparation. Inevitably, the teacher would end up teaching all the material that should have been read...punishing those of us that did actually read it and were ready for more depth and discussion.

As I understand flipped classrooms, they assign the "lecture" as homework, either in the form of reading or videos. Then the students are expected to come to class ready to ask questions, clear up misunderstandings, and put that learning into practice under the guidance of the teacher. But what I imagine happening (based on my experiences) is the teacher assigning problems during class only to find that most students haven't read/watched the lecture and therefore need to be taught the method from square one before they can undertake the practice problems. So the teacher ends up spending most of the class time teaching students the lecture material individually or in small groups, while the students who actually did read/watch the lecture at home are left to fend for themselves.

I think it is a great philosophy, but ultimately, it is much easier to verify, grade and enforce students doing problems at home rather than reading, watching or thinking about new concepts. 

[Not_a_Number]

6 minutes ago, wendyroo said:

I like the idea of flipped classrooms, but all of my experience as a student makes me question how well it actually works in most schools.

In my experience, the vast majority of students do the absolute minimum of work (especially outside of school hours) that they can possibly get away with. When I was in high school, probably 75%ish of students did the homework, but only 10-15% of students did any reading assignments, especially pre-reading of material that would be then covered in class.

Unfortunately, these percentages did not seem to change much when I went to community college and then the local university for dual enrollment. The class would be assigned reading in either a novel or textbook, but during class discussion it quickly became crystal clear that most students had not even cracked the book in preparation. Inevitably, the teacher would end up teaching all the material that should have been read...punishing those of us that did actually read it and were ready for more depth and discussion.

As I understand flipped classrooms, they assign the "lecture" as homework, either in the form of reading or videos. Then the students are expected to come to class ready to ask questions, clear up misunderstandings, and put that learning into practice under the guidance of the teacher. But what I imagine happening (based on my experiences) is the teacher assigning problems during class only to find that most students haven't read/watched the lecture and therefore need to be taught the method from square one before they can undertake the practice problems. So the teacher ends up spending most of the class time teaching students the lecture material individually or in small groups, while the students who actually did read/watch the lecture at home are left to fend for themselves.

I think it is a great philosophy, but ultimately, it is much easier to verify, grade and enforce students doing problems at home rather than reading, watching or thinking about new concepts. 

I've seen the model where the kids are supposed to read before. I didn't really do that, although I knew people that did... I just taught for half the class and that was the lecture. I don't tend to think lecture teaches nearly as much as the practice, anyway, so doing a shorter lecture wasn't a big problem. 

I would assume you could set up an incentive system for reading and preparing at home? Short quizzes or something? Pop quizzes at the beginning of class? You'd have to tinker to make it work... but anyway, I've never tried. I only ever tried it the way I described. 

I can't tell you whether it was useful or not, though, because these classes required me to teach calculus to kids who had largely never really understood what a function was in the first place, so it was a totally impossible task. 

Edited November 24, 2020 by Not_a_Number