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Comment by chongli | Hacker News Reader

I find myself endlessly frustrated by these discussions around the teaching of mathematics. People seem to want to make it more conversational, more interactive, more engaging. It doesn't work that way! Learning mathematics is like learning to play a musical instrument. No amount of 1-on-1 discussion with a teacher will get you to mastery.

You just need to practice. For hours and hours and hours.

The teacher's job is to help guide you to what you might want to study next. The teacher cannot replace individual practice time.

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Agree. A common theme in the science of learning is that effective learning centers on deliberate practice, where activities are done entirely for the purpose of pushing one's limits and improving performance. Of course, these activities tend to be more effortful and less enjoyable.

Classroom activities that are enjoyable, collaborative, and non-repetitive (such as group discussions) can sometimes be useful for increasing student motivation and softening the discomfort associated with deliberate practice. However, these activities are only supplements, not substitutes, for deliberate practice. Unlike deliberate practice, they do not directly move the needle on student performance – rather, they "grease the wheels" and reduce psychological friction during the process of deliberate practice.

> You just need to practice.

The need for practice notwithstanding, suspect both practice and narrative/conversational exposition are needed. I’ve worked with students who become very good at applied procedural problem solving through practice but who can only get just so far because they lack foundational constructs to guide them. More time in discussion with an expert could potentially fill in those gaps. An expert who understands the material well enough to adapt it to the learner in front of them.

> More time in discussion with an expert could potentially fill in those gaps.

Those gaps were left by the removal, and lack of coverage towards critical thinking and reasoning which normally preceded these classes.

You don't see these issues in cohorts that have received classical education focused on first-principles.

Coverage of Descartes rules of Method; and a priori deduction are mandatory. The trouble these students are having is primarily because they've been taught by rote(authority and dynamics based in Marxism), instead of by rational first principles. Its infected most publicly funded education.

In all fairness, I don't think more time with an 'expert' would fill those gaps, I've personally observed several contradictions where the supposed expert fails to perform as a result of incentives.

Most teachers at that level aren't actually experts. They have their lesson plans, but the moment a problem comes up outside of that (such as when the salt the earth/burning bridges strategy is used between Algebra->Geometry->Trigonometry courses), they'll just enforce a struggle session making the student suffer so unimaginably that they succumb and believe that they simply are not good at math.

The student as a result of that torture (yes it is that) gets PTSD which few ever recover from.

The structure is by systemic design since math is used to gatekeep all hard science and technology. If you can't understand Math, you won't be able to follow any of the subsequent lessons that use it as the primary tool to impart knowledge (eschewing intuitive or deductive understandings or relationships as was previously done pre-1970s).

> In all fairness, I don't think more time with an 'expert' would fill those gaps, I've personally observed several contradictions where the supposed expert fails to perform as a result of incentives.

Or, as a result of objectives? If the ‘expert’ in this regard knows the material well enough, then the objective and constraints seem relevant. When you are referring to an expert, what is the context? An expert teacher for a class maintains different skill set from an expert tutor.

One on one time with an expert tutor does help tremendously, but realistic goals have to be determined. I made my way through undergrad and grad school as a private tutor for math from elementary school to math majors. The student’s objective, time scale, and current state determine how thorough a tutor can be.

>The trouble these students are having is primarily because they've been taught by rote(authority and dynamics based in Marxism), instead of by rational first principles. It’s infected most publicly funded education.

Yes, but the Marxism aside makes little sense here. Crudely quantifying education outcomes into metrics and making the teachers responsible to said metrics is a primary reason for rote learning (i.e., teaching to the test). The ideas which inspire this type of metricizing come from Taylorism and Fordism, both of which are American schools of thought.

> I find myself endlessly frustrated by these discussions around the teaching of mathematics. People seem to want to make it more conversational, more interactive, more engaging. It doesn't work that way! Learning mathematics is like learning to play a musical instrument. No amount of 1-on-1 discussion with a teacher will get you to mastery.

That's flat out untrue. I speak from experience of teaching math all the way from grade school to teaching undergrads and grad students to teaching adults.

I can take a failing student who is completely confused and get them on track in a few hours over a few weeks. 1-on-1 discussions are not just really important, they're astronomically more effective than anything else.

Making math engaging is the number one thing that matters. Being an engaging teacher matters. Adapting to a specific student and their needs matters. Someone can practice something they hate for 10 hours and get nowhere. Someone else can practice something they love for 10 minutes and make amazing progress. You remember things that are engaging, you hardly remember anything about things that are boring.

This is what all of these AI tools get wrong. Even if they understood the problem domain and could explain it --- which they cannot.

I've had this experience with adults switching jobs into programming too. So many avoided STEM because they hated math. After like two hours together they not only got math, they started to enjoy it. Happens all the time.

> You just need to practice. For hours and hours and hours.

This is probably the worst brain damage that Gladwell and others have done with the whole nonsense of 10,000 hours of practice.

No. No amount of poor quality practice where you're hitting your head against the wall will be productive.

You need to learn how to practice math. Practice is worth it after you've gained some mathematical sophistication and some understanding of the problem domain. Most people need peers or a teacher or a tutor to help get them unstuck and point you in the right direction.

> The teacher cannot replace individual practice time.

No amount of practice for 99% of people can replace a tutor. Yes, people with mathematical aptitude don't need one; I didn't. But most people need to have some guidance and then they can practice.

What kind of career do you have that you're teaching both elementary school and graduate-level mathematics? Are you a mathematician and a parent?

I had an English teacher in my K-12 school district that also taught classes at the local university. I don't think that is unheard of.

It makes sense for an English teacher. For mathematics it makes no sense at all. Elementary school arithmetic is nothing like graduate level mathematics.

In the morning you teach fourth graders about fractions, and in the evening you teach first year graduate students about localization. Math is just progressively more complicated fraction problems each year. Sounds about right to me.

Not intended to be a full rebuttal, but also speaking from experience: I've gotten a front-row seat to multiple dumpster fires where a well-intentioned teacher focuses all their energy on class discussions about mathematical beauty and cool applications, thinks that because they're so engaging they don't have to optimize instruction strategies and hold students accountable, and graduates students who are excited about math yet can't solve even the most basic kinds of problems that they were supposed to have learned in the class.

As someone whose parents were Bay Area workaholics and was left to teach myself, was yelled at for asking questions and not knowing how to do it, and physically punished if homework wasn't done by the time they came home, 1 on 1 discussion would have been incredibly helpful and probably life changing in more areas than just math (homework instilling fear doesn't translate to dealing well with tasks later in life).

I can't imagine how nice it would have been not to have to hide out of fear that I was having problems figuring out calculus on my own.

I agree in principle, but I must stress that conceptual understanding of what the problem or functional area of math is and how you perform the task to solve and know that you are solving it correctly should be understood.

Simply being shown it once on a whiteboard or in a book isn’t working for the majority of humans.

Then yes. Practice, practice

Really? I struggled with calc in school because lectures were dry proofs (without ever formally teaching how proofs work!!) and then they just told us to read the textbook. Then I found mit ocw and ripped through the material because it was much more engaging. Never would have passed calc I-III, linear algebra, and diff eq without those lecture recordings.

It's a sad fact that teachers are often not very good. This often has to do with the fact that they do not know what you already know, and proscribe inappropriate things for you to try next. You struggled in school, it seems, because you had a teacher very focused on proofs and you never learned proofs. This is a common problem I find in mathematics, where someone never learned one key element, and struggles to pin down why everything is so difficult for them. For many people, it can be calculus, or linear algebra, or bayeseian statistics, or fourier transforms, to name a few topics myself and people I know have had aha moments in when they finally learned.

> This is a common problem I find in mathematics, where someone never learned one key element, and struggles to pin down why everything is so difficult for them.

Reminds me of Ben Orlin's "law of the broken futon": a single missing part can, over time, warp an entire futon and render it unusable.

https://horizonsaftermath.blogspot.com/2016/02/the-law-of-br...

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