March 15, 2011

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A collection of articles and arguments for and against course acceleration.

- Mathematics > General

- Higher Education
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- Undergraduate-Upper Division
- Undergraduate-Lower Division

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Mike McKean

March 28, 2011

Kathy, for what it's worth, I don't think there is a protocol yet. To me, it would seem that using the comments tab for extended discussions is probably easier than using the main "content" tab. That also makes it easier to distinguish contributions by different participants, and to see when they made their remarks.

Kathy Yoshiwara

March 17, 2011

I also put my comments in the main thread -- I didn't realize that was the wrong protocol. (Still learning how to navigate this site...)

Ruben Arenas

March 17, 2011

Hi Tom, I put a David's response in the main thread -- I felt it fit better there.

Tom Carey

March 16, 2011

1. I don't think the term "course acceleration" is the most accurate one to use, because it does not seem that the work within a course is accelerated. I think "accelerated completion" if more accurate - which begs the question of 'completion of what' that David implicitly raises.

2. David suggests that the picture might look better if students not aiming to complete a transfer-level course are not counted as incompletes, making their success rate more like those who start 3 levels short of their goal rather than 4 levels.

Well, sure, students who start closer to their goal should show a higher success rate - I don't think that invalidates the point that fewer steps in the sequence (however 'step' is defined) will result in higher completion rates. There is fuller data on this from multiple colleges in a research report at http://ccrc.tc.columbia.edu/Publication.asp?UID=734, and a peer-reviewed version in Economics of Education Review, vol. 29, pp. 255-270. March 2010.

3. I didn't get the point about probabilities: these are success rates from past classes, so that makes them proportions, yes? Not sure where randomness etc. comes in. If the argument is that these numbers may represent outlier data, the report cited in point 2 indicates otherwise.

4. I would love to see a deeper analysis of why we lose students "between the steps", assuming as per David's point that they are stopping before completing their goal. Do they fail to enroll in the next course due to neglect or lack of information? Do they run out of energy because they still seem far from their goal, or do they realize that their finances and part-time jobs won't carry them through the planned sequence? Shortening the sequence will help with any of these, but maybe there are other solutions which would address one or more of these problems if the students really need a large amount of content doled out in our current course-size bites?

Here is an article by Hern with contributions from Snell arguing for course acceleration because of the "multiplication principle" -- basically multiplying success rates at each course in a sequence to argue for the low success rate at the transfer level.

I'm the facilitator for the Pierce College Statway team, so I'm writing to correct some of the errors and confusion evident in David Senensieb's e-mail. Let's start with the math, since that is the only objective concern in his letter. David seems to be confused about the relevance of independent events in Myra's analysis. If we choose to regard the percentages as probabilities instead of plain statistics, then the events they describe are certainly not independent: they describe a sequence of conditional probabilities, e.g. 75% of students elect to take Math 125 GIVEN THAT they passed Math 115. Then P(both A and B) = P(B given A) x P(A) We also assume that students placed into a course have the same probability of passing as those who took the previous course. Theoretically, that is how placement cut-offs are set. Of course, the individual pass rates, etc. may be different on different campuses, but the multiplication principle is still valid.

While I personally agree with the need for course acceleration, it seems that the "multiplication principle" argument is fundamentally flawed.

Here is the reasoning from David Senensieb, ELAC Math (edited for content relevant to this discussion):

Now for those of you who are interested in the faulty reasoning used to present as abysmal of situation as possible, consider this somewhat simplified problem:Hi Ruben,

Lets say 100 students start at Math 115 (Algebra I) and 80 pass. Rate 80%

60 of the 80 choose to take Math 125 (Algebra II). Rate 75%

42 of the 60 pass Math 125. Rate 70%

32 of the 42 choose to take a transfer level course. Rate 76+%

24 of the 32 pass a transfer level course. Rate 75%

The calculation of success/persistence rate was given as (in the "multiplication principle")

0.80 x 0.75 x 0.70 x 0.76 x 0.75 = .24 or 24%.

Yet her analysis includes as “failures” those students who choose not to move forward. For instance, those students who enroll in the nursing program need Math 125, but not a transfer level course. They should not be included in the statistics. So if we only look at those who wish to go on, the success rate (using her analysis) would be 0.80 x 0.70 x 0.75 = .42 or 42% (significantly higher).

But …

Her analysis also assumes randomness and independent events (this is what allows her to multiply probabilities). But does randomness and independent events hold when taking a sequence of classes. The answer is no.

The student who passes one class in the sequence is presumably prepared to succeed in the next class. This student is no longer a random student in the class but rather in the class based on demonstrated performance, i.e. dependent on

passing the prerequisite class.

(By the way, her analysis does not address the manner in which the student gets into a class – placement exam, passing a prerequisite, re-taking etc.)

This gets to the next point - her epiphany. Success rate improves by shortening the sequence of classes.

Using her logic that we assign a probability of passing to each course (I will exclude the ones we opt not to take - unlike her analysis), and that all course are independent, then you come to the illogical conclusion that the the probability of

passing a set of courses needed for any program will tend to zero. Enough said.

While you may argue that some sequences can be shortened due to redundancy or unnecessary material, this is an argument separate of success rate. If you want to argue that there are natural paths for mathematical sequences and we should tailor those accordingly, that is separate of success rate. (Many schools already do this for Calculus. Following the body of knowledge gained through Pre-calculus, the Calculus sequence is tailored for physical science majors, life science majors, liberal arts or business majors). If the question is when the branching should begin, then lets have that discussion. But lets have that discussion based on the educational merits and the body of knowledge that is needed to succeed in the chosen path. (Let's also keep in mind that a person who starts in one branch may later wish to switch to another. It would be quite a setback for that individual if the branches diverged early on and they must then go back to fill in the body of knowledge they missed.)

I'm the facilitator for the Pierce College Statway team, so I'm writing to correct some of the errors and confusion evident in David Senensieb's e-mail. Let's start with the math, since that is the only objective concern in his letter. David seems to be confused about the relevance of independent events in Myra's analysis. If we choose to regard the percentages as probabilities instead of plain statistics, then the events they describe are certainly not independent: they describe a sequence of conditional probabilities, e.g. 75% of students elect to take Math 125 GIVEN THAT they passed Math 115. Then P(both A and B) = P(B given A) x P(A) We also assume that students placed into a course have the same probability of passing as those who took the previous course. Theoretically, that is how placement cut-offs are set. Of course, the individual pass rates, etc. may be different on different campuses, but the multiplication principle is still valid.

And yes, this sequence of events does depict a "worst case scenario," but it is also the most frequent scenario on most of our campuses. What percent of our incoming students place into Math 115 -- or below? Some of our students do not intend to transfer, but the Statway program is intended to address those who do. Myra's analysis shows how few of our students even have a chance at that goal.

The Statway course is not for STEM majors or business majors. However, at Pierce at least 75% of the students in our developmental math classes want to major in non-STEM fields that require statistics, such as psychology or social sciences. The Carnegie Institute doesn't have to pitch Statway to our administrators: the numbers speak for themselves. That is why the MAA, AMS, ASA, and AMATYC have all endorsed Statway, why the Gates Foundation is funding it, and why Cal State has already approved the course for transfer.

I should also point out that Statway was only one strand of the Math Summit. Speakers from several local campuses described new developmental math programs being tested in their departments. Perhaps David did not attend any of those sessions. A more balanced view of the work going on numerous fronts might alleviate the fear that motivates the rest of the e-mail: that people will be forced to teach Statway.

I can tell you without reservation that nothing is further from the truth. Last year the Carnegie Institute selected and invited teams from 19 colleges across the country to participate in the pilot program. They have invested significant resources in training the teams and in developing both the research and the practical support for the Statway curriculum. If you have not already done so, I encourage you to visit the Statway website to read more about the development of the program. The course will be class-tested by the 19 teams in the 2011-2012 academic year. The developers expect that there will be significant revision of the materials during and after the pilot year. They do not expect to invite new teams until after the pilot year, and even then new participants will need training and mentoring. They will certainly not be recruiting faculty who are opposed to the principles or the pedagogy embodied in the program.

Statway is not intended to replace the traditional statistics course for STEM majors, but to provide an alternative pathway for students who are interested in the growing number of non-STEM disciplines that use statistics.

If people have other questions or concerns about Statway, please feel free to contact me and I will try to provide answers.

Kathy Yoshiwara

David wrote a response to Tom's questions that brings up a very interesting point (though, I don't think they answer Tom's questions). I think this might be one of the first arugments I've heard that really makes me think twice about acceleration that omits content, at least in a traditional math department. It also makes me think about how math is taught at the community colleges. David writes:

My main point is that the discussion/program should be based on its educational merits. Just like any other field, be it Chemistry or English or History, there is a certain body of knowledge in Math that is needed to understand higher level or more abstract ideas. It is our "job" to deliver education, not to deliver a goal. Chemists design their courses to train new chemists. Engineers design their courses to train new engineers. Yet in math, we are treated differently. Since other fields utilize math (e.g. stats in psychology), we are not viewed as a field unto itself but rather as just one step in their sequences. This is what leads to the skewed idea that we are somehow holding people back from their goals. So is it really that we are putting obstacles in peoples way (content they claim they don't need) or is it really that they don't understand or care to understand that we are being true to our own field. Maybe it's all a matter of perspective.

Kathy writes:

We need to acknowledge that at community colleges mathematics is in fact largely a service discipline. How many of your students are math majors? How many of your developmental math students are math majors? It is our job to help our students achieve their educational goals, not to impose our own vision on them. I'd like everyone to learn Greek so that they can read the classics in the original. I'd like everyone to play a musical instrument and appreciate classical music. Maybe someone responds by saying "Everyone needs mathematics in their lives, but not languages or music," but perhaps we can agree that it's a question of how much. How much Homer, how much Brahms, how much algebra.

Statistics is a part of mathematics, and not an easy branch. Hypothesis testing and experimental design are as intellectually challenging as adding rational fractions -- many people would say more challenging. We are not short-changing students by teaching them substantive mathematics that is relevant to their future careers (and their future lives, whereas the rational fractions in most intermediate algebra books are not useful to anyone.)

Instead of designing a course by looking at existing courses and deciding what to leave out and what must be retained in order to be "true to our field" (LOL), let us try looking at the goals of the course and the program of which it is a part and deciding what to put in. There will still be mathematicians in the world, and the fate of mathematics will not be harmed if community colleges do not insist that every student who comes through their doors knows what a determinant is.

Ruben writes:

I think David has a point that might eventually need to be addressed. Just because we have a certain situation in the community colleges, doesn't mean that's how it should be. I've been wondering since David made that comment if following a model that splits Basic Skills courses into their own department (away from their parent departments such as Math and English) might better serve both the students and math as a discipline. Perhaps math as a "service discipline" would be more effective with other disciplines that have the same basic purpose.

When I think of hiring, community college math is often in a bit of a bind -- we need to hire instructors who can teach addition and fractions with the same effectiveness as they can teach differential equations and multivariable calculus. I suspect we often don't meet this goal (speaking only for myself and my campus). Part of the reason is that trying to satisfy both ends of the spectrum is nearly impossible given how a 4-year math degree and subsequent graduate degrees work. A math education major might be great for teaching Math 105, but terrible at teaching DE's (seeing as how many math education degrees do not require a course in DE's, or science courses beyond the basics). On the other hand, a pure math major might be great with our calculus students, but may be terrible at teaching addition since that's been off such a person's radar for a long time, and they've probably never had formal training in teaching except to undergrads taking precalc or calculus. And yet, when we hire, we look for someone who can excel at doing both.

I wonder if developmental students would be better served by faculty with backgrounds in math education, and if the pure math folks would serve our students better by focusing on what most mathematicians think of as the modern discipline (namely, preparation for upper division work in algebra, analysis, topology, differential geometry, algorithms, etc).

In that sense, I agree with David in that a certain body of knowledge comprises what mathematics is, and this body of knowledge should be treated with the same respect as any other discipline. And yes, I think its a shame that some disciplines have gone underrepresented in the community colleges. I would personally love to see community college linguistics courses... In the end, I think our students have been short-changed by a lack of respect for a diversity of fields. Part of the first few years of college for many students is discovering what's out there -- most folks who take a college chemistry course don't become chemists, and the situation is certainly true for math as well. But I think being able to explore your options, for $36 a unit, is one of the great strengths of the California community college system. With that said, an enomorous amount of funds and faculty hours goes into Basic Skills, and most students who enter the community college system can't benefit from all the options we're offering anyways… so we're in a bind there.

I wonder if I should start a second discussion page on "Should Basic Skills be it's own department? Arguments for and against". Thoughts? I realize some colleges already follow that model -- and I would love to hear from those colleges.

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