Yes your rationale makes perfect sense for why to show those 3 examples - first trial being potentially a bit too hard allows you to get to efficient communication much more quickly than starting with examples that are too easy and potentially are slower to communicate the rule and potentially more likely to induce stipulation (E.g. that you can only substitute for numbers). And certainly it will come down to a judgement of the class too and you can respond in the moment if you have gone too hard. I think I will change some of my sequences based on this conversation.
I still don't yet believe that the sameness or difference principles apply here or to any transformation as every change in the input results in a change to the output. The difference principle is that when a minimal change results in a change in label then we narrow down what causes a change in the label and by extrapolation can rule out as many possible reasons for the change in label as possible. Whereas the sameness principle is that a great change in input resulting in no change to the label implies a large number of possible cases for which the label still applies by interpolation. But perhaps my understanding of the principles is limited to (categoricals)?
Yeah exactly - how the principles apply to transformations is actually a bit different to categoricals, which are a lot more intuitive.
Took me a long time to make any sense of it, especially since the language is confusing (‘sameness principle means make things very different’ 🤔)
It sounds like you’ve read Theory, so Engelmann and Carnine talk through this around loc 3602 (kindle version,) though maybe not much more helpful.
Have you come across the book Could John Stewart Mill Have Saved Our Schools?
It’s very short, and in it Engelmann talks through how their five principles appear identical to the five methods of experimental inquiry of Mill.
For categoricals we’re trying to draw boundary lines acting what the concept can and cannot be. Easy to picture.
For transformations we’re not doing that. We’re trying to show what must change, at a minimum, and then something about how it might change, at a maximum. Harder to grasp intuitively.
To me it seems that the examples you present seem reasonable in showing the transformation rule here (replace the letter with its given value in brackets) and show a good range of variation which would hopefully be sufficient to induce generalisation.
I don't quite understand what you say about the sameness and difference prompts. If changing from x = 3 to x = 31 leads to a difference in the way we treat the example, then changing from x = 31 to y = 6a also leads to a difference in the way we treat the example.
It seems you've deviated from Theory of Instruction in this which suggests it is preferable to model the first 2 to 5 with minimum difference variations, to include at least 2 test examples as minimum difference variations, and then potentially have maximum difference variations that learners attempt (which forms part of the expansion sequence). My guess is that you've done that to a) reinforce important idea of minimal and maximal difference prompts , b) make it more consistent with approach for ‘Categoricals' and hence make it easier for teachers to remember and c) to reduce the time spent on the modelled examples, hoping that the rule has already been made sufficiently clear so showing the maximal variation in your example is more likely to induce the desired generalisation. Is that correct? Under what conditions would you err more on doing it in the manner prescribed in ToI?
Hey Jack. Thanks for the comment, and the kind words!
“If changing from x = 3 to x = 31 leads to a difference in the way we treat the example, then changing from x = 31 to y = 6a also leads to a difference in the way we treat the example.”
Yes, but the change from 3 to 31 is a minimal difference. The difference principle is about showing small, minimal differences between examples, and then showing how that leads to a difference in how we respond. It shows that ‘even the smallest of changes results in a change to our response,’ and how.
So yes, switching to y = 6a also shows difference and change, but the change is so big it is much more difficult to attend to; lots of things changed, what was I supposed to pay attention to? It might be okay, but greater risk of cog overload. So the difference principle makes it much more likely that *every* child will successfully understand and follow what is happening.
Engelmann puts it like this: "By beginning with differences, we show how a small change in the example controls a corresponding small change in the response." - not sure if that's any clearer.
"It seems you've deviated from Theory of Instruction in this which suggests it is preferable to model the first 2 to 5 with minimum difference variations, to include at least 2 test examples as minimum difference variations, and then potentially have maximum difference variations that learners attempt (which forms part of the expansion sequence)"
You're right, I haven't followed Theory perfectly with this example, which says:
"Follow the first examples with 3 to 6 minimum-difference examples arranged progressively. Create each by making only one small change from the preceding example. Do not repeatedly change the preceding example in the same way.
Follow the minimum differences with a series of juxtaposed examples that are as different as possible within the constraints of the setup"
Mainly it's just that I haven't introduced testing and expansion sequences yet, so I need a sequence here that shows the principles at work.
It's also succinct, which is probably a good idea for most readers - enough to get across the core idea.
A bit more detail below:
This is a part of the 'nuance' I talked about. In general, we have two goals:
(1) success
(2) efficiency
A communication that achieves the same success in less time is more efficient, and better.
Engelmann says it's sometimes better not to 'overatomise,' to start with a communication less likely to be 100% successful, if it's more efficient. If it turns out that it works, that you just achieved a more efficient communication. If it doesn't, you can adjust it and try again.
Go the other way and start with something more likely to definitely succeed, but maybe a bit slower, and assuming it succeeds, you'll never know if a more efficient communication might have worked.
In general Engelmann and those who worked closely with him all hew close to the maxim 'if it works, it works; don't overanalyse it.'
In this case, I've found that the instructional component of a transformation sequence might need only a single example - very efficient. Three tends to be a decent rule of thumb. If I look at the sequence above and ask 'do I think it'll work for 100% of kids,' I'm very confident it will. If it doesn't, it could always be adjusted.
So there's something in here around efficiency. There are definitely times I do have more minimal difference before moving to maximal difference (to show sameness.)
Some of it's down to knowing the group. If I were working with kids I was confident would get it from the actual sequence above, then we could move into expansion very quickly, substituting horrible looking things like x = (a/b)^2/3 or changing the expressions in various ways (including multiple instances of the same letter.)
Yes your rationale makes perfect sense for why to show those 3 examples - first trial being potentially a bit too hard allows you to get to efficient communication much more quickly than starting with examples that are too easy and potentially are slower to communicate the rule and potentially more likely to induce stipulation (E.g. that you can only substitute for numbers). And certainly it will come down to a judgement of the class too and you can respond in the moment if you have gone too hard. I think I will change some of my sequences based on this conversation.
I still don't yet believe that the sameness or difference principles apply here or to any transformation as every change in the input results in a change to the output. The difference principle is that when a minimal change results in a change in label then we narrow down what causes a change in the label and by extrapolation can rule out as many possible reasons for the change in label as possible. Whereas the sameness principle is that a great change in input resulting in no change to the label implies a large number of possible cases for which the label still applies by interpolation. But perhaps my understanding of the principles is limited to (categoricals)?
Yeah exactly - how the principles apply to transformations is actually a bit different to categoricals, which are a lot more intuitive.
Took me a long time to make any sense of it, especially since the language is confusing (‘sameness principle means make things very different’ 🤔)
It sounds like you’ve read Theory, so Engelmann and Carnine talk through this around loc 3602 (kindle version,) though maybe not much more helpful.
Have you come across the book Could John Stewart Mill Have Saved Our Schools?
It’s very short, and in it Engelmann talks through how their five principles appear identical to the five methods of experimental inquiry of Mill.
For categoricals we’re trying to draw boundary lines acting what the concept can and cannot be. Easy to picture.
For transformations we’re not doing that. We’re trying to show what must change, at a minimum, and then something about how it might change, at a maximum. Harder to grasp intuitively.
Any help?
Also I’d be very keen to see how you change your seating designs, if you’re happy to share?
Thanks for this.
To me it seems that the examples you present seem reasonable in showing the transformation rule here (replace the letter with its given value in brackets) and show a good range of variation which would hopefully be sufficient to induce generalisation.
I don't quite understand what you say about the sameness and difference prompts. If changing from x = 3 to x = 31 leads to a difference in the way we treat the example, then changing from x = 31 to y = 6a also leads to a difference in the way we treat the example.
It seems you've deviated from Theory of Instruction in this which suggests it is preferable to model the first 2 to 5 with minimum difference variations, to include at least 2 test examples as minimum difference variations, and then potentially have maximum difference variations that learners attempt (which forms part of the expansion sequence). My guess is that you've done that to a) reinforce important idea of minimal and maximal difference prompts , b) make it more consistent with approach for ‘Categoricals' and hence make it easier for teachers to remember and c) to reduce the time spent on the modelled examples, hoping that the rule has already been made sufficiently clear so showing the maximal variation in your example is more likely to induce the desired generalisation. Is that correct? Under what conditions would you err more on doing it in the manner prescribed in ToI?
Thanks, very much enjoying the blog.
Hey Jack. Thanks for the comment, and the kind words!
“If changing from x = 3 to x = 31 leads to a difference in the way we treat the example, then changing from x = 31 to y = 6a also leads to a difference in the way we treat the example.”
Yes, but the change from 3 to 31 is a minimal difference. The difference principle is about showing small, minimal differences between examples, and then showing how that leads to a difference in how we respond. It shows that ‘even the smallest of changes results in a change to our response,’ and how.
So yes, switching to y = 6a also shows difference and change, but the change is so big it is much more difficult to attend to; lots of things changed, what was I supposed to pay attention to? It might be okay, but greater risk of cog overload. So the difference principle makes it much more likely that *every* child will successfully understand and follow what is happening.
Engelmann puts it like this: "By beginning with differences, we show how a small change in the example controls a corresponding small change in the response." - not sure if that's any clearer.
"It seems you've deviated from Theory of Instruction in this which suggests it is preferable to model the first 2 to 5 with minimum difference variations, to include at least 2 test examples as minimum difference variations, and then potentially have maximum difference variations that learners attempt (which forms part of the expansion sequence)"
You're right, I haven't followed Theory perfectly with this example, which says:
"Follow the first examples with 3 to 6 minimum-difference examples arranged progressively. Create each by making only one small change from the preceding example. Do not repeatedly change the preceding example in the same way.
Follow the minimum differences with a series of juxtaposed examples that are as different as possible within the constraints of the setup"
Mainly it's just that I haven't introduced testing and expansion sequences yet, so I need a sequence here that shows the principles at work.
It's also succinct, which is probably a good idea for most readers - enough to get across the core idea.
A bit more detail below:
This is a part of the 'nuance' I talked about. In general, we have two goals:
(1) success
(2) efficiency
A communication that achieves the same success in less time is more efficient, and better.
Engelmann says it's sometimes better not to 'overatomise,' to start with a communication less likely to be 100% successful, if it's more efficient. If it turns out that it works, that you just achieved a more efficient communication. If it doesn't, you can adjust it and try again.
Go the other way and start with something more likely to definitely succeed, but maybe a bit slower, and assuming it succeeds, you'll never know if a more efficient communication might have worked.
In general Engelmann and those who worked closely with him all hew close to the maxim 'if it works, it works; don't overanalyse it.'
In this case, I've found that the instructional component of a transformation sequence might need only a single example - very efficient. Three tends to be a decent rule of thumb. If I look at the sequence above and ask 'do I think it'll work for 100% of kids,' I'm very confident it will. If it doesn't, it could always be adjusted.
So there's something in here around efficiency. There are definitely times I do have more minimal difference before moving to maximal difference (to show sameness.)
Some of it's down to knowing the group. If I were working with kids I was confident would get it from the actual sequence above, then we could move into expansion very quickly, substituting horrible looking things like x = (a/b)^2/3 or changing the expressions in various ways (including multiple instances of the same letter.)
Am I making any sense?