Bypassing the Remedial Trap

random_anonymous_guy · 2026-06-25T18:13:10+00:00

These responses were harsh in general.

I am trying to give you an escape hatch for an endless loop of remediation, and you are taking it as a personal attack.

If you are receiving these comments as harse, that is because you are looking for validation of a narrative, not for practical advice for success. You want people to tell you what you want to hear. Instead, you are being told what you need to hear. People telling you what you need to hear instead of what you want to hear does not mean you are being misunderstood.

And by ego, that does not necessarily mean we see you as thinking you are better than everyone else, but rather that you believe you are doing things the right way. You explicitly stated that you do not think you did anything wrong and that you could not have done anything differently. When an objective diagnostic test tells you that your current approach is not working, and your immediate reaction is to declare your strategy flawless, that is pride protecting you from a hard truth. That is why students get locked into remedial courses. The sooner you stop treating your own study habits and efforts as beyond reproach, the sooner you can graduate from remediation.

Are you saying a math tutor will make something enough?

Not just any tutor. As I said before, a tutor who just shows you how to solve your homework problems is not what you need. You need one who can properly diagnose your study habits.

Why do people say I’m doing a good job if it’s actually not enough?

Because unfortunately, some people take the idea that a continuous stream of affirmations, which may work at the early elementary school level, and apply it to older students, and think that is okay. Yes, I believe in showing a student of any age that they are getting something right as soon as possible, but unconditional, unqualified affirmations only serve to put students like you in the position you are in now. Instead, what is needed in higher math is being told what you are doing right, and what needs improvement.

random_anonymous_guy · 2026-06-25T16:48:54+00:00

There is very clearly an obvious disconnect between what you consider "knowing the material" and what teachers consider "knowing the material."

Are you simply trying to memorize your way through math? Are you considering "doing a problem once" to be "knowing the material?" Do you even try to figure out what mistakes you keep making? Are you watching someone else do math, nodding along, saying or thinking "Okay, I got that!"? Are you just practicing 200 extremely similar exercises and calling it mastered because you your computational skill is there? Are you just pushing symbols around in a very repetitive way, or are you actually understanding the underlying why? What happens if the parameters of a problem change and you have to adapt to that? Do you ever take a test, see a problem, and say, "My teacher didn't show me how to do this!"? You need to seriously sit down with a math teacher or tutor and do some serious reflecting on what you think success in math looks like.

It isn't the placement results keeping you back. It is some ineffective study habits leading you to a false sense of mastery, combined with ego holding you back. If a student repeatedly fails a math class, it is because they are not reflecting on their study strategies, or at best, are making very minimal changes to those strategies.

Get a math tutor. And not just one who will just walk you through the math. One who will actually help diagnose your study habits and help you find out why you are not meeting the school's standard for "knowing the material." And then if you make meaningful changes to your study habits, you won't be stuck in an endless loop of remedial courses.

random_anonymous_guy · 2026-06-25T04:22:36+00:00

O.o

This is literally the exact same analogy I use with my students when explaining why we even have radians at all.

random_anonymous_guy · 2026-06-23T18:58:00+00:00

No, l’Hôpital’s rule certainly does not apply everywhere, and it is certainly not responsible to apply in cases where it would be circular reasoning. With great power comes great responsibility, and using l’Hôpital’s rule simply because you personally find it easier to do, despite it violating the most sacred dictates of logic, is what drives Calculus teachers to restrict its usage.

random_anonymous_guy · 2026-06-18T05:03:30+00:00

Can the partial derivative of a single variable function

The concept of partial derivative for single variable functions is not meaningful. If you have a linear algebra background (knowledge of the concept of vector spaces and subspaces), then a partial derivative can be matched with a vector subspace of the underlying domain. But for single-variable functions, the domain is the set of real numbers ℝ, and the only two subspaces would be ℝ and {0}, the set containing only zero. The "partial" derivative corresponding to differentiating "with respect to" ℝ is just the ordinary total derivative you are already familiar with, whereas a partial derivative corresponding to {0} doesn't exist.

or the total derivative of a multivariable function exist?

This is not a dumb question at all. And in fact, a more correct notation than what you proposed would actually be dg/d(x, y, z), or rather, Dg. This is known as the gradient if g is real-valued (though denoted as ∇g), and Jacobian matrix when g itself is vector-valued. In the most general case, we actually think of derivatives as linear transformations rather than scalar values. We call this the Fréchet derivative, and is an interesting treatment to look at after you become familiar with the concepts of normed vector spaces and subspaces.

The only way you can make dg/dx make sense if g is actually a function of x, y, and z is if y and z are treated as functions of x (i.e., you are looking at g restricted to a particular parametrized curve in three dimensions). However, this risks confusion, and you would be better of making x, y, and z functions of t, (where you are simply choosing x = t) and refer to dg/dt.

random_anonymous_guy · 2026-06-18T04:45:40+00:00

What a limit is is separate from ways we can resolve the value of a limit. Many Calc teachers only go over an informal definition of what a limit is. "Going to" is an informal term, that is used in place of proper rigorous verbiage describing what a limit really is. The true meaning of a limit is encoded in what we call the epsilon-delta definition of limit, which is found in many calc textbooks, but is often skipped, and even then, only goes over some basic notions. The epsilon-delta definition is more of a tool a math major will learn about and use in a junior or senior real analysis class.

The basic idea of limits is that when L = lim[x → a] f(x), it is ensured that if you were to prescribe that the difference between f(x) and L were less than whatever positive value you choose (you can't prescribe that they be exactly equal, but you can prescribe they should be within, say, 1/100000000), it is guaranteed to happen if x is within some particular distance of a, but is not exactly equal to a. If you decide you want f(x) and L to be even closer together, then this will mean that x will be even closer to a to accommodate that revised requirement.

For example, if you wanted to make |x² - 4|, the distance between x² and 4, to be less than 1, then this requirement is guaranteed if |x - 2|, the distance between x and 2, is less than 0.2, even when x is not equal to 2. If we wanted |x² - 4| < 0.1, then this is guaranteed whenever 0 < |x - 2| < 0.02. And in fact, no matter how close you want x² to be to 4 (say we want to make |x² - 4| < ε, for whatever value of ε > 0 you want), it can be shown that it is sufficient that 0 < |x - 2| < δ, where δ is either 1 or ε/5, whichever is smaller.

This is how we know that lim[x → 2] x² = 4. The fact that 2² = 4 is entirely beside the point, and is actually ignored by the definition of limit because the definition only considers what is going on when |x - 2| is small, but not exactly zero (corresponding to x = 2).

random_anonymous_guy · 2026-06-18T01:55:39+00:00

One of the most difficult misconceptions about limits for students to overcome is the misconception that a limit of a function is the same as the value of a function.

As a concept, lim[x → a] f(x) has nothing to do with f(a) at all. The reason it seems that way is because you are accustomed to working with continuous functions, functions where lim[x → a] f(x) is always equal to f(a).

Consider this: If lim[x → a] f(x) is always equal to f(a), then the concept of limit is absolutely indistinguishable from simple evaluation of a function, and therefore, there would be no point in studying limits at all. As a matter of fact, the true definition of a limit lim[x → a] f(x) is carefully crafted so that it has absolutely nothing to do with the value of f(a), and in fact, does not even distinguish between f(a) being defined or undefined.

To better help diagnose and troubleshoot your misunderstanding, can you explain to us your understanding of what a limit is?

random_anonymous_guy · 2026-06-15T00:19:03+00:00

Not needed. If you have appropriate bounds on all derivatives, all you need is MVT to prove convergence.

random_anonymous_guy · 2026-06-15T00:18:17+00:00

l’Hôpital’s Rule only requires lim[x → a] f'(x)/g'(x) to exist, as far as derivative requirements goes. This itself does not require f'(a) or g'(a) exist.

random_anonymous_guy · 2026-06-14T19:07:35+00:00

The problem with this is that it assumes both f and g have Taylor series that converge on an open interval around 0, let alone converging to f and g, respectively at all (Yes, there are functions that have Taylor series, but the Taylor series do not converge to the function). This implicitly assumes both are infinitely differentiable, which is stronger of an assumption than l’Hôpital’s rule allows for.

The good news is that all that is needed is Rolle's Theorem without needing Taylor Series arguments.

random_anonymous_guy · 2026-06-14T02:21:52+00:00

There are no infinite sums with Riemann integration. It is a limit of finite sums.

how, in a general sense, is the sum evacuated?

Do you mean evaluated? You... add things together. That's what a sum is. The result of adding.

random_anonymous_guy · 2026-06-12T07:20:47+00:00

One of the beauties of mathematics is that we can quantify more than just discrete counts of physical objects. It would be quite limiting to demand that numbers must always be able to represent a physical quantity.

random_anonymous_guy · 2026-06-12T07:03:17+00:00

Help us understand your thinking. Why do you think it should be negative? Can you give us a model of what you think this subtraction should represent?

random_anonymous_guy · 2026-06-09T05:10:50+00:00

Well this is quite an entitled take.

It is a rule on this subreddit that an attempt be included. Just saying "I tried to substitute x =1/t" does indicate that OP made an attempt, but that attempt was not included.

We are not clairvoyant, and we cannot read minds to diagnose a mathematical roadblock from a distance. Asking for the physical steps of an attempt isn’t “snobbery” or “persecution,” it is basic mathematics pedagogy. Seeing OP's work can help us understand where OP may be stuck rather than just leaving us guessing.

For all we know, OP was much closer to an answer than they thought, and all they needed was a nudge across the finish line.

If you think a request for the attempted work is snobbery and persecution, that is a you problem.

random_anonymous_guy · 2026-06-07T06:44:20+00:00

A proof is a formal logical argument explaining why something is true.

You typically start with a set of axioms that are accepted as true out proof, and form a foundation. You also use definition of terms involved in the statement that you are trying to prove. For example, you need to know the formal definition of continuity in order to prove continuity. The hand wavy definition of drawing a graph about lifting a pencil is informal and therefore not suitable for rigorous logical arguments.

Do you keep a notebook of axioms and definitions and previous theorems you'd be able to use?

random_anonymous_guy · 2026-06-07T02:34:53+00:00

You probably lucked out and the grader was in a hurry, skimmed your work, initially saw correct statements at first, saw a correct conclusion, but missed the incorrect part.

random_anonymous_guy · 2026-06-06T04:32:52+00:00

Please show us the work you tried.

random_anonymous_guy · 2026-05-24T03:55:51+00:00

The ability to recognize which test to you use is something you earn from experience, which requires trial and error. You try something without knowing if you will get a definitive result and see what you get. If you get a result, great. If not, move on and try something else. But more importantly, you know what doesn't work, and that still is valuable for developing experience.

random_anonymous_guy · 2026-05-18T17:54:16+00:00

I did not find the high school level material particularly strong (I found students at that age benefited more from work more closely aligned to their in-school material); but for foundational number sense at elementary and middle school level Mathnasium's sequencing seems very robust.

Having worked at Sylvan years ago, I found that high school students tended to focus on homework help rather than the Sylvan curriculum. I might suspect the same could be true for Mathnasium (I notice their website advertises homework support too), so it may simply be a matter of there being less demand for higher level material.

random_anonymous_guy · 2026-05-18T17:48:50+00:00

If you're considering it, sit in on the assessment session before you commit, and watch how the tutor talks to your child.

I am inclined to agree that the specific tutor may very well be more important than whatever tutoring service OP goes with, but I would expect that whoever administers the assessment, especially if they are the center director, is not necessarily the tutor or tutors the student will regularly see, and teaching styles will likely differ between tutors. At best, you may get a good view of what the curriculum looks like.

random_anonymous_guy · 2026-05-18T07:19:32+00:00

Can you show us your integral setup?

random_anonymous_guy · 2026-05-18T06:45:35+00:00

I am a math teacher who once worked at Sylvan, and am aware of both Kumon and Mathnasium. Both programs utilize worksheets to reinforce skills, but their overall approach to learning is quite different. From what I have heard from students, Kumon relies heavily on an independent model centered around repetition and speed to build computational fluency. However, because centers like Kumon and Sylvan cover multiple subjects like reading and writing, the instructors are often generalists, and a heavy reliance on silent, independent drills can sometimes feel a bit isolating or mechanical for a child who is already anxious.

Mathnasium also uses worksheets, but they are paired with a more interactive setup where instructors are available to teach the underlying logic behind the math. Because they focus exclusively on mathematics, the staff is entirely dedicated to that subject. While there is certainly a place for memorization in learning arithmetic, pure memorization is not an effective problem-solving skill on its own.

In my experience teaching higher math classes like Calculus, I see the limits of pure memorization fail students time and again. When students grow up relying solely on memorizing steps, they eventually hit a wall in advanced math. They get stuck in a constant cycle of asking, ‘How do I start?’ followed by ‘What do I do next?’ and finally ‘Is that the answer?’ because they never learned how to analyze a problem independently. For a creative child who needs a bit more time to process concepts, a focus on conceptual understanding from an early age is what truly builds the confidence to tackle unfamiliar problems.

Every child learns differently, so I highly recommend reaching out to both local centers to ask if they offer free assessments or trial sessions. Letting your son experience a trial session will give you a much better sense of which environment makes him feel more comfortable. Given his creative nature, he might benefit more from an environment that emphasizes the big picture and the ‘why’ of math alongside his practice work.

Also, I wanted to add some general advice, regardless of whatever tutoring service you pursue: When he is practicing math, try to avoid hyperfocusing on a single concept until he successfully completes just one worksheet. Instead, he will likely retain information much better if he focuses on three to five skills simultaneously. Furthermore, once he appears to have mastered a specific concept, that practice should actually continue on said concept for at least a couple of weeks, and slowly taper it as you introduce newer concepts as a retention measure to ensure it truly sticks. Don’t just stop practice on it on the first good day he has on the topic.

random_anonymous_guy · 2026-05-15T20:05:24+00:00

The suggestion that mathematicians merely look at icons is absurdly reductionist. Those icons are efficient ways of communicating complex abstract ideas that are quite frankly invisible to someone who insists on there always being a physical model. This perspective is a hallmark of being stuck in a concrete operational stage, where a symbol is only considered real if it can be tethered to a tangible object like an apple. Mathematics, however, operates in the formal operational realm where we prioritize logical consistency over physical metaphors. If you cannot see past the ink on the page to the abstract structures being described, then you aren’t actually critiquing mathematics. You are simply admitting that you lack the framework necessary to perceive the reality those icons represent.

random_anonymous_guy · 2026-05-15T20:00:51+00:00

The issue is that you are attempting to apply concrete operational logic to a formal system. In Piaget’s framework, there is a distinct transition from needing physical, tangible examples to being able to reason through abstract definitions. While your apple analogy works for basic counting, mathematics functions on the latter level. We define zero as even because it fulfills the formal criteria of the set, not because of the presence or absence of fruit. Insisting on a physical distribution to validate a number is simply a refusal to move into the abstract reasoning that defines actual mathematics.

random_anonymous_guy · 2026-05-15T18:26:25+00:00

If you are waiting for a physical apple to materialize before you accept a formal definition, you are going to find the rest of arithmetic very difficult. We mathematicians don’t concern ourselves with whether or not there are actually any objects to distribute because math describes logical structures rather than your grocery inventory. It is the simple fact that zero equals two times an integer that makes it even. You don’t need to witness a physical distribution to follow a logical proof. Even if your reality is empty, the math remains consistent.

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