New Trig Mnemonic for H.S.
Richard Alvarez
be both easier and more instructive just to draw a diagram showing the
two angles, and figure it out from the diagram?
If you memorize something, but later forget some little detail, then
you are in trouble. But if you understand where it came from, then you
can figure it out when you need it, even if it takes a bit longer than
mentally reciting a story.
And a story tells you one fact (in this case, two formulas). But a
diagram can open a whole world of understanding.
I don't mean to berate the story. Just a principle that I have
found useful.
Dick Alvarez
Michael A. Stueben
A New Trigonometry Mnemonic
I recently created the following story to help my
students memorize the sum and difference formulas for
the sine and cosine. Also, I like to introduce math
with observations of life. This mnemonic gave me the
chance.
"As you all know, the people to whom we are
attracted are often not attracted to us. And it is not
unusual for a person of the opposite sex who has shown
interest in us to later lose interest. Maybe that is a
good thing, because it forces us to date a lot of
people before we marry. Anyway, this is the story of
Sinbad and Cosette. Sinbad loved Cosette, but Cosette
did not feel the same way about Sinbad. Naturally, when
Sinbad was in charge of their double date, he put
himself with Cosette and he put his brother with her
sister:"
sin(A + B) = sin A cos B + sin B cos A.
sin(A - B) = sin A cos B - sin B cos A.
"Sinbad loved to tell people that his and Cosette's
signs were the same. However, when Cosette was in
charge of the double date she placed herself with her
sister and put Sinbad with his brother. She made sure
everyone knew that their signs were NOT the same."
cos(A + B) = cos A cos B - sin A sin B.
cos(A - B) = cos A cos B + sin A sin B.
"Also, notice that Cosette placed herself and her
sister BEFORE Sinbad and his brother. This detail is
important. She was very snobby you know."
Most students checked the formulas just to be sure
that the story made sense. Other students were
motivated to show that the story didn't account for all
the details in the formulas. Even for the weaker
students, this was presentation was fun.
---
+----------------------------------------------------------+\
| --From Michael Stueben: high school math/C.S. teacher ||
| collector of mathematical humor and education theories ||
| E-mail address: mstu...@pen.k12.va.us ||
+----------------------------------------------------------||
\----------------------------------------------------------\|
sjmil...@cc.memphis.edu
I agree with Dick, mnemonics tend to confuse the issue and hide true
understanding of the topic from the student. Simply remembering a few
definitions and a standard picture, gives more insight and usually requires
LESS MEMORIZATION than committing rymes, stories, etc to memory. Moreover,
understanding a fundamental idea allows one to recall the important aspects of
a subject later on in life, while a mnemonic, even if remember 20 years down
the road, only allows one to recall an empty formula.
For instance, my high school teacher wanted us to committ some rhyme to memory
that would tell us the signs of sin and cos in the various quadrants. But this
was silly...all one has to do is remember that cos is the x-coordinate and sin
is the y-coordinate and then the signs are obvious. It takes far less to
committ these basic definitions to memory than it takes to committ her rhyme
(which I can't remember!). Moreover, you get more for your effort, for now a
simple picture of the unit circle tells you that the cos is even and the sin is
odd. Moreover, it is clear from teh picture that we have
sin(theta+pi/2)=cos(theta) and that cos(theta+pi/2)=-sin(theta).
Now, from the picture it is obvious that cos(pi/2)=0 and that sin(pi/2)=1.
Given that I vaguely remember that the sin and cos addition formulas are
linear combinations of products, I can use the last two pictorial
observations to find the sina nd cos addition formulas. Let's see,
Since sin(pi/2)=1, to make sin(theta+pi/2)=cos(theta), I would
guess that I should write sin(theta+pi/2)=cos(theta)(sin(pi/2). Ah,
I guess that sin(A+B)=cos(A)sin(B)+/- cos(B)sin(A). Should it
be + or -? Hmm, let's let B=0 and A=pi/2. Then from the pciture
it is obvious that 1=sin(pi/2)=sin(A+B)=cos(pi/2)sin(0)+/-cos(0)sin(pi/2)
=0+/-1. Ah, the sign should be positive.
Of course, you can make similar reasoning to recall the cos addition
formula.
If a student is forced to deal with the basic definitions, derive the
elementary consequences, remember formuals by being able to deduce them in an
elementary way (or RECALL rather than DEDUCE them by by making arguments as
above), then they'll develop a true feel for the subject and not see
mathematics as a mystery. Give them cute rhymes to remember formulas and
they'll only see empty formulas and view mathematics as nothing more than a
colelction of facts to be memorized.
Regards,
bob
PS: I find that beginning students have trouble remembering is
the derivative of sin(x)=+ or - cos(x) and similarily if the
derivative of cos(x)=+ or - sinx. Again, there is no need to make a rhyme or
spend some time trying to memorize these formulas. If you are stuck, draw a
picture. It is clear that the slope of the tangent at 0 for sin is 1. Sin
cos(0)=1, we must have that the derivative of sin is cos. Remembering that one
of these formulas has a minus, we now know that the other formula says that the
derivative of cos is -sin.
The upshot of all of this is: Understand the fundamental ideas and when in
doubt over a formula, draw a picture.
>
sjmil...@cc.memphis.edu
> gars...@nyc.pipeline.com (Gary Scott Simon) wrote:
>
>>In article <4e4ej2$6...@shellx.best.com>, Richard Alvarez writes:
>>
>>> Neat story story about Sinbad and Cosette. But...
>>
>
>>
>> Amen!
>
> Hallelujah and praise gee to beesus.
>
> It would be a shame if learning were just a bit fun. Learn, remember, and
> understand the fundamentals. No doubt that is the most important thing. But
> adding a little spice, especially at the hs level increases the accessibility of
> the subject.
I doubt that any of those that replied expressing concern over the use of
mnemonics is against "adding spice" to the classroom. For instance, I throw in
some humor in every class period. It is generally the case that at least one
during every class, the entire class will be rolling with laughter (or groaning
a failed joke :}). It is rare that when I teach, the entire class period
is spent in a solemn state with only the sound of the chalk to break
the silence.
My point is that I'm all for adding a little humor and spice, but the
use of mnemonics can mislead students and that is what I (and the others that
have replied if I may speak for them) are concerned about. Certainly a little
humor and spics can make a nice transistion from one topic to another. I find
after completing a difficult problem or concept, I need to have a short break
before starting something new and I'm fairly sure the students would agree.
Moreover, a little humor breaks down some of the teacher-student barriers.
On the other hand, even good things can be abused. The problem with relying on
mnemonics in one's teaching is that student tend to IGNORE the fundamentals.
All they are left with at the end at empty rhymes and mnemonics that allow them
to recall formulas...but they have missed the fundamental ideas. For instance,
any student that needs All Sailors Take Chances to remember the signs ont he
sin and cos function, simply doesn't understand the sin and cos functions.
Moreover, as I pointed out in another note, you've forced him to do extra work
for little benefit...knowing one picture not only gives you this information
but much more info about the trig functions AND it requires no more work to
committ it to memory than your 4 words.
So why give the student more work for
less benefit? Moreover, it doesn't take much experience in working with
students to know that most students will ignore the
elementary mathematical principles since your 4 words will give
them the answer to this particular problem.
Finally, I mention again that forcing students to memorize formulas, whether by
forcing them to directly memorize formulas or by having them remember cute
mnemonics, gives them the WRONG impression of mathematics. In doing this, they
will not be turned on to mathematics...they will see it as nothing more than
memorization and symbol pushing. Perhaps this is why it is so hard to get
students at the colelge level to think about the deep ideas, the concepts and
the theory of mathematics.
Regards,
Bob
> The kids you reach with stuff like All Sailors Take Chances may
> some day mature into quality mathematicians, scientists, or engineers.
>
> Keep up the good work, Mike.
>
> Montani Semper Liberi
>
> ds
>
Don Specht
>In article <4e4ej2$6...@shellx.best.com>, Richard Alvarez writes:
>
>> Neat story story about Sinbad and Cosette. But...
>
<snip>
>
> Amen!
Hallelujah and praise gee to beesus.
It would be a shame if learning were just a bit fun. Learn, remember, and
understand the fundamentals. No doubt that is the most important thing. But
adding a little spice, especially at the hs level increases the accessibility of
the subject. The kids you reach with stuff like All Sailors Take Chances may
Brian M. Scott
Stueben) says:
A New Trigonometry Mnemonic
I recently created the following story to help my
students memorize the sum and difference formulas for
the sine and cosine. Also, I like to introduce math
with observations of life. This mnemonic gave me the chance.
I've deleted the story. It's cute, and I can imagine that it might
encourage some students to check that the implied mnemonic actually
works. This extra contact with the identities certainly can't hurt.
Nevertheless, I can't imagine that the story would actually help
anyone memorize the things *except* by encouraging an extra investment
in active examination of them. Nor should such a mnemonic be necessary:
if one knows their general form, checking a few elementary values will
sort out the signs easily enough. And even if one relies on rote memory,
it seems to me a good deal easier just to memorize the formulae than
to remember a complicated tale like this.
Frankly, there are only two reliable ways to commit these trig identities
to long-term memory. One is to understand where they come from; since
the derivations aren't transparent, that's asking a lot of most students.
The other is to use them until they're second nature, a method that is
available to most. However, the will is usually weak, if not altogether
absent. If tales like this reinforce it, well and good; but I seriously
doubt that they are useful mnemonics, unless the goal is merely to get
the identities into short-term memory long enough to use them on an exam.
Brian M. Scott
Cleveland State Univ.
sc...@math.csuohio.edu
Michael A. Stueben
1). People learn in different ways. Mnemonics are good for
the people who like them and not good for the people who don't
like them.
2) This is not really a good mnemonic, because it has too
many ways it can be misinterpreted. But what it does do very
well is to help a class focus on the details of the formulas.
3) It is also valuable because it is a fun way to introduce
the formulas.
4) Just because I like it does not mean that it is going to
be valuable to other teachers who have different teaching
styles. And it may not be much good with classes that are
different from mine.
In short, a suggestion is a choice, not a claim that the
BEST METHOD has been found.
Gary Scott Simon
> 1). People learn in different ways. Mnemonics are good for
>the people who like them and not good for the people who don't
>like them.
humor is an excellent teaching tool (I've performed, and written, stand-up
comedy professionally, but I am never as funny as when I teach math).
Still, I disagree strenuously with your claim that "mnemonics are good
for the people who like them". Let's consider a hypothetical mnemonic that
"works" (i.e. actually helps a student remember a formula), and the
students who might "like it" (i.e. cramming for a test, desperate to
remember the formula). In what sense is the mnemonic "good for" these
students, other than as a crutch?
What skills we have to entertain and excite our students can and should
be devoted to enhancing their understanding and appreciation of the beauty
and poetry of mathematics. I love the look in students' eyes when, having
been introduced to the unit circle as a "trigonometric tour guide", they
finally realize just why the quadrants of the Cartesian plane are numbered
I - IV counter-clockwise.
Stephen P. Sorkin
>In article <4e4ej2$6...@shellx.best.com>, alv...@shellx.best.com (Richard
memory
>that would tell us the signs of sin and cos in the various quadrants. But
this
But it's sooo much fun to remember that "Aunt Sally Takes Cocaine" or maybe a
more tame "All Students Take Calculus." :)
Stephen
sso...@casltatela.edu