Remediation - and Calculators

Dean Dad put together a fantastic article Friday about the remediation "death march". Timely as well as interesting. Timely, because the part of our college that has this as its mission has been working hard for the last two years to revisit the entire system they use, and interesting because I'm not in an area where I learn much about how others address this challenge.

I only have one thing to add on the main theme (beyond what I already said in my comments on his blog), but will expand on two other issues: the "tough sell", and issues related to algebra preparation (remediation) for pre-engineering students.

Math Remediation:

Math is really where the death march takes place. Math is handled so badly at the K-12 level (with most of the damage taking place in the 3-8 territory based on my limited contact with a range of students who enter our CC as math cripples) that students are math phobic - putting off their remedial math classes and often barely giving them a chance to work.

I already commented on the structure of our system, which mirrors the one DD described. That is pretty typical, although ours is going to have some major internal changes (involving more targeted remediation) in the near future. I only know the broad outlines of the plan, which will be implemented over the next couple of years, but I think it builds on something we learned about some higher-level math classes as a result of a major effort by one of our best low-level algebra teachers. We learned that the usual process of starting with "review" topics was fatal. It bored the ones who had actually learned the material in the previous course, and led all of the students to believe there would be nothing new in this next class. The new approach is to present new material on the very first day !!! and work the review skills in on the fly. It appears to be working, although it still works best in the hands of experienced professors.

The Tough Sell:

Our success rate with remedial courses is much higher with students who return to school after many years working. They are under no illusions that their HS education has prepared them to take college classes, because they know they have forgotten what little they learned in HS. They are ready to start over, and often thrive in our environment.

The challenge is to reach students who have just left high school and have high self esteem and little else. The ones in the middle third, the ones who can't get into university unless they can play football or basketball, are a big problem. They got coached well enough to eventually pass the math exit exam so they could earn a diploma. They think that this should mean that they are ready to move from HS math classes to college math classes, just as they moved from middle school math to HS math. Unfortunately, no one told them that the HS exit exam only proved that they were ready to leave middle school.

I am not joking. It is not enough to look at those exams and notice a few problems at the level of 9th grade Algebra 1, as one commenter did on DD's blog. You don't need to get every problem right to pass the test. If you look at the score needed to pass, it is immediately clear that they don't need to know any HS math at all. If you factor in the detail that they have a calculator when taking that test, and have been coached in how to test answers against questions, etc, etc. I could not disagree more with what Sherman Dorn wrote on this subject. The confusion is between taking something called "algebra" and the sad fact that such a course in HS merely prepares a student to place into a remedial class, particularly if the next two years are spent taking "consumer math". They certainly are not ready for college algebra.

As I commented, my old article presenting an idea for Freshman Orientation at a CC suggests telling them they were lied to in HS. I have no idea if this would work. They probably would not believe any adult. It would have to come from a student. The same goes for the reality of failure in college.

This problem is deeply ingrained because of the massive amount of propaganda related to passing rates and No Child Left Behind. (None left behind? Ha! Read Sherman Dorn about "graduation rate statistics". Eye opener.) They could very well have improved math skills in our local high school grads. I have little doubt they used to be worse based on stories from my elders. But students and their parents (plus taxpayers and legislators) have been led to believe they know some math when they don't. Only the best local schools produce an average grad who does not need remediation.

Algebra for Physics and Calculus:

Finally, the promised remarks concerning what mthgeek wrote in the comments about calculators and other technology in the calculus classroom - and what the expectations are by the customers of calculus.

Mthgeek wrote (first comment):

At my university we proudly outlaw calculators from the Calc sequence even though all the disciplines we "serve" want their students to be proficient with technology including calculators, spreadsheets, and computer-packages.
Maybe, just maybe, the conversation about remediation should also be expanded to include discussion of the credit-bearing courses as well.

We already did a first step in that expansion at our CC. Other changes appear to be working their way down from the one mentioned earlier, and there is some hope that other changes will work their way up. I am particularly sensitive to the low level of algebra skills in the students who enter my physics class. They can do basic algebra, but they can't follow algebra being done at the board at anywhere near the rate expected in a calculus or physics class. So I hope the use of calculators in algebra classes gets looked at.

I don't know if this reflects the fact that our CC proudly requires a specific calculator and spends weeks teaching them how to do algebra with it. Weeks! How to graph. How to identify discontinuities and poles. How to "trace" to a zero. None of this time does anything to increase the chance that they can move symbols around or substitute an entire expression into another one. Yet, despite all of this experience, a large fraction still don't know how to use a calculator correctly. In addition to entry errors (some related to not knowing how scientific notation works), they round intermediate results and can't round answers correctly to the relevant significant figures.

As I wrote on DD's blog, the engineering school attended by most of my grads has its very own indoctrination program for everything from computer drafting to computer algebra. All classes use the same set of programs, and these are taught in conjunction with other basic engineering skills in some set of intro "gateway" courses taken by all entering juniors. Reports from my grads indicate that experience with programs like Maple or Matlab in some of their calculus classes has made that transition easy, but they find it far more significant that I expected correct free-body diagrams along with correctly calculated answers.

But the use of symbols rather than numbers in problems is something we often think about and talk about. There was an excellent article on this subject from Chad over at Uncertain Principles, including the other article he links to (by Excited State) and the ones linked from the comments. I will single out the ones in comment #16 from "Gerry R", chair of the mechanical engineering department at Portland State, as worthy of particular attention. I will have to spend some time this summer thinking about how to write problems, like ones from his MechE fluids class, that combine conceptual and quantitative skills within the same wrapper.

Mthgeek also wrote (later comment, selectively editted):

And, if all of our examples have nice answers so that the arithmetic is simple what happens when it's not?
And if all of the examples are nicely segregated into sections of text based on the methods that they use what happens when they run into an ill-defined problem?
But, I am saying that the way that many such courses are constituted only imparts a very small set of skills that students only know how to apply in nicely-formulated problems. I hope you expect more from us, seriously, I do.

Journal of Mathematical Behavior 26 (2007) 348–370:
The results also show that about 70% of the tasks were solvable by imitative reasoning and that 15 of the exams could be passed using only imitative reasoning.

Oh, we expect more, because we definitely give comprehensive midterm and final exams that require analysis and retention of much more than the least memorizable unit. Yet, even then, I know many of my exam problems require only imitative reasoning. I hate to say that I set my goals low, but I am rarely teaching math majors or even physics majors, and even most physics majors don't need to be creative theoreticians. Most of them need to be creative experimentalists (ditto for the engineers), and this does not require much beyond imitative reasoning applied (over and over again, with extremely high levels of reliability) to creatively constructed scenarios or designs.

And I agree with the concern about what happens when the numbers are not simple. Ditto for when the functions are not simple. (Do you teach them about integral tables, like G+R, or numerical tools? Those are used a lot by people who know how to set up problems that contain an integral as part of the solution. Few real problems produce easily integrated functions.) I can tell you that my problems contain non-trivial numbers as well as symbols, but more of the former than the latter. However, a correct answer that does not start from a correct symbolic presentation of the solution does not get full marks. Sometimes it gets less than half marks if the answer comes from numerology rather than physics.

I could write more, but I have already chewed on my first draft enough times that it is time to let this go.

PS -
A recent article by mthgeek really needs a response. I can't believe there is a grad program in any field where students don't learn to write papers by working with faculty who are writing papers, but maybe I am naive about areas outside of the sciences.