Short story short: I went back to high school in my late 20s and next year I'll graduate. Over the years I went from hating math to really enjoying it, especially calculus, but there's still one thing that keeps bugging me, and that's my command of its language. I have no problem solving most of algebra or calculus problems, but if you asked me to explain what I'm doing I wouldn't be able to tell you, at least not using mathematical constructs/words. It doesn't help that I can't find any resources like you could for any natural language.
When I asked my teacher she said to practice over and over, but that's all I've been doing these years and it didn't do much to improve my oral presentation. So, I'm asking here in the hope that someone who went through the same struggle or understands the mental process very well and knows what it takes to "fix the computation-presentation bridge", could help me or guide me in the right direction, to overcome this obstacle.
Thank y'all for your time.
One way to making thinking visible is to apply thinking routines, like: - what are given?
- what are the unknowns?
- what do we know?
- what do we need, how i can get there? (strategies)
- choose a strategy and explore it
- rethink the strategy and optimize it
- connect and reflex
Once we practice thinking routines, we will find it increasingly easier to explain our thinking and even be able to debug our work. If your class also applies thinking routines, it’ll get even easier for everyone to learn and understand each other.
I recommend the following books:
- Polya, How to Solve It
- Velleman, How to Prove It: A Structured Approach
- Ritchhart: Make Thinking Visible
Mathematical work is basically a series of equivalent transformations and logical reasonings, starting from the given informations to get where we want, thus once we knew the why, it should not be too difficult to explain our work because most are self-explanatory.
Hope it can help.
I would say the absolute best thing you can do is take the problems you DO know how to solve and write out the solutions, using complete sentences. If you can't explain some part, then you will quickly discover which parts you are following by rote.
At that point you need to go back and understand the process. It will be a process of writing and writing, revising, and understanding. It's not an instantaneous process and may take a few months to a year, but it is a rewarding process.
You may find you will have to think about certain concepts you thought you knew, even basic ones. Don't worry about that. It took centuries for mathematicians to even write down precisely what a set is for example.
Either you cannot do maths, and you're only good at solving repetitive problems of the same genre from school textbooks. Or you can do maths, and you struggle to follow the generally accepted set of notation.
If your problem is the first, you need to explore a wider range of mathematical problems, which will fundamentally require you to think, apply your skills to the problem in various new ways. Maths is a subject where learning to think and learning to solve a specific set of problems are two very different things. Good resources for this are the Art of Problem Solving, and various Olympiad questions. You can also just try to apply your mathematical knowledge to real life examples.
If your problem is the second, I don't know a solution, but this is a far better problem to have than the first. I'd rather have the problem solving ability and use my own set of notation that nobody else understands, than have less problem solving ability and understand the notation. But, in short, that'll just come down to practice, seeing the general notation, and somehow forcing yourself to use it. Alternatively, just try to define your own set of notation at the start of your solutions. It may depend on what kind of professor you have as to how many credits you'll get, though. A good professor will not penalise you that much for making up your own notation, if your solutions are correct, but may try to encourage you to use standard notation (as this is required for any kind of collaborative mathematics).
It spends a lot of time on the language and how to express things. The ebook is also "pay what you want," so you can see if you like it before deciding I deserve your money :)
I suggest you find some maths videos on Youtube which explain maths you already know with lots of verbal detail narrating each step. Pay attention to the words and description (you already know the maths), then try talking through your own working in the same way. You could record yourself and watch it back (camera pointing straight down at paper is good, and phone camera held up on a ruler and stack of books is easy). As others have suggested, once you can do that to some extent, explaining things to others can be good practice, but if you're stumped as to how to describe anything, that's not going to be a good first step.
I teach university maths, and when I'm doing a problem in a lecture, writing down the steps to demonstrate, I am constantly talking, narrating everything I'm doing and why. E.g. "Okay, we need to cancel out that 3, so let's divide both sides by 3, and then we can use log rules to bring that 2 on the right inside the log as a power. Now to cancel out these logs we need to use an exponential on both sides, and if we move these terms over we'll have a quadratic to solve. We know how to do that! Hm, it doesn't seem easy to factorise, so let's complete the square. We need half of this coefficient to go inside the brackets, and don't forget the minus is part of it! Now if we think about expanding this out we'll get the x squared and minus 4 x that we want, but also a plus 4, so let's take that away and add on the 3 we need instead ..."
I'll also be referring to the context of the problem ("that makes sense, because ..." etc.) and earlier examples to compare and contrast with.
Book Of Proof by Richard Hammack
https://www.people.vcu.edu/~rhammack/BookOfProof/
[0] Obviously, I am not talking about the theory of algos where you design algos and consider questions of optimality and correctness.
OK, not English specifically - what I mean is that mathematics is not a bunch of abstract squiggle notation on paper, it’s a series of sentences written in a natural language. The abstract squiggles are shorthand for those sentences.
Some might say the squiggles are more precise than the natural language words. Not true. The notation really is just a shorthand for the words, and mathematics is the study of words that have precise meanings.
Now, don’t get me wrong, notation is great and can help make connections that would otherwise be obscured by wordy prose - but it is not primal.
So, to become better at expressing mathematics, you need to rediscover the connection between the notation and the underlying sentences that they represent.
Take the following “statement”:
a = b & b = c => a = c
Translate it into:
“If a number a is equal to another number b, and b is equal to another number c, then a is equal to c”
Note that this wasn’t even a rote translation - I added in the premise that the variables are numbers. This is the kind of information which can get forgotten or assumed in notation, which could make it harder for an uninitiated reader to follow. Of course, you may also need to remind the reader what a number is and what the “equals” relation means... knowing what to assume about your audience’s level of knowledge is part of the art.
The “connective tissue” between the symbols is the language of logic. Words like “if”, “then”, “and”, “or”, “not”, and “implies”. These are the words that allow you to form statements with precise meaning, and you can improve your use of them by improving your language skills, not just your mathematical skills.
Now, I’m not advocating for writing everything out in long form, because some notation is just too good and too expressive to pass up. The point is to remember that mathematics is a form of writing, intended to be read by other humans, that they will be reading it through the lens of natural language, and that all the rules of good writing still apply to mathematics exposition.
Introduce your topic. Motivate your definitions. Clearly and precisely lay out your reasoning. State your assumptions. Emphasise important points. Give illuminating examples. Explain your thinking. Avoid trying to demonstrate your own cleverness. Summarize your conclusions.
And all of this applies doubly for oral presentation instead of written.
Try to write down solutions to known, quite easy problems (that you are able to solve easily by hand) step-by-step (line by line) using SymPy (https://www.sympy.org/) library (or any other computer algebra system; but SymPy is most well documented, free and most beginner friendly IMO).
It may seem counterintuitive at first (you want to do learn mathematics by heart and I’m proposing to offload it to computer...) but carry on.
Contrarily to hand written math, it’s harder to abuse notation using computer algebra system. Don’t use pre-made functions (such as Integrate, Simplify, Solve) right at the start, but manually work your way (via simple operations like expand, substitute, and your hand-written functions working on expression parts) to the point when expression is in form you would put it by hand.
Each step, substitution, transformation etc has to be stated explicitly in order to be understood by SymPy! That way if you are able to “explain” it to a computer (that has no intuition about objects that it’s working with, maybe beyond some basic algebra and fact that they are symbols) it will be much easier to explain and present it more clearly.
That said the obvious con of this method is learning SymPy, but believe me it will pay back both by understanding math more deeply, as well as in ability to solve problems that are too large/tedious to do by hand in the future.
Btw I believe this is basically the way the applied math (computational, non-proof part of math) should be presented nowadays. When I (hopefully) finish writing my phd thesis I hope to to write CAS dedicated to this purpose (doing math in by-hand like way, but with help form computer) from scratch. Until then SymPy is IMO your best bet for that ;)
You don't have to be able to explain how you got the answer, but you do have to be able to explain *a* way to get the answer.
In proving theorems in mathematics, there's often an intuition that it's true, and even why it's true, but still it needs to be put into a formal proof.
But it's very hard to arrive at a proof, by thinking in formal terms, because the search space grows exponentially. Intuition is a shortcut. But you still need to go back and work out the formal proof, to show you're right - to yourself and also to communicate it to others.
And, once you know the answer, and have an idea of why it's true, this guidance makes it much easier to find your way.
To come back to your question: you can learn another way to get the answer by watching videos, where they explain what they are doing as they go. So, this won't explain how you do it, but you can learn another way of doing it that you can explain.
BTW IMHO the most insidious difficulty in learning mathematics is gaps - an incomplete or subtly incorrect understanding. It's insidious because the gap is not apparent at later material, because it assumed, so you won't know what you're missing, and therefore can't diagnose or fix it. You need either an excellent teacher/tutor/mentor/coach to pick it up (but they often don't), or to methodically go through all the previous work. The sooner you do it, the easier the easier it is.
That can come later, the more important point is to explain "why" you are doing a specific step and not "what" and you are free to use just English words.
For example, say you want to find the maximum of f(x)= -x^2 + 4 x + 10. A bad explanation would be some mumbling about taking down the 2, ... , f'(x) =-2x +4, so x=2, f''(x)=-2, is negative, done.
Better: We know that at a maximum the slope of f has to be zero (otherwise we could go left or right to get larger), so let's find such points first. We thus compute the derivative function as f'(x)=-2x+4 and find that it is zero if x=2. But that does not give an answer yet, because it could still be a min/max or saddle point. .. draw a small picture of each. We thus compute the second derivative to see in which of these three cases we might be. f'(x)=-2 is negative so we locally look like a parabola opened downward, so this is indeed a maximum.
So I'd recommend some college textbooks. If you live near a college, their library should have a ton of them to browse through and lend. You should also be able to find accompanying lectures on YouTube or college web sites.
I've got a collection of texts that I can go through, and I've found lately that starting on a random key of the day I roll a dice - has really helped me on both sides - in that I've got a bunch of kinds of exercises I do for practice, but the piece of randomness helps me with finding things to practice, that help me broaden my existing knowledge and practice.
What does this mean for you? Well maybe you need to pick a random page of a suitable book each day, find the start of the section for that page, go through it and then see if you can think of any exercises or things you need to do to help you with your knowledge consolidation from that.
Try Playing with Infinity by Rózsa Péter. https://www.maa.org/press/maa-reviews/playing-with-infinity
[1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...
You may also find doing mathematical proofs to help your thinking, although it is a higher level than explanation. It might help with clarifying the thought process.
Language is a medium to communicate. Actual message is more important than medium.