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I am a postgraduate in biology. I haven’t had an opportunity to learn mathematics beyond secondary school. In my current project I have to use differential equations, ordinary differential equations (ODE), and trajectories to understand kinetics and modeling mechanisms. Unfortunately, I have zero experience in these mathematical topics.

What is the best way to learn and refresh these mathematical skills? Are there any user friendly math books — with simple examples and figures (rather than a lot of text)?

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  • Comments are not for extended discussion; this conversation has been moved to chat. Please read this FAQ before posting another comment.
    – Massimo Ortolano
    Dec 9 '21 at 14:20
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    Is a collaborator an option? I imagine there are some applied mathematicians who would love the opportunity to co-author. It sounds like some calculus needs to be done weighing the opportunity cost of using your time focusing on a weak spot rather than capitalizing on your strengths.
    – Brickman
    Dec 9 '21 at 17:26

12 Answers 12

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I recommend that you take a look at

Brown, C. (2007). Differential equations: A modeling approach. Thousand Oaks, CA: Sage Publications.

It's targeted at social science undergraduates, so I think it would be understandable to a postgraduate student of biology. It uses more words than equations and even has some examples that might apply to you (predator-prey systems). Its focus is on qualitative analyses of the behavior of two-dimensional systems of ordinary differential equations. It's quite a thin book, and covers the basics.

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    Thank you very much Joel
    – Dendrobium
    Dec 15 '21 at 4:19
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tl;dr Consider a session with a tutor – not to learn math from them, but rather to get some perspective. Sorta like asking them to sketch a map of Math-Land, including pinpointing where you're currently at, helping you to identify where you'd like to go, and how to get there. Plus bonus if they can suggest helpful software/tools that could speed things up.


Hire a tutor.

Tutors are great for learning a new topic.

I mean, by the time you're a grad-student, presumably you ought to be able to self-teach. So maybe you don't need a tutor to really teach you the subject.

Instead, you'd probably want to get a session or two with a tutor just to get advice. They can read where you're at, what you know, etc., and then kinda sketch out the field and discuss the topics you'll want to study. Once they sketch a basic roadmap and suggest some potential resources (such as software and learning-resources that might be helpful to you), then you'd be predisposed toward an easier learning-pathway.

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    I think tutors in general are not a good idea and serve mostly as a crutch for learning how to learn yourself. It's certainly not a bad idea to get someone who knows math to help guide you in a general sense, but not to learn the material for you.
    – CSSTUDENT
    Dec 7 '21 at 10:03
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    @CSSTUDENT Why? A good tutor teaches the material at the students own pace. Dec 7 '21 at 14:58
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    Tutors are generally very expensive, so probably not a viable option for a lot of students.
    – Tom
    Dec 8 '21 at 16:37
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    @Tom: Might vary with context. Some places have free tutoring centers, while some folks have friends who might tutor them.
    – Nat
    Dec 8 '21 at 21:17
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    You might be able to get a start by dropping by the lounge in the math department, or talking to a professor during office hours. Ask someone for advice on getting a tutor or other resources.
    – ThomasW
    Dec 9 '21 at 2:04
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I fear that there is no easy way. My advice would be that you should try to get your hands on course notes or textbooks on dynamics for undergraduate engineers (for example), as in these the basic principles (time derivatives, kinematics, the basics to vector algebra, differential equations, etc.) are often included to some extent. From then on, however, you have no other option in my eyes than to try and work through such a textbook/lecture, tackling mathematics problems in a target-oriented manner when they arise.

It's certainly not very easy to get into the topic without prior knowledge, especially since (in my personal experience) engineering students also tend to struggle with the topic of dynamics very much.

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  • Thank you very much @pbaer
    – Dendrobium
    Dec 15 '21 at 4:20
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Books make a great reference, and others have listed just a few of the many great ones on calculus, but they can get a bit dry. So I'd recommend supplementing your reading with some MOOCs (massive open online courses), the grand-daddy of which is Khan Academy, which is free (donations are encouraged but not required), has a logical structure and progression, and has quizzes and achievements. Brilliant is pretty good I believe, but is a paid-for subscription. On top of that, I'd say get a good calculator and lots of paper, and just solve problems the old-fashioned way.

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    I used Khan Academy all the way through my math undergrad, even for classes like multivariable calculus. As a teacher, I also recommend it to my students and have them complete units from it as homework. It's a fantastic resource. Dec 7 '21 at 17:11
  • Thank you very much @Alan & Robin
    – Dendrobium
    Dec 15 '21 at 4:21
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    It was @MerseyViking who wrote the answer. I just made minor edits to the post (fixed spelling of "Khan Academy", etc.). But thank you. :)
    – Alan
    Dec 15 '21 at 13:55
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    Thank you very much @MerseyViking I really appreciate your answer
    – Dendrobium
    Dec 21 '21 at 7:15
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There are some good courses at Complexity Explorer. I'd recommend the two differential equation tutorials, followed by Liz Bradley's Non-linear dynamics, provided the latter fits your schedule. I second Nick Alger's recommendation for Steve Strogatz. There is also a very good book

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  • Thank you very much Simon
    – Dendrobium
    Dec 15 '21 at 4:21
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Another suggestion that could help is to learn to code. Some languages are better for Math than others. I could highly recommend R. Compared to other languages, it focusses a little less on software engineering and a little more on statistics. It's also completely free and open source.

Some things you'll be able to do very quickly:

  • Run statistical simulations
  • Plot the results
  • Work out summary statistics (e.g. 'average', 'median', 'mode') for small or large datasets easily
  • Machine learning

You could browse kaggle for some inspiration!


Afterthought: I just realised you already code in R! I'll leave the answer here in case it can be helpful to others.

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    I never thought about learning programming would help to learn mathematics. This is a great suggestion Stevec. Yes I know R a bit. Thank you very much.
    – Dendrobium
    Dec 7 '21 at 11:47
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    I would suggest Mathematica instead. Comes for free with every Raspberry Pi. Dec 7 '21 at 14:59
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    @ThorbjørnRavnAndersen The initiative must be democratising mathematical computing (and hence computational physics, bioinformatics and many other sciences). I'm a big fan of SW's work, although I had really no idea of that initiative.
    – stevec
    Dec 7 '21 at 15:12
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    Practically speaking, it's unlikely these equations will need to be done by hand anyway, so programming is the way to go. OR it might require a deeper cut into dynamical systems in which case, frankly, it's easier and better to find an outside expert: in physics programs, that'd take 2-3 years of fairly terse education...
    – Lodinn
    Dec 8 '21 at 1:21
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    This was what really helped make math stick for me after doing poorly (relative to other subjects) in it in high school and college. Once I could code, math just started making intuitive sense to me Dec 9 '21 at 5:42
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This is going to perhaps sound "un-academic" to some, but with your background and stated goals:

  • No mathematics beyond secondary school
  • Need to learn to use differential equations
  • Looking for user friendly math books — with simple examples and figures rather than a lot of text...

When I have been in similar situations, or have had students that are, I always recommend a book from one of the For Dummies, Idiot's Guide, or Demystified series.

These books are very accessible, thorough enough to give you a "working level" understanding of the mathematics and tecniques, and are designed for rapid self-study. They also have good practice problems.

They aren't designed for a rigorous treatment of the subject (you won't be deriving the proofs of any theorems with these), but that doesn't sound like it's your goal.

Here's one for Differential Equations: https://amzn.to/31DZ3Xw

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  • Thank you very much @Ifalin
    – Dendrobium
    Dec 15 '21 at 4:21
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Step 1 Obtain a note-taking device. It can be your notebook and pencil, tablet and stylus, or your laptop. You will need to take notes and study them in order to learn about the subject.

Step 2 Choose your learning environment. It can be a private tutor or an online course. If you choose to consult with a paid tutor, then I believe that nobody here can give you proper advice.

If, on the other hand, you intent to go with online material, then there are some decent options. These options are, and not limited to, the following.

Free
Taking a MIT Open Courseware course. Completing this course has a perk. Upon finishing all the assignments, you obtain a digitally signed certificate from MIT.

Watching Differential Equations Playlist from Khan Academy on Youtube. Unlike the MIT course, this course starts from very basics and gradually goes into the complicated stuff. No certificates, though.

Paid

Taking the course offered by Math Sorcerer on Udemy. Not everyone will agree, but I really find this particular content creator's teaching very user-friendly. You will get a complimentary certificate -- probably of no use in your case after completing this course.

Step 3 Study. Probably the step which seems the most trivial but ignored by a lot of individuals. Unfortunately, learning a skill (in your case being able to identify and solve the problems which require differential equations) takes some time and effort. You may need to take vigorous notes and study those notes periodically.

On a side note, I want to appreciate your desire to learn. Instead of saying "that's not my area of expertise," trying to learn a brand new topic deserves respect, in my opinion.

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  • Thank you very much @Padawan
    – Dendrobium
    Dec 15 '21 at 4:22
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For a well organised and comprehensive introduction, I warmly recommend you to take a look at this book:

Otto, Sarah P.; Day, Troy, 2007. A Biologist's Guide to Mathematical Modeling in Ecology and Evolution. Princeton University Press, Princeton.

It doesn't just teach you the mathematical tools you can use for modelling, but it shows how to actually use them for modelling. It was like a Time-Turner for me. I was pretty much in the same situation as you. My feeling is that it has saved me long years of building a systematic mathematical modelling knowledge from small pieces of information here and there on the Internet. I will defeat the (sky-high) barrier to entry of mathematical content much earlier than I would have without this book. The flip side of this is that it has a good amount of text. So it doesn't really fit all your requirements.

One doesn't need any serious mathematical training to understand it other than basic algebra and maybe very basic calculus, for which I can recommend this website (even for just refreshing):
https://www.mathsisfun.com/algebra/index.html
https://www.mathsisfun.com/calculus/index.html
But it's very rare for the book to take it for granted that you know something. It keeps on holding your hand. So – just to smuggle in a funny little quote here – it's not like Alfred J. Lotka:
"Like most mathematicians, he takes the hopeful biologist to the edge of a pond, points out that a good swim will help his work, and then pushes him in and leaves him to drown."
(Charles Elton, 1935, in reference to a work by Lotka)

And a final piece of advice (however unoriginal it is): work on the problems in the book! Studying maths is like studying a language or a musical instrument. You can gain a certain level of understanding by reading about it, but in order to apply/speak/play it, you have to practice doing it.

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  • Thank you very much benjamin
    – Dendrobium
    Dec 15 '21 at 4:14
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If you want to "understand" the subject watch 3Blue1Brown's DE series. Just brilliant.

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  • Thank you very much Krishnan.
    – Dendrobium
    Dec 20 '21 at 6:43
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In my experience, as other commenters have mentioned, Khan Academy is pretty decent and constantly embiggening its resources. From what I've seen, Brilliant.org is good in higher-level maths, but I do not have personal experience that I can use to vouch for it. In my Discrete Math, Beginning-level Engineering, and C++ programming classes, I use Zybooks.

Zybooks is very interactive and tells you how to do the problems in ways to where one can actually understand why things occur the way they do - in contrast to the typical textbook approach where they provide methods that never seem to be helpful (to me at least). Here's a Zybooks book catalog. Yearly subscription is around $50 USD - much more affordable than $120+ for a physical textbook that you use one time in your life. If you want to keep notes since it's a subscription, I recommend the Snipping Tool (should already be installed on your computer) in addition to Google Docs/Ctrl+C/Ctrl+V.

I agree with others about getting a tutor to assess where you're currently at, and where to advance, but I personally feel that a long-term personal tutor for continuous learning will probably not accomplish what you're seeking.

I love learning and I've learned that the most difficult thing to self-learning is knowing what words to use to search (usually esoteric language...), which is where the temporary tutor should help you a significant amount. E.g. "How does electricity work?" versus "What is the difference between Wattage and Amperage?". The latter quotes will give you significantly broader and more helpful information. Lastly, not only are Wikipedia pages helpful for info through the articles themselves and the sources listed at the bottom, but the articles usually have many hyperlinks of prerequisite/corequisite topics that can lead you down to what you need to know. Don't forget about YouTube and public PDFs that some universities post online.

In summation: Khan Academy, Brilliant, Zybooks, a tutor only to assess current knowledge and what else there is to learn (so you can get those keywords to use for self teaching), Wikipedia (article, in-text references, sources), YouTube, university PDFs

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    Yes! Learning some keywords and terminology is a great idea - you can't search for something if you don't know what you're searching for. For strict academic stuff I'd steer clear of Wikipedia, but for a pragmatic approach it's a great resource for at least learning the jargon. Dec 8 '21 at 10:59
  • Thank you very much @Steven and Mersey
    – Dendrobium
    Dec 15 '21 at 4:22
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I would recommend Paul's Online Notes as it was very down to earth while also being rigorous.

Similarly I would recommend the Demystified series of books.

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  • Thank you very much Laurel
    – Dendrobium
    Dec 15 '21 at 4:22

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