Math on the Other Side

A recent post here dealt with the metaphor of developmental mathematics as a bridge, designed to help students reach the other side. The ‘other side’ is not just mathematics (would we really want that?), with a diverse collection of courses … some of which are called ‘gateway courses’, while others are ‘just’ college courses. So, the question today is “What about the math on the other side”?

Is the math ‘on the other side’ the good stuff (important mathematics)? Do courses ‘on the other side’ place a high priority on student success? If we reform developmental mathematics in to a program which makes a difference in the mathematical learning of students, will their ‘college math courses’ have the same vitality?

These are questions which I can not answer; I am not immersed in the world of college-credit math classes (just parts of it). However, I do know that our profession is rather silent on this component of our curriculum … we are talking a great deal about developmental mathematics, and I hear quite a bit about STEM and calculus. Not so much about college algebra … pre-calculus … liberal arts math … or math for elementary education majors.

The easy target in this list is college algebra. Pre-calculus … at least we know what the goal is (calculus), and students taking pre-calculus can be assumed to have that goal (even if incorrectly assumed). However, we have absolutely no agreement on what ‘college algebra’ is. For some of us, college algebra is what we happen to call our pre-calculus course; for this group, I would say “Hey, be honest … call if pre-calculus!” For others, college algebra is actually a prerequisite to pre-calculus; on this … “how much time is needed getting ready for calculus?” [Perhaps we place additional steps in between to make sure that only the best survive; I hope not.] For still others, college algebra is a course outside of the pre-calculus sequence, perhaps used as a preparation for symbolic-based science courses; this is a good reason to have a course … though I question whether ‘algebra’ is the majority of what the students need. Some use ‘college algebra’ as a general education course; I suggest to you that a course could be either college algebra OR general education … but not both. One of the problems with the ‘college algebra’ label is that the traditional developmental math courses generally have ‘algebra’ in the titles; is ‘college algebra’ more of that developmental stuff?

Perhaps my worries here are just due to my extensive ignorance of some aspects of our curriculum. Perhaps, outside of the college algebra mess … perhaps we have generally sound mathematics and important ideas in our curriculum. Perhaps my problem is that I look at textbooks. If most of my colleagues who specialize in these courses tell me that ‘things are okay’ on the other side, I would certainly be relieved. However, with all of the current focus on developmental mathematics, it is possible that we are ignoring something equally important.

In our bridge metaphor, are we working on improving the bridge … just so that students can be delivered to a great wasteland of college mathematics on ‘the other side’?

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Math curriculum in general | Jack Rotman | April 24, 2012 8:16 am

7 Comments

By David Thomas, April 28, 2012 @ 7:32 am

Jack, hello. Just found your blog and I like it. Wanted to suggest one possible purpose for college algebra, or perhaps for basic algebra in general – preparation for calculus. As an undergrad math ed major, I got to be an instructor’s assistant in a college calculus course. It didn’t take long for me to observe that the problem many students were having in calculus was not calculus, but weak algebra skills. Getting the formula set up was ok, but then the formula has to be solved with algebra and there is where the breakdown happened. For example, not knowing when to subtract or divide, as you mentioned in the 4500=4000(1+r) problem earlier, is crucial and instantly defeating if not understood. I think we could reduce the dread of calculus if we could convince students that their algebra skills make a tremendous, important difference and set the stage for calculus success.
By Jack Rotman, April 28, 2012 @ 3:26 pm

Thanks, David.
The ‘calculus prep question’ is all about a balance between understanding procedures and problem solving (reasoning). A perception of the current pre-calculus course (as a normative reference) is unbalanced towards the procedures. I’d like to hear people deeply immersed in calculus instruction to discuss what the best balance is between procedural understanding and problem solving/reasoning.
By David Thomas, April 28, 2012 @ 4:35 pm

John,

What do you think of the idea that procedural competence can form a basis from which a student can launch into the problem solving/reasoning realm?
By Jack Rotman, April 28, 2012 @ 4:42 pm

David:
I do not believe that there is a prerequisite relationship between procedural compentence and problem solving/reasoning — there is a relationship, but a complex one. Smarter people than me have studied this area, usually as an analysis of ‘novice to expert transition’. Some level of procedural understanding is required (prerequisite); I believe that the most powerful way to develop both areas is to have situations involving both, including variations and novel problems. One thing for sure: I have never seen evidence that high procedural competence is sufficient for good problem solving.
Jack
By David Thomas, April 30, 2012 @ 7:45 am

Perhaps my interpration of what I saw in the calculus students was a little too narrow. I looked at their weakness in algebraic procedures and saw only that – a procedural weakness, with the remedy, in my mind at least, being more focus on teaching or re-teaching the procedures.

You certainly have more experience with Developmental Math students than I do, my only exposure to that world being, again, as an undergraduate teaching assistant in a Developmental Math course for college students. Is your approach to Developmental Math students primarily as a detective to discover where problems lie,and then plan your instruction to include extra focus on those problem areas? I am afraid that would be my approach, since I admit I have a perception that these students do present evidence of weaknesses in some area(s) of mathematics that have landed them in Developmental classes. With that perception, my approach would be to discover and address specific problems.

I believe what you are wanting me to see, on the other hand, is that things are not that cut and dried or that simple, and that what is needed is a broader approach, or maybe we do need to diagnose and correct specific problems, but the corrections are achieved through a broader approach?
By Jack Rotman, April 30, 2012 @ 1:09 pm

David:
Yes, I am thinking of any mathematics course as needing a broader approach. The current courses often have faculty being the ‘detective’ you mention … and sometimes students feel like they are the ‘person of interest’. Presenting mathematics as a positive area of work or learning, along with focusing on core concepts and connections, will result in our courses being more motivating (as well ss being more helpful to students).
Jack
By David Kephart, May 24, 2012 @ 3:03 pm

This is a question close to the center of math education — thanks, Jack! And (hello!) David Thomas is right, too; a critical mass of procedural expertise and practice is needed to prevent a relapse into pre-calculus questions at the time that passing calculus (or whatever) is the pressing question.

Basically, algebraic concept lost hold of by calculus students are less trivial than we presume. They are not even absolute — the core concepts and connections, however necessary, come with a caveat or two. When masters of college algebra come up against the limit in calculus, they encounter this issue. With respect to “the other side,” limits are part of it. Science itself approximates truth through a limiting and cumulative approach.

It seems the bridge is not the subject in itself, maybe, and much less the textbook. The bridge is something perhaps that the student must construct for himself or herself. We are more like guides than forepeople in this, too. Tools like technology are not gimmicks but offer ways to present math and its learning goals as part of real life. If we advertise core concepts and connections, it is to equip students to do the same, and not be taken unawares when the rules seem to change.

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