How do you place incoming mathematics students?

Our department is looking for a better method of placing incoming students in mathematics courses.

Currently we have a placement exam that determines whether a student should begin in a calculus I course or in a calculus/precalculus hybrid course (our lowest-level math class). The exam consists of 25 precalculus questions. It does a pretty good job.

Recently we’ve had an increase in the number of students who have taken a calculus class in high school. If it is an AP class, then we can use their AP score to determine if they should skip the first semester (or more) of calculus. However, students who have taken a good non-AP course slip through the cracks. This is especially true for international students. If those students identify themselves, then we use ad-hoc methods to determine their placement (talking to them, and looking at the syllabus from their course, their exams, and their textbook—which is sometimes not in English).

Because this is time consuming and because it requires the students to come forward to ask for it, it is not ideal. We are now thinking about constructing a placement exam that would determine if a student should place out of first semester calculus. I thought it would be interesting and informative to hear how you do mathematics placement at your school. As I said, we’re especially interested in the Calc I vs. Calc II placement, but other readers may be interested in other math placement issues. Please share your placement procedures in the comments below. 

If you have a calculus placement exam and are willing to share it with us, please email me. That would be fantastic. We would keep it confidential and would not post it where it would be publicly available.

If you are interested in how we place students, see our online placement guide.

Note: I know many schools allow students to place themselves. We would rather not go that route. We’ve found that students are not always good at judging their mathematical ability. And we’ve found that it is problematic for the students, for the professor, and for the students’ classmates if a student ends up in the wrong class (either too high or too low). Our placement exam forces the students who do well to go into Calculus I and students who do poorly to go into the lower-level class. Students with scores in the middle can place themselves.


  1. Hitchman says:

    I teach at a comprehensive university, rather than a liberal arts college, so our needs and students are a bit different than yours.

    We have just started using a placement test and online remediation module set-up from the ALEKS company. I wasn’t part of the decision making process on this choice, though it was painfully obvious that we needed something.

    We sometimes have the same trouble with calc 1 vs 2 placement. It is really hard to know where to put a student who scores a 3 on the AP calc AB exam… ALEKS is not meant to solve that particular problem, though.

    Too early to tell if it is working as intended.

  2. Dave says:

    I teach at a very small interdisciplinary college, College of the Atlantic. I do placement just by talking to students and learning about their background. I used to give a test that was developed at UCDavis, but I found that talking to students was more useful and much less stressful for the students.

  3. I teach at a small liberal arts college as well – and if you come up with a great solution, I’d love to hear it. We have the same issues (our problems also include figuring out which students need to be in precalculus vs. calculus I).

    1. We use the MAA’s placement exam to do precalculus/calculus placement. It works pretty well.

  4. Ronald says:

    I think one problem you’re having is identifying the difference between experience and ability.

    International students (in particular) may test poorly if they’re not familiar with the local system, or if they have a different body of knowledge due to the syllabus they’ve studied, or if they haven’t fully crossed a mathematical language barrier.
    Perhaps they could catch up quickly, but haven’t got familiarity with the type of questions you are asking.
    I have anecdotal evidence that this happens often, and it seems to match with your experience as well.

    I had a couple of suggestions for dealing with this:
    – offer the students a complete syllabus for the exam with example questions, as close to the exam as possible, and fully worked answers, or perhaps
    – set the exam as a take-home quiz rather than a formal exam

    This will ensure all students understand what they are expected to know, and in the correct format, without the pressure of an exam for incoming students.

    As a simple example (I’m not sure if this is relevant): your syllabus says ‘inequalities’ – but it is not clear whether you mean ‘linear inequalities’, or ‘quadratic inequalities’ – a student with the former skill presented as ‘inequalities’ might not have had the opportunity to realise that the latter skill is more subtle; and an international student might not even be clear exactly what this word means.

    Then, you have to face the danger that students will ‘cheat’, and over-represent their ability.
    This can be diminished by making it clear to the students how they benefit from being honest: put into the appropriate class for them where they get appropriate help.

    I would also assume that there’s a group of students who are proactive enough to research and study questions for a quiz, who might score at a higher level than they would otherwise. I would hope they would be motivated to do the same in a classroom situation – i.e. this is a group of students worth having in the higher class, so you should give them the opportunity to prove themselves.

    1. Thanks! We’ve actually had pretty good results with our lower-level placement. We “require” all incoming students to take the math placement exam (the summer before their first year), so hopefully the material is fresh in their minds. Unfortunately, some students ignore the requirement thinking they’ll never take a math class. Then we run into trouble with placing them when they decide in their sophomore or junior or senior year that they need to take a math class. Then they tend to score poorly on the placement exam, but sometimes only because they are rusty and not because they have a weak background in math.

  5. Iwan Praton says:

    At F&M we’ve done calculus placement for a very long time (longer than I’ve been in Lancaster). I’ll see if I can get our calculus placement czar to provide more details.

    1. That would be fantastic Iwan. Thanks! We’d love to hear what you do at F&M.

  6. Mimi says:

    I teach in a secondary school, but one idea I had (never yet implemented) for placement is to have a short “math bootcamp” in the summer for the students you feel that are gray in terms of placement (in your case, that would be the international students and/or non-AP calc students). In the bootcamp, you’d go over some fundamental calculus concepts and see how the students do. Nothing does placement as well as first-hand observations, obviously, and this has the added benefit of bringing those students up to speed on their fundamental skills prior to start of school.

  7. videos4math says:

    I work at a secondary school, but I think one solution to this is to institute a “math bootcamp” at the end of summer for the students who are in the gray zone for placement (ie. non-AP calc or international students, in your case). This way you can both sharpen their fundamental skills AND see first hand how well-equipped they are for the classes they intend on entering.

  8. Greg Miller says:

    This comment is posted by an independent boarding secondary school science dept. head; we have about fifty new international students metriculate to our school each year (grades 9-12). While not math, my hope is that these thoughts may be helpful.

    For placing students we look at the following:
    – Standardized testing to get a sense of math and language skills (math for chemistry and physics, language for biology).
    – The student application essay (another language marker, but also the ability to express oneself in an organized way – serves as something of a math marker)
    – Our own placement test, of six parts, each measuring different skills:
    * basic science background,
    * algebra skills,
    * interpret graphs,
    * interpret data tables,
    * process [that is, list the sequence of steps to complete a compound process such as a simple experiment],
    * a problem involving high-level critical thinking,
    * reading a scientific document

    Each section reveals a little bit about the student, taken in both its parts and viewed as a whole has proven itself for new student placement purposes.

    I would encourage you to design a test that measures both
    math readiness as well as
    the ability to think (including non-math type questions to get at the student’s critical/analytical skills- that is, ID the kids who may be simply lacking the “language” of the subject [which can be quickly learned if the basic skills are in place]).

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