[heymijo]

Durable and flexible knowledge...

After reading your comment and ansible's reply [0] I wanted to pause and comment on this.

The United States Air Force Academy found that its cadets who took their first calculus class with a professor who focused on conceptual understanding helped those cadets create a durable and flexible understanding of the math [1].

The kicker is that the cadets got worse scores in Calculus I and gave professors who taught in this way worse ratings.

Ansible's anecdotal reply is what a lot of students experience. A feeling of initial success with the material, but they later find that their knowledge of it was fleeting and inflexible. What the Air Force Academy study found was that professors who taught in the manner ansible described, that resulted in fleeting and inflexible knowledge, were rated higher by their students. Those students got better initial scores in Calculus I, but went on to do worse in later calculus courses and related courses.

I encourage you to read the study. It is as good of a study design and execution you can get in the social sciences.

David Epstein also discusses the study in Chapter 4 of his book, Range [2].

[0] https://news.ycombinator.com/item?id=23154241 [1] http://faculty.econ.ucdavis.edu/faculty/scarrell/profqual2.p... [2] https://www.goodreads.com/book/show/41795733-range

[romwell]

When I taught Calculus, I taught understanding over memorizing steps to arrive at a solution for a particular type of problem.

The very best students loved it, but most of the people didn't like it at all.

With mathematics, like with gym, you gain when you put in effort. Most people don't enjoy either.

[thr0w__4w4y]

Yes indeed. Outside of work, I'm an endurance sports person, so basically performance is correlated strongly with training hard and suffering. There is a saying, "Pain is weakness leaving the body", I first heard it in high school (team went on to win a state championship in a highly competitive state). When I was suffering on workouts I just pictured myself getting stronger.

OK hopefully I didn't get too far afield. To me, the analogous concept in learning, particularly in technical fields, is that "learning is ignorance leaving the mind".

In college, particularly math and physics, I /always/ focused on understanding the underlying principles. Initially it was out of fear that if I forgot the formulas, I could re-derive them. But a strange thing happened... through that process, I developed an intuition and an ability to "see" what formulas and concepts to apply when. Once I got to that point in a problem, "seeing it for what it was", finishing to the solution became busywork.

[heymijo]

You're between a rock and a hard place.

The rock are the incentives, how your performance is measured, and the short duration you will have teaching these students.

The hard place is students who have likely spent 13 years in K-12 learning without understanding and are now being asked to do engage in practices they have little to no experience with.* They also have incentives to get good grades and a good GPA, which can be at odds with actual learning.

*To get more concrete, the practices have a name--Standards for Mathematical Practice (SMPs). The National Council of Teachers of Mathematics developed them and considers them the "heart and soul" of the Common Core Mathematics Standards. Not only are these practices absent from most classrooms, all too many teachers are not even aware of them! (see my Notch Generation reply to Sriram to understand why)

https://www.teacherstep.com/breaking-down-the-common-cores-8...

[jkmcf]

Did you happen to explain why you were teaching this way?

[sriram_malhar]

Very interesting. I don't understand why a teaching system cannot incorporate both a conceptual understanding as well as hands-on applied knowledge. Is it a matter of the time available?

[heymijo]

Hi Sriram, a teaching system can incorporate both!

Apologies if my original reply made it seem like it can't.

Why don't teaching systems in America incorporate both the majority of the time?

Two major reasons:

1. Cultural inertia. Most teachers emulate the pedagogy that they experienced in their schooling. Some are aware that you can try to mix conceptual+procedural and try to. I call them the "notch generation"- trying to teach in a way that is different than they were taught. It's hard to do because...

2. The system is not designed to accommodate it. Incentives and higher order effects all conspire with cultural inertia to thwart it.

[ZeikJT]

#2 bothered me so much in school. The system gauges success via tests that check short term learning. It really, really isn't good at measuring learning.

[apricot]

I always did very well on tests at school, but I wasn't really learning anything, or more precisely, I wasn't learning how to learn. I was learning how to pass tests, but that's a rather useless skill to have. I had to learn learning as an adult, and it was more difficult than if I had to learn it as a child.

[heymijo]

Hey man, that really sucks, and I'm sorry to hear it. I have a bunch of follow-up questions I'm curious about. I know HN isn't the best way to track replies. I've got heymijo.hn at gmail set up if you want to shoot me an e-mail.

I've worked in both K-12 and post-secondary education, studied the history of education reform in the United States, and visited schools/met teachers/students/etc that I've connected with across the U.S.

I'm always interested in hearing someone's story about school, how it did/didn't meet their needs, and how it has impacted them.