I’m replying just so I don’t have to scroll through the whole thread to read the end. 
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Speaking of ROI, my son is visiting and spent an evening with a number of his college classmates at an elite LAC. Several of them came from poorer families and are URMs. They were often the strongest kids in their high school and got great financial aid to attend. But, he said they didn’t really get the kind of guidance they needed on careers/finances and also were dissuaded from certain majors because of the weakness of their high schools. Three of the five kids he got together were not prepared for the difficulty of the math and were rebuffed and hence didn’t take computer science classes. They majored in psych or English and then joined TFA or went immediately into low-ish paying teaching jobs. The combination of student loans and low-paying jobs. After a few years those three joined a coding bootcamp and now have jobs in tech. He thinks that when the school accepts low SES kids from weaker high schools, they really need to get them up to speed to participate in the full curriculum. What they do get is a summer academy before school, but it may require an extra year. In addition, they would really benefit from coaching that upper middle class kids get from their parents about what majors might be better economically and what career paths enable them to get the ROI on their decision to accept said elite school’s offer. He said this is the first time since college where the conversation hasn’t been about the stress and tension of not being able to easily make ends meet financially. And at my son’s graduation, the president announced how happy she was that the proportion of people taking a poetry class had increased over the last few years. Nothing against people taking poetry, but I wish the college would see the relevance of helping the humanities folks to also learn something about the modern technological world.
I met a similar kid who had grown up in Venezuela before her parents moved to Houston, where she attended an inner-city HS. Clearly very bright and she wanted to major in physics but discovered that the other folks hoping to major in physics were two years ahead in math. She realized she could never catch up. When I met her, she was in grad school in a social science-y area but she also felt a bit let down by a system that admitted her but in a way where she would not be prepared to pursue her interests. In discussing her thesis and career prospects, I got a sense of a very bright very determined woman who did have some regrets about not being able to pursue her interests. She’ll be successful, but maybe in a way that isn’t appealing to her as alternate directions she couldn’t pursue. Recently, I spoke to another low SES grad of a (different) highly regarded LAC, who said she had difficulty with calculus and thus couldn’t take computer science. This young woman is really impressive. She was born to a young mother who ignored her (father largely out of the picture) – put her in daycare immediately, pre-school, school, pretty much paid her no attention. On her own, she applied to a private school and took public transportation to get to the interview. Similarly, she applied to college entirely on her own, got a full scholarship at said highly-ranked LAC, took a job in tech, went to HBS, has a very good career. But she too regrets that she wasn’t really prepared to take calculus and thus computer science.
Those colleges could really boost ROI and serve their students better by really offering some academic catchup and providing the kind of career guidance CC parents likely provide to their kids.
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Almost all HYPSM… type colleges do this for math/science. For example, Harvard requires that all incoming freshman take a math placement exam to help better understand HS math background and help decide what math starting point is most appropriate. Based on placement exam score, HS course background, AP scores, personal goals, planned major, feedback from placement officer, and other factors; the student decides on which math course to choose. Harvard offers any of the following intro math starting points and sequence options – Math Ma,b; 1a,b; 19a,b; 20; 21a,b; 23a,b; 25a,b; and 55a,b. The lowest level (MA) is a half normal speed calc/pre-calc type class , while Harvard’s website describes math 55 as “probably the most difficult undergraduate math class in the country”.
Stanford also requires a math placement test and long list of possible starting points and possible paces… In addition to these options, Stanford offers a summer bridge program. As summarized at Log in , some of the summer online bridge programs target students from less resourced backgrounds that you mention.
…which gets to an important point, namely that calculus really isn’t very useful in computer science. Personally, I like calculus and if I had to, I could try to argue that it comes up in numerical analysis (if you include that with CS) and just brushes some theory (integral approximation for a summation that may come up in a complexity bound). But, eh, not really. If you’re good at discrete math and combinatorics you can go far in CS theory, and if you just want to program, you don’t need any of it.
So what is it but an irrelevant weed-out course? It’s no exaggeration to say I’m angry that difficulty with calculus would discourage anyone from studying computer science.
As you stated early in your comment:
This is a perfectly reasonable route to a tech job for the self-motivated. The bigger question is what was the point of their college degree?
My point is the general use of even ivy when talking about colleges since the eight are pretty different. Especially in discussing economic mobility that Chetty does, public colleges and STEM careers will do a lot more to address income inequality than private colleges focused more on keeping the status quo than changing it, which is understandable. Not having LACs, or public colleges like Georgia Tech or Michigan, or stem focused schools like Cal Tech or CMU makes the ivy plus term arbitrary. Why would you keep MIT but not Cal Tech, Duke but not Hopkins, Chicago but not Northwestern? Stanford, sure, but if the ivies are so great, why not just stick with them.
“So what is it but an irrelevant weed-out course”
CS is pretty math intensive and if you have a lot of kids entering as CS and not enough spots in jr or sr year, Calculus is the first weed out course, then typically something else, typically switching theory. It’s weed out because it’s not easy and professors aren’t shy about giving Cs and lower. But this typically happens in larger public colleges, not LACs, so that is surprising.
If you wanted to give As and Bs to everyone in the class and not make it a weed-out class, I’m sure professors could do it, but they will still have to get through even harder classes, and outside of say Linear Algebra, most will need to have some knowledge of Calculus.
“Did Frank Bruni mention that?”
He didn’t mention it, not sure why you’re getting so defensive about his book, anyway you don’t have to give it any credence. But I will leave you with one more Bruni tidbit, he left his post at NYT a couple days back and one of the reasons was his merciless takedown of Ted Cruz during the presidential primary campaign.
Ted Cruz refused to study with anyone other than Harvard, Yale or Princeton grads in law school, he referred to Penn and Brown as minor ivies. That’s elitism for you!
Below please find the entire excerpt from Esquire magazine detailing Cruz’s supposed study group elitism at HBS
“The elite academic circles that Cruz was now traveling in began to rub off. As a law student at Harvard, he refused to study with anyone who hadn’t been an undergrad at Harvard, Princeton, or Yale. Says Damon Watson, one of Cruz’s law-school roommates: “He said he didn’t want anybody from ’minor Ivies’ like Penn or Brown.”
The link below provides a summary of the disbarment status of the single source (Damon Watson) for that narrative that was repeated throughout hundreds of media outlets without any further witnesses or confirmation ever coming forward publicly. Please also note that this disbarment and the criminal conviction were public record but never mentioned in the article (you choose between sloppy or malicious journalism).
In summary the Ted Cruz article you reference was a single source “hit piece” that was refuted by numerous of Cruze’s classmates and picked up nationally with no further corroboration or first hand witnesses. The only source was Damon Watson who was censured several times and ultimately disbarred (see above). He never provides any context for how he became aware of Cruz’s supposed stance or for that matter any detail. Instead he is just sighted as “says Damon”.
Not saying it didn’t happen or that Cruz isn’t an elitist but I am suggesting that a certain degree of cynicism is required when considering the accuracy of today’s journalism from all political perspectives and that includes Frank Bruni (who didn’t author the Esquire piece FYI).
You commented “ not sure why you’re getting so defensive about his book, anyway you don’t have to give it any credence”. I don’t give it any credence but object to the tendency of some to believe whatever is written is truthful if it confirms their biases, the Cruz lore being a perfect case in point.
Frank Bruni in his final piece before moving on to a teaching role at Duke stated, “ Too many columnists generalize too broadly. I know I did”. Never had he published a more truthful statement then this his last.
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“Best” in this link is defined as having grads with the highest pay.
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There’re branches of computer science that require calculus. Machine learning and data science, for example. Calculus is the basic entry point to all the math and statistics that belong to applied math or mathematical analysis.
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Purdue had a similar summer “catch up”program, as does Cornell.
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The LAC had a summer catch up session as well, but that is definitely insufficient from someone trying to enter physics just have taken calculus.
I fully agree that calculus is unnecessary for learning how to code. Back in the Dark Ages, I took calculus in my senior year of HS. But, I had learned to write software in BASIC and FORTRAN a few years before that and in the summer before my senior year in HS, I had a job writing software on a language called b for Berkeley Computer Software Language (b was a predecessor to c which was a predecessor to c++) on the original Unix machine. For various jobs and school courses, I probably learned about 10 languages as varied as assembly language, lisp, c++ and apl. I didn’t need calculus for any of those languages.
My son argues that his school should create a coding minor or certificate so that the folks for whom calculus and CS are not a calling can learn basic skills.
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At most colleges, the physics sequence for physics majors can be started after the first semester of calculus.
Is the problem here that some students had weaker high school math so that calculus in the first semester is too difficult for them? Or is it the relative competition factor where seeing other students coming in far advanced already is discouraging to students at the standard level?
I can tell you from experience how little I used it all the way up to a PhD (I did a physics minor and it was useful there). Indeed, it can come up in particular specialties (graphics is another). It is treated as an entry point, but it is inaccurate to suggest it is needed for all the math in “applied math.” The traditional operations research problems (e.g. linear programming, maximum flow, and matching) are numerical but do not require any calculus.
Of course, you can assess a student’s overall math aptitude by their performance in calculus and it may work as a predictor, but it is unclear to me why most students ever need to learn, e.g., how to integrate using trigonometric substitutions, or solve what few differential equations have a closed form. Some of these techniques are fascinating in their own right, but rarely done outside of class. A conceptual understanding of derivatives and integrals is useful, though I do not believe that should be a blocker for someone who is looking at software development as a career.
(Sorry this is off-topic, but it’s a peeve of mine. I think mathematics education in general has serious problems.)
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Calculus (multivariate) is most certainly used in some CS specialization like AI/ML and computer vision. Error minimization, gradients and gradient descent, local and global minima and maxima in N-space. Students need a strong grasp of calculus in many dimensions to be successful. Not needing calculus is one of several assumptions about CS that may have been close to true back “in the day”, but not so true today.
Another assumption is that with CS only an Undergrad degree is needed. While it is true that undergrads are extremely employable, many employers in certain specializations still require MS and even PhD for certain roles.
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I am not inclined to believe my degree is as obsolete as all that. Has the core of computer science shifted away from: algorithms and data structures, operating systems, databases (relational or otherwise), programming languages, and distributed systems? Note that I omit AI, though some classic results such as A* search or resolution proof might fit into other areas such as algorithms or optimization. But fine, include AI too. I think there’s already plenty of material to pack into a 4 year degree.
Machine learning is an increasingly useful tool, but it’s a particular application of computers and despite its prominence today isn’t even what most developers work on. If quantum computing becomes practical, that will open up another area where continuous math is a prerequisite. My point is that for the vast body of computer science, it is not.
Computer science should IMO focus on what do computers do, not what can you do with computers, which has a short answer “Just about anything.”
Second, calculus as taught (unless that’s changed) is certainly not useful for much beyond teaching calculus. We are decades beyond the point where symbolic math packages can outperform top students at finding closed forms of integrals when possible. While these kinds of mental challenges surely function as a winnowing process, I think it is dishonest to suggest that they are “useful” or “needed.” It is useful (in some fields) to know the meaning of integrals, derivatives, partial derivatives, curl, gradient, etc.
It is a fascinating curiosity, e.g., that certain non-trigonometric functions can be integrated by transforming to trigonometric functions and back again. But is it possible that the time spent teaching non-motivated high school seniors how to do this could be better applied, even for those going into highly mathematical careers?
Just curious… How can the time be better applied?
Data science, for one thing.
Putting more focus on the analytical basis of calculus rather than a grab bag of recipes for solving arbitrary puzzles. Since the puzzles themselves can be solved by a symbolic math package, symbolic math packages should be available to students. Needless to say, computers are also excellent at graphing and solving these problems numerically, and those tools should be available. You should leave with an intuition about what is an integral or derivative. How, for instance, does calculus relate the surface area and volume of a solid (of course, a good HS math teacher will do their best to convey this).
In an ideal world, I would love to see students working on their own symbolic math programs to apply a technique such as trig substitution. This kind of crossover would help a lot more in computer science than memorizing a bag of tricks and forgetting them a year later. I don’t recommend that as a requirement either, but I think it would be a great improvement.
What is “data science”?
Students in American high schools don’t have (almost) any experience in the techniques required to understand calculus at theoretical level. I added “(almost)”, because there is some rudimentary knowledge of proofs taught in Geometry, but it IS rudimentary.
That would be great, but don’t the students need to first LEARN about these techniques, before they start writing code that applies them?