Math and memorization conversation in education (from another thread)

[DeusAbscondus]

Moving beyond memorizing formulas on faith and starting to prove them or follow a proof is a great feeling. Whenever I tutor math at any level I almost never suggest memorizing something, instead reinforce concepts that apply to many formulas and situations. Most of the time it doesn't work that well - either the student wants a list of "tricks" or the approach needs more time to sink in. I've seriously read through solution manuals for a university entrance exam that say something like "Using tactic D-15 from this book the solution is ________". Anyway, I digress.

The last bit of your integral calculation 2 and 2/3 u-squared. Two things:

1) Where did the u^2 come from?
2) Why did you convert the fraction to a mixed number? Leaving it as an "improper fraction" is perfectly fine and the standard way to write a fraction unless told otherwise. Maybe you wrote it that way to show the area was between 2 and 3 square units.

I tried writing a mixed number in Latex just now but the spacing wasn't right.

Jameson,
1. I thought that because what I have integrated is essentially an area that it should be written as squared units (since the unit of measurement isn't otherwise stipulated)

2. thanks for the tip regarding mixed fractions; duly noted!

"Using Tactic D-15 ..." You encapsulate my distaste for modern teaching methods so nicely! (perhaps D-13, for Doom-13 would have been even better)
I'm sure I'm not telling you anything you don't know here: words carry not only meaning but the history of the meaning(s) as this/these evolve in time. If one throws out words and names of theorems for an algorithmized (pardon the neologism, but it is almost apt to my point here) for a computerized way of communicating, let alone teaching!, well then, the memory of civilization itself is imperilled by a kind of cultural and intellectual Altzheimer's.

Momentarily gloomy ("this too shall pass" )
Regs,
Deus Absconditus
(Dieu_nous_fuit)

 

[MarkFL]

Re: first use of geogebra to do a simple integration

Moving beyond memorizing formulas on faith and starting to prove them or follow a proof is a great feeling. Whenever I tutor math at any level I almost never suggest memorizing something, instead reinforce concepts that apply to many formulas and situations. Most of the time it doesn't work that well - either the student wants a list of "tricks" or the approach needs more time to sink in...

I agree with this wholeheartedly.

I have told students I've tutored that until they can derive a formula, I don't want them to use it. I tell them when they can derive it, then they have earned the "right" to use it.(Nerd)

I have had limited success with such an approach.:D

 

[Jameson]

Re: first use of geogebra to do a simple integration

This thread is about to be derailed but that's fine. I can make a new one and move the posts as needed. :)

Even more so than just memorizing or not memorizing formulas, the whole approach to solving math problems is taught as independent situations that have no connection to each other. I used to tutor for the SAT math and many kids worry about the geometry content even though the amount of formulas one needs to have memorized is quite small. Whenever we started a problem I would ask "What is this problem about in the most general sense?" as well as "Even if you have no idea how to do this problem, what are some thoughts that you had after reading it? Can you make any connections to problems you've seen before or is this question totally new?" and so on.

Unfortunately I only had 10 hours or so over a month to work with this group and this approach was not very successful and a couple of kids called me rude in my evaluation (I'm not rude but being rude is far less worse than being incompetent and nice). They were really put off by me not giving them any hints or formulaic steps, instead asking them lots of questions.

 

[DeusAbscondus]

Re: first use of geogebra to do a simple integration

This thread is about to be derailed but that's fine. I can make a new one and move the posts as needed. :)

Even more so than just memorizing or not memorizing formulas, the whole approach to solving math problems is taught as independent situations that have no connection to each other. I used to tutor for the SAT math and many kids worry about the geometry content even though the amount of formulas one needs to have memorized is quite small. Whenever we started a problem I would ask "What is this problem about in the most general sense?" as well as "Even if you have no idea how to do this problem, what are some thoughts that you had after reading it? Can you make any connections to problems you've seen before or is this question totally new?" and so on.

Unfortunately I only had 10 hours or so over a month to work with this group and this approach was not very successful and a couple of kids called me rude in my evaluation (I'm not rude but being rude is far less worse than being incompetent and nice). They were really put off by me not giving them any hints or formulaic steps, instead asking them lots of questions.

I am studying in a small group which includes just one other student, also a widely-read, experienced man of mature-age; but we are pretty much left to our own devices, with some helpful input from time to time from our instructress, who, as she is the first to admit, has not done much with her calculus for the past 20 years, and so is pretty rusty. Still, she has presided over the biggest challenge to my intellectual life, the greatest fillip to my experience of the "life of the mind" for 20 years and I count myself lucky to have any teacher at all.

But I know I am ripe for some really engaging, dynamic -socratic?- style teaching; perhaps I will be lucky enough to find that on the next leg of my journey, when I graduate to university level study.

Dominus_est_Abscondus
(and He yawns with an eternal, Divine Apathia at our wretched plight down here)

"To exist is to be in a cosmic jam!"
-Michael Fitzgerald, mumbled to his dog, Clyde

 

[Matrix0]

I completely agree with your sentiments about the importance of understanding concepts and not just memorizing formulas. It is crucial for students to not just know how to solve a problem, but to also understand why and how the solution works. This not only leads to a deeper understanding of the subject, but also helps students apply their knowledge to new and different situations.

As for the use of mixed fractions and proper units, I believe it is important to follow standard mathematical conventions and not just rely on shortcuts or tricks. This not only helps with clarity and consistency in communication, but also reinforces the importance of precision and accuracy in scientific work.

I also share your concern about the increasing reliance on algorithms and computerized methods in education. While technology can certainly enhance learning, it should not replace critical thinking and understanding. As you mentioned, the memory of civilization and our collective knowledge is at risk if we become too reliant on shortcuts and forget the importance of truly understanding and mastering concepts.

Thank you for bringing up these important points and for your thoughtful response. As scientists, it is our responsibility to not only advance knowledge, but also to promote a deeper understanding and appreciation of the subjects we study.