The use of a graphing calculator is considered an integral part of the AP Calculus course, and is permissible on parts of the AP Calculus Exams. Students should use this technology on a regular basis so that they become adept at using their graphing calculators. Students should also have experience with the basic paper-and-pencil techniques of calculus and be able to apply them when technological tools are unavailable or inappropriate.

(From The College Board's AP Calculus calculator policy.)

Yesterday's post was about a sticky tech issue. This one's more about ideology.

Fierce debate raged in the 1990s over whether graphing calculators should be used in mathematics education at all. Proponents thought graphing calculators opened new vistas of understanding, as students could play and experiment, and see instantly how functions are affected by different tweaks. Detractors said graphing calculators killed the ability to work things out by hand; that they're a crux, and true understanding is only obtained by pencil-and-paper repetition of the proper methods.

Arguably, the issue ended (in the United States) with the AP test adoption of the graphing calculator. What resulted was a deeper and more difficult test; not only did students still need pencil and paper methods, but they had to apply the calculator to answer higher-level questions.

(No, not every student will make it to AP classes, but I believe curriculum should be written with the assumption that some will.)

However, the issue never truly died, because there was (and still is) a hidden ideological struggle going on:

*Should our primary focus in algebra be on symbolic manipulation, or is visualization and synthesis an important aspect?*

In other words, are graphical methods and applications just an afterthought?

While this doesn't seem to relate directly to social technology, the interesting enhancements computers can offer don't attract the interest of a symbolic-manipulation-only teacher. Alternately, graphing calculators can be thought of as the gateway application -- take a teacher comfortable with them, and it's easy to hook them on related Internet apps. For the Internet to be truly useful, teachers need to see there is a world beyond factoring binomials.

So, if you're a technology coordinator with a resistant math department, there's one question you might ask: are there teachers who haven't taken their class set of calculators out of their boxes? If so, there might be more going on than mere tech resistance.

Jason Dyer, Guest Blogger

I believe that teachers should use the graphing calculators especially if it is part of the curriculum. However, I don't believe that is the only way the students should learn. The calculators should be used as a "check". Students should first learn how to solve the problem and then punch in the formula into the calculator to prove if their answer is right. Using the calculator can be just as hard as solving a problem using paper and pencil. I believe it is important for a student to use both methods.

Posted by: Laura | February 27, 2008 at 11:52 AM

My quote above agrees with you: "Students should also have experience with the basic paper-and-pencil techniques of calculus and be able to apply them when technological tools are unavailable or inappropriate."

Technological tools are for augmenting, not supplanting.

Posted by: Jason Dyer | February 27, 2008 at 11:55 AM

I feel guilty reading this. I've stayed away from calculators in my algebra class this year.

One reason for my "tech resistance" is because of supplies--we haven't had enough functional calculators to go around the school most of the year. (I recently got a set through Donor's Choose.)

A larger reason is that my students are drastically below level, so I've been teaching basic addition and subtraction with my algebra. I want students to practice those skills at every opportunity and calculators make it too easy to not learn those skills. (Perhaps one reason why they haven't mastered second grade math skills when by the time they've entered high school?)

It's not so much that I don't want to use the technology. I'm just not sure how to allow parts of a tool and not the entirety of it. At what level do you bring in what tools? At what level do you expect students to be able to do something without the tools?

Posted by: Sarah | February 27, 2008 at 02:08 PM

Glad to hear Donor's Choose came through for you!

Don't feel guilty -- graphing calculators are hard to integrate at earlier levels, and really only come into their own once the students are far enough to do actual graphing.

However, with the right software, you may still find the calculators useful as a miniature computer lab. If they're TIs, shop around their education site for something useful.

Just be sure to emphasize that the graphing calculator cannot be a substitute for knowing the mathematics. Something simple to try once they start graphing lines is to give a function that lands outside the standard window, and ask them to figure out what's going on.

Posted by: Jason Dyer | February 27, 2008 at 04:56 PM

Jason, you say, "Students should also have experience with the basic paper-and-pencil techniques of calculus and be able to apply them when technological tools are unavailable or inappropriate." When would this ever be the case?

Posted by: Scott McLeod | February 27, 2008 at 06:54 PM

Although I can't imagine someone sitting down on a project with paper and pencil to complete a calculus-based problem, I can beleive that it is a more tactile approach to understanding the concepts. It is much like clicking the "translate" button on a document and claiming to read Spanish or French unless the understanding of input vs. output is achieved.

Posted by: Lewshall | February 27, 2008 at 09:36 PM

This is a great post. We are currently in a curriculum review cycle in my school district. I am on the committee and we are examining the use of calculators from elementary up to H.S..

We seem to have more discussion regarding elementary calculator use. Everyday Mathematics, the curriculum our Elementary schools use, recommends calculators. However, the calculator that they use is a TI-15. Here is a link to it: http://education.ti.com/educationportal/sites/US/productDetail/us_ti15_explorer.html

The issue we are seeing is with adding, subtracting fractions and common denominators. When these kids then get to algebra and are required to manipulate algebraic fractions to evaluate and solve, they have real trouble. It might suggest that the TI-15 and it's "fraction button" contribute to very poor skill development.

For those of you unfamiliar with current calculators, there are fraction buttons on them and they spit out answers in fraction form. So one enters 1/2 + 3/4 and it produces 5/4 or 1 and 1/4. So students in early elementary school can "skip" skills that are necessary to be successful with algebra.

So my question is...are we making robots that can push the right buttons or are we making people that can solve problems? It almost feels like we are indirectly creating an intelligence class system. A class for people that push buttons and a class for people that can think freely? Maybe that is already here...maybe not.

Posted by: Erik | February 28, 2008 at 08:57 AM

Erik: Calculators can be used at lower levels, but it must be strongly emphasized that curriculum must include a significant non-calculator portion. Education has difficulty with mixed curriculum. Note the phonics/whole word wars -- is there anyone who does *both*?

I have given tests where a graphing calculator was required for the entire test. I have given tests with no calculator allowed. I have given tests where the first page has no calculator allowed and the rest of the test requires a calculator.

I have given these tests all to the same class in the same year.

To be honest, though, if I thought the faculty was using calculators as a crutch, I'd be comfortable omitting calculators from the elementary level and only phasing them in at the middle school level.

Posted by: Jason Dyer | February 28, 2008 at 09:40 AM

Scott: Let me list some instances where a calculator won't help.

The most obvious instance is in a "thinking problem" where numbers are not involved. A lower level example might be: why is subtracting a negative number the same as adding the positive version? A higher level example might be: a pendulum can be started by either pushing it from a resting position, or lifting it up to a height and letting go. Which one is best represented by a sine curve, which by a cosine, and why?

An example the AP test uses is analyzing graphs with no numbers attached; they'll have a picture of the derivative of a function and ask the student to reconstruct the original. It's purely pictorial, meaning the student has to really understand the meaning of a derivative.

In some cases the calculator is dumb the student has to compensate. For example, the graph of a logarithm often gets fouled up on graphing calculators, because it approaches a vertical asymptote and the picture just cuts off.

Your manipulation might be purely symbolic; while there's software that handles symbolic manipulation (and also the TI-92, not allowed on the AP test) it only works if you know which way you want to go with the problem. You may be just trying to "simplify" in which case it's like a puzzle, so human intervention is required.

You may be wanting an *exact* answer, like knowing the answer isn't just around 1.299 but sqrt(3) * 3 / 4. Perhaps eventually the sqrt(3) cancels, but the calculator won't be smart enough and it gets some strange approximation like 0.754745 rather than the true answer of 3/4. (Calculators have more issues with rounding problems than you might realize -- computer scientists go through all sorts of convolutions in complex calculations to keep errors from cropping up.)

Finally, when the AP test has problems that absolutely require a graphing calculator, usually they are asking for higher level understanding, so it's sort of a hybrid problem. They're really not asking the student to integrate between 0 and 18 (because for the particular equation they picked there may be no method other than approximating) but they want the student to understand when the proper moment is to integrate from 0 to 18.

I will try to get up another example on my blog later.

Posted by: Jason Dyer | February 28, 2008 at 09:55 AM

I thought you might get a kick of my calculator rant from last November: http://tcmtechnologyblog.blogspot.com/2007/11/marias-calculator-rant-and-throwing.html

This is MY problem with these calculators. Why ARE graphing calculators so expensive?

Posted by: Maria H. Andersen | February 28, 2008 at 04:45 PM

Jason: I really appreciate your comments. I connect very well to your final paragraph:

"To be honest, though, if I thought the faculty was using calculators as a crutch, I'd be comfortable omitting calculators from the elementary level and only phasing them in at the middle school level."

Here is where I am going with this...do we need math specialists in the Elementary school? My thoughts are starting at 4th grade. Elementary teachers with math minors or even middle school teachers with at least math minors/math licensure teaching in the upper elementary.

Especially with future trends pointing more towards algebra in elementary schools:

http://www.mathcurriculumcenter.org/conferences/CSMC/index.php

Might it be safe to say: Teachers that are not comfortable with math are more likely to use the calculator as a crutch? Thus, leading to missed teaching opportunities to teach children proper algebraic thinking skills early on in the development process. Especially since CAS is around. (I assume it was created because the kids "already know the algebra" and need not bother with it in H.S.???)

So where are students "learning" the math? If in H.S., it is assumed that they learn it in middle, and in the middle school, we assume it happens in the elementary...now with TI-15s and such, does it have a chance to be learned? Maybe it isn't being learned at all...

Posted by: Erik | February 28, 2008 at 09:15 PM

Erik, I'm not qualified enough with elementary ed to know the situation with math specialists. It is possible if you have a good enough math coordinator they'd be able to pull of the same effect with training. Has anyone directly advised the teachers just how the calculators ought to be used?

Still, with the introduction of algebra at elementary levels, a math specialist may be the only way to go.

The main issue I've noticed is teachers that teach fractions yet are scared of fractions themselves. This seems to match "teachers that are not comfortable with math are more likely to use the calculator as a crutch" if as you say the main issue is their use of the fraction button.

Posted by: Jason Dyer | February 29, 2008 at 09:34 AM

I agree with the use of graphing calculators at the AP Calculus level. I think by this time in a students school career, he or she deserves to use a calculator. If I would have made it to AP Calculus in high school then I would have felt pretty proud of myself and felt that I have worked hard enough to get to this point that I should be able to use it. Most jobs, in my opinion, that use high level math skills and formulas, I would assume require the use of such tools to minimize mathematical error and increase productivity.

I think for lower grades, there needs to be a healthy balance between the two. I think some students may prefer to do their computations on paper, while others may be hindered by not being allowed to. You really have to know some math to be able to use a calculator. By not allowing calculators at any level, is hindering students performance.

Posted by: Shannon | March 02, 2008 at 10:55 PM

Shannon:

You have struck onto the tune of performance. We all struggle to balance performance and mastery of knowledge. However, I have a difficult time believing that seamless calculator use in Elementary school delivers maximum performance. Especially since on these state assessments that are attached to NCLB funding has sections on the test that do not allow the calculator.

You are right, at some point, it makes sense to graph your lines on a Ti-84 to find the intersections or y intercepts, but to take away the ability to perform basic algebra (adding fractions, common denominators, etc...) is not okay.

So then...do we give parents a choice? I imagine it might look like this on registration night:

___check this box if you want your child to perform tasks like a robot

or

___check this box if you want your child to learn how to think for themself, creatively, and solve problems on their own.

I think most parents would overwhelming pick the latter...

Posted by: Erik | March 03, 2008 at 01:11 PM

I don't know about this situation. I have never been able to decide. On one hand i see my stepdaughter who relies so much on a calculator to do simple things like division, yet i see the need for technology. As I've always believed that calculator can't make you get the right answer if you don't know what to put in. So maybe it just isn't needed until the information is embedded. However, i have never taken calculus and have never graphed on a calculator maybe it really does give you the answer. I have one more example. My stepdaughter always used the internet to look up the meanings of her Spanish words. One night our internet was out and she was lost. She relied so much on the computer to do the work that she couldn't think of any other way to solve the problem (i.e. borrow a Spanish dictionary from her aunt, ask us to take her to the store, or use her textbook that she doesn't even bring home because it is on the net). So we are back to the same position.+

Posted by: colette | March 04, 2008 at 02:01 PM

Why are you even having this debate? I took high school calculus 30 years ago. As a matter of point, my class was the first to use calculators vs. slide rulers. I don't see anyone wanting to go to back to the slide rule any time soon.

When you flip on a light you do not have to know how electricity works. Flip the switch and start graphing your calculus problems away. Then, when you get 30 years removed from that class, ask yourself why in the hell did you take it anyway.

Posted by: Joe | March 06, 2008 at 11:31 AM

Okay everyone, I am back. I refuse to let this die. Please check out this npr story done on math and the importance of algebra and fraction "understanding".

http://www.npr.org/templates/story/story.php?storyId=88154049#share

Posted by: Erik Hanson | March 14, 2008 at 09:57 AM