Saving Mathematics, Part III: It’s Not Just Intermediate Algebra

It’s true (in my view) that intermediate algebra must die; that was discussed in a recent post. We need to look for other places for basic change in the mathematical curriculum. #STEM_Path #Pre-calculus

In response to that post, a long-term critic of our work with some good ideas (Schremmer) made this statement in part of a comment:

In fact, intermediate Algebra cannot be killed, as long as Precalculus, the reincarnation of College Algebra, has not been killed too. And Precalculus is not going to die either as long as it has not been reunited with the Differential Calculus. And, in spite of the few millions it spent in the late 80s, even the NSF was not able to reconstruct the Calculus

There are certainly challenges to changing these courses on the STEM-path (articulation being the paramount issue). However, we have done little to work on the known problems. Whether you think we can create a more efficient curriculum of 5 courses as I do (1 reformed precalculus course, 2 reformed calculus courses, 1 reformed differential equation course, 1 reformed linear algebra course) … or 3 courses as some others do (3 courses encompassing all of those topics) … nothing excuses our continuing past practice in the year 2016 or beyond.

The stakes are high. If we do not fix this problem, our client disciplines will teach all of the mathematics they really need (much of which is already happening) — and they will stop using our courses in their programs whenever they have the option. Most of our enrollment are from programs in these client disciplines.

If we do not fix this problem, we continue a curriculum that hides the modern nature of our work from students; who do we expect to become tomorrow’s mathematicians? Using cool software to teach awful mathematics is a terrible trick to play on students; I compare that to putting a GPS on a 1975 Pinto … it looks, in a very small part of reality, like we have modernized but the body of the work is mostly useless material.

This is our greatest challenge. Will our legacy be that we had an opportunity to modernize the curriculum but wasted it … or will people see that the profession can work together to achieve something great?

We must step up; we must respond to the challenge with hard work and collaboration. The rewards are too great, the risks too great, for us to take the easy path of ‘change nothing’.

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Math curriculum in general, pre-calculus | Jack Rotman | December 15, 2015 3:03 pm

4 Comments

By Kenneth Tilton, December 15, 2015 @ 4:00 pm

“Using cool software to teach awful mathematics is a terrible trick to play on students.”

I heard that! As Mr. Miyagi might have said, “There are no bad mathematics, only bad teachers.” YouTube has the videos to prove it.

Classic Algebra is where we build the fundamentals of mathematical thinking and notation, climbing off the concrete of arithmetic to our first abstractions about quantity.

Consider my treatment of the mechanics of flipping an inequality when multiplying or dividing by a negative: https://youtu.be/4hBPuVs1E5s?t=5m55s (It ends with “How cool is that?” ).

If you consider my pedagogy inadequate, no problem! Record your own explanation and drop it right in the app alongside mine. We’ll let the students “mod up” the ones they find most helpful, and over time produce a deep treatment of Algebra that would make any CCSS fan happy.

But don’t get me wrong. I am all for whatever we have in mind for “wonderful mathematics”. That will breathe life and meaning into Algebra, which students can actually master now that they have cool software. 🙂
By schremmer, December 15, 2015 @ 7:42 pm

1. Re “long-term critic of our work”. My criticism has been based on, and only on, two aspects of said work.

—I have not been able to see any textbook product thereof. All I have seen are general purpose descriptions that are not different from any general purpose description. Back in the times when I used commercial textbooks and some of them were actual “different”, it would not have come to my mind to order one without having spent a couple of hours on it.

—Yes, language is most important. I wrote about that in my AMATYC column. But the language has to be integrated with the mathematical progression. I cannot see how working on, say, mathematical literacy in isolation is going to do much. For whom and for what? Yes, I have read the literature but I would need to see what happens to the successful students afterwards.

2. I do not know what

“create a more efficient curriculum of 5 courses as I do (1 reformed precalculus course, 2 reformed calculus courses, 1 reformed differential equation course, 1 reformed linear algebra course) … or 3 courses as some others do (3 courses encompassing all of those topics)”

refers to. Certainly not to what I wrote which was:

“a three semester sequence, say 5-4-4, starting from Arithmetic up to and including the Differential Calculus”

and then

“probably followed, instead of by the Integral Calculus, by Differential Equations, perhaps in the shape of Dynamical Systems.”

3. My point, then, if I had to put it in one single word, would be “integration”.
By Jack Rotman, December 18, 2015 @ 11:32 am

#1: The ‘long term critic’ phrase was meant to encompass your commentary over the past 2 decades (perhaps more) about college mathematics … not the current reform efforts in particular.
The mathematical literacy work is intended to start students on a ‘path’ of mathematical reasoning, though with limited algebraic symbolism
#2: Actually, my intent was to include your ‘5-4-4 three semester sequence’ within the category of ‘3 courses’; I know that this is a poor approximation, but I wanted to get the 3-course idea out there.
#3: Integration is a good word, and receives considerable support among our colleagues; the disagreements tend to be in the details of what that means. Since we have such a lean set of experience on integration, those disagreements are normal and healthy.
By schremmer, December 15, 2015 @ 7:57 pm

I should have explained why:

–Re. my “5-4-4 [sequence], starting from Arithmetic up to and including the Differential Calculus”

Because, in a two-year college, no “sequence” should be longer than three semesters to leave at least one semesster for specialization. For instance:

–Re. my “Differential Equations, perhaps in the shape of Dynamical Systems” because Dynamical Systems are a lot more appropriate in the life sciences. So, a choice would be a good thing but, in any case, I do not see that the Integral Calculus has much to offer at this stage. The real question is what to do with Calculus III.

By the way, I would be all for Two-Dimensional Calculus, not because I wrote a book starting with 2 dimensional calculus (I did) but because of Osserman’s book (an alternative to Calculus III)—which inspired me to write mine.

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