I am in a research project where one of our jobs is improving the first year university experience for our students. One of the topics we are looking into is changing the way we teach our introductory course in mathematics.

In this regard I have two questions -- that I hope are not too broad

  1. If you were given 100% control of how the first semester mathematics course were taught, what would you do and why.

  2. Are there any sources, research articles, papers etc of what works? I have found plenty that talk about the difficulties in designing such a course, but not many "solutions". Is PBL a good way to go, designing oral exams? Projects in stead of exams?

EDIT: This is a course taken by all of our engineers, and also a handfew mathematicians. Usually about 140 engineers and 10 mathematicians and some physics students.

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    $\begingroup$ I'd try yo emphasize the discrete over the continuous. $\endgroup$ – Peter Saveliev Jan 17 '20 at 13:45
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    $\begingroup$ This is going to be very difficult to answer unless you tell us something about what you perceive to be wrong with the class as it exists. Are the class sizes too huge to allow students to ask questions? Are the students getting their homework answers from chegg? Is the success rate lower than it should be, and if so, how do you know what it should be? Are the math majors getting too little rigor and proof writing? Are the engineers getting too much of those things? $\endgroup$ – Ben Crowell Jan 17 '20 at 16:25
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    $\begingroup$ Define "improving the first year university experience" and why what you have now is not good enough. $\endgroup$ – Rusty Core Jan 17 '20 at 16:59
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    $\begingroup$ @DavidTonhofer I doubt that many "numerical mathematicians" would agree with your claim that numerical integration is computer science and not mathematics. But in any case, it's good for engineering students to understand that most differential equations in practice cannot be solved by separation of variables but need numerical methods. $\endgroup$ – Michael Bächtold Jan 18 '20 at 13:28
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    $\begingroup$ Is it an introductory course in mathematics or Calculus I? (kind of a big difference). "This is a course taken by all of our engineers" - so you can't place out of it? Introductory courses should match their placement results; Calculus I should just be Calculus I. If you don't understand arithmetic, algebra, geometry and trigonometry, you shouldn't be taking Calculus. $\endgroup$ – Mazura Jan 18 '20 at 19:54
  1. Given 100% control, I would have one-to-one instruction. One instructor meeting individually with each student. That instructor can change the approach, the speed, the order of topics, the method of instruction, based on that individual student.

But of course hiring (and training) enough instructors for that is probably way beyond any reasonable budget. (It is possible that in the futture such quality instruction will be provided for us by computers; but I do not think we are there yet.)

Probably no single method of instruction is "the best" for all of your 140 students.

[The question 1 may be closed as "opinion based", but here are my opinions. Question 2 deserves some answers, though.]


Without directly answering your question, you don't seem to have the background you need to be "improving" the undergraduate experience yet, and have some work to do. I think you're right in sensing that your question is too general

Don't talk about the "difficulties in forming such a course", and spend your initial time finding out what aspects of the first year experience you're trying to fix. Is it too many students fail calculus I? Is it that students don't know calc well enough when they're done to support the engineering curriculum? Is is that too many students drop engineering as a major and go to other disciplines? Is it that attendance in the class is poor?

You might have a number of stakeholders that you need to question -- maybe students, advisors, faculty,...

You can spend an awful lot of effort revamping a curriculum that doesn't touch the real issues. You minimize wasted resources by going through the exercises of characterizing the issues, and prioritizing what problems need to be fixed.

Now, let me, as an engineering prof, try to guess about the kinds of situations that you might be trying to improve. The biggest issues I see concerning our calc sequence is that some of our first year students have plenty of on ramp issues, and might not be ready to take calculus. I think we owe these students the support that they need to thrive in our programs.

My own experience is that there isn't going to be ONE course that can address the needs of all the first year students at the same time. If I was trying to create ONE class to support all the students, it would have multiple scaffolding options that students can select, according to their needs, but students that don't have a good foundation would need to put in an awful lot of extra work, right when they're transitioning from a high school environment. Tough path, but maybe some of the newer tools available through learning management systems can be of some help (not holding my breath).

We've landed on replacing our Calc I/II sequence with a somewhat decelerated three course sequence for students we feel would benefit, based upon placement calculations from their applications packages, with many students completing the third course during the summer. This allows a fairly usual engineering sequence, though there are some issues with the timing of Physics.

This works to some extent. I always wonder if the better option would be a semester of pre-calc for students that would benefit, followed by Calc I and Calc II

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    $\begingroup$ As a practicing engineer, “What does success look like?” is definitely the engineering answer. $\endgroup$ – fectin Jan 17 '20 at 21:42
  • $\begingroup$ No, @fectin-freeMonica, the engineering answer is not a practicing engineer. Likewise, Scott, "let me" is not an engineering prof. "Many students completing the third course during the summer" - and paying for it? $\endgroup$ – Rusty Core Jan 17 '20 at 22:40
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    $\begingroup$ @RustyCore -- actually, as there is somewhat of an institutional agreement with how important it is to support these students, the School of Engineering has found a way to pick up the expense of the summer course. $\endgroup$ – Scott Seidman Jan 17 '20 at 22:40
  • $\begingroup$ ... proud to say. $\endgroup$ – Scott Seidman Jan 17 '20 at 22:42
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    $\begingroup$ The Naval Academy had/has a semester long precalc course for weaker students. SM005 or "smoo five". Seemed to work OK for them. All of the students there have to take math through diffyQs, even English majors, because of the naval engineering courses. Since the science requirement in first year was chem, and then physics next year, the students still had what they needed to support normal plebe track classes. I think the SMOOs caught up by doing Calc2 in the summer after plebe year. $\endgroup$ – guest Jan 18 '20 at 0:21

It's about the learning experience.

An issue with project based learning is the need for objective grading isn't going away. Since the first semester of mathematics is what serves to filter out under performing students there is a desire to have an unmovable bar. Without it you get grade inflation.

This isn't to say there can't be projects. You just have to ensure that the student work is objectively and individually gradable.

So rather then do anything to change how the students are graded I'd emphasize changes to their learning experience.

  1. In the age of youTube any lecturer who doesn't encourage questions will find themselves outclassed. Be interactive or die.

  2. The best tutor isn't the teacher. It's someone who just took the same class. If you don't have a math lab create one. Populate it with second semester tutors. Encourage everyone, regardless of performace, to seek tutoring

  3. Teach study skills aimed directly at math itself, the class in question, and the test.

    3.1 If the tests are closed book and closed notes teach the students the fine art of recreating a sheet of formulas from memory on a blank sheet of paper.

    3.2 If the tests are open notes teach the students the fine art of organizing all the formulas on a blank sheet of paper so they're easy to use.

    3.3 If the tests are open notes and open book teach them 3.2 anyway so they don't spend forever flipping through the book.

    3.4 As for project based learning, consider this story: way back in my statistics days our instructor gave fantastically hard quizzes for 5 minutes in every class. The whole class was bombing them. So a group of us poured over the quiz we just failed for an hour after each class. After that hour we usually had the quiz figured out which we checked with the instructor. Our group got the only A's in the class because we worked together on quizzes that gave us no credit.

    3.5 Encourage your tutors to go a step beyond. When I worked as a math lab tutor I was asked to lecture a number of test preparation sessions. As an undergrad this is wonderful experience that teaches you the material like nothing else, because you know you're gonna get asked.

  • $\begingroup$ How does one ask questions of a You Tube professor ? Yet, students use You Tube lectures to gain insight in their classes. I then question the necessity of universal interaction. In fact, over-emphasis on interaction is a big minus for some students. $\endgroup$ – James S. Cook Jan 20 '20 at 18:21
  • $\begingroup$ @JamesS.Cook my point was if you don't interact then you're directly competing with YouTube professors who also don't interact and will likely outclass you in other areas. $\endgroup$ – candied_orange Jan 20 '20 at 18:23
  • $\begingroup$ I understand, but, I think almost all in-person lectures are interactive. The joy of teasing students is one of the main job benefits. In any event, I do agree with your assertion that Math Lab is important. Both for current students and for the creation of new teachers. $\endgroup$ – James S. Cook Jan 20 '20 at 18:32

I'm an engineer who really likes and uses mathematics on work and daily life.

Myself, I would try to make the course as much self-contained in classes, but still stick to some written material the students have access to. This allows the subject to be more complex and extensive while keeping down the difficulty settings.

I'd prove theorems as much as feasible in class, and show both synthetic and practical examples, i.e. both a simple equation that showcases the point you are making and a rather complex but realistic problem. epsilon-delta proofs are important for creating a sound basis, but engineers often dislike them.

Also, students need to learn how to use computers. So a few classes on software that computes limits, derivatives and integrals is handy as well. Mathematica comes to mind, but there are free tools for this purpose. Symbolic integration warrants the comment "learn this first, so you'll not get addicted to numeric integration".

Even better, Automated Differentiation has even better applications in industry and science (AdiMat is one example). Maybe your students don't need to know how this works, but they should know it exists and what it can or cannot achieve. I've once used one of these tools to differentiate a function defined by over 300 lines of code, with respect to 7 variables.

Many students would thank you for providing them with these tools early in their graduation. Others might just consider this "extra work", forget how to use them and move on with life. The challenge is that your students are likely not to know anything about programming at this point in their courses.

Even if I taught all of this, usual examples of derivation and integration should be shown as "this is simple enough you need to get a hang of it".

Also a few functions such as the error function, should be introduced as well.


Select and use an open textbook. Students who wish to use a physical textbook will be able to get a copy run off and hole-punched at a local print shop, then put in a binder, for far less than the cost of a conventional textbook.

You will save your students a pile of money and there are enough available options that you should be able to find something that suits your needs. This has benefits to student learning other than supplying beer money; financial stress is real, and impedes scholarship. Steps can and should be taken to alleviate said stress where possible. I don't have sources off-hand, but I suspect research is available to back that up.

This is what's used at my alma mater; the program has been quite successful, and textbooks for derivative and integral calculus are available. All questions have hints, answers, and fully-worked solutions. It is available under a CC BY-NC-SA 4.0 license, which I believe will permit usage at other institutions: their contact information is available on the page linked for clarification.

  • $\begingroup$ I understand, it is a free book, so "we are not going to be precise here, and to give definitions there", but stuff like this is too sloppy. $\endgroup$ – Rusty Core Jan 17 '20 at 22:56
  • $\begingroup$ The great thing about open textbooks is that they're much more responsive to notes like that than traditional publishers: they maintain errata and, given that the next images lack the strange artifact you noted, this would probably be a simple switch for the next time they release. I'll probably submit that as an issue by the end of the week. I would challenge the notion that the image in question would impede learning in any meaningful way. $\endgroup$ – johnharveybc Jan 17 '20 at 23:08
  • $\begingroup$ @johnharveybc I just want to point out that it doesn't have to be an "open" textbook to be be able to correct quickly in respond to criticism. It is generally applies to digital publishing. $\endgroup$ – Peter Saveliev Jan 19 '20 at 14:42

For a Calculus I class, I usually:

  1. Don't teach limits first. Limits, as they are currently taught, lead to confusion. You don't do anything with them except prove the derivative. After that they are largely forgotten, basically leaving students feeling they wasted their time, and they are quickly forgotten. Instead, teach limits at the end to justify the the fundamentals of calculus, and show additional uses with infinity.
  2. Focus on differentials instead of derivatives. Most of calculus is simplified if your focus is on differentials. It unifies single-variable, implicit, and multivariable versions of calculus in an intuitive way, and leverages students' knowledge of algebra.
  3. Get rid of logarithmic differentiation. There is a simple rule that can be used in place of logarithmic differentiation. Using this rule makes all of Calculus I derivatives a simple application of rules. If you aren't familiar, the rule is $d(x^y) = yx^{y - 1} dx + \ln(x)x^y\,dy$. Its a combination of the power rule and exponent rule (and actually points towards an interesting property of partial derivatives).
  4. Treat the integral as primarily an infinite sum of infinitely small values. Areas under the curve are an application area of the integral, not the main thing. This unifies the usage of integration across all the different geometric uses. You are simply finding an infinitely small geometric figure and summing all the values. In "area under the curve", the figures are boxes, $y\cdot dx$, in revolutions around $x$, the figures are circles, $\pi\,y^2dx$. A lot of curriculum tries to get tricky, by moving parts of this (like $\pi$) outside of the integral. The problem is that this makes it less obvious to students what is happening - a simple sum of regular geometric shapes/formulas.
  5. When you do introduce limits, the hyperreal version of limits greatly reduces the cognitive load compared to epsilon-delta proofs, and the students are more likely to remember how to do them.

See the paper "Simplifying and Refactoring Introductory Calculus" for more details.


I'm not in education, so take my answer with a grain of salt. I am merely approaching this from a perspective of someone who is really frustrated with how mathematics is taught.

Mathematics is a tool we use to understand the universe. If you want to make math more engaging, it would be useful to actually show how one could use math to solve problems.

For example, say you're teaching a class simple rates. I think in the current paradigm a teacher/prof would ask a question like:

  • Given a tank of 5L, at a rate of 100mL/min, how long will it take to fill the tank?

then go on to explain how the units work with this problem and whatnot.

Pretty boring right? I think a better way to approach people with this question is to actually bring a tank of water and start to fill it with water. This will naturally make people wonder how long it will take to fill the tank. From there, the teacher/prof could ask:

  • How would you solve this problem? Any thoughts?

The educator would then guide the class through to the solution.

To me that's captivating. I believe that by the end of that class, you will definitely understand rates, and see how math is useful.

I think most concepts in math are not taught this way because it is challenging to come up with examples, and you may require physical examples.

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    $\begingroup$ "I think most concepts in math are not taught this way because it is challenging to come up with examples, and you may require physical examples." Pretty much. Plus, even if I came up with the idea of a tank of water, do I own such a tank? If not, what budget will be spent? Even if I do have one, is it unused? Do I feel like it is worth the effort to transport it? Clean-up afterwards? Whiteboard markers are much easier to deal with (and chalk, back in the day). Or reliance on pre-created videos (or books). Just soooo much simpler from the instructors' perspective. $\endgroup$ – TOOGAM Jan 18 '20 at 13:51
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    $\begingroup$ Stein/Barcellos Calculus and Analytic Geometry, To the Instructor: "At the Tulane conference on 'Lean and Lively Calculus' in 1986 we heard the engineers say, 'Teach the concepts. We'll take care of the applications.'... Oddly, the mathematicians suggest that we emphasize applications, and the applied people suggest that we emphasize concepts." $\endgroup$ – Daniel R. Collins Jan 20 '20 at 9:13
  • $\begingroup$ " do I own such a tank? If not, what budget will be spent? Even if I do have one, is it unused? Do I feel like it is worth the effort to transport it? Clean-up afterwards?" Look, if someone if going to pay 60k to go to school I think the math department can afford a tank of water. " Or reliance on pre-created videos" Yeah exactly, this is all freely available on YouTube... so why should someone go to you to learn? people are suffering with your lazy and expensive attitude for a piece of paper. If you want to be a good educator, you need to put in WORK. $\endgroup$ – Anon Jan 28 '20 at 20:06

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