Please give me some advice about math self-study

[Douglas321]

Hello,

This is my first post, so my backgrounds is basically that I've studied biochemistry and neuroscience in university, and I want to continue working in science in some capacity, although that's not my profession at the moment. I've come to the realization that the all of the cutting-edge research in biology (and really every field of science) heavily relies on math so I've decided that it's in my best interest to develop a deeper understanding of mathematical thinking beyond the basic calculus and statistics that I've already studied. I don't have any interest in going back to school at the moment so I've decided to teach myself math. I am generally good at self-study, and have taught myself various things that some people consider difficult to learn (Chinese, computer programming, etc)

The problem is that I (along with most biologists) am terrible at math. I've bought some books that are supposed to be an introduction to "real" math, such as "introduction to mathematical thinking" and "the nuts and bolts of proofs", and when I try to do the practice problems, I struggle with them for several minutes and then get a feeling like my brain is melting and just give up. After this, I usually lose motivation to study for several days. I think the problem is I'm not used to thinking in such an abstract way, as most problems in biology are easily visualized. I feel like mathematical thinking is just too alien to me.

Anyway, I would appreciate any advice on how to develop a mathematical sense and how to self study effectively at the beginner level. Also if have any textbook recommendations, that would also be appreciated. My goal is to eventually be able to understand more advanced mathematical ideas such as abstract algebra, analysis, etc, and be able to write proofs on my own. I'm not entirely sure which fields of math I want to focus on, I feel I should wait until my ability level is higher to decide.

Thanks.

 

[jasonRF]

I've bought some books that are supposed to be an introduction to "real" math, such as "introduction to mathematical thinking" and "the nuts and bolts of proofs", and when I try to do the practice problems, I struggle with them for several minutes and then get a feeling like my brain is melting and just give up. After this, I usually lose motivation to study for several days. I think the problem is I'm not used to thinking in such an abstract way, as most problems in biology are easily visualized. I feel like mathematical thinking is just too alien to me.

Although I had a little stronger math background than you do, I too basically taught myself how to read and write proofs and taught myself a little abstract math. Clearly I am not a mathematician, but here are my thoughts:

1. Start at the right place. I don't know where this is for you - a mathematician may have a better idea. But I would guess that learning something like linear algebra (even at an intro level) where every theorem is proved can help. Work through the proofs, write up your own versions, do a mix of problems and proofs for the homework exercises. Make sure you are understanding how the different definitions, theorems, and concepts fit together. This will give you experience with a little abstraction (vector spaces, inner product spaces) in a more familiar context. Plus, linear algebra is SO useful in science; as important as calculus in my opinion. So you will know that you are learning truly useful math, not just "mathematical thinking," which may make it easier to be motivated.

2. Seriously work through the proofs that are given in the book - write up your own version in your own words such that you understand every step. This takes time but I haven't really found a substitute. Likewise make sure you really understand all of the definitions. For me this required writing up each definition myself, and sometimes coming up with my own examples.

3. Giving up after a few minutes of working on a proof is WAY too quick. Especially if you are new to this kind of thinking. Some of these things need to be poked at from various angles and thought about for awhile until the key idea comes together. Some need to be set aside and returned to later.

4. Think of proofs as fun logic puzzles to figure out - that is exactly what they are!

Jason

 

[epenguin]

I'd say first look at your motivation. If you think of doing math because you think you ought to, or must, because you think biology is going that way, not because you can have any interest in math or mathematical biology as such, then when you hit the textbook you will still have the problem you recount - you might as well join the company of those others who are terrible at it. It is not going to be indispensable in all branches of biology yet awhile.

But if you have a flicker of interest I might advise - start with the biological end. Look at the math parts of the biology that interests you and see what math they are using and work back, e.g. from their references.

Then there are different areas of biomath that use different math but I'd say everyone needs a little grounding in differential equations. In biology you mostly don't and can't solve them analytically, you analyse them qualitatively and draw solutions for the parameters of interest by computer. A rather small bag of tricks is used repeatedly for a variety of models. I recommend the biologist-oriented book (various editions) by Lee A Segel. There are exercises to do but at least you see the point of them.

But if you are still getting the same feelings like you describe now from exercises in that book, this whole area may not be for you.

As well as d.e.'s there are other possible areas. What you study may turn out to not have strict direct application in what you eventually do, that's unavoidable.

If you make good progress with that, within the year I'd recommend these things to look at. Try and develop an interest in some problem, area, theme, of your choice (I'd think neuroscience supplies some). Check out your nearest Universities' websites to see if anyone there is working on things that could interest you. Later on a contact with someone there for at least some advice on where how what on to start. At some point a summer school can be very good - there are some fantastic ones.

When last I heard there was a good employment market for biologists with good modelling experience and skills even well outside biology.

 

[mal4mac]

Try reading Tim Gowers "Mathematics: A Very Short Introduction". He's very good on abstract thinking in mathematics, so he might get you thinking that way.

 

[EE4life]

If there is one thing I learned from (sometimes boring) seminar's in academia is that almost no one stays in the exact sub-field they got their PhD in. It is absolutely normal to delve into new fields and this is actually where breakthroughs come from. You have trained your mind like a biologist, and so you and others like you are closed minded to possible solutions because of the narrowness of your knowledge. I have a Mech Eng bachelors, and I am doing a PhD in elec eng, but I do material science research and I am taking my PhD qual exam in Semiconductor devices (unrelated to my research). I have only good hopes that this diverse experience will benefit me, as it already has.

However, it is difficult to just say, "Hey, everything is math, so let me study everything that is math!" you should make sure what you study has some immediate application or you will lose heart.

About the difficulty part, boy, math is difficult. It may take years(or decades) to become the master that you expect to be overnight. However, in a matter of a few months, or even now that you have shown interest, you are ahead of your colleagues, and I think that is what matters. You are not a mathematician, but you can be a biologist who knows math.

We (and me) expect learning to be some sort of comfortable exercise. You may spend hours or even years to understand concepts and ideas, eventually feeling you wasted time, but you actually obtained mastery during those grudging hours. You know your learning when your brain is melting, so don't worry about that. Keep to what is practical to your field, and slowly start digging deeper. That's what I would say.

 

[Douglas321]

Thank you to everyone that replied, You were all very helpful. I do have a genuine interest in math and mathematical biology, your posts made me realize I need to be a lot more patient and spend a lot of time studying other people's proofs carefully. I will check out the books that you've recommended.

Thanks again.