on understanding

[ajschumacher]

mnemonics are a bad "code smell" https://en.wikipedia.org/wiki/Mnemonics_in_trigonometry#All_Students_Take_Calculus is gross; SOH CAH TOA is maybe okay

in the case of ASTC, why even know that as an independent thing to know? why spend time memorizing it rather than just thinking it through from what sine etc. are? much better to strengthen that connection rather than create isolated facts

[ajschumacher]

"understanding is the best mnemonic"

yeah, more examples of dumb mnemonics here: https://en.wikipedia.org/wiki/Mnemonic

[ajschumacher]

If there's a reason, that should be the guide. If something is really arbitrary (a convention, etc.) then maybe a mnemonic is useful. But even "arbitrary" things can have more connections/reasons: historical context, like "this happened after this other thing" can help position dates, for example.

[ajschumacher]

relation to FOIL, in A Mathematician's Lament:

"I was made to learn by heart: “The square of the sum of two numbers is equal to the sum of their squares increased by twice their product.” I had not the vaguest idea what this meant and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way." (quoting Bertrand Russell, pages 61-62)

aha! the original is easy to find online: page 31 of the 1998 edition of Russell's 1951 "Autobiography"

[ajschumacher]

others who have written on this:

• https://yourawesomememory.com/the-dark-side-of-mnemonics/
• https://music.stackexchange.com/questions/788/mnemonics-for-memorizing-the-staff-useful-or-harmful
• https://universeofmemory.com/effectiveness-and-usefulness-of-mnemonics/
  • refs: https://www.psychologicalscience.org/publications/journals/pspi/learning-techniques.html

from that last ref:

"Practice testing and distributed practice received high utility assessments" ... "Elaborative interrogation, self-explanation, and interleaved practice received moderate utility assessments" ... " low utility assessment: summarization, highlighting, the keyword mnemonic, imagery use for text learning, and rereading"

[ajschumacher]

possibly related ideas:

"Catechism of Rules without Reasons" - I'm not sure "catechism" is really the best word, but I thought it sounded catchy: the idea of learning things as if they don't have reasons, and therefore not really knowing when to apply, when there are exceptions, etc.

"Another Kind of Imposter Syndrome" - actually, maybe multiple kinds:

• when you think nobody understands, you don't try to understand
• when you don't understand but feel like you have to fake it, you might be defensive or attack people who question the "catechism" (from above) that you've "learned"

[ajschumacher]

"the frontiers of scientific knowledge, where we do not have intuition to guide us." (page 16, The Book of Why)

sort of an interesting thing to consider... to what extent is that true? why is that?

[ajschumacher]

related: doing simulations (e.g., like https://planspace.org/20200912-what_should_be_in_your_regression/) to check/develop intuition/conclusions

[ajschumacher]

(from Statistical Rethinking)

[ajschumacher]

Thought: Misunderstanding math as a set of rules and procedures to follow without understanding leads to related problems in (especially) statistics: what's the "right" answer, without thinking about what you're doing.

https://planspace.org/20200909-mathematicians_lament_by_lockhart/

[ajschumacher]

"We tend to teach mathematics as a long list of rules. You learn them in order and you have to obey them, because if you don’t obey them you get a C-. This is not mathematics. Mathematics is the study of things that come out a certain way because there is no other way they could possibly be." (page 12)

"The specialized language in which mathematicians converse with one another is a magnificent tool for conveying complex ideas precisely and swiftly. But its foreignness can create among outsiders the impression of a sphere of thought totally alien to ordinary thinking. That's exactly wrong." (page 12)

Ellenberg, in How Not to Be Wrong

[ajschumacher]

"School mathematics is largely made up of a sequence of facts and rules, facts which are certain, rules which come from a higher authority and cannot be questioned." (page 14)

Ellenberg, in How Not to Be Wrong

[ajschumacher]

"Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense—or deciding whether the method is the right one to use in the first place—requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel." (page 56, Ellenberg)

[ajschumacher]

"It has come to my attention that hardly anybody knows what the logarithm is. Let me take a step toward fixing this. The logarithm of a positive number N, called log N, is the number of digits it has." (page 139, Ellenberg)

He goes on to call this the "fake logarithm," of "flogarithm."

Kind of fun to connect this to different bases...

[ajschumacher]

"As always, it can be easier to see what's going on with the math if we make the numbers small enough that we can draw pictures." (page 258, Ellenberg)

[ajschumacher]

"It seems to me then, in repeating a reasoning learned, that I could have invented it. This is often only an illusion; but even then, even if I am not so gifted as to create it by myself, I myself re-invent it in so far as I repeat it."

Poincaré in Mathematical Creation https://www.google.com/books/edition/_/qgkeAAAAMAAJ?hl=en&gbpv=1

[ajschumacher]

Wolfram suggesting that lots of stuff doesn't have good ways for humans to understand:

https://writings.stephenwolfram.com/2018/11/logic-explainability-and-the-future-of-understanding/

[ajschumacher]

"conceptual foundations" (as opposed to axiomatic foundations)

[ajschumacher]

"There is ample evidence, however, that knowledge taught by drill and practice in one format does not transfer. My favorite example is from the New Jersey assessment of 1975-76. When students were asked to add decimals in vertical format, the state percent passing the items was 86%. In horizontal format for identically difficult decimals, the percent passing was 46%. For subtraction of decimals in the two formats the passing rates were 78% and 30%, respectively (New Jersey Department of Education, 1976). Which result is true? My conception of learning requires that students be able to add or subtract in either of these ways and a half-dozen other ways besides."

Should Instruction Be Measurement Driven? A Debate (Lorrie Shepard, 1988)

https://nepc.colorado.edu/publication/should-instruction-be-measurement-driven-a-debate re

ref'ed in The Promise and Pitfalls of Using Imprecise School Accountability Measures (Kane and Staiger, 2002)

https://www.aeaweb.org/articles?id=10.1257/089533002320950993

[ajschumacher]

"In my experience, any learning environment that encourages students to be fast learners is setting its students up for failure." (page xi, Bad Choices by Almossawi)

[ajschumacher]

Mnemonic system: https://en.wikipedia.org/wiki/Mnemonic_peg_system

[ajschumacher]

Learning with Understanding: Seven Principles https://www.nap.edu/read/10129/chapter/8#119 in Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools (2002)

1. Learning with understanding is facilitated when new and existing knowledge is structured around the major concepts and principles of the discipline.
2. Learners use what they already know to construct new understandings.
3. Learning is facilitated through the use of metacognitive strategies that identify, monitor, and regulate cognitive processes.
4. Learners have different strategies, approaches, patterns of abilities, and learning styles that are a function of the interaction between their heredity and their prior experiences.
5. Learners’ motivation to learn and sense of self affects what is learned, how much is learned, and how much effort will be put into the learning process.
6. The practices and activities in which people engage while learning shape what is learned.
7. Learning is enhanced through socially supported interactions.