Sometime early this week , I finished listening to the excellent video lectures from MIT OCW’s **Calculus Revisited course** . I had been meaning to listen to MIT OCW’s Single Variable Calculus course for quite sometime as my background in Calculus is a bit flaky. My interests are in machine learning, data mining and AI where Calculus has a nasty habit of making surprise entries 🙂

I somehow finished my Master’s using my old Calculus knowledge. I took a course on Numerical Methods which kind of exposed my weaknesses. I kept getting confused with the error approximations which used ideas from infinite series. Other advanced ideas like multivariable optimization were also problematic to me. Once that course was over, I swore myself to refresh my Calculus stuff and also learn multivariable calculus.

I started listening to MIT OCW’s Single Variable Calculus lecture videos and felt two things – The course was a bit slow for my pace and the course jumped right away into the mechanics without spending much time on the intuitive explanations of the Calculus. In other words, I felt 18.01 was more focused on the analytic part which emphasized proofs and derivations whereas for my purposes an intuitive explanation of the concept would have sufficed. In fact, I remembered almost all of the Calculus formulas from undergrad – My only problem was the lack of “sense” in how to apply it to the problem I faced (say in machine learning or some optimization).

Then I found the **Calculus Revisited course** from MIT OCW. It consists of a series of lectures on Calculus but also assumes that students have had prior exposure to it. This assumption had some interesting consequences and I fit the bill perfectly. I downloaded the set of videos and started listening to them. Interestingly, all the lectures were between 20-40 minutes which allowed for maximum focus and also allowed you to listen to multiple lectures in the same day. In fact, Arlington had a heavy snow this week and my university had to be closed for the entire week. I completed around 16 lectures in 3 days and was able to finish it ahead of my target date of Feb 15.

The course starts with the absolute basic ideas of sets, functions, induction and other stuff. If you are from CS and had taken discrete math, you can feel free to skip the first section. But I would suggest you to still take a look as it , in a sense, sets the stage for the entire course. Do take some time to listen to the lecture on limits. (Part 1 , lecture 4). Here, the discussion of limits effortlessly leads to the derivation of the formula for instantaneous speed and hence differentiation.

Part 2 forms the crux of the course and covers differentiation. Professor Herbert Gross had a beautiful way of teaching stuff about derivatives. In particular, he extensively used the idea of geometric proofs or visualizations to expound basic ideas. The way he brought out the tight relation between analysis (as in Math) and geometry was enthralling. He had a huge emphasis on geometric intuition which helped me to “grasp” the key concepts.

Part 3 had some nice discussion on Circular functions. He joked about how teachers don’t provide good motivation for learning trigonometry which I felt very true to me. He also explained some concepts that were new to me – that you do not really need triangles to define cosine and sine. Previously, I was aware of the radian concept but never put it all together. He also explained how sine and cosine tend to come up in unexpected places – like as the solution of the differential equation for harmonic motion 🙂 He also masterfully showed the close relation between circular and hyperbolic functions with a playful title of ‘What a difference a sign makes’ (in Part 5).

Part 4 discussed about integration and how it can be used to calculate 2 and 3 dimensional areas (and volumes). This part also had a great discussion on how differential and integral calculus are related. That being said, I was a bit dissatisfied with the discussion on the two fundamental theorems of Calculus. The discussion on Mean Value Theorem also felt a bit rushed. I got a bit lost on the discussion on 1 dimensional arc length calculations. May be I should revisit the lecture notes for the same when I get some free time.

Part 6 was my favorite part for two reasons – This had a discussion of infinite series and my favorite quip of the course. When discussing about the non intuitiveness and the intellectual challenges posed by infinity , professor Herbert Gross playfully quips (which goes something like this)– ‘ of course, one thing to do is to not study it. I can call it as the right wing conservative educational philosophy’ – Ouch 🙂 I think I mostly understood the idea of infinite series even though there was not much explanation of “why” it works that way. I also felt the topic of Uniform Convergence to be way beyond my comprehension level.

Overall, it is a great course and acted as a fast paced refresher for those who had already taken Calculus. The course slowly starts from basic pre-calculus ideas and rapidly gains speed and covers a huge list of calculus topics. I felt few of important Calculus topics were not covered or rushed up – First and second fundamental theorem of Calculus, Mean Value theorem, Taylor series, L’Hospital rule, discussion of exponents and logarithms etc.

But that being said, I feel the course more than makes it up for the way the basic ideas were covered. I had fun learning the ideas of limits, infinitesimals , intuitive ideas of differentiation/integration, geometric explanation of differentiation/integration, how the concept of inverse functions pervades Calculus etc. Prof. Herbert Gross had a jovial air around him and occasionally delved into philosophical discussions which made listening to the lectures more interesting.

He also had an extensive set of supplementary notes and huge amount of problems with solutions. I had to skip the problems part to conserve time. But if you have some time do spend some on it.

Lastly, I found that one of the lectures in the series was missing. Lecture 5 in Part 2 on Implicit Differentiation was the same as the one on Lecture 4. I had sent a mail to MIT OCW about this and got an reply saying they will fix it soon. Hopefully, it will be fixed soon.

In conclusion, this is a great , fast paced course on Calculus that emphasizes geometric intuition of the major ideas in Calculus. Listen to it if you already know Calculus and want a fast refresher ! I am currently listening to the lectures on Multi Variable calculus. I do intend to listen to Single variable Calculus again , may be in Summer. I will put out another post on how it went 🙂

on February 8, 2011 at 7:46 am |Herb GrossHi Saravanan,

Thank you for such an in-depth review of my “Calculus Revisited” course. I think it’s remarkable that you got so much from the course in such a relatively short period of time. The course was designed to be a refresher course for scientists and engineers who were moving into management positions and had come to MIT on sabbatical leave from ther companies; and the course was taught live by me 5 days a week for 6 weeks.. The attendees usually stayed at MIT for from 1 to 4 demesters to receive education in fields that had developed since they obtained their degrees.

The course was so well received that MIT decided to make a video-based version of i,t to be used by industries for any of their employees who were interested in reviewing calculus. Several industries commented that if the course was presented at a slower pace it could be used as a first-time calculus course for their technicians. The videos simply took the place fo the live lecturer and most of the work by the studentrs was to be done by using the textbook and the study guides.

However it wasn’t until recently that the videos were digitized and made available on the OCW website. It is an amazing feeling to me that something I developed 40 years ago had been resurrected and has allowed me to teach calculus to a new generation of students, most of whom weren’t yet born when the course was developed.

if you’d like to chat with me, my email address is hgross3@comcast.net

Thanks again!!

Warmest regards and best wishes,

Herb Gross

PS

I am hoping that OCW will eventually publish the rest of “Calculus Revisited”. What they published was just Part 1. Part 2 deals with the calculus of several variables and Part 3 deals with selected topics in linear algebra, complex variables and differential equations. If OCW doesn’t do this I will eventually publish them on my own website (www.adjectivenounmath.com).

on February 8, 2011 at 11:34 am |Saravanan ThirumuruganathanProf.Herb Gross,

It was an honor to have your comment on my blog. I will surely be in touch with you. Looking forward for more of your lectures on MIT OCW !

on March 27, 2011 at 7:26 am |Larry ShapoffDear Professor Gross,

I have just finished watching the first video of your “Calculus

Revisited” course.

Thank you for your very adroit way of explaining the first

concepts in a very meaningful and understandable way.oo

Anyone with your teaching talent owes the students of this

world a full blown textbook that includes the epsilon-delta

definition of limits and all the rigorous proofs of continuty,

the mean value theorem,etc.

In any event, thank you for a lovely set of lectures.

Very thankfully yours,

Larry Shapoff

on May 13, 2011 at 11:17 pm |envelopyHi Saravanan,

I have a question. In my school calc is divided into a sequence of 3 courses where the first 2 covers single variable calc. In MIT videos this part is covered with only 1 semester of calc course (the 18.01 for e.g.). So do you happen to know what’s going on? Specifically, I’m worried watching the MIT videos to self-study may make me miss something.

Is it b/c MIT kids are just smart enough to absorb stuff others need 2 semester, or is the 18.01 course itself is a condensed/shortened version of the 2-semester version of single variable calc?

on May 16, 2011 at 5:57 pm |Saravanan Thirumuruganathan@envelopy,

That is something I have wondered too. I am a graduate student now and took 18.01 with ease. I dont think I would have been able to handle 18.01 as an undergrad.

I think one thing that might help is if the student took some calculus AP courses in their high school or so. This way they will have a basic intro to calculus and they will be ready for the rigor of 18.01.

on May 20, 2011 at 3:50 pmenvelopyThanks for letting me know. So I take it that 18.01 does actually cover the usual 2-semester calculus sequence? Btw I’m a grad student too, and I’m trying to use summer to make up for some calc background I lacked.

on May 13, 2011 at 11:22 pm |envelopyI clicked Post button too quick!! But, regardless, please let me know what you think. I checked the video’s table of contents and looks like 18.01 alone does cover all topics of the 2-semester-long single variable calc used in my school (and in some others too I know).

Envelopy

on May 22, 2011 at 3:53 pm |Math StudentDoes anyone else get the sense that mathematics education at the time these videos were recorded was based around a foundational and conceptual understanding of the material rather than a computational approach?(assuming that the approach Prof. Gross took in these lectures was not a lone occurrence)

Of course, I may have simply had the misfortune of having poor mathematics professors in my calc courses (Analysis was better, but calculus of a single variable up through multivariate seemed this way), or it may have been the result of the course having a (roughly) 22:1 engineer to math major ratio. (I’m a math major by the way.)

If anyone has had similar or completely different experiences, I’d appreciate hearing about them.

on May 22, 2011 at 8:36 pm |Saravanan Thirumuruganathan@Math Student,

I have a slightly diff experience – Most of the lectures I had were focused on using Calculus to solving some artificially constructed problem rather than encouraging a conceptual understanding or how it is practically used in the real world. That is one of the reason why Prof. Herb gross’s lectures came as a whiff of fresh air as he focused on the conceptual and geometric intuition rather than on derivation. I am also trying to find few lectures on “applied calculus” where the ideas are used practically – say using infinite series for error bounding or maxima/minima for optimization etc. If you found any interesting ones , let me know !

on April 1, 2012 at 9:40 am |pk1976I was thinking the same thing. I work as an engineer and in the industry we essentially have templates if you will. For example if you want to simulate a beam pattern response of a sinusoidal signal that was previously transmitted you use Euler’s relation (e^I*theta) and plug in the specifics for your problem.

I was never content with this and wanted to dig deeper and find out exactly how everything worked together. I found much of what I was looking for when I looked at the history of mathematics. For example what motiviated Newton and Leibnitz to invent the calculus in the first place? Where did the number ‘e’ come from? What is the history of the number pi (let’s forget that pi is the ratio of the circumference of a circle to it’s diameter)?

on February 13, 2012 at 8:56 pm |Sunny MarellaVanakkam Saravanan Sir,you put in exactly what everyone who watched the lectures would think.The lectures by Professor Gross are awesome!I think the fact that Prof.Gross managed to do all this in the period of 1968-1973 is a achievement in itself.The supplementary notes are so beautifully crafted and the section on limits is like no other book i have ever encountered.’DON’T “MONKEY” WITH CONDITIONAL CONVERGENCE”.I will always remember that quote.I wish Prof.Gross posted lectures on some topic in pure mathematics.

on February 15, 2012 at 9:17 am |john vossCurrently I am retired. I received a MS in Math in 1966. How i wish that I had a teacher like Proffessor Gross. I now enjoy your lectures thoroughly, Thank you. John Voss.

on April 10, 2012 at 1:54 pm |Herb GrossHi All,

After many months of not looking at this site, I came across it today and I want to thank everyone for the positive comments they made. When OCW decided to digitize and upload the videos I had made forty years earlier I was nervous about whether the sophisticated viewers of today would “look down” at the primitive production values in the videos. I was relieved (perhaps “exhilarated” would be a better word) to find that this was not at all the case. Parts 2 and 3 of the course are now also posted and I am anxiously looking forward to seeing how these videos are being received.

What might seem sort of surprising is that the five years I spent producing the course at MIT were a small but enjoyable part of my career. More specifically, for the ten years before coming to MIT I was teaching at a community college in Corning New York and for the thirty years following my stay at MIT I taught at a community college in Boston MA. In all those years I specialized in teaching arithmetic and basic algebra to “mathephobic” adult learners. Three years ago at the age of 80, I came to grips with the possibility that death might not be optional; so I began to develop my own website (www.adjectivenounmth.com) where I am uploading all of my videos, power point presentations and written material on arithmetic and basic algebra for anyone to use free of charge. If you had the time I would be delighted if were to visit my website and let me know what you think of it. I am very pleased with it but I worry that the folks who might benefit the most form it might not even know that it exists. So if you do like the website, please help me spread the word.

I wish all of you the very best in all of your endeavors. I feel blessed that I was given the opportunity to develop the “Calculus Revisited” course for MIT.

on October 8, 2013 at 2:37 am |AnonymousSir,

http://www.adjectivenounmth.com is not right. The right URL is http://www.adjectivenounmath.com

And thanks for the wonderful lectures 🙂

.A

on May 28, 2012 at 9:09 pm |P.JohnstonThank you, Sir! I am now transitioning to another career (Operations Research) and am stepping through your classes via Apple’s iTunes. You have a wonderful classroom persona and I feel blessed to time travel 40 years ago to sit in your class. Good health and God bless you! -Paul Johnston

on June 25, 2012 at 11:35 pm |MikeWI just finished Calculus Revisited 1 from iTunes U. Excellent! I had searched for calc courses when I saw the great reviews for this course. Even though it was in B&W and 40 years old, all the reviewers raved about it. I went through the course watching about 2 lectures per week. Learned more about the hyperbolic and exponential functions in 5 minutes than I did in my calc course in college. Also have to agree about the section on radians. The way it was explained, it was like a light bulb went off and it seemed clear cut. Professor Gross uses a great balance of theory and examples to help you understand the material. Also appreciate the sly humor and the obvious love of the subjects and teaching. I was saddened that the course was coming to an end, but I just found out through this site that there are other courses and just subscribed to Part 2. Awesome! I had a bunch of other courses ready to go, but this will go to the front of the line. Thank you Professor Gross!

on October 23, 2012 at 5:42 pm |pedrapgwilymPedroRobertoI looked at being able to download the videos into Real Player format but it seems that this facility is not available – just i-Tunes, uggh!

on November 21, 2012 at 11:36 am |Matt DubuqueProfessor Gross, having a background in film, I am compelled to point out that the “production values” you were concerned about in these videos were superb!

Beautifully framed, superb focus, excellent camera work and sound. It’s obvious to me that you and your crew spent a LOT of time trying to get that aspect correct when you made it and that hard work shines through 45 years later.

ONE thing that puts your presentation head and shoulders above all the others is that your blackboards were already beautifully prepared beforehand for these videos with immaculate explanations and graphs.

I contrast this with other well known examples in this genre where the professor’s failure to completely erase his rather messy blackboard writings makes the blackboard virtually unreadable after 45 minutes, because the previous layers of writing remain visible to varying degrees.

So in that sense your “production values” aced the competition.

Watching these lectures in superb black and white just reminds me of watching an old Hitchcock film in Panavision, a thoroughly enriching experience.

Thanks for such a superb course!

on August 13, 2013 at 11:09 pm |Larry WinklerHerb Gross and Calculus Revisited is the standard by which all live and online presentations should be measured. No other course I’ve seen comes close.

on October 8, 2013 at 2:36 am |Anonymoushttp://www.adjectivenounmth.com is not right. The right URL is http://www.adjectivenounmath.com