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Here is an overview of the practice of "gateway testing", which explains it much better than I could:

https://sites.lsa.umich.edu/michigan-math-in-action/2015/09/24/25-years-gateway-testing-at-michigan/

But for completeness I'll quote the basic idea:

What’s a gateway test? Our definition is that it is a test of basic skills. These may be skills that are prerequisite for success in a course, or may be skills which every student in a course should develop.... However, while these skills are intrinsic to or essential for success in the courses, they are not our educational focus. Our courses focus on conceptual understanding... Our gateway tests are 7 or 10 question tests, administered on-line... which students may take multiple times and which they must complete with almost no errors to pass.... Students may practice the tests as many times as they like, but to get credit for having passed the test they must take the test in a proctored lab where their identity is verified and where they are not allowed to use outside resources.... completion of the test is not a part of the students’ course averages; instead, any student who doesn’t pass a gateway by the specified deadline... has her/his final grade reduced by 1/3 to a full letter grade at the end of the semester.

The above link includes a bit of evidence that the gateway tests work at Michigan, at least that they correlate with students learning the gateway material. But I would like to know if any more serious data analysis has been done on gateway tests (anywhere)? In particular, I would like to know whether implementing gateway tests has a positive effect on overall student performance.

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    $\begingroup$ I did some TAing for a professor who was practising something very similar and it worked rather well in my experience. I do not have any more substantiated evidence, however. Should this question not get an answer in due time, I will ask him whether he knows something (ping me in case I forget). $\endgroup$
    – Wrzlprmft
    May 9, 2017 at 7:35
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    $\begingroup$ I have done something similar to this in the past in my freshman mechanics course. Topics included entry skills (conversions and proportionalities) and skills taught in the course (vector addition). I called them MOB tests, for "mastery of the basics." It was online, and they could retry as many times as they wished. I didn't have resources available to make it proctored. I discontinued it because many students seemed to be copying answers from someone else, and one way I could tell that was happening was that they couldn't do the same stuff on exams. $\endgroup$
    – user507
    May 9, 2017 at 20:07
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    $\begingroup$ Regardless of who has the webwork data (and I expect it would be the UMich sysadmins, since I suspect they are running their own webwork server rather than using the MAA's hosted version), I don't think it would answer the question. I want to know about the effect of gateway tests on overall performance in the class, e.g. on final exams, not just about the performance of students on the gateway tests themselves. $\endgroup$
    – Mike Shulman
    May 10, 2017 at 4:56
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    $\begingroup$ I have done a gateway exam (mostly trigonometry) in a Calculus class and anecdotally it definitely raises the quality of the course -- setting "prerequisite expectations" early and clearly does a lot for the classroom environment. But this isn't data. I hope to see answers posted to this question someday. $\endgroup$
    – Chris Cunningham
    May 11, 2017 at 15:04
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    $\begingroup$ I teach at Michigan, including I taught Applied Linear Algebra which instituted a Gateway on row reduction and other matrix operations while I was there. My post-gateway syllabus has an extra week of material in it, because I spend so much less time on these basics. But I don't have the sort of formal data you want. $\endgroup$
    – David E Speyer
    May 18, 2017 at 13:48

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I apologize for not leaving this as a comment, but being new to this StackExchange I will reply in the way in which I am able.

I do not have the hard data you are seeking, but I'll share my personal experience.

I am a math PhD student, and each semester I teach a 100 level math course (in our department these are (primarily) retreads of things the students should have learned in high school). Examples include Algebra, trigonometry and Pre-Calculus. Incoming students are placed in these courses based on a placement test taken during orientation (provided they do not have credit for AP Calculus, or transfer credits from a community college, etc.)

However, many students still struggle with basic things like exponent rules, fraction addition, and knowing the 'legal' algebraic moves they may make to solve an equation. So in this sense I'm not sure how useful our placement exams are.

Personally, I address this issue in the following way. If I'm about to teach something, for example an introduction to exponential functions, I will do something like the following:

  1. Motivate with some discrete examples

  2. Give the students problems to try that require a good understanding of exponent rules

  3. circulate and give differentiated help

  4. work thorough the problem while explaining my reasoning

  5. tell the students that if they struggled with exponent rules that that is something they are assumed to know, so it is beyond the scope of the lecture

  6. direct them to resources for additional practice and offer to help in office hours

  7. give a general algorithm for solving similar problems

  8. give word problems to try and help students to develop basic number sense

    I do this for two reasons. First, is that these courses are prerequisites for others in the department, and the instructors of those courses will assume a certain set of material has been covered. Second, I want my students to take ownership of the learning process. I will do everything reasonable to help them to succeed, but I also make them earn their success. To use a trite saying: I will lead them to water, but I will not force said water down their throats.

Having taught high school before university, I find that many students in high school, and indeed in the first couple years of college, have been conditioned to think that it is the instructors job to make them pass. I do my best to break them of this notion.

While it is true that I occasionally get bad reviews from individual students, and indeed more students fail my course (roughly 30-40% are likely to fail this semester) than might be ideal to pass them on when they haven't demonstrated some level of mastery does no one favors. This is how we have students in multivariate calculus that still cannot correctly do basic algebra.

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