What to do if all students lack prerequisites?

Question

I am teaching a calculus class for business this summer (6 students) and all of them do not have the math background needed for the class. We are supposed to cover derivatives and integrals, but they don't even know how to evaluate functions. For example, I gave them $f(x)=x^2+x$ and the following happened:

• 4 of them had no idea what $f(2+h)$ was supposed to be. They tried $f(2+h)=2^2+h$, or $2^2+x$ or similar incorrect expressions.
• 2 of them had no idea what $f(2)$ was supposed to be. They tried $f(2)=(x^2+x)\cdot 2$ (yes, $x^2+x$ times 2), or $2x^2+2$ or $f(2)=fx^2+x2$, or similar incorrect expressions. After spending nearly the whole class explaining how evaluation works, they understood how to compute $f(2)$ and had some idea of $f(2+h)$. Then, I changed to another function $g(x)=-x^2+1$ and they were stuck again.

NOTES

• The example above is just an example. Please do not give suggestions about how to explain evaluation to a student.
• I have taught the course before, but this problem has been getting worse and worse over the years
• I have no idea how they passed previous classes
• This is the last math class the students will take
• My chair and dean verbally told me students are getting worse (math-background-wise) and that nobody cares what I do with this particular class. They won't say that in writing, of course.

My questions are:

• How to deal with the problem of ALL students having a very poor background?
• Do I just teach the required topics and likely fail all students?
• Do I spend time making sure some students understand even if I only cover 50% of the material?
• Do I just curve the grades even if they do not understand the material?

Answer 1

I'm going to take the crummy-feeling position of upholding the legitimacy of the course (possibly) at the expense of the students' education.

I presume there is a department-approved syllabus on what topics should be covered in this University course, right? Those topics are the topics that students who pass the course are expected to have understood by the end of it. Assuming I accurately understand your position in this, unfortunately you are more than just an educator in this situation, but you are also a gatekeeper as you have to grade them and decide if they pass. If it shows on their transcript that a student passed this course, it means that you've signed off on the fact that the student has a sufficient understanding of the material listed in the University's (or Math Department's) description of the course.

So teach the material that's usually taught, and examine them on that material. You can still teach it to them softly with the understanding that they're lacking some basic prerequisite knowledge (go slowly, write out more details that you would normally think are necessary, etc). But at the end of the day, it's got to be a calculus course. Heck, maybe they'll surprise you at how much calculus they've learned by the end of the course. :)

All of this of course is pending any further word from the dean and chair.

Answer 2

You said the chair and dean don't care what you do with this class, so I wouldn't worry too much about upholding historical standards for rigor or what you think a calculus class should look like. Also, with six students you can and should customize the course for the students you have.

In ten years, what impact do you hope this course will have on them? It will probably be useful for them to understand standard mathematics notation, and reason about core concepts from calculus, like slopes and areas. Actually doing symbolic manipulation and simple calculation is (I would guess) going to be irrelevant. I wouldn't cut that material though, but use it only as a means to an end. They don't have to be good at taking derivatives and integrals, they just need enough exposure for some of the big ideas to sink in.

For concrete advice, I would start from the basics, recognizing where they are now, moving along very quickly but also giving them frequent reminders for basic material. I would focus only on very simple computational problems, and/or make traditional problems strictly homework or open-book assessment. Go heavy on conceptual problems, even challenging ones. Maybe try some discussions problems in class, ask for short essays, or do applied projects. I would pass them if they get some core concept and might have a hint of a benefit in ten years. I would assign grades A-D in line with recent standards, adjusted for changes to the course.

Answer 3

I'm going to take a contrarian stance and say you should avoid the drama of failing them and just get by. This institution is not Harvard. And these kids are not little Feynmans. Everyone gets that they are not real calculus students. Just deal with it. It's not your place to challenge it.

Do some basic evaluation, but don't kill yourself (like going back to teach them trig). If you can cover basics of compound interest (exponentials or the like), this would be helpful for business.

Avoid teaching theory of differentiation and integration.

Emphasize simple, graphical understanding of the topic and mechanical problem solving (like differentiation and antidifferentiation of polynomials).

Use frequent business examples (e.g. rate of production growth)

Curve them and pass them if they are trying.

P.s. Learning how to use Excel shortcuts would do them more good than calculus anyways. It's not like you have to worry about them taking physics or diffyQs.