Am a parent of a 7th Grader (of Cambridge curriculum) in India :) Your posts are wonderful and helps me in the little coaching I do to my daughter. One specific thing that has aroused my curiosity recently is the "need for understanding" vs "let the intuition figure it out as shortcuts". For example, in your circumference case - Would it be OK to get it wrong a few times with "Multiply the number given with 3.14" instead of squeeze the concept of 3.14 times the diameter?
In a recent podcast interview, a mathematician suggests to let intuition take over and figure out over time - He calls it System-3. He of course didn't have the specific circumference example and I am overlaying that. I am keen to hear to your thoughts on this. The podcast is: https://www.econtalk.org/a-mind-blowing-way-of-looking-at-math-with-david-bessis/
Thanks! I would agree with the sentiment that it's ok to multiply by 3.14 without thinking too hard about it while a student is first getting a sense for circumference. I don't think understanding always needs to come first, and it can feel overwhelming to juggle too many ideas at once.
I do think that should sit in a larger trajectory. I review area and perimeter of rectangles and triangles before doing circles, so students have some solid prior knowledge to build on (circumference is like perimeter, etc). Then we do a bunch of practice drawing circles with different radii and diameters and converting between radius and diameter so that little piece is solid. Then when we first get to circumference, what you describe is totally fine. The key, as soon as students are solid with the procedure, to add more variety to the problems. Find circumference given diameter. Find diameter given circumference. Solve a word problem where you have to distinguish whether it's asking for circumference or area. Eventually solve tougher problems, like the distance a wheel rolls in 4 revolutions (so you have to multiply circumference by 4) or a runner who goes 2 1/2 laps on a circular track. Move between circumference and area. Etc.
To me, introducing a procedure without worrying about understanding is fine because understanding happens gradually over time, as part of a much larger sequence. The key is not to settle for "multiply by 3.14" as sufficient understanding. Get that procedure solid, then have students apply the idea lots of different ways.
Yup, agree with this. It isn’t a mutually exclusive thing. I like the “multi-stranded” curriculum way you had described earlier. Applying something like that at a micro-level here could help too. It gets overwhelming as a parent, but I am super glad teachers like you deliberately think through these and share notes :)
Another great post, Dylan. Laughing because I just posted on Substack about how so much about retrieval practice uses math examples. I'm always looking for more examples in history/social studies!
Yea my uninformed take from the outside is that retrieval is important in all subjects, but the other strategies (especially connections) are especially important in history because historical knowledge is so much about the relationships between facts and ideas.
One other tactic that is really good for increasing retention is teaching. I don’t know why, but it feels like the combination of retrieval, attention, connection to relevant concepts, and repetition all make teaching a subject a very effective retention tool.
Unfortunately I don’t really know of an organic way to get students to teach; perhaps if, after you’ve done enough review on a subject that you think students have got a good grasp, some system where they function as TAs for the previous grade could work (obviously this is hard/impossible to implement in a normal school)
Yea I agree - I think teaching forces students to think about the meaning of the material which is the active ingredient. But I agree it's a tough thing to make happen. I don't know if this is ideal, but lots of quick turn and talks can be a good way to get small chunks of reciprocal teaching in class without spending tons of time setting up a complicated system that may or may not work.
I'm a huge believer in the power of flashcards. Plenty of research supporting their effectiveness, both directly (students using flashcards) or indirectly (students using either of recall or spacing - the two ingredients of flashcards).
What's most alarming I think is how the majority of students rely on the least effective techniques for studying. With the rise of everything being a ChatGPT away, there's a very large risk here that students studying techniques will become even worse - or nonexistent.
I agree that schools can do a better job of teaching students how to study. But flashcards are one of those things that are easier said than done in schools. Physical flashcards are easy to lose. Digital flashcards bring in all the challenges of technology. And lots of procedural skills like solving equations, multi-digit arithmetic, etc don't lend themselves well to flashcards. At some point, the teacher needs to structure some effective spaced practice for students.
Oh yeah I definitely wouldn't recommend physical flashcards in this day and age for several reasons.
Out of interest though how come you don't think flashcards are suited to equations etc? All they are are spaced questions. Funnily enough I actually wrote a short post on this specific point just the other day.
Two reasons. First, solving an equation is a procedure that is best learned by doing the procedure. Flashcards would either a) ask about the generalization (how do you solve an equation like...?) or b) ask about a specific equation, and I want students solving lots of equations. I could have a whole bunch of flashcards for different equations, but that becomes inefficient. Second, I want students to both solve a lot of equations and to generalize what they're doing to new situations. That means gradually increasing the difficulty, from predictable one-step equations to interleaved one-step equations to one-step equations with fractions and decimals to simple two-step equations, etc. I want to vary the order terms are written in, increasingly use more challenging numbers, work in negatives, etc. I think flashcards could be used to do that but I think good old-fashioned pencil-and-paper practice designed by a teacher is a better approach.
A flashcard is just a question. That question can be whatever the user/student/teacher wants it to be. It could be those formats above or it could also be other formats. Some of those formats I think will be more effective, and others less so.
Genuinely curious as to whether you'd still find the above cards 'bad(?)', and if so, why? All of these would be asked in exams, by teachers - or for the latter in interviews etc.
--
(PS: I agree that having a tutor alongside is better - of course. But even better than that would be having, say, a tenured professor from Harvard. Unfortunately these things aren't always possible. Everything comes down to context and tradeoffs)
My point is that if I want students to be fluent in solving equations like 4x = 12, there's a risk that they focus on the specific equations in the flashcards rather than generalizing to any equation. I could include a bunch of those types of equations, but there are hundreds of possibilities and it's overwhelming and inefficient to include them all for spaced repetition. I think if I was designing materials for someone to learn on their own, I would be more interested in solving that problem but to me the variation and sequencing in questions is a key feature that I can manage in a classroom with paper and pencil.
>>> "there's a risk that they focus on the specific equations in the flashcards"
- Indeed if no New cards are ever made - this applies if the teacher were to use the same questions (ironically many students still would not be able to remember). Digital flashcards with todays technology allow for a single-click to "~update/change/reword this question" whilst maintaining the scheduling. The same cannot be said without many extra steps or points of friction for paper cards and/or asking a teacher in class etc.
>>> "it's overwhelming and inefficient to include them all for spaced repetition"
This would apply to any non-flashcard study. It's not an all-else-equal comparison.
>>> "the variation and sequencing in questions is a key feature that I can manage in a classroom"
See earlier point re the optimal environment (in a classroom with a teacher) not always being possible. In fact, a student solely studying/learning when in the classroom with a teacher engaging them would be counterproductive.
Remember that once the student has progressed enough and the question becomes trivial to solve it can be discarded with one-click. One problem with traditional class-based progression is that this is actually assumed for all students (equally). This could though be remedied somewhat via appropriate flashcards.
--
If MathAcademy were to fix a few things re how their system works then they may be a good option for some financially-comfortable families/individuals - but only primarily because they maintain scheduling whilst changing the questions.
I'd very strongly recommend digital over physical for a few reasons, personally:
- Huge advances in algorithmic scheduling (user-optimised FSRS being the current 'best*')
- Easier to keep track of Review Cards
- Easier to create and update New Cards
- Ability to look things up in-app
- Ability for audio control (voice, speed, background sounds etc)
- Easier to do whilst out (on the train, queuing, morning walk etc)
So I should update my earlier reply: I'd recommend over nothing, but not over digital.
--
*There's some debate (read: algorithm wars) currently between whether SM19 is better or not but it's closed source and I think only compatible with the latest SM (?)
Aha. Well, that's the point: you don't need to know. You just use FSRS and relax knowing it's running in the background - similar to how most of us view plane/car engines. With paper cards, the longer one uses the more time one is wasting with inefficient scheduling (along with what I believe to be other substantial drawbacks)
Re the handwriting, I touched on that in the "Arguments Against Flashcards" near the bottom. I agree that the actual process of making the cards (be it hand-written or typed) oneself is likely offering an additive effect, I'm just quite certain after being quite deep in this space that the squeeze is not worth the juice - and especially for practical language learning.
--
SM = SuperMemo. An *incredibly* complex piece of SRS developed by Piotr Wozniak. Piotr created the very first scheduling algorithm used in most apps today so is kind of the pioneer behind all this. Since his initial algorithm of course there's been loads of new ones by different researchers trying to find the best.
Am a parent of a 7th Grader (of Cambridge curriculum) in India :) Your posts are wonderful and helps me in the little coaching I do to my daughter. One specific thing that has aroused my curiosity recently is the "need for understanding" vs "let the intuition figure it out as shortcuts". For example, in your circumference case - Would it be OK to get it wrong a few times with "Multiply the number given with 3.14" instead of squeeze the concept of 3.14 times the diameter?
In a recent podcast interview, a mathematician suggests to let intuition take over and figure out over time - He calls it System-3. He of course didn't have the specific circumference example and I am overlaying that. I am keen to hear to your thoughts on this. The podcast is: https://www.econtalk.org/a-mind-blowing-way-of-looking-at-math-with-david-bessis/
Thanks! I would agree with the sentiment that it's ok to multiply by 3.14 without thinking too hard about it while a student is first getting a sense for circumference. I don't think understanding always needs to come first, and it can feel overwhelming to juggle too many ideas at once.
I do think that should sit in a larger trajectory. I review area and perimeter of rectangles and triangles before doing circles, so students have some solid prior knowledge to build on (circumference is like perimeter, etc). Then we do a bunch of practice drawing circles with different radii and diameters and converting between radius and diameter so that little piece is solid. Then when we first get to circumference, what you describe is totally fine. The key, as soon as students are solid with the procedure, to add more variety to the problems. Find circumference given diameter. Find diameter given circumference. Solve a word problem where you have to distinguish whether it's asking for circumference or area. Eventually solve tougher problems, like the distance a wheel rolls in 4 revolutions (so you have to multiply circumference by 4) or a runner who goes 2 1/2 laps on a circular track. Move between circumference and area. Etc.
To me, introducing a procedure without worrying about understanding is fine because understanding happens gradually over time, as part of a much larger sequence. The key is not to settle for "multiply by 3.14" as sufficient understanding. Get that procedure solid, then have students apply the idea lots of different ways.
Yup, agree with this. It isn’t a mutually exclusive thing. I like the “multi-stranded” curriculum way you had described earlier. Applying something like that at a micro-level here could help too. It gets overwhelming as a parent, but I am super glad teachers like you deliberately think through these and share notes :)
Thanks for this post. I've been thinking a lot about thinking. Here's my take, Daniel Willingham Reminded Me That Memory Is the Residue of Thought (https://harriettjanetos.substack.com/p/daniel-willingham-reminded-me-that?r=5spuf).
Love it. Memory is the residue of thought is a classic. So much of teaching boils down to that idea.
Another great post, Dylan. Laughing because I just posted on Substack about how so much about retrieval practice uses math examples. I'm always looking for more examples in history/social studies!
Yea my uninformed take from the outside is that retrieval is important in all subjects, but the other strategies (especially connections) are especially important in history because historical knowledge is so much about the relationships between facts and ideas.
One other tactic that is really good for increasing retention is teaching. I don’t know why, but it feels like the combination of retrieval, attention, connection to relevant concepts, and repetition all make teaching a subject a very effective retention tool.
Unfortunately I don’t really know of an organic way to get students to teach; perhaps if, after you’ve done enough review on a subject that you think students have got a good grasp, some system where they function as TAs for the previous grade could work (obviously this is hard/impossible to implement in a normal school)
Yea I agree - I think teaching forces students to think about the meaning of the material which is the active ingredient. But I agree it's a tough thing to make happen. I don't know if this is ideal, but lots of quick turn and talks can be a good way to get small chunks of reciprocal teaching in class without spending tons of time setting up a complicated system that may or may not work.
I'm a huge believer in the power of flashcards. Plenty of research supporting their effectiveness, both directly (students using flashcards) or indirectly (students using either of recall or spacing - the two ingredients of flashcards).
What's most alarming I think is how the majority of students rely on the least effective techniques for studying. With the rise of everything being a ChatGPT away, there's a very large risk here that students studying techniques will become even worse - or nonexistent.
I agree that schools can do a better job of teaching students how to study. But flashcards are one of those things that are easier said than done in schools. Physical flashcards are easy to lose. Digital flashcards bring in all the challenges of technology. And lots of procedural skills like solving equations, multi-digit arithmetic, etc don't lend themselves well to flashcards. At some point, the teacher needs to structure some effective spaced practice for students.
Oh yeah I definitely wouldn't recommend physical flashcards in this day and age for several reasons.
Out of interest though how come you don't think flashcards are suited to equations etc? All they are are spaced questions. Funnily enough I actually wrote a short post on this specific point just the other day.
Two reasons. First, solving an equation is a procedure that is best learned by doing the procedure. Flashcards would either a) ask about the generalization (how do you solve an equation like...?) or b) ask about a specific equation, and I want students solving lots of equations. I could have a whole bunch of flashcards for different equations, but that becomes inefficient. Second, I want students to both solve a lot of equations and to generalize what they're doing to new situations. That means gradually increasing the difficulty, from predictable one-step equations to interleaved one-step equations to one-step equations with fractions and decimals to simple two-step equations, etc. I want to vary the order terms are written in, increasingly use more challenging numbers, work in negatives, etc. I think flashcards could be used to do that but I think good old-fashioned pencil-and-paper practice designed by a teacher is a better approach.
There might be some confusion here.
A flashcard is just a question. That question can be whatever the user/student/teacher wants it to be. It could be those formats above or it could also be other formats. Some of those formats I think will be more effective, and others less so.
For kids, potentially: https://snipboard.io/IdBM9X.jpg
For older students, potentially: https://snipboard.io/9daWPl.jpg
Genuinely curious as to whether you'd still find the above cards 'bad(?)', and if so, why? All of these would be asked in exams, by teachers - or for the latter in interviews etc.
--
(PS: I agree that having a tutor alongside is better - of course. But even better than that would be having, say, a tenured professor from Harvard. Unfortunately these things aren't always possible. Everything comes down to context and tradeoffs)
My point is that if I want students to be fluent in solving equations like 4x = 12, there's a risk that they focus on the specific equations in the flashcards rather than generalizing to any equation. I could include a bunch of those types of equations, but there are hundreds of possibilities and it's overwhelming and inefficient to include them all for spaced repetition. I think if I was designing materials for someone to learn on their own, I would be more interested in solving that problem but to me the variation and sequencing in questions is a key feature that I can manage in a classroom with paper and pencil.
>>> "there's a risk that they focus on the specific equations in the flashcards"
- Indeed if no New cards are ever made - this applies if the teacher were to use the same questions (ironically many students still would not be able to remember). Digital flashcards with todays technology allow for a single-click to "~update/change/reword this question" whilst maintaining the scheduling. The same cannot be said without many extra steps or points of friction for paper cards and/or asking a teacher in class etc.
>>> "it's overwhelming and inefficient to include them all for spaced repetition"
This would apply to any non-flashcard study. It's not an all-else-equal comparison.
>>> "the variation and sequencing in questions is a key feature that I can manage in a classroom"
See earlier point re the optimal environment (in a classroom with a teacher) not always being possible. In fact, a student solely studying/learning when in the classroom with a teacher engaging them would be counterproductive.
Remember that once the student has progressed enough and the question becomes trivial to solve it can be discarded with one-click. One problem with traditional class-based progression is that this is actually assumed for all students (equally). This could though be remedied somewhat via appropriate flashcards.
--
If MathAcademy were to fix a few things re how their system works then they may be a good option for some financially-comfortable families/individuals - but only primarily because they maintain scheduling whilst changing the questions.
I think physical flashcards are fine! Curious why you don't recommend them?
https://laurenbrownoned.substack.com/p/what-im-learning-about-learning-from
I'd very strongly recommend digital over physical for a few reasons, personally:
- Huge advances in algorithmic scheduling (user-optimised FSRS being the current 'best*')
- Easier to keep track of Review Cards
- Easier to create and update New Cards
- Ability to look things up in-app
- Ability for audio control (voice, speed, background sounds etc)
- Easier to do whilst out (on the train, queuing, morning walk etc)
So I should update my earlier reply: I'd recommend over nothing, but not over digital.
--
*There's some debate (read: algorithm wars) currently between whether SM19 is better or not but it's closed source and I think only compatible with the latest SM (?)
Got it. Don’t know what SM19 or SM is?? And had to look up FSRS, so maybe that tells you why I prefer physical ones. 🤣
Also, for me, MAKING the flashcards by hand helps is another retrieval point.
Aha. Well, that's the point: you don't need to know. You just use FSRS and relax knowing it's running in the background - similar to how most of us view plane/car engines. With paper cards, the longer one uses the more time one is wasting with inefficient scheduling (along with what I believe to be other substantial drawbacks)
Re the handwriting, I touched on that in the "Arguments Against Flashcards" near the bottom. I agree that the actual process of making the cards (be it hand-written or typed) oneself is likely offering an additive effect, I'm just quite certain after being quite deep in this space that the squeeze is not worth the juice - and especially for practical language learning.
--
SM = SuperMemo. An *incredibly* complex piece of SRS developed by Piotr Wozniak. Piotr created the very first scheduling algorithm used in most apps today so is kind of the pioneer behind all this. Since his initial algorithm of course there's been loads of new ones by different researchers trying to find the best.
SM: https://supermemo.store/products/supermemo-19-for-windows
Piotr: https://supermemo.guru/wiki/Piotr_Wozniak
Blog: https://supermemo.guru/wiki/SuperMemo_Guru
FSRS (older version) vs Others: https://imgur.com/a/benchmark-fVxiJvx