How to lower one's expectation in teaching college algebra

[symbolipoint]

That's a very small sample.

Narrow region. The choices are picked for practicality. Where can a person go to find Geometry at a community college for PRACTICAL purposes. I can pick a slightly larger radius and find maybe 6 colleges, but beyond that, less practical. The FOUR colleges identified in the narrow region ARE the colleges (community colleges) IN THAT REGION. The sample size there is exactly 4. They are the only colleges which are important. Basically a change from driving up to 6 to 8 miles; or going over 15 miles, to attend a c.c.

 

[mathwonk]

I also checked online info for about 10 different local colleges and community colleges vis a vis geometry courses. about half had courses for elementary school teachers that included geometry, others had calculus courses that included analytic geometry. my impression therefore is that people think of euclidean geometry as an elementary course that is taught and learned in high school or earlier, and hence is assumed in some form in all college courses.

my experience shows this is false, and that most US secondary schools offer very inadequate geometry instruction and most college students have little grasp of it. but i only learned that late in my career by teaching the geometry courses for elementary and high school teachers.

at my former uni, univ of georgia, there is a very strong school of math education that coordinates with the math dept and thus there are numerous geometry courses aimed at teachers of all levels, at least elem. school, middle school, and high school. there are also courses in differential geometry, topology and algebraic geometry, as well as courses showing how modern abstract algebra (group theory and linear algebra) can be used in euclidean as well as projective geometry.

the basic fact that euclidean geometry is not well taught or learned by most americans except in advanced university courses seems still somewhat ignored. As an example of this ignorance, when preparing to teach the class I read on page 8 of Hartshorne that he would use a fact he hoped was familiar to most readers: that any two angles in a circle which subtend the same arc, are equal even if their vertices are at different points of the circle, I did not myself recall that fact. And I was then a university professor and researcher specializing in (algebraic) geometry for several decades. Teaching and learning geometry from the book of Euclid, in my 60's, was one of my greatest intellectual pleasures, and I have studied thoroughly only books 1-4 and parts of 5 and 6.

 

[mathwonk]

Interestingly, the one local college that did offer a (euclidean and non euclidean) geometry course, evergreen state college, apparently last tried offering it for summer 2019, but it was canceled for low enrollment. One pleasurable outcome for me was noticing the instructor for that course was to have been Richard Weiss, the gentleman who was credited by Michael Spivak for supplying the answers to exercises in his original Calculus book, pub. 1967. Weiss was then a Brandeis undergraduate and went on to a PhD at Harvard on chern classes for foliations, with the great Raoul Bott. Thus students at that local state college certainly have adequate resources for any geometry instruction they might desire.

 

[TeethWhitener]

that any two angles in a circle which subtend the same arc, are equal even if their vertices are at different points of the circle,

Ref Euclid Elements Book 3, Prop 21, in case anyone is interested. The meat of the proof is in Prop 20, and it’s just lovely. Off topic, but I have a fantasy that a forward-seeing university would offer a crossover course between the math and art departments where Euclid (as well as some other stunningly beautiful proofs) are taught.

On topic, OP, keep in mind that in the eyes of the students who don’t meet your expectations, you likely also don’t meet theirs.

 

[mathwonk]

at least he is trying to meet theirs. hopefully they will do likewise.

 

[Office_Shredder]

.my impression therefore is that people think of euclidean geometry as an elementary course that is taught and learned in high school or earlier, and hence is assumed in some form in all college courses.

my experience shows this is false, and that most US secondary schools offer very inadequate geometry instruction and most college students have little grasp of it.

At this point I think most people assume that Euclidean geometry is not necessary for a well rounded education.

 

[mathwonk]

wow. i was going to remark that most americans also do not believe in evolution, but it seems that has changed as of 2021.

here is an answer to the question of what is needed for a well rounded education, from a professional tutor, on quora:

"You will always need to have taken English Composition and Grammar no matter what career you choose. Writing well opens doors and helps keep them open.

United States and World history are important so that you can understand our country’s changes over the years, as well as follow what is happening in the world.

Mathematics, particularly multiplication and division, but only up to and including 7th grade. I have never found algebra and above, to be needed or practical for most people."

so maybe you are right.

but wouldn't a well rounded education include practice in reasoning and detecting obvious falsehoods? back in my day, euclidean geometry was the only course that actually addressed the question of what a statement means and how to demonstrate whether it is true or false. so i guess i am arguing that the important part of the course is not the list of triangle facts that many people think is most relevant. hence the tendency of modern courses to eliminate the proofs and emphasize the trivial facts, guts the value of the course.

but i digress. still this suggests the OP would do well to try to accumulate some motivational arguments for the students to be interested in why they should learn the material he/she offers them.

 

[Office_Shredder]

@mathwonk this is what I was getting at in an earlier post. Most colleges say they require a math class because it teaches logic and rigor, even though most of us look at the lower level classes and think that they don't do that.

 

[swampwiz]

Here's what my alma mater (LSU) has in the catalog for liberal arts majors. I guess there it is possible to not even take college algebra.

1029 Introduction to Contemporary Mathematics (3) Ge, F, S, Su
Prerequisites: Primarily for students in liberal arts and social sciences.
Here is the description of this course in the 2020-2021 and subsequent catalogs:
"Mathematical approaches to practical life problems. Topics include counting techniques and probability, statistics, graph theory, and linear programming."

1100 The Nature of Mathematics (3) Ge, F, S, Su
Prerequisites: Not for science, engineering, or mathematics majors. For students who desire an exposure to mathematics as part of a liberal education.
Here is the description of this course in the 2020-2021 and subsequent catalogs:
Using mathematics to solve problems and reason quantitatively. Topics include set theory, logic, personal finance, and elementary number theory.

 

[Office_Shredder]

If they really hit all the topics in those courses they are doing better than forcing someone to take algebra.

 

[vela]

Here's what my alma mater (LSU) has in the catalog for liberal arts majors. I guess there it is possible to not even take college algebra.

I'd expect the majority of four-year colleges and universities expect incoming students to have already learned algebra in high school, and don't offer college algebra. If they do offer such a remedial course, it shouldn't count toward meeting a degree requirement.

 

[symbolipoint]

I'd expect the majority of four-year colleges and universities expect incoming students to have already learned algebra in high school, and don't offer college algebra. If they do offer such a remedial course, it shouldn't count toward meeting a degree requirement.

And in fact it would not. Introductory and Intermediate Algebra each are not college level. They are remedial. At least for College and University purposes. College Algebra is college level course.

 

[vela]

And in fact it would not. Introductory and Intermediate Algebra each are not college level. They are remedial. At least for College and University purposes. College Algebra is college level course.

The university I went to considered college algebra a remedial course. In fact, when I was there, some students were complaining that the school didn't offer such a course. The math department responded that: (1) students supposedly already know the material as it was an entrance requirement and (2) no one in the math department would want to teach such a course. The department said students should go take a course at a community college if they needed to.

 

[symbolipoint]

The university I went to considered college algebra a remedial course. In fact, when I was there, some students were complaining that the school didn't offer such a course. The math department responded that: (1) students supposedly already know the material as it was an entrance requirement and (2) no one in the math department would want to teach such a course. The department said students should go take a course at a community college if they needed to.

That there is the great advantage of the community college systems. Very likely a student who goes through the college prep. Math courses in high schools for all four years is not assured of being prepared , mostly because the high school Mathematics courses are likely deficient about "College Algebra" instruction.

Best I recall, so long ago, H.S. offered Algebra 1, Algebra 2, and "Mathematical Analysis"; and maybe something for more advanced college preparatory students. What I recall of "Mathematical Analysis" was NOT up to the level of College Algebra.

 

[swampwiz]

I'd expect the majority of four-year colleges and universities expect incoming students to have already learned algebra in high school, and don't offer college algebra. If they do offer such a remedial course, it shouldn't count toward meeting a degree requirement.

Yes, LSU has the requirement of having taken Algebra II (but not Calc).

 

[swampwiz]

As an example of this ignorance, when preparing to teach the class I read on page 8 of Hartshorne that he would use a fact he hoped was familiar to most readers: that any two angles in a circle which subtend the same arc, are equal even if their vertices are at different points of the circle, I did not myself recall that fact.

I learned it and remembered it, and I am pretty sure that the PSAT, SAT, ACT & GRE had problems based on this mathematical fact. And this fact also seems to be the basis behind Mohr's Circle.