Anyone proposing discovery learning in groups has to explain how they prevent only one member of the group discovering and the rest learning from that person’s exposition.
They might claim everyone spent some time attempting to discover the thing to be learned but they have no way to establish this- it’s just what they hope happened.
The second I heard the "Vertical Non-Permanent Surfaces" my inner skeptic triggered. I read the book and visited some classrooms, and completely agree with you. Daniel Buck wrote an article for the Fordham Institute that summarizes it nicely, and reminds readers of the embarassing fact in math education, which is that we've know what worked for decades. We just don't do it:
"The What Works Clearing House, a research consortium housed within the federal Institute of Education Sciences, published a 2021 practice guide on the most effective, research-backed methods for math instruction. It includes systematic instruction, representation and models, and timed activities, all of which Liljedhal denigrates for no other reason than they don’t constitute “thinking”—which by the end of the book, in a textbook example of circular logic, he seems to only define as activities that he already approves of."
Fads an myths have been plaguing education for too long. Keep up the good fight.
I'm just curious: have you actually seen a BTC classroom in action? If not, would you be willing to actually check out a couple? Because I don't think it is what you think it is.
Well, here's a short list. I'm not really willing to spend a ton of time explaining, because you've already explained why you can't see how it can possibly work. But if you decide that it's possible that it's not all smoke and mirrors and want to learn more, you can let me know.
1. I think your quibbles about Jason's thin slicing example is missing the forest for the trees. Thin-slicing, done well (and I am at no point suggesting that BTC is easy to do well, but neither is anything else in teaching) takes kids through a logical learning progression from more concrete to more abstract, from simple to more complicated. And not only do kids seem to do just fine under this approach, it is also very efficient, with teachers reporting that they are able to move through the curriculum more quickly than using their previous methods. I also think that thin-slicing is the hardest to do and the hardest to explain without seeing it, because it does indeed sound a little woo-woo.
2. The injunction against writing while learning: I think this is over-stating the amount of writing that happens on the VNPSs. I think that it's likely that you, as do I, frequently put pen to paper to explain what I mean or what I'm thinking.
3. Discovery learning. I am not completely sure the point you are making, but I think it's along the lines of "BTC may not be pure discovery learning but it's still discovery learning and therefore it's bad." I wouldn't classify BTC as discovery learning, or minimally guided instruction, at all. The progression of tasks during a lesson (not all lessons involve thin-slicing) is designed to take kids pretty quickly from what is inside their current competence to the next step in the learning progression, with support from the teacher. Peter is pretty clear in Chapter 9 (I think it's chapter 9, but I don't have my copy of the book to hand) that the job of the teacher is to make sure that the work that students are doing at any given moment is a good balance between their ability and an appropriate level of challenge. A good BTC teacher is on the move for the whole lesson, listening to what kids are thinking and providing appropriate teaching points.
4. I think there are other factors that you (and other critics) fail to take into account. For example: a. Motivation: a lot of kids are already turned off math by the time they reach middle school, and to go into a classroom where the teacher has had a hard time getting kids to try or keeping them on task and to see a room full of kids fully engaged in learning is quite impressive; b. Second language learners: the way that the tasks and the group work are set up in BTC make the doing of mathematics much more accessible for second language learners.
5. Data. I don't have huge implementation studies to show that BTC is successful. But I think to dismiss BTC on the grounds that there aren't gold standard studies is disingenuous. I do have classroom-level growth data showing big cohort gains with quite limited implementation of BTC. There are people all over the place collecting data on implementation for their own schools and districts, and to argue a priori that BTC can't possibly work while ignore those data seems like confirmation bias to me.
1. I think that anything ‘done well’ is preferrable to it being done badly. However, I worry about these arguments because they can lead to the ‘no true Scotsman’ fallacy. If a teacher is struggling to achieve the supposed gains of the program, is it because they have not done it well? This demonstrates that it is not good enough to have a highly effective program that cannot be implemented at scale — it needs to be effective and implementable. Another key question is whether the program ‘done well’ is better than an alternative ‘done well.’ If you have any evidence addressing this question, I would be interested in it. I am deeply sceptical.
2. This is not a relevant point. Jason To drew on research on collaborative learning where a key protocol was that groups do not write things down to support collaborative learning in BTC where the groups write things down on vertical surfaces. The research therefore does not apply. It is not about the *amount* of writing.
3. This is one of those semantic traps that people like to set. Maybe it does not seem like discovery learning to you. However, this would seem at odds with Liljedahl’s research interest in ‘Instances of creativity and discovery in mathematics teaching’ (https://www.sfu.ca/education/faculty-profiles/pliljedahl.html) or a paper he wrote on ‘Mathematics discovery’ and the ‘AHA!’ effect. (https://peterliljedahl.com/wp-content/uploads/JA-IJMEST-2005.pdf). The injunctions in the BTC book against answering student questions and instead, walking away and smiling, do not suggest concepts are meant to be fully explained to students. An absence of full explanation, where students have to figure out part of a concept or procedure for themselves, is the definition of discovery I am applying.
4. Motivation is absolutely critical and it always surprises me that advocates of discovery learning (my definition) think they own it. I can imagine if a student asked a question and the teacher walked away smiling without answering it, that would be quite frustrating. Parents of kids in BTC classrooms have contacted me and told me how demotivated their kids have become from the approach. Perhaps the teachers in these classrooms are not doing it well? Motivation is complex but one aspect is very clear — long-term motivation (or ‘personal interest’) is strongly associated with success. Getting better at maths is motivating. The stories I hear of students not understanding the maths are not consistent with a motivating method of teaching.
5. I don’t dismiss BTC on the grounds that there aren’t gold standard studies, although such studies would help. I dismiss it on the grounds that it is at odds with some very basic principles of learning. Kids should not be experimented on in this way, particularly since gimmicks aside, it is nothing new.
I kept this away from Jason's paper, as I don't actually think he's right about everything. I don't even agree with everything Peter says about his own research--I think he says some of the things he says with a particular audience in mind, and honestly I'm not confident he completely has explanations for everything that goes on in BTC classrooms. I'm trying to suggest to you that you are making an academic argument based on abstractions when what you should do is entertain the possibility that those abstractions do not necessarily represent the complexity of what happens in a BTC classroom, and we would all be better served if people were curious and not judgmental.
I also think the "don't experiment on kids" argument is disingenuous. First, there is so little control over actual practice in classrooms that in effect kids are being experimented on all the time. Second, thousands and thousands of teachers in the US are using BTC and finding it to be more successful than their previous teaching. Whether or not they should have been allowed to do that is no longer salient. That ship has sailed. For those of them for whom the experiment has worked, shouldn't we be interested in why it's been successful rather than continuing to dismiss it as gimmicks?
I just find it so ironic when you say that it is at odds with some very basic principles of learning. Well, I can take you places where kids are making more growth than they've ever done. So either those basic principles are wrong, or the methods aren't as deviant from those basic principles as you think they are, or there are factors in play that make some less "pure" methods more powerful than those pure methods in certain circumstances. The fact that it doesn't make sense to you becomes irrelevant at some point, don't you think, if kids are learning?
"I kept this away from Jason's paper, as I don't actually think he's right about everything."
It seems odd, then, that you are commenting on this post which is a direct response to the arguments in Jason's paper.
You can assert all you like that BTC is tremendous, but given its flawed logic, I doubt that and would need hard evidence. Extraordinary claims do, indeed, need extraordinary evidence. I shall await the hard evidence.
Well actually, I am commenting on what I think are misperceptions you have about BTC, which does not seem to require that I comment on Jason's paper. And I have not asserted that BTC is tremendous, merely that some teachers appear to be having more success with it than they have had with other methods they have tried, including Direct Instruction. Nor am I making extraordinary claims. I am suggesting that perhaps it would behoove you, and everyone who reads your work, which is a lot of people, for you to try and understand why that might be the case. Your confidence that the logic of BTC is flawed is very interesting, given that you don't want to go see it in action or talk to teachers who have data on student achievement. I am not so sure that you understand it as well as you think you do, and that makes me sad.
It sounds like you think To has it quite wrong. Keep in mind Greg is addressing To’s paper here and if To is misrepresenting BTC you might point that out more explicitly.
Also I’d take another read of Greg’s post. For example his point about writing is merely addressing the details of an experimental condition used to ensure the experiment examined the effects of working memory. No where is he talking about the merits of writing for learning.
You spend a lot of time writing about stuff no one is saying.
A third example. Do you really mean to suggest Greg and everyone else not teaching using BTC has never seen a class really engaged in learning? If not what is your point about motivation?
Yes. We have a lot of visitors to our school and they all comment on how engaged our students are because they are usually answering a question every couple of minutes on a mini whiteboard, especially in maths. However, I would not point to this as evidence of anything. I would point to the evidence that we drew on, such as Rosenshine's Principles, when we introduced the approach. The fact that there are some engaged kids in a class somewhere tells us little.
As another example Greg is not objecting to going from concrete to abstract or simple to more complex. He is objecting to the ban on mimicking and full explanation by worked examples. If he has this wrong and BTC does involve fully explained worked examples with students doing the same type of examples as the teacher demonstrates you should point this out.
But To does not seem to think that is what BTC is about.
You don’t seem to be engaging with Greg’s views on BTC here.
Fascinating. Thomas Kuhn would say the actual workings of scientists is not in actively seeking disinformation of existing theories, instead Kuhn argues (and provides lots of historical examples) that "scientists do not typically try to disconfirm their own theories. Instead, they work within the prevailing paradigm and focus on solving puzzles that fit within that framework. Kuhn believes that scientists aim to extend and refine the paradigm rather than challenge it. Anomalies or data that don't fit are often set aside as problems to be solved later or as errors in experimentation. It's only when anomalies accumulate to a significant degree and can no longer be ignored or explained within the existing paradigm that scientists may begin to question and consider alternatives, leading to a paradigm shift. This approach contrasts with the idea of falsification, where scientists actively seek to test and potentially disprove their theories."
You nailed it. Groping to discover techniques and group projects both waste a lot of time the students could spend practicing, honing basic skills, getting fast and accurate. Once students own those skills, they become chunks, like vocabulary, that the students can use creatively. It’s easier to get the creative challenge at the end, as with the tradition of extra credit problems.
Interesting article! Clearly you value the contribution cognitive science can make to instructional practice. What do you think about the role neuroscience (and especially educational neuroscience) can play? A lot of educational neuroscience work focuses on numerical and mathematical cognition and learning disabilities (e.g., https://open.substack.com/pub/neuroeducation/p/understanding-dyscalculia-how-the?r=27wrx5&utm_campaign=post&utm_medium=web). Although the history of educational focus is much shorter than for cognitive science, Im optimistic that useful bridges can be built!
I think it can be valuable if note taking is the skill you are teaching eg taking prose and reducing it into notes. It may have other value, depending how it’s done. To aid learning, it would require some processing rather than just copying. I don’t use note taking much in my own teaching. In maths, I think it’s better to get kids answering questions.
What do you know about the effectiveness of note taking in other areas, for example, history or science? I've been going around in circles on this for years. To save time I now print them the notes with blanks they fill in as I do the initial explanations, but I am still not convinced that is the best way.
I teach high school history and what works for me is printing "lecture" notes that I will use to discuss but I insert check in questions each subsection that we answer as a class. This helps to keep them engaged in the notes, but they have to write the responses down, I don't write them on the board. I also assign point values to make sure they do it.
Dan Willingham covers some of the science on note taking:
Outsmart Your Brain: Why Learning is Hard and How You Can Make It Easyhttps://g.co/kgs/7B8HD73
I think a lot of forcing note taking is just crude incentives to do something rather than nothing. What would help is a more methodical approach by the students.
It’s interesting how with all the discussion of meta cognition there is so little explicit teaching on the best science of how to study. Lots of competing theories on the best way to teach get airtime but you have to go elsewhere to discover the best practices for studying.
A learners toolkit from Churchie has the most student-friendly version of it, I think. So I've read more about note-taking now and have discovered that lengthy note taking for high school aged students is not great... Better to use "guided note taking" (thanks Bronwyn Ryrie Jones for that one)
Anyone proposing discovery learning in groups has to explain how they prevent only one member of the group discovering and the rest learning from that person’s exposition.
They might claim everyone spent some time attempting to discover the thing to be learned but they have no way to establish this- it’s just what they hope happened.
Hey Greg,
The second I heard the "Vertical Non-Permanent Surfaces" my inner skeptic triggered. I read the book and visited some classrooms, and completely agree with you. Daniel Buck wrote an article for the Fordham Institute that summarizes it nicely, and reminds readers of the embarassing fact in math education, which is that we've know what worked for decades. We just don't do it:
"The What Works Clearing House, a research consortium housed within the federal Institute of Education Sciences, published a 2021 practice guide on the most effective, research-backed methods for math instruction. It includes systematic instruction, representation and models, and timed activities, all of which Liljedhal denigrates for no other reason than they don’t constitute “thinking”—which by the end of the book, in a textbook example of circular logic, he seems to only define as activities that he already approves of."
Fads an myths have been plaguing education for too long. Keep up the good fight.
I'm just curious: have you actually seen a BTC classroom in action? If not, would you be willing to actually check out a couple? Because I don't think it is what you think it is.
No? What have I got wrong about it?
Well, here's a short list. I'm not really willing to spend a ton of time explaining, because you've already explained why you can't see how it can possibly work. But if you decide that it's possible that it's not all smoke and mirrors and want to learn more, you can let me know.
1. I think your quibbles about Jason's thin slicing example is missing the forest for the trees. Thin-slicing, done well (and I am at no point suggesting that BTC is easy to do well, but neither is anything else in teaching) takes kids through a logical learning progression from more concrete to more abstract, from simple to more complicated. And not only do kids seem to do just fine under this approach, it is also very efficient, with teachers reporting that they are able to move through the curriculum more quickly than using their previous methods. I also think that thin-slicing is the hardest to do and the hardest to explain without seeing it, because it does indeed sound a little woo-woo.
2. The injunction against writing while learning: I think this is over-stating the amount of writing that happens on the VNPSs. I think that it's likely that you, as do I, frequently put pen to paper to explain what I mean or what I'm thinking.
3. Discovery learning. I am not completely sure the point you are making, but I think it's along the lines of "BTC may not be pure discovery learning but it's still discovery learning and therefore it's bad." I wouldn't classify BTC as discovery learning, or minimally guided instruction, at all. The progression of tasks during a lesson (not all lessons involve thin-slicing) is designed to take kids pretty quickly from what is inside their current competence to the next step in the learning progression, with support from the teacher. Peter is pretty clear in Chapter 9 (I think it's chapter 9, but I don't have my copy of the book to hand) that the job of the teacher is to make sure that the work that students are doing at any given moment is a good balance between their ability and an appropriate level of challenge. A good BTC teacher is on the move for the whole lesson, listening to what kids are thinking and providing appropriate teaching points.
4. I think there are other factors that you (and other critics) fail to take into account. For example: a. Motivation: a lot of kids are already turned off math by the time they reach middle school, and to go into a classroom where the teacher has had a hard time getting kids to try or keeping them on task and to see a room full of kids fully engaged in learning is quite impressive; b. Second language learners: the way that the tasks and the group work are set up in BTC make the doing of mathematics much more accessible for second language learners.
5. Data. I don't have huge implementation studies to show that BTC is successful. But I think to dismiss BTC on the grounds that there aren't gold standard studies is disingenuous. I do have classroom-level growth data showing big cohort gains with quite limited implementation of BTC. There are people all over the place collecting data on implementation for their own schools and districts, and to argue a priori that BTC can't possibly work while ignore those data seems like confirmation bias to me.
Thanks for engaging on this. Happy to talk more.
Hi Isobel
Comments below:
1. I think that anything ‘done well’ is preferrable to it being done badly. However, I worry about these arguments because they can lead to the ‘no true Scotsman’ fallacy. If a teacher is struggling to achieve the supposed gains of the program, is it because they have not done it well? This demonstrates that it is not good enough to have a highly effective program that cannot be implemented at scale — it needs to be effective and implementable. Another key question is whether the program ‘done well’ is better than an alternative ‘done well.’ If you have any evidence addressing this question, I would be interested in it. I am deeply sceptical.
2. This is not a relevant point. Jason To drew on research on collaborative learning where a key protocol was that groups do not write things down to support collaborative learning in BTC where the groups write things down on vertical surfaces. The research therefore does not apply. It is not about the *amount* of writing.
3. This is one of those semantic traps that people like to set. Maybe it does not seem like discovery learning to you. However, this would seem at odds with Liljedahl’s research interest in ‘Instances of creativity and discovery in mathematics teaching’ (https://www.sfu.ca/education/faculty-profiles/pliljedahl.html) or a paper he wrote on ‘Mathematics discovery’ and the ‘AHA!’ effect. (https://peterliljedahl.com/wp-content/uploads/JA-IJMEST-2005.pdf). The injunctions in the BTC book against answering student questions and instead, walking away and smiling, do not suggest concepts are meant to be fully explained to students. An absence of full explanation, where students have to figure out part of a concept or procedure for themselves, is the definition of discovery I am applying.
4. Motivation is absolutely critical and it always surprises me that advocates of discovery learning (my definition) think they own it. I can imagine if a student asked a question and the teacher walked away smiling without answering it, that would be quite frustrating. Parents of kids in BTC classrooms have contacted me and told me how demotivated their kids have become from the approach. Perhaps the teachers in these classrooms are not doing it well? Motivation is complex but one aspect is very clear — long-term motivation (or ‘personal interest’) is strongly associated with success. Getting better at maths is motivating. The stories I hear of students not understanding the maths are not consistent with a motivating method of teaching.
5. I don’t dismiss BTC on the grounds that there aren’t gold standard studies, although such studies would help. I dismiss it on the grounds that it is at odds with some very basic principles of learning. Kids should not be experimented on in this way, particularly since gimmicks aside, it is nothing new.
Best wishes
I kept this away from Jason's paper, as I don't actually think he's right about everything. I don't even agree with everything Peter says about his own research--I think he says some of the things he says with a particular audience in mind, and honestly I'm not confident he completely has explanations for everything that goes on in BTC classrooms. I'm trying to suggest to you that you are making an academic argument based on abstractions when what you should do is entertain the possibility that those abstractions do not necessarily represent the complexity of what happens in a BTC classroom, and we would all be better served if people were curious and not judgmental.
I also think the "don't experiment on kids" argument is disingenuous. First, there is so little control over actual practice in classrooms that in effect kids are being experimented on all the time. Second, thousands and thousands of teachers in the US are using BTC and finding it to be more successful than their previous teaching. Whether or not they should have been allowed to do that is no longer salient. That ship has sailed. For those of them for whom the experiment has worked, shouldn't we be interested in why it's been successful rather than continuing to dismiss it as gimmicks?
I just find it so ironic when you say that it is at odds with some very basic principles of learning. Well, I can take you places where kids are making more growth than they've ever done. So either those basic principles are wrong, or the methods aren't as deviant from those basic principles as you think they are, or there are factors in play that make some less "pure" methods more powerful than those pure methods in certain circumstances. The fact that it doesn't make sense to you becomes irrelevant at some point, don't you think, if kids are learning?
"I kept this away from Jason's paper, as I don't actually think he's right about everything."
It seems odd, then, that you are commenting on this post which is a direct response to the arguments in Jason's paper.
You can assert all you like that BTC is tremendous, but given its flawed logic, I doubt that and would need hard evidence. Extraordinary claims do, indeed, need extraordinary evidence. I shall await the hard evidence.
Well actually, I am commenting on what I think are misperceptions you have about BTC, which does not seem to require that I comment on Jason's paper. And I have not asserted that BTC is tremendous, merely that some teachers appear to be having more success with it than they have had with other methods they have tried, including Direct Instruction. Nor am I making extraordinary claims. I am suggesting that perhaps it would behoove you, and everyone who reads your work, which is a lot of people, for you to try and understand why that might be the case. Your confidence that the logic of BTC is flawed is very interesting, given that you don't want to go see it in action or talk to teachers who have data on student achievement. I am not so sure that you understand it as well as you think you do, and that makes me sad.
It sounds like you think To has it quite wrong. Keep in mind Greg is addressing To’s paper here and if To is misrepresenting BTC you might point that out more explicitly.
Also I’d take another read of Greg’s post. For example his point about writing is merely addressing the details of an experimental condition used to ensure the experiment examined the effects of working memory. No where is he talking about the merits of writing for learning.
You spend a lot of time writing about stuff no one is saying.
Well actually, if you read the thread, I'm responding to Greg's question about what I think he's got wrong about BTC.
A third example. Do you really mean to suggest Greg and everyone else not teaching using BTC has never seen a class really engaged in learning? If not what is your point about motivation?
Yes. We have a lot of visitors to our school and they all comment on how engaged our students are because they are usually answering a question every couple of minutes on a mini whiteboard, especially in maths. However, I would not point to this as evidence of anything. I would point to the evidence that we drew on, such as Rosenshine's Principles, when we introduced the approach. The fact that there are some engaged kids in a class somewhere tells us little.
As another example Greg is not objecting to going from concrete to abstract or simple to more complex. He is objecting to the ban on mimicking and full explanation by worked examples. If he has this wrong and BTC does involve fully explained worked examples with students doing the same type of examples as the teacher demonstrates you should point this out.
But To does not seem to think that is what BTC is about.
You don’t seem to be engaging with Greg’s views on BTC here.
I understood that which is why your writing on writing seems odd as Greg was not writing about btc there but about To’s paper.
Fascinating. Thomas Kuhn would say the actual workings of scientists is not in actively seeking disinformation of existing theories, instead Kuhn argues (and provides lots of historical examples) that "scientists do not typically try to disconfirm their own theories. Instead, they work within the prevailing paradigm and focus on solving puzzles that fit within that framework. Kuhn believes that scientists aim to extend and refine the paradigm rather than challenge it. Anomalies or data that don't fit are often set aside as problems to be solved later or as errors in experimentation. It's only when anomalies accumulate to a significant degree and can no longer be ignored or explained within the existing paradigm that scientists may begin to question and consider alternatives, leading to a paradigm shift. This approach contrasts with the idea of falsification, where scientists actively seek to test and potentially disprove their theories."
You nailed it. Groping to discover techniques and group projects both waste a lot of time the students could spend practicing, honing basic skills, getting fast and accurate. Once students own those skills, they become chunks, like vocabulary, that the students can use creatively. It’s easier to get the creative challenge at the end, as with the tradition of extra credit problems.
Interesting article! Clearly you value the contribution cognitive science can make to instructional practice. What do you think about the role neuroscience (and especially educational neuroscience) can play? A lot of educational neuroscience work focuses on numerical and mathematical cognition and learning disabilities (e.g., https://open.substack.com/pub/neuroeducation/p/understanding-dyscalculia-how-the?r=27wrx5&utm_campaign=post&utm_medium=web). Although the history of educational focus is much shorter than for cognitive science, Im optimistic that useful bridges can be built!
How many educational parasites do we need to keep putting down. It’s exhausting
Dr. Ashman, do you have any thoughts about the value of note-taking?
I think it can be valuable if note taking is the skill you are teaching eg taking prose and reducing it into notes. It may have other value, depending how it’s done. To aid learning, it would require some processing rather than just copying. I don’t use note taking much in my own teaching. In maths, I think it’s better to get kids answering questions.
What do you know about the effectiveness of note taking in other areas, for example, history or science? I've been going around in circles on this for years. To save time I now print them the notes with blanks they fill in as I do the initial explanations, but I am still not convinced that is the best way.
I teach high school history and what works for me is printing "lecture" notes that I will use to discuss but I insert check in questions each subsection that we answer as a class. This helps to keep them engaged in the notes, but they have to write the responses down, I don't write them on the board. I also assign point values to make sure they do it.
Dan Willingham covers some of the science on note taking:
Outsmart Your Brain: Why Learning is Hard and How You Can Make It Easyhttps://g.co/kgs/7B8HD73
I think a lot of forcing note taking is just crude incentives to do something rather than nothing. What would help is a more methodical approach by the students.
It’s interesting how with all the discussion of meta cognition there is so little explicit teaching on the best science of how to study. Lots of competing theories on the best way to teach get airtime but you have to go elsewhere to discover the best practices for studying.
A learners toolkit from Churchie has the most student-friendly version of it, I think. So I've read more about note-taking now and have discovered that lengthy note taking for high school aged students is not great... Better to use "guided note taking" (thanks Bronwyn Ryrie Jones for that one)
Yeah nice. I have skim read that book. It was slightly too tertiary-education directed for my liking.