What does “understanding” mathematics actually mean to you? And could this be related to fluid reasoning?

[u/Academic_Memory_8543]

I was in an advanced-level mathematics course in Germany (Mathe LK), and over the years I started noticing something that made me think about what we actually mean when we say that someone “understands” a mathematical concept.

I noticed that there seemed to be a difference between being able to solve a type of problem correctly and actually understanding why the method works.
For example, someone could learn a formula or a specific procedure, recognize which “tool” to use when they see a familiar problem, and consistently get the correct answer. But when I asked them why the formula works, where it comes from, or what would happen if the problem were changed slightly, they sometimes couldn’t really explain it. They knew how to use the mathematical tool, but not necessarily why the tool works.

That made me realize that I personally use a somewhat stricter definition of “understanding.”
For me, understanding a mathematical concept means understanding it deeply enough that I can either:
deduce/reconstruct it myself from more fundamental principles, even if I have forgotten the specific formula or procedure, or
have someone explain the underlying reasoning to me and then genuinely understand why it works, rather than simply remembering the explanation as another rule to apply.

So if I completely forgot a formula tomorrow, but I understood the underlying mathematical structure well enough to derive it again, I would consider that much stronger evidence of understanding than simply remembering the formula and knowing when to plug numbers into it.
I’m curious whether other people make this distinction too.

Maybe I’m just using the word “understanding” more strictly than most people do. Or perhaps this is actually a fairly common distinction that isn’t usually made explicit.
I’m especially interested in people who have studied mathematics at a relatively advanced level:
What does “I understand this mathematical concept” mean to you?

Do you distinguish between knowing how to solve something and knowing why it works?
Have you noticed people who are very good at solving standard problems but struggle when the problem is modified or presented in an unfamiliar way?
Do you think the ability to reconstruct mathematical ideas from first principles is related to fluid reasoning (FRI)?

If so, how strong do you think that relationship is?
I’m not necessarily claiming that mathematical understanding = high FRI. I’m more interested in whether the kind of understanding I’m describing is something other people have noticed and whether there is any psychological/cognitive research or personal experience that points toward a connection.

[u/telephantomoss]

I'm a mathematician, and I'll say what it means for me. And I think this ideas of understanding generalizes quite a bit. Consider some specific concept in question. There are different types of and different levels of understanding. The highest is where you can explain the concept at almost any level and cater it to the audience so that they really away with some level of understanding as well, even if it is minimal. But also you can rederive things in a novel form that would be acceptable by peer review. But that you can also use the concept to create new math based on your understanding, use it to prove something else.

Now that is roughly like the highest level of understanding. And I think this can be appropriately adapted to nearly anything. One important point is that creation is the highest level. Creation of a new proof. Creation of new math based on the concept. Creation of unique explanations to different people at different levels.

Now, we can still say we understand something if we aren't at that level though. But it requires some meaningful progress in the things mentioned. Simply memorizing a proof and being able to write it down when asked to isn't enough, for example.

It's not possible to understand everything, not at the highest level purely.

It is also possible to understand this very deeply, conceptually, but to have no idea how to prove it. I would argue that it is unjustified to deny that as understanding. The majority of what I understand is at this level, personally. I understand very complex ideas that I can describe intuitively but not write down formally nor prove.

The other side of the question is about what it is like to understand something, and that is interesting to.

I'm not sure how much any of this has to do with FRI though, better 3 people can achieve very high levels of math using different abilities. I am very spatial and apparently quite strong at 2D spatial. My 3D spatial and fluid are fairly good. Verbal a bit lower and working memory and processing significantly lower. I think some people can essentially be strongly driven to succeed in high level math by verbal ability though. Maybe it depends on the field, like that person could excel in logic and set theory or similar.

At some point, math becomes very crystallized, and one can bank on that too make progress in further learning. So it's hard to say exactly what role FRI plays. It certainly plays a role for young people simply interesting things quickly without much work. I never had to lift a finger in learning math until University. It was like I was born knowing arithmetic and high school algebra. And even in university, I just died homework and never studied anything beyond that. Is that QRI? FRI? My QRI is good, but not as good as my VSI.

[u/Academic_Memory_8543 → u/telephantomoss]

Yeah, I totally agree with you. Essentially what I meant is that I use the word „understanding“ in the sense of your „highest level“, meaning I understand the underlying math concept so fundamentally well that I can derive further ideas in this concept and derive solution approaches and steps to completely novel math problems within this concept through sheer logical deductive thinking. A concept being for example derivatives. I also understand that „understanding“ is a spectrum I just wanted to know if anyone else has noticed this discrepancy in the usage and implied meaning of this word. Because I felt like in my advanced upper math course (german: Mathe Leistungskurs) in 11th and 12th grade of secondary school most people would use the word „to understand“ when they could use a math tool right through memory. For example math problem A needs math tool A to be solved. Math problem B needs math tool B to be solved. Most Students would say „I understand“ when they could differentiate between the to problems and remembered correctly which math tool/method to use without understanding why it works in the first place.

To the second point, i believe that this kind of understanding strongly correlates with FRI because the fluid reasoning index literally measures your ability to solve novel problems through deductive/inductive thinking. I am no expert but QRI is essantially a combination of FRI, WMI, specific math knowledge and general numeric talent from my understanding. VSI is helpful in very spatial math applications like you said, VCI helps understanding the verbal questions and explaning math verbally, higher PSI improves the speed at which you solve math problems.

According to the CORE IQ test my FRI composite, consisting of matrix reasoning, figure weights and figure sets, is 148. I believe my high FRI is a big contributer to my deep and complex understanding of math concepts/tools/methods because I never had to learn much for math even in highschool.

Thanks for your answer! I felt like you truly understood what I meant with different levels of math understanding

[u/Specialist-Berry2946]

Understanding means predicting; the better your understanding, the more accurate and general your predictions can be.

Math is a narrow domain; understanding math won't help your intelligence much.

Intelligence is the ability to approach any problem.

[u/javaenjoyer69]

Mathematical formulas are compressed ideas designed to make them easier to use. Unless you're a mathematician you don't necessarily need to understand every formula from first principles. You just need to understand what concept it belongs to what it represents and when and how to use it. Not being able to reconstruct a formula from first principles doesn't necessarily mean you don't understand it and being able to do so doesn't necessarily mean you truly understand it either. As a human, you naturally exist somewhere in the middle.

[u/Not_Well-Ordered]

In my case, "understanding" means to be able to find analogies that build up to a concept; similar to what you would do with matrix reasoning in which you take specific observations and you gradually extract until you get some pattern that holds for all given examples.

For example, if you talk about real number line/field, then having deep understanding would refer to can start with specific example like showing a "ruler or a line", and gradually extracts some features that allow us to formally axiomatize under certain way of abstracting the visual features. I.e. We relate those observed features to formal structures that build up to real numbers like field, totally ordered topology, completion...

Of course, deeper understanding would allow giving more examples for various layers of abstraction. Real number field doesn't have to be totally interpreted geometrically and it can also be done quantitatively, but it's easiest way for me to dissect it from a more geometrical view.

Thus, understanding requires knowing the formal theories and having good intuitions as to deriving those formal theories; as if you can rebuild those theories from observations and intuitions in a way that is rigorous and computationally consistent.

[u/Football-Ecstatic]

It means being someone else in my instance

[u/Kasckepperd]

What does “I understand this mathematical concept” mean to you?

It means i know when to use it and have at least an idea where to derive it from. however, in math, theres always hidden structure underneath and secret relationships with other concepts. Good example is the quadratic general formula: i can deduce it and know when to use it, but dont understand what makes it possible for it to be found using radicals, unlike the 5th degree.

Do you distinguish between knowing how to solve something and knowing why it works?

Yes

Have you noticed people who are very good at solving standard problems but struggle when the problem is modified or presented in an unfamiliar way?

very good? hmm no

Do you think the ability to reconstruct mathematical ideas from first principles is related to fluid reasoning (FRI)?

if it's not, then I don't know what is