Mathematics (and Physics) and Calculators

This is the third of three articles concerning calculators and mathematics triggered by a blogspot and IHE blog article by Dean Dad, a community college dean who appears to be writing from another part of the country yet has the same problems we have at our CC. The original article concerned calculator use in Developmental math classes that typically cover fractions and 7th grade algebra, but the comments spanned a range from that topic through mathematics and its applications beyond calculus. My first article merely laid out a common set of definitions, but does include a few assertions about various types of calculators and levels of mathematics that might deserve comment. The second article tried to focus on Developmental math but also included some comments about Algebra. In between these, I posted a shorter article that included a more polemical set of comments about the "modern" Z80-based Graphing calculators. Comments on the second article made me realize I also owe the community a long-deferred article about the math preparation of elementary ed teachers.

My second article limited itself to classes that are remedial in the sense that their goal is to get students to finally learn skills that were supposed to be taught in elementary and middle school as well as the first year or so of high school. College Algebra occupies a fuzzy territory because it is sometimes learned in high school (where it would be Algebra II) but is considered a college-level math class that is sometimes a general education requirement. I included it in my previous article because it is not the only gen-ed math option at our CC and serves many masters. In this article, I will take up the issue of most interest to me: whether students are prepared to use calculators and algebra to do physics, calculus, and (perhaps) engineering problems.

My expectations

As noted earlier, I allow my students to use a Scientific calculator and I expect them to have a decent one and be fairly fluent in its use. I do not allow them to use a Graphing calculator or one that is capable of doing computer algebra. The former is excluded because I do not have time to police all of them for cheat sheets, the latter is excluded because I want a level playing field. They can use MathCAD or Maple or Mathematica when they get into upper division classes where everyone will be using equivalent tools on any given assignment. I expect them to do algebra with pencil and paper in a freshman physics class.

The calculus teachers here have a similar expectation. Many (but not all) give exams where no calculators are allowed on part of the test, but a Graphing calculator (mainly for the numerical integration feature that is on some Scientific calculators as well) is allowed on others. Sometimes they even use a computer algebra program on an exam, but that is rare.

One thing I mentioned in a comment on Dean Dad's blog was the importance of defining outcomes. I forgot to mention that outcomes are best defined so the match the desired inputs for a subsequent class. It is for that reason that our calculus faculty require that students actually know certain derivatives cold, like times tables, and why they were stunned into disbelief when a student transferred here from a school where they used an Algebraic calculator that can do all of the basic derivatives and integrals symbolically. That outcome (being able to take a derivative with a calculator) is mismatched to the requirements of physics and engineering. (True, an engineer taking the "fundamentals" exam has a reference book handy that contains the basic derivatives, but the few minutes you are given to answer each question does not give you enough time to look up every basic result.)

Physics

In general terms, my views on calculators are similar to what Chad Orzel wrote in response to Dean Dad's blog. Real math (meaning math major math classes) have no need at all for calculators unless the topic is numerical analysis, and then you are better off with a programmable computer. Ditto for upper division physics majors classes, although they can have a computational component as well (that is, arithmetic rather than the symbolic mathematics of algebra or calculus). My impression from former students is that engineering expects correct computation as well as algebra, so exams require computation as well as the proper setup of the problem.

I should add that the exam security issues inherent in larger classes, where students are unavoidably sitting within copying range, also requires numerical variations between problems. (Exam fairness has, so far, kept me from putting totally different problems on versions used in the same class.) Most on-line homework systems also do this, although some have symbolic variations as well as numerical ones. This leads to an emphasis on problems with numerical values.

Further, because my students tell me what they do in their first engineering classes, I know computation is only part of it. Setting up the problem algebraically and simplifying before computing is ALSO part of it. For this reason, I require them to state the problem symbolically before plugging in the numbers. However, primarily because of their comfort level, I do not take off if they do the algebra with numbers present rather than keep the symbols until the end. (Having numbers and unknowns makes it easier for most of them to keep track of what is unknown and needs to be isolated or eliminated.) I'll let someone else break them of that habit later on, but I will encourage them to work on it in my class. That said, I do sometimes give exam problems where a symbol like L has to be in the final answer. See below.

Computing

What has surprised me is the degree to which students either cannot compute efficiently or use their calculators inappropriately when solving a problem.

The first problem has only become evident to me recently. I don't think it is a new development; I just happened to see a particularly egregious case last year where the student would evaluate something like A*B*C/D by doing A*B, write down the answer, enter the answer*C, write down that answer, then enter that answer/D. Painful. And slow. And prone to error. I should have suspected this sort of problem because the other version, entering ((A*B)*C)/(D), is a bit of craziness not uncommon in Algebra classes. They don't know order of operations and, even if they do, some have used bad calculators that violate those rules and been burned.

This is, however, a real handicap. They need to use one calculator type and use it enough to understand what it does under different circumstances, but might never have been taught that it is OK (and even necessary) to hit lots of buttons and see what they do under different circumstances. I'm going to mention that this year, going beyond such simple things as whether your calculator does -3^2 correctly or whether it knows automatically that the arcsin of 2 (or the ln of -1) is imaginary.

The second problem is doing algebra with long messy numbers in the equations. This came up in an earlier blog post about algebra, with some nice observations in the comments. This summer I've been thinking about where this comes from, and I am convinced it is because they never use realistic numbers in Algebra classes. Their equations all have numerical coefficients that are small whole numbers, not the 10 digit value for the y component of the velocity, v*sin(theta). There is no penalty for using 3 as a coefficient. There is a penalty for using 34.5619288 as a coefficient. They also seem to have not been exposed much to subscripts, so they are initially quite uncomfortable using Vx as a symbolic replacement for that nasty number.

My preferred solution would be to have pre-calc and trig classes use symbols with subscripts so they get comfortable with that math skill, just as I would like them to work with functions like g(y) or x(t) or even x(y). As we talk more about outcomes at my college, I have to see where those skills fit into the goals of our math curriculum. It might be that these are one-and-done skills (like some skills in physics) because instructors at one level don't know how important it is when you do kinematics in physics or power series in calculus and how much students struggle with those concepts. However, I also know that this is overly optimistic. Instead, I am thinking about ways to work those in from the beginning in my class, perhaps by starting with y(t) motion rather than x(t) motion and using v_y and a_y even when they aren't really required at that point.

Finally, there is the way I model doing problems in class. Comment number 4 on Chad's article mentioned math exams where you could only use a calculator on part two, something some of our math teachers do, but then came up with a nice insight:

it also could be used to introduce the concept of only taking out your calculator when you reach the stage where you've gotten the problem to its simplest state, and need only put in the numbers.

I've seen students do exactly that while taking an exam, just as I do, but I've never thought about really making a SHOW of pulling out the calculator at that point of the problem. I need to model that step as clearly and explicitly as I model algebraic steps when solving a problem. I also need to find or invent more problems where a symbol is in the final answer, like it would be if you were writing a program where a few values are fed in by the user but others are fixed by material properties or whatever.

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Calculators and Basic Math

This is the second of three articles concerning calculators and mathematics triggered by a blog article by Dean Dad, a community college dean who appears to be writing from another part of the country yet has the same problems we have at our CC. I have already commented on the blogspot version of this blog (more than once), which has collected a huge number of comments, but there are also a large number of comments on the IHE version of the same posting. I'll write as if you have at least read Dean Dad's article. The discussion has been quite wide ranging, often not bothering to make a distinction between the various levels of "calculator" available to students or the many levels of math classes they might be used in. (Definitions are given here to provide a common reference.)

In this article, I will take up the specific issue raised in Dean Dad's blog and the one I know the least about as an instructor - calculators and developmental mathematics - but also look at college algebra. I'm mainly interested in putting some of my thoughts on paper and seeing feedback I get from others about their opinions of the problem.

I'll start with a particularly telling comment from Dean Dad:

...part of me wonders if we’re sacrificing too much on the altar of pencil and paper. It’s great to be able to do addition in your head and long division on paper -- yes, I know, I’m old -- but is it worth flunking out huge cohorts of students because their high schools let them use calculators and we don’t?

This isn't about LETTING them use calculators. Part of it is about spending time teaching them how to use a specific brand of calculator rather than how to do algebra. However, as I wrote in my first comment on DD's blog (9:11 AM time stamp about halfway into the 50 or so comments that are there now), I think this is mostly correlation without causation. The real problem lies elsewhere.

I think the problem starts in K-5 and gets compounded by pushing kids along into the next class and lying about the content level of that class. That is, I don't believe for a minute that a student I advised had passed a REAL pre-calc class in HS just a week before I talked to her. DD writes about similar cases:

...students who have passed algebra and even pre-calc in high school frequently crash and burn when they hit our developmental math, because the high schools let them use calculators and we don’t.

I don't buy it, and here is why: Our placement test will put them into Intermediate even if they can't do arithmetic, provided their algebra score is high enough. And you can't work with logarithms and exponentials or trig identities (a given if it is really pre-calc) if you can't solve a simple linear equation written symbolically like I asked the student to do.

You might forget 6 months of math in a week, the newest stuff, but not 3 years of it. And if you do forget that much that fast, you should have failed that pre-calc class. You can't get to much new material if you spend most of the year re-teaching three years of previously-taught material de novo.

K-5 Curriculum:

I don't want to belabor this, but one reason they can't do arithmetic might be that they never learned it. I'm convinced this is the biggest problem we face because it also lies behind the existence of pre-calc classes that are really teaching basic algebra. I'm sure part of it is that teachers who never understood math and hate it with a passion are teaching it by-the-book following a curriculum none of you could possibly imagine anyone would use.

My analogy is to the "look say" approach to reading, where guessing replaced phonetic decoding of words and Johnny (not to mention a cousin with a high IQ who is now an senior engineer at the VP level) couldn't read. Using guessing to construct your own mathematics might work with someone like me (I feel eternal guilt for, AFAICT, being an unwitting subject in a math ed research project that was run before the days of IRB and informed consent where they deduced that this curriculum worked really well), but it is unlikely to work with someone who was not going to get a PhD in physics. Really good algorithms were developed 12 centuries ago and survived for a reason. As is illustrated here, for the Everyday Math curriculum, the most efficient methods are not taught first or (in some cases) are not taught at all in some schools. The starting point for Dean Dad might be to get out into the feeder systems for his CC and find what they are doing in fourth and fifth grade.

If I know anything about learning, it is that students always favor the first method they get taught. (That is one reason you have to really emphasize when conservation of energy or momentum should be used instead of Newton's Laws: they learned F=ma first so it is the first thing they want to try. I'm the same way.) That means it is a really bad idea to start with an inefficient method, but which some people find useful when doing 'mental math', and teach the more efficient one last.

Now think about how to teach synthetic division or multiplication of polynomials to someone who only knows partial quotients division or the lattice method for multiplication. Not pretty. Then consider that they might never have even heard about "invert and multiply". Not pretty at all.

Outcomes:

The answer to Dean Dad's fundamental question, whether students should be allowed to use calculators in an Arithmetic class, starts at the beginning - the first step of course design. What are the desired outcomes for this math course? If the outcome is to be able to do a certain amount of arithmetic with pencil and paper (not in their heads), then the only use of a calculator is to check your own work as you make up your own problem and solve it. Ditto if the purpose is to simplify fractions involving simple whole numbers as preparation for a similar skill with symbols. You need to change the outcomes before changing what you do in the class.

Personally, I'd be happy if they taught them how to do arithmetic on their Basic calculator. Seriously! My biggest complaint when teaching physics isn't that they can't do arithmetic (they can't), it is that they can't calculate worth a damn. Digital natives my ass. I was 22 when I got my first calculator and I am faster than most of them are, and I'm slower than I used to be. (Sure, I've been using one longer than they have been alive, but that only proves they haven't used the thing enough to be competent with it.) I mean, I've watched a student work out a product by multiplying two numbers, writing it down times all of the others, entering it again !!! and multiplying it by the next, etc etc.

I know they use their calculators a lot in our Algebra (meaning College Algebra) classes, but it must all be with simple whole numbers like we used back when there were no calculators. That is the only possible explanation for their struggles with 3 and 4 digit decimal or scientific notation numbers or their mysterious belief in rounding intermediate answers.

Algebra:

There really shouldn't be many numbers in an algebra class, IMHO. Somehow the appearance of Graphing calculators changed the curriculum to emphasize numbers and, curiously, de-emphasize graphing. Since you can't actually read a graph on a TI display screen, let alone interpolate on it using a ruler, they don't appear to know how to make or read an actual graph rather than a cartoon of a graph. This is a nightmare in the physics lab, but also in the classroom when data are supplied in a graphical representation.

I don't believe anyone has tried teaching algebra with a Scientific calculator and graph paper in decades. I doubt if anyone other than the textbook and calculator companies have studied it, and studies like that are notorious for the difficulty in controlling the student mix or the instructor effect. However, stories about students who finally got algebra in a class where only symbols were used - no calculators needed - are common enough to make one wonder how it would work. The studies (see some comments toward the end) seem to lack a smoking gun in favor of the primitive Graphing calculators used today.

There is another side effect. Since they don't know how to use their calculators, particularly concerning order of operations, they use parentheses like they were the only operator known to man. ((3)(2))/((5)). One result is they don't see the key role of the parenthesis to denote "function of". I've seen calculus students who think x(t) means x*t, although this could be partly due to the fact that x is never a function in calculator-based Algebra. I have to wonder out loud if this would improve if they all used HP calculators instead of TI calculators. Also see my next comment below.

Commenting on the comments:

Some Anonymous, writing at 5:56 PM, about 3/4 of the way into the comments, writes:

A kid that is getting good marks in algebra screws up their physics equations. i check with their math teacher, and they don't make those mistakes in math class. So I test them myself, and they can manage algebra just fine when x, y, and z are variables and a, b, c, and d are constants. Anything else and they're lost.

This could be a result of using graphing calculators. The TI-83 will only plot Y(X) unless it is in one of the other modes (where it is similarly limited). Parametric mode, the only place where you can do X(T), does not appear to be used at all until they get to Calc III. This really bugs a chemistry colleague, because they are always plotting the log of this versus the sqrt of that, neither of which is X or Y. Similarly, we start out in physics by plotting x on the Y axis and t on the X axis and it blows their minds. I'm tempted to start by doing only y(t) problems at the start of fall, then moving to x(t).

Mthgeek, aka timfc, writing at 7:45AM of the second day of comments, listed several references. The first of these was

The Arithmetic Gap
Educational Leadership, v61 n5 p55 Feb 2004
Summary: The students using calculators in school classrooms result in lower math scores than students who never use them.

I'd like to know what grade level this was, but it sounds like K-8 from the title. As for the second one that was listed, I don't ever pay attention to something like a meta analysis of 42 other papers that span middle school through calculus. Apples, oranges, confounding variables, design differences, and systematic errors make a tasty goulash but don't help with teaching Basic algebra. One other reference, discussing "computer assisted instruction" would appear to be irrelevant to this discussion. You can use computers as an instructional aid (instant HW feedback, for example) without using a graphing calculator - or any calculator at all.

However, the last reference from Mthgeek is rather interesting. It is a link to an article (Refocusing Introductory College Mathematics Courses) that has a link to a new textbook (Contemporary College Algebra: Data, Functions, Modeling) that implements some of the ideas from the study. That is, the study and the textbook are closely coupled, but I recognize some things in that report that are reflected in what we do in our Intermediate class with a different book. (I'll have to ask around, but we might have made this choice because Intermediate is a pseudo-terminal course for many majors in our curriculum. The situation discussed in the article does not apply as much to our college Algebra course, because it normally leads to business calculus or trig. The statement in that report that biological sciences don't go beyond Algebra is patently false in our curriculum. They have to take Calculus even if they don't ever use it.) That said, I strongly criticize the textbook author for conflating a graph on a Graphing calculator with a graph produced on a computer. There is no comparison in detail or quality.

Some Anonymous, writing at 11:21AM on the first day, said (in part):

1) middle school math is more focused on algebra as early as 7th grade. So students don't have enough mastery of fractions, percents etc
...
3) More students attending college- so the lack of good high school prep is more evident.
4) Content of dev math courses in college are aimed at preparing students for a precalc/calculus track. But those going into sociology or psychology ...

Developmental math at my CC is about preparing students for 9th grade math, not pre-calc. The Intermediate course barely prepares them for real Algebra, and certainly not for the calculus track. (Our failure rate is spectacular at every one of those steps.) Besides. students going into Psychology need a real, college-level statistics class that has college Algebra as a pre-req. Criminal Justice, on the other hand, has no real math requirements and our statistics show that the combination of our Developmental and Intermediate classes does a GREAT job of preparing them to pass the basic financial math class that constitutes their "college level" math requirement while teaching them about compound interest.

The fact that they have not learned arithmetic or fractions by the time they get to 7th grade (which is when we started Basic algebra when I was growing up) is the real problem. Three years should be enough if the curriculum and teachers were any good, but if they aren't or the kids don't learn it in 3 years, our schools track those kids away from Basic algebra for another year, or more. But this does help strengthen my point that the problem is really in the K-5 classroom.

I don't buy the "more students" argument because the fraction going to college has not changed that much in the last few decades.

Finally ...

If you have read this far, thank you. I want to close by saying that the problem really is deeply rooted in our educational system and very frustrating for all involved. The high failure rate in Developmental classes is a major problem that no one is ignoring at our CC.

However, many students fail because they never attend class, or don't attend frequently enough to engage with the instructor. With any instructor, no matter how talented ze might be. I've written about that in an old bit of wishful thinking about new student orientation. Coming straight out of HS, they believe they were taught pre-calculus or Algebra II, so they just don't believe they need to go to class and actually learn math. Older students, out of school for years or decades, know they don't remember anything from school so they take it seriously and often do quite well. An age-based breakdown of performance in Developmental classes might be worth looking at, Dean Dad.

Or Dean Dad might only need to walk by a classroom or three on a regular basis and take a sort of visual attendance. Is the room still full after 4 weeks? Maybe that, rather than calculators, is the real problem.

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Interlude - Calculator history

This cartoon from last week really captured my view of "modern" calculators.


Click on the image to see the entire cartoon from XKCD, including the highly relevant punch line.

I was originally going to riff off of this cartoon to discuss "modern technology" in the classroom, but then Dean Dad's article came along. Just for perspective, the current model (TI-83 Plus) shows up priced between $89.99 (on sale at Staples for the new school year) and just under $100 (at Walmart and Amazon). For comparison, the CPI says $110 in 1996 will buy about $150 of normal goods today, but computer prices have been going down even as performance increases. For many decades.

Further, the cartoon is not exaggerating the connection to 1996. Today's TI-83 Plus is still running on a 6 MHz Zilog Z80 microprocessor, an 8-bit cpu that dates to the mid 1970s (as an upgrade to the legendary Intel 8080 chip). The Z80 was used in such memorable machines as the Kaypro II (running CP/M), the TRS-80, and the Sinclair and Timex notebook-sized computers. [The Kaypro, like the Osborne, was a "luggable" computer that would have to be sent in checked baggage today. I still remember using both of those.]

Not exactly MODERN technology, particularly when you consider the limitations of the 96x64 screen compared to, say, a (much smaller) iPhone. This has practical effects in that the calculator has great trouble graphing certain kinds of functions and the interface for "tracing" to a zero is really crude. More importantly, for whatever reason, I see no improvement in algebra skills associated with the month or more of time spent specifically on using this technology. Students do not use the graphs to check their answers, but that is a topic for my other postings on this topic.

Other observations:

The Plus indicates it has 512 kB of flash memory rather than 32 kB of RAM on the original model. The Silver Edition has a 15 MHz cpu and even more memory, but the added speed is one reason why you can clear its memory so much faster than on the Plus. AFAIK, the main difference is that you can clear uploaded programs on the 83 Plus but cannot clear the equivalent programs that are installed OEM on the Silver Edition. The main advantage for students is that you can connect any of these to a computer and download modestly sophisticated applications into Flash memory that are run with the Apps key.

Naive instructors believe that students have to laboriously type in crude crib sheets listing, say, trig identities or chemistry and physics formulas as fake programs. Many do this, but TI provides sophisticated, indexed crib cards - and similar tools are also available from others on the internets. Anyone who "limits" students to a note card of notes but allows a TI-83 without clearing it is laughably naive. Might as well let them bring in a notebook.

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Calculators - Background Info

This is the first of three articles concerning calculators and mathematics triggered by a blogspot and IHE blog article by Dean Dad, a community college dean who appears to be writing from another part of the country yet has the same problems we have at our CC. The original article concerned calculator use in "developmental" math classes that typically cover fractions and 7th grade algebra. I have already commented on the blogspot version of this blog (more than once) and the two together have generated more than 80 comments. I added some more in my second article of this series.

I won't actually comment on this topic here. My purpose is solely to set the terms of the debate, as it were, because the wide-ranging discussions of this topic by Dean Dad and others are seldom clear about which of the four or more levels of "calculator" available to students are being discussed and/or which of the three or more levels of math classes (plus physics and chemistry) provides the context for the discussion.

The divisions I make are somewhat arbitrary and perhaps idiosyncratic, so I want to spell them out somewhere without cluttering up a discussion of the teaching and learning issues as I see them. That way I can link here for future discussions of this topic and not have to repeat myself.

Although I think three levels of "calculator" suffice for most classroom use, and hence for later discussion, I think I need to list at least five to make the definitions as sharp as possible.

• Basic - Here I have in mind a wide range of very cheap calculators that can do arithmetic, including parentheses and scientific notation, but cannot deal with trig functions.
  
  
• SCIENTIFIC - These calculators can evaluate all of the basic functions (trig, hyperbolic, log, exponential, power) but cannot store text or programs. Some can work with complex numbers and/or hexadecimal numbers. At the high end, some can numerically evaluate definite integrals or derivatives or solve simple equations, but they cannot show any intermediate algebraic steps or work purely with symbols.
  
  
• GRAPHING - Here I have in mind several calculators that are similar in capability to the TI-83Plus. They can do all of the calculations of a top end "Scientific" calculator, but can also make graphs and store programs (including large amounts of text that can serve as a sophisticated crib sheet). They can store text, but cannot work with symbols. Functions are limited to y(x) except in the rarely-used parametric or polar modes.
  
  
• ALGEBRAIC - These calculators can solve equations written symbolically and can, in some cases, even show step-by-step the algebra or calculus used in the solution. They are typically somewhat limited in how much calculus they can do symbolically, but they make it unnecessary to learn any of the derivatives typically encountered in calculus.
  
  
• Computer Algebra - Here I have in mind small computers that can run computer algebra programs like Maple, Mathematica, MathCAD, etc. Now you might say "a laptop is not a calculator", but there is actually a rather modest size difference between a notebook-sized laptop and the top end TI "calculator" that comes with a full keyboard and a wide screen. Besides, these are widely used in classes at the Junior level and above so they help frame the discussion.

The three in the middle, in all caps, are the ones I will refer to most often within the context of lower division classes taught at a community college.

For the record, I allow Scientific calculators in my introductory physics classes but do not allow formula sheets or cell phones or Graphing calculators to be used on exams. I encourage students to get one of the high-end Scientific calculators that can be used throughout their engineering career, including on licensing exams, so they become fluent in its use.

The four levels of mathematics classes are defined as follows:

• Developmental - The content here ranges from arithmetic and fractions (what I characterize as 4th and 5th grade math) to basic algebra (the first class where "x" is used, taught in 7th grade when I was in school). These do not carry college credit. A well-calibrated placement test determines where a student starts, and some have an exit exam to verify competency at a certain level.
  
  
• Intermediate - The content here is algebra through what I knew as the 9th grade level (the quadratic formula, for example, but no logarithms). This might earn college credit at a community college, but not at a university. It is not considered to be at the college level. A well-calibrated placement test is used to place students in or through this level of math.
  
  
• College Algebra and Trig - I group all of the pre-calculus "college level" courses here but exclude other "college level" classes that exist mainly to ensure that liberal arts majors can graduate even if they can't do college algebra. (Those other classes usually cover enough about exponential behavior to understand compound interest on credit cards and enough probability so you should know better than to play the lottery, both very valuable life skills!) At our college, College Algebra serves many masters so skills not needed for the pre-business curriculum are put in an "advanced" college algebra class (pre-calc) and a trig class. (I know that some colleges, like my high school and undergrad university, combine these into a single course but I will use our curriculum as my reference point.) A different, also well calibrated, test is used to place students above this level although most students take the class.
  
  
• Calculus - Although my students will usually take everything through differential equations and linear algebra, I'm mainly thinking about first semester calculus because that is where the bulk of students fail.

The distinction between Developmental and Intermediate might seem unnecessary to some readers, because both levels are usually non-credit classes at a university. Indeed, some universities define college algebra as a remedial course. I make the distinction because our math department teaches classes at the Intermediate level and above, while the Developmental classes are taught by a separate department that specializes in teaching those skills. I know that smaller colleges do not make this distinction, but we are not a small college. (We have more t-t faculty in our Developmental math department than a private school like Union College has in its regular Math department.)

If I just say "Algebra", I mean College Algebra. I will say "Basic Algebra" or "Arithmetic" when I am talking about Developmental skills classes.

For the record, our Developmental classes use a Basic calculator for some things but some exams must be taken without any calculator. (The placement test and exit exam do not allow use of a calculator.) I believe they allow the use of any calculator up through a Graphing calculator when they allow a Basic one, but that might depend on the instructor. Our Intermediate classes all use calculators. Our Algebra classes require a specific Graphing calculator that is also required for statistics. Our calculus classes are a bit less picky about which Graphing calculator students can use, but ban Algebraic calculators and computers except in some special situations.

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