## College Algebra is Not Pre-Calculus, and Neither is Pre-calc

“Everybody knows what college algebra is!” This was said by a math chair from a university in my state, as we worked though our state’s new transfer requirement for mathematics. Of course, he was wrong … though he has a lot of company. Today’s main question is this: Is college algebra a subset of pre-calculus?

The original college algebra course developed in the 19th century at the universities of the day (Harvard, Yale, Bowdain, etc), with a focus on meeting a math requirement for their degree. Of course, those times were very different … the Yale Catalog listed every student, and every student had the same default schedule. College algebra was everybody’s math course as a freshman; those ‘desiring’ calculus took it as a Junior. See http://elischolar.library.yale.edu/cgi/viewcontent.cgi?article=1069&context=yale_catalogue

That tradition carries forward to the present day, in the work of the MAA. The MAA College Algebra guidelines remain a narrative for a general education class, not a pre-calculus course. See http://www.maa.org/sites/default/files/pdf/CUPM/crafty/CRAFTY-Coll-Alg-Guidelines.pdf

The use of the name ‘college algebra’ for a calculus prerequisite appears to be a regional variation. In states use ‘college algebra’ as a prerequisite for pre-calculus; other states use college algebra as the first semester of pre-calculus … or as their one-semester pre-calculus (as in “college algebra and trig”).

Our situation has become illogical and disfunctional.

When publishers market their textbooks, sometimes the key difference between college algebra and pre-calculus is this: pre-calc emphasizes a unit circle for trig functions, while college algebra uses right triangles. Other than that, the pre-calculus book has more complicated problems, but no substantive differences. Both courses trace their ancestry back to the 19th century mathematics course later known as ‘college algebra’. [Search for Jeff Suzuki’s talk on college algebra.]

Neither course is really pre-calculus.

Of course, I don’t mean “students can not take these prior to calculus”; they do, though the benefits are small and accidental. A pre-calculus course would be designed to prepare students for the work of a calculus course. We make the fatal mistake of equating the ability to solve complicated symbolic problems with the capacity to reason with those objects.

A good preparation for calculus begins much earlier for many students. “Developmental” mathematics is being re-formed to focus on understanding and reasoning, with a de-emphasis on artificially complex symbolic work. A mathematical literacy course is a better preparation for calculus than the traditional algebra course.

More importantly, Algebraic Literacy is where we can begin the serious work of preparation for calculus. Intermediate algebra is a documented failure as preparation for college mathematics; algebraic literacy is designed deliberately for these purposes. The Algebraic Literacy course has learning outcomes backward-designed to meet the needs of calculus preparation … to be followed by a well-designed course at the next level which completes that preparation.

Here are some conditions necessary for good calculus preparation, based on the available information:

- diversity of content (algebra, geometry, trig as minimum)
- non-trivial reasoning about mathematical objects
- concrete (context) and abstract situations
- properties of functions, and relationships between types
- reasoning and visualization involving related quantities (2, 3, or 4 at a time)
- procedural expertise and flexibility

I do not intend for this list to be exhaustive. The intent is to focus on key outcomes so we can determine when we have a real pre-calculus experience that will work for our students. It is my belief that the great majority (>99%) of our current courses used as a calculus prerequisite are not reasonable preparations for the demands of such a course.

Some of our colleagues are beginning the work of correcting the curriculum; we need to support that work when possible. If you’d like to explore what the new curriculum would look like, the Algebraic Literacy course provides a good starting point; I’ll be doing a session at the AMATYC conference (Nov 21, 11:55am) in New Orleans.

We can solve this problem, together.

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By schremmer, October 19, 2015 @ 9:35 amRe. “some conditions necessary for good calculus preparation”

With that list, hardly anybody will ever make it. In my long experience—about as long as yours, “non-trivial reasoning about mathematical objects” is the only necessary mathematical condition. An open mind and a readiness to commit are the only metamathematical ones. The real issue is the choice of “mathematical objects” and of their architecture.

P.S. In my recollection, pre-calculus was sold as a “function approach” version of college algebra. The first precalculus text I ever saw was Munem and Yizze.

By Jack Rotman, October 19, 2015 @ 9:50 amInteresting comment. I know of one very famous mathematician who confessed that he thought, for years, that reasoning was so important for calculus that he did not worry about symbolic skills … and then concluded that he was wrong. Based on what I learn from people who teach calculus, a combination of reasoning and procedural fluency is critical for calculus, along with a deep understanding of functions. Those conversations resulted in my list.

By schremmer, October 19, 2015 @ 10:00 amAs an example, here is something I recently wrote on Research Gate:

“And indeed, at least some of these students can be brought to consider issues and to start thinking about how they might deal with a “problem”. Here is an example in so-called “Developmental Math”. When the data set is counting numbers, x<5 has four solutions: 1, 2, 3, 4. When the data set is decimal numbers, I want the students to come to a proof that there is an infinite number of solutions. That draws of course a completely blank stare from the class. So, now I claim that there are only a million solutions. Prove me wrong. Quite often, at this stage, students get very angry. So, I back down: how about if I claim that there are only ten solutions? After a usually long while, someone invariably comes up with 0.5, 1.0, etc. How about if I now claim that there are only a hundred solutions? After still a surprising long while, one finally gets 0.1, 0.2, etc.

By schremmer, October 19, 2015 @ 8:18 pmRe. “Based on what I learn from people who teach calculus, a combination of reasoning and procedural fluency is critical for calculus, along with a deep understanding of functions.”

Based on what I learn from people who **learn** calculus, what are necessary are a willingness to ponder a question and to keep one’s goal in mind, an ability not to be obsessed by “getting the answer”, a recognition that without a very precise language, we have no idea of what we are talking about, and a respect of logic.

A Developmental Algebra student emailed me the following last night:

-3x-12=-7x

-3x=-7x+12

Also how is it possible to divide 7 divided by 3

X=2.3x+12

? And then do you divide 2.3 by 1??

Here is my response:

Since I want a basic equation (something like x = #), I want to get rid of the -12 on the left which I do as follows:

-3x -12 oplus +12 = -7x oplus +12

-3x = -7x oplus +12

Now I want to get rid of the -7x on the right which I do as follows:

-3x oplus +7x = -7x oplus +7x +12

+4x = +12

To get a basic equation, I want 1x on the left. So I divide both sides by 4:

x = +3

(As for your other question, can’t you divide 7 dollars among 3 people? How much does each person get?)

By schremmer, October 19, 2015 @ 8:22 pmP.S. “Symbolic skills” are a series of shortcuts one starts using once one knows what one is doing. But, in order not to depend on memory, one should be confident that one is able to revert to making one’s case step by step.

In the above example, once you are completely comfortable you will probably write only one step, at most two.

By schremmer, October 19, 2015 @ 8:30 pmP.P.S Actually, the above was the second exchange. Here is the first.

Hello,

On the question 10-6. The value of X is in both sides of the equal sign. Do we need to pass

What does “pass” mean? (It is not in the index of the book.)

the X to the left side of the equal sign or do we just have the value X in both sides and we either add it or subtract it?

Not too clear, at least to me.

Keep your goal in mind: you want a **basic equation**, that is you want one **x** on the left and **a number** on the right.

So, first you need to “get rid” both of the -12 which is on the left and of the -7x which is on the right. (But one at a time.)

What is the only tool that you can use to do that?

Answer: ominus what you want to get rid of—but ominus the same thing on the other side to keep things balanced.

Once you have a number of **x’s** on the left and **a number** on the right, you know what to do: divide both sides by the number of **x’s** you have.

Then, you will have 1x = **a number**