CAS vs nonCAS

[mc]

Hi all - I know I've seen posts that have danced around this - I
searched for them and could only find a couple and those meandered
from what does CAS look like in the classroom to arguments pertaining
to the constitutionality of censoring posts (*smile*). Now, I have
encouraged my school district to buy the nonCAS NSpire simply because
students cannot use the CAS calculators on state assessments.
Yesterday I led PD in my district on using the NSpire. I mentioned
that the CAS version seemingly gets more play in a science classroom
(but I could not answer why until reading something about accepting
undeclared variables?) Anyway, here is the question I was asked
yesterday:

What is the advantage of using a CAS handheld vs. a nonCAS handheld in
the foundational mathematics classrooms (thinking 9th and 10th grades,
or even 7th to 11th if necessary)?

Follow-up question from same teacher: Does it even belong there?
Scenario: Teaching students to graph linear and quadratic equations,
by hand and then by calc (regardless of TI-8x or NSpire). As soon as
you show students to graph on calc - pffft! Who needs to know how to
do it by hand, I have the calc? (*I know this question was alluded to
in the "What does CAS look like in the classroom post, but, fresh
responses?*)

I appreciate something TI Instructor Vince Doty said: "Put the math up
front, support and enrich with technology." and am trying to implement
that in my classroom, but, are there other ideas?

Thank you for your reading & responses!
~
Marc Coffie
Math Teacher
Spencerport High School

[Michael Houston]

Hi Marc-

I have encountered a similar problem in my home district, as well as many PD's I've led.  Here's my take on CAS.  A computer algebra system allows students (especially those in 9th and 10th grade) to explore algebra.  I found it hard to believe until I witnessed it occur in my own classroom, with my own kids (many of whom are special needs).  I used many of the lessons that have been created using AlgebraNspired, which I believe, is geared towards that goal.  Students don't need to know many of the features of Nspire, and those documents help them focus less on button pushing, and more on learning the mathematics.

One of my favorite things to show teachers who are skeptical or who don't use CAS is to have the Nspire CAS help a student solve a linear equation.  For example, let's say that a student is looking to solve the equation 2x=5.  Many of my students when prompted to perform the first step, would multiply both sides by 2, because that's how the 2 and the x are "connected".  The cool thing about CAS is that it will perform that step.  So, the CAS returns the equivalent equation, 4x=10.  At this point, I can ask my students if they feel they are getting any closer to solving for x, or do they want to UNDO (an awesome feature of Nspire) and try again.  Having my students work through this, they come up with their own steps and reasoning to solving equations.  It's pretty cool when a special needs student can't multiply 3*2, but can take a problem situation, create a linear equation to model the situation, and solve the equation to arrive at an answer, and it's all done with CAS.

Mike Houston

Riverside HS

Ellwood City, PA

> Date: Tue, 18 Aug 2009 04:47:55 -0700
> Subject: [tinspire] CAS vs nonCAS
> From: mrco...@gmail.com
> To: tins...@googlegroups.com

[Nelson Sousa]

I also like to use CAS especially with younger students. Rather than
using CAS to compute derivatives or limits, which we want them to be
able to do by hand, CAS allows the teacher to demonstrate a lot of the
basic rules of algebra in a more interactive way.

See attached file about solving an equation on CAS. We start with the
equation and then just type in the operation we want to do. It goes to
what Michael was saying.

After typing in "2x+1=-7" and pressing ENTER the next step is obtained
by pressing the minus key, 1 and then ENTER; the last step is obtained
pressing the division key, 2, then ENTER.

I think this is one of the best examples to show how CAS can be used
in the classroom. Of course, how well will students grasp the concepts
depends a lot more on the teacher than on the calculator. No CAS will
turn a lazy student into a hard working one, nor will it allow
students to have some miraculous insight over maths topics. What it
does is providing the teacher yet another powerful tool to explore
concepts and provide some topics of discussion to the class.


Cheers,
Nelson

[mathletics]

I would offer that the teaching of many common procedural things
becomes a pattern detection activity with CAS.

I think of...
The factoring pattern for a difference of two cubes,
The pattern for the square of a binomial,
Simplifying radicals (does anyone do this anymore?)
The connection between the zeros of a polynomial (using the zeros()
function) and the solutions of p(x)=0 using the solve() function.

[mrcoffie]

I appreciate the responses (thus far - please give more!)

Steve - agree with you about the pattern recognition possibilities
with the CAS - my only hesitation is the disallowance of it on formal
(mainly state) tests which my students will want to use it on. I know,
I know: "If you teach them good math..." but even the 'good' math
students recognize when to use a calculator vs. when to 'solve by
hand' for lack of a better terminology is faster (more efficient?)

And Nelson and Mike - I wholeheartedly agree with your responses - I
tell my students I consider them the 'video-game' generation of
students because they learn by [*drum roll*] Action/Consequence - it
makes sense to them! However, Nelson, in the second 'step' of your
example:

2*x+1=-7 2*x+1=-7
------------------------------------------------------------------------------
(2*x+1=-7)-1 <-- this step here 2*x=-8
------------------------------------------------------------------------------
((2*x=-8)/(2))
x=-4
------------------------------------------------------------------------------

seems to imply 'multiply both sides by -1' instead of 'subtract both
sides by 1 - I know, I know, I have heard this discussed before, but
is it wrong teaching or just asking our students to expand their (and
my) understanding?

As I have taught math these past 10 years, I have recognized a shift
in my math teaching philosophies, and one of them has become to have
students seek a pattern (and then see if it generalizes reason
inductively, if you will) . While not all students 'get' this, it does
encourage problem-solving and I'm all for that, because it is a LIFE
SKILL!

Please continue to respond/rebut - I am thankful for your thoughts!

~marc

[John Hanna]

There is a distinct difference between the 'negative' and the 'subtraction'
signs and this is why there are two keys on calculators. There is no
confusion below.

Try *(-)1 and see the difference.

Re:

[Nelson Sousa]

Indeed, the two "minus" signs are slightly different; one for negative
numbers, the other one for subtraction; thanks John!

Nelson

[Eric Findlay]

Also in this case (and similar ones), would it not be conventional to
use parentheses to indicate multiplication by a negative?

Ex: (2*x+1=-7)(-1)

The lack of parentheses should imply (or you should infer) subtraction,
not multiplication.

--Eric

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AKA Eagle-Man

[Andy Kemp]

Back to original question:

I am increasingly of the opinion that CAS the very valuable tool with all students, but issues related to assessment do complicate matters...

A question I would ask:

Will the students be using your calculators in the exams - i.e. do you have enough to lend them out to every student?

I am unlikely to ever be in the situation where I own enough calcs to lend out one to every student sitting the exam, so for this reason I am not too worried about the assessment side of things - If they want to use a calc in the exam then they need to own their own... (which is a requirement for some of my courses)

To this end I plan on buying CAS calc for class use, and encouraging non-CAS handhelds for personal and examination use.  

Andy

[mrcoffie]

Good question, Andy - and the answer is yes, I have class sets of N-
Spires for lending during exams, and actually, I encourage students to
buy the nonCAS because it has the 84 faceplate. However, maybe it
would be worth it to 'borrow' a set of CAS calcs from TI to see if it
helps students?

And I am aware of the difference between the two symbols 'subtract' (m
-- n for lack of the subtract symbol in a text box) and 'negate' (-n)
- in fact I always include that nuance in my PD and classroom for
users of both the NSpire and TI-8x. I was asking the question should

(2*x+1=-7)--1 (again -- = 'subtract')

be readily interpreted as 'subtract one from both sides of the equal
sign' versus 'multiply both sides by -1' and it is simply a
misunderstanding on my part? And from that premise, then the statement

(2*x+1=-7)+1

should also be interpreted to mean 'add one to both sides of the equal
sign'?

(These questions by the by are for clarification, not for fomenting an
argument :o) )

Thank you all again for your input! It provides food for thought! Any
other insights/preferences/experiences?

~m

Marc Coffie
Math Teacher
Spencerport HS

[Nelson Sousa]

as far as expression syntax goes, the Nspires write

(2*x+1=-7)--1
meaning subtract 1 to either side and

(2*x+1=-7)*-1
meaning multiply both sides by -1

-- is the subtraction sign and - is the negative number sign.

A parenthesis would be advisable in the second situation, but the
Nspires don't print them (I thought they did, actually).

Cheers,
Nelson

[Corey Andreasen]

This brings up a question that occasionally bothers me. What exactly does your colleague claim is the value in knowing how to graph by hand? Graphs are important, don't get me wrong. To me, the important thing is to understand what a graph is, what it tells you about the relationship between variables, etc. Is there some inherent benefit to being able to construct a graph by hand?

I also know that, if a student understands the concept of what a graph is, creating one by hand isn't difficult. But graphing by hand was at one time needed so we could get the graph. Now we can get the graph easily with technology.

Some math teachers seem to think the value of everything that's ever been taught in mathematics is obvious and needs no justification. And many have never thought about why we teach things or whether there's really any value in it. Or, at least, enough value in it to take priority over other things we could spend time on.

Corey

----- Original Message ----
From: mc <mrco...@gmail.com>
To: tinspire <tins...@googlegroups.com>
Sent: Tuesday, August 18, 2009 6:47:55 AM
Subject: [tinspire] CAS vs nonCAS

[mrcoffie]

Good point, Corey - I do not teach freshman algebra, nor did I ask
that question, so I do not know my colleague's response.

I *assume* the reason to graph by hand is it is the basis for graphing
other curves by hand. For example, graphing a quadrantal/inverse
hyperbola, oh, xy = 12. Now, if you graph y=12/x on a calc and get a
table of values, only the integer values are of use to the students
(they hate fractions and most decimals...) versus asking the students
to create a table and list all of the factors of 12:

x | y
1 | 12
2 | 6
3 | 4
4 | 3
6 | 2
12 | 1
-1 | -12
-2 | -6
-3 | -4
..
and graphing those (x,y) pairs instead.

I would think graphing by hand also practices the important concept of
replacing x (or whatever variable) with its value and evaluating the
resulting expression - I know many students need that reinforcement,
and in 9th grade algebra, that is a fundamental concept.

I also would think that graphing by hand reinforces the concept of
"What do I do if I don't have a calculator?" Many students come in
still not knowing their times tables because, why should they? They
have a calculator, whether an actual calculator (in school) or their
cell phone (not in school). And to many students, the calculator is
gospel. I call it the Magic Answer Box and tell my students not to
become Magic Answer Box Addicts. I explain to them that an addict will
do anything and accept anything to get their 'fix' - in this case the
right answer - even if they know what they do to get the 'fix' is
wrong. Sometimes their Magic Answer Box receives the wrong input and
their 'fix' isn't correct. I always illustrate by asking them what is
(-3)^2 (verbally - I don't say the parentheses) and they all respond
'9' which I praise, "Correct!" and then ask them, "So why is it when
your calculator tells you -3^2 is -9, you automatically assume it is
correct and use it? (Now I know it is and I explain the 'missing' part
of PEMDAS to them, but they still will revert back to (-5)^2 = -25) I
know an obvious and easy solution to this is to teach students to use
parentheses for value substitution, but perhaps they were never taught/
never learned that. I practice it a lot in my classes, but again, I do
not teach 9th grade.

So in response to my question, here is another question: Does anyone
give two portions of their test? A calculator portion and a non-
calculator portion (like the AP)? Our pre-calc and calc finals are
given that way, perhaps that would be a better way to validate student
knowledge - what they are expected to 'know' and what they are
expected to 'apply' or problem solve. [I know that this should be the
topic of a different thread in a different forum, but, hey, we're
teachers, right?]

Thanks again, one and all - keep 'em coming!
~m

[Andy Kemp]

To add a little to the debate about technology use in the classroom and the teaching of paper and pencil skills I wanted to add one of my favourite quotes about technology in the maths classroom:

“Look around you in the tree of Mathematics today, and you will see 

some new kids playing around in the branches.  They’re exploring 

parts of the tree that have not seen this kind of action in centuries, 

and they didn’t even climb the trunk to get there.  You know how they 

got there?  They cheated: they used a ladder.  They climbed directly 

into the branches using a prosthetic extension of their brains known 

in the Ed Biz as technology.  They got up there with graphing 

calculators.  You can argue all you want about whether they deserve 

to be there, and about whether or not they might fall, but that won’t 

change the fact that they are there, straddled alongside the best 

trunk-climbers in the tree – and most of them are glad to be there.  

Now I ask you: Is that beautiful, or is that bad?”  (Kennedy 1995)

The full article by Kennedy (from 1995) can be found here:

http://mail.baylorschool.org/~dkennedy/treeofmath

For me this sums up the state of tech use in the classroom - It is there and possible whether we like it or not, the question becomes how we react to it...

I think this is especially try when we start exploring the role of CAS in the classroom...

[Joe]

Sorry but I just can't resist commenting on this subject. The cas
calculator makes teaching math so much easier that surely it is here
to stay, it's not simply a fad, so why dance around the issue.
Embrace it. Buy stock in TI and provide every student a cas nspire
calculator. Then give them a card that lists the button pushing
sequences for the tests, and call it done. Job over. Sure they won't
understand a thing about math, and they won't gain any of the insite
and understanding that working problems by hand provides, but they
"will" come up with answers. Isn't that what it's all about?
Besides, the average student just needs to do a little arithematic
during his/her lifetime such as figuring out if the paycheck (assuming
they can get a job) will cover the new car payment. And yes of
course, the student that goes to college will get the shock of their
life when the instructor announces that this is a math class so
calculators won't be used. Yeh, at that point they are screwed but
lets face it, they can switch to another field, so why bother learning
math anyway? Who needs engineers or the stuff they develope? Not
us. We can buy those things from foreign countries.

How many of your algebra students actually know the quadratic formula
by heart, and can they use it if you take away the calculators? Do a
one problem quadratic equation pop quiz tomorrow and find out. Then
collect the papers and give them calculators and ask them to write
down on another piece of paper the quadratic formula and what it is
used for. Based on the lack of math knowledge, of the entering
college students that I get, I believe you will be lucky if half of
your students can do those tasks correctly. So tell me, is that a
passing grade for their algebra teacher, and for the technology that
was employed? And is going from graphing calculators to cas
calculators going to fix the situation and improve the students
knowledge of mathematics any? Perhaps high school math classes should
focus on teaching the basics and leave the cas calculators for college
engineering courses. Or to put it another way, you can spend more
money on math technology, and most certainly that is what calculator
companies want, but that doesn't mean that the student is going to get
a better math education, and that is what the student needs.
Best regards,
Just plain Joe

[Andy Kemp]

There seem to be some confusion in Maths teaching that 'getting answers to questions' is the same doing Mathematics.  Also who says that when a 'student goes to college [they] will get the shock of their life when the instructor announces that this is a math class so calculators won't be used', just becuase calculators are not currently used frequently in college classrooms at the moment doesn't mean this will (or should) always be the case...

At the moment, in my opinion, far too much of what is taught at high school and college level mathematics is nothing more than learning algortihms and applying them repeatedly...  What understanding of Quaratic Equations does a student gain by being able to recite the Quadratic Formula by heart?  I for one look forward to the day when High School and College leve Mathematics more closely resembles research level mathematics, in that it, it become more exploritory and investigative with students looking to explore and discover 'new' (either actually new or at least new to them) Maths.


No one seems to be complaining about the fact that we no longer teach students how to calculate square roots by hand, or how to work with trigonometric and log tables.  Technology has rendered these mathematical skills redundant - who is to say which skills we currently treasure will be considered redundant 20-30 years from now?

Why should Maths continue to be taught and learnt in exactly the same way as it was pre-technology (or at least only allowing technology to be used in a highly controlled manner and only to support the learning of paper and pencil methods)?  It is time for Mathematics education to think radically about what it is we teach (or want to teach), what is important (i.e. what is Maths?), and how do we best get students to where we want them to be...  This doesn't meant using technology for everything or replacing all algebraic skills with CAS (anymore than use of scientific calculator implies replacing all numerical skills)...

However it will require change (something that is always difficult and painful) at all levels from primary through to degree level and beyond, this is already happening at the top end where computer-aided proofs are becoming increasingly common...  but until there is some coherence in the system we will continue to come across arguments like you can't teach them to do integration using CAS as they will have to do it by hand when they get to college (why?  If there is a good reason to do it by hand then great - but if you are just doing it by hand becuase that's how it 'always' been done then...)

Much of Mathematics is done the way it is becuase even 20-30 years ago there was no other option, but maintaining that those methods are still the best when there are other options available to use now requires research, justification and evidence...

I know not everyone will agree with me and I appreciate this is a very emotive topic amoungst Maths teachers, but I wanted to explain where my previous comment comes from...  I have not interest in starting a debate on this (and so will not reply to any posts on this topic), as I don't think I will change anyones opinion but wanted to put my views out there for people to consider...

[mathletics]

Marc,

All five of the math teachers at my school have class sets of CAS
units. We value CAS as a teaching tool, regardless of the calculator
they use on our state test or end of course exam. We teach with CAS,
and our students are all the better for it.

We ask our students to purchase the non-CAS units for use on the ACT
and ACT end-of-course tests.

It is ironic that the students who NEED CAS the least--our AP Calculus
class--are provided CAS units supplied by the school which are theirs
to use for the year. No shared class sets there! I wish my school
would do the same for my basic algebra class as they did for the AP
Calc class!!

Steve

[-TJ]

Dear Marc,

Simply said, I would personally choose the TI-Nspire (non-CAS) over
the TI-Nspire CAS for the sake of simplicity, and ease of use. I don't
think students are supposed to "learn" through calculators; it is more
of a means of computing calculations to save time. Allowing students
to use the TI-Nspire and switch over to the TI-84 mode increases
flexibility within the infrastructure. Most compatibility issues will
also be eliminated with this choice. You must consider the fact that
students may tend to become "lazy" while using the TI-Nspire CAS.

The overall decision is based on your implications of the technologies
that Texas Instruments provides in the TI-Nspire family of
calculators. How frequently will you allow the students to use the
calculators? For what purpose are the calculators for; to learn with
the calculator or to save time by executing various calculations?

The safest choice would be getting the TI-Nspire (non-CAS).

Sincerely,
-TJ

[John Hanna]

-TJ writes: "I don't think students are supposed to "learn" through calculators; it is more of a means of computing calculations to save time."

 

Sorry, TJ, but… SAY WHAT???!!! Actually, there's nothing written in the first paragraph below that I can ignore. I always recommend a CAS machine over a ‘calculator’. In workshops where users have numeric calculators and I use a CAS version, most regret having selected the numeric.

- John Hanna

[mathletics]

"You must consider the fact that students may tend to become "lazy"
while using the TI-Nspire CAS."

Only a lazy teacher will let their students become lazy using CAS.

Steve

[Joe]

Thank you Steve for bringing that up. I've been hesitant about
bringing up such considerations (I don't enjoy getting hateful e-
mail), but we all know math teachers that are mathematically
challenged and should be teaching something else. For those
individuals, nothing can ever be said that will cause them to give up
their calculator crutch (cas or otherwise) and start teaching math,
because they can't. And because they can't, the argument will go
on for the next decade skirting the real issue which is competence.
It must occur to the poor teacher that cas calculators are banned from
tests because the purpose of the test is to determine what the student
knows and not what their calculator knows, but they can't care about
such things. For them it is a matter of doing what ever it takes to
hang onto their job so they will do what ever it takes including
spreading false information and ignoring the obvious. Ok, now that
I've exposed the real issue, I'll just have to endure the ugly
responses. Who knows, maybe I'd do the same thing if I were in their
position.
Just plain Joe

[Daniel Dudley]

Your comment makes no sense. There are very good teachers who advocate CAS and very poor teachers who only teach algorithms that they learned in school. While I have met teachers with various levels of skills or "competence" I would say that the teachers with the lower skills tend to avoid any technology other than the basic operations and rarely ventured to try to use the calculator as a tool for exploration and only stick to what they were taught in school. I am not implying that there are no good teachers that do not use calculators, I have certainly met many very good teachers who do not use much technology in their course, but the opposite is much rarer, I think. How on earth can you equate the desire to use CAS with competence? I have noticed that the teachers who understand calculators very well often have explored much more mathematics than those who don't. That doesn't equate at all with their competence.

Dan

[Joe]

Hi Daniel Dudley,
First let me say that I was addressing mathematical competence as
apposed to teaching savy. And if we live on the same planet, I am
sure that you are well aware that there are some teachers who cannot
work demonstration problems on the board without making mistakes and
getting into trouble, so they compensate by focusing their students
efforts on working problems using calculators. But you probably
already knew that.

A student in an introductory calculus course once asked, how do you
integrate something like x*sin(x). The teacher demonstrated using a
cas calculator. Did that teacher recognize the problem as a trivial
one that cries out for treatment using integration by parts? Probably
not. Did the student get his question answered? No. The students
question was how do you "do" the integration, not what is the answer.

Years ago when I used my first graphing calculator, I was wildly
enthusiastic about it and the first time I graphed a z=f(x,y) surface
on Maple and rotated it, again I was wildly enthusiastic. Since then
however, I've come to the conclusion that demonstating technology is
not in the same league as teaching. Technology doesn't look around
the class and spot a student that has a confused look. Technology
doesn't look that student in the eye, and ask what is troubling you.
And from the response, technology doesn't tailor a reply that is
exactly what the student needed. And finally, technology doesn't
consistantly result in a student with a big grin that says "I got
it." That is not to say that I am anti-math technology. It is to say
that the power of technology is not flexable enough to compete with a
good teacher, a black board, and some chalk (or white board and
marker, or smart board and stylus). I teach math what do you do?

joe

[mrcoffie]

First off, thank you again Daniel and Joe for your responses!

Even so, once again, we resort to a different topic. *sigh*

If we examine the quality of teachers, regardless of discipline, I am
sure we will find [to wax mathematic] a normal distribution of
teachers, where hopefully the mean is a 'good' mean in that the
quality of the 'mean' teachers is sufficient and appropriate. The low
percentile teachers are what they are, regardless of technology use or
lack thereof. The upper percentile teachers are those that are the
perhaps perfect unison of teaching skills, knowledge, caring, empathy,
rigor, etc. (the list is endless) and they strive to not only 'teach'
their students, but their colleagues as well (?).

Looking more closely at mathematics teachers, I'm sure we would see a
similar distribution - one in which the lower quality teachers may
lean on technology or eschew it for 'kill and drill' tactics. The
'mean' teachers are there hopefully because of their love of teaching
mathematics while perhaps recognizing there is always another way to
solve any problem. But they have heard their call and and responded
wholeheartedly.

I agree in part with you Joe:

"..demonst[r]ating technology is not in the same league as teaching."

While technology demonstration is not in the same league as teaching,
properly employed, it provides further exploration of some topics that
are too abstract or complex to create multiple simple physical models.

However:

"Technology doesn't look around the class and spot a student that has
a confused look. Technology doesn't look that student in the eye, and

ask what is troubling you. ... It is to say that the power of
technology is not flex[i]ble enough to compete with a good teacher, a

black board, and some chalk (or white board and marker, or smart board
and stylus)."

I would have to again examine the *quality* of the teacher, not the
quality of the technology. Again, using technology properly and
appropriately will not spy the confused student, but give the teacher
time to spot the student. I don't know where this idea that technology
will replace teachers originates, but is a frustrating notion for both
in the non-educational and educational audiences. For the non-
educational, using technology in education [to me anyways] is just
'another way society is progressing - why stop it?' and, I would
posit, provides a 'false hope' that 'it is going to help my child ace
that exam!' For the educational, it removes the hope of teaching and
the beauty and art of teaching. Technology will never replace a good
teacher, in that, you are correct. But, it makes a GOOD teacher
better! (heh heh - technology as the BASF of teaching? heh heh)

I return to Vince Doty's premise: "Put the math forward - teach them
correctly first using pencil, paper, etc.; then use the technology to
reinforce, supplement, and enrich the instruction." This can be
generalized to any discipline - simply replace 'math' with the
discipline. Look at science where students are exploring and creating
virtual chemical models and virtually dissecting animals. Look at
English where the Internet has allowed collaboration across county-,
state-, and even international borders! Look at social studies where
students can explore their genealogies and conduct important research
unheard of a decade ago. Were these 'types' of things accomplished a
decade ago? YES! By the same GOOD teachers who, regardless of
technology, had similar lessons and activities, albeit not as
advanced?

My intent for this thread was simply to find other users of the TI-
NSpire technology's opinion of using a CAS handheld vs. a nonCAS
handheld in mathematics instruction. I'm sure, if we wanted to start a
new thread on math teacher quality, it would speedily and easily be
filled with stories and suggestions. But that is not the scope of this
thread.

Thank you again for your input - it is good discourse and leaves room
for consideration.

Regards,
Marc

[Joe]

Lets see, step one is: "Put the math forward - teach them
correctly first using pencil, paper, etc. (I assume etc.,
includes black board and white board)
Then step two is: then use the technology to

reinforce, supplement, and enrich the instruction."

Now that is an approach that I whole heartedly support. I was just
objecting to the notion of some, who perhaps because of a personal
fasination with technology, or an inability to use the board
effectively, or perhaps for financial reasons, want to eliminate step
one. Oh well, I'll fade into the back ground now.

[lee kucera]

Joe:
Sometimes the steps are more complicated--say step two, then one, then
two again. An example that comes to mind is least squares
regression. I introduce this with the movable line one a graph--
working to get the smallest sum-of-squares, then we go to hand
calculation, and then back for more technology. I think you need to
have more faith in teachers and teaching.
lk

lee kucera
a.p. statistics
leek...@gmail.com

[Eric Findlay]

Another example that comes to mind is learning about parabolas. With
the Nspire, you could have the students discover for themselves what
happens to the equation when you manipulate the parabola. Then, once
they have some ideas, teach them what the formula looks like and what
all the variables represent. Teach them how to graph from the equation
and find the equation from the graph, etc. Then once they're good at
it, allow them to use the calculators again, or use it to further
explore/reinforce the ideas.

--Eric

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[pam]

Lee, I have done the same thing. It is very powerful for students to
manipulate their own best-fitting lines.

Eric, even simpler: manipulating a line to see the impact of the
movement on the equation. How do you change the slope and how do you
change the y-intercept? What does this look like in the table of
values?

But none of these examples needs a CAS. I have used CAS as described
above - following the steps of solving an equation, pattern
recognition, etc. I have even had situations where using the CAS
became so complicated (solving logarithmic equations, for example)
that my students opted for "old school" methods!

Marc, I think you made the appropriate choice given the constraints of
your testing situation.

Pam

[John Hanna]

Three quotes (pre-handheld-CAS) from Drs. Bert Waits and Frank Demana:

"Some Mathematics is less important because technology replaces it. Some
mathematics is more important because technology requires it. Some
mathematics is possible because technology allows it."

"Do algebraically, confirm graphically; do graphically, prove algebraically;
do graphically because there is no other way."

When asked "When do you see CAS as an accepted tool in the HS mathematics
classroom?" Bert replied "Not in our lifetime."

Enjoy the beach,
John Hanna

-----Original Message-----
From: tins...@googlegroups.com [mailto:tins...@googlegroups.com] On Behalf
Of lee kucera
Sent: Saturday, August 22, 2009 4:11 PM
To: tins...@googlegroups.com
Subject: [tinspire] Re: CAS vs nonCAS

[George Elizondo]

Hi Nelson,
Regarding your (excellent) example solving a linear equation using
CAS [cas_example.tns], you mixed lines of text with lines of
calculations on a calculator page. How’d you do that!

Do you have any other CAS examples like the one from cas_examples.tns?
George Elizondo


On 18 Aug, 07:19, Nelson Sousa <nso...@gmail.com> wrote:
> I also like to use CAS especially with younger students. Rather than
> using CAS to compute derivatives or limits, which we want them to be
> able to do by hand, CAS allows the teacher to demonstrate a lot of the
> basic rules of algebra in a more interactive way.
>
> See attached file about solving an equation on CAS. We start with the
> equation and then just type in the operation we want to do. It goes to
> what Michael was saying.
>
> After typing in "2x+1=-7" and pressing ENTER the next step is obtained
> by pressing the minus key, 1 and then ENTER; the last step is obtained
> pressing the division key, 2, then ENTER.
>
> I think this is one of the best examples to show how CAS can be used
> in the classroom. Of course, how well will students grasp the concepts
> depends a lot more on the teacher than on the calculator. No CAS will
> turn a lazy student into a hard working one, nor will it allow
> students to have some miraculous insight over maths topics. What it
> does is providing the teacher yet another powerful tool to explore
> concepts and provide some topics of discussion to the class.
>
> Cheers,
> Nelson
>
>
>
> On Tue, Aug 18, 2009 at 13:39, Michael Houston<msh...@hotmail.com> wrote:
> > Hi Marc-
> > I have encountered a similar problem in my home district, as well as many
> > PD's I've led.  Here's my take on CAS.  A computer algebra system allows
> > students (especially those in 9th and 10th grade) to explore algebra.  I
> > found it hard to believe until I witnessed it occur in my own classroom,
> > with my own kids (many of whom are special needs).  I used many of the
> > lessons that have been created using AlgebraNspired, which I believe, is
> > geared towards that goal.  Students don't need to know many of the features
> > of Nspire, and those documents help them focus less on button pushing, and
> > more on learning the mathematics.
> > One of my favorite things to show teachers who are skeptical or who don't
> > use CAS is to have the Nspire CAS help a student solve a linear equation.
> >  For example, let's say that a student is looking to solve the equation
> > 2x=5.  Many of my students when prompted to perform the first step, would
> > multiply both sides by 2, because that's how the 2 and the x are
> > "connected".  The cool thing about CAS is that it will perform that step.
> >  So, the CAS returns the equivalent equation, 4x=10.  At this point, I can
> > ask my students if they feel they are getting any closer to solving for x,
> > or do they want to UNDO (an awesome feature of Nspire) and try again.
> >  Having my students work through this, they come up with their own steps and
> > reasoning to solving equations.  It's pretty cool when a special needs
> > student can't multiply 3*2, but can take a problem situation, create a
> > linear equation to model the situation, and solve the equation to arrive at
> > an answer, and it's all done with CAS.
> > Mike Houston
> > Riverside HS
> > Ellwood City, PA
> >> Date: Tue, 18 Aug 2009 04:47:55 -0700

> >> Subject: [tinspire] CAS vs nonCAS

> >> From: mrcof...@gmail.com
> >> To: tins...@googlegroups.com

>  cas_example.tns
> 1KViewDownload- Hide quoted text -
>
> - Show quoted text -

[George Elizondo]

Oops! I see. The copywrite symbol allows you to do that. Nice. I would
still like to see more of these kinds of CAS examples!
George

> > - Show quoted text -- Hide quoted text -

[Nelson Sousa]

Hi George!

Yes, the copyright sign defines comments. So anything after the
copyright sign won't be calculated/processed, but is useful to comment
programs or, in this case, leave small text messages in calculator
pages.

Cheers,
Nelson

[John Hanna]

The 'copyright' sign is inserted when you use menu>Actions>Insert Comment and can also be used at the end of a command to document it:

--------------------------------------------

2+3 © this adds 2 and 3                5

--------------------------------------------

 

Comments are also useful in programs for documentation and for ‘commenting out’ parts of code that you do not want processed for debugging purposes. Just put a © at the beginning of each line you do not want to be executed when you run the program.

 

Sail Upwind,

    John Hanna

    973.398.3815

    jeh...@optonline.net

    www.johnhanna.us

The future is not what it used to be. - Paul Valery

 

 

 

 

 

-----Original Message-----
From: tins...@googlegroups.com [mailto:tins...@googlegroups.com] On Behalf Of Nelson Sousa
Sent: Monday, August 31, 2009 6:03 PM
To: tins...@googlegroups.com
Subject: [tinspire] Re: CAS vs nonCAS

 

 

Hi George!