Mostly fields outside math try hard to teach math within their fields. Examples of such fields include physics, chemistry, economics, psychology, sociology, electronic engineering, and, yes, computer science, In all cases, the results are at best poor. Of course, those fields are helped by the fact that the math departments don't want to do well teaching service courses, say, in second order stationary stochastic processes for signal processing in electronic engineering with the fast Fourier transform!
But you noticed the problem: The CS profs make a mess out of math.
For the book of this thread from MIT, that fact is a bit surprising: For one, among the authors, at least Rivest has done some good math. For another, for each math topic in that book, MIT has some high expertise. E.g., in optimization, Bertsekas is there and long there have been others.
Once the basic ugrad lessons in math are learned, one still has to learn how to communicate in less severe forms. So, the book failed there, too: Their algebra for the core steps of the simplex algorithm was too long. And, their intuitive explanation, as I objected to here, really was not correct. Sadly they just wrote their chapter on the simplex algorithm too quickly from too little study. This is especially sad since there are many highly polished presentations of the simplex algorithm.
They got pushed into attention to linear programming, and the simplex algorithm, because of some important connections with some problems, e.g., flows on networks that they want to regard as part of computer science, where linear programming is a very important tool.
So, really, it's a strain to call 'flows on networks' computer science. Instead, it's a topic in applied math, back to Ford and Fulkerson, yes, with some by Dantzig, but more from Cunningham and Bertsekas, in particular, part of operations research. So, CS wants to hijack flows on networks. Not really good.
I omitted a more serious objection to the book: It makes a mess, a really bad mess, out of probability. Yes, in the study of the performance of algorithms, probability is important. So, once again, CS wants to borrow and use or just hijack some applied math and makes a mess. That at MIT they make such a big mess out of probability is something of a surprise. Still, in the US, the good courses in probability require both measure theory and functional analysis and are rare, even in math departments. Sad situation.
In using probability for performance of algorithms, in TACP Knuth did better: He stayed with a more elementary approach to probability and made it powerful enough.
E.g., in the book, I saw where they mentioned that the need for probability was that the data had a 'distribution'. Mostly that's not what have to pay attention to: E.g., don't much care about Gaussian, uniform, exponential, etc. Instead, what care about most of all is independence and, then, sometimes, lack of ties. Sad situation.
For both teaching and research, CS needs to clean up its act in math.
Still better, math should just open up, welcome, take in, and take over all the more important fields that are heavily math -- physics, economics, social science, electronic engineering, computing, operations research, etc. Instead of being one of the smallest departments on campus, teaching a few service courses, and otherwise struggling, math could be one of the biggest departments on campus, awash in both funding and students, and one of the most important pillars of growth in our economy.
I have an example: Some years ago I took a problem in practical computing and derived some new probabilistic math for a solution. Right, I used measure theory. I wrote prototype software, ran it on some real data, etc., and wrote a paper. So that I could get the paper in front of people in computing, where the applications are, I looked for journals in computer science. From one editor in chief of one of the best journals, a chaired prof of computer science at one of the best research universities, I got back "Neither I nor anyone on my board of editors has the prerequisites to review your paper.". Later I got almost the same statement from a second such person. For a third such person, at MIT, I wrote custom tutorials for two weeks before they gave up. But in the end the editor in chief of a good Elsevier journal seems to have walked the paper around campus and gotten feedback enough to get the paper reviewed -- the editor couldn't review it!
The paper was a nice contribution to computer science. Generally nice contributions will want some math. For all concerned, CS needs to learn some math.
Here's what they need to do: After calculus, get a good book on abstract algebra, pay full attention to the proofs, and learn some 'mathematical maturity'. Then get a good, first book on linear algebra and learn some important material, still paying attention to the proofs, and get more maturity. Then get Halmos, 'Finite Dimensional Vector Spaces' and learn from one of the best writers of math ever and get an introduction to Hilbert space theory. Then go carefully through Rudin's 'Principles'. Then do some more in ordinary differential equations where get to use both Rudin and Halmos. Then do some optimization -- e.g., pick from linear programming and some of Bertsekas. Then learn some measure theory, functional analysis, and probability based on measure theory. Cover basic stochastic processes, e.g., Markov, Gaussian, and martingales. Then learn some mathematical statistics. THEN return to computer science.