
A Randomized Controlled Trial of Interleaved Mathematics Practice [pdf] - mpweiher
http://uweb.cas.usf.edu/~drohrer/pdfs/Rohrer_et_al_InPressJEdPsych.pdf
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RobertoG
Somehow related to this: I think that the linear way of teaching, first week
you teach a, second week b.. it's inferior to showing everything to the
student, in a very shallow way, the sooner the better, and then keep adding
deepness.

In the same way that muscle need to rest to grow, knowledge need to rest to
stay with you. Even if subjectively seems that nothing has been learned the
first time that you are in contact with a subject, that is hardly ever truth.
Even if only the name of the concepts are familiar that's some progress from
where it's possible to build solid knowledge.

~~~
Arun2009
I too am a huge believer in breadth-first learning rather than depth-first,
and my reasoning is the same: in my experience, ideas need to stay with you
for a while before they take root in your mind and you get used to them.

The big ideas in any Mathematics subject are usually very few. They can be
covered at a high-level in a few classes before going into them in detail.
This also gives you a broader context for you to place the details as you come
across them.

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bjourne
I think the best way to practice math has already been discovered. Get a book,
pen and paper, and practice math problems that are very challenging to hard
for you. Do this day out and day in.

Problem? ... It's not very fun. You always feel frustrated and dumb because
you don't understand it. So you'll eventually stop practicing at all because
you feel like you don't make any progress.

Same thing with chess; play only with Grand masters (if you can find them) and
after a few years of getting your ass kicked, you'll eventually get good.
(Disclaimer: I only skimmed the article)

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whatshisface
> _Same thing with chess; play only with Grand masters (if you can find them)
> and after a few years of getting your ass kicked, you 'll eventually get
> good._

Wouldn't the grand masters not enjoy beating you over and over again? That's
why Go has a handicap system and why online games have matchmaking systems. I
think it would be more courteous to try and play with people that could
potentially learn something from you, instead of you only learning from them.

~~~
saagarjha
> Wouldn't the grand masters not enjoy beating you over and over again?

I mean, a couple times maybe? At some point I feel like they’d think this is a
waste of time.

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moosey
This is aligned with everything I learned in Dr.'s Oakley and Sejnowski class
entitled "Learning How to Learn" on Coursera. Conceptually, the class is
ridiculously simple,but with some of the most powerful mental tools available
for learning subjects. This interleaving tool is included, and it should be
noted that this is part of a self-testing phase of learning, which is
something that they want you to leap into as quickly as possible.

I would strongly recommend the course to anyone interested in learning and who
doesn't have a good learning process, because that's all that it ultimately
is: a set of steps that you use to train your brain to remember a body of
knowledge.

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nathanasmith
I've seen interleaving research like this before and it's my understanding
interleaving trains not just the ability to work a particular problem but the
ability to better recognize the _class_ of problem.

Anecdotally, after reading a paper on this, I made a Python script to drill
guitar pieces, chords, progressions, etc. in an interleaved fashion and I
believe it has helped me become a better player. Before doing this, when I
would practice, e.g., scales, I might start in A then B, C, etc. but what I
noticed was after doing the first scale, the rest were just minor variations
so my training went from deliberate to auto-pilot. Now, with interleaving, I
have to give some brief thought not just to how to mechanically perform the
piece but what exactly the piece being asked for is. Consequently, during a
performance, my ability to go from an intention of doing something on the
guitar to actually plucking the first string is much more comfortable and
faster.

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busyant
In the mid-1980s, I used my trusty Commodore 64 to build "test banks" for my
math, chem, physics, and bio courses in high school and early college.

My rationale was the same as in this paper: I wanted to handicap myself from
knowing the "type" of question I was trying to answer and force myself to
think, "Ok. Where is this question 'located' on the map of everything I've
been taught? And once I know its location, where do I need to go to get to its
answer?"

So, I hand-entered every question I could get my hands on and I had the
computer throw random problems/questions at me before an exam.

The problem w/ my approach was that I had to hand-enter problems from my
textbooks, so there was a decent chance that I would still remember the type
of question simply from the act of typing it into my 'database.' But it was
the best I could do.

I teach some college chemistry now and I think interleaving would be helpful
to my students. I say this because one of the biggest issues on exams is that
the students will have no idea what _type_ of problem they are looking at. If
they correctly determine the type of problem, the remaining work is relatively
easy. And I think the problem stems from students practicing the work by
answering multiple problems that belong to the same category.

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jbotz
Interleaving different types of problems is one of several "evidence based"
study techniques described in the excellent book "Make it stick" by Peter
Brown. Highly recommended; this book is based on the latest and most solid
research on learning and dispels a lot of long-time misconceptions.

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zby
I am affraid this might be
[https://en.wikipedia.org/wiki/Hawthorne_effect](https://en.wikipedia.org/wiki/Hawthorne_effect)

I don't see anything controlling for that in the setup:

"The present study examined interleaved practice in a large number of classes
at multiple schools over a period of five months, and all instruction was
delivered solely by teachers who had no prior association with the
intervention or the authors."

~~~
0xab
There's a control group. Both groups were watched by the experimenters. Not
only that, the same teachers had students in both groups in different classes.

~~~
zby
But the control group would know that nothing has changed. The group testing
the new method would feel the excitement - the control group would not.

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vbuwivbiu
I believe that interleaving is a principle to applied to life in general

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faitswulff
Say you want to learn something - let's just say math or a new language - how
do you go about interleaving the learning materials?

~~~
joshvm
Spaced repetition aids are fairly common. If you can store your curriculum as
flash cards, then apps like Memrise and Anki are designed specifically for
this.

Spaced repetition is not the same as interleaving, but the apps will force you
to interleave knowledge anwyay.

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projectramo
I think this is why practicing on the actual test (SAT, GMAT, CFA etc) is so
much more effective than exercises on particular sections.

~~~
kwhitefoot
> so much more effective than exercises on particular sections

Effective at what though. Just passing the test?

Do the students actually understand the material? Can they apply it to novel
(to them) situations?

When I was studying physics ('74-'77) a significant number of students who all
had just as good test results as I did in senior high school dropped out after
the first term because they just couldn't manage the material. They had learnt
enough to pass the exams (British A-levels) but had not really understood and
had not practised enough. I suspect that rise of modular courses might make
this situation even worse but that might be prejudice on my part.

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visarga
It somehow makes me remember about spaced repetition and Anki.

~~~
gwern
The testing effect, the spacing effect, and the interleaving effect are all
closely related. For example, if you use spacing to learn various items of a
subject, they will naturally be interleaved over time.

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luckydata
I keep finding most of those papers obvious on their face and quite simple to
derive from first principles and some observation. What happened to teaching
that so many patently obvious concepts have to be "rediscovered" so often?

~~~
jimhefferon
Let's say I'm teaching 50 min periods. 15 mins for questions. Half hour to
introduce new material, with a couple of examples. Five mins left. Interleave
now?

Everyone has sat through 12 or 16 years of school. That can lead them to think
they know all about it. I've been teaching since 79 and I'm still learning a
lot,all the time. It ain't obvious.

~~~
aethertap
I've only been actively teaching (at home, no less) for a couple of years, but
I can definitely identify with this. I make index cards with each kind of
question we encounter in the curriculum, then I draw some random cards from
the deck in order to pick questions going all the way back to the start. We do
a quiz every week for every subject that's made up of those questions.

It always comes down to time though, it seems like you can use all these ideas
and teach for mastery, but it costs about 30% in terms of forward motion
through the material. I'm lucky in that that I can just decide to pay that
price, but it would be a real stretch to put all this into practice in a place
where you have a coverage deadline that's already hard to hit.

