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Someone I know recently took an online intermediate algebra course to prepare for college algebra.

Thus course had 70+ sections, each with 10-30 poblems, beginning with set-builder notation and going on to associative/distributive/commutative laws, etc. and in general proceeding axiomatically.

Most college math courses proceed axiomatically, to maintain rigor.

But in the same high school courses this is intended to replace, the emphasis is on 'learn this useful formula and apply it 30 times until you get it'.

In my experience, the best students, who would actually appreciate the rigor, never take the rigorous college courses, but instead do great in high school and move directly into college-only courses such as linear algebra and vector calculus. While those who take intermediate or college algebra in college tend to be put off by the overly rigorous approach.

My questions are:

Is there a significant difference in the rigor of these courses as taught in college and high school (i.e., is it not just my imagination)?

If there is a difference, is there any published research supporting such a difference, or statements by those in charge of curriculum at an institution giving an explanation?

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Part of the explanation is just that the first builds on the second... – vonbrand Apr 15 '14 at 0:08
    
High school algebra is about 180 course-hours long. College algebra is about 36 course-hours long. They cover (mostly) the same material but leisure is not an option any more. Rigor is simply the most efficient way to go. Supposedly, college students are adults and possess analytical and synthetic skills. – Steven Gregory Nov 4 '15 at 13:53
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@StevenGregory: College algebra is about 36 course-hours long. At the community college where I teach it's 64 hours. You're also making an apples-to-oranges comparison, because in a college course, students are typically expected to do 2 hours outside of class for every hour in class, which would make a total of 192 hours in my example. High schools where kids have 6 or 7 hours of instruction per day can't possibly require twice that many hours outside of class. – Ben Crowell Dec 21 '15 at 0:45
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@BenCrowell And how many college algebra students actually spend two hours per credit hour studying? My point is that the material is the same, but much more is expected of the students. Hence they complain about how much harder the class is. – Steven Gregory Dec 21 '15 at 2:11
    
I've never come across a College Algebra course taught axiomatically, and I haven't seen any College Algebra textbook that attempts to present the course as rigorous (in the U.S.). Can you give some examples or clarify? What do you mean by "axiomatically" and "rigorous"- you can't possibly mean that all of the theorems used in elementary algebra are proven- are many not far beyond the scope of the course? – Andrew Dec 21 '15 at 22:51
up vote 14 down vote accepted

Although someone else might agree that "difference in the rigor" is the crucial aspect, my own observation is that this is not the issue, although one can argue that there is some sort of "difference of rigor".

Rather, the best students in today's high schools are treated very gently grade-wise, which they have learned is the only truly relevant currency as a measure of success. Thus, they subliminally infer a level of effort, a "good intention", that suffices.

Upon arrival at a substantial university in the U.S., there are several upsets. First, although the students think this is a good thing, the structure to their personal lives created by living at their parents' house is gone, and this makes everything else more volatile. Second, instead of being high-school seniors perhaps at the top of their graduating class, they've become the lowest-status creatures on the whole campus... which has a population 20 times greater than their high school, etc. On top of all of that, suddenly their Pre-Calc instructors don't seem to have their own self-esteem wrapped up in the success of their students... but almost the opposite, in many cases.

In many cases, the use of low-level math courses is not to educate anyone, but to filter, so that otherwise-alarming failure rates are not immediately considered a bad thing.

By this point, seriously, issues of "rigor" are not the critical ones... but general ambience, adversarial relationship of student and instructor, and so on.

(I do not have literal stats, but have been a concerned observer for 30+ years...)

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Similar observations here. – vonbrand Apr 13 '14 at 3:14
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I'd also like to point out that students who start with college algebra are to some degree playing catchup. Perhaps only one or two percent might desire to become a math major, but these students deserve a chance. A rigorous college algebra course can allow underprepared students to gain "mathematical maturity" as they catch up to speed. If they succeed in Algebra, they can now succeed in there next class, calculus, where many students have probably already taken AP calculus. If Algebra was watered down, those that succeeded may not do very well in future courses. – MHH Jun 15 '14 at 1:32
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The question is about a specific college math course; almost all of this answer is just about college as opposed to high school in general. In many cases, the use of low-level math courses is not to educate anyone, but to filter, The course being described in the question isn't designed to filter. It's a remedial ("developmental") course covering material that here in California should have been learned in either 8th or 9th grade. – Ben Crowell Jun 15 '14 at 1:55
    
Belatedly, I notice the comment that the course in question is purportedly not to filter, but to remediate. Then I have to wonder "to what end?" @MHH's comment that a tiny fraction of the students may truly be well-served by such a course is surely not a genuine explanation for the existence of these courses (which exist in my university, too, but I don't believe there is any pretense that most of the students will do anything with the material subsequently). – paul garrett Dec 20 '15 at 22:04
    
As a frequent teacher of these courses, I agree with MHH's comment as about the best justification, IMO. The alternative is to be awarding college degrees to students who don't even have general 7th-grade math skills, and are unable to interact with any science or computer writing in the future (where "these courses" are the remedial elementary algebra courses). – Daniel R. Collins Dec 21 '15 at 5:08

The actual level of rigor is going to depend on the school and the teacher, and has very little to do with the level of rigor claimed in the official course outline or in the textbook. Colleges are required by accrediting bodies to use college-level textbooks, but accrediting bodies can't tell whether teachers are actually applying college-level standards.

It depends on the country we're talking about, but in the US, the material you're talking about is stuff that students are supposed to be exposed to for the first time at an age somewhere between 13 and 15. At this age, they aren't adults intellectually. At age 13, many kids are barely able to read, so they need a textbook pitched to a very low reading level.

The reality is that remedial math courses at the college level have extremely low success rates. This fact has very little to do with the level of rigor of the courses and a lot more to do with the fact that the students who place into these courses are students who have already built up a record of failure in math. Past failure correlates with present failure.

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High school algebra courses have to cater to both serious and "nonserious" math students (the latter are taking their last course in math). As such, they are not in a position to match college levels of rigor, even for the same class.

It's only beginning with (senior year) calculus, where all the students are "serious," that math is taught a true college level.

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I don't think it is literally true that most senior high-school calculus courses, "AP" or not, are taught at a "true college level", based upon extensive observation... – paul garrett Dec 20 '15 at 22:05
    
@paulgarrett: didn't say that "most" high school calculus courses are taught at a college level, only that they "begin" to be. But a good test of whether or not a high school calculus course is taught at a college level is the number of 4s and 5s scored on the AP test (even 3s for BC). i – Tom Au Dec 20 '15 at 23:02

Here's a late response, but this question keeps taunting me for a direct and difficult-to-craft answer.

Is there a significant difference in the rigor of these courses as taught in college and high school (i.e., is it not just my imagination)?

Likely yes; although this will vary by location, institution, and instructor.

If there is a difference, is there any published research supporting such a difference, or statements by those in charge of curriculum at an institution giving an explanation?

My best reply is this: The published research supporting this difference is the entire body of literature for the discipline of mathematics since Ancient Greece.

To expand: College professors are likely to be active mathematicians and have a perspective on the structure and development of mathematics that high school instructors are less likely to have. Moreover, college professors are principally incentivized and promoted on the basis of doing math, whereas modern high school teachers are judged on the basis of passing students. At the high school level, the algebra course is likely terminal for most students; whereas at the college level, it will be viewed as only the beginning of a much larger body of knowledge and exploration. The college faculty will want this foundation to be firmly set, with correct explanations and formulations, so as to support later work throughout the discipline. At high school this is unlikely to be a goal or even an observation; if the system can get the student past a given standardized exam, it is considered "job done".

"But perhaps the greatest accomplishment during the first three hundred years of Greek mathematics was the Greek notion of a logical discourse as a sequence of statements obtained by deductive reasoning from an accepted set of initial statements assumed at the onset of the discourse... Certainly the most outstanding contribution of the early Greeks to mathematics was the formulation of the pattern of material axiomatics and the insistence that mathematics be systematized according to this pattern. The concept of axiomatic development in mathematics must be ranked as one of the very greatest of the GREAT MOMENTS IN MATHEMATICS." -- Howard Eves, Great Moments in Mathematics (Before 1650), Lecture 7.

To the college professor, math without a logical axiomatic development doesn't even count as mathematics at all. Personally, I refer to the memorize-this-formula-without-proof approach as "faith-based mathematics".

Now, if you look for some modern literature on teaching algebra at the college level, it will be very easy to find publications arguing exactly the opposite. That is: Most students won't "use" algebra, practical applications are the only thing we need to consider, deep understanding is extraneous, taken-for-faith procedures are acceptable, college professors should be removed from remediation efforts, etc. But this overlooks the whole point of mathematics -- the search and sharing of proper justifications of our knowledge. This oversight is exactly how rigorous math has been removed from high schools in the last few decades, resulting in the jarring wake-up call when students jump the chasm to college-level work. There is definitely a movement to remove real mathematics from college academics for the general student in the future, because of this ongoing difficulty; but the idea that most students will never even be introduced to real mathematics is almost overwhelmingly difficult for the dedicated math practitioner to swallow.

In short: Math without logical proof is not really mathematics at all. The fact that today's high school programs leave the majority of students completely unaware of that fact is indeed an ongoing tragedy, one that we can only hope won't be forced as a contamination into our higher education programs.

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